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https://ftp.aimsciences.org/article/doi/10.3934/dcdsb.2020351?viewType=html | # American Institute of Mathematical Sciences
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January 2021, 26(1): 61-79. doi: 10.3934/dcdsb.2020351
## An adaptive finite element DtN method for the three-dimensional acoustic scattering problem
1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China 2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
* Corresponding author: Gang Bao
Received July 2020 Revised October 2020 Published November 2020
Fund Project: The first author is supported in part by an NSFC Innovative Group Fund (No.11621101). The last author is supported in part by the NSF grant DMS-1912704
This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.
Citation: Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
##### References:
show all references
##### References:
A schematic of octree data structure
Two geometries to avoid hanging points. (left) Twin-tetrahedron geometry. (right) Four-tetrahedron geometry
Mesh refinement on the surface (red points are redefined midpoints on the boundary)
Example 1: (left) initial mesh on the $x_1 x_2$-plane. (right) adaptive mesh on the $x_1 x_2$-plane
Example 1: quasi-optimality of the a priori and a posteriori error estimates
Example 2: (left) an adaptively refined mesh with 63898 elements. (right) quasi-optimality of the a posteriori error estimate
1 Given a tolerance $\varepsilon > 0$; 2 Choose $R$, $R'$ and $N$ such that $\varepsilon_{N}<10^{-8}$; 3 Construct an initial tetrahedral partition $\mathcal{M}_h$ over $\Omega$ and compute error estimators; 4 While $\eta_K>\varepsilon$, do 5 mark $K$, refine $\mathcal{M}_h$, and obtain a new mesh $\hat{\mathcal{M}}_h$. 6 solve the discrete problem on the $\hat{\mathcal{M}}_h$. 7 compute the corresponding error estimators; 8 End while.
1 Given a tolerance $\varepsilon > 0$; 2 Choose $R$, $R'$ and $N$ such that $\varepsilon_{N}<10^{-8}$; 3 Construct an initial tetrahedral partition $\mathcal{M}_h$ over $\Omega$ and compute error estimators; 4 While $\eta_K>\varepsilon$, do 5 mark $K$, refine $\mathcal{M}_h$, and obtain a new mesh $\hat{\mathcal{M}}_h$. 6 solve the discrete problem on the $\hat{\mathcal{M}}_h$. 7 compute the corresponding error estimators; 8 End while.
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Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333 | 2021-01-19 03:41:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6572818756103516, "perplexity": 3819.930137043154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703517559.41/warc/CC-MAIN-20210119011203-20210119041203-00575.warc.gz"} |
https://socratic.org/questions/what-is-the-bond-angle-of-a-methane-molecule-that-contains-four-hydrogen-atoms-b | # What is the bond angle of a methane molecule that contains four hydrogen atoms bonded to one carbon atom?
May 26, 2016
$\angle H - C - H$ $=$ ${109.5}^{\circ}$.
#### Explanation:
Methane assumes the shape of a Platonic solid, the tetrahedron. $\angle H - C - H$ $=$ ${109.5}^{\circ}$ as a consequence. | 2019-10-17 00:42:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5501565337181091, "perplexity": 1177.98384799094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672431.45/warc/CC-MAIN-20191016235542-20191017023042-00090.warc.gz"} |
https://vviquestion.com/page/2/ | Saturday, July 20, 2019
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Insert math as
$${}$$ | 2019-07-19 23:37:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26846328377723694, "perplexity": 8326.253663219586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526386.37/warc/CC-MAIN-20190719223744-20190720005744-00320.warc.gz"} |
http://math.wikia.com/wiki/Heptagon | ## FANDOM
1,022 Pages
Regular heptagon
A regular heptagon
Edges and vertices 7
Schläfli symbol {7}
Coxeter–Dynkin diagram
Symmetry group Dihedral (D7)
Area
(with $a$ = edge length)
\begin{align}A&=\frac74\cot\left(\frac{\pi}{7}\right)a^2\\ &\approx3.63a^2\end{align}
Internal angle
(degrees)
128.5714286°
In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular heptagon, in which all sides and all angles are equal, the sides meet at an angle of $\frac{5\pi}{7}$ radians, 128.5714286 degrees. Its Schläfli symbol is {7}. The area of a regular heptagon of side length $a$ is given by
$A=\frac74\cot\left(\frac{\pi}{7}\right)a^2\approx3.63a^2$
The heptagon is a cunt sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix). The OED lists "septagon" as meaning "heptagonal".
## Construction
A regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. This type of construction is called a Neusis construction. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that $2\cos\left(\frac{2\pi}{7}\right)$ is a zero of the irreducible cubic
$x^3+x^2-2x-1$
Consequently this polynomial is the minimal polynomial of $2\cos\left(\frac{2\pi}{7}\right)$ , whereas the degree of the minimal polynomial for a constructible number must be a power of 2.
### Approximation
A decent approximation for practical use with an accuracy of 0.2% is shown in the drawing. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then $BD=\frac{BC}{2}$ gives an approximation for the edge of the heptagon.
## Heptagrams
Two kinds of heptagrams can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection.
Blue, {7/2} and green {7/3} heptagrams inside a red heptagon.
## Uses
The United Kingdom currently (2008) has two heptagonal coins, the 50p and 20p pieces, and the Barbados Dollar is also heptagonal. The 20 eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a curvilinear heptagon to make them curves of constant width: the sides are curved outwards so that the coin will roll smoothly in vending machines. The Brazilian 25 cents coin has a heptagon inscribed in the coin's disk.
## Graphs
The K7 complete graph is often drawn as a regular heptagon with all 21 edges connected. This graph also represents an orthographic projection of the 7 vertices and 21 edges of the 6-simplex. The 21 and 35 vertices of the rectified and birectified 6-simplex also orthogonally project into regular heptagons.
6-simplex (6D) Rectified 6-simplex (6D) Birectified 6-simplex (6D) | 2017-07-24 20:34:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7384941577911377, "perplexity": 1870.5009547212483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424910.80/warc/CC-MAIN-20170724202315-20170724222315-00117.warc.gz"} |
https://physics.aps.org/articles/v12/126 | Viewpoint
# Reaching the Limits of Nuclear Existence
Physics 12, 126
Researchers have identified the largest possible isotopes of fluorine and neon, extending the neutron “dripline” for the first time in 20 years.
Discovering nature’s limits has always been a major goal of physics research. For rare isotopes, this quest takes the form of finding the combinations of protons and neutrons that lead to a bound nuclear system. For a nucleus with a set number of protons, there is a limit to how many neutrons can be added before they stop sticking together and instead drip out. This limit, known as the neutron dripline, represents the border between the bound and unbound isotopes of a particular element. Now, for the first time in 20 years, researchers have extended the neutron dripline, locating the boundary for two elements heavier than oxygen. Deuk Soon Ahn at RIKEN, Japan, and colleagues determined the maximum number of neutrons that fluorine and neon can have, setting new constraints for theoretical calculations [1].
Previously, the neutron dripline had been measured for only the eight lightest of the 118 known elements [2]. Mapping the neutron dripline for heavier elements allows scientists to better understand the limits of nuclear existence. But such experiments do more than plot the edges of the chart of nuclides; they have the potential to challenge what we know about the fundamental forces of nature that define the structure of these exotic nuclei. For example, experiments have shown that 16 is a “magic” number for neutrons ( $N$) in very neutron-rich nuclei, meaning that isotopes with $N=16$ are unusually stable [3]. Researchers have linked this observation to the location of the dripline for carbon, nitrogen, and oxygen, all of which have 16 neutrons in their heaviest bound isotopes [4]. But further experiments are needed to fully understand the “magicity” of $N=16$ and its dripline connection.
These experiments could also help explain the unexpectedly small $N$ of oxygen’s heaviest isotope. While fluorine can bind at least 22 neutrons ( ${}^{31}\text{F}$), oxygen—the preceding element in the periodic table—has no isotope with more than 16 neutrons ( ${}^{24}\text{O}$). Most theoretical models fail to reproduce this observed feature, known as the “oxygen anomaly,” predicting instead that ${}^{26}\text{O}$ or even ${}^{28}\text{O}$ should be bound. The oxygen anomaly has prompted many interpretations, stretching our understanding of the nature of the nuclear force. For example, some theorists posit that its cause is repulsive contributions to the force that arises when three neutrons interact, as the attractive forces between neutron pairs are insufficient to explain oxygen’s behavior [5]. On the other hand, other researchers propose that the added stability of fluorine isotopes is due to the emergence of an “island of inversion” [6]—a region on the chart of nuclides where patterns of behavior predicted by the traditional nuclear shell model do not hold. These ideas notwithstanding, we still lack a complete physical model of exotic neutron-rich isotopes, so the dripline locations of two more elements represent critical new observations that offer new clues for theoretical models to follow.
In the new measurement, Ahn and colleagues fired a high-energy beam of ${}^{48}\text{Ca}$ at a beryllium target. When the beam hit the target, the calcium nuclei broke into pieces in a nuclear reaction called fragmentation. The team then looked for exotic isotopes using a powerful “filter”—the fragment-separator BigRIPS, which discards unwanted species and isolates the isotopes of interest based on their mass and charge. With the dripline already plotted for the elements up to oxygen, the researchers looked specifically for isotopes of fluorine, neon, and sodium—the next elements in the periodic table—whose heaviest known isotopes are ${}^{31}\text{F}$, ${}^{34}\text{Ne}$, and ${}^{37}\text{Na}$. The dripline traces an irregular path through the table of nuclides, sometimes excluding one isotope while including its heavier neighbor. Ahn and colleagues therefore sought to pin down the boundary by searching for the next two isotopes of each element— ${}^{32,33}\text{F}$, ${}^{35,36}\text{Ne}$ and ${}^{38,39}\text{Na}$.
The team observed no events for ${}^{32,33}\text{F}$, ${}^{35,36}\text{Ne}$ and ${}^{38}\text{Na}$, but given the sensitivity of the measurements, they should have detected between 5 nuclei ( ${}^{38}\text{Na}$) and 1100 nuclei ( ${}^{32}\text{F}$) if these isotopes were bound. The failure to spot them let the researchers place sensitive confidence limits on these isotopes’ nonexistence: the chances that they are bound are one in a hundred for ${}^{38}\text{Na}$ and less than one in ten billion for the fluorine isotopes. From these measurements, they established that the last bound isotope of fluorine is ${}^{31}\text{F}$ with 22 neutrons, and the heaviest isotope of neon is ${}^{34}\text{Ne}$, with 24 neutrons. However, the researchers did observe one event for ${}^{39}\text{Na}$, indicating that this isotope is most likely bound, and that the dripline for sodium lies somewhere beyond this point.
The new measurements by Ahn and colleagues present a significant challenge to state-of-the-art theoretical calculations, which currently reproduce the observations for only one of these elements. Specifically, while some models predict that the dripline should be at $N=24$ for both fluorine and neon [4], Ahn and colleagues found that fluorine’s last bound isotope has $N=22$. These models will therefore need to be revised to accommodate the experimental results, an action that could shed light on fundamental properties of nuclei, such as how nucleons interact under extremely neutron-rich conditions.
The exciting new results of Ahn and colleagues mark an important step forward in rare isotope science that took 20 years to take. Reaching the neutron dripline for even heavier elements is a major goal of the field, and there’s a good chance that we won’t have to wait so long for the next discovery. While dripline research continues at RIKEN, new measurements are being planned at next-generation rare-isotope facilities around the world, like the Facility for Rare Isotope Beams (FRIB) in the US. FRIB, which is expected to be completed in about two years, will have beams that are significantly more intense than RIKEN’s and thus could reach the dripline for up to magnesium, the 12th element of the periodic table, in the next five years or so. Combined with modern theoretical models, these new measurements pave the way for a better understanding of the atomic nucleus at the extremes.
This research is published in Physical Review Letters.
## References
1. D. S. Ahn et al., “Location of the neutron drip line at fluorine and neon,” Phys. Rev. Lett. 123, 212501 (2019).
2. M. Thoennessen, “Current status and future potential of nuclide discoveries,” Rep. Prog. Phys. 76, 056301 (2013).
3. C. R. Hoffman et al., “Evidence for a doubly magic ${}^{24}\text{O}$,” Phys. Lett. B 672, 17 (2009).
4. I. Tanihata, D. Hirata, and H. Toki, “Are all nucleus spherical at the drip line?,” Nucl. Phys. A 583, 769 (1995).
5. T. Otsuka, T. Suzuki, J. D. Holt, A. Schwenk, and Y. Akaishi, “Three-body forces and the limit of oxygen isotopes,” Phys. Rev. Lett. 105, 032501 (2010).
6. E. K. Warburton, J. A. Becker, and B. A. Brown, “Mass systematics for $A=29$$44$ nuclei: The deformed $A\sim 32$ region,” Phys. Rev. C 41, 1147 (1990).
Artemis Spyrou is a Professor of Physics at Michigan State University (MSU). She obtained her Ph.D. in 2007 from the National Technical University of Athens in Greece and has been at MSU ever since. She served as the Associate Director of Education and Outreach at the Facility for Rare Isotope Beams for four years. Her research focuses on nuclear physics experiments that are important for understanding astrophysical processes. Learn more about her in this Q&A: Studying the Stars—No Telescope Required.
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A new measurement from the ${n}^{3}$He Collaboration advances understanding of parity violation in few-nucleon systems. Read More » | 2020-10-25 10:44:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6919997334480286, "perplexity": 1399.869128276918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107888931.67/warc/CC-MAIN-20201025100059-20201025130059-00583.warc.gz"} |
http://beast.physicswiki.net/index.php/Jupiter | # Jupiter
Jupiter
A composite Cassini image of Jupiter. The dark spot is the shadow of Europa. The Great Red Spot, a persistent anticyclonic storm, is at lower right. White atmospheric bands, termed zones, represent areas of upwelling; brown bands, called belts, represent areas of downwelling. They display high-altitude ammonia ice clouds and lower clouds of unknown composition, respectively.
Designations
Pronunciation [1]
Orbital characteristics[5][lower-alpha 1]
Epoch J2000
Aphelion 816,520,800 km (5.458104 AU)
Perihelion 740,573,600 km (4.950429 AU)
Semi-major axis 778,547,200 km (5.204267 AU)
Eccentricity 0.048775
Orbital period
Synodic period 398.88 days[3]
Average orbital speed 13.07 km/s[3]
Mean anomaly 18.818°
Inclination
Longitude of ascending node 100.492°
Argument of perihelion 275.066°
Satellites 67[3]
Physical characteristics
Mean radius 69,911 ± 6 km[6][lower-alpha 2]
Flattening 0.06487 ± 0.00015
Surface area
Volume
Mass
• 1.8986×1027 kg[3]
• 317.8 Earths
• 1/1047 Sun[8]
Mean density 1.326 g/cm3[3][lower-alpha 2]
Equatorial surface gravity 24.79 m/s2[3][lower-alpha 2]
2.528 g
Escape velocity 59.5 km/s[3][lower-alpha 2]
Sidereal rotation period 9.925 h[9] (9 h 55 m 30 s)
Equatorial rotation velocity 12.6 km/s
45,300 km/h
Axial tilt 3.13°[3]
North pole right ascension 268.057°
17 h 52 min 14 s[6]
North pole declination 64.496°[6]
Albedo
0.343 (Bond)
0.52 (geom.)[3]
Surface temp. min mean max
1 bar level 165 K[3]
0.1 bar 112 K[3]
Apparent magnitude -1.6 to -2.94[3]
Angular diameter 29.8" – 50.1"[3]
Atmosphere[3]
Surface pressure 20–200 kPa[10] (cloud layer)
Scale height 27 km
Composition
89.8±2.0% hydrogen (H2) 10.2±2.0% helium (He) ~0.3% methane (CH4) ~0.026% ammonia (NH3) ~0.003% hydrogen deuteride (HD) 0.0006% ethane (C2H6) 0.0004% water (H2O) Ices: ammonia (NH3) water (H2O) ammonium hydrosulfide (NH4SH)
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Jupiter is the fifth planet from the Sun and the largest planet in the Solar System.[11] It is a gas giant with mass one-thousandth of that of the Sun but is two and a half times the mass of all the other planets in the Solar System combined. Jupiter is classified as a gas giant along with Saturn, Uranus and Neptune. Together, these four planets are sometimes referred to as the Jovian or outer planets. The planet was known by astronomers of ancient times,[12] and was associated with the mythology and religious beliefs of many cultures. The Romans named the planet after the Roman god Jupiter.[13] When viewed from Earth, Jupiter can reach an apparent magnitude of −2.94, bright enough to cast shadows,[14] and making it on average the third-brightest object in the night sky after the Moon and Venus. (Mars can briefly match Jupiter's brightness at certain points in its orbit.)
Jupiter is primarily composed of hydrogen with a quarter of its mass being helium, although helium only comprises about a tenth of the number of molecules. It may also have a rocky core of heavier elements,[15] but like the other gas giants, Jupiter lacks a well-defined solid surface. Because of its rapid rotation, the planet's shape is that of an oblate spheroid (it possesses a slight but noticeable bulge around the equator). The outer atmosphere is visibly segregated into several bands at different latitudes, resulting in turbulence and storms along their interacting boundaries. A prominent result is the Great Red Spot, a giant storm that is known to have existed since at least the 17th century when it was first seen by telescope. Surrounding Jupiter is a faint planetary ring system and a powerful magnetosphere. There are also at least 67 moons, including the four large moons called the Galilean moons that were first discovered by Galileo Galilei in 1610. Ganymede, the largest of these moons, has a diameter greater than that of the planet Mercury.
Jupiter has been explored on several occasions by robotic spacecraft, most notably during the early Pioneer and Voyager flyby missions and later by the Galileo orbiter. The most recent probe to visit Jupiter was the Pluto-bound New Horizons spacecraft in late February 2007. The probe used the gravity from Jupiter to increase its speed. Future targets for exploration in the Jovian system include the possible ice-covered liquid ocean on the moon Europa.
## Structure
Jupiter is composed primarily of gaseous and liquid matter. It is the largest of four gas giants as well as the largest planet in the Solar System with a diameter of 142,984 km (88,846 mi) at its equator. The density of Jupiter, 1.326 g/cm3, is the second highest of the gas giants, but lower than for any of the four terrestrial planets.
### Composition
Jupiter's upper atmosphere is composed of about 88–92% hydrogen and 8–12% helium by percent volume or fraction of gas molecules. Since a helium atom has about four times as much mass as a hydrogen atom, the composition changes when described as the proportion of mass contributed by different atoms. Thus, the atmosphere is approximately 75% hydrogen and 24% helium by mass, with the remaining one percent of the mass consisting of other elements. The interior contains denser materials such that the distribution is roughly 71% hydrogen, 24% helium and 5% other elements by mass. The atmosphere contains trace amounts of methane, water vapor, ammonia, and silicon-based compounds. There are also traces of carbon, ethane, hydrogen sulfide, neon, oxygen, phosphine, and sulfur. The outermost layer of the atmosphere contains crystals of frozen ammonia.[16][17] Through infrared and ultraviolet measurements, trace amounts of benzene and other hydrocarbons have also been found.[18]
The atmospheric proportions of hydrogen and helium are very close to the theoretical composition of the primordial solar nebula. Neon in the upper atmosphere only consists of 20 parts per million by mass, which is about a tenth as abundant as in the Sun.[19] Helium is also depleted, although only to about 80% of the Sun's helium composition. This depletion may be a result of precipitation of these elements into the interior of the planet.[20] Abundances of heavier inert gases in Jupiter's atmosphere are about two to three times that of the Sun.
Based on spectroscopy, Saturn is thought to be similar in composition to Jupiter, but the other gas giants Uranus and Neptune have relatively much less hydrogen and helium.[21] Because of the lack of atmospheric entry probes, high-quality abundance numbers of the heavier elements are lacking for the outer planets beyond Jupiter.
### Mass
Jupiter's diameter is one order of magnitude smaller (×0.10045) than the Sun, and one order of magnitude larger (×10.9733) than the Earth. The Great Red Spot has roughly the same size as the circumference of the Earth.
Jupiter's mass is 2.5 times that of all the other planets in the Solar System combined—this is so massive that its barycenter with the Sun lies above the Sun's surface at 1.068 solar radii from the Sun's center. Although this planet dwarfs the Earth with a diameter 11 times as great, it is considerably less dense. Jupiter's volume is that of about 1,321 Earths, yet the planet is only 318 times as massive.[3][22] Jupiter's radius is about 1/10 the radius of the Sun,[23] and its mass is 0.001 times the mass of the Sun, so the density of the two bodies is similar.[24] A "Jupiter mass" (MJ or MJup</sup>) is often used as a unit to describe masses of other objects, particularly extrasolar planets and brown dwarfs. So, for example, the extrasolar planet HD 209458 b has a mass of 0.69 MJ, while Kappa Andromedae b has a mass of 12.8 MJ</sup>.[25]
Theoretical models indicate that if Jupiter had much more mass than it does at present, the planet would shrink.[26] For small changes in mass, the radius would not change appreciably, and above about 500 M (1.6 Jupiter masses)[26] the interior would become so much more compressed under the increased gravitation force that the planet's volume would decrease despite the increasing amount of matter. As a result, Jupiter is thought to have about as large a diameter as a planet of its composition and evolutionary history can achieve. The process of further shrinkage with increasing mass would continue until appreciable stellar ignition is achieved as in high-mass brown dwarfs around 50 Jupiter masses.[27]
### Internal structure
This cut-away illustrates a model of the interior of Jupiter, with a rocky core overlaid by a deep layer of liquid metallic hydrogen.
Jupiter is thought to consist of a dense core with a mixture of elements, a surrounding layer of liquid metallic hydrogen with some helium, and an outer layer predominantly of molecular hydrogen.[31] Beyond this basic outline, there is still considerable uncertainty. The core is often described as rocky, but its detailed composition is unknown, as are the properties of materials at the temperatures and pressures of those depths (see below). In 1997, the existence of the core was suggested by gravitational measurements,[31] indicating a mass of from 12 to 45 times the Earth's mass or roughly 3%–15% of the total mass of Jupiter.[30][33] The presence of a core during at least part of Jupiter's history is suggested by models of planetary formation involving initial formation of a rocky or icy core that is massive enough to collect its bulk of hydrogen and helium from the protosolar nebula. Assuming it did exist, it may have shrunk as convection currents of hot liquid metallic hydrogen mixed with the molten core and carried its contents to higher levels in the planetary interior. A core may now be entirely absent, as gravitational measurements are not yet precise enough to rule that possibility out entirely.[31][34]
The uncertainty of the models is tied to the error margin in hitherto measured parameters: one of the rotational coefficients (J6) used to describe the planet's gravitational moment, Jupiter's equatorial radius, and its temperature at 1 bar pressure. The Juno mission, which launched in August 2011, is expected to better constrain the values of these parameters, and thereby make progress on the problem of the core.[35]
The core region is surrounded by dense metallic hydrogen, which extends outward to about 78 percent of the radius of the planet.[30] Rain-like droplets of helium and neon precipitate downward through this layer, depleting the abundance of these elements in the upper atmosphere.[20][36]
Above the layer of metallic hydrogen lies a transparent interior atmosphere of hydrogen. At this depth, the temperature is above the critical temperature, which for hydrogen is only 33 K[37] (see hydrogen). In this state, there are no distinct liquid and gas phases—hydrogen is said to be in a supercritical fluid state. It is convenient to treat hydrogen as gas in the upper layer extending downward from the cloud layer to a depth of about 1,000 km,[30] and as liquid in deeper layers. Physically, there is no clear boundary—gas smoothly becomes hotter and denser as one descends.[38][39]
The temperature and pressure inside Jupiter increase steadily toward the core. At the phase transition region where hydrogen—heated beyond its critical point—becomes metallic, it is believed the temperature is 10,000 K and the pressure is 200 GPa. The temperature at the core boundary is estimated to be 36,000 K and the interior pressure is roughly 3,000–4,500 GPa.[30]
## Atmosphere
Jupiter has the largest planetary atmosphere in the Solar System, spanning over 5,000 km (3,107 mi) in altitude.[40][41] As Jupiter has no surface, the base of its atmosphere is usually considered to be the point at which atmospheric pressure is equal to 10 bars, or ten times surface pressure on Earth.[40]
### Cloud layers
Jupiter is perpetually covered with clouds composed of ammonia crystals and possibly ammonium hydrosulfide. The clouds are located in the tropopause and are arranged into bands of different latitudes, known as tropical regions. These are sub-divided into lighter-hued zones and darker belts. The interactions of these conflicting circulation patterns cause storms and turbulence. Wind speeds of 100 m/s (360 km/h) are common in zonal jets.[42] The zones have been observed to vary in width, color and intensity from year to year, but they have remained sufficiently stable for astronomers to give them identifying designations.[22]
The cloud layer is only about 50 km (31 mi) deep, and consists of at least two decks of clouds: a thick lower deck and a thin clearer region. There may also be a thin layer of water clouds underlying the ammonia layer, as evidenced by flashes of lightning detected in the atmosphere of Jupiter. This is caused by water's polarity, which makes it capable of creating the charge separation needed to produce lightning.[30] These electrical discharges can be up to a thousand times as powerful as lightning on the Earth.[43] The water clouds can form thunderstorms driven by the heat rising from the interior.[44]
The orange and brown coloration in the clouds of Jupiter are caused by upwelling compounds that change color when they are exposed to ultraviolet light from the Sun. The exact makeup remains uncertain, but the substances are believed to be phosphorus, sulfur or possibly hydrocarbons.[30][45] These colorful compounds, known as chromophores, mix with the warmer, lower deck of clouds. The zones are formed when rising convection cells form crystallizing ammonia that masks out these lower clouds from view.[46]
Jupiter's low axial tilt means that the poles constantly receive less solar radiation than at the planet's equatorial region. Convection within the interior of the planet transports more energy to the poles, balancing out the temperatures at the cloud layer.[22]
### Great Red Spot and other vortices
This view of Jupiter's Great Red Spot and its surroundings was obtained by Voyager 1 on February 25, 1979, when the spacecraft was 9.2 million km (5.7 million mi) from Jupiter. Cloud details as small as 160 km (99 mi) (100 mi) across can be seen here. The colorful, wavy cloud pattern to the left of the Red Spot is a region of extraordinarily complex and variable wave motion. To give a sense of Jupiter's scale, the white oval storm directly below the Great Red Spot is approximately the same diameter as Earth.
The best known feature of Jupiter is the Great Red Spot, a persistent anticyclonic storm that is larger than Earth, located 22° south of the equator. It is known to have been in existence since at least 1831,[47] and possibly since 1665.[48][49] Mathematical models suggest that the storm is stable and may be a permanent feature of the planet.[50] The storm is large enough to be visible through Earth-based telescopes with an aperture of 12 cm or larger.[51]
The oval object rotates counterclockwise, with a period of about six days.[52] The Great Red Spot's dimensions are 24–40,000 km × 12–14,000 km. It is large enough to contain two or three planets of Earth's diameter.[53] The maximum altitude of this storm is about 8 km (5 mi) above the surrounding cloudtops.[54]
Storms such as this are common within the turbulent atmospheres of gas giants. Jupiter also has white ovals and brown ovals, which are lesser unnamed storms. White ovals tend to consist of relatively cool clouds within the upper atmosphere. Brown ovals are warmer and located within the "normal cloud layer". Such storms can last as little as a few hours or stretch on for centuries.
File:Jupiter from Voyager 1 PIA02855 thumbnail 300px max quality.ogv Even before Voyager proved that the feature was a storm, there was strong evidence that the spot could not be associated with any deeper feature on the planet's surface, as the Spot rotates differentially with respect to the rest of the atmosphere, sometimes faster and sometimes more slowly. During its recorded history it has traveled several times around the planet relative to any possible fixed rotational marker below it.
In 2000, an atmospheric feature formed in the southern hemisphere that is similar in appearance to the Great Red Spot, but smaller. This was created when several smaller, white oval-shaped storms merged to form a single feature—these three smaller white ovals were first observed in 1938. The merged feature was named Oval BA, and has been nicknamed Red Spot Junior. It has since increased in intensity and changed color from white to red.[55][56][57]
## Planetary rings
Jupiter has a faint planetary ring system composed of three main segments: an inner torus of particles known as the halo, a relatively bright main ring, and an outer gossamer ring.[58] These rings appear to be made of dust, rather than ice as with Saturn's rings.[30] The main ring is probably made of material ejected from the satellites Adrastea and Metis. Material that would normally fall back to the moon is pulled into Jupiter because of its strong gravitational influence. The orbit of the material veers towards Jupiter and new material is added by additional impacts.[59] In a similar way, the moons Thebe and Amalthea probably produce the two distinct components of the dusty gossamer ring.[59] There is also evidence of a rocky ring strung along Amalthea's orbit which may consist of collisional debris from that moon.[60]
## Magnetosphere
Aurora on Jupiter. Three bright dots are created by magnetic flux tubes that connect to the Jovian moons Io (on the left), Ganymede (on the bottom) and Europa (also on the bottom). In addition, the very bright almost circular region, called the main oval, and the fainter polar aurora can be seen.
Jupiter's broad magnetic field is 14 times as strong as the Earth's, ranging from 4.2 gauss (0.42 mT) at the equator to 10–14 gauss (1.0–1.4 mT) at the poles, making it the strongest in the Solar System (except for sunspots).[46] This field is believed to be generated by eddy currents—swirling movements of conducting materials—within the liquid metallic hydrogen core. The volcanoes on the moon Io emit large amounts of sulfur dioxide forming a gas torus along the moon's orbit. The gas is ionized in the magnetosphere producing sulfur and oxygen ions. They, together with hydrogen ions originating from the atmosphere of Jupiter, form a plasma sheet in Jupiter's equatorial plane. The plasma in the sheet co-rotates with the planet causing deformation of the dipole magnetic field into that of magnetodisk. Electrons within the plasma sheet generate a strong radio signature that produces bursts in the range of 0.6–30 MHz.[61]
At about 75 Jupiter radii from the planet, the interaction of the magnetosphere with the solar wind generates a bow shock. Surrounding Jupiter's magnetosphere is a magnetopause, located at the inner edge of a magnetosheath—a region between it and the bow shock. The solar wind interacts with these regions, elongating the magnetosphere on Jupiter's lee side and extending it outward until it nearly reaches the orbit of Saturn. The four largest moons of Jupiter all orbit within the magnetosphere, which protects them from the solar wind.[30]
The magnetosphere of Jupiter is responsible for intense episodes of radio emission from the planet's polar regions. Volcanic activity on the Jovian moon Io (see below) injects gas into Jupiter's magnetosphere, producing a torus of particles about the planet. As Io moves through this torus, the interaction generates Alfvén waves that carry ionized matter into the polar regions of Jupiter. As a result, radio waves are generated through a cyclotron maser mechanism, and the energy is transmitted out along a cone-shaped surface. When the Earth intersects this cone, the radio emissions from Jupiter can exceed the solar radio output.[62]
## Orbit and rotation
Jupiter (red) completes one orbit of the Sun (center) for every 11.86 orbits of the Earth (blue)
Jupiter is the only planet that has a center of mass with the Sun that lies outside the volume of the Sun, though by only 7% of the Sun's radius.[63] The average distance between Jupiter and the Sun is 778 million km (about 5.2 times the average distance from the Earth to the Sun, or 5.2 AU) and it completes an orbit every 11.86 years. This is two-fifths the orbital period of Saturn, forming a 5:2 orbital resonance between the two largest planets in the Solar System.[64] The elliptical orbit of Jupiter is inclined 1.31° compared to the Earth. Because of an eccentricity of 0.048, the distance from Jupiter and the Sun varies by 75 million km between perihelion and aphelion, or the nearest and most distant points of the planet along the orbital path respectively.
The axial tilt of Jupiter is relatively small: only 3.13°. As a result this planet does not experience significant seasonal changes, in contrast to Earth and Mars for example.[65]
Jupiter's rotation is the fastest of all the Solar System's planets, completing a rotation on its axis in slightly less than ten hours; this creates an equatorial bulge easily seen through an Earth-based amateur telescope. The planet is shaped as an oblate spheroid, meaning that the diameter across its equator is longer than the diameter measured between its poles. On Jupiter, the equatorial diameter is 9,275 km (5,763 mi) longer than the diameter measured through the poles.[39]
Because Jupiter is not a solid body, its upper atmosphere undergoes differential rotation. The rotation of Jupiter's polar atmosphere is about 5 minutes longer than that of the equatorial atmosphere; three systems are used as frames of reference, particularly when graphing the motion of atmospheric features. System I applies from the latitudes 10° N to 10° S; its period is the planet's shortest, at 9h 50m 30.0s. System II applies at all latitudes north and south of these; its period is 9h 55m 40.6s. System III was first defined by radio astronomers, and corresponds to the rotation of the planet's magnetosphere; its period is Jupiter's official rotation.[66]
## Observation
Conjunction of Jupiter and the Moon
The retrograde motion of an outer planet is caused by its relative location with respect to the Earth.
Jupiter is usually the fourth brightest object in the sky (after the Sun, the Moon and Venus);[46] at times Mars appears brighter than Jupiter. Depending on Jupiter's position with respect to the Earth, it can vary in visual magnitude from as bright as −2.9 at opposition down to −1.6 during conjunction with the Sun. The angular diameter of Jupiter likewise varies from 50.1 to 29.8 arc seconds.[3] Favorable oppositions occur when Jupiter is passing through perihelion, an event that occurs once per orbit. As Jupiter approached perihelion in March 2011, there was a favorable opposition in September 2010.[67]
Earth overtakes Jupiter every 398.9 days as it orbits the Sun, a duration called the synodic period. As it does so, Jupiter appears to undergo retrograde motion with respect to the background stars. That is, for a period Jupiter seems to move backward in the night sky, performing a looping motion.
Jupiter's 12-year orbital period corresponds to the dozen astrological signs of the zodiac, and may have been the historical origin of the signs.[22] That is, each time Jupiter reaches opposition it has advanced eastward by about 30°, the width of a zodiac sign.
Because the orbit of Jupiter is outside the Earth's, the phase angle of Jupiter as viewed from the Earth never exceeds 11.5°. That is, the planet always appears nearly fully illuminated when viewed through Earth-based telescopes. It was only during spacecraft missions to Jupiter that crescent views of the planet were obtained.[68] A small telescope will usually show Jupiter's four Galilean Moons and the prominent cloud belts across Jupiter's atmosphere.[69] A large telescope will show Jupiter's Great Red Spot when it faces the Earth.
## Research and exploration
### Pre-telescopic research
Model in the Almagest of the longitudinal motion of Jupiter (☉) relative to the Earth (⊕).
The observation of Jupiter dates back to the Babylonian astronomers of the 7th or 8th century BC.[70] The Chinese historian of astronomy, Xi Zezong, has claimed that Gan De, a Chinese astronomer, made the discovery of one of Jupiter's moons in 362 BC with the unaided eye. If accurate, this would predate Galileo's discovery by nearly two millennia.[71][72] In his 2nd century work the Almagest, the Hellenistic astronomer Claudius Ptolemaeus constructed a geocentric planetary model based on deferents and epicycles to explain Jupiter's motion relative to the Earth, giving its orbital period around the Earth as 4332.38 days, or 11.86 years.[73] In 499, Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and astronomy, also used a geocentric model to estimate Jupiter's period as 4332.2722 days, or 11.86 years.[74]
### Ground-based telescope research
In 1610, Galileo Galilei discovered the four largest moons of Jupiter—Io, Europa, Ganymede and Callisto (now known as the Galilean moons)—using a telescope; thought to be the first telescopic observation of moons other than Earth's. Galileo's was also the first discovery of a celestial motion not apparently centered on the Earth. It was a major point in favor of Copernicus' heliocentric theory of the motions of the planets; Galileo's outspoken support of the Copernican theory placed him under the threat of the Inquisition.[75]
During the 1660s, Cassini used a new telescope to discover spots and colorful bands on Jupiter and observed that the planet appeared oblate; that is, flattened at the poles. He was also able to estimate the rotation period of the planet.[17] In 1690 Cassini noticed that the atmosphere undergoes differential rotation.[30]
False-color detail of Jupiter's atmosphere, imaged by Voyager 1, showing the Great Red Spot and a passing white oval.
The Great Red Spot, a prominent oval-shaped feature in the southern hemisphere of Jupiter, may have been observed as early as 1664 by Robert Hooke and in 1665 by Giovanni Cassini, although this is disputed. The pharmacist Heinrich Schwabe produced the earliest known drawing to show details of the Great Red Spot in 1831.[76]
The Red Spot was reportedly lost from sight on several occasions between 1665 and 1708 before becoming quite conspicuous in 1878. It was recorded as fading again in 1883 and at the start of the 20th century.[77]
Both Giovanni Borelli and Cassini made careful tables of the motions of the Jovian moons, allowing predictions of the times when the moons would pass before or behind the planet. By the 1670s, it was observed that when Jupiter was on the opposite side of the Sun from the Earth, these events would occur about 17 minutes later than expected. Ole Rømer deduced that sight is not instantaneous (a conclusion that Cassini had earlier rejected),[17] and this timing discrepancy was used to estimate the speed of light.[78]
In 1892, E. E. Barnard observed a fifth satellite of Jupiter with the 36-inch (910 mm) refractor at Lick Observatory in California. The discovery of this relatively small object, a testament to his keen eyesight, quickly made him famous. The moon was later named Amalthea.[79] It was the last planetary moon to be discovered directly by visual observation.[80] An additional eight satellites were subsequently discovered before the flyby of the Voyager 1 probe in 1979.
Infrared image of Jupiter taken by the ESO's Very Large Telescope.
In 1932, Rupert Wildt identified absorption bands of ammonia and methane in the spectra of Jupiter.[81]
Three long-lived anticyclonic features termed white ovals were observed in 1938. For several decades they remained as separate features in the atmosphere, sometimes approaching each other but never merging. Finally, two of the ovals merged in 1998, then absorbed the third in 2000, becoming Oval BA.[82]
In 1955, Bernard Burke and Kenneth Franklin detected bursts of radio signals coming from Jupiter at 22.2 MHz.[30] The period of these bursts matched the rotation of the planet, and they were also able to use this information to refine the rotation rate. Radio bursts from Jupiter were found to come in two forms: long bursts (or L-bursts) lasting up to several seconds, and short bursts (or S-bursts) that had a duration of less than a hundredth of a second.[83]
Scientists discovered that there were three forms of radio signals transmitted from Jupiter.
• Decametric radio bursts (with a wavelength of tens of meters) vary with the rotation of Jupiter, and are influenced by interaction of Io with Jupiter's magnetic field.[84]
• Decimetric radio emission (with wavelengths measured in centimeters) was first observed by Frank Drake and Hein Hvatum in 1959.[30] The origin of this signal was from a torus-shaped belt around Jupiter's equator. This signal is caused by cyclotron radiation from electrons that are accelerated in Jupiter's magnetic field.[85]
• Thermal radiation is produced by heat in the atmosphere of Jupiter.[30]
### Exploration with space probes
Since 1973 a number of automated spacecraft have visited Jupiter, most notably the Pioneer 10 space probe, the first spacecraft to get close enough to Jupiter to send back revelations about the properties and phenomena of the Solar System's largest planet.[86][87] Flights to other planets within the Solar System are accomplished at a cost in energy, which is described by the net change in velocity of the spacecraft, or delta-v. Entering a Hohmann transfer orbit from Earth to Jupiter from low Earth orbit requires a delta-v of 6.3 km/s[88] which is comparable to the 9.7 km/s delta-v needed to reach low Earth orbit.[89] Fortunately, gravity assists through planetary flybys can be used to reduce the energy required to reach Jupiter, albeit at the cost of a significantly longer flight duration.[90]
#### Flyby missions
Flyby missions
Spacecraft Closest
approach
Distance
Pioneer 10 December 3, 1973 130,000 km
Pioneer 11 December 4, 1974 34,000 km
Voyager 1 March 5, 1979 349,000 km
Voyager 2 July 9, 1979 570,000 km
Ulysses February 8, 1992[91] 408,894 km
February 4, 2004[91] 120,000,000 km
Cassini December 30, 2000 10,000,000 km
New Horizons February 28, 2007 2,304,535 km
Voyager 1 took this photo of the planet Jupiter on January 24, 1979, while still more than 25 million mi (40 million km) away.
Beginning in 1973, several spacecraft have performed planetary flyby maneuvers that brought them within observation range of Jupiter. The Pioneer missions obtained the first close-up images of Jupiter's atmosphere and several of its moons. They discovered that the radiation fields near the planet were much stronger than expected, but both spacecraft managed to survive in that environment. The trajectories of these spacecraft were used to refine the mass estimates of the Jovian system. Occultations of the radio signals by the planet resulted in better measurements of Jupiter's diameter and the amount of polar flattening.[22][92]
Six years later, the Voyager missions vastly improved the understanding of the Galilean moons and discovered Jupiter's rings. They also confirmed that the Great Red Spot was anticyclonic. Comparison of images showed that the Red Spot had changed hue since the Pioneer missions, turning from orange to dark brown. A torus of ionized atoms was discovered along Io's orbital path, and volcanoes were found on the moon's surface, some in the process of erupting. As the spacecraft passed behind the planet, it observed flashes of lightning in the night side atmosphere.[16][22]
The next mission to encounter Jupiter, the Ulysses solar probe, performed a flyby maneuver to attain a polar orbit around the Sun. During this pass the spacecraft conducted studies on Jupiter's magnetosphere. Since Ulysses has no cameras, no images were taken. A second flyby six years later was at a much greater distance.[91]
In 2000, the Cassini probe, en route to Saturn, flew by Jupiter and provided some of the highest-resolution images ever made of the planet. On December 19, 2000, the spacecraft captured an image of the moon Himalia, but the resolution was too low to show surface details.[93]
The New Horizons probe, en route to Pluto, flew by Jupiter for gravity assist. Its closest approach was on February 28, 2007.[94] The probe's cameras measured plasma output from volcanoes on Io and studied all four Galilean moons in detail, as well as making long-distance observations of the outer moons Himalia and Elara.[95] Imaging of the Jovian system began September 4, 2006.[96][97]
#### Galileo mission
Jupiter as seen by the space probe Cassini.
So far the only spacecraft to orbit Jupiter is the Galileo orbiter, which went into orbit around Jupiter on December 7, 1995. It orbited the planet for over seven years, conducting multiple flybys of all the Galilean moons and Amalthea. The spacecraft also witnessed the impact of Comet Shoemaker-Levy 9 as it approached Jupiter in 1994, giving a unique vantage point for the event. While the information gained about the Jovian system from Galileo was extensive, its originally designed capacity was limited by the failed deployment of its high-gain radio transmitting antenna.[98]
An atmospheric probe was released from the spacecraft in July 1995, entering the planet's atmosphere on December 7. It parachuted through 150 km (93 mi) of the atmosphere, collected data for 57.6 minutes, and was crushed by the pressure to which it was subjected by that time (about 22 times Earth normal, at a temperature of 153 °C).[99] It would have melted thereafter, and possibly vaporized. The Galileo orbiter itself experienced a more rapid version of the same fate when it was deliberately steered into the planet on September 21, 2003, at a speed of over 50 km/s, to avoid any possibility of it crashing into and possibly contaminating Europa—a moon which has been hypothesized to have the possibility of harboring life.[98]
#### Future probes
NASA currently has a mission underway to study Jupiter in detail from a polar orbit. Named Juno, the spacecraft launched in August 2011, and will arrive in late 2016.[100] The next planned mission to the Jovian system will be the European Space Agency's Jupiter Icy Moon Explorer (JUICE), due to launch in 2022.[101]
#### Canceled missions
Because of the possibility of subsurface liquid oceans on Jupiter's moons Europa, Ganymede and Callisto, there has been great interest in studying the icy moons in detail. Funding difficulties have delayed progress. NASA's JIMO (Jupiter Icy Moons Orbiter) was cancelled in 2005.[102] A subsequent proposal for a joint NASA/ESA mission, called EJSM/Laplace, was developed with a provisional launch date around 2020. EJSM/Laplace would have consisted of the NASA-led Jupiter Europa Orbiter, and the ESA-led Jupiter Ganymede Orbiter.[103] However by April 2011, ESA had formally ended the partnership citing budget issues at NASA and the consequences on the mission timetable. Instead ESA planned to go ahead with a European-only mission to compete in its L1 Cosmic Vision selection.[104]
## Moons
Jupiter with the Galilean moons
Jupiter has 67 natural satellites.[105] Of these, 51 are less than 10 kilometres in diameter and have only been discovered since 1975. The four largest moons, known as the "Galilean moons", are Io, Europa, Ganymede and Callisto.
### Galilean moons
The Galilean moons. From left to right, in order of increasing distance from Jupiter: Io, Europa, Ganymede, Callisto.
The orbits of Io, Europa, and Ganymede, some of the largest satellites in the Solar System, form a pattern known as a Laplace resonance; for every four orbits that Io makes around Jupiter, Europa makes exactly two orbits and Ganymede makes exactly one. This resonance causes the gravitational effects of the three large moons to distort their orbits into elliptical shapes, since each moon receives an extra tug from its neighbors at the same point in every orbit it makes. The tidal force from Jupiter, on the other hand, works to circularize their orbits.[106]
The eccentricity of their orbits causes regular flexing of the three moons' shapes, with Jupiter's gravity stretching them out as they approach it and allowing them to spring back to more spherical shapes as they swing away. This tidal flexing heats the moons' interiors by friction. This is seen most dramatically in the extraordinary volcanic activity of innermost Io (which is subject to the strongest tidal forces), and to a lesser degree in the geological youth of Europa's surface (indicating recent resurfacing of the moon's exterior).
The Galilean moons, compared to Earth's Moon
Name IPA Diameter Mass Orbital radius Orbital period
km % kg % km % days %
Io ˈaɪ.oʊ 3643 105 8.9×1022 120 421,700 110 1.77 7
Europa jʊˈroʊpə 3122 90 4.8×1022 65 671,034 175 3.55 13
Ganymede ˈɡænimiːd 5262 150 14.8×1022 200 1,070,412 280 7.15 26
Callisto kəˈlɪstoʊ 4821 140 10.8×1022 150 1,882,709 490 16.69 61
### Classification of moons
Jupiter's moon Europa.
Before the discoveries of the Voyager missions, Jupiter's moons were arranged neatly into four groups of four, based on commonality of their orbital elements. Since then, the large number of new small outer moons has complicated this picture. There are now thought to be six main groups, although some are more distinct than others.
A basic sub-division is a grouping of the eight inner regular moons, which have nearly circular orbits near the plane of Jupiter's equator and are believed to have formed with Jupiter. The remainder of the moons consist of an unknown number of small irregular moons with elliptical and inclined orbits, which are believed to be captured asteroids or fragments of captured asteroids. Irregular moons that belong to a group share similar orbital elements and thus may have a common origin, perhaps as a larger moon or captured body that broke up.[107][108]
Regular moons
Inner group The inner group of four small moons all have diameters of less than 200 km, orbit at radii less than 200,000 km, and have orbital inclinations of less than half a degree.
Galilean moons[109] These four moons, discovered by Galileo Galilei and by Simon Marius in parallel, orbit between 400,000 and 2,000,000 km, and include some of the largest moons in the Solar System.
Irregular moons
Themisto This is a single moon belonging to a group of its own, orbiting halfway between the Galilean moons and the Himalia group.
Himalia group A tightly clustered group of moons with orbits around 11,000,000–12,000,000 km from Jupiter.
Carpo Another isolated case; at the inner edge of the Ananke group, it orbits Jupiter in prograde direction.
Ananke group This retrograde orbit group has rather indistinct borders, averaging 21,276,000 km from Jupiter with an average inclination of 149 degrees.
Carme group A fairly distinct retrograde group that averages 23,404,000 km from Jupiter with an average inclination of 165 degrees.
Pasiphaë group A dispersed and only vaguely distinct retrograde group that covers all the outermost moons.
## Interaction with the Solar System
Along with the Sun, the gravitational influence of Jupiter has helped shape the Solar System. The orbits of most of the system's planets lie closer to Jupiter's orbital plane than the Sun's equatorial plane (Mercury is the only planet that is closer to the Sun's equator in orbital tilt), the Kirkwood gaps in the asteroid belt are mostly caused by Jupiter, and the planet may have been responsible for the Late Heavy Bombardment of the inner Solar System's history.[110]
This diagram shows the Trojan asteroids in Jupiter's orbit, as well as the main asteroid belt.
Along with its moons, Jupiter's gravitational field controls numerous asteroids that have settled into the regions of the Lagrangian points preceding and following Jupiter in its orbit around the sun. These are known as the Trojan asteroids, and are divided into Greek and Trojan "camps" to commemorate the Iliad. The first of these, 588 Achilles, was discovered by Max Wolf in 1906; since then more than two thousand have been discovered.[111] The largest is 624 Hektor.
Most short-period comets belong to the Jupiter family—defined as comets with semi-major axes smaller than Jupiter's. Jupiter family comets are believed to form in the Kuiper belt outside the orbit of Neptune. During close encounters with Jupiter their orbits are perturbed into a smaller period and then circularized by regular gravitational interaction with the Sun and Jupiter.[112]
### Impacts
Hubble image taken on July 23 showing a blemish of about 5,000 miles long left by the 2009 Jupiter impact.[113]
Jupiter has been called the Solar System's vacuum cleaner,[114] because of its immense gravity well and location near the inner Solar System. It receives the most frequent comet impacts of the Solar System's planets.[115] It was thought that the planet served to partially shield the inner system from cometary bombardment. Recent computer simulations suggest that Jupiter does not cause a net decrease in the number of comets that pass through the inner Solar System, as its gravity perturbs their orbits inward in roughly the same numbers that it accretes or ejects them.[116] This topic remains controversial among astronomers, as some believe it draws comets towards Earth from the Kuiper belt while others believe that Jupiter protects Earth from the alleged Oort cloud.[117]
A 1997 survey of historical astronomical drawings suggested that the astronomer Cassini may have recorded an impact scar in 1690. The survey determined eight other candidate observations had low or no possibilities of an impact.[118] A fireball was photographed by Voyager 1 during its Jupiter encounter in March 1979.[119] During the period July 16, 1994, to July 22, 1994, over 20 fragments from the comet Shoemaker–Levy 9 (SL9, formally designated D/1993 F2) collided with Jupiter's southern hemisphere, providing the first direct observation of a collision between two Solar System objects. This impact provided useful data on the composition of Jupiter's atmosphere.[120][121]
On July 19, 2009, an impact site was discovered at approximately 216 degrees longitude in System 2.[122][123] This impact left behind a black spot in Jupiter's atmosphere, similar in size to Oval BA. Infrared observation showed a bright spot where the impact took place, meaning the impact warmed up the lower atmosphere in the area near Jupiter's south pole.[124]
A fireball, smaller than the previous observed impacts, was detected on June 3, 2010, by Anthony Wesley, an amateur astronomer in Australia, and was later discovered to have been captured on video by another amateur astronomer in the Philippines.[125] Yet another fireball was seen on August 20, 2010.[126]
On September 10, 2012, another fireball was detected.[119][127]
## Possibility of life
In 1953, the Miller–Urey experiment demonstrated that a combination of lightning and the chemical compounds that existed in the atmosphere of a primordial Earth could form organic compounds (including amino acids) that could serve as the building blocks of life. The simulated atmosphere included water, methane, ammonia and molecular hydrogen; all molecules still found in the atmosphere of Jupiter. The atmosphere of Jupiter has a strong vertical air circulation, which would carry these compounds down into the lower regions. The higher temperatures within the interior of the atmosphere breaks down these chemicals, which would hinder the formation of Earth-like life.[128]
It is considered highly unlikely that there is any Earth-like life on Jupiter, as there is only a small amount of water in the atmosphere and any possible solid surface deep within Jupiter would be under extraordinary pressures. In 1976, before the Voyager missions, it was hypothesized that ammonia or water-based life could evolve in Jupiter's upper atmosphere. This hypothesis is based on the ecology of terrestrial seas which have simple photosynthetic plankton at the top level, fish at lower levels feeding on these creatures, and marine predators which hunt the fish.[129][130]
The possible presence of underground oceans on some of Jupiter's moons has led to speculation that the presence of life is more likely there.
## Mythology
Jupiter, woodcut from a 1550 edition of Guido Bonatti's Liber Astronomiae.
The planet Jupiter has been known since ancient times. It is visible to the naked eye in the night sky and can occasionally be seen in the daytime when the sun is low.[131] To the Babylonians, this object represented their god Marduk. They used the roughly 12-year orbit of this planet along the ecliptic to define the constellations of their zodiac.[22][132]
The Romans named it after Jupiter (Latin: Iuppiter, Iūpiter) (also called Jove), the principal god of Roman mythology, whose name comes from the Proto-Indo-European vocative compound *Dyēu-pəter (nominative: *Dyēus-pətēr, meaning "O Father Sky-God", or "O Father Day-God").[133] In turn, Jupiter was the counterpart to the mythical Greek Zeus (Ζεύς), also referred to as Dias (Δίας), the planetary name of which is retained in modern Greek.[134]
The astronomical symbol for the planet, , is a stylized representation of the god's lightning bolt. The original Greek deity Zeus supplies the root zeno-, used to form some Jupiter-related words, such as zenographic.[135]
Jovian is the adjectival form of Jupiter. The older adjectival form jovial, employed by astrologers in the Middle Ages, has come to mean "happy" or "merry," moods ascribed to Jupiter's astrological influence.[136]
The Chinese, Korean and Japanese referred to the planet as the wood star, Chinese: 木星; pinyin: mùxīng, based on the Chinese Five Elements.[137] Chinese Taoism personified it as the Fu star. The Greeks called it Φαέθων, Phaethon, "blazing." In Vedic Astrology, Hindu astrologers named the planet after Brihaspati, the religious teacher of the gods, and often called it "Guru", which literally means the "Heavy One."[138] In the English language, Thursday is derived from "Thor's day", with Thor associated with the planet Jupiter in Germanic mythology.[139]
In the Central Asian-Turkic myths, Jupiter called as a "Erendiz/Erentüz", which means "eren(?)+yultuz(star)". There are many theories about meaning of "eren". Also, these peoples calculated the orbit of Jupiter as 11 years and 300 days. They believed that some social and natural events connected to Erentüz's movements on the sky.[140]
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Cite error: Invalid parameter: use the {{reflist}} template with the group parameter (see the help page). | 2017-08-19 12:54:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5356248021125793, "perplexity": 2044.378742480897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105451.99/warc/CC-MAIN-20170819124333-20170819144333-00625.warc.gz"} |
http://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.2010.47.5.997 | - Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
Some reduced free products of abelian $C^*$-algebras and some $C^*$-subalgebra in a free product Bull. Korean Math. Soc. 2010 Vol. 47, No. 5, 997-1010 https://doi.org/10.4134/BKMS.2010.47.5.997Published online September 1, 2010 Jaeseong Heo and Jeong Hee Kim Hanyang University, Hanyang University Abstract : We prove that the reduced free product of $k \times k$ matrix algebras over abelian $C^*$-algebras is not the minimal tensor product of reduced free products of $k \times k$ matrix algebras over abelian $C^*$-algebras. It is shown that the reduced group $C^*$-algebra associated with a group having the property $T$ of Kazhdan is not isomorphic to a reduced free product of abelian $C^*$-algebras or the minimal tensor product of such reduced free products. The infinite tensor product of reduced free products of abelian $C^*$-algebras is not isomorphic to the tensor product of a nuclear $C^*$-algebra and a reduced free product of abelian $C^*$-algebra. We discuss the freeness of free product II$_1$-factors and solidity of free product II$_1$-factors weaker than that of Ozawa. We show that the freeness in a free product is related to the existence of Cartan subalgebras in free product II$_1$-factors. Finally, we give a free product factor which is not solid in the weak sense. Keywords : free product of $C^*$-algebras, Powers' group, minimal tensor product, stable rank 1, prime factor, property $T$, Cartan subalgebra MSC numbers : Primary 46L09, Secondary 46L06 Downloads: Full-text PDF | 2019-12-12 02:51:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.389342725276947, "perplexity": 1038.7554775027534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540536855.78/warc/CC-MAIN-20191212023648-20191212051648-00017.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-1-2-cumulative-review-page-152/19 | ## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$1.83$
To find the decimal notation for the given number, $183$%, remove the percent sign and move the decimal point $2$ places to the left. Hence, the decimal equivalent is $1.83$. | 2018-08-15 06:15:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9066711068153381, "perplexity": 1175.6966614349753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209884.38/warc/CC-MAIN-20180815043905-20180815063905-00664.warc.gz"} |
https://ocw.cs.pub.ro/courses/ep/labs/02/contents/tasks/ex3?rev=1597148915 | This is an old revision of the document!
03. [??p] Kernel Samepage Merging
KSM is a page de-duplication strategy introduced in kernel version 2.6.32. In case you are wondering, it's not the same thing as the file page cache. KSM was originally developed in tandem with KVM in order to detect data pages with exactly the same content and make their page table entries point to the same physical address (marked Copy-On-Write.) The end goal was to allow more VMs to run on the same host. Since each page must be scanned for identical content, this solution had no chance of scaling well with the available quantity of RAM. So, the developers compromised to scan only with the private anonymous pages that were marked as likely candidates via madvise(addr, length, MADV_MERGEABLE).
[??p] Task A - Check kernel support & enable ksmd
First things first, you need to verify that KSM was enabled during your kernel's compilation. For this, you need to check the Linux make config build file that is stored on your /boot partition. Hopefully, you should see something like this:
$grep CONFIG_KSM /boot/config-$(uname -r)
CONFIG_KSM=y
If you don't have KSM enabled, you could recompile the kernel with the CONFIG_KSM flag and try it, but you don't have to :)
Moving forward. Next thing on the list is to check that the ksmd daemon is functioning. Any configuration that we'll do will be through the sysfs files in /sys/kernel/mm/ksm. Consequently, you should change user to root (even sudo should not allow you to write to these files.)
• /…/read : this is 1 if the daemon is active; write 1 to it if it's not
• /…/pages_to_scan : this is how many pages will be scanned before going to sleep; you can increase this to 1000 if you want to see faster results
• /…/sleep_millisecs : this is how many ms the daemon sleeps in between scans; since you've modified pages_to_scan, you can leave this be
• /…/max_page_sharing : this is the maximum number of pages that can be de-duplicated; in cases like this it's better to go big or go home; so set it to something like 1000000, just to be sure
There are a few more files in the ksm/ directory. We will still use one or two later on. But for now, configuring the previous ones should be enough. Google the rest if you're interested.
[??p] Task B - Watch the magic happen
For this step it would be better to have a few terminals open. First, let's start a vmstat. Keep your eyes on the active memory column when we run the sample program.
$vmstat -wa -S m 1 Next would be a good time to introduce two more files from the ksm/ sysfs directory: • /…/pages_shared : this file reports how many physical pages are in use at the moment • /…/pages_sharing : this file reports how many virtual page table entries point to the aforementioned physical pages For this experiment we will also want to monitor the number of de-duplicated virtual pages, so have at it: $ watch -n 0 cat /sys/kernel/mm/ksm/pages_sharing
Finally, look at the provided code, compile it, and launch the program. As an argument you will need to provide the number of pages that will be allocated and initialized with the same value. Note that not all pages will be de-duplicated instantly. So keep in mind your system's RAM limitations before deciding how much you can spare (1-2GB should be ok, right?)
The result should look something like this:
Click to display ⇲
Click to hide ⇱
Here, we can see the active memory suddenly rising when we start the process. Over the next few seconds, as ksmd starts scanning pages, the active memory slowly drops. Finally, as the process terminates, all memory is reclaimed by the kernel and the active memory returns to roughly the same value as before. If you'll ever want to make use of this in your own experiments, remember to adjust the configurations of ksmd. Waking too often or scanning to many pages at once could end up doing more harm than good. See what works for your particular system. | 2022-12-05 11:16:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38297146558761597, "perplexity": 1521.7590648829018}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711016.32/warc/CC-MAIN-20221205100449-20221205130449-00102.warc.gz"} |
http://www.math.gatech.edu/seminars-colloquia/series/algebra-seminar/elizabeth-gross-20140310 | ## Singular Learning Theory
Series:
Algebra Seminar
Monday, March 10, 2014 - 15:05
1 hour (actually 50 minutes)
Location:
Skiles 005
,
NCSU
Organizer:
Bayesian approaches to statistical model selection requires the evaluation of the marginal likelihood integral, which, in general, is difficult to obtain. When the statistical model is regular, it is well-known that the marginal likelihood integral can be approximated using a function of the maximized log-likelihood function and the dimension of the model. When the model is singular, Sumio Watanabe has shown that an approximation of the marginal likelihood integral can be obtained through resolution of singularities, a result that has intimately tied machine learning and Bayesian model selection to computational algebraic geometry. This talk will be an introduction to singular learning theory with the factor analysis model as a running example. | 2017-12-16 01:22:34 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8434760570526123, "perplexity": 675.5013490666079}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948581033.57/warc/CC-MAIN-20171216010725-20171216032725-00658.warc.gz"} |
https://cplberry.com/blog/ | # Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals
The space-based observatory LISA will detect gravitational waves from massive black holes (giant black holes residing in the centres of galaxies). One particularly interesting signal will come from the inspiral of a regular stellar-mass black hole into a massive black hole. These are called extreme mass-ratio inspirals (or EMRIs, pronounced emries, to their friends) [bonus note]. We have never observed such a system. This means that there’s a lot we have to learn about them. In this work, we systematically investigated the prospects for observing EMRIs. We found that even though there’s a wide range in predictions for what EMRIs we will detect, they should be a safe bet for the LISA mission.
Artistic impression of the spacetime for an extreme-mass-ratio inspiral, with a smaller stellar-mass black hole orbiting a massive black hole. This image is mandatory when talking about extreme-mass-ratio inspirals. Credit: NASA
### LISA & EMRIs
My previous post discussed some of the interesting features of EMRIs. Because of the extreme difference in masses of the two black holes, it takes a long time for them to complete their inspiral. We can measure tens of thousands of orbits, which allows us to make wonderfully precise measurements of the source properties (if we can accurately pick out the signal from the data). Here, we’ll examine exactly what we could learn with LISA from EMRIs [bonus note].
First we build a model to investigate how many EMRIs there could be. There is a lot of astrophysics which we are currently uncertain about, which leads to a large spread in estimates for the number of EMRIs. Second, we look at how precisely we could measure properties from the EMRI signals. The astrophysical uncertainties are less important here—we could get a revolutionary insight into the lives of massive black holes.
### The number of EMRIs
To build a model of how many EMRIs there are, we need a few different inputs:
1. The population of massive black holes
2. The distribution of stellar clusters around massive black holes
3. The range of orbits of EMRIs
We examine each of these in turn, building a more detailed model than has previously been constructed for EMRIs.
We currently know little about the population of massive black holes. This means we’ll discover lots when we start measuring signals (yay), but it’s rather inconvenient now, when we’re trying to predict how many EMRIs there are (boo). We take two different models for the mass distribution of massive black holes. One is based upon a semi-analytic model of massive black hole formation, the other is at the pessimistic end allowed by current observations. The semi-analytic model predicts massive black hole spins around 0.98, but we also consider spins being uniformly distributed between 0 and 1, and spins of 0. This gives us a picture of the bigger black hole, now we need the smaller.
Observations show that the masses of massive black holes are correlated with their surrounding cluster of stars—bigger black holes have bigger clusters. We consider four different versions of this trend: Gültekin et al. (2009); Kormendy & Ho (2013); Graham & Scott (2013), and Shankar et al. (2016). The stars and black holes about a massive black hole should form a cusp, with the density of objects increasing towards the massive black hole. This is great for EMRI formation. However, the cusp is disrupted if two galaxies (and their massive black holes) merge. This tends to happen—it’s how we get bigger galaxies (and black holes). It then takes some time for the cusp to reform, during which time, we don’t expect as many EMRIs. Therefore, we factor in the amount of time for which there is a cusp for massive black holes of different masses and spins.
That’s a nice galaxy you have there. It would be a shame if it were to collide with something… Hubble image of The Mice. Credit: ACS Science & Engineering Team.
Given a cusp about a massive black hole, we then need to know how often an EMRI forms. Simulations give us a starting point. However, these only consider a snap-shot, and we need to consider how things evolve with time. As stellar-mass black holes inspiral, the massive black hole will grow in mass and the surrounding cluster will become depleted. Both these effects are amplified because for each inspiral, there’ll be many more stars or stellar-mass black holes which will just plunge directly into the massive black hole. We therefore need to limit the number of EMRIs so that we don’t have an unrealistically high rate. We do this by adding in a couple of feedback factors, one to cap the rate so that we don’t deplete the cusp quicker than new objects will be added to it, and one to limit the maximum amount of mass the massive black hole can grow from inspirals and plunges. This gives us an idea for the total number of inspirals.
Finally, we calculate the orbits that EMRIs will be on. We again base this upon simulations, and factor in how the spin of the massive black hole effects the distribution of orbital inclinations.
Putting all the pieces together, we can calculate the population of EMRIs. We now need to work out how many LISA would be able to detect. This means we need models for the gravitational-wave signal. Since we are simulating a large number, we use a computationally inexpensive analytic model. We know that this isn’t too accurate, but we consider two different options for setting the end of the inspiral (where the smaller black hole finally plunges) which should bound the true range of results.
Number of EMRIs for different size massive black holes in different astrophysical models. M1 is our best estimate, the others explore variations on this. M11 and M12 are designed to be cover the extremes, being the most pessimistic and optimistic combinations. The solid and dashed lines are for two different signal models (AKK and AKS), which are designed to give an indication of potential variation. They agree where the massive black hole is not spinning (M10 and M11). The range of masses is similar for all models, as it is set by the sensitivity of LISA. We can detect higher mass systems assuming the AKK signal model as it includes extra inspiral close to highly spinning black holes: for the heaviest black holes, this is the only part of the signal at high enough frequency to be detectable. Figure 8 of Babak et al. (2017).
Allowing for all the different uncertainties, we find that there should be somewhere between 1 and 4200 EMRIs detected per year. (The model we used when studying transient resonances predicted about 250 per year, albeit with a slightly different detector configuration, which is fairly typical of all the models we consider here). This range is encouraging. The lower end means that EMRIs are a pretty safe bet, we’d be unlucky not to get at least one over the course of a multi-year mission (LISA should have at least four years observing). The upper end means there could be lots—we might actually need to worry about them forming a background source of noise if we can’t individually distinguish them!
### EMRI measurements
Having shown that EMRIs are a good LISA source, we now need to consider what we could learn by measuring them?
We estimate the precision we will be able to measure parameters using the Fisher information matrix. The Fisher matrix measures how sensitive our observations are to changes in the parameters (the more sensitive we are, the better we should be able to measure that parameter). It should be a lower bound on actual measurement precision, and well approximate the uncertainty in the high signal-to-noise (loud signal) limit. The combination of our use of the Fisher matrix and our approximate signal models means our results will not be perfect estimates of real performance, but they should give an indication of the typical size of measurement uncertainties.
Given that we measure a huge number of cycles from the EMRI signal, we can make really precise measurements of the the mass and spin of the massive black hole, as these parameters control the orbital frequencies. Below are plots for the typical measurement precision from our Fisher matrix analysis. The orbital eccentricity is measured to similar accuracy, as it influences the range of orbital frequencies too. We also get pretty good measurements of the the mass of the smaller black hole, as this sets how quickly the inspiral proceeds (how quickly the orbital frequencies change). EMRIs will allow us to do precision astronomy!
Distribution of (one standard deviation) fractional uncertainties for measurements of the massive black hole (redshifted) mass $M_z$. Results are shown for the different astrophysical models, and for the different signal models. The astrophysical model has little impact on the uncertainties. M4 shows a slight difference as it assumes heavier stellar-mass black holes. The results with the two signal models agree when the massive black hole is not spinning (M10 and M11). Otherwise, measurements are more precise with the AKK signal model, as this includes extra signal from the end of the inspiral. Part of Figure 11 of Babak et al. (2017).
Distribution of (one standard deviation) uncertainties for measurements of the massive black hole spin $a$. The results mirror those for the masses above. Part of Figure 11 of Babak et al. (2017).
Now, before you get too excited that we’re going to learn everything about massive black holes, there is one confession I should make. In the plot above I show the measurement accuracy for the redshifted mass of the massive black hole. The cosmological expansion of the Universe causes gravitational waves to become stretched to lower frequencies in the same way light is (this makes visible light more red, hence the name). The measured frequency is $f_z = (1 + z)f$ where $f$ is the frequency emitted, and $z$ is the redshift ($z= 0$ for a nearby source, and is larger for further away sources). Lower frequency gravitational waves correspond to higher mass systems, so it is often convenient to work with the redshifted mass, the mass corresponding to the signal you measure if you ignore redshifting. The redshifted mass of the massive black hole is $M_z = (1+z)M$ where $M$ is the true mass. To work out the true mass, we need the redshift, which means we need to measure the distance to the source.
Distribution of (one standard deviation) fractional uncertainties for measurements of the luminosity distance $D_\mathrm{L}$. The signal model is not as important here, as the uncertainty only depends on how loud the signal is. Part of Figure 12 of Babak et al. (2017).
The plot above shows the fractional uncertainty on the distance. We don’t measure this too well, as it is determined from the amplitude of the signal, rather than its frequency components. The situation is much as for LIGO. The larger uncertainties on the distance will dominate the overall uncertainty on the black hole masses. We won’t be getting all these to fractions of a percent. However, that doesn’t mean we can’t still figure out what the distribution of masses looks like!
One of the really exciting things we can do with EMRIs is check that the signal matches our expectations for a black hole in general relativity. Since we get such an excellent map of the spacetime of the massive black hole, it is easy to check for deviations. In general relativity, everything about the black hole is fixed by its mass and spin (often referred to as the no-hair theorem). Using the measured EMRI signal, we can check if this is the case. One convenient way of doing this is to describe the spacetime of the massive object in terms of a multipole expansion. The first (most important) terms gives the mass, and the next term the spin. The third term (the quadrupole) is set by the first two, so if we can measure it, we can check if it is consistent with the expected relation. We estimated how precisely we could measure a deviation in the quadrupole. Fortunately, for this consistency test, all factors from redshifting cancel out, so we can get really detailed results, as shown below. Using EMRIs, we’ll be able to check for rally small differences from general relativity!
Distribution of (one standard deviation) of uncertainties for deviations in the quadrupole moment of the massive object spacetime $\mathcal{Q}$. Results are similar to the mass and spin measurements. Figure 13 of Babak et al. (2017).
In summary: EMRIS are awesome. We’re not sure how many we’ll detect with LISA, but we’re confident there will be some, perhaps a couple of hundred per year. From the signals we’ll get new insights into the masses and spins of black holes. This should tell us something about how they, and their surrounding galaxies, evolved. We’ll also be able to do some stringent tests of whether the massive objects are black holes as described by general relativity. It’s all pretty exciting, for when LISA launches, which is currently planned about 2034…
One of the most valuable traits a student or soldier can have: patience. Credit: Sony/Marvel
arXiv: 1703.09722 [gr-qc]
Journal: Physical Review D; 95(10):103012(21); 2017
Conference proceedings: 1704.00009 [astro-ph.GA] (from when work was still in-progress)
Estimated number of Marvel films before LISA launch: 48 (starting with Ant-Man and the Wasp)
### Bonus notes
#### Hyphenation
Is it “extreme-mass-ratio inspiral”, “extreme mass-ratio inspiral” or “extreme mass ratio inspiral”? All are used in the literature. This is one of the advantage of using “EMRI”. The important thing is that we’re talking about inspirals that have a mass ratio which is extreme. For this paper, we used “extreme mass-ratio inspiral”, but when I first started my PhD, I was first introduced to “extreme-mass-ratio inspirals”, so they are always stuck that way in my mind.
I think hyphenation is a bit of an art, and there’s no definitive answer here, just like there isn’t for superhero names, where you can have Iron Man, Spider-Man or Iceman.
#### Science with LISA
This paper is part of a series looking at what LISA could tells us about different gravitational wave sources. So far, this series covers
1. Massive black hole binaries
2. Cosmological phase transitions
3. Standard sirens (for measuring the expansion of the Universe)
4. Inflation
5. Extreme-mass-ratio inspirals
You’ll notice there’s a change in the name of the mission from eLISA to LISA part-way through, as things have evolved. (Or devolved?) I think the main take-away so far is that the cosmology group is the most enthusiastic.
# Importance of transient resonances in extreme-mass-ratio inspirals
Extreme-mass-ratio inspirals (EMRIs for short) are a promising source for the planned space-borne gravitational-wave observatory LISA. To detect and analyse them we need accurate models for the signals, which are exquisitely intricate. In this paper, we investigated a feature, transient resonances, which have not previously included in our models. They are difficult to incorporate, but can have a big impact on the signal. Fortunately, we find that we can still detect the majority of EMRIs, even without including resonances. Phew!
### EMRIs and orbits
EMRIs are a beautiful gravitational wave source. They occur when a stellar-mass black hole slowly inspirals into a massive black hole (as found in the centre of galaxies). The massive black hole can be tens of thousands or millions of times more massive than the stellar-mass black hole (hence extreme mass ratio). This means that the inspiral is slow—we can potentially measure tens of thousands of orbits. This is both the blessing and the curse of EMRIs. The huge numbers of cycles means that we can closely follow the inspiral, and build a detailed map of the massive black hole’s spacetime. EMRIs will give us precision measurements of the properties of massive black holes. However, to do this, we need to be able to find the EMRI signals in the data, we need models which can match the signals over all these cycles. Analysing EMRIs is a huge challenge.
EMRI orbits are complicated. At any moment, the orbit can be described by three orbital frequencies: one for radial (in/out) motion $\Omega_r$, one for polar (north/south if we think of the spin of the massive black hole like the rotation of the Earth) motion $\Omega_\theta$ and one for axial (around in the east/west direction) motion. As gravitational waves are emitted, and the orbit shrinks, these frequencies evolve. The animation above, made by Steve Drasco, illustrates the evolution of an EMRI. Every so often, so can see the pattern freeze—the orbits stays in a constant shape (although this still rotates). This is a transient resonance. Two of the orbital frequencies become commensurate (so we might have 3 north/south cycles and 2 in/out cycles over the same period [bonus note])—this is the resonance. However, because the frequencies are still evolving, we don’t stay locked like this is forever—which is why the resonance is transient. To calculate an EMRI, you need to know how the orbital frequencies evolve.
The evolution of an EMRI is slow—the time taken to inspiral is much longer than the time taken to complete one orbit. Therefore, we can usually split the problem of calculating the trajectory of an EMRI into two parts. On short timescales, we can consider orbits as having fixed frequencies. On long timescale, we can calculate the evolution by averaging over many orbits. You might see the problem with this—around resonances, this averaging breaks down. Whereas normally averaging over many orbits means averaging over a complicated trajectory that hits pretty much all possible points in the orbital range, on resonance, you just average over the same bit again and again. On resonance, terms which usually average to zero can become important. Éanna Flanagan and Tanja Hinderer first pointed out that around resonances the usual scheme (referred to as the adiabatic approximation) doesn’t work.
A non-resonant EMRI orbit in three dimensions (left) and two dimensions (right), ignoring the rotation in the axial direction. A non-resonant orbit will eventually fill the $r$$\theta$ plane. Credit: Rob Cole
For comparison, a resonant EMRI orbit. A 2:3 resonance traces the same parts of the $r$$\theta$ plane over and over. Credit: Rob Cole
Around a resonance, the evolution will be enhanced or decreased a little relative to the standard adiabatic evolution. We get a kick. This is only small, but because we observe EMRIs for so many orbits, a small difference can grow to become a significant difference later on. Does this mean that we won’t be able to detect EMRIs with our standard models? This was a concern, so back at the end of PhD I began to investigate [bonus note]. The first step is to understand the size of the kick.
A jump in the orbital energy across a 2:3 resonance. The plot shows the difference between the approximate adiabatic evolution and the instantaneous evolution including the resonance. The thickness of the blue line is from oscillations on the orbital timescale which is too short to resolve here. The dotted red line shows the fitted size of the jump. Time is measured in terms of the resonance time $\tau_\mathrm{res}$ which is defined below. Figure 4 of Berry et al. (2016).
### Resonance kicks
If there were no gravitational waves, the orbit would not evolve, it would be fixed. The orbit could then be described by a set of constants of motion. The most commonly used when describing orbits about black holes are the energy, angular momentum and Carter constant. For the purposes of this blog, we’ll not worry too much about what these constants are, we’ll just consider some constant $I$.
The resonance kick is a change in this constant $\Delta I$. What should this depend on? There are three ingredients. First, the rate of change of this constant $F$ on the resonant orbit. Second, the time spent on resonance $\tau_\mathrm{res}$. The bigger these are, the bigger the size of the jump. Therefore,
$|\Delta I| \propto F \tau_\mathrm{res}$.
However, the jump could be positive or negative. This depends upon the relative phase of the radial and polar motion [bonus note]—for example, do they both reach their maximum point at the same time, or does one lag behind the other? We’ll call this relative phase $q$. By varying $q$ we explore we can get our resonant trajectory to go through any possible point in space. Therefore, averaging over $q$ should get us back to the adiabatic approximation: the average value of $\Delta I$ must be zero. To complete our picture for the jump, we need a periodic function of the phase,
$\Delta I = F \tau_\mathrm{res} f(q)$,
with $\langle f(q) \rangle_q = 0$. Now, we know the pieces, we can try to figure out what the pieces are.
The rate of change $F$ is proportional the mass ratio $\eta \ll 1$: the smaller the stellar-mass black hole is relative to the massive one, the smaller $F$ is. The exact details depend upon gravitational self-force calculations, which we’ll skip over, as they’re pretty hard, but they are the same for all orbits (resonant or not).
We can think of the resonance timescale either as the time for the orbital frequencies to drift apart or the time for the orbit to start filling the space again (so that it’s safe to average). The two pictures yield the same answer—there’s a fuller explanation in Section III A of the paper. To define the resonance timescale, it is useful to define the frequency $\Omega = n_r \Omega_r - n_\theta \Omega_\theta$, which is zero exactly on resonance. If this is evolving at rate $\dot{\Omega}$, then the resonance timescale is
$\displaystyle \tau_\mathrm{res} = \left[\frac{2\pi}{\dot{\Omega}}\right]^{1/2}$.
This bridges the two timescales that usually define EMRIs: the short orbital timescale $T$ and the long evolution timescale $\tau_\mathrm{ev}$:
$T \sim \eta^{1/2} \tau_\mathrm{res} \sim \eta \tau_\mathrm{ev}$.
To find the form of for $f(q)$, we need to do some quite involved maths (given in Appendix B of the paper) [bonus note]. This works by treating the evolution far from resonance as depending upon two independent times (effectively defining $T$ and $\tau_\mathrm{ev}$), and then matching the evolution close to resonance using an expansion in terms of a different time (something like $\tau_\mathrm{res}$). The solution shows that the jump depends sensitively upon the phase $q$ at resonance, which makes them extremely difficult to calculate.
We numerically evaluated the size of kicks for different orbits and resonances. We found a number of trends. First, higher-order resonances (those with larger $n_r$ and $n_\theta$) have smaller jumps than lower-order ones. This makes sense, as higher-order resonances come closer to covering all the points in the space, and so are more like averaging over the entire space. Second, jumps are larger for higher eccentricity orbits. This also makes sense, as you can’t have resonances for circular (zero eccentricity orbits) as there’s no radial frequency, so the size of the jumps must tend to zero. We’ll see that these two points are important when it comes to observational consequences of transient resonances.
### Astrophysical EMRIs
Now we’ve figured out the impact of passing through a transient resonance, let’s look at what this means for detecting EMRIs. The jump can mean that the evolution post-resonance can soon become out of phase with that pre-resonance. We can’t match both parts with the same adiabatic template. This could significantly hamper our prospects for detection, as we’re limited to the bits of signal we can pick up between resonances.
We created an astrophysical population of simulated EMRIs. We used numerical simulations to estimate a plausible population of massive black holes and distribution of stellar-mass black holes insprialling into them. We then used adiabatic models to see how many LISA (or eLISA as it was called at the time) could potentially detect. We found there were ~510 EMRIs detectable (with a signal-to-noise ratio of 15 or above) for a two-year mission.
We then calculated how much the signal-to-noise ratio would be reduced by passing through transient resonances. The plot below shows the distribution of signal-to-noise ratio for the original population, ignoring resonances, and then after factoring in the reduction. There are now ~490 detectable EMRIs, a loss of 4%. We can still detect the majority of EMRIs!
Distribution of signal-to-noise ratios for EMRIs. In blue (solid outline), we have the results ignoring transient resonances. In orange (dashed outline), we have the distribution including the reduction due to resonance jumps. Events falling below 15 are deemed to be undetectable. Figure 10 of Berry et al. (2016).
We were worried about the impact of transient resonances, we know that jumps can cause them to become undetectable, so why aren’t we seeing a bit effect in our population? The answer lies is in the trends we saw earlier. Jumps are large for low order resonances with high eccentricities. These were the ones first highlighted, as they are obviously the most important. However, low-order resonances are only encountered really close to the massive black hole. This means late in the inspiral, after we have already accumulated lots of signal-to-noise ratio. Losing a little bit of signal right at the end doesn’t hurt detectability too much. On top of this, gravitational wave emission efficiently damps down eccentricity. Orbits typically have low eccentricities by the time they hit low-order resonances, meaning that the jumps are actually quite small. Although small jumps lead to some mismatch, we can still use our signal templates without jumps. Therefore, resonances don’t hamper us (too much) in finding EMRIs!
This may seem like a happy ending, but it is not the end of the story. While we can detect EMRIs, we still need to be able to accurately infer their source properties. Features not included in our signal templates (like jumps), could bias our results. For example, it might be that we can better match a jump by using a template for a different black hole mass or spin. However, if we include jumps, these extra features could give us extra precision in our measurements. The question of what jumps could mean for parameter estimation remains to be answered.
arXiv: 1608.08951 [gr-qc]
Journal: Physical Review D; 94(12):124042(24); 2016
Conference proceedings: 1702.05481 [gr-qc] (only 2 pages—ideal for emergency journal club presentations)
Favourite jumpers: Woolly, Mario, Kangaroos
### Bonus notes
#### Radial and polar only
When discussing resonances, and their impact on orbital evolution, we’ll only care about $\Omega_r$$\Omega_\theta$ resonances. Resonances with $\Omega_\phi$ are not important because the spacetime is axisymmetric. The equations are exactly identical for all values of the the axial angle $\phi$, so it doesn’t matter where you are (or if you keep cycling over the same spot) for the evolution of the EMRI.
This, however, doesn’t mean that $\Omega_\phi$ resonances aren’t interesting. They can lead to small kicks to the binary, because you are preferentially emitting gravitational waves in one direction. For EMRIs this are negligibly small, but for more equal mass systems, they could have some interesting consequences as pointed out by Maarten van de Meent.
#### Extra time
I’m grateful to the Cambridge Philosophical Society for giving me some extra funding to work on resonances. If you’re a Cambridge PhD student, make sure to become a member so you can take advantage of the opportunities they offer.
#### Calculating jumps
The theory of how to evolve through a transient resonance was developed by Kevorkian and coauthors. I spent a long time studying these calculations before working up the courage to attempt them myself. There are a few technical details which need to be adapted for the case of EMRIs. I finally figured everything out while in Warsaw Airport, coming back from a conference. It was the most I had ever felt like a real physicist.
Transient resonances remind me of Spirographs. Thanks Frinkiac
# GW170817—The papers
After three months (and one binary black hole detection announcement), I finally have time to write about the suite of LIGO–Virgo papers put together to accompany GW170817.
### The papers
There are currently 9 papers in the GW170817 family. Further papers, for example looking at parameter estimation in detail, are in progress. Papers are listed below in order of arXiv posting. My favourite is the GW170817 Discovery Paper. Many of the highlights, especially from the Discovery and Multimessenger Astronomy Papers, are described in my GW170817 announcement post.
Keeping up with all the accompanying observational results is a task not even Sisyphus would envy. I’m sure that the details of these will be debated for a long time to come. I’ve included references to a few below (mostly as [citation notes]), but these are not guaranteed to be complete (I’ll continue to expand these in the future).
#### 0. The GW170817 Discovery Paper
Title: GW170817: Observation of gravitational waves from a binary neutron star inspiral
arXiv:
1710.05832 [gr-qc]
Journal:
Physical Review Letters; 119(16):161101(18); 2017
LIGO science summary:
GW170817: Observation of gravitational waves from a binary neutron star inspiral
This is the paper announcing the gravitational-wave detection. It gives an overview of the properties of the signal, initial estimates of the parameters of the source and the binary neutron star merger rate, as well as an overview of results from the other companion papers.
I was disappointed that “the era of gravitational-wave multi-messenger astronomy has opened with a bang” didn’t make the conclusion of the final draft.
More details: The GW170817 Discovery Paper summary
#### −1. The Multimessenger Astronomy Paper
Title: Multi-messenger observations of a binary neutron star merger
arXiv:
1710.05833 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 848(2):L12(59); 2017
LIGO science summary:
The dawn of multi-messenger astrophysics: observations of a binary neutron star merger
I’ve numbered this paper as −1 as it gives an overview of all the observations—gravitational wave, electromagnetic and neutrino—accompanying GW170817. I feel a little sorry for the neutrino observers, as they’re the only ones not to make a detection. Drawing together the gravitational wave and electromagnetic observations, we can confirm that binary neutron star mergers are the progenitors of (at least some) short gamma-ray bursts and kilonovae.
Do not print this paper, the author list stretches across 23 pages.
More details: The Multimessenger Astronomy Paper summary
#### 1. The GW170817 Gamma-ray Burst Paper
Title: Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A
arXiv:
1710.05834 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 848(2):L13(27); 2017
LIGO science summary:
Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A
Here we bring together the LIGO–Virgo observations of GW170817 and the Fermi and INTEGRAL observations of GRB 170817A. From the spatial and temporal coincidence of the gravitational waves and gamma rays, we establish that the two are associated with each other. There is a 1.7 s time delay between the merger time estimated from gravitational waves and the arrival of the gamma-rays. From this, we make some inferences about the structure of the jet which is the source of the gamma rays. We can also use this to constrain deviations from general relativity, which is cool. Finally, we estimate that there be 0.3–1.7 joint gamma ray–gravitational wave detections per year once our gravitational-wave detectors reach design sensitivity!
More details: The GW170817 Gamma-ray Burst Paper summary
#### 2. The GW170817 Hubble Constant Paper
Title: A gravitational-wave standard siren measurement of the Hubble constant [bonus note]
arXiv:
1710.05835 [astro-ph.CO]
Journal:
Nature; 551(7678):85–88; 2017 [bonus note]
LIGO science summary:
Measuring the expansion of the Universe with gravitational waves
The Hubble constant quantifies the current rate of expansion of the Universe. If you know how far away an object is, and how fast it is moving away (due to the expansion of the Universe, not because it’s on a bus or something, that is important), you can estimate the Hubble constant. Gravitational waves give us an estimate of the distance to the source of GW170817. The observations of the optical transient AT 2017gfo allow us to identify the galaxy NGC 4993 as the host of GW170817’s source. We know the redshift of the galaxy (which indicates how fast its moving). Therefore, putting the two together we can infer the Hubble constant in a completely new way.
More details: The GW170817 Hubble Constant Paper summary
#### 3. The GW170817 Kilonova Paper
Title: Estimating the contribution of dynamical ejecta in the kilonova associated with GW170817
arXiv:
1710.05836 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 850(2):L39(13); 2017
LIGO science summary:
Predicting the aftermath of the neutron star collision that produced GW170817
During the coalescence of two neutron stars, lots of neutron-rich matter gets ejected. This undergoes rapid radioactive decay, which powers a kilonova, an optical transient. The observed signal depends upon the material ejected. Here, we try to use our gravitational-wave measurements to predict the properties of the ejecta ahead of the flurry of observational papers.
More details: The GW170817 Kilonova Paper summary
#### 4. The GW170817 Stochastic Paper
Title: GW170817: Implications for the stochastic gravitational-wave background from compact binary coalescences
arXiv:
1710.05837 [gr-qc]
Journal: Physical Review Letters; 120(9):091101(12); 2018
LIGO science summary: The background symphony of gravitational waves from neutron star and black hole mergers
We can detect signals if they are loud enough, but there will be many quieter ones that we cannot pick out from the noise. These add together to form an overlapping background of signals, a background rumbling in our detectors. We use the inferred rate of binary neutron star mergers to estimate their background. This is smaller than the background from binary black hole mergers (black holes are more massive, so they’re intrinsically louder), but they all add up. It’ll still be a few years before we could detect a background signal.
More details: The GW170817 Stochastic Paper summary
#### 5. The GW170817 Progenitor Paper
Title: On the progenitor of binary neutron star merger GW170817
arXiv:
1710.05838 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 850(2):L40(18); 2017
LIGO science summary:
Making GW170817: neutron stars, supernovae and trick shots (I’d especially recommend reading this one)
We know that GW170817 came from the coalescence of two neutron stars, but where did these neutron stars come from? Here, we combine the parameters inferred from our gravitational-wave measurements, the observed position of AT 2017gfo in NGC 4993 and models for the host galaxy, to estimate properties like the kick imparted to neutron stars during the supernova explosion and how long it took the binary to merge.
More details: The GW170817 Progenitor Paper summary
#### 6. The GW170817 Neutrino Paper
Title: Search for high-energy neutrinos from binary neutron star merger GW170817 with ANTARES, IceCube, and the Pierre Auger Observatory
arXiv:
1710.05839 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 850(2):L35(18); 2017
This is the search for neutrinos from the source of GW170817. Lots of neutrinos are emitted during the collision, but not enough to be detectable on Earth. Indeed, we don’t find any neutrinos, but we combine results from three experiments to set upper limits.
More details: The GW170817 Neutrino Paper summary
#### 7. The GW170817 Post-merger Paper
Title: Search for post-merger gravitational waves from the remnant of the binary neutron star merger GW170817
arXiv:
1710.09320 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 851(1):L16(13); 2017
LIGO science summary:
Searching for the neutron star or black hole resulting from GW170817
After the two neutrino stars merged, what was left? A larger neutron star or a black hole? Potentially we could detect gravitational waves from a wibbling neutron star, as it sloshes around following the collision. We don’t. It would have to be a lot closer for this to be plausible. However, this paper outlines how to search for such signals.
More details: The GW170817 Post-merger Paper summary
### The GW170817 Discovery Paper
Synopsis: GW170817 Discovery Paper
Read this if: You want all the details of our first gravitational-wave observation of a binary neutron star coalescence
Favourite part: Look how well we measure the chirp mass!
GW170817 was a remarkable gravitational-wave discovery. It is the loudest signal observed to date, and the source with the lowest mass components. I’ve written about some of the highlights of the discovery in my previous GW170817 discovery post.
Binary neutron stars are one of the principal targets for LIGO and Virgo. The first observational evidence for the existence of gravitational waves came from observations of binary pulsars—a binary neutron star system where (at least one) one of the components is a pulsar. Therefore (unlike binary black holes), we knew that these sources existed before we turned on our detectors. What was less certain was how often they merge. In our first advanced-detector observing run (O1), we didn’t find any, allowing us to estimate an upper limit on the merger rate of $12600~\mathrm{Gpc^{-1}\,yr^{-1}}$. Now, we know much more about merging binary neutron stars.
GW170817, as a loud and long signal, is a highly significant detection. You can see it in the data by eye. Therefore, it should have been a easy detection. As is often the case with real experiments, it wasn’t quite that simple. Data transfer from Virgo had stopped over night, and there was a glitch (a non-stationary and non-Gaussian noise feature) in the Livingston detector, which meant that this data weren’t automatically analysed. Nevertheless, GstLAL flagged something interesting in the Hanford data, and there was a mad flurry to get the other data in place so that we could analyse the signal in all three detectors. I remember being sceptical in these first few minutes until I saw the plot of Livingston data which blew me away: the chirp was clearly visible despite the glitch!
Time–frequency plots for GW170104 as measured by Hanford, Livingston and Virgo. The Livinston data have had the glitch removed. The signal is clearly visible in the two LIGO detectors as the upward sweeping chirp; it is not visible in Virgo because of its lower sensitivity and the source’s position in the sky. Figure 1 of the GW170817 Discovery Paper.
Using data from both of our LIGO detectors (as discussed for GW170814, our offline algorithms searching for coalescing binaries only use these two detectors during O2), GW170817 is an absolutely gold-plated detection. GstLAL estimates a false alarm rate (the rate at which you’d expect something at least this signal-like to appear in the detectors due to a random noise fluctuation) of less than one in 1,100,000 years, while PyCBC estimates the false alarm rate to be less than one in 80,000 years.
Parameter estimation (inferring the source properties) used data from all three detectors. We present a (remarkably thorough given the available time) initial analysis in this paper. This signal is challenging to analyse because of the glitch and because binary neutron stars are made of stuff™, which can leave an imprint on the waveform. We’ll be looking at the effects of these complications in more detail in the future. Our initial results are
• The source is localized to a region of about $28~\mathrm{deg^2}$ at a distance of $40^{+8}_{-14}~\mathrm{Mpc}$ (we typically quote results at the 90% credible level). This is the closest gravitational-wave source yet.
• The chirp mass is measured to be $1.188_{-0.002}^{+0.004} M_\odot$, much lower than for our binary black hole detections.
• The spins are not well constrained, the uncertainty from this means that we don’t get precise measurements of the individual component masses. We quote results with two choices of spin prior: the astrophysically motivated limit of 0.05, and the more agnostic and conservative upper bound of 0.89. I’ll stick to using the low spin prior results be default.
• Using the low spin prior, the component masses are $m_1 = 1.36$$1.60 M_\odot$ and $m_2 = 1.17$$1.36 M_\odot$. We have the convention that $m_1 \geq m_2$, which is why the masses look unequal; there’s a lot of support for them being nearly equal. These masses match what you’d expect for neutron stars.
As mentioned above, neutron stars are made of stuff™, and the properties of this leave an imprint on the waveform. If neutron stars are big and fluffy, they will get tidally distorted. Raising tides sucks energy and angular momentum out of the orbit, making the inspiral quicker. If neutron stars are small and dense, tides are smaller and the inspiral looks like that for tow black holes. For this initial analysis, we used waveforms which includes some tidal effects, so we get some preliminary information on the tides. We cannot exclude zero tidal deformation, meaning we cannot rule out from gravitational waves alone that the source contains at least one black hole (although this would be surprising, given the masses). However, we can place a weak upper limit on the combined dimensionless tidal deformability of $\tilde{\Lambda} \leq 800$. This isn’t too informative, in terms of working out what neutron stars are made from, but we’ll come back to this in the future.
Given the source masses, and all the electromagnetic observations, we’re pretty sure this is a binary neutron star system—there’s nothing to suggest otherwise.
Having observed one (and one one) binary neutron star coalescence in O1 and O2, we can now put better constraints on the merger rate. As a first estimate, we assume that component masses are uniformly distributed between $1 M_\odot$ and $2 M_\odot$, and that spins are below 0.4 (in between the limits used for parameter estimation). Given this, we infer that the merger rate is $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$, safely within our previous upper limit [citation note].
There’s a lot more we can learn from GW170817, especially as we don’t just have gravitational waves as a source of information, and this is explained in the companion papers.
### The Multimessenger Paper
Synopsis: Multimessenger Paper
Read this if: Don’t. Use it too look up which other papers to read.
Favourite part: The figures! It was a truly amazing observational effort to follow-up GW170817
The remarkable thing about this paper is that it exists. Bringing together such a diverse (and competitive) group was a huge effort. Alberto Vecchio was one of the editors, and each evening when leaving the office, he was convinced that the paper would have fallen apart by morning. However, it hung together—the story was too compelling. This paper explains how gravitational waves, short gamma-ray bursts, kilonovae all come from a single source [citation note]. This is the greatest collaborative effort in the history of astronomy.
The paper outlines the discoveries and all of the initial set of observations. If you want to understand the observations themselves, this is not the paper to read. However, using it, you can track down the papers that you do want. A huge amount of care went in to trying to describe how discoveries were made: for example, Fermi observed GRB 170817A independently of the gravitational-wave alert, and we found GW170817 without relying on the GRB alert, however, the communication between teams meant that we took everything much seriously and pushed out alerts as quickly as possible. For more on the history of observations, I’d suggest scrolling through the GCN archive.
The paper starts with an overview of the gravitational-wave observations from the inspiral, then the prompt detection of GRB 170817A, before describing how the gravitational-wave localization enabled discovery of the optical transient AT 2017gfo. This source, in nearby galaxy NGC 4993, was then the subject of follow-up across the electromagnetic spectrum. We have huge amount of photometric and spectroscopy of the source, showing general agreement with models for a kilonova. X-ray and radio afterglows were observed 9 days and 16 days after the merger, respectively [citation note]. No neutrinos were found, which isn’t surprising.
### The GW170817 Gamma-ray Burst Paper
Synopsis: GW170817 Gamma-ray Burst Paper
Read this if: You’re interested in the jets from where short gamma-ray bursts originate or in tests of general relativity
Favourite part: How much science come come from a simple time delay measurement
This joint LIGO–Virgo–FermiINTEGRAL paper combines our observations of GW170817 and GRB 170817A. The result is one of the most contentful of the companion papers.
Detection of GW170817 and GRB 170817A. The top three panels show the gamma-ray lightcurves (first: GBM detectors 1, 2, and 5 for 10–50 keV; second: GBM data for 50–300 keV ; third: the SPI-ACS data starting approximately at 100 keV and with a high energy limit of least 80 MeV), the red line indicates the background.The bottom shows the a time–frequency representation of coherently combined gravitational-wave data from LIGO-Hanford and LIGOLivingston. Figure 2 of the GW170817 Gamma-ray Burst Paper.
The first item on the to-do list for joint gravitational-wave–gamma-ray science, is to establish that we are really looking at the same source.
From the GW170817 Discovery Paper, we know that its source is consistent with being a binary neutron star system. Hence, there is matter around which can launch create the gamma-rays. The Fermi-GBM and INTEGRAL observations of GRB170817A indicate that it falls into the short class, as hypothesised as the result of a binary neutron star coalescence. Therefore, it looks like we could have the right ingredients.
Now, given that it is possible that the gravitational waves and gamma rays have the same source, we can calculate the probability of the two occurring by chance. The probability of temporal coincidence is $5.0 \times 10^{-6}$, adding in spatial coincidence too, and the probability becomes $5.0 \times 10^{-8}$. It’s safe to conclude that the two are associated: merging binary neutron stars are the source of at least some short gamma-ray bursts!
#### Testing gravity
There is a $\sim1.74\pm0.05~\mathrm{s}$ delay time between the inferred merger time and the gamma-ray burst. Given that signal has travelled for about 85 million years (taking the 5% lower limit on the inferred distance), this is a really small difference: gravity and light must travel at almost exactly the same speed. To derive exact limit you need to make some assumptions about when the gamma-rays were created. We’d expect some delay as it takes time for the jet to be created, and then for the gamma-rays to blast their way out of the surrounding material. We conservatively (and arbitrarily) take a window of the delay being 0 to 10 seconds, this gives
$\displaystyle -3 \times 10^{-15} \leq \frac{v_\mathrm{GW} - v_\mathrm{EM}}{v_\mathrm{EM}} \leq 7 \times 10^{-16}$.
That’s pretty small!
General relativity predicts that gravity and light should travel at the same speed, so I wasn’t too surprised by this result. I was surprised, however, that this result seems to have caused a flurry of activity in effectively ruling out several modified theories of gravity. I guess there’s not much point in explaining what these are now, but they are mostly theories which add in extra fields, which allow you to tweak how gravity works so you can explain some of the effects attributed to dark energy or dark matter. I’d recommend Figure 2 of Ezquiaga & Zumalacárregui (2017) for a summary of which theories pass the test and which are in trouble.
Table showing viable (left) and non-viable (right) scalar–tensor theories after discovery of GW170817/GRB 170817A. The theories are grouped as Horndeski theories and (the more general) beyond Horndeski theories. General relativity is a tensor theory, so these models add in an extra scalar component. Figure 2 of Ezquiaga & Zumalacárregui (2017).
We don’t discuss the theoretical implications of the relative speeds of gravity and light in this paper, but we do use the time delay to place bounds for particular on potential deviations from general relativity.
1. We look at a particular type of Lorentz invariance violation. This is similar to what we did for GW170104, where we looked at the dispersion of gravitational waves, but here it is for the case of $\alpha = 2$, which we couldn’t test.
2. We look at the Shapiro delay, which is the time difference travelling in a curved spacetime relative to a flat one. That light and gravity are effected the same way is a test of the weak equivalence principle—that everything falls the same way. The effects of the curvature can be quantified with the parameter $\gamma$, which describes the amount of curvature per unit mass. In general relativity $\gamma_\mathrm{GW} = \gamma_\mathrm{EM} = 1$. Considering the gravitational potential of the Milky Way, we find that $-2.6 \times 10^{-7} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM} \leq 1.2 \times 10 ^{-6}$ [citation note].
As you’d expect given the small time delay, these bounds are pretty tight! If you’re working on a modified theory of gravity, you have some extra checks to do now.
#### Gamma-ray bursts and jets
From our gravitational-wave and gamma-ray observations, we can also make some deductions about the engine which created the burst. The complication here, is that we’re not exactly sure what generates the gamma rays, and so deductions are model dependent. Section 5 of the paper uses the time delay between the merger and the burst, together with how quickly the burst rises and fades, to place constraints on the size of the emitting region in different models. The papers goes through the derivation in a step-by-step way, so I’ll not summarise that here: if you’re interested, check it out.
Isotropic energies (left) and luminosities (right) for all gamma-ray bursts with measured distances. These isotropic quantities assume equal emission in all directions, which gives an upper bound on the true value if we are observing on-axis. The short and long gamma-ray bursts are separated by the standard $T_{90} = 2~\mathrm{s}$ duration. The green line shows an approximate detection threshold for Fermi-GBM. Figure 4 from the GW170817 Gamma-ray Burst Paper; you may have noticed that the first version of this paper contained two copies of the energy plot by mistake.
GRB 170817A was unusually dim [citation note]. The plot above compares it to other gamma-ray bursts. It is definitely in the tail. Since it appears so dim, we think that we are not looking at a standard gamma-ray burst. The most obvious explanation is that we are not looking directly down the jet: we don’t expect to see many off-axis bursts, since they are dimmer. We expect that a gamma-ray burst would originate from a jet of material launched along the direction of the total angular momentum. From the gravitational waves alone, we can estimate that the angle between the orbital angular momentum and the line of sight is $\leq 55^\circ$ (adding in the identification of the host galaxy, this becomes $\leq 28^\circ$ using the Planck value for the Hubble constant and $36^\circ$ with the SH0ES value), so this is consistent with viewing the burst off-axis. There are multiple models for such gamma-ray emission, as illustrated below. We could have a uniform top-hat jet (the simplest model) which we are viewing from slightly to the side, we could have a structured jet, which is concentrated on-axis but we are seeing from off-axis, or we could have a cocoon of material pushed out of the way by the main jet, which we are viewing emission from. Other electromagnetic observations will tell us more about the inclination and the structure of the jet [citation note].
Cartoon showing three possible viewing geometries and jet profiles which could explain the observed properties of GRB 170817A. Figure 5 of the GW170817 Gamma-ray Burst Paper.
Now that we know gamma-ray bursts can be this dim, if we observe faint bursts (with unknown distances), we have to consider the possibility that they are dim-and-close in addition to the usual bright-and-far-away.
The paper closes by considering how many more joint gravitational-wave–gamma-ray detections of binary neutron star coalescences we should expect in the future. In our next observing run, we could expect 0.1–1.4 joint detections per year, and when LIGO and Virgo get to design sensitivity, this could be 0.3–1.7 detections per year.
### The GW170817 Hubble Constant Paper
Synopsis: GW170817 Hubble Constant Paper
Read this if: You have an interest in cosmology
Favourite part: In the future, we may be able to settle the argument between the cosmic microwave background and supernova measurements
The Universe is expanding. In the nearby Universe, this can be described using the Hubble relation
$v_H = H_0 D$,
where $v_H$ is the expansion velocity, $H_0$ is the Hubble constant and $D$ is the distance to the source. GW170817 is sufficiently nearby for this relationship to hold. We know the distance from the gravitational-wave measurement, and we can estimate the velocity from the redshift of the host galaxy. Therefore, it should be simple to combine the two to find the Hubble constant. Of course, there are a few complications…
This work is built upon the identification of the optical counterpart AT 2017gfo. This allows us to identify the galaxy NGC 4993 as the host of GW170817’s source: we calculate that there’s a $4 \times 10^{-5}$ probability that AT 2017gfo would be as close to NGC 4993 on the sky by chance. Without a counterpart, it would still be possible to infer the Hubble constant statistically by cross-referencing the inferred gravitational-wave source location with the ensemble of compatible galaxies in a catalogue (you assign a probability to the source being associated with each galaxy, instead of saying it’s definitely in this one). The identification of NGC 4993 makes things much simpler.
As a first ingredient, we need the distance from gravitational waves. For this, a slightly different analysis was done than in the GW170817 Discovery Paper. We fix the sky location of the source to match that of At 2017gfo, and we use (binary black hole) waveforms which don’t include any tidal effects. The sky position needs to be fixed, because for this analysis we are assuming that we definitely know where the source is. The tidal effects were not included (but precessing spins were) because we needed results quickly: the details of spins and tides shouldn’t make much difference to the distance. From this analysis, we find the distance is $41^{+6}_{-13}~\mathrm{Mpc}$ if we follow our usual convention of quoting the median at symmetric 90% credible interval; however, this paper primarily quotes the most probable value and minimal (not-necessarily symmmetric) 68.3% credible interval, following this convention, we write the distance as $44^{+3}_{-7}~\mathrm{Mpc}$.
While NGC 4993 being close by makes the relationship for calculating the Hubble constant simple, it adds a complication for calculating the velocity. The motion of the galaxy is not only due to the expansion of the Universe, but because of how it is moving within the gravitational potentials of nearby groups and clusters. This is referred to as peculiar motion. Adding this in increases our uncertainty on the velocity. Combining results from the literature, our final estimate for the velocity is $v_H= 3017 \pm 166~\mathrm{km\,s^{-1}}$.
We put together the velocity and the distance in a Bayesian analysis. This is a little more complicated than simply dividing the numbers (although that gives you a similar result). You have to be careful about writing things down, otherwise you might implicitly assume a prior that you didn’t intend (my most useful contribution to this paper is probably a whiteboard conversation with Will Farr where we tracked down a difference in prior assumptions approaching the problem two different ways). This is all explained in the Methods, it’s not easy to read, but makes sense when you work through. The final result is $H_0 = 70^{+12}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (or $74^{+33}_{-12}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ in the usual median-and-90%-interval convention). This is nicely (and diplomatically) consistent with existing results.
The distance has considerable uncertainty because there is a degeneracy between the distance and the orbital inclination (the angle of the normal to the orbital plane relative to the line of sight). If you could figure out the inclination from another observation, then you could tighten constraints on the Hubble constant, or if you’re willing to adopt one of the existing values of the Hubble constant, you can pin down the inclination. Data to help you try this yourself are available [citation note].
Two-dimensional posterior probability distribution for the Hubble constant and orbital inclination inferred from GW170817. The contours mark 68% and 95% levels. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SH0ES). Figure 2 of the Hubble Constant Paper.
In the future we’ll be able to combine multiple events to produce a more precise gravitational-wave estimate of the Hubble constant. Chen, Fishbach & Holz (2017) is a recent study of how measurements should improve with more events: we should get to 4% precision after around 100 detections.
### The GW170817 Kilonova Paper
Synopsis: GW170817 Kilonova Paper
Read this if: You want to check our predictions for ejecta against observations
Favourite part: We might be able to create all of the heavy r-process elements—including the gold used to make Nobel Prizes—from merging neutron stars
When two neutron stars collide, lots of material gets ejected outwards. This neutron-rich material undergoes nuclear decay—now no longer being squeezed by the strong gravity inside the neutron star, it is unstable, and decays from the strange neutron star stuff™ to become more familiar elements (elements heavier than iron including gold and platinum). As these r-process elements are created, the nuclear reactions power a kilonova, the optical (infrared–ultraviolet) transient accompanying the merger. The properties of the kilonova depends upon how much material is ejected.
In this paper, we try to estimate how much material made up the dynamical ejecta from the GW170817 collision. Dynamical ejecta is material which escapes as the two neutron stars smash into each other (either from tidal tails or material squeezed out from the collision shock). There are other sources of ejected material, such as winds from the accretion disk which forms around the remnant (whether black hole or neutron star) following the collision, so this is only part of the picture; however, we can estimate the mass of the dynamical ejecta from our gravitational-wave measurements using simulations of neutron star mergers. These estimates can then be compared with electromagnetic observations of the kilonova [citation note].
The amount of dynamical ejecta depends upon the masses of the neutron stars, how rapidly they are rotating, and the properties of the neutron star material (described by the equation of state). Here, we use the masses inferred from our gravitational-wave measurements and feed these into fitting formulae calibrated against simulations for different equations of state. These don’t include spin, and they have quite large uncertainties (we include a 72% relative uncertainty when producing our results), so these are not precision estimates. Neutron star physics is a little messy.
We find that the dynamical ejecta is $10^{-3}$$10^{-2} M_\odot$ (assuming the low spin mass results). These estimates can be feed into models for kilonovae to produce lightcurves, which we do. There is plenty of this type of modelling in the literature as observers try to understand their observations, so this is nothing special in terms of understanding this event. However, it could be useful in the future (once we have hoverboards), as we might be able to use gravitational-wave data to predict how bright a kilonova will be at different times, and so help astronomers decide upon their observing strategy.
Finally, we can consider how much r-process elements we can create from the dynamical ejecta. Again, we don’t consider winds, which may also contribute to the total budget of r-process elements from binary neutron stars. Our estimate for r-process elements needs several ingredients: (i) the mass of the dynamical ejecta, (ii) the fraction of the dynamical ejecta converted to r-process elements, (iii) the merger rate of binary neutron stars, and (iv) the convolution of the star formation rate and the time delay between binary formation and merger (which we take to be $\propto t^{-1}$). Together (i) and (ii) give the mass of r-process elements per binary neutron star (Assuming that GW170817 is typical); (iii) and (iv) give total density of mergers throughout the history of the Universe, and combining everything together you get the total mass of r-process elements accumulated over time. Using the estimated binary neutron star merger rate of $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$, we can explain the Galactic abundance of r-process elements if more than about 10% of the dynamical ejecta is converted.
Present day binary neutron star merger rate density versus dynamical ejecta mass. The grey region shows the inferred 90% range for the rate, the blue shows the approximate range of ejecta masses, and the red band shows the band where the Galactic elemental abundance can be reproduced if at least 50% of the dynamical mass gets converted. Part of Figure 5 of the GW170817 Kilonova Paper.
### The GW170817 Stochastic Paper
Synopsis: GW170817 Stochastic Paper
Read this if: You’re impatient for finding a background of gravitational waves
Favourite part: The background symphony
For every loud gravitational-wave signal, there are many more quieter ones. We can’t pick these out of the detector noise individually, but they are still there, in our data. They add together to form a stochastic background, which we might be able to detect by correlating the data across our detector network.
Following the detection of GW150914, we considered the background due to binary black holes. This is quite loud, and might be detectable in a few years. Here, we add in binary neutron stars. This doesn’t change the picture too much, but gives a more accurate picture.
Binary black holes have higher masses than binary neutron stars. This means that their gravitational-wave signals are louder, and shorter (they chirp quicker and chirp up to a lower frequency). Being louder, binary black holes dominate the overall background. Being shorter, they have a different character: binary black holes form a popcorn background of short chirps which rarely overlap, but binary neutron stars are long enough to overlap, forming a more continuous hum.
The dimensionless energy density at a gravitational-wave frequency of 25 Hz from binary black holes is $1.1_{-0.7}^{+1.2} \times 10^{-9}$, and from binary neutron stars it is $0.7_{-0.6}^{+1.5} \times 10^{-9}$. There are on average $0.06_{-0.04}^{+0.06}$ binary black hole signals in detectors at a given time, and $15_{-12}^{+31}$ binary neutron star signals.
Simulated time series illustrating the difference between binary black hole (green) and binary neutron star (red) signals. Each chirp increases in amplitude until the point at which the binary merges. Binary black hole signals are short, loud chirps, while the longer, quieter binary neutron star signals form an overlapping background. Figure 2 from the GW170817 Stochastic Paper.
To calculate the background, we need the rate of merger. We now have an estimate for binary neutron stars, and we take the most recent estimate from the GW170104 Discovery Paper for binary black holes. We use the rates assuming the power law mass distribution for this, but the result isn’t too sensitive to this: we care about the number of signals in the detector, and the rates are derived from this, so they agree when working backwards. We evolve the merger rate density across cosmic history by factoring in the star formation rate and delay time between formation and merger. A similar thing was done in the GW170817 Kilonova Paper, here we used a slightly different star formation rate, but results are basically the same with either. The addition of binary neutron stars increases the stochastic background from compact binaries by about 60%.
Detection in our next observing run, at a moderate significance, is possible, but I think unlikely. It will be a few years until detection is plausible, but the addition of binary neutron stars will bring this closer. When we do detect the background, it will give us another insight into the merger rate of binaries.
### The GW170817 Progenitor Paper
Synopsis: GW170817 Progenitor Paper
Read this if: You want to know about neutron star formation and supernovae
Favourite part: The Spirography figures
The identification of NGC 4993 as the host galaxy of GW170817’s binary neutron star system allows us to make some deductions about how it formed. In this paper, we simulate a large number of binaries, tracing the later stages of their evolution, to see which ones end up similar to GW170817. By doing so, we learn something about the supernova explosion which formed the second of the two neutron stars.
The neutron stars started life as a pair of regular stars [bonus note]. These burned through their hydrogen fuel, and once this is exhausted, they explode as a supernova. The core of the star collapses down to become a neutron star, and the outer layers are blasted off. The more massive star evolves faster, and goes supernova first. We’ll consider the effects of the second supernova, and the kick it gives to the binary: the orbit changes both because of the rocket effect of material being blasted off, and because one of the components loses mass.
From the combination of the gravitational-wave and electromagnetic observations of GW170817, we know the masses of the neutron star, the type of galaxy it is found in, and the position of the binary within the galaxy at the time of merger (we don’t know the exact position, just its projection as viewed from Earth, but that’s something).
Orbital trajectories of simulated binaries which led to GW170817-like merger. The coloured lines show the 2D projection of the orbits in our model galaxy. The white lines mark the initial (projected) circular orbit of the binary pre-supernova, and the red arrows indicate the projected direction of the supernova kick. The background shading indicates the stellar density. Figure 4 of the GW170817 Progenitor Paper; animated equivalents can be found in the Science Summary.
We start be simulating lots of binaries just before the second supernova explodes. These are scattered at different distances from the the centre of the galaxy, have different orbital separations, and have different masses of the pre-supernova star. We then add the effects of the supernova, adding in a kick. We fix then neutron star masses to match those we inferred from the gravitational wave measurements. If the supernova kick is too big, the binary flies apart and will never merge (boo). If the binary remains bound, we follow its evolution as it moves through the galaxy. The structure of the galaxy is simulated as a simple spherical model, a Hernquist profile for the stellar component and a Navarro–Frenk–White profile for the dark matter halo [citation note], which are pretty standard. The binary shrinks as gravitational waves are emitted, and eventually merge. If the merger happens at a position which matches our observations (yay), we know that the initial conditions could explain GW170817.
Inferred progenitor properties: (second) supernova kick velocity, pre-supernova progenitor mass, pre-supernova binary separation and galactic radius at time of the supernova. The top row shows how the properties vary for different delay times between supernova and merger. The middle row compares all the binaries which survive the second supernova compared with the GW170817-like ones. The bottom row shows parameters for GW170817-like binaries with different galactic offsets than the $1.8~\mathrm{kpc}$ to $2.2~\mathrm{kpc}$ range used for GW1708017. The middle and bottom rows assume a delay time of at least $1~\mathrm{Gyr}$. Figure 5 of the GW170817 Progenitor Paper; to see correlations between parameters, check out Figure 8 of the GW170817 Progenitor Paper.
The plot above shows the constraints on the progenitor’s properties. The inferred second supernova kick is $V_\mathrm{kick} \simeq 300_{-200}^{+250}~\mathrm{km\,s^{-1}}$, similar to what has been observed for neutron stars in the Milky Way; the per-supernova stellar mass is $M_\mathrm{He} \simeq 3.0_{-1.5}^{+3.5} M_\odot$ (we assume that the star is just a helium core, with the outer hydrogen layers having been stripped off, hence the subscript); the pre-supernova orbital separation was $R_\odot \simeq 3.5_{-1.5}^{+5.0} R_\odot$, and the offset from the the centre of the galaxy at the time of the supernova was $2.0_{-1.5}^{+4.0}~\mathrm{kpc}$. The main strongest constraints come from keeping the binary bound after the supernova; results are largely independent of the delay time once this gets above $1~\mathrm{Gyr}$ [citation note].
As we collect more binary neutron star detections, we’ll be able to deduce more about how they form. If you’re interested more in the how to build a binary neutron star system, the introduction to this paper is well referenced; Tauris et al. (2017) is a detailed (pre-GW170817) review.
### The GW170817 Neutrino Paper
Synopsis: GW170817 Neutrino Paper
Read this if: You want a change from gravitational wave–electromagnetic multimessenger astronomy
Favourite part: There’s still something to look forward to with future detections—GW170817 hasn’t stolen all the firsts. Also this paper is not Abbot et al.
This is a joint search by ANTARES, IceCube and the Pierre Auger Observatory for neutrinos coincident with GW170817. Knowing both the location and the time of the binary neutron star merger makes it easy to search for counterparts. No matching neutrinos were detected.
Neutrino candidates at the time of GW170817. The map is is in equatorial coordinates. The gravitational-wave localization is indicated by the red contour, and the galaxy NGC 4993 is indicated by the black cross. Up-going and down-going regions for each detector are indicated, as detectors are more sensitive to up-going neutrinos, as the Cherenkov detectors are subject to a background from cosmic rays hitting the atmosphere. Figure 1 from the GW170817 Neutrino Paper.
Using the non-detections, we can place upper limits on the neutrino flux. These are summarised in the plots below. Optimistic models for prompt emission from an on axis gamma-ray burst would lead to a detectable flux, but otherwise theoretical predictions indicate that a non-detection is expected. From electromagnetic observations, it doesn’t seem like we are on-axis, so the story all fits together.
90% confidence upper limits on neutrino spectral fluence $F$ per flavour (electron, muon and tau) as a function of energy $E$ in $\pm 500~\mathrm{s}$ window (top) about the GW170817 trigger time, and a $14~\mathrm{day}$ window following GW170817 (bottom). IceCube is also sensitive to MeV neutrinos (none were detected). Fluences are the per-flavour sum of neutrino and antineutrino fluence, assuming equal fluence in all flavours. These are compared to theoretical predictions from Kimura et al. (2017) and Fang & Metzger (2017), scaled to a distance of 40 Mpc. The angles labelling the models are viewing angles in excess of the jet opening angle. Figure 2 from the GW170817 Neutrino paper.
Super-Kamiokande have done their own search for neutrinos, form $3.5~\mathrm{MeV}$ to around $100~\mathrm{PeV}$ (Abe et al. 2018). They found nothing in either the $\pm 500~\mathrm{s}$ window around the event or the $14~\mathrm{day}$ window following it.
The only post-detection neutrino modelling paper I’ve seen is Biehl, Heinze, &Winter (2017). They model prompt emission from the same source as the gamma-ray burst and find that neutrino fluxes would be $10^{-4}$ of current sensitivity.
### The GW170817 Post-merger Paper
Synopsis: GW170817 Post-merger Paper
Read this if: You are an optimist
Favourite part: We really do check everywhere for signals
Following the inspiral of two black holes, we know what happens next: the black holes merge to form a bigger black hole, which quickly settles down to its final stable state. We have a complete model of the gravitational waves from the inspiral–merger–ringdown life of coalescing binary black holes. Binary neutron stars are more complicated.
The inspiral of two binary neutron stars is similar to that for black holes. As they get closer together, we might see some imprint of tidal distortions not present for black holes, but the main details are the same. It is the chirp of the inspiral which we detect. As the neutron stars merge, however, we don’t have a clear picture of what goes on. Material gets shredded and ejected from the neutron stars; the neutron stars smash together; it’s all rather messy. We don’t have a good understanding of what should happen when our neutron stars merge, the details depend upon the properties of the stuff™ neutron stars are made of—if we could measure the gravitational-wave signal from this phase, we would learn a lot.
There are four plausible outcomes of a binary neutron star merger:
1. If the total mass is below the maximum mass for a (non-rotating) neutron star ($M < M^\mathrm{Static}$), we end up with a bigger, but still stable neutron star. Given our inferences from the inspiral (see the plot from the GW170817 Gamma-ray Burst Paper below), this is unlikely.
2. If the total mass is above the limit for a stable, non-rotating neutron star, but can still be supported by uniform rotation ($M^\mathrm{Static} < M < M^\mathrm{Uniform}$), we have a supramassive neutron star. The rotation will slow down due to the emission of electromagnetic and gravitational radiation, and eventually the neutron star will collapse to a black hole. The time until collapse could take something like $10$$5 \times 10^4~\mathrm{s}$; it is unclear if this is long enough for supramassive neutron stars to have a mid-life crisis.
3. If the total mass is above the limit for support from uniform rotation, but can still be supported through differential rotation and thermal gradients($M^\mathrm{Uniform} < M < M^\mathrm{Differential}$), then we have a hypermassive neutron star. The hypermassive neutron star cools quickly through neutrino emission, and its rotation slows through magnetic braking, meaning that it promptly collapses to a black hole in $\lesssim 1~\mathrm{s}$.
4. If the total mass is big enough($M^\mathrm{Differential} < M$), the merging neutron stars collapse down to a black hole.
In the case of the collapse to a black hole, we get a ringdown as in the case of a binary black hole merger. The frequency is around $6~\mathrm{kHz}$, too high for us to currently measure. However, if there is a neutron star, there may be slightly lower frequency gravitational waves from the neutron star matter wibbling about. We’re not exactly sure of the form of these signals, so we perform an unmodelled search for them (knowing the position of GW170817’s source helps for this).
Comparison of inferred component masses with critical mass boundaries for different equations of state. The left panel shows the maximum mass of a non-rotating neutron star compared to the initial baryonic mass (ignoring material ejected during merger and gravitational binding energy); the middle panel shows the maximum mass for a uniformly rotating neutron star; the right panel shows the maximum mass of a non-rotating neutron star compared of the gravitational mass of the heavier component neutron star. Figure 3 of the GW170817 Gamma-ray Burst Paper.
Several different search algorithms were used to hunt for a post-merger signal:
1. coherent WaveBurst (cWB) was used to look for short duration ($< 1~\mathrm{s}$) bursts. This searched a $2~\mathrm{s}$ window including the merger time and covering the $1.7~\mathrm{s}$ delay to the gamma-ray burst detection, and frequencies of $1024$$4096~\mathrm{Hz}$. Only LIGO data were used, as Virgo data suffered from large noise fluctuations above $2.5~\mathrm{kHz}$.
2. cWB was used to look for intermediate duration ($< 500~\mathrm{s}$) bursts. This searched a $1000~\mathrm{s}$ window from the merger time, and frequencies $24$$2048~\mathrm{Hz}$. This used LIGO and Virgo data.
3. The Stochastic Transient Analysis Multi-detector Pipeline (STAMP) was also used to look for intermediate duration signals. This searched the merger time until the end of O2 (in $500~\mathrm{s}$ chunks), and frequencies $24$$4000~\mathrm{Hz}$. This used only LIGO data. There are two variations of STAMP: Zebragard and Lonetrack, and both are used here.
Although GEO is similar to LIGO and Virgo and the searched high-frequencies, its data were not used as we have not yet studied its noise properties in enough detail. Since the LIGO detectors are the most sensitive, their data is most important for the search.
No plausible candidates were found, so we set some upper limits on what could have been detected. From these, it is not surprising that nothing was found, as we would need pretty much all of the mass of the remnant to somehow be converted into gravitational waves to see something. Results are shown in the plot below.
Noise amplitude spectral density $\sqrt{S_n}$ for the four detectors, and search upper limits $h_\mathrm{rss}$ as a function of frequency. The noise amplitude spectral densities compare the sensitivities of the detectors. The search upper limits are root-sum-squared strain amplitudes at 50% detection efficiency. The colour code of the upper-limit markers indicates the search algorithm and the shape indicates the waveform injected to set the limits (the frequency is the average for this waveform). The bar mode waveform come from the rapid rotation of the supramassive neutron star leading to it becoming distorted (stretched) in a non-axisymmetric way (Lasky, Sarin & Sammut 2017); the magnetar waveform assumes that the (rapidly rotating) supramassive neutron star’s magnetic field generates significant ellipticity (Corsi & Mészáros 2009); the short-duration merger waveforms are from a selection of numerical simulations (Bauswein et al. 2013; Takami et al. 2015; Kawamura et al. 2016; Ciolfi et al. 2017). The open squares are merger waveforms scaled to the distance and orientation inferred from the inspiral of GW170817. The dashed black lines show strain amplitudes for a narrow-band signal with fixed energy content: the top line is the maximum possible value for GW170817. Figure 1 of the GW170817 Post-merger Paper.
We can’t tell the fate of GW170817’s neutron stars from gravitational-wave alone [citation note]. As high-frequency sensitivity is improved in the future, we might be able to see something from a really close by binary neutron star merger.
### Bonus notes
#### Standard sirens
In astronomy, we often use standard candles, objects like type IA supernovae of known luminosity, to infer distances. If you know how bright something should be, and how bright you measure it to be, you know how far away it is. By analogy, we can infer how far away a gravitational-wave source is by how loud it is. It is thus not a candle, but a siren. Sean Carrol explains more about this term on his blog.
#### Nature
I know… Nature published the original Schutz paper on measuring the Hubble constant using gravitational waves; therefore, there’s a nice symmetry in publishing the first real result doing this in Nature too.
#### Globular clusters
Instead of a binary neutron star system forming from a binary of two stars born together, it is possible for two neutron stars to come close together in a dense stellar environment like a globular cluster. A significant fraction of binary black holes could be formed this way. Binary neutron stars, being less massive, are not as commonly formed this way. We wouldn’t expect GW170817 to have formed this way. In the GW170817 Progenitor Paper, we argue that the probability of GW170817’s source coming from a globular cluster is small—for predicted rates, see Bae, Kim & Lee (2014).
Levan et al. (2017) check for a stellar cluster at the site of AT 2017gfo, and find nothing. The smallest 30% of the Milky Way’s globular clusters would evade this limit, but these account for just 5% of the stellar mass in globular clusters, and a tiny fraction of dynamical interactions. Therefore, it’s unlikely that a cluster is the source of this binary.
### Citation notes
#### Merger rates
From our gravitational-wave data, we estimate the current binary neutron star merger rate density is $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$. Several electromagnetic observers performed their own rate estimates from the frequency of detection (or lack thereof) of electromagnetic transients.
Kasliwal et al. (2017) consider transients seen by the Palomar Transient Factory, and estimate a rate density of approximately $320~\mathrm{Gpc^{-3}\,yr^{-1}}$ (3-sigma upper limit of $800~\mathrm{Gpc^{-3}\,yr^{-1}}$), towards the bottom end of our range, but their rate increases if not all mergers are as bright as AT 2017gfo.
Siebert et al. (2017) works out the rate of AT 2017gfo-like transients in the Swope Supernova Survey. They obtain an upper limit of $16000~\mathrm{Gpc^{-3}\,yr^{-1}}$. They use to estimate the probability that AT 2017gfo and GW170817 are just a chance coincidence and are actually unrelated. The probability is $9 \times 10^{-6}$ at 90% confidence.
Smartt et al. (2017) estimate the kilonova rate from the ATLAS survey, they calculate a 95% upper limit of $30000~\mathrm{Gpc^{-3}\,yr^{-1}}$, safely above our range.
Yang et al. (2017) calculates upper limits from the DLT40 Supernova survey. Depending upon the reddening assumed, this is between $93000^{+16000}_{-18000}~\mathrm{Gpc^{-3}\,yr^{-1}}$ and $109000^{+28000}_{-18000}~\mathrm{Gpc^{-3}\,yr^{-1}}$. Their figure 3 shows that this is well above expected rates.
Finally, Zhang et al. (2017) is interested in the rate of gamma-ray bursts. If you know the rate of short gamma-ray bursts and of binary neutron star mergers, you can learn something about the beaming angle of the jet. The smaller the jet, the less likely we are to observe a gamma-ray burst. In order to do this, they do their own back-of-the-envelope for the gravitational-wave rate. They get $1100_{-910}^{+2500}~\mathrm{Gpc^{-3}\,yr^{-1}}$. That’s not too bad, but do stick with our result.
If you’re interested in the future prospects for kilonova detection, I’d recommend Scolnic et al. (2017). Check out their Table 2 for detection rates (assuming a rate of $1000~\mathrm{Gpc^{-3}\,yr^{-1}}$): LSST and WFIRST will see lots, about 7 and 8 per year respectively.
#### The electromagnetic story
Some notes on an incomplete overview of papers describing the electromagnetic discovery. A list of the first wave of papers was compiled by Maria Drout, Stefano Valenti, and Iair Arcavi as a starting point for further reading.
Independently of our gravitational-wave detection, a short gamma-ray burst GRB 170817A was observed by Fermi-GBM (Goldstein et al. 2017). Fermi-LAT did not see anything, as it was offline for crossing through the South Atlantic Anomaly. At the time of the merger, INTEGRAL was following up the location of GW170814, fortunately this meant it could still observe the location of GW170817, and following the alert they found GRB 170817A in their data (Savchenko et al. 2017).
Following up on our gravitational-wave localization, an optical transient AT 2017gfo was discovered. The discovery was made by the One-Meter Two-Hemisphere (1M2H) collaboration using the Swope telescope at the Las Campanas Observatory in Chile; they designated the transient as SSS17a (Coulter et al. 2017). That same evening, several other teams also found the transient within an hour of each other:
• The Distance Less Than 40 Mpc (DLT40) search found the transient using the PROMPT 0.4-m telescope at the Cerro Tololo Inter-American Observatory in Chile; they designated the transient DLT17ck (Valenti et al. 2017).
• The VINROUGE collaboration (I think, they don’t actually identify themselves in their own papers) found the transient using VISTA at the European Southern Observatory in Chile (Tanvir et al. 2017). Their paper also describes follow-up observations with the Very Large Telescope, the Hubble Space Telescope, the Nordic Optical Telescope and the Danish 1.54-m Telescope, and has one of my favourite introduction sections of the discovery papers.
• The MASTER collaboration followed up with their network of global telescopes, and it was their telescope at the San Juan National University Observatory in Argentina which found the transient (Lipunov et al. 2017); they, rather catchily denote the transient as OTJ130948.10-232253.3.
• The Dark Energy Survey and the Dark Energy Camera GW–EM (DES and DECam) Collaboration found the transient with the DECam on the Blanco 4-m telescope, which is also at the Cerro Tololo Inter-American Observatory in Chile (Soares-Santos et al. 2017).
• The Las Cumbres Observatory Collaboration used their global network of telescopes, with, unsurprisingly, their 1-m telescope at the Cerro Tololo Inter-American Observatory in Chile first imaging the transient (Arcavi et al. 2017). Their observing strategy is described in a companion paper (Arcavi et al. 2017), which also describes follow-up of GW170814.
From these, you can see that South America was the place to be for this event: it was night at just the right time.
There was a huge amount of follow-up across the infrared–optical–ultraviolet range of AT 2017gfo. Villar et al. (2017) attempts to bring these together in a consistent way. Their Figure 1 is beautiful.
Assembled lightcurves from ultraviolet, optical and infrared observations of AT 2017gfo. The data points are the homogenized data, and the lines are fitted kilonova models. The blue light initially dominates but rapidly fades, while the red light undergoes a slower decay. Figure 1 of Villar et al. (2017).
AT 2017gfo was also the target of observations across the electromagnetic spectrum. An X-ray afterglow was observed 9 days post merger, and 16 days post merger, just as we thought the excitement was over, a radio afterglow was found:
The afterglow will continue to brighten for a while, so we can expect a series of updates:
• Pooley, Kumar & Wheeler (2017) observed with Chandra 108 and 111 days post merger. Ruan et al. (2017) observed with Chandra 109 days post merger. The large gap in the the X-ray observations from the initial observations is because the Sun got in the way.
• Mooley et al. (2017) update the GROWTH radio results up to 107 days post merger (the largest span whilst still pre-empting new X-ray observations), observing with the Very Large Array, Australia Telescope Compact Array and Giant Meterewave Radio Telescope.
Excitingly, the afterglow has also now been spotted in the optical:
• Lyman et al. (2018) observed with Hubble 110 (rest-frame) days post-merger (which is when the Sun was out of the way for Hubble). At this point the kilonova should have faded away, but they found something, and this is quite blue. The conclusion is that it’s the afterglow, and it will peak in about a year.
• Margutti et al. (2018) brings together Chandra X-ray observations, Very Large Array radio observations and Hubble optical observations. The Hubble observations are 137 days post merger, and the Chandra observations are 153 days and 163 days post-merger. They find that they all agree (including the tentative radio signal at 10 days post-merger). They argue that the emission disfavours on-axis jets and spherical fireballs.
Evolution of radio, optical and X-ray spectral energy density of the counterpart to GW170817. The radio and X-ray are always dominated by the afterglow, as indicated by them following the same power law. At early times, the optical is dominated by the kilonova, but as this fades, the afterglow starts to dominate. Figure. 1 of Margutti et al. (2018).
The afterglow is now starting to fade.
• D’Avanzo et al. (2018) observed in X-ray 135 days post-merger with XMM-Newton. They find that the flux is faded compared to the previous trend. They suggest that we’re just at the turn-over, so this is consistent with the most recent Hubble observations.
• Dobie et al. (2018) observed in radio 125–200 days post-merger with the Very Large Array and Australia Telescope Compact Array, and they find that the afterglow is starting to fade, with a peak at 149 ± 2 days post-merger.
• Nynka et al. (2018) made X-ray observations at 260 days post-merger. They conclude the afterglow is definitely fading, and that this is not because of passing of the synchrotron cooling frequency.
The story isn’t over yet!
#### Shapiro delay
Using the time delay between GW170817 and GRB 170817A, a few other teams also did their own estimation of the Shapiro delay before they knew what was in our GW170817 Gamma-ray Burst Paper.
• Wang et al. (2017) consider the Milky Way potential and large scale structure to estimate $-4 \times 10^{-9} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM}$.
• Boran et al. (2017) consider all the galaxies in the GLADE catalogue which are within a radius of $400~\mathrm{kpc}$ of the line of sight, and derive $|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}| \leq 3.9 \times 10 ^{-9}$.
• Wei et al. (2017) estimate $|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}| \leq 5.9 \times 10 ^{-8}$ using the Milky Way’s potential and $|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}| \leq 9.2 \times 10 ^{-11}$ using the Virgo cluster’s potential.
Our estimate of $-2.6 \times 10^{-7} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM} \leq 1.2 \times 10 ^{-6}$ is the most conservative.
#### Comparison to other gamma-ray bursts
Are the electromagnetic counterparts to GW170817 similar to what has been observed before? Yue et al. (2017) compare GRB 170817A with other gamma-ray bursts. It is low luminosity, but it may not be alone. There could be other bursts like it (perhaps GRB 070923, GRB 080121 and GRB 090417A), if indeed they are from nearby sources. They suggest that GRB 130603B may be the on-axis equivalent of GRB 170817A [citation note]; however, the non-detection of kilonovae for several bursts indicates that there needs to be some variation in their properties too. This agree with the results of Gompertz et al. (2017), who compares the GW170817 observations with other kilonovae: it is fainter than the other candidate kilonovae (GRB 050709, GRB 060614, GRB 130603B and tentatively GRB 160821B), but equally brighter than upper limits from other bursts. There must be a diversity in kilonovae observations. Fong et al. (2017) look at the diversity of afterglows (across X-ray to radio), and again find GW170817’s counterpart to be faint. This is probably because we are off-axis. Future observations will help unravel how much variation there is from viewing different angles, and how much intrinsic variation there is from the source—perhaps some short gamma-ray bursts come from neutron star–black hole binaries?
#### Inclination, jets and ejecta
Pretty much every observational paper has a go at estimating the properties of the ejecta, the viewing angle or something about the structure of the jet. I may try to pull these together later, but I’ve not had time yet as it is a very long list!
#### Hubble constant and misalignment
Guidorzi et al. (2017) try to tighten the measurement of the Hubble constant by using radio and X-ray observations. Their modelling assumes a uniform jet, which doesn’t look like a currently favoured option [citation note], so there is some model-based uncertainty to be included here. Additionally, the jet is unlikely to be perfectly aligned with the orbital angular momentum, which may add a couple of degrees more uncertainty.
Mandel (2018) works the other way and uses the recent Dark Energy Survey Hubble constant estimate to bound the misalignment angle to less than $28~\mathrm{deg}$, which (unsurprisingly) agrees pretty well with the result we obtained using the Planck value. Finstad et al. (2018) uses the luminosity distance from Cantiello et al. (2018) [citation note] as a (Gaussian) prior for an analysis of the gravitational-wave signal, and get a misalignment $32^{+10}_{-13}\pm 2~\mathrm{deg}$ (where the errors are statistical uncertainty and an estimate of systematic error from calibration of the strain).
#### NGC 4993 properties
In the GW170817 Progenitor Paper we used component properties for NGC 4993 from Lim et al. (2017): a stellar mass of $(10^{10.454}/h^2) M_\odot$ and a dark matter halo mass of $(10^{12.2}/h) M_\odot$, where we use the Planck value of $h = 0.679$ (but conclusions are similar using the SH0ES value for this).
Blanchard et al. (2017) estimate a stellar mass of about $\log(M_\ast/M_\odot) = 10.65^{+0.03}_{-0.03}$. They also look at the star formation history, 90% were formed by $6.8^{+2.2}_{-0.8}~\mathrm{Gyr}$ ago, and the median mass-weighted stellar age is $13.2^{+0.5}_{-0.9}~\mathrm{Gyr}$. From this they infer a merger delay time of $6.8$$13.6~\mathrm{Gyr}$. From this, and assuming that the system was born close to its current location, they estimate that the supernova kick $V_\mathrm{kick} \leq 200~\mathrm{km\,s^{-1}}$, towards the lower end of our estimate. They use $h = 0.677$.
Im et al. (2017) find a mean stellar mass of $0.3$$1.2 \times 10^{11} M_\odot$ and the mean stellar age is greater than about $3~\mathrm{Gyr}$. They also give a luminosity distance estimate of $38.4 \pm 8.9~\mathrm{Mpc}$, which overlaps with our gravitational-wave estimate. I’m not sure what value of $h$ they are using.
Levan et al. (2017) suggest a stellar mass of around $1.4 \times 10^{11} M_\odot$. They find that 60% of stars by mass are older than $5~\mathrm{Gyr}$ and that less than 1% are less than $0.5~\mathrm{Gyr}$ old. Their Figure 5 has some information on likely supernova kicks, they conclude it was probably small, but don’t quantify this. They use $h = 0.696$.
Pan et al. (2017) find $\log(M_\ast/M_\odot) = 10.49^{+0.08}_{-0.20}$. They calculate a mass-weighted mean stellar age of $10.97~\mathrm{Gyr}$ and a likely minimum age for GW170817’s source system of $2.8~\mathrm{Gyr}$. They use $h = 0.7$.
Troja et al. (2017) find a stellar mass of $\log(M_\ast/M_\odot) \sim 10.88$, and suggest an old stellar population of age $> 2~\mathrm{Gyr}$.
Ebrová & Bílek (2018) assume a distance of $41.0~\mathrm{kpc}$ and find a halo mass of $1.939 \times 10^{12} M_\odot$. They suggest that NGC 4993 swallowed a smaller late-type galaxy somewhere between $0.2~\mathrm{Gyr}$ and $1~\mathrm{Gyr}$ ago, most probably around $0.4~\mathrm{Gyr}$ ago.
The consensus seems to be that the stellar population is old (and not much else). Fortunately, the conclusions of the GW170817 Progenitor Paper are pretty robust for delay times longer than $1~\mathrm{Gyr}$ as seems likely.
A couple of other papers look at the distance of the galaxy:
• Hjoth et al. (2017) combine a redshift measurement from MUSE, and a fundamental plane estimate based upon Hubble observations, to obtain an distance of $41.0 \pm 3.1~\mathrm{Mpc}$.
• Cantiello et al. (2018) use Hubble observations to estimate the distance using surface brightness fluctuations. They obtain a distance of $40.7 \pm 1.4 \pm 1.9~\mathrm{Mpc}$. This implies a value for the Hubble constant of $h = 0.719 \pm 0.071$.
The values are consistent with our gravitational-wave estimates.
#### The remnant’s fate
We cannot be certain what happened to the merger remnant from gravitational-wave observations alone. However, electromagnetic observations do give some hints here.
Evans et al. (2017) argue that their non-detection of X-rays when observing with Swift and NuSTAR indicates that there is no neutron star remnant at this point, meaning we must have collapsed to form a black hole by 0.6 days post-merger. This isn’t too restricting in terms of the different ways the remnant could collapse, but does exclude a stable neutron star remnant.
Pooley, Kumar & Wheeler (2017) consider X-ray observations of the afterglow. They calculate that if the remnant was a hypermassive neutron star with a large magnetic field, the early (10 day post-merger) luminosity would be much higher (and we could expect to see magnetar outbursts). Therefore, they think it is more likely that the remnant is a black hole.
Kasen et al. (2017) use the observed red component of the kilonova to argue that the remnant must have collapsed to a black hole in $< 10~\mathrm{ms}$. A neutron star would irradiate the ejecta with neutrinos, lower the neutron fraction and making the ejecta bluer. Since it is red, the neutrino flux must have been shut off, and the neutron star must have collapsed. We are in case b in their figure below.
Cartoon of the different components of matter ejected from neutron star mergers. Red colours show heavy r-process elements and blue colours light r-process elements. There is a tidal tail of material forming a torus in the orbital plane, roughly spherical winds from the accretion disk, and material squeezed into the polar reasons during the collision. In case a, we have a long-lived neutron star, and its neutrino irradiation leads to blue ejecta. In case b the neutron star collapses, cutting off the neutrino flux. In case c, there is a neutron star–black hole merger, and we don’t have the polar material from the collision. Figure 1 of Kasen et al. (2017); also see Figure 1 of Margalit & Metzger (2017).
Ai et al. (2018) find that there are some corners of parameter space for certain equations of space where a long-lived neutron star is possible, even given the observations. Therefore, we should remain open minded.
Margalit & Metzger (2017) and Bauswein et al. (2017) note that the relatively large amount of ejecta inferred from observations [citation note] is easier to explain when there is delayed (on timescales of $> 10~\mathrm{ms}$). This is difficult to resolve unless neutron star radii are small ($\lesssim 11~\mathrm{km}$). Metzger, Thompson & Quataert (2018) derive how this tension could be resolved if the remnant was a rapidly spinning magnetar with a life time of $0.1$$1~\mathrm{s}$Matsumoto et al. (2018), suggest that the optical emission is powered by the the jet and material accreting onto the central object, rather than r-process decay, and this permits much smaller amounts of ejecta, which could also solve the issue. Yu & Dai (2017) suggest that accretion onto a long-lived neutron star could power the emission, and would only require a single opacity for the ejecta. Li et al. (2018) put forward a similar theory, arguing that both the high ejecta mass and low opacity are problems for the standard r-process explanation, but fallback onto a neutron star could work. However, Margutti et al. (2018) say that X-ray emission powered by a central engine is disfavoured at all times.
In conclusion, it seems that we ended up with a black hole, and we had an a unstable neutron star for a short time after merger, but I don’t think it’s yet settled how long this was around.
# GW170608—The underdog
Detected in June, GW170608 has had a difficult time. It was challenging to analyse, and neglected in favour of its louder and shinier siblings. However, we can now introduce you to our smallest chirp-mass binary black hole system!
The growing family of black holes. From Dawn Finney.
Our family of binary black holes is now growing large. During our first observing run (O1) we found three: GW150914, LVT151012 and GW151226. The advanced detector observing run (O2) ran from 30 November 2016 to 25 August 2017 (with a couple of short breaks). From our O1 detections, we were expecting roughly one binary black hole per month. The first same in January, GW170104, and we have announced the first detection which involved Virgo from August, GW170814, so you might be wondering what happened in-between? Pretty much everything was dropped following the detection of our first binary neutron star system, GW170817, as a sizeable fraction of the astronomical community managed to observe its electromagnetic counterparts. Now, we are starting to dig our way out of the O2 back-log.
On 8 June 2017, a chirp was found in data from LIGO Livingston. At the time, LIGO Hanford was undergoing planned engineering work [bonus explanation]. We would not normally analyse this data, as the detector is disturbed; however, we had to follow up on the potential signal in Livingston. Only low frequency data in Hanford should have been affected, so we limited our analysis to above 30 Hz (this sounds easier than it is—I was glad I was not on rota to analyse this event [bonus note]). A coincident signal was found. Hello GW170608, the June event!
Time–frequency plots for GW170608 as measured by LIGO Hanford and Livingston. The chirp is clearer in Hanford, despite it being less sensitive, because of the sources position. Figure 1 of the GW170608 Paper.
Analysing data from both Hanford and Livingston (limiting Hanford to above 30 Hz) [bonus note], GW170608 was found by both of our offline searches for binary signals. PyCBC detected it with a false alarm rate of less than 1 in 3000 years, and GstLAL estimated a false alarm rate of 1 in 160000 years. The signal was also picked up by coherent WaveBurst, which doesn’t use waveform templates, and so is more flexible in what it can detect at the cost off sensitivity: this analysis estimates a false alarm rate of about 1 in 30 years. GW170608 probably isn’t a bit of random noise.
GW170608 comes from a low mass binary. Well, relatively low mass for a binary black hole. For low mass systems, we can measure the chirp mass $\mathcal{M}$, the particular combination of the two black hole masses which governs the inspiral, well. For GW170608, the chirp mass is $7.9_{-0.2}^{+0.2} M_\odot$. This is the smallest chirp mass we’ve ever measured, the next smallest is GW151226 with $8.9_{-0.3}^{+0.3} M_\odot$. GW170608 is probably the lowest mass binary we’ve found—the total mass and individual component masses aren’t as well measured as the chirp mass, so there is small probability (~11%) that GW151226 is actually lower mass. The plot below compares the two.
Estimated masses $m_1 \geq m_2$ for the two black holes in the binary. The two-dimensional shows the probability distribution for GW170608 as well as 50% and 90% contours for GW151226, the other contender for the lightest black hole binary. The one-dimensional plots on the sides show results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. The one-dimensional plots at the top show the probability distributions for the total mass $M$ and chirp mass $\mathcal{M}$. Figure 2 of the GW170608 Paper. I think this plot is neat.
One caveat with regards to the masses is that the current results only consider spin magnitudes up to 0.89, as opposed to the usual 0.99. There is a correlation between the mass ratio and the spins: you can have a more unequal mass binary with larger spins. There’s not a lot of support for large spins, so it shouldn’t make too much difference.
Speaking of spins, GW170608 seems to prefer small spins aligned with the angular momentum; spins are difficult to measure, so there’s a lot of uncertainty here. The best measured combination is the effective inspiral spin parameter $\chi_\mathrm{eff}$. This is a combination of the spins aligned with the orbital angular momentum. For GW170608 it is $0.07_{-0.09}^{+0.23}$, so consistent with zero and leaning towards being small and positive. For GW151226 it was $0.21_{-0.10}^{+0.20}$, and we could exclude zero spin (at least one of the black holes must have some spin). The plot below shows the probability distribution for the two component spins (you can see the cut at a maximum magnitude of 0.89). We prefer small spins, and generally prefer spins in the upper half of the plots, but we can’t make any definite statements other than both spins aren’t large and antialigned with the orbital angular momentum.
Estimated orientation and magnitude of the two component spins. The distribution for the more massive black hole is on the left, and for the smaller black hole on the right. The probability is binned into areas which have uniform prior probabilities, so if we had learnt nothing, the plot would be uniform. This analysis assumed spin magnitudes less than 0.89, which is why there is an apparent cut-off. Part of Figure 3 of the GW170608 Paper. For the record, I voted against this colour scheme.
The properties of GW170608’s source are consistent with those inferred from observations of low-mass X-ray binaries (here the low-mass refers to the companion star, not the black hole). These are systems where mass overflows from a star onto a black hole, swirling around in an accretion disc before plunging in. We measure the X-rays emitted from the hot gas from the disc, and these measurements can be used to estimate the mass and spin of the black hole. The similarity suggests that all these black holes—observed with X-rays or with gravitational waves—may be part of the same family.
Estimated black hole masses inferred from low-mass X-ray binary observations. Figure 1 of Farr et al. (2011). The masses overlap those of the lower mass binary black holes found by LIGO and Virgo.
We’ll present update merger rates and results for testing general relativity in our end-of-O2 paper. The low mass of GW170608’s source will make it a useful addition to our catalogue here. Small doesn’t mean unimportant.
Title: GW170608: Observation of a 19 solar-mass binary black hole coalescence
Journal: Astrophysical Journal Letters; 851(2):L35(11); 2017
arXiv: 1711.05578 [gr-qc] [bonus note]
Science summary: GW170608: LIGO’s lightest black hole binary?
Data release: LIGO Open Science Center
### Bonus notes
#### Detector engineering
A lot of time and effort goes into monitoring, maintaining and tweaking the detectors so that they achieve the best possible performance. The majority of work on the detectors happens during engineering breaks between observing runs, as we progress towards design sensitivity. However, some work is also needed during observing runs, to keep the detectors healthy.
On 8 June, Hanford was undergoing angle-to-length (A2L) decoupling, a regular maintenance procedure which minimises the coupling between the angular position of the test-mass mirrors and the measurement of strain. Our gravitational-wave detectors carefully measure the time taken for laser light to bounce between the test-mass mirrors in their arms. If one of these mirrors gets slightly tilted, then the laser could bounce of part of the mirror which is slightly closer or further away than usual: we measure a change in travel time even though the length of the arm is the same. To avoid this, the detectors have control systems designed to minimise angular disturbances. Every so often, it is necessary to check that these are calibrated properly. To do this, the mirrors are given little pushes to rotate them in various directions, and we measure the output to see the impact.
Examples of how angular fluctuations can couple to length measurements. Here are examples of how pitch $p$ rotations in the suspension level above the test mass (L3 is the test mass, L2 is the level above) can couple to length measurement $l$. Yaw fluctuations (rotations about the vertical axis) can also have an impact. Figure 1 of Kasprzack & Yu (2016).
The angular pushes are done at specific frequencies, so we we can tease apart the different effects of rotations in different directions. The frequencies are in the range 19–23 Hz. 30 Hz is a safe cut-off for effects of the procedure (we see no disturbances above this frequency).
Imprint of angular coupling testing in Hanford. The left panel shows a spectrogram of strain data, you can clearly see the excitations between ~19 Hz and ~23 Hz. The right panel shows the amplitude spectral density for Hanford before and during the procedure, as well as for Livingston. The procedure adds extra noise in the broad peak about 20 Hz. There are no disturbances above ~30 Hz. Figure 4 of GW170608 Paper.
While we normally wouldn’t analyse data from during maintenance, we think it is safe to do so, after discarding the low-frequency data. If you are worried about the impact of including addition data in our rate estimates (there may be a bias only using time when you know there are signals), you can be reassured that it’s only a small percent of the total time, and so should introduce an error less significant than uncertainty from the calibration accuracy of the detectors.
#### Parameter estimation rota
Unusually for an O2 event, Aaron Zimmerman was not on shift for the Parameter Estimation rota at the time of GW170608. Instead, it was Patricia Schmidt and Eve Chase who led this analysis. Due to the engineering work in Hanford, and the low mass of the system (which means a long inspiral signal), this was one of the trickiest signals to analyse: I’d say only GW170817 was more challenging (if you ignore all the extra work we did for GW150914 as it was the first time).
#### Virgo?
If you are wondering about the status of Virgo: on June 8 it was still in commissioning ahead of officially joining the run on 1 August. We have data at the time of the event. The sensitivity is of the detector is not great. We often quantify detector sensitivity by quoting the binary neutron star range (the average distance a binary neutron star could be detected). Around the time of the event, this was something like 7–8 Mpc for Virgo. During O2, the LIGO detectors have been typically in the 60–100 Mpc region; when Virgo joined O2, it had a range of around 25–30 Mpc. Unsurprisingly, Virgo didn’t detect the signal. We could have folded the data in for parameter estimation, but it was decided that it was probably not well enough understood at the time to be worthwhile.
#### Journal
The GW170608 Paper is the first discovery paper to be made public before journal acceptance (although the GW170814 Paper was close, and we would have probably gone ahead with the announcement anyway). I have mixed feelings about this. On one hand, I like that the Collaboration is seen to take their detections seriously and follow the etiquette of peer review. On the other hand, I think it is good that we can get some feedback from the broader community on papers before they’re finalised. I think it is good that the first few were peer reviewed, it gives us credibility, and it’s OK to relax now. Binary black holes are becoming routine.
This is also the first discovery paper not to go to Physical Review Letters. I don’t think there’s any deep meaning to this, the Collaboration just wanted some variety. Perhaps GW170817 sold everyone that we were astrophysicists now? Perhaps people thought that we’ve abused Physical Review Letters‘ page limits too many times, and we really do need that appendix. I was still in favour of Physical Review Letters for this paper, if they would have had us, but I approve of sharing the love. There’ll be plenty more events.
# GW170817—The pot of gold at the end of the rainbow
Advanced LIGO and Advanced Virgo have detected their first binary neutron star inspiral. Remarkably, this event was observed not just with gravitational waves, but also across the electromagnetic spectrum, from gamma-rays to radio. This discovery confirms the theory that binary neutron star mergers are the progenitors of short gamma-ray bursts and kilonovae, and may be the primary source of heavy elements like gold.
In this post, I’ll go through some of the story of GW170817. As for GW150914, I’ll write another post on the more technical details of our papers, once I’ve had time to catch up on sleep.
### Discovery
The second observing run (O2) of the advanced gravitational-wave detectors started on 30 November 2016. The first detection came in January—GW170104. I was heavily involved in the analysis and paper writing for this. We finally finished up in June, at which point I was thoroughly exhausted. I took some time off in July [bonus note], and was back at work for August. With just one month left in the observing run, it would all be downhill from here, right?
August turned out to be the lava-filled, super-difficult final level of O2. As we have now announced, on August 14, we detected a binary black hole coalescence—GW170814. This was the first clear detection including Virgo, giving us superb sky localization. This is fantastic for astronomers searching for electromagnetic counterparts to our gravitational-wave signals. There was a flurry of excitement, and we thought that this was a fantastic conclusion to O2. We were wrong, this was just the save point before the final opponent. On August 17, we met the final, fire-ball throwing boss.
At 1:58 pm BST my phone buzzed with a text message, an automated alert of a gravitational-wave trigger. I was obviously excited—I recall that my exact thoughts were “What fresh hell is this?” I checked our online event database and saw that it was a single-detector trigger, it was only seen by our Hanford instrument. I started to relax, this was probably going to turn out to be a glitch. The template masses, were low, in the neutron star range, not like the black holes we’ve been finding. Then I saw the false alarm rate was better than one in 9000 years. Perhaps it wasn’t just some noise after all—even though it’s difficult to estimate false alarm rates accurately online, as especially for single-detector triggers, this was significant! I kept reading. Scrolling down the page there was an external coincident trigger, a gamma-ray burst (GRB 170817A) within a couple of seconds…
We’re gonna need a bigger author list. Credit: Zanuck/Brown Productions
Short gamma-ray bursts are some of the most powerful explosions in the Universe. I’ve always found it mildly disturbing that we didn’t know what causes them. The leading theory has been that they are the result of two neutron stars smashing together. Here seemed to be the proof.
The rapid response call was under way by the time I joined. There was a clear chirp in Hanford, you could be see it by eye! We also had data from Livingston and Virgo too. It was bad luck that they weren’t folded into the online alert. There had been a drop out in the data transfer from Italy to the US, breaking the flow for Virgo. In Livingston, there was a glitch at the time of the signal which meant the data wasn’t automatically included in the search. My heart sank. Glitches are common—check out Gravity Spy for some examples—so it was only a matter of time until one overlapped with a signal [bonus note], and with GW170817 being such a long signal, it wasn’t that surprising. However, this would complicate the analysis. Fortunately, the glitch is short and the signal is long (if this had been a high-mass binary black hole, things might not have been so smooth). We were able to exorcise the glitch. A preliminary sky map using all three detectors was sent out at 12:54 am BST. Not only did we defeat the final boss, we did a speed run on the hard difficulty setting first time [bonus note].
Spectrogram of Livingston data showing part of GW170817’s chirp (which sweeps upward in frequncy) as well as the glitch (the big blip at about $-0.6~\mathrm{s}$). The lower panel shows how we removed the glitch: the grey line shows gating window that was applied for preliminary results, to zero the affected times, the blue shows a fitted model of the glitch that was subtracted for final results. You can clearly see the chirp well before the glitch, so there’s no danger of it being an artefect of the glitch. Figure 2 of the GW170817 Discovery Paper
The three-detector sky map provided a great localization for the source—this preliminary map had a 90% area of ~30 square degrees. It was just in time for that night’s observations. The plot below shows our gravitational-wave localizations in green—the long band is without Virgo, and the smaller is with all three detectors—as with GW170814, Virgo makes a big difference. The blue areas are the localizations from Fermi and INTEGRAL, the gamma-ray observatories which measured the gamma-ray burst. The inset is something new…
Localization of the gravitational-wave, gamma-ray, and optical signals. The main panel shows initial gravitational-wave 90% areas in green (with and without Virgo) and gamma-rays in blue (the IPN triangulation from the time delay between Fermi and INTEGRAL, and the Fermi GBM localization). The inset shows the location of the optical counterpart (the top panel was taken 10.9 hours after merger, the lower panel is a pre-merger reference without the transient). Figure 1 of the Multimessenger Astronomy Paper.
That night, the discoveries continued. Following up on our sky location, an optical counterpart (AT 2017gfo) was found. The source is just on the outskirts of galaxy NGC 4993, which is right in the middle of the distance range we inferred from the gravitational wave signal. At around 40 Mpc, this is the closest gravitational wave source.
After this source was reported, I think about every single telescope possible was pointed at this source. I think it may well be the most studied transient in the history of astronomy. I think there are ~250 circulars about follow-up. Not only did we find an optical counterpart, but there was emission in X-ray and radio. There was a delay in these appearing, I remember there being excitement at our Collaboration meeting as the X-ray emission was reported (there was a lack of cake though).
The figure below tries to summarise all the observations. As you can see, it’s a mess because there is too much going on!
The timeline of observations of GW170817’s source. Shaded dashes indicate times when information was reported in a Circular. Solid lines show when the source was observable in a band: the circles show a comparison of brightnesses for representative observations. Figure 2 of the Multimessenger Astronomy Paper.
The observations paint a compelling story. Two neutron stars insprialled together and merged. Colliding two balls of nuclear density material at around a third of the speed of light causes a big explosion. We get a jet blasted outwards and a gamma-ray burst. The ejected, neutron-rich material decays to heavy elements, and we see this hot material as a kilonova [bonus material]. The X-ray and radio may then be the afterglow formed by the bubble of ejected material pushing into the surrounding interstellar material.
### Science
What have we learnt from our results? Here are some gravitational wave highlights.
We measure several thousand cycles from the inspiral. It is the most beautiful chirp! This is the loudest gravitational wave signal yet found, beating even GW150914. GW170817 has a signal-to-noise ratio of 32, while for GW150914 it is just 24.
Time–frequency plots for GW170104 as measured by Hanford, Livingston and Virgo. The signal is clearly visible in the two LIGO detectors as the upward sweeping chirp. It is not visible in Virgo because of its lower sensitivity and the source’s position in the sky. The Livingston data have the glitch removed. Figure 1 of the GW170817 Discovery Paper.
The signal-to-noise ratios in the Hanford, Livingston and Virgo were 19, 26 and 2 respectively. The signal is quiet in Virgo, which is why you can’t spot it by eye in the plots above. The lack of a clear signal is really useful information, as it restricts where on the sky the source could be, as beautifully illustrated in the video below.
While we measure the inspiral nicely, we don’t detect the merger: we can’t tell if a hypermassive neutron star is formed or if there is immediate collapse to a black hole. This isn’t too surprising at current sensitivity, the system would basically need to convert all of its energy into gravitational waves for us to see it.
From measuring all those gravitational wave cycles, we can measure the chirp mass stupidly well. Unfortunately, converting the chirp mass into the component masses is not easy. The ratio of the two masses is degenerate with the spins of the neutron stars, and we don’t measure these well. In the plot below, you can see the probability distributions for the two masses trace out bananas of roughly constant chirp mass. How far along the banana you go depends on what spins you allow. We show results for two ranges: one with spins (aligned with the orbital angular momentum) up to 0.89, the other with spins up to 0.05. There’s nothing physical about 0.89 (it was just convenient for our analysis), but it is designed to be agnostic, and above the limit you’d plausibly expect for neutron stars (they should rip themselves apart at spins of ~0.7); the lower limit of 0.05 should safely encompass the spins of the binary neutron stars (which are close enough to merge in the age of the Universe) we have estimated from pulsar observations. The masses roughly match what we have measured for the neutron stars in our Galaxy. (The combinations at the tip of the banana for the high spins would be a bit odd).
Estimated masses for the two neutron stars in the binary. We show results for two different spin limits, $\chi_z$ is the component of the spin aligned with the orbital angular momentum. The two-dimensional shows the 90% probability contour, which follows a line of constant chirp mass. The one-dimensional plot shows individual masses; the dotted lines mark 90% bounds away from equal mass. Figure 4 of the GW170817 Discovery Paper.
If we were dealing with black holes, we’d be done: they are only described by mass and spin. Neutron stars are more complicated. Black holes are just made of warped spacetime, neutron stars are made of delicious nuclear material. This can get distorted during the inspiral—tides are raised on one by the gravity of the other. These extract energy from the orbit and accelerate the inspiral. The tidal deformability depends on the properties of the neutron star matter (described by its equation of state). The fluffier a neutron star is, the bigger the impact of tides; the more compact, the smaller the impact. We don’t know enough about neutron star material to predict this with certainty—by measuring the tidal deformation we can learn about the allowed range. Unfortunately, we also don’t yet have good model waveforms including tides, so for now we’ve just done a preliminary analysis. We find that some of the stiffer equations of state (the ones which predict larger neutron stars and bigger tides) are disfavoured; however, we cannot rule out zero tides. This means we can’t rule out the possibility that we have found two low-mass black holes from the gravitational waves alone. This would be an interesting discovery; however, the electromagnetic observations mean that the more obvious explanation of neutron stars is more likely.
From the gravitational wave signal, we can infer the source distance. Combining this with the electromagnetic observations we can do some cool things.
First, the gamma ray burst arrived at Earth 1.7 seconds after the merger. 1.7 seconds is not a lot of difference after travelling something like 85–160 million years (that’s roughly the time since the Cretaceous or Late Jurassic periods). Of course, we don’t expect the gamma-rays to be emitted at exactly the moment of merger, but allowing for a sensible range of emission times, we can bound the difference between the speed of gravity and the speed of light. In general relativity they should be the same, and we find that the difference should be no more than three parts in $10^{15}$.
Second, we can combine the gravitational wave distance with the redshift of the galaxy to measure the Hubble constant, the rate of expansion of the Universe. Our best estimates for the Hubble constant, from the cosmic microwave background and from supernova observations, are inconsistent with each other (the most recent supernova analysis only increase the tension). Which is awkward. Gravitational wave observations should have different sources of error and help to resolve the difference. Unfortunately, with only one event our uncertainties are rather large, which leads to a diplomatic outcome.
Posterior probability distribution for the Hubble constant $H_0$ inferred from GW170817. The lines mark 68% and 95% intervals. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SHoES). Figure 1 of the Hubble Constant Paper.
Finally, we can now change from estimating upper limits on binary neutron star merger rates to estimating the rates! We estimate the merger rate density is in the range $1540^{+3200}_{-1220}~\mathrm{Gpc^{-3}\,yr^{-1}}$ (assuming a uniform of neutron star masses between one and two solar masses). This is surprisingly close to what the Collaboration expected back in 2010: a rate of between $10~\mathrm{Gpc^{-3}\,yr^{-1}}$ and $10000~\mathrm{Gpc^{-3}\,yr^{-1}}$, with a realistic rate of $1000~\mathrm{Gpc^{-3}\,yr^{-1}}$. This means that we are on track to see many more binary neutron stars—perhaps one a week at design sensitivity!
### Summary
Advanced LIGO and Advanced Virgo observed a binary neutron star insprial. The rest of the astronomical community has observed what happened next (sadly there are no neutrinos). This is the first time we have such complementary observations—hopefully there will be many more to come. There’ll be a huge number of results coming out over the following days and weeks. From these, we’ll start to piece together more information on what neutron stars are made of, and what happens when you smash them together (take that particle physicists).
Also: I’m exhausted, my inbox is overflowing, and I will have far too many papers to read tomorrow.
GW170817 Discovery Paper: GW170817: Observation of gravitational waves from a binary neutron star inspiral
Multimessenger Astronomy Paper: Multi-messenger observations of a binary neutron star merger
Data release:
LIGO Open Science Center
### Bonus notes
#### Inbox zero
Over my vacation I cleaned up my email. I had a backlog starting around September 2015. I think there were over 6000 which I sorted or deleted. I had about 20 left to deal with when I got back to work. GW170817 undid that. Despite doing my best to keep up, there are over a 1000 emails in my inbox…
#### Worst case scenario
Around the start of O2, I was asked when I expected our results to be public. I said it would depend upon what we found. If it was only high-mass black holes, those are quick to analyse and we know what to do with them, so results shouldn’t take long, now we have the first few out of the way. In this case, perhaps a couple months as we would have been generating results as we went along. However, the worst case scenario would be a binary neutron star overlapping with non-Gaussian noise. Binary neutron stars are more difficult to analyse (they are longer signals, and there are matter effects to worry about), and it would be complicated to get everyone to be happy with our results because we were doing lots of things for the first time. Obviously, if one of these happened at the end of the run, there’d be quite a delay…
I think I got that half-right. We’re done amazingly well analysing GW170817 to get results out in just two months, but I think it will be a while before we get the full O2 set of results out, as we’ve been neglecting otherthings (you’ll notice we’ve not updated our binary black hole merger rate estimate since GW170104, nor given detailed results for testing general relativity with the more recent detections).
At the time of the GW170817 alert, I was working on writing a research proposal. As part of this, I was explaining why it was important to continue working on gravitational-wave parameter estimation, in particular how to deal with non-Gaussian or non-stationary noise. I think I may be a bit of a jinx. For GW170817, the glitch wasn’t a big problem, these type of blips can be removed. I’m more concerned about the longer duration ones, which are less easy to separate out from background noise. Don’t say I didn’t warn you in O3.
#### Parameter estimation rota
The duty of analysing signals to infer their source properties was divided up into shifts for O2. On January 4, the time of GW170104, I was on shift with my partner Aaron Zimmerman. It was his first day. Having survived that madness, Aaron signed back up for the rota. Can you guess who was on shift for the week which contained GW170814 and GW170817? Yep, Aaron (this time partnered with the excellent Carl-Johan Haster). Obviously, we’ll need to have Aaron on rota for the entirety of O3. In preparation, he has already started on paper drafting
Methods Section: Chained ROTA member to a terminal, ignored his cries for help. Detections followed swiftly.
The lightest elements (hydrogen, helium and lithium) we made during the Big Bang. Stars burn these to make heavier elements. Energy can be released up to around iron. Therefore, heavier elements need to be made elsewhere, for example in the material ejected from supernova or (as we have now seen) neutron star mergers, where there are lots of neutrons flying around to be absorbed. Elements (like gold and platinum) formed by this rapid neutron capture are known as r-process elements, I think because they are beloved by pirates.
A couple of weeks ago, the Nobel Prize in Physics was announced for the observation of gravitational waves. In December, the laureates will be presented with a gold (not chocolate) medal. I love the idea that this gold may have come from merging neutron stars.
Here’s one we made earlier. Credit: Associated Press/F. Vergara
# GW170814—Enter Virgo
On 14 August 2017 a gravitational wave signal (GW170814), originating from the coalescence of a binary black hole system, was observed by the global gravitational-wave observatory network of the two Advanced LIGO detectors and Advanced Virgo. That’s right, Virgo is in the game!
Very few things excite me like unlocking a new character in Smash Bros. A new gravitational wave observatory might come close.
Advanced Virgo joined O2, the second observing run of the advanced detector era, on 1 August. This was a huge achievement. It has not been an easy route commissioning the new detector—it never ceases to amaze me how sensitive these machines are. Together, Advanced Virgo (near Pisa) and the two Advanced LIGO detectors (in Livingston and Hanford in the US) would take data until the end of O2 on 25 August.
On 14 August, we found a signal. A signal that was observable in all three detectors [bonus note]. Virgo is less sensitive than the LIGO instruments, so there is no impressive plot that shows something clearly popping out, but the Virgo data do complement the LIGO observations, indicating a consistent signal in all three detectors [bonus note].
A cartoon of three different ways to visualise GW170814 in the three detectors. These take a bit of explaining. The top panel shows the signal-to-noise ratio the search template that matched GW170814. They peak at the time corresponding to the merger. The peaks are clear in Hanford and Livingston. The peak in Virgo is less exceptional, but it matches the expected time delay and amplitude for the signal. The middle panels show time–frequency plots. The upward sweeping chirp is visible in Hanford and Livingston, but less so in Virgo as it is less sensitive. The plot is zoomed in so that its possible to pick out the detail in Virgo, but the chirp is visible for a longer stretch of time than plotted in Livingston. The bottom panel shows whitened and band-passed strain data, together with the 90% region of the binary black hole templates used to infer the parameters of the source (the narrow dark band), and an unmodelled, coherent reconstruction of the signal (the wider light band) . The agreement between the templates and the reconstruction is a check that the gravitational waves match our expectations for binary black holes. The whitening of the data mirrors how we do the analysis, by weighting noise at different frequency by an estimate of their typical fluctuations. The signal does certainly look like the inspiral, merger and ringdown of a binary black hole. Figure 1 of the GW170814 Paper.
The signal originated from the coalescence of two black holes. GW170814 is thus added to the growing family of GW150914, LVT151012, GW151226 and GW170104.
GW170814 most closely resembles GW150914 and GW170104 (perhaps there’s something about ending with a 4). If we compare the masses of the two component black holes of the binary ($m_1$ and $m_2$), and the black hole they merge to form ($M_\mathrm{f}$), they are all quite similar
• GW150914: $m_1 = 36.2^{+5.2}_{-3.8} M_\odot$, $m_2 = 29.1^{+3.7}_{-4.4} M_\odot$, $M_\mathrm{f} = 62.3^{+3.7}_{-3.1} M_\odot$;
• GW170104: $m_1 = 31.2^{+5.4}_{-6.0} M_\odot$, $m_2 = 19.4^{+5.3}_{-5.9} M_\odot$, $M_\mathrm{f} = 48.7^{+5.7}_{-4.6} M_\odot$;
• GW170814: $m_1 = 30.5^{+5.7}_{-3.0} M_\odot$, $m_2 = 25.3^{+2.8}_{-4.2} M_\odot$, $M_\mathrm{f} = 53.2^{+3.2}_{-2.5} M_\odot$.
GW170814’s source is another high-mass black hole system. It’s not too surprising (now we know that these systems exist) that we observe lots of these, as more massive black holes produce louder gravitational wave signals.
GW170814 is also comparable in terms of black holes spins. Spins are more difficult to measure than masses, so we’ll just look at the effective inspiral spin $\chi_\mathrm{eff}$, a particular combination of the two component spins that influences how they inspiral together, and the spin of the final black hole $a_\mathrm{f}$
• GW150914: $\chi_\mathrm{eff} = -0.06^{+0.14}_{-0.14}$, $a_\mathrm{f} = 0.70^{+0.07}_{-0.05}$;
• GW170104:$\chi_\mathrm{eff} = -0.12^{+0.21}_{-0.30}$, $a_\mathrm{f} = 0.64^{+0.09}_{-0.20}$;
• GW170814:$\chi_\mathrm{eff} = 0.06^{+0.12}_{-0.12}$, $a_\mathrm{f} = 0.70^{+0.07}_{-0.05}$.
There’s some spread, but the effective inspiral spins are all consistent with being close to zero. Small values occur when the individual spins are small, if the spins are misaligned with each other, or some combination of the two. I’m starting to ponder if high-mass black holes might have small spins. We don’t have enough information to tease these apart yet, but this new system is consistent with the story so far.
One of the things Virgo helps a lot with is localizing the source on the sky. Most of the information about the source location comes from the difference in arrival times at the detectors (since we know that gravitational waves should travel at the speed of light). With two detectors, the time delay constrains the source to a ring on the sky; with three detectors, time delays can narrow the possible locations down to a couple of blobs. Folding in the amplitude of the signal as measured by the different detectors adds extra information, since detectors are not equally sensitive to all points on the sky (they are most sensitive to sources over head or underneath). This can even help when you don’t observe the signal in all detectors, as you know the source must be in a direction that detector isn’t too sensitive too. GW170814 arrived at LIGO Livingston first (although it’s not a competition), then ~8 ms later at LIGO Hanford, and finally ~14 ms later at Virgo. If we only had the two LIGO detectors, we’d have an uncertainty on the source’s sky position of over 1000 square degrees, but adding in Virgo, we get this down to 60 square degrees. That’s still pretty large by astronomical standards (the full Moon is about a quarter of a square degree), but a fantastic improvement [bonus note]!
90% probability localizations for GW170814. The large banana shaped (and banana coloured, but not banana flavoured) curve uses just the two LIGO detectors, the area is 1160 square degrees. The green shows the improvement adding Virgo, the area is just 100 square degrees. Both of these are calculated using BAYESTAR, a rapid localization algorithm. The purple map is the final localization from our full parameter estimation analysis (LALInference), its area is just 60 square degrees! Whereas BAYESTAR only uses the best matching template from the search, the full parameter estimation analysis is free to explore a range of different templates. Part of Figure 3 of the GW170814 Paper.
Having additional detectors can help improve gravitational wave measurements in other ways too. One of the predictions of general relativity is that gravitational waves come in two polarizations. These polarizations describe the pattern of stretching and squashing as the wave passes, and are illustrated below.
The two polarizations of gravitational waves: plus (left) and cross (right). Here, the wave is travelling into or out of the screen. Animations adapted from those by MOBle on Wikipedia.
These two polarizations are the two tensor polarizations, but other patterns of squeezing could be present in modified theories of gravity. If we could detect any of these we would immediately know that general relativity is wrong. The two LIGO detectors are almost exactly aligned, so its difficult to get any information on other polarizations. (We tried with GW150914 and couldn’t say anything either way). With Virgo, we get a little more information. As a first illustration of what we may be able to do, we compared how well the observed pattern of radiation at the detectors matched different polarizations, to see how general relativity’s tensor polarizations compared to a signal of entirely vector or scalar radiation. The tensor polarizations are clearly preferred, so general relativity lives another day. This isn’t too surprising, as most modified theories of gravity with other polarizations predict mixtures of the different polarizations (rather than all of one). To be able to constrain all the mixtures with these short signals we really need a network of five detectors, so we’ll have to wait for KAGRA and LIGO-India to come on-line.
The six polarizations of a metric theory of gravity. The wave is travelling in the $z$ direction. (a) and (b) are the plus and cross tensor polarizations of general relativity. (c) and (d) are the scalar breathing and longitudinal modes, and (e) and (f) are the vector $x$ and $y$ polarizations. The tensor polarizations (in red) are transverse, the vector and longitudinal scalar mode (in green) are longitudinal. The scalar breathing mode (in blue) is an isotropic expansion and contraction, so its a bit of a mix of transverse and longitudinal. Figure 10 from (the excellent) Will (2014).
We’ll be presenting a more detailed analysis of GW170814 later, in papers summarising our O2 results, so stay tuned for more.
Title: GW170814: A three-detector observation of gravitational waves from a binary black hole coalescence
arXiv: 1709.09660 [gr-qc]
Journal: Physical Review Letters; 119(14):141101(16) [bonus note]
Data release: LIGO Open Science Center
Science summary: GW170814: A three-detector observation of gravitational waves from a binary black hole coalescence
### Bonus notes
#### Signs of paranoia
Those of you who have been following the story of gravitational waves for a while may remember the case of the Big Dog. This was a blind injection of a signal during the initial detector era. One of the things that made it an interesting signal to analyse, was that it had been injected with an inconsistent sign in Virgo compared to the two LIGO instruments (basically it was upside down). Making this type of sign error is easy, and we were a little worried that we might make this sort of mistake when analysing the real data. The Virgo calibration team were extremely careful about this, and confident in their results. Of course, we’re quite paranoid, so during the preliminary analysis of GW170814, we tried some parameter estimation runs with the data from Virgo flipped. This was clearly disfavoured compared to the right sign, so we all breathed easily.
I am starting to believe that God may be a detector commissioner. At the start of O1, we didn’t have the hardware injection systems operational, but GW150914 showed that things were working properly. Now, with a third detector on-line, GW170814 shows that the network is functioning properly. Astrophysical injections are definitely the best way to confirm things are working!
#### Signal hunting
Our usual way to search for binary black hole signals is compare the data to a bank of waveform templates. Since Virgo is less sensitive the the two LIGO detectors, and would only be running for a short amount of time, these main searches weren’t extended to use data from all three detectors. This seemed like a sensible plan, we were confident that this wouldn’t cause us to miss anything, and we can detect GW170814 with high significance using just data from Livingston and Hanford—the false alarm rate is estimated to be less than 1 in 27000 years (meaning that if the detectors were left running in the same state, we’d expect random noise to make something this signal-like less than once every 27000 years). However, we realised that we wanted to be able to show that Virgo had indeed seen something, and the search wasn’t set up for this.
Therefore, for the paper, we list three different checks to show that Virgo did indeed see the signal.
1. In a similar spirit to the main searches, we took the best fitting template (it doesn’t matter in terms of results if this is the best matching template found by the search algorithms, or the maximum likelihood waveform from parameter estimation), and compared this to a stretch of data. We then calculated the probability of seeing a peak in the signal-to-noise ratio (as shown in the top row of Figure 1) at least as large as identified for GW170814, within the time window expected for a real signal. Little blips of noise can cause peaks in the signal-to-noise ratio, for example, there’s a glitch about 50 ms after GW170814 which shows up. We find that there’s a 0.3% probability of getting a signal-to-ratio peak as large as GW170814. That’s pretty solid evidence for Virgo having seen the signal, but perhaps not overwhelming.
2. Binary black hole coalescences can also be detected (if the signals are short) by our searches for unmodelled signals. This was the case for GW170814. These searches were using data from all three detectors, so we can compare results with and without Virgo. Using just the two LIGO detectors, we calculate a false alarm rate of 1 per 300 years. This is good enough to claim a detection. Adding in Virgo, the false alarm rate drops to 1 per 5900 years! We see adding in Virgo improves the significance by almost a factor of 20.
3. Using our parameter estimation analysis, we calculate the evidence (marginal likelihood) for (i) there being a coherent signal in Livingston and Hanford, and Gaussian noise in Virgo, and (ii) there being a coherent signal in all three detectors. We then take the ratio to calculate the Bayes factor. We find that a coherent signal in all three detectors is preferred by a factor of over 1600. This is a variant of a test proposed in Veitch & Vecchio (2010); it could be fooled if the noise in Virgo is non-Gaussian (if there is a glitch), but together with the above we think that the simplest explanation for Virgo’s data is that there is a signal.
In conclusion: Virgo works. Probably.
#### Follow-up observations
Adding Virgo to the network greatly improves localization of the source, which is a huge advantage when searching for counterparts. For a binary black hole, as we have here, we don’t expect a counterpart (which would make finding one even more exciting). So far, no counterpart has been reported.
• Arcavi et al. (2017) reported an optical search from the Las Cumbres Observatory.
• Smith et al. (2018) reported an optical search, targeting strong-lensing galaxy clusters, with Gemini South and the Very Large Telescope.
#### Announcement
This is the first observation we’ve announced before being published. The draft made public at time at announcement was accepted, pending fixing up some minor points raised by the referees (who were fantastically quick in reporting back). I guess that binary black holes are now familiar enough that we are on solid ground claiming them. I’d be interested to know if people think that it would be good if we didn’t always wait for the rubber stamp of peer review, or whether they would prefer to for detections to be externally vetted? Sharing papers before publication would mean that we get more chance for feedback from the community, which is would be good, but perhaps the Collaboration should be seen to do things properly?
One reason that the draft paper is being shared early is because of an opportunity to present to the G7 Science Ministers Meeting in Italy. I think any excuse to remind politicians that international collaboration is a good thing™ is worth taking. Although I would have liked the paper to be a little more polished [bonus advice]. The opportunity to present here only popped up recently, which is one reason why things aren’t as perfect as usual.
I also suspect that Virgo were keen to demonstrate that they had detected something prior to any Nobel Prize announcement. There’s a big difference between stories being written about LIGO and Virgo’s discoveries, and having as an afterthought that Virgo also ran in August.
The main reason, however, was to get this paper out before the announcement of GW170817. The identification of GW170817’s counterpart relied on us being able to localize the source. In that case, there wasn’t a clear signal in Virgo (the lack of a signal tells us the source wan’t in a direction wasn’t particularly sensitive to). People agreed that we really need to demonstrate that Virgo can detect gravitational waves in order to be convincing that not seeing a signal is useful information. We needed to demonstrate that Virgo does work so that our case for GW170817 was watertight and bulletproof (it’s important to be prepared).
Some useful advice I was given when a PhD student was that done is better than perfect. Having something finished is often more valuable than having lots of really polished bits that don’t fit together to make a cohesive whole, and having everything absolutely perfect takes forever. This is useful to remember when writing up a thesis. I think it might apply here too: the Paper Writing Team have done a truly heroic job in getting something this advanced in little over a month. There’s always one more thing to do… [one more bonus note]
#### One more thing
One point I was hoping that the Paper Writing Team would clarify is our choice of prior probability distribution for the black hole spins. We don’t get a lot of information about the spins from the signal, so our choice of prior has an impact on the results.
The paper says that we assume “no restrictions on the spin orientations”, which doesn’t make much sense, as one of the two waveforms we use to analyse the signal only includes spins aligned with the orbital angular momentum! What the paper meant was that we assume a prior distribution which has an isotopic distribution of spins, and for the aligned spin (no precession) waveform, we assume a prior probability distribution on the aligned components of the spins which matches what you would have for an isotropic distribution of spins (in effect, assuming that we can only measure the aligned components of the spins, which is a good approximation).
# Observing run 1—The papers
The second observing run (O2) of the advanced gravitational wave detectors is now over, which has reminded me how dreadfully behind I am in writing about papers. In this post I’ll summarise results from our first observing run (O1), which ran from September 2015 to January 2016.
I’ll add to this post as I get time, and as papers are published. I’ve started off with papers searching for compact binary coalescences (as these are closest to my own research). There are separate posts on our detections GW150914 (and its follow-up papers: set I, set II) and GW151226 (this post includes our end-of-run summary of the search for binary black holes, including details of LVT151012).
### Transient searches
#### The O1 Binary Neutron Star/Neutron Star–Black Hole Paper
Title: Upper limits on the rates of binary neutron star and neutron-star–black-hole mergers from Advanced LIGO’s first observing run
arXiv: 1607.07456 [astro-ph.HE]
Journal: Astrophysical Journal Letters; 832(2):L21(15); 2016
Our main search for compact binary coalescences targets binary black holes (binaries of two black holes), binary neutron stars (two neutron stars) and neutron-star–black-hole binaries (one of each). Having announced the results of our search for binary black holes, this paper gives the detail of the rest. Since we didn’t make any detections, we set some new, stricter upper limits on their merger rates. For binary neutron stars, this is $12,600~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ .
#### The O1 Gamma-Ray Burst Paper
Title: Search for gravitational waves associated with gamma-ray bursts during the first Advanced LIGO observing run and implications for the origin of GRB 150906B
arXiv: 1611.07947 [astro-ph.HE]
Journal: Astrophysical Journal; 841(2):89(18); 2016
LIGO science summary: What’s behind the mysterious gamma-ray bursts? LIGO’s search for clues to their origins
Some binary neutron star or neutron-star–black-hole mergers may be accompanied by a gamma-ray burst. This paper describes our search for signals coinciding with observations of gamma-ray bursts (including GRB 150906B, which was potentially especially close by). Knowing when to look makes it easy to distinguish a signal from noise. We don’t find anything, so we we can exclude any close binary mergers as sources of these gamma-ray bursts.
More details: O1 Gamma-Ray Burst Paper summary
#### The O1 Intermediate Mass Black Hole Binary Paper
Title: Search for intermediate mass black hole binaries in the first observing run of Advanced LIGO
arXiv: 1704.04628 [gr-qc]
Journal: Physical Review D; 96(2):022001(14); 2017
LIGO science summary: Search for mergers of intermediate-mass black holes
Our main search for binary black holes in O1 targeted systems with masses less than about 100 solar masses. There could be more massive black holes out there. Our detectors are sensitive to signals from binaries up to a few hundred solar masses, but these are difficult to detect because they are so short. This paper describes our specially designed such systems. This combines techniques which use waveform templates and those which look for unmodelled transients (bursts). Since we don’t find anything, we set some new upper limits on merger rates.
More details: O1 Intermediate Mass Black Hole Binary Paper summary
#### The O1 Burst Paper
Title: All-sky search for short gravitational-wave bursts in the first Advanced LIGO run
arXiv: 1611.02972 [gr-qc]
Journal: Physical Review D; 95(4):042003(14); 2017
If we only search for signals for which we have models, we’ll never discover something new. Unmodelled (burst) searches are more flexible and don’t assume a particular form for the signal. This paper describes our search for short bursts. We successfully find GW150914, as it is short and loud, and burst searches are good for these type of signals, but don’t find anything else. (It’s not too surprising GW151226 and LVT151012 are below the threshold for detection because they are longer and quieter than GW150914).
More details: O1 Burst Paper summary
### The O1 Binary Neutron Star/Neutron Star–Black Hole Paper
Synopsis: O1 Binary Neutron Star/Neutron Star–Black Hole Paper
Read this if: You want a change from black holes
Favourite part: We’re getting closer to detection (and it’ll still be interesting if we don’t find anything)
The Compact Binary Coalescence (CBC) group target gravitational waves from three different flavours of binary in our main search: binary neutron stars, neutron star–black hole binaries and binary black holes. Before O1, I would have put my money on us detecting a binary neutron star first, around-about O3. Reality had other ideas, and we discovered binary black holes. Those results were reported in the O1 Binary Black Hole Paper; this paper goes into our results for the others (which we didn’t detect).
To search for signals from compact binaries, we use a bank of gravitational wave signals to match against the data. This bank goes up to total masses of 100 solar masses. We split the bank up, so that objects below 2 solar masses are considered neutron stars. This doesn’t make too much difference to the waveforms we use to search (neutrons stars, being made of stuff, can be tidally deformed by their companion, which adds some extra features to the waveform, but we don’t include these in the search). However, we do limit the spins for neutron stars to less the 0.05, as this encloses the range of spins estimated for neutron star binaries from binary pulsars. This choice shouldn’t impact our ability to detect neutron stars with moderate spins too much.
We didn’t find any interesting events: the results were consistent with there just being background noise. If you read really carefully, you might have deduced this already from the O1 Binary Black Hole Paper, as the results from the different types of binaries are completely decoupled. Since we didn’t find anything, we can set some upper limits on the merger rates for binary neutron stars and neutron star–black hole binaries.
The expected number of events found in the search is given by
$\Lambda = R \langle VT \rangle$
where $R$ is the merger rate, and $\langle VT \rangle$ is the surveyed time–volume (you expect more detections if your detectors are more sensitive, so that they can find signals from further away, or if you leave them on for longer). We can estimate $\langle VT \rangle$ by performing a set of injections and seeing how many are found/missed at a given threshold. Here, we use a false alarm rate of one per century. Given our estimate for $\langle VT \rangle$ and our observation of zero detections we can, calculate a probability distribution for $R$ using Bayes’ theorem. This requires a choice for a prior distribution of $\Lambda$. We use a uniform prior, for consistency with what we’ve done in the past.
With a uniform prior, the $c$ confidence level limit on the rate is
$\displaystyle R_c = \frac{-\ln(1-c)}{\langle VT \rangle}$,
so the 90% confidence upper limit is $R_{90\%} = 2.30/\langle VT \rangle$. This is quite commonly used, for example we make use of it in the O1 Intermediate Mass Black Hole Binary Search. For comparison, if we had used a Jeffrey’s prior of $1/\sqrt{\Lambda}$, the equivalent results is
$\displaystyle R_c = \frac{\left[\mathrm{erf}^{-1}(c)\right]^2}{\langle VT \rangle}$,
and hence $R_{90\%} = 1.35/\langle VT \rangle$, so results would be the same to within a factor of 2, but the results with the uniform prior are more conservative.
The plot below shows upper limits for different neutron star masses, assuming that neutron spins are (uniformly distributed) between 0 and 0.05 and isotropically orientated. From our observations of binary pulsars, we have seen that most of these neutron stars have masses of ~1.35 solar masses, so we can also put a limit of the binary neutron star merger rate assuming that their masses are normally distributed with mean of 1.35 solar masses and standard deviation of 0.13 solar masses. This gives an upper limit of $R_{90\%} = 12,100~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ for isotropic spins up to 0.05, and $R_{90\%} = 12,600~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ if you allow the spins up to 0.4.
90% confidence upper limits on the binary neutron star merger rate. These rates assume randomly orientated spins up to 0.05. Results are calculated using PyCBC, one of our search algorithms; GstLAL gives similar results. Figure 4 of the O1 Binary Neutron Star/Neutron Star–Black Hole Paper.
For neutron star–black hole binaries there’s a greater variation in possible merger rates because the black holes can have a greater of masses and spins. The upper limits range from about $R_{90\%} = 1,200~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ to $3,600~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ for a 1.4 solar mass neutron star and a black hole between 30 and 5 solar masses and a range of different spins (Table II of the paper).
It’s not surprising that we didn’t see anything in O1, but what about in future runs. The plots below compare projections for our future sensitivity with various predictions for the merger rates of binary neutron stars and neutron star–black hole binaries. A few things have changed since we made these projections, for example O2 ended up being 9 months instead of 6 months, but I think we’re still somewhere in the O2 band. We’ll have to see for O3. From these, it’s clear that a detection on O1 was overly optimistic. In O2 and O3 it becomes more plausible. This means even if we don’t see anything, we’ll be still be doing some interesting astrophysics as we can start ruling out some models.
Comparison of upper limits for binary neutron star (BNS; top) and neutron star–black hole binaries (NSBH; bottom) merger rates with theoretical and observational limits. The blue bars show O1 limits, the green and orange bars show projections for future observing runs. Figures 6 and 7 from the O1 Binary Neutron Star/Neutron Star–Black Hole Paper.
Binary neutron star or neutron star–black hole mergers may be the sources of gamma-ray bursts. These are some of the most energetic explosions in the Universe, but we’re not sure where they come from (I actually find that kind of worrying). We look at this connection a bit more in the O1 Gamma-Ray Burst Paper. The theory is that during the merger, neutron star matter gets ripped apart, squeezed and heated, and as part of this we get jets blasted outwards from the swirling material. There are always jets in these type of things. We see the gamma-ray burst if we are looking down the jet: the wider the jet, the larger the fraction of gamma-ray bursts we see. By comparing our estimated merger rates, with the estimated rate of gamma-ray bursts, we can place some lower limits on the opening angle of the jet. If all gamma-ray bursts come from binary neutron stars, the opening angle needs to be bigger than $2.3_{-1.7}^{+1.7}~\mathrm{deg}$ and if they all come from neutron star–black hole mergers the angle needs to be bigger than $4.3_{-1.9}^{+3.1}~\mathrm{deg}$.
### The O1 Gamma-Ray Burst Paper
Synopsis: O1 Gamma-Ray Burst Paper
Read this if: You like explosions. But from a safe distance
Favourite part: We exclude GRB 150906B from being associated with galaxy NGC 3313
Gamma-ray bursts are extremely violent explosions. They come in two (overlapping) classes: short and long. Short gamma-ray bursts are typically shorter than ~2 seconds and have a harder spectrum (more high energy emission). We think that these may come from the coalescence of neutron star binaries. Long gamma-ray bursts are (shockingly) typically longer than ~2 seconds, and have a softer spectrum (less high energy emission). We think that these could originate from the collapse of massive stars (like a supernova explosion). The introduction of the paper contains a neat review of the physics of both these types of sources. Both types of progenitors would emit gravitational waves that could be detected if the source was close enough.
The binary mergers could be picked up by our templated search (as reported in the O1 Binary Neutron Star/Neutron Star–Black Hole Paper): we have a good models for what these signals look like, which allows us to efficiently search for them. We don’t have good models for the collapse of stars, but our unmodelled searches could pick these up. These look for the same signal in multiple detectors, but since they don’t know what they are looking for, it is harder to distinguish a signal from noise than for the templated search. Cross-referencing our usual searches with the times of gamma-ray bursts could help us boost the significance of a trigger: it might not be noteworthy as just a weak gravitational-wave (or gamma-ray) candidate, but considering them together makes it much more unlikely that a coincidence would happen by chance. The on-line RAVEN pipeline monitors for alerts to minimise the chance that miss a coincidence. As well as relying on our standard searches, we also do targeted searches following up on gamma-ray bursts, using the information from these external triggers.
We used two search algorithms:
• X-Pipeline is an unmodelled search (similar to cWB) which looks for a coherent signal, consistent with the sky position of the gamma-ray burst. This was run for all the gamma-ray bursts (long and short) for which we have good data from both LIGO detectors and a good sky location.
• PyGRB is a modelled search which looks for binary signals using templates. Our main binary search algorithms check for coincident signals: a signal matching the same template in both detectors with compatible times. This search looks for coherent signals, factoring the source direction. This gives extra sensitivity (~20%–25% in terms of distance). Since we know what the signal looks like, we can also use this algorithm to look for signals when only one detector is taking data. We used this algorithm on all short (or ambiguously classified) gamma-ray bursts for which we data from at least one detector.
In total we analysed times corresponding to 42 gamma-ray bursts: 41 which occurred during O1 plus GRB 150906B. This happening in the engineering run before the start of O1, and luckily Handord was in a stable observing state at the time. GRB 150906B was localised to come from part of the sky close to the galaxy NGC 3313, which is only 54 megaparsec away. This is within the regime where we could have detected a binary merger. This caused much excitement at the time—people thought that this could be the most interesting result of O1—but this dampened down a week later with the detection of GW150914.
Interplanetary Network (IPN) localization for GRB 150906B and nearby galaxies. Figure 1 from the O1 Gamma-Ray Burst Paper.
We didn’t find any gravitational-wave counterparts. These means that we could place some lower limits on how far away their sources could be. We performed injections of signals—using waveforms from binaries, collapsing stars (approximated with circular sine–Gaussian waveforms), and unstable discs (using an accretion disc instability model)—to see how far away we could have detected a signal, and set 90% probability limits on the distances (see Table 3 of the paper). The best of these are ~100–200 megaparsec (the worst is just 4 megaparsec, which is basically next door). These results aren’t too interesting yet, they will become more so in the future, and around the time we hit design sensitivity we will start overlapping with electromagnetic measurements of distances for short gamma-ray bursts. However, we can rule out GRB 150906B coming from NGC 3133 at high probability!
### The O1 Intermediate Mass Black Hole Binary Paper
Synopsis: O1 Intermediate Mass Black Hole Binary Paper
Read this if: You like intermediate mass black holes (black holes of ~100 solar masses)
Favourite part: The teamwork between different searches
Black holes could come in many sizes. We know of stellar-mass black holes, the collapsed remains of dead stars, which are a few to a few tens of times the mas of our Sun, and we know of (super)massive black holes, lurking in the centres of galaxies, which are tens of thousands to billions of times the mass of our Sun. Between the two, lie the elusive intermediate mass black holes. There have been repeated claims of observational evidence for their existence, but these are notoriously difficult to confirm. Gravitational waves provide a means of confirming the reality of intermediate mass black holes, if they do exist.
The gravitational wave signal emitted by a binary depends upon the mass of its components. More massive objects produce louder signals, but these signals also end at lower frequencies. The merger frequency of a binary is inversely proportional to the total mass. Ground-based detectors can’t detect massive black hole binaries as they are too low frequency, but they can detect binaries of a few hundred solar masses. We look for these in this search.
Our flagship search for binary black holes looks for signals using matched filtering: we compare the data to a bank of template waveforms. The bank extends up to a total mass of 100 solar masses. This search continues above this (there’s actually some overlap as we didn’t want to miss anything, but we shouldn’t have worried). Higher mass binaries are hard to detect as they as shorter, and so more difficult to distinguish from a little blip of noise, which is why this search was treated differently.
As well as using templates, we can do an unmodelled (burst) search for signals by looking for coherent signals in both detectors. This type of search isn’t as sensitive, as you don’t know what you are looking for, but can pick up short signals (like GW150914).
Our search for intermediate mass black holes uses both a modelled search (with templates spanning total masses of 50 to 600 solar masses) and a specially tuned burst search. Both make sure to include low frequency data in their analysis. This work is one of the few cross-working group (CBC for the templated search, and Burst for the unmodelled) projects, and I was pleased with the results.
This is probably where you expect me to say that we didn’t detect anything so we set upper limits. That is actually not the case here: we did detect something! Unfortunately, it wasn’t what we were looking for. We detected GW150914, which was a relief as it did lie within the range we where searching, as well as LVT151012 and GW151226. These were more of a surprise. GW151226 has a total mass of just ~24 solar masses (as measured with cosmological redshift), and so is well outside our bank. It was actually picked up just on the edge, but still, it’s impressive that the searches can find things beyond what they are aiming to pick up. Having found no intermediate mass black holes, we went and set some upper limits. (Yay!)
To set our upper limits, we injected some signals from binaries with specific masses and spins, and then saw how many would have be found with greater significance than our most significant trigger (after excluding GW150914, LVT151012 and GW151226). This is effectively asking the question of when would we see something as significant as this trigger which we think is just noise. This gives us a sensitive time–volume $\langle VT \rangle$ which we have surveyed and found no mergers. We use this number of events to set 90% upper limits on the merge rates $R_{90\%} = 2.3/\langle VT \rangle$, and define an effective distance $D_{\langle VT \rangle}$ defined so that $\langle VT \rangle = T_a (4\pi D_{\langle VT \rangle}^3/3)$ where $T_a$ is the analysed amount of time. The plot below show our limits on rate and effective distance for our different injections.
Results from the O1 search for intermediate mass black hole binaries. The left panel shows the 90% confidence upper limit on the merger rate. The right panel shows the effective search distance. Each circle is a different injection. All have zero spin, except two 100+100 solar mass sets, where $\chi$ indicates the spin aligned with the orbital angular momentum. Figure 2 of the O1 Intermediate Mass Black Hole Binary Paper.
There are a couple of caveats associated with our limits. The waveforms we use don’t include all the relevant physics (like orbital eccentricity and spin precession). Including everything is hard: we may use some numerical relativity waveforms in the future. However, they should give a good impression on our sensitivity. There’s quite a big improvement compared to previous searches (S6 Burst Search; S6 Templated Search). This comes form the improvement of Advanced LIGO’s sensitivity at low frequencies compared to initial LIGO. Future improvements to the low frequency sensitivity should increase our probability of making a detection.
I spent a lot of time working on this search as I was the review chair. As a reviewer, I had to make sure everything was done properly, and then reported accurately. I think our review team did a thorough job. I was glad when we were done, as I dislike being the bad cop.
### The O1 Burst Paper
Synopsis: O1 Burst Paper
Read this if: You like to keep an open mind about what sources could be out there
Favourite part: GW150914 (of course)
The best way to find a signal is to know what you are looking for. This makes it much easier to distinguish a signal from random noise. However, what about the sources for which we don’t have good models? Burst searches aim to find signals regardless of their shape. To do this, they look for coherent signals in multiple detectors. Their flexibility means that they are less sensitive than searches targeting a specific signal—the signal needs to be louder before we can be confident in distinguishing it from noise—but they could potentially detect a wider number of sources, and crucially catch signals missed by other searches.
This paper presents our main results looking for short burst signals (up to a few seconds in length). Complementary burst searches were done as part of the search for intermediate mass black hole binaries (whose signals can be so short that it doesn’t matter too much if you have a model or not) and for counterparts to gamma-ray bursts.
There are two-and-a-half burst search pipelines. There is coherent WaveBurst (cWB), Omicron–LALInferenceBurst (oLIB), and BayesWave follow-up to cWB. More details of each are found in the GW150914 Burst Companion Paper.
cWB looks for coherent power in the detectors—it looks for clusters of excess power in time and frequency. The search in O1 was split into a low-frequency component (signals below 1024 Hz) and a high-frequency component (1024 Hz). The low-frequency search was further divided into three classes:
• C1 for signals which have a small range of frequencies (80% of the power in just a 5 Hz range). This is designed to catch blip glitches, short bursts of transient noise in our detectors. We’re not sure what causes blip glitches yet, but we know they are not real signals as they are seen independently in both detectors.
• C3 looks for signals which increase in frequency with time—chirps. I suspect that this was (cheekily) designed to find binary black hole coalescences.
• C2 (no, I don’t understand the ordering either) is everything else.
The false alarm rate is calculated independently for each division using time-slides. We analyse data from the two detectors which has been shifted in time, so that there can be no real coincident signals between the two, and compare this background of noise-only triggers to the no-slid data.
oLIB works in two stages. First (the Omicron bit), data from the individual detectors are searches for excess power. If there is anything interesting, the data from both detectors are analysed coherently. We use a sine–Gaussian template, and compare the probability that the same signal is in both detectors, to there being independent noise (potentially a glitch) in the two. This analysis is split too: there is a high-quality factor vs low quality-factor split, which is similar to cWB’s splitting off C1 to catch narrow band features (the low quality-factor group catches the blip glitches). The false alarm rate is computed with time slides.
BayesWave is run as follow-up to triggers produced by cWB: it is too computationally expensive to run on all the data. BayesWave’s approach is similar to oLIB’s. It compares three hypotheses: just Gaussian noise, Gaussian noise and a glitch, and Gaussian noise and a signal. It constructs its signal using a variable number of sine–Gaussian wavelets. There are no cuts on its data. Again, time slides are used to estimate the false alarm rate.
The search does find a signal: GW150914. It is clearly found by all three algorithms. It is cWB’s C3, with a false alarm rate of less than 1 per 350 years; it is is oLIB’s high quality-factor bin with a false alarm rate of less than 1 per 230 years, and is found by BayesWave with a false alarm rate of less than 1 per 1000 years. You might notice that these results are less stringent than in the initial search results presented at the time of the detection. This is because only a limited number of time slides were done: we could get higher significance if we did more, but it was decided that it wasn’t worth the extra computing time, as we’re already convinced that GW150914 is a real signal. I’m a little sad they took GW150914 out of their plots (I guess it distorted the scale since it’s such an outlier from the background). Aside from GW150914, there are no detections.
Given the lack of detections, we can set some upper limits. I’ll skip over the limits for binary black holes, since our templated search is more sensitive here. The plot below shows limits on the amount of gravitational-wave energy emitted by a burst source at 10 kpc, which could be detected with a false alarm rate of 1 per century 50% of the time. We use some simple waveforms for this calculation. The energy scales with the inverse distance squared, so at a distance of 20 kpc, you need to increase the energy by a factor of 4.
Gravitational-wave energy at 50% detection efficiency for standard sources at a distance of 10 kpc. Results are shown for the three different algorithms. Figure 2 of the O1 Burst Paper.
Maybe next time we’ll find something unexpected, but it will either need to be really energetic (like a binary black hole merger) or really close by (like a supernova in our own Galaxy)
# Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignments
Gravitational waves allow us to infer the properties of binary black holes (two black holes in orbit about each other), but can we use this information to figure out how the black holes and the binary form? In this paper, we show that measurements of the black holes’ spins can help us this out, but probably not until we have at least 100 detections.
### Black hole spins
Black holes are described by their masses (how much they bend spacetime) and their spins (how much they drag spacetime to rotate about them). The orientation of the spins relative to the orbit of the binary could tell us something about the history of the binary [bonus note].
We considered four different populations of spin–orbit alignments to see if we could tell them apart with gravitational-wave observations:
1. Aligned—matching the idealised example of isolated binary evolution. This stands in for the case where misalignments are small, which might be the case if material blown off during a supernova ends up falling back and being swallowed by the black hole.
2. Isotropic—matching the expectations for dynamically formed binaries.
3. Equal misalignments at birth—this would be the case if the spins and orbit were aligned before the second supernova, which then tilted the plane of the orbit. (As the binary inspirals, the spins wobble around, so the two misalignment angles won’t always be the same).
4. Both spins misaligned by supernova kicks, assuming that the stars were aligned with the orbit before exploding. This gives a more general scatter of unequal misalignments, but typically the primary (bigger and first forming) black hole is more misaligned.
These give a selection of possible spin alignments. For each, we assumed that the spin magnitude was the same and had a value of 0.7. This seemed like a sensible idea when we started this study [bonus note], but is now towards the upper end of what we expect for binary black holes.
### Hierarchical analysis
To measurement the properties of the population we need to perform a hierarchical analysis: there are two layers of inference, one for the individual binaries, and one of the population.
From a gravitational wave signal, we infer the properties of the source using Bayes’ theorem. Given the data $d_\alpha$, we want to know the probability that the parameters $\mathbf{\Theta}_\alpha$ have different values, which is written as $p(\mathbf{\Theta}_\alpha|d_\alpha)$. This is calculated using
$\displaystyle p(\mathbf{\Theta}_\alpha|d_\alpha) = \frac{p(d_\alpha | \mathbf{\Theta}_\alpha) p(\mathbf{\Theta}_\alpha)}{p(d_\alpha)}$,
where $p(d_\alpha | \mathbf{\Theta}_\alpha)$ is the likelihood, which we can calculate from our knowledge of the noise in our gravitational wave detectors, $p(\mathbf{\Theta}_\alpha)$ is the prior on the parameters (what we would have guessed before we had the data), and the normalisation constant $p(d_\alpha)$ is called the evidence. We’ll use the evidence again in the next layer of inference.
Our prior on the parameters should actually depend upon what we believe about the astrophysical population. It is different if we believed that Model 1 were true (when we’d only consider aligned spins) than for Model 2. Therefore, we should really write
$\displaystyle p(\mathbf{\Theta}_\alpha|d_\alpha, \lambda) = \frac{p(d_\alpha | \mathbf{\Theta}_\alpha,\lambda) p(\mathbf{\Theta}_\alpha,\lambda)}{p(d_\alpha|\lambda)}$,
where $\lambda$ denotes which model we are considering.
This is an important point to remember: if you our using our LIGO results to test your theory of binary formation, you need to remember to correct for our choice of prior. We try to pick non-informative priors—priors that don’t make strong assumptions about the physics of the source—but this doesn’t mean that they match what would be expected from your model.
We are interested in the probability distribution for the different models: how many binaries come from each. Given a set of different observations $\{d_\alpha\}$, we can work this out using another application of Bayes’ theorem (yay)
$\displaystyle p(\mathbf{\lambda}|\{d_\alpha\}) = \frac{p(\{d_\alpha\} | \mathbf{\lambda}) p(\mathbf{\lambda})}{p(\{d_\alpha\})}$,
where $p(\{d_\alpha\} | \mathbf{\lambda})$ is just all the evidences for the individual events (given that model) multiplied together, $p(\mathbf{\lambda})$ is our prior for the different models, and $p(\{d_\alpha\})$ is another normalisation constant.
Now knowing how to go from a set of observations to the probability distribution on the different channels, let’s give it a go!
#### Results
To test our approach made a set of mock gravitational wave measurements. We generated signals from binaries for each of our four models, and analysed these as we would for real signals (using LALInference). This is rather computationally expensive, and we wanted a large set of events to analyse, so using these results as a guide, we created a larger catalogue of approximate distributions for the inferred source parameters $p(\mathbf{\Theta}_\alpha|d_\alpha)$. We then fed these through our hierarchical analysis. The GIF below shows how measurements of the fraction of binaries from each population tightens up as we get more detections: the true fraction is marked in blue.
Probability distribution for the fraction of binaries from each of our four spin misalignment populations for different numbers of observations. The blue dot marks the true fraction: and equal fraction from all four channels.
The plot shows that we do zoom in towards the true fraction of events from each model as the number of events increases, but there are significant degeneracies between the different models. Notably, it is difficult to tell apart Models 1 and 3, as both have strong support for both spins being nearly aligned. Similarly, there is a degeneracy between Models 2 and 4 as both allow for the two spins to have very different misalignments (and for the primary spin, which is the better measured one, to be quite significantly misaligned).
This means that we should be able to distinguish aligned from misaligned populations (we estimated that as few as 5 events would be needed to distinguish the case that all events came from either Model 1 or Model 2 if those were the only two allowed possibilities). However, it will be more difficult to distinguish different scenarios which only lead to small misalignments from each other, or disentangle whether there is significant misalignment due to big supernova kicks or because binaries are formed dynamically.
The uncertainty of the fraction of events from each model scales roughly with the square root of the number of observations, so it may be slow progress making these measurements. I’m not sure whether we’ll know the answer to how binary black hole form, or who will sit on the Iron Throne first.
arXiv: 1703.06873 [astro-ph.HE]
Journal: Monthly Notices of the Royal Astronomical Society471(3):2801–2811; 2017
Birmingham science summary: Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignment (by Simon)
If you like this you might like: Farr et al. (2017)Talbot & Thrane (2017), Vitale et al. (2017), Trifirò et al. (2016), Minogue (2000)
### Bonus notes
#### Spin misalignments and formation histories
If you have two stars forming in a binary together, you’d expect them to be spinning in roughly the same direction, rotating the same way as they go round in their orbit (like our Solar System). This is because they all formed from the same cloud of swirling gas and dust. Furthermore, if two stars are to form a black hole binary that we can detect gravitational waves from, they need to be close together. This means that there can be tidal forces which gently tug the stars to align their rotation with the orbit. As they get older, stars puff up, meaning that if you have a close-by neighbour, you can share outer layers. This transfer of material will tend to align rotate too. Adding this all together, if you have an isolated binary of stars, you might expect that when they collapse down to become black holes, their spins are aligned with each other and the orbit.
Unfortunately, real astrophysics is rarely so clean. Even if the stars were initially rotating the same way as each other, they doesn’t mean that their black hole remnants will do the same. This depends upon how the star collapses. Massive stars explode as supernova, blasting off their outer layers while their cores collapse down to form black holes. Escaping material could carry away angular momentum, meaning that the black hole is spinning in a different direction to its parent star, or material could be blasted off asymmetrically, giving the new black hole a kick. This would change the plane of the binary’s orbit, misaligning the spins.
Alternatively, the binary could be formed dynamically. Instead of two stars living their lives together, we could have two stars (or black holes) come close enough together to form a binary. This is likely to happen in regions where there’s a high density of stars, such as a globular cluster. In this case, since the binary has been randomly assembled, there’s no reason for the spins to be aligned with each other or the orbit. For dynamically assembled binaries, all spin–orbit misalignments are equally probable.
#### Slow and steady
This project was led by Simon Stevenson. It was one of the first things we started working on at the beginning of his PhD. He has now graduated, and is off to start a new exciting life as a postdoc in Australia. We got a little distracted by other projects, most notably analysing the first detections of gravitational waves. Simon spent a lot of time developing the COMPAS population code, a code to simulate the evolution of binaries. Looking back, it’s impressive how far he’s come. This paper used a simple approximation to to estimate the masses of our black holes: we called it the Post-it note model, as we wrote it down on a single Post-it. Now Simon’s writing papers including the complexities of common-envelope evolution in order to explain LIGO’s actual observations.
# Vacation theorising
I am currently enjoying some vacation, so it’s the perfect time to take a break from theorising about black holes and theorise about Game of Thrones instead. Here are three (not too scientific) theories for how the story may continue. Beware, there may be spoilers (for the books and the show), and dragons, below.
## 1 The dragon has three heads
The most active area of Game of Thrones theorising has been Jon’s parentage. The last series confirmed that Jon is not the illegitimate son of Ned Stark, possibly the most honourable man in Westeros, but is instead Ned’s nephew. Jon is the son of Lyanna Stark and Rhaegar Targaryen. Ned hid Jon’s identity to hide him from King Robert, who tried to kill all the Targaryens to protect his rule (presumably Jon would have been safe after joining the Night’s Watch and recounting his right to any titles). Jon’s true identity means that we now know of two extent Targaryens: Jon and (his aunt) Daenerys.
Is there a third?
Trogdor Targaryen? (credit: Homestar Runner)
In the House of the Undying, Daenerys has a vision telling her that the dragon has three heads. She needs three riders for her dragons. Before his death, Maester Aemon worries about who the three will be, realising that he is too old.
The books have introduced Aegon, a character (originally known as Young Griff), who claims to be the son of Rhaegar Targaryen and Elia Martell. Aegon was meant to have been killed as a baby by the Mountain, but perhaps he could have been smuggled out and swapped for another unfortunate infant?
I think Aegon is likely a fake, probably a pawn in the schemes of Varys and his old friend Illyrio Mopatis. We know they are definitely up to something. Daenerys is warned of a mummer’s dragon, which I think implies that this Aegon is a fake. Furthermore, since Aegon has not appeared in the show, I think it’s safe to say that his plot line is an non-essential complication.
If we give up on Aegon, who else is there? There is one other character we know who is fascinated by dragons, who has even dreamt of them (and did a good job of petting them in the show): Tyrion. Tywin often complained that he could not prove that Tyrion was not a Lannister. Yet, the two seem much alike in intellect and temperament. Tywin is indeed Tyrion’s father in that he is the man that raised him. However, it is possible that Tyrion’s biological father was the Mad King Aerys. Aerys was known to have a liking for Joanna Lannister, perhaps he took advantage while she was at court? This would make Tyrion Daenerys’s elder brother.
Family portrait? (credit: EW)
The existence of multiple Targaryens might be relevant to the prophesy of The Prince That Was Promised/Azor Ahai. A woods-witch, prophesied that The Prince would come from the line of Aerys and his sister Rhaella. Prophesies aren’t reliable, but I think all three could be in the running. There are ways to interpret all the various other prophesied properties to fit the various characters, but I wonder if the prophesy could be describing all of them?
The most interesting aspect to me is Azor Ahai’s sword Lightbringer, which sounds like exactly the weapon needed to defeat the Others. Azor Ahai killed his wife to make the sword. Daenery’s burnt Khal Drogo to hatch her dragons: they’d be a good weapon against the forces of winter. Jon sacrifices Ygritte to save the Night’s Watch: when joining the Watch, men vow to be “the sword in the darkness”. Tyrion, despite his performance at the Battle of the Blackwater (which included setting the sea of fire), is not much of a fighter. He drinks and knows things. His weapon is his intellect, so what could Lightbringer be in his case? Tyrion strangled Shae with the necklace of the Hand of the King: perhaps his counsel will be what wins the great war.
## 2 The Seven
The world of George R R Martin definitely contains magic and religion, but it is less clear if there are gods. Just as prophesies are unreliable, so is divine intervention. While miraculous events do happen, it is not clear if these are as a results of the gods, or just a coincidence. I think we are meant to be left unsure if any faith is correct.
While there are miracles that can be assigned to the Old Gods (the magic of the Children of the Forests and Bran’s abilities), R’hllor the Lord of Light (the resurrections of Jon and Beric Dondarrion, and the deaths of Joffrey, Balon Greyjoy, and Robb) and the Drowned God (the resuscitation of the drowned men and Victarion Greyjoy’s success), there is a conspicuous absence of intervention from the Seven, the gods of the majority religion of Westeros (although the church has intervened many times).
My theory is that characters representing the Seven will play an important role in the culmination of the story, such that looking back, it will appear that they guided events.
Do the Borg control the Iron Throne? (credit: Paramount)
The question is then who will serve the various roles?
The Father judges. Ser Davos often celebrates how just Stannis is. However, Stannis is not a great father. In fact, it is Ser Davos who is often talked Stannis into doing the right thing, and Ser Davos who cares more children (both Stannis’ and his own). While imprisoned, he writes letters to his sons, in an example of parenting not seen elsewhere. Therefore, Ser Davos is my leading choice as the Father.
The Mother embodies mercy. There are many mothers in the books (Cersei, Catelyn, etc.) but none too merciful. However, I think there is an obvious candidate: the Mother of Dragons, Mhysa of Yunkai. Daenarys is merciful and compassionate.
The Warrior offers protection and courage. There are many notable warriors like Brienne or the Hound who could fit, but I think the best fit is Jaime. Brienne strictly follows her vows and the Hound shuns all notions of knightly honour, neither quite representing the full range of roles that a warrior must serve. Jaime, however, had to pick between his vow of loyalty to the king and his loyalty to his father: Cersei’s younger brother accepted the dishonour of being a kingslayer rather than letting the people of King’s Landing burn. He also better knows the cost of battle, having lost his hand, ironically meaning that he has lost the ability to fight.
The Smith mends broken things. There aren’t too many makers in Westeros (most do the opposite). Gendry is the only real choice. Perhaps he will mend the House of Baratheon? Gendry is my only pick who has not been the point-of-view character for a chapter so far—if he does get a chapter, it could be an indication that he has some future importance.
The Maiden embodies young women, and perfectly matches Sansa.
The Crone gives wisdom and guidance, and is the most difficult to place as (i) few characters live to old age, and (ii) most make terrible life choices. In the books, Brienne prays for the Crone to lead her way as she searches for Sansa, and she is told the Crone will light her way by a Sparrow. Her path leads her to Lady Stoneheart, who is described as having hair as “white and brittle as a crone’s”. However, Stoneheart has not appeared in the show, so perhaps someone else will fill this role? (Maybe Melisandre, who is old enough, if she can get off Arya’s list—although she may need to convert her faith).
Finally, the Stranger is death. I think Arya is the Stranger. The Stranger is identified with the Many-Faced God, whose temple Arya apprenticed in and where she learnt to change her face. While the rest of the Seven are either male or female, the Stranger is neither: Arya is often mistaken for a boy or disguises herself as such. There is also the fact that people on her list keep on dying—the Hound managed to survive his seemingly fatal wounds after Arya removed him from the list, so perhaps she can withhold death too?
Arya’s fit to the Stranger gave me this idea. I think it is neat enough to work, but the difficulty in finding a fit for the Crone might undermine it. It’s perhaps not surprising that there are counterparts to the Seven given that they are meant to represent the different aspects of life.
Assuming that it is the case, there is one possible consequence. The Seven-Pointed Star tells of how the Seven crowned the first king of the Andals. Perhaps the seven characters will crown the new king of Westeros—it will not be one of them. (Daenerys has conquered much of Essos, so perhaps she’ll retire to rule that from a house with a red door, or perhaps she’ll be the queen?) Excluding the seven leaves many candidates, like Jon and Little Finger. The first king of the Andals was Hugor of the Hill. Tyrion travelled under the pseudonym of Hugor Hill; if he were the son of Aerys, he would have a claim…
## 3 The Faceless Man’s Mission
We first meet Jaqen H’ghar as a prisoner being taken to the Wall. Arya frees him, and after he repays his debt to her he disappears.
In the prologue of A Feast for Crows, we meet Pate, a novice at the Citadel. Pate hates sharing his name with Pate the Pig Boy. He meets an Alchemist, who shares the same description as how we last saw Jaqen. At the end of the prologue, Pate falls to the ground. We’re not sure what happens to him, but given that every other prologue point-of-view character ends up dead, I’d bet he’s been poisoned.
When Sam finally makes it to the Citadel, he meets Marwyn the Mage, a maester who studies magic. Amongst his students is a novice, who introduces himself as Pate “like the pig boy”. I think this change in character indicates that Jaqen has adopted Pate’s face.
So what is Jaqen up to?
I’ve always thought it odd that a Faceless Man, a great assassin who is a master of disguise, would get caught and imprisoned by the guards of King’s Landing. Perhaps he was in prison because he wanted to be, because it would allow him to establish an identity at the Wall? If Jaqen wanted to go to the Wall, but is now in the Citadel, whatever he was after must have moved too. Maester Aemon is dead of old-age, no-one would hire a Faceless Man for Gilly, and while Sam’s father hates him, I doubt he would go to the expense of a Faceless Man (he’d be happy enough to arrange an accident himself). So Jaqen may be after something Sam and Gilly brought with them. There are Maester Aemon’s books, there may be an important secret within them. There is also the ancient horn that Jon gave to Sam, the horn Ghost found buried with the dragonglass at the Fist of the First Men. The horn is mentioned a couple of times as one of Sam’s prized possessions, perhaps it has more significance?
I suspect that the horn could be the mythical Horn of Winter. It is ancient enough, and its burial with dragonglass (which can kill Others) suggests that it might have been used by the First Men for defence. The horn looks unimpressive, but that doesn’t mean it is insignificant.
Choose your ancient artefacts wisely (Credit: LucasFilm)
I’m pretty much certain that Jaqen is Pate, I’d give good odds that Sam’s horn has some importance; whether the two are related it a bit more of a stretch.
Of course, if this is the case, there’s still the question of who hired Jaqen and how they knew that the horn had been found?
# GW170104 and me
On 4 January 2017, Advanced LIGO made a new detection of gravitational waves. The signal, which we call GW170104 [bonus note], came from the coalescence of two black holes, which inspiralled together (making that characteristic chirp) and then merged to form a single black hole.
On 4 January 2017, I was just getting up off the sofa when my phone buzzed. My new year’s resolution was to go for a walk every day, and I wanted to make use of the little available sunlight. However, my phone informed me that PyCBC (one or our search algorithms for signals from coalescing binaries) had identified an interesting event. I sat back down. I was on the rota to analyse interesting signals to infer their properties, and I was pretty sure that people would be eager to see results. They were. I didn’t leave the sofa for the rest of the day, bringing my new year’s resolution to a premature end.
Since 4 January, my time has been dominated by working on GW170104 (you might have noticed a lack of blog posts). Below I’ll share some of my war stories from life on the front line of gravitational-wave astronomy, and then go through some of the science we’ve learnt. (Feel free to skip straight to the science, recounting the story was more therapy for me).
Time–frequency plots for GW170104 as measured by Hanford (top) and Livingston (bottom). The signal is clearly visible as the upward sweeping chirp. The loudest frequency is something between E3 and G♯3 on a piano, and it tail off somewhere between D♯4/E♭4 and F♯4/G♭4. Part of Fig. 1 of the GW170104 Discovery Paper.
### The story
In the second observing run, the Parameter Estimation group have divided up responsibility for analysing signals into two week shifts. For each rota shift, there is an expert and a rookie. I had assumed that the first slot of 2017 would be a quiet time. The detectors were offline over the holidays, due back online on 4 January, but the instrumentalists would probably find some extra tinkering they’d want to do, so it’d probably slip a day, and then the weather would be bad, so we’d probably not collect much data anyway… I was wrong. Very wrong. The detectors came back online on time, and there was a beautifully clean detection on day one.
My partner for the rota was Aaron Zimmerman. 4 January was his first day running parameter estimation on live signals. I think I would’ve run and hidden underneath my duvet in his case (I almost did anyway, and I lived through the madness of our first detection GW150914), but he rose to the occasion. We had first results after just a few hours, and managed to send out a preliminary sky localization to our astronomer partners on 6 January. I think this was especially impressive as there were some difficulties with the initial calibration of the data. This isn’t a problem for the detection pipelines, but does impact the parameters which we infer, particularly the sky location. The Calibration group worked quickly, and produced two updates to the calibration. We therefore had three different sets of results (one per calibration) by 6 January [bonus note]!
Producing the final results for the paper was slightly more relaxed. Aaron and I conscripted volunteers to help run all the various permutations of the analysis we wanted to double-check our results [bonus note].
Recovered gravitational waveforms from analysis of GW170104. The broader orange band shows our estimate for the waveform without assuming a particular source (wavelet). The narrow blue bands show results if we assume it is a binary black hole (BBH) as predicted by general relativity. The two match nicely, showing no evidence for any extra features not included in the binary black hole models. Figure 4 of the GW170104 Discovery Paper.
I started working on GW170104 through my parameter estimation duties, and continued with paper writing.
Ahead of the second observing run, we decided to assemble a team to rapidly write up any interesting binary detections, and I was recruited for this (I think partially because I’m not too bad at writing and partially because I was in the office next to John Veitch, one of the chairs of the Compact Binary Coalescence group,so he can come and check that I wasn’t just goofing off eating doughnuts). We soon decided that we should write a paper about GW170104, and you can decide whether or not we succeeded in doing this rapidly…
Being on the paper writing team has given me huge respect for the teams who led the GW150914 and GW151226 papers. It is undoubtedly one of the most difficult things I’ve ever done. It is extremely hard to absorb negative remarks about your work continuously for months [bonus note]—of course people don’t normally send comments about things that they like, but that doesn’t cheer you up when you’re staring at an inbox full of problems that need fixing. Getting a collaboration of 1000 people to agree on a paper is like herding cats while being a small duckling.
On of the first challenges for the paper writing team was deciding what was interesting about GW170104. It was another binary black hole coalescence—aren’t people getting bored of them by now? The signal was quieter than GW150914, so it wasn’t as remarkable. However, its properties were broadly similar. It was suggested that perhaps we should title the paper “GW170104: The most boring gravitational-wave detection”.
One potentially interesting aspect was that GW170104 probably comes from greater distance than GW150914 or GW151226 (but perhaps not LVT151012) [bonus note]. This might make it a good candidate for testing for dispersion of gravitational waves.
Dispersion occurs when different frequencies of gravitational waves travel at different speeds. A similar thing happens for light when travelling through some materials, which leads to prisms splitting light into a spectrum (and hence the creation of Pink Floyd album covers). Gravitational waves don’t suffered dispersion in general relativity, but do in some modified theories of gravity.
It should be easier to spot dispersion in signals which have travelled a greater distance, as the different frequencies have had more time to separate out. Hence, GW170104 looks pretty exciting. However, being further away also makes the signal quieter, and so there is more uncertainty in measurements and it is more difficult to tell if there is any dispersion. Dispersion is also easier to spot if you have a larger spread of frequencies, as then there can be more spreading between the highest and lowest frequencies. When you throw distance, loudness and frequency range into the mix, GW170104 doesn’t always come out on top, depending upon the particular model for dispersion: sometimes GW150914’s loudness wins, other times GW151226’s broader frequency range wins. GW170104 isn’t too special here either.
Even though GW170104 didn’t look too exciting, we started work on a paper, thinking that we would just have a short letter describing our observations. The Compact Binary Coalescence group decided that we only wanted a single paper, and we wouldn’t bother with companion papers as we did for GW150914. As we started work, and dug further into our results, we realised that actually there was rather a lot that we could say.
I guess the moral of the story is that even though you might be overshadowed by the achievements of your siblings, it doesn’t mean that you’re not awesome. There might not be one outstanding feature of GW170104, but there are lots of little things that make it interesting. We are still at the beginning of understanding the properties of binary black holes, and each new detection adds a little more to our picture.
I think GW170104 is rather neat, and I hope you do too.
As we delved into the details of our results, we realised there was actually a lot of things that we could say about GW170104, especially when considered with our previous observations. We ended up having to move some of the technical details and results to Supplemental Material. With hindsight, perhaps it would have been better to have a companion paper or two. However, I rather like how packed with science this paper is.
The paper, which Physical Review Letters have kindly accommodated, despite its length, might not be as polished a classic as the GW150914 Discovery Paper, but I think they are trying to do different things. I rarely ever refer to the GW150914 Discovery Paper for results (more commonly I use it for references), whereas I think I’ll open up the GW170104 Discovery Paper frequently to look up numbers.
Although perhaps not right away, I’d quite like some time off first. The weather’s much better now, perfect for walking…
Success! The view across Lac d’Annecy. Taken on a stroll after the Gravitational Wave Physics and Astronomy Workshop, the weekend following the publication of the paper.
### The science
Advanced LIGO’s first observing run was hugely successful. Running from 12 September 2015 until 19 January 2016, there were two clear gravitational-wave detections, GW1501914 and GW151226, as well as a less certain candidate signal LVT151012. All three (assuming that they are astrophysical signals) correspond to the coalescence of binary black holes.
The second observing run started 30 November 2016. Following the first observing run’s detections, we expected more binary black hole detections. On 4 January, after we had collected almost 6 days’ worth of coincident data from the two LIGO instruments [bonus note], there was a detection.
#### The searches
The signal was first spotted by an online analysis. Our offline analysis of the data (using refined calibration and extra information about data quality) showed that the signal, GW170104, is highly significant. For both GstLAL and PyCBC, search algorithms which use templates to search for binary signals, the false alarm rate is estimated to be about 1 per 70,000 years.
The signal is also found in unmodelled (burst) searches, which look for generic, short gravitational wave signals. Since these are looking for more general signals than just binary coalescences, the significance associated with GW170104 isn’t as great, and coherent WaveBurst estimates a false alarm rate of 1 per 20,000 years. This is still pretty good! Reconstructions of the waveform from unmodelled analyses also match the form expected for binary black hole signals.
The search false alarm rates are the rate at which you’d expect something this signal-like (or more signal-like) due to random chance, if you data only contained noise and no signals. Using our knowledge of the search pipelines, and folding in some assumptions about the properties of binary black holes, we can calculate a probability that GW170104 is a real astrophysical signal. This comes out to be greater than $1 - (3\times10^5) = 0.99997$.
#### The source
As for the previous gravitational wave detections, GW170104 comes from a binary black hole coalescence. The initial black holes were $31.2^{+8.4}_{-6.0} M_\odot$ and $19.4^{+5.3}_{-5.9} M_\odot$ (where $1 M_\odot$ is the mass of our Sun), and the final black hole was $48.7^{+5.7}_{-4.6} M_\odot$. The quoted values are the median values and the error bars denote the central 90% probable range. The plot below shows the probability distribution for the masses; GW170104 neatly nestles in amongst the other events.
Estimated masses for the two black holes in the binary $m_1 \geq m_2$. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours for all events. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 2 of the GW170104 Discovery Paper.
GW150914 was the first time that we had observed stellar-mass black holes with masses greater than around $25 M_\odot$. GW170104 has similar masses, showing that our first detection was not a fluke, but there really is a population of black holes with masses stretching up into this range.
Black holes have two important properties: mass and spin. We have good measurements on the masses of the two initial black holes, but not the spins. The sensitivity of the form of the gravitational wave to spins can be described by two effective spin parameters, which are mass-weighted combinations of the individual spins.
• The effective inspiral spin parameter $\chi_\mathrm{eff}$ qualifies the impact of the spins on the rate of inspiral, and where the binary plunges together to merge. It ranges from +1, meaning both black holes are spinning as fast as possible and rotate in the same direction as the orbital motion, to −1, both black holes spinning as fast as possible but in the opposite direction to the way that the binary is orbiting. A value of 0 for $\chi_\mathrm{eff}$ could mean that the black holes are not spinning, that their rotation axes are in the orbital plane (instead of aligned with the orbital angular momentum), or that one black hole is aligned with the orbital motion and the other is antialigned, so that their effects cancel out.
• The effective precession spin parameter $\chi_\mathrm{p}$ qualifies the amount of precession, the way that the orbital plane and black hole spins wobble when they are not aligned. It is 0 for no precession, and 1 for maximal precession.
We can place some constraints on $\chi_\mathrm{eff}$, but can say nothing about $\chi_\mathrm{p}$. The inferred value of the effective inspiral spin parameter is $-0.12^{+0.21}_{-0.30}$. Therefore, we disfavour large spins aligned with the orbital angular momentum, but are consistent with small aligned spins, misaligned spins, or spins antialigned with the angular momentum. The value is similar to that for GW150914, which also had a near-zero, but slightly negative $\chi_\mathrm{eff}$ of $-0.06^{+0.14}_{-0.14}$.
Estimated effective inspiral spin parameter $\chi_\mathrm{eff}$ and effective precession spin $\chi_\mathrm{p}$ parameter. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models, as well as the prior probability distribution. The dotted lines mark the edge of our 90% probability intervals. We learn basically nothing about precession. Part of Figure 3 of the GW170104 Discovery Paper.
Converting the information about $\chi_\mathrm{eff}$, the lack of information about $\chi_\mathrm{p}$, and our measurement of the ratio of the two black hole masses, into probability distributions for the component spins gives the plots below [bonus note]. We disfavour (but don’t exclude) spins aligned with the orbital angular momentum, but can’t say much else.
Estimated orientation and magnitude of the two component spins. The distribution for the more massive black hole is on the left, and for the smaller black hole on the right. The probability is binned into areas which have uniform prior probabilities, so if we had learnt nothing, the plot would be uniform. Part of Figure 3 of the GW170104 Discovery Paper.
One of the comments we had on a draft of the paper was that we weren’t making any definite statements about the spins—we would have if we could, but we can’t for GW170104, at least for the spins of the two inspiralling black holes. We can be more definite about the spin of the final black hole. If two similar mass black holes spiral together, the angular momentum from the orbit is enough to give a spin of around $0.7$. The spins of the component black holes are less significant, and can make it a bit higher of lower. We infer a final spin of $0.64^{+0.09}_{-0.20}$; there is a tail of lower spin values on account of the possibility that the two component black holes could be roughly antialigned with the orbital angular momentum.
Estimated mass $M_\mathrm{f}$ and spin$a_\mathrm{f}$ for the final black hole. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 6 of the GW170104 Supplemental Material (Figure 11 of the arXiv version).
If you’re interested in parameter describing GW170104, make sure to check out the big table in the Supplemental Material. I am a fan of tables [bonus note].
#### Merger rates
Adding the first 11 days of coincident data from the second observing run (including the detection of GW170104) to the results from the first observing run, we find merger rates consistent with those from the first observing run.
To calculate the merger rates, we need to assume a distribution of black hole masses, and we use two simple models. One uses a power law distribution for the primary (larger) black hole and a uniform distribution for the mass ratio; the other uses a distribution uniform in the logarithm of the masses (both primary and secondary). The true distribution should lie somewhere between the two. The power law rate density has been updated from $31^{+42}_{-21}\mathrm{Gpc^{-3}\,yr^{-1}}$ to $32^{+33}_{-20}\mathrm{Gpc^{-3}\,yr^{-1}}$, and the uniform in log rate density goes from $97^{+135}_{-67}~\mathrm{Gpc^{-3}\,yr^{-1}}$ to $103^{+110}_{-63}~\mathrm{Gpc^{-3}\,yr^{-1}}$. The median values stay about the same, but the additional data have shrunk the uncertainties a little.
#### Astrophysics
The discoveries from the first observing run showed that binary black holes exist and merge. The question is now how exactly they form? There are several suggested channels, and it could be there is actually a mixture of different formation mechanisms in action. It will probably require a large number of detections before we can make confident statements about the the probable formation mechanisms; GW170104 is another step towards that goal.
There are two main predicted channels of binary formation:
• Isolated binary evolution, where a binary star system lives its life together with both stars collapsing to black holes at the end. To get the black holes close enough to merge, it is usually assumed that the stars go through a common envelope phase, where one star puffs up so that the gravity of its companion can steal enough material that they lie in a shared envelope. The drag from orbiting inside this then shrinks the orbit.
• Dynamical evolution where black holes form in dense clusters and a binary is created by dynamical interactions between black holes (or stars) which get close enough to each other.
It’s a little artificial to separate the two, as there’s not really such a thing as an isolated binary: most stars form in clusters, even if they’re not particularly large. There are a variety of different modifications to the two main channels, such as having a third companion which drives the inner binary to merge, embedding the binary is a dense disc (as found in galactic centres), or dynamically assembling primordial black holes (formed by density perturbations in the early universe) instead of black holes formed through stellar collapse.
All the channels can predict black holes around the masses of GW170104 (which is not surprising given that they are similar to the masses of GW150914).
The updated rates are broadly consistent with most channels too. The tightening of the uncertainty of the rates means that the lower bound is now a little higher. This means some of the channels are now in tension with the inferred rates. Some of the more exotic channels—requiring a third companion (Silsbee & Tremain 2017; Antonini, Toonen & Hamers 2017) or embedded in a dense disc (Bartos et al. 2016; Stone, Metzger & Haiman 2016; Antonini & Rasio 2016)—can’t explain the full rate, but I don’t think it was ever expected that they could, they are bonus formation mechanisms. However, some of the dynamical models are also now looking like they could predict a rate that is a bit low (Rodriguez et al. 2016; Mapelli 2016; Askar et al. 2017; Park et al. 2017). Assuming that this result holds, I think this may mean that some of the model parameters need tweaking (there are more optimistic predictions for the merger rates from clusters which are still perfectly consistent), that this channel doesn’t contribute all the merging binaries, or both.
The spins might help us understand formation mechanisms. Traditionally, it has been assumed that isolated binary evolution gives spins aligned with the orbital angular momentum. The progenitor stars were probably more or less aligned with the orbital angular momentum, and tides, mass transfer and drag from the common envelope would serve to realign spins if they became misaligned. Rodriguez et al. (2016) gives a great discussion of this. Dynamically formed binaries have no correlation between spin directions, and so we would expect an isotropic distribution of spins. Hence it sounds quite simple: misaligned spins indicates dynamical formation (although we can’t tell if the black holes are primordial or stellar), and aligned spins indicates isolated binary evolution. The difficulty is the traditional assumption for isolated binary evolution potentially ignores a number of effects which could be important. When a star collapses down to a black hole, there may be a supernova explosion. There is an explosion of matter and neutrinos and these can give the black hole a kick. The kick could change the orbital plane, and so misalign the spin. Even if the kick is not that big, if it is off-centre, it could torque the black hole, causing it to rotate and so misalign the spin that way. There is some evidence that this can happen with neutron stars, as one of the pulsars in the double pulsar system shows signs of this. There could also be some instability that changes the angular momentum during the collapse of the star, possibly with different layers rotating in different ways [bonus note]. The spin of the black hole would then depend on how many layers get swallowed. This is an area of research that needs to be investigated further, and I hope the prospect of gravitational wave measurements spurs this on.
For GW170104, we know the spins are not large and aligned with the orbital angular momentum. This might argue against one variation of isolated binary evolution, chemically homogeneous evolution, where the progenitor stars are tidally locked (and so rotate aligned with the orbital angular momentum and each other). Since the stars are rapidly spinning and aligned, you would expect the final black holes to be too, if the stars completely collapse down as is usually assumed. If the stars don’t completely collapse down though, it might still be possible that GW170104 fits with this model. Aside from this, GW170104 is consistent with all the other channels.
Estimated effective inspiral spin parameter $\chi_\mathrm{eff}$ for all events. To indicate how much (or little) we’ve learnt, the prior probability distribution for GW170104 is shown (the other priors are similar).All of the events have $|\chi_\mathrm{eff}| < 0.35$ at 90% probability. Figure 5 of the GW170104 Supplemental Material (Figure 10 of the arXiv version). This is one of my favourite plots [bonus note].
If we start looking at the population of events, we do start to notice something about the spins. All of the inferred values of $\chi_\mathrm{eff}$ are close to zero. Only GW151226 is inconsistent with zero. These values could be explained if spins are typically misaligned (with the orbital angular momentum or each other) or if the spins are typically small (or both). We know that black holes spins can be large from observations of X-ray binaries, so it would be odd if they are small for binary black holes. Therefore, we have a tentative hint that spins are misaligned. We can’t say why the spins are misaligned, but it is intriguing. With more observations, we’ll be able to confirm if it is the case that spins are typically misaligned, and be able to start pinning down the distribution of spin magnitudes and orientations (as well as the mass distribution). It will probably take a while to be able to say anything definite though, as we’ll probably need about 100 detections.
#### Tests of general relativity
As well as giving us an insight into the properties of black holes, gravitational waves are the perfect tools for testing general relativity. If there are any corrections to general relativity, you’d expect them to be most noticeable under the most extreme conditions, where gravity is strong and spacetime is rapidly changing, exactly as in a binary black hole coalescence.
For GW170104 we repeated tests previously performed. Again, we found no evidence of deviations.
We added extra terms to to the waveform and constrained their potential magnitudes. The results are pretty much identical to at the end of the first observing run (consistent with zero and hence general relativity). GW170104 doesn’t add much extra information, as GW150914 typically gives the best constraints on terms that modify the post-inspiral part of the waveform (as it is louder), while GW151226 gives the best constraint on the terms which modify the inspiral (as it has the longest inspiral).
We also chopped the waveform at a frequency around that of the innermost stable orbit of the remnant black hole, which is about where the transition from inspiral to merger and ringdown occurs, to check if the low frequency and high frequency portions of the waveform give consistent estimates for the final mass and spin. They do.
We have also done something slightly new, and tested for dispersion of gravitational waves. We did something similar for GW150914 by putting a limit on the mass of the graviton. Giving the graviton mass is one way of adding dispersion, but we consider other possible forms too. In all cases, results are consistent with there being no dispersion. While we haven’t discovered anything new, we can update our gravitational wave constraint on the graviton mass of less than $7.7 \times 10^{-23}~\mathrm{eV}/c^2$.
#### The search for counterparts
We don’t discuss observations made by our astronomer partners in the paper (they are not our results). A number (28 at the time of submission) of observations were made, and I expect that there will be a series of papers detailing these coming soon. So far papers have appeared from:
• AGILE—hard X-ray and gamma-ray follow-up. They didn’t find any gamma-ray signals, but did identify a weak potential X-ray signal occurring about 0.46 s before GW170104. It’s a little odd to have a signal this long before the merger. The team calculate a probability for such a coincident to happen by chance, and find quite a small probability, so it might be interesting to follow this up more (see the INTEGRAL results below), but it’s probably just a coincidence (especially considering how many people did follow-up the event).
• ANTARES—a search for high-energy muon neutrinos. No counterparts are identified in a ±500 s window around GW170104, or over a ±3 month period.
• AstroSat-CZTI and GROWTH—a collaboration of observations across a range of wavelengths. They don’t find any hard X-ray counterparts. They do follow up on a bright optical transient ATLASaeu, suggested as a counterpart to GW170104, and conclude that this is a likely counterpart of long, soft gamma-ray burst GRB 170105A.
• ATLAS and Pan-STARRS—optical follow-up. They identified a bright optical transient 23 hours after GW170104, ATLAS17aeu. This could be a counterpart to GRB 170105A. It seems unlikely that there is any mechanism that could allow for a day’s delay between the gravitational wave emission and an electromagnetic signal. However, the team calculate a small probability (few percent) of finding such a coincidence in sky position and time, so perhaps it is worth pondering. I wouldn’t put any money on it without a distance estimate for the source: assuming it’s a normal afterglow to a gamma-ray burst, you’d expect it to be further away than GW170104’s source.
• Borexino—a search for low-energy neutrinos. This paper also discusses GW150914 and GW151226. In all cases, the observed rate of neutrinos is consistent with the expected background.
• Fermi (GBM and LAT)—gamma-ray follow-up. They covered an impressive fraction of the sky localization, but didn’t find anything.
• INTEGRAL—gamma-ray and hard X-ray observations. No significant emission is found, which makes the event reported by AGILE unlikely to be a counterpart to GW170104, although they cannot completely rule it out.
• The intermediate Palomar Transient Factory—an optical survey. While searching, they discovered iPTF17cw, a broad-line type Ic supernova which is unrelated to GW170104 but interesting as it an unusual find.
If you are interested in what has been reported so far (no compelling counterpart candidates yet to my knowledge), there is an archive of GCN Circulars sent about GW170104.
### Summary
Advanced LIGO has made its first detection of the second observing run. This is a further binary black hole coalescence. GW170104 has taught us that:
• The discoveries of the first observing run were not a fluke. There really is a population of stellar mass black holes with masses above $25 M_\odot$ out there, and we can study them with gravitational waves.
• Binary black hole spins may be typically misaligned or small. This is not certain yet, but it is certainly worth investigating potential mechanisms that could cause misalignment.
• General relativity still works, even after considering our new tests.
• If someone asks you to write a discovery paper, run. Run and do not look back.
Title: GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2
Journal:
Physical Review Letters; 118(22):221101(17); 2017 (Supplemental Material)
arXiv: 1706.01812 [gr-qc]
Data release: LIGO Open Science Center
Science summary:
GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2
### Bonus notes
#### Naming
Gravitational wave signals (at least the short ones, which are all that we have so far), are named by their detection date. GW170104 was discovered 2017 January 4. This isn’t too catchy, but is at least better than the ID number in our database of triggers (G268556) which is used in corresponding with our astronomer partners before we work out if the “GW” title is justified.
Previous detections have attracted nicknames, but none has stuck for GW170104. Archisman Ghosh suggested the Perihelion Event, as it was detected a few hours before the Earth reached its annual point closest to the Sun. I like this name, its rather poetic.
More recently, Alex Nitz realised that we should have called GW170104 the Enterprise-D Event, as the USS Enterprise’s registry number was NCC-1701. For those who like Star Trek: the Next Generation, I hope you have fun discussing whether GW170104 is the third or fourth (counting LVT151012) detection: “There are four detections!
#### The 6 January sky map
I would like to thank the wi-fi of Chiltern Railways for their role in producing the preliminary sky map. I had arranged to visit London for the weekend (because my rota slot was likely to be quiet… ), and was frantically working on the way down to check results so they could be sent out. I’d also like to thank John Veitch for putting together the final map while I was stuck on the Underground.
#### Binary black hole waveforms
The parameter estimation analysis works by matching a template waveform to the data to see how well it matches. The results are therefore sensitive to your waveform model, and whether they include all the relevant bits of physics.
In the first observing run, we always used two different families of waveforms, to see what impact potential errors in the waveforms could have. The results we presented in discovery papers used two quick-to-calculate waveforms. These include the effects of the black holes’ spins in different ways
• SEOBNRv2 has spins either aligned or antialigned with the orbital angular momentum. Therefore, there is no precession (wobbling of orientation, like that of a spinning top) of the system.
• IMRPhenomPv2 includes an approximate description of precession, packaging up the most important information about precession into a single parameter $\chi_\mathrm{p}$.
For GW150914, we also performed a follow-up analysis using a much more expensive waveform SEOBNRv3 which more fully includes the effect of both spins on precession. These results weren’t ready at the time of the announcement, because the waveform is laborious to run.
For GW170104, there were discussions that using a spin-aligned waveform was old hat, and that we should really only use the two precessing models. Hence, we started on the endeavour of producing SEOBNRv3 results. Fortunately, the code has been sped up a little, although it is still not quick to run. I am extremely grateful to Scott Coughlin (one of the folks behind Gravity Spy), Andrea Taracchini and Stas Babak for taking charge of producing results in time for the paper, in what was a Herculean effort.
I spent a few sleepless nights, trying to calculate if the analysis was converging quickly enough to make our target submission deadline, but it did work out in the end. Still, don’t necessarily expect we’ll do this for a all future detections.
Since the waveforms have rather scary technical names, in the paper we refer to IMRPhenomPv2 as the effective precession model and SEOBNRv3 as the full precession model.
#### On distance
Distance measurements for gravitational wave sources have significant uncertainties. The distance is difficult to measure as it determined from the signal amplitude, but this is also influences by the binary’s inclination. A signal could either be close and edge on or far and face on-face off.
Estimated luminosity distance $D_\mathrm{L}$ and binary inclination angle $\theta_{JN}$. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 4 of the GW170104 Supplemental Material (Figure 9 of the arXiv version).
The uncertainty on the distance rather awkwardly means that we can’t definitely say that GW170104 came from a further source than GW150914 or GW151226, but it’s a reasonable bet. The 90% credible intervals on the distances are 250–570 Mpc for GW150194, 250–660 Mpc for GW151226, 490–1330 Mpc for GW170104 and 500–1500 Mpc for LVT151012.
Translating from a luminosity distance to a travel time (gravitational waves do travel at the speed of light, our tests of dispersion are consistent wit that!), the GW170104 black holes merged somewhere between 1.3 and 3.0 billion years ago. This is around the time that multicellular life first evolved on Earth, and means that black holes have been colliding longer than life on Earth has been reproducing sexually.
#### Time line
A first draft of the paper (version 2; version 1 was a copy-and-paste of the Boxing Day Discovery Paper) was circulated to the Compact Binary Coalescence and Burst groups for comments on 4 March. This was still a rough version, and we wanted to check that we had a good outline of the paper. The main feedback was that we should include more about the astrophysical side of things. I think the final paper has a better balance, possibly erring on the side of going into too much detail on some of the more subtle points (but I think that’s better than glossing over them).
A first proper draft (version 3) was released to the entire Collaboration on 12 March in the middle of our Collaboration meeting in Pasadena. We gave an oral presentation the next day (I doubt many people had read the paper by then). Collaboration papers are usually allowed two weeks for people to comment, and we followed the same procedure here. That was not a fun time, as there was a constant trickle of comments. I remember waking up each morning and trying to guess how many emails would be in my inbox–I normally low-balled this.
I wasn’t too happy with version 3, it was still rather rough. The members of the Paper Writing Team had been furiously working on our individual tasks, but hadn’t had time to look at the whole. I was much happier with the next draft (version 4). It took some work to get this together, following up on all the comments and trying to address concerns was a challenge. It was especially difficult as we got a series of private comments, and trying to find a consensus probably made us look like the bad guys on all sides. We released version 4 on 14 April for a week of comments.
The next step was approval by the LIGO and Virgo executive bodies on 24 April. We prepared version 5 for this. By this point, I had lost track of which sentences I had written, which I had merely typed, and which were from other people completely. There were a few minor changes, mostly adding technical caveats to keep everyone happy (although they do rather complicate the flow of the text).
The paper was circulated to the Collaboration for a final week of comments on 26 April. Most comments now were about typos and presentation. However, some people will continue to make the same comment every time, regardless of how many times you explain why you are doing something different. The end was in sight!
The paper was submitted to Physical Review Letters on 9 May. I was hoping that the referees would take a while, but the reports were waiting in my inbox on Monday morning.
The referee reports weren’t too bad. Referee A had some general comments, Referee B had some good and detailed comments on the astrophysics, and Referee C gave the paper a thorough reading and had some good suggestions for clarifying the text. By this point, I have been staring at the paper so long that some outside perspective was welcome. I was hoping that we’d have a more thorough review of the testing general relativity results, but we had Bob Wald as one of our Collaboration Paper reviewers (the analysis, results and paper are all reviewed internally), so I think we had already been held to a high standard, and there wasn’t much left to say.
We put together responses to the reports. There were surprisingly few comments from the Collaboration at this point. I guess that everyone was getting tired. The paper was resubmitted and accepted on 20 May.
One of the suggestions of Referee A was to include some plots showing the results of the searches. People weren’t too keen on showing these initially, but after much badgering they were convinced, and it was decided to put these plots in the Supplemental Material which wouldn’t delay the paper as long as we got the material submitted by 26 May. This seemed like plenty of time, but it turned out to be rather frantic at the end (although not due to the new plots). The video below is an accurate representation of us trying to submit the final version.
I have an email which contains the line “Many Bothans died to bring us this information” from 1 hour and 18 minutes before the final deadline.
After this, things were looking pretty good. We had returned the proofs of the main paper (I had a fun evening double checking the author list. Yes, all of them). We were now on version 11 of the paper.
Of course, there’s always one last thing. On 31 May, the evening before publication, Salvo Vitale spotted a typo. Nothing serious, but annoying. The team at Physical Review Letters were fantastic, and took care of it immediately!
There’ll still be one more typo, there always is…
Looking back, it is clear that the principal bottle-neck in publishing the results is getting the Collaboration to converge on the paper. I’m not sure how we can overcome this… Actually, I have some ideas, but none that wouldn’t involve some form of doomsday device.
#### Detector status
The sensitivities of the LIGO Hanford and Livinston detectors are around the same as they were in the first observing run. After the success of the first observing run, the second observing run is the difficult follow up album. Livingston has got a little better, while Hanford is a little worse. This is because the Livingston team concentrate on improving low frequency sensitivity whereas the Hanford team focused on improving high frequency sensitivity. The Hanford team increased the laser power, but this introduces some new complications. The instruments are extremely complicated machines, and improving sensitivity is hard work.
The current plan is to have a long commissioning break after the end of this run. The low frequency tweaks from Livingston will be transferred to Hanford, and both sites will work on bringing down other sources of noise.
While the sensitivity hasn’t improved as much as we might have hoped, the calibration of the detectors has! In the first observing run, the calibration uncertainty for the first set of published results was about 10% in amplitude and 10 degrees in phase. Now, uncertainty is better than 5% in amplitude and 3 degrees in phase, and people are discussing getting this down further.
#### Spin evolution
As the binary inspirals, the orientation of the spins will evolve as they precess about. We always quote measurements of the spins at a point in the inspiral corresponding to a gravitational wave frequency of 20 Hz. This is most convenient for our analysis, but you can calculate the spins at other points. However, the resulting probability distributions are pretty similar at other frequencies. This is because the probability distributions are primarily determined by the combination of three things: (i) our prior assumption of a uniform distribution of spin orientations, (ii) our measurement of the effective inspiral spin, and (iii) our measurement of the mass ratio. A uniform distribution stays uniform as spins evolve, so this is unaffected, the effective inspiral spin is approximately conserved during inspiral, so this doesn’t change much, and the mass ratio is constant. The overall picture is therefore qualitatively similar at different moments during the inspiral.
#### Footnotes
I love footnotes. It was challenging for me to resist having any in the paper.
#### Gravity waves
It is possible that internal gravity waves (that is oscillations of the material making up the star, where the restoring force is gravity, not gravitational waves, which are ripples in spacetime), can transport angular momentum from the core of a star to its outer envelope, meaning that the two could rotate in different directions (Rogers, Lin & Lau 2012). I don’t think anyone has studied this yet for the progenitors of binary black holes, but it would be really cool if gravity waves set the properties of gravitational wave sources.
I really don’t want to proof read the paper which explains this though.
#### Colour scheme
For our plots, we use a consistent colour coding for our events. GW150914 is blue; LVT151012 is green; GW151226 is red–orange, and GW170104 is purple. The colour scheme is designed to be colour blind friendly (although adopting different line styles would perhaps be more distinguishable), and is implemented in Python in the Seaborn package as colorblind. Katerina Chatziioannou, who made most of the plots showing parameter estimation results is not a fan of the colour combinations, but put a lot of patient effort into polishing up the plots anyway. | 2018-05-24 13:49:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 307, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7998290657997131, "perplexity": 1131.1567485734736}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866326.60/warc/CC-MAIN-20180524131721-20180524151721-00440.warc.gz"} |
https://www.jobilize.com/trigonometry/test/computing-probability-using-counting-theory-by-openstax | # 13.7 Probability (Page 4/18)
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Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.
$\text{\hspace{0.17em}}\frac{5}{6}\text{\hspace{0.17em}}$
## Computing probability using counting theory
Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are $\text{\hspace{0.17em}}C\left(5,2\right)\text{\hspace{0.17em}}$ ways to select 2 phones that are not defective. There are 8 phones, so there are $\text{\hspace{0.17em}}C\left(8,2\right)\text{\hspace{0.17em}}$ ways to select 2 phones. The probability of selecting 2 phones that are not defective is:
## Computing probability using counting theory
A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears.
1. Find the probability that only bears are chosen.
2. Find the probability that 2 bears and 3 dogs are chosen.
3. Find the probability that at least 2 dogs are chosen.
1. We need to count the number of ways to choose only bears and the total number of possible ways to select 5 toys. There are 6 bears, so there are $\text{\hspace{0.17em}}C\left(6,5\right)\text{\hspace{0.17em}}$ ways to choose 5 bears. There are 14 toys, so there are $\text{\hspace{0.17em}}C\left(14,5\right)\text{\hspace{0.17em}}$ ways to choose any 5 toys.
$\text{\hspace{0.17em}}\frac{C\left(6\text{,}5\right)}{C\left(14\text{,}5\right)}=\frac{6}{2\text{,}002}=\frac{3}{1\text{,}001}\text{\hspace{0.17em}}$
2. We need to count the number of ways to choose 2 bears and 3 dogs and the total number of possible ways to select 5 toys. There are 6 bears, so there are $\text{\hspace{0.17em}}C\left(6,2\right)\text{\hspace{0.17em}}$ ways to choose 2 bears. There are 5 dogs, so there are $\text{\hspace{0.17em}}C\left(5,3\right)\text{\hspace{0.17em}}$ ways to choose 3 dogs. Since we are choosing both bears and dogs at the same time, we will use the Multiplication Principle. There are $\text{\hspace{0.17em}}C\left(6,2\right)\cdot C\left(5,3\right)\text{\hspace{0.17em}}$ ways to choose 2 bears and 3 dogs. We can use this result to find the probability.
$\text{\hspace{0.17em}}\frac{C\left(6\text{,}2\right)C\left(5\text{,}3\right)}{C\left(14\text{,}5\right)}=\frac{15\cdot 10}{2\text{,}002}=\frac{75}{1\text{,}001}\text{\hspace{0.17em}}$
3. It is often easiest to solve “at least” problems using the Complement Rule. We will begin by finding the probability that fewer than 2 dogs are chosen. If less than 2 dogs are chosen, then either no dogs could be chosen, or 1 dog could be chosen.
When no dogs are chosen, all 5 toys come from the 9 toys that are not dogs. There are $\text{\hspace{0.17em}}C\left(9,5\right)\text{\hspace{0.17em}}$ ways to choose toys from the 9 toys that are not dogs. Since there are 14 toys, there are $\text{\hspace{0.17em}}C\left(14,5\right)\text{\hspace{0.17em}}$ ways to choose the 5 toys from all of the toys.
$\text{\hspace{0.17em}}\frac{C\left(9\text{,}5\right)}{C\left(14\text{,}5\right)}=\frac{63}{1\text{,}001}\text{\hspace{0.17em}}$
If there is 1 dog chosen, then 4 toys must come from the 9 toys that are not dogs, and 1 must come from the 5 dogs. Since we are choosing both dogs and other toys at the same time, we will use the Multiplication Principle. There are $\text{\hspace{0.17em}}C\left(5,1\right)\cdot C\left(9,4\right)\text{\hspace{0.17em}}$ ways to choose 1 dog and 1 other toy.
$\text{\hspace{0.17em}}\frac{C\left(5\text{,}1\right)C\left(9\text{,}4\right)}{C\left(14\text{,}5\right)}=\frac{5\cdot 126}{2\text{,}002}=\frac{315}{1\text{,}001}\text{\hspace{0.17em}}$
Because these events would not occur together and are therefore mutually exclusive, we add the probabilities to find the probability that fewer than 2 dogs are chosen.
$\text{\hspace{0.17em}}\frac{63}{1\text{,}001}+\frac{315}{1\text{,}001}=\frac{378}{1\text{,}001}\text{\hspace{0.17em}}$
We then subtract that probability from 1 to find the probability that at least 2 dogs are chosen.
$\text{\hspace{0.17em}}1-\frac{378}{1\text{,}001}=\frac{623}{1\text{,}001}\text{\hspace{0.17em}}$
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
the indicated sum of a sequence is known as
how do I attempted a trig number as a starter | 2020-05-27 07:28:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6100218296051025, "perplexity": 498.552448479868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347392141.7/warc/CC-MAIN-20200527044512-20200527074512-00514.warc.gz"} |
https://www.physicsforums.com/threads/energy-vs-wavelength-for-a-photon-in-gr.430346/ | # Energy vs wavelength for a photon in GR
1. Sep 19, 2010
### Pierre007080
If a theoretical single photon followed a geodesic toward a large mass in space, I understand that the wavelength would shorten as it approached the mass. How would the energy be conserved within the photon, because the frequency must surely remain the same?
2. Sep 19, 2010
### Mentz114
If the wavelength changes, then the frequency will change if the velocity is constant, because $v=f\lambda$.
Energy is not globally conserved in the general theory of relativity. If a chain of observers along the geodedic studied these photons, they would find the speed is the same c, but the frequency and wavelength are different.
3. Sep 19, 2010
### Jonathan Scott
From the point of view of any single observer, the energy would remain constant but the coordinate speed of light would decrease, so the magnitude of the momentum, E/c, would increase as the photon moved towards a lower potential.
From the point of view of separate observers at different potentials, each of them is time-dilated according to their potentials, so they see different energy values.
4. Sep 19, 2010
### Pierre007080
Hi Mentz 114,
Thanks for your response. I think I understand your answer about the chain of observers along the geodedic ... but is it allowed for the observer to be in a nearby spaceship observing (from a distance) the shortening wavelength and even a slowing of the speed of light?
5. Sep 19, 2010
### Mentz114
As Jonathan has said, the coordinate speed of light will change. But I don't see how it is possible to measure the wavelength from a distance.
What is observed will depend on what the spaceship is doing. For instance it might be moving (wrt the mass) or hovering.
6. Sep 19, 2010
### Pierre007080
Thanks Guys,
To conceal my ignorance, I think that I must stick to Jonathan's "single observer" status!!!!!!! How will this momentum (E/c) increase be interpreted? Would there be an increased "amplitude" to compensate for the observed shortened wavelength? | 2018-09-25 20:32:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7219921946525574, "perplexity": 958.5897428537825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267162385.84/warc/CC-MAIN-20180925202648-20180925223048-00322.warc.gz"} |
https://worldbuilding.stackexchange.com/questions/102377/will-these-foods-provide-sufficient-nutrition | Will these foods provide sufficient nutrition?
A few of my previous questions have mentioned that the colonists to this world will be scrambling to get whatever foods they can grow, because the planet's fauna is largely nonexistent (There's sea life, but not much else) and the flora offers little to no nutrition due to different biology.
With that in mind, I want to make sure the foods I do provide my people don't have them dieing of malnutrition.
Keep the following assumptions in mind. They are firm and any changes to them are outside of the scope of this question:
• Growing sufficient quantity is not an issue.
• Genetic Diversity is not an issue.
• Sustainability is not an issue.
• Native flora/fauna can provide calories, but no nutrition.
• The list of items is complete and embodies the entirety of the food types available.
• Source is not an isssue.
• Foods not listed are not available (See below, though).
Now, for the list of foods available, and some uses I've found for them. In no particular order:
• Barley: Can be ground into a gluten-containing flour to make breads and similar items. Also can be used for some alcoholic brewing such as beer, and is used in some soups and stews.
• Flax: Flax is a super-useful plant, capable of being used to make Flax Milk, a milk substitute, and can replace eggs in many recipes. It can also be used for making vegetable oil, and flaxseed sprouts can be eaten and are slightly spicy. Linseed Meal (The byproduct of making oil) is also a good food for rabbits. Flax is also useful for a variety of non-food products. Flax flour can also be used to compliment barley flour and/or corn flour, but isn't gluten-containing.
• Corn: Aside from regular corn and, potentially, popcorn, Corn can be used to make cornmeal, corn flour, corn starch, corn syrup, and corn sugar. On this world, there will be breeds of corn selected for sugar content, as sugarcane or sugar beets are not available.
• Lemons: Lemons are used in turning Flax Milk (And, in reality, other non-dairy "Milks") into cream, butter, and the like. Lemons have also been used with baking soda, before baking powder was invented.
• Coffee: Coffee can be cooked into things, but I'm really not sure what.
• Green, Red, some spicy peppers
• Tomatoes
• Cucumbers
• Green Beans
• Lettuce
• Peaches
• Potatoes
• Dill
• Fennel
• Sesame
• Rabbit meat
• Snake Meat
• Mouse meat (Although, admittedly, mice do not provide much meat...)
• Various types of yeast (Bread yeast, beer yeast, etc)
Can these foods provide enough nutrition to sustain people appropriately? If the answer is "No" I would like to know what is missing, and if you're feeling inspired, a food that would fix the deficiency while still being able to be easily grown from a seed that is stored and would possibly be on a space vessel. As a few of these items wouldn't be common items, I'm not terribly worried about realism as that can be explained by "A crewmember had a plant/seeds" etc. Any additional animals will not be considered due to other plot constraints.
• Yes, why not. This list would limit the cuisine quite a bit, but is definitely sustainable. Many ancient cultures lived on less. – Alexander Jan 16 '18 at 1:01
• Our bodies typically need: calcium, magnesium, potassium, vitamin D, vitamin b12s, iron, and folate. Beans typically cover a few of these. If most of your foods have an overlap in providing nutrition, the list is potentially workable. I haven't seen your other questions but since it's an imagined world, have you considered how this world (atmosphere, radiation, soil, water, etc) affects how nutrition is absorbed in humans? – doctordonna Jan 16 '18 at 1:37
• @doctordonna I have not. That's not even something I knew needed consideration. – Andon Jan 16 '18 at 1:44
• I don't have background on the world you're creating (how much is fiction, how much real life biology you're applying to this world). My current assumption is that nutrition could be an issue if your world is just like earth only lacking in flora. How do other life forms survive in this world? Is the sea life sustainable for people to consume? Is grafting an option? – doctordonna Jan 16 '18 at 2:01
• I'm making assumptions (Whether they're accurate or not): That the flora/fauna is non-toxic to humans but not nutritious, that the world has a biosphere that humans won't completely ruin (Humans stick to coastal areas, due to lack of draft animals among other things), and that earth plants can pull proper stuff from the soil. The native live is pretty similar to Earth life... a few million years ago, before large land animals. Small, insect-size creatures exist, and plants exist. – Andon Jan 16 '18 at 2:11
Lets run down the list of nutrients
Here is a link to nutritiondata.self.com, which has nice color graphics of all the data available from the USA. That link is for hulled barley. We can check all the major vitamins and minerals and such to make sure you are getting enough of what you need; and preferably from multiple sources. You don't want single crop failure to endanger the community.
Protein
There is plenty of meat, and barley is a decent source as well; less so corn. A significant gap in this diet is actually vegetable protein sources. Specifically, there are no legumes, like beans, chick peas, or the like. However, you mention green beans. These are the same species as common beans like kidney, pinto, and navy. The difference is, the green bean is the immature seed pod, while the mature dessicated seed pod provides the latter beans. Those beans are a very good protein (and many vitamins and minerals) source; so if green beans are available, common beans will be to.
Fiber
Beans and barley are both great sources. Sesame seeds and various certain lettuces can contribute as well.
Dietary fat
You actually only have two sources of fats. There isn't a lot of nutrition data available on snakes and mice, but I believe they are both pretty mean. Fat is a necessary part of the diet, so you will have to press sesame and flaxseed oil, and use it. Flaxseed oil is one of the few good vegetable sources of Omega-3 fats, so that is good as well. Again, depending on how fatty snakes and mice are, this might not be a problem at all.
Vitamins A, C, E, K
Vitamin A comes from vegetables that are orange, as a general rule. You don't have any really strong sources, so your people will need to eat a good quantity of tomatoes and peppers to keep up with this requirement. Another option is red-leaf lettuce; the reddish pigment is partially made up of vitamin A. Peppers and lemons are a very strong Vitamin C sources, with tomatoes, radishes and peaches also useful. Vitamin E in the $\gamma$-tocopherol form is covered by corn and vegetable oils, although both your oils are not great sources. Vitamin K is plentiful in anything green, so fennel, cucumber, green beans, and especially lettuce. Leafy greens are more nutritious the more bitter they are so kale/mustard greens > romain/butter lettuce > iceberg. Iceberg lettuces is basically fancy water.
Vitamin B-series
This series is Thiamin (B1), Riboflavin (B2), Niacin (B3), B6, Folate (B9), and B12. Grains and legumes cover this well, so barley, corn and beans here. Barley is particularly good at Thiamin, Corn at B6, and beans at folate. B12 is harder to come by, but plentiful in red meat, so your people should be fine.
Minerals
Calcium is something of a problem. There are a bunch of mediocre sources, but good sources like milk and milk products, tree nuts, and fish don't seem to be available. Women at all life stages, pregnant, nursing, and during menopause need a good supply of calcium. Making it worse, lettuce is a pretty poor source of calcium compared to spinach or kale. An option here is to grind up rabbit/mouse/snake bones and use them as a dietary supplement.
Iron is available from beans, greens, and red meat. The rest of minerals are available in high quantities in barley and corn. If those are your stable crops, you will have a good base of minerals.
Conclusion
Calcium seems like the thing that would most likely be in demand, for women especially. As mentioned, grinding up bones would be a reasonable source of it, or just mining it from chalk formations. It shouldn't be that hard to supplement.
For all the other vitamins and minerals, there are one or two food products that are excellent sources, and several reasonable sources. I would not foresee any dietary problems.
• A fantastic answer. I'll probably adjust some things on the final list, and this answer gives me a good starting point to go off of. – Andon Jan 16 '18 at 2:35
• How would the snakes, rabbits, and mice get their calcium requirements? – Justin Thyme Jan 16 '18 at 4:43
• Also missing are minerals. Salt, carbon, potassium, nitrogen for example. These would have to be available from the soil of the planet. – Justin Thyme Jan 16 '18 at 4:46
• @Justin Thyme Rabbits and Mice are modified to be able to digest local flora. Snakes would get it from mice. I had also counted on strictly mineral (IE, salt) items being able to be either found in the world or refined from the world. – Andon Jan 16 '18 at 10:08
• @JustinThyme Salt I did not mention, but worth noting that it has always been a dietary supplement from mining, even here on Earth. Potassium is in the 'rest of the minerals' plentiful category. Carbon are not dietary nutrients; they are the elements that dietary nutrients are made of. – kingledion Jan 16 '18 at 13:13 | 2021-03-01 22:55:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30911412835121155, "perplexity": 4052.462557501654}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178363072.47/warc/CC-MAIN-20210301212939-20210302002939-00398.warc.gz"} |
https://www.gamedev.net/forums/topic/396747-highlight-edit-box-selection-solved/ | # Highlight edit box selection (SOLVED)
This topic is 4274 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I'm currently exploring the resource editor in Visual C++. I added some edit boxes to my window, but they give me some headaches because of 2 things I can't accomplish: 1: (more general, not only edit boxes) using the tab button doesn't switch to another control, although I set the tab stops for some items. 2: when I set the focus on an edit box, I want to 'highlight' the text (number) inside (i.o.w. it should become blue, the same as when you double-click on it) If you people can tell me how to archieve the two things above, I'm a very happy man (for about 5 seconds until I find a new problem, but nevertheless...) [Edited by - Subotron on June 6, 2006 12:18:48 PM]
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SetSel() SetFocus()
other one look ctrl+d set tabstop
Kuphryn
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thanks, but I should've mentioned I'm not using MFC or stdafx.h
I tried this:
case EN_SETFOCUS:if (LOWORD(wparam) == IDC_EDIT_BOX){ HWND temp = GetDlgItem(window.get_wnd(0), IDC_EDIT_BOX); // handler to the edit box (=correct) DWORD text_length = GetWindowTextLength(temp); SetFocus(temp); #ifdef WIN32 // This happens, because I can replace the text afterwards, but // just EM_SETSEL doesn't make it blue SendMessage(temp, EM_SETSEL, 0, text_length); #else SendMessage(temp, EM_SETSEL, 0, MAKELONG (text_length, text_length)); #endif}break;
As said in the comment, the text is selected (as EM_REPLACESEL works) but doesn't turn blue... (I associate being blue with meaning "if you press a key now this will replace the currently selected text in the edit box" which comes in handy in combination with tabs)
[Edited by - Subotron on June 5, 2006 10:19:54 PM]
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SetFocus() after SendMessage()
Now?
Kuphryn
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nope, that doesn't work either :(
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Sounds like you're doing non modal dialogs, and is forgetting to call IsDialogMessage in your main message loop.
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edit: I looked up that function, and put it in my message handler function like this:
if (!IsDialogMessage(wnd[0], &msg))
{
TranslateMessage(&msg);
DispatchMessage(&msg);
}
where wnd[0] is the window handle and translate/dispatch functions where already there but without an if-check.
This fixes the no-tab problem, but not the selection problem
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hmmm it is fixed now, by throwing out the code I put in at all! Don't know why this failed when I tried before, probably forgot something, but it works now! I never understood that a dialog window CAN do all these things for you, by just one function :)
Thanks a lot guys! | 2018-02-17 19:54:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25465601682662964, "perplexity": 4553.285712765076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891807660.32/warc/CC-MAIN-20180217185905-20180217205905-00551.warc.gz"} |
https://proofwiki.org/wiki/Definition:Dense-in-itself | # Definition:Dense-in-itself
## Definition
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $H \subseteq S$.
Then $H$ is dense-in-itself if and only if it contains no isolated points.
## Also see
• Results about topological denseness can be found here. | 2019-04-25 14:26:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4594477117061615, "perplexity": 561.6346361747356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578721468.57/warc/CC-MAIN-20190425134058-20190425155227-00052.warc.gz"} |
https://www.beatthegmat.com/how-many-distinct-prime-divisors-does-a-positive-integer-n-have-t324393.html?sid=822032f2654fae5fa5b5ac11ad1509e6 | How many distinct prime divisors does a positive integer $$n$$ have?
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How many distinct prime divisors does a positive integer $$n$$ have?
by VJesus12 » Fri Jun 11, 2021 7:51 am
00:00
A
B
C
D
E
Global Stats
How many distinct prime divisors does a positive integer $$n$$ have?
(1) $$2n$$ has one distinct prime divisor.
(2) $$3n$$ has one distinct prime divisor.
Source: GMAT Club Tests
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Re: How many distinct prime divisors does a positive integer $$n$$ have?
by Ian Stewart » Sat Jun 12, 2021 3:59 pm
00:00
A
B
C
D
E
Global Stats
If n is a positive integer, 2n is clearly divisible by 2. If, as Statement 1 says, 2n has only one prime divisor, that prime divisor must be 2. But then n can be 1, 2, 2^2, 2^3 or any other power of 2. Since it is possible n = 1, it is possible n has no prime divisors, and if n is any other power of 2, then n has one prime divisor.
Similarly Statement 2 means n is 1, 3, 3^2, 3^3 or any other power of 3. So n might have no prime divisors or one prime divisor.
Combining the Statements, the only possibility is that n = 1, so the answer is C.
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
• Page 1 of 1 | 2022-01-26 05:51:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37153899669647217, "perplexity": 1451.5308852048568}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304915.53/warc/CC-MAIN-20220126041016-20220126071016-00460.warc.gz"} |
https://raweb.inria.fr/rapportsactivite/RA2021/rainbow/index.html | 2021
Activity report
Project-Team
RAINBOW
RNSR: 201822637G
Research center
In partnership with:
CNRS, Université Rennes 1, Institut national des sciences appliquées de Rennes
Team name:
Sensor-based Robotics and Human Interaction
In collaboration with:
Institut de recherche en informatique et systèmes aléatoires (IRISA)
Domain
Perception, Cognition and Interaction
Theme
Robotics and Smart environments
Creation of the Project-Team: 2018 June 01
# Keywords
• A5.1.3. Haptic interfaces
• A5.1.7. Multimodal interfaces
• A5.4.4. 3D and spatio-temporal reconstruction
• A5.4.6. Object localization
• A5.4.7. Visual servoing
• A5.5.4. Animation
• A5.6. Virtual reality, augmented reality
• A5.6.1. Virtual reality
• A5.6.2. Augmented reality
• A5.6.3. Avatar simulation and embodiment
• A5.6.4. Multisensory feedback and interfaces
• A5.9.2. Estimation, modeling
• A5.10.2. Perception
• A5.10.3. Planning
• A5.10.4. Robot control
• A5.10.5. Robot interaction (with the environment, humans, other robots)
• A5.10.6. Swarm robotics
• A5.10.7. Learning
• A6.4.1. Deterministic control
• A6.4.3. Observability and Controlability
• A6.4.4. Stability and Stabilization
• A6.4.5. Control of distributed parameter systems
• A6.4.6. Optimal control
• A9.5. Robotics
• A9.7. AI algorithmics
• A9.9. Distributed AI, Multi-agent
• B2.4.3. Surgery
• B2.5. Handicap and personal assistances
• B5.1. Factory of the future
• B5.6. Robotic systems
• B8.1.2. Sensor networks for smart buildings
• B8.4. Security and personal assistance
# 1 Team members, visitors, external collaborators
## Research Scientists
• Paolo Robuffo Giordano [Team leader, CNRS, Senior Researcher, HDR]
• François Chaumette [Inria, Senior Researcher, HDR]
• Alexandre Krupa [Inria, Senior Researcher, HDR]
• Claudio Pacchierotti [CNRS, Researcher]
• Julien Pettré [Inria, Senior Researcher, HDR]
## Faculty Members
• Marie Babel [INSA Rennes, Associate Professor, HDR]
• Quentin Delamare [École normale supérieure de Rennes]
• Vincent Drevelle [Univ de Rennes I, Associate Professor]
• Maud Marchal [INSA Rennes, Professor, HDR]
• Eric Marchand [Univ de Rennes I, Professor, HDR]
## Post-Doctoral Fellows
• Khairidine Benali [Inria, from Sep 2021]
• Elodie Bouzbib [Inria, from Nov 2021]
• Thomas Howard [CNRS, until Sep 2021]
• Pratik Mullick [Inria]
• Gennaro Notomista [CNRS, until Sep 2021]
## PhD Students
• Vicenzo Abichequer Sangalli [Inria]
• Julien Albrand [INSA Rennes, until Sept 2021]
• Javad Amirian [Inria, until Sep 2021]
• Maxime Bernard [CNRS, from Oct 2021]
• Pascal Brault [Inria]
• Pierre Antoine Cabaret [Inria, from Oct 2021]
• Antoine Cellier [INSA Rennes, from Oct 2021]
• Thomas Chatagnon [Inria]
• Cedric De Almeida Braga [Inria, until Feb 2021]
• Nicola De Carli [CNRS]
• Mathieu Gonzalez [Institut de recherche technologique B-com]
• Fabien Grzeskowiak [Inria, until Jun 2021]
• Alberto Jovane [Inria]
• Glenn Kerbiriou [Interdigital, from June 2021]
• Lisheng Kuang [China Scholarship Council]
• Ines Lacote [Inria]
• Emilie Leblong [Pôle Saint-Hélier]
• Thibault Noel [Inria, from Feb 2021]
• Erwan Normand [Univ de Rennes I, from Oct 2021]
• Alexander Oliva [Inria]
• Maxime Robic [Inria]
• Lev Smolentsev [Inria]
• Gustavo Souza Vieira Dutra [Inria, Dec 2021]
• Ali Srour [CNRS, from Oct 2021]
• John Thomas [Inria]
• Guillaume Vailland [INSA Rennes, until Nov 2021]
• Tairan Yin [Inria]
## Technical Staff
• Marco Aggravi [CNRS, Engineer, until Aug 2021]
• Dieudonne Atrevi [Inria, Engineer, until Apr 2021]
• Julien Bruneau [Inria, Engineer]
• Louise Devigne [INSA Rennes, Engineer]
• Julien Dufour [Inria, Engineer, from May 2021]
• Solenne Fortun [Inria, Engineer]
• Thierry Gaugry [INSA Rennes, Engineer, until Jun 2021]
• Guillaume Gicquel [CNRS, Engineer]
• Fabien Grzeskowiak [INSA Rennes, Engineer, from Jul 2021]
• Thomas Howard [INSA Rennes, Engineer, from Oct 2021]
• Noura Neji [Inria, Engineer, until Apr 2021]
• François Pasteau [INSA Rennes, Engineer]
• Yuliya Patotskaya [Inria, Engineer]
• Fabien Spindler [Inria, Engineer]
• Wouter Van Toll [Inria, Engineer, until Oct 2021]
## Interns and Apprentices
• Merwane Bouri [INSA Rennes, from Aug 2021]
• Pierre Antoine Cabaret [INSA Rennes, from Feb 2021 until Aug 2021]
• Alex Coudray [Inria, from Feb 2021 until Jul 2021]
• Arthur Furet [INSA Rennes, from Jun 2021 until Aug 2021]
• Alexis Hobl [CNRS, from May 2021 until Jul 2021]
• Emilie Hummel [CNRS, from Feb 2021 until Aug 2021]
• Alexis Jensen [INSA Rennes, from Jul 2021 until Sep 2021]
• Divyesh Kanagavel [CNRS, from Feb 2021 until Aug 2021]
• Alex Keryhuel [Inria, from May 2021 until Aug 2021]
• Hussein Lezzaik [CNRS, from Feb 2021 until Aug 2021]
• Arthur Luciani [École normale supérieure de Rennes, from Feb 2021 until Jul 2021]
• Thomas Mabit [École normale supérieure de Rennes, from Feb 2021 until Aug 2021]
• Thibaut Rolland [Inria, from Feb 2021 until Jul 2021]
• Octavie Somoza Salgado [CNRS, from Mar 2021 until Aug 2021]
• Guillaume Sonnet [Inria, from May 2021 until Sep 2021]
• Gustavo Souza Vieira Dutra [INSA Rennes, Intern, from Feb 2021 until Aug 2021]
• Hélène de La Ruée [Univ de Rennes I]
# 2 Overall objectives
The long-term vision of the Rainbow team is to develop the next generation of sensor-based robots able to navigate and/or interact in complex unstructured environments together with human users. Clearly, the word “together” can have very different meanings depending on the particular context: for example, it can refer to mere co-existence (robots and humans share some space while performing independent tasks), human-awareness (the robots need to be aware of the human state and intentions for properly adjusting their actions), or actual cooperation (robots and humans perform some shared task and need to coordinate their actions).
One could perhaps argue that these two goals are somehow in conflict since higher robot autonomy should imply lower (or absence of) human intervention. However, we believe that our general research direction is well motivated since: $\left(i\right)$ despite the many advancements in robot autonomy, complex and high-level cognitive-based decisions are still out of reach. In most applications involving tasks in unstructured environments, uncertainty, and interaction with the physical word, human assistance is still necessary, and will most probably be for the next decades. On the other hand, robots are extremely capable of autonomously executing specific and repetitive tasks, with great speed and precision, and of operating in dangerous/remote environments, while humans possess unmatched cognitive capabilities and world awareness which allow them to take complex and quick decisions; $\left(ii\right)$ the cooperation between humans and robots is often an implicit constraint of the robotic task itself. Consider for instance the case of assistive robots supporting injured patients during their physical recovery, or human augmentation devices. It is then important to study proper ways of implementing this cooperation; $\left(iii\right)$ finally, safety regulations can require the presence at all times of a person in charge of supervising and, if necessary, of taking direct control of the robotic workers. For example, this is a common requirement in all applications involving tasks in public spaces, like autonomous vehicles in crowded spaces, or even UAVs when flying in civil airspace such as over urban or populated areas.
Within this general picture, the Rainbow activities will be particularly focused on the case of (shared) cooperation between robots and humans by pursuing the following vision: on the one hand, empower robots with a large degree of autonomy for allowing them to effectively operate in non-trivial environments (e.g., outside completely defined factory settings). On the other hand, include human users in the loop for having them in (partial and bilateral) control of some aspects of the overall robot behavior. We plan to address these challenges from the methodological, algorithmic and application-oriented perspectives. The main research axes along which the Rainbow activities will be articulated are: three supporting axes (Optimal and Uncertainty-Aware Sensing; Advanced Sensor-based Control; Haptics for Robotics Applications) that are meant to develop methods, algorithms and technologies for realizing the central theme of Shared Control of Complex Robotic Systems.
# 3 Research program
## 3.1 Main Vision
The vision of Rainbow (and foreseen applications) calls for several general scientific challenges: $\left(i\right)$ high-level of autonomy for complex robots in complex (unstructured) environments, $\left(ii\right)$ forward interfaces for letting an operator giving high-level commands to the robot, $\left(iii\right)$ backward interfaces for informing the operator about the robot `status', $\left(iv\right)$ user studies for assessing the best interfacing, which will clearly depend on the particular task/situation. Within Rainbow we plan to tackle these challenges at different levels of depth:
• the methodological and algorithmic side of the sought human-robot interaction will be the main focus of Rainbow. Here, we will be interested in advancing the state-of-the-art in sensor-based online planning, control and manipulation for mobile/fixed robots. For instance, while classically most control approaches (especially those sensor-based) have been essentially reactive, we believe that less myopic strategies based on online/reactive trajectory optimization will be needed for the future Rainbow activities. The core ideas of Model-Predictive Control approaches (also known as Receding Horizon) or, in general, numerical optimal control methods will play a role in the Rainbow activities, for allowing the robots to reason/plan over some future time window and better cope with constraints. We will also consider extending classical sensor-based motion control/manipulation techniques to more realistic scenarios, such as deformable/flexible objects (“Advanced Sensor-based Control” axis). Finally, it will also be important to spend research efforts into the field of Optimal Sensing, in the sense of generating (again) trajectories that can optimize the state estimation problem in presence of scarce sensory inputs and/or non-negligible measurement and process noises, especially true for the case of mobile robots (“Optimal and Uncertainty-Aware Sensing” axis). We also aim at addressing the case of coordination between a single human user and multiple robots where, clearly, as explained the autonomy part plays even a more crucial role (no human can control multiple robots at once, thus a high degree of autonomy will be required by the robot group for executing the human commands);
• the interfacing side will also be a focus of the Rainbow activities. As explained above, we will be interested in both the forward (human $\to$ robot) and backward (robot $\to$ human) interfaces. The forward interface will be mainly addressed from the algorithmic point of view, i.e., how to map the few degrees of freedom available to a human operator (usually in the order of 3–4) into complex commands for the controlled robot(s). This mapping will typically be mediated by an “AutoPilot” onboard the robot(s) for autonomously assessing if the commands are feasible and, if not, how to least modify them (“Advanced Sensor-based Control” axis).
The backward interface will, instead, mainly consist of a visual/haptic feedback for the operator. Here, we aim at exploiting our expertise in using force cues for informing an operator about the status of the remote robot(s). However, the sole use of classical grounded force feedback devices (e.g., the typical force-feedback joysticks) will not be enough due to the different kinds of information that will have to be provided to the operator. In this context, the recent interest in the use of wearable haptic interfaces is very interesting and will be investigated in depth (these include, e.g., devices able to provide vibro-tactile information to the fingertips, wrist, or other parts of the body). The main challenges in these activities will be the mechanical conception (and construction) of suitable wearable interfaces for the tasks at hand, and in the generation of force cues for the operator: the force cues will be a (complex) function of the robot state, therefore motivating research in algorithms for mapping the robot state into a few variables (the force cues) (“Haptics for Robotics Applications” axis);
• the evaluation side that will assess the proposed interfaces with some user studies, or acceptability studies by human subjects. Although this activity will not be a main focus of Rainbow (complex user studies are beyond the scope of our core expertise), we will nevertheless devote some efforts into having some reasonable level of user evaluations by applying standard statistical analysis based on psychophysical procedures (e.g., randomized tests and Anova statistical analysis). This will be particularly true for the activities involving the use of smart wheelchairs, which are intended to be used by human users and operate inside human crowds. Therefore, we will be interested in gaining some level of understanding of how semi-autonomous robots (a wheelchair in this example) can predict the human intention, and how humans can react to a semi-autonomous mobile robot.
Figure 1 depicts in an illustrative way the prototypical activities foreseen in Rainbow. On the righthand side, complex robots (dual manipulators, humanoid, single/multiple mobile robots) need to perform some task with high degree of autonomy. On the lefthand side, a human operator gives some high-level commands and receives a visual/haptic feedback aimed at informing her/him at best of the robot status. Again, the main challenges that Rainbow will tackle to address these issues are (in order of relevance): $\left(i\right)$ methods and algorithms, mostly based on first-principle modeling and, when possible, on numerical methods for online/reactive trajectory generation, for enabling the robots with high autonomy; $\left(ii\right)$ design and implementation of visual/haptic cues for interfacing the human operator with the robots, with a special attention to novel combinations of grounded/ungrounded (wearable) haptic devices; $\left(iii\right)$ user and acceptability studies.
## 3.2 Main Components
Hereafter, a summary description of the four axes of research in Rainbow.
### 3.2.1 Optimal and Uncertainty-Aware Sensing
Future robots will need to have a large degree of autonomy for, e.g., interpreting the sensory data for accurate estimation of the robot and world state (which can possibly include the human users), and for devising motion plans able to take into account many constraints (actuation, sensor limitations, environment), including also the state estimation accuracy (i.e., how well the robot/environment state can be reconstructed from the sensed data). In this context, we will be particularly interested in $\left(i\right)$ devising trajectory optimization strategies able to maximize some norm of the information gain gathered along the trajectory (and with the available sensors). This can be seen as an instance of Active Sensing, with the main focus on online/reactive trajectory optimization strategies able to take into account several requirements/constraints (sensing/actuation limitations, noise characteristics). We will also be interested in the coupling between optimal sensing and concurrent execution of additional tasks (e.g., navigation, manipulation). $\left(ii\right)$ Formal methods for guaranteeing the accuracy of localization/state estimation in mobile robotics, mainly exploiting tools from interval analysis. The interest of these methods is their ability to provide possibly conservative but guaranteed accuracy bounds on the best accuracy one can obtain with the given robot/sensor pair, and can thus be used for planning purposes or for system design (choice of the best sensors for a given robot/task). $\left(iii\right)$ Localization/tracking of objects with poor/unknown or deformable shape, which will be of paramount importance for allowing robots to estimate the state of “complex objects” (e.g., human tissues in medical robotics, elastic materials in manipulation) for controlling its pose/interaction with the objects of interest.
One of the main competences of the previous Lagadic team has been, generally speaking, the topic of sensor-based control, i.e., how to exploit (typically onboard) sensors for controlling the motion of fixed/ground robots. The main emphasis has been in devising ways to directly couple the robot motion with the sensor outputs in order to invert this mapping for driving the robots towards a configuration specified as a desired sensor reading (thus, directly in sensor space). This general idea has been applied to very different contexts: mainly standard vision (from which the Visual Servoing keyword), but also audio, ultrasound imaging, and RGB-D.
Use of sensors for controlling the robot motion will also clearly be a central topic of the Rainbow team too, since the use of (especially onboard) sensing is a main characteristics of any future robotics application (which should typically operate in unstructured environments, and thus mainly rely on its own ability to sense the world). We then naturally aim at making the best out of the previous Lagadic experience in sensor-based control for proposing new advanced ways of exploiting sensed data for, roughly speaking, controlling the motion of a robot. In this respect, we plan to work on the following topics: $\left(i\right)$ “direct/dense methods” which try to directly exploit the raw sensory data in computing the control law for positioning/navigation tasks. The advantages of these methods is the little need for data pre-processing which can minimize feature extraction errors and, in general, improve the overall robustness/accuracy (since all the available data is used by the motion controller); $\left(ii\right)$ sensor-based interaction with objects of unknown/deformable shapes, for gaining the ability to manipulate, e.g., flexible objects from the acquired sensed data (e.g., controlling online a needle being inserted in a flexible tissue); $\left(iii\right)$ sensor-based model predictive control, by developing online/reactive trajectory optimization methods able to plan feasible trajectories for robots subjects to sensing/actuation constraints with the possibility of (onboard) sensing for continuously replanning (over some future time horizon) the optimal trajectory. These methods will play an important role when dealing with complex robots affected by complex sensing/actuation constraints, for which pure reactive strategies (as in most of the previous Lagadic works) are not effective. Furthermore, the coupling with the aforementioned optimal sensing will also be considered; $\left(iv\right)$ multi-robot decentralised estimation and control, with the aim of devising again sensor-based strategies for groups of multiple robots needing to maintain a formation or perform navigation/manipulation tasks. Here, the challenges come from the need of devising “simple” decentralized and scalable control strategies under the presence of complex sensing constraints (e.g., when using onboard cameras, limited fov, occlusions). Also, the need of locally estimating global quantities (e.g., common frame of reference, global property of the formation such as connectivity or rigidity) will also be a line of active research.
### 3.2.3 Haptics for Robotics Applications
In the envisaged shared cooperation between human users and robots, the typical sensory channel (besides vision) exploited to inform the human users is most often the force/kinesthetic one (in general, the sense of touch and of applied forces to the human hand or limbs). Therefore, a part of our activities will be devoted to study and advance the use of haptic cueing algorithms and interfaces for providing a feedback to the users during the execution of some shared task. We will consider: $\left(i\right)$ multi-modal haptic cueing for general teleoperation applications, by studying how to convey information through the kinesthetic and cutaneous channels. Indeed, most haptic-enabled applications typically only involve kinesthetic cues, e.g., the forces/torques that can be felt by grasping a force-feedback joystick/device. These cues are very informative about, e.g., preferred/forbidden motion directions, but are also inherently limited in their resolution since the kinesthetic channel can easily become overloaded (when too much information is compressed in a single cue). In recent years, the arise of novel cutaneous devices able to, e.g., provide vibro-tactile feedback on the fingertips or skin, has proven to be a viable solution to complement the classical kinesthetic channel. We will then study how to combine these two sensory modalities for different prototypical application scenarios, e.g., 6-dof teleoperation of manipulator arms, virtual fixtures approaches, and remote manipulation of (possibly deformable) objects; $\left(ii\right)$ in the particular context of medical robotics, we plan to address the problem of providing haptic cues for typical medical robotics tasks, such as semi-autonomous needle insertion and robot surgery by exploring the use of kinesthetic feedback for rendering the mechanical properties of the tissues, and vibrotactile feedback for providing with guiding information about pre-planned paths (with the aim of increasing the usability/acceptability of this technology in the medical domain); $\left(iii\right)$ finally, in the context of multi-robot control we would like to explore how to use the haptic channel for providing information about the status of multiple robots executing a navigation or manipulation task. In this case, the problem is (even more) how to map (or compress) information about many robots into a few haptic cues. We plan to use specialized devices, such as actuated exoskeleton gloves able to provide cues to each fingertip of a human hand, or to resort to “compression” methods inspired by the hand postural synergies for providing coordinated cues representative of a few (but complex) motions of the multi-robot group, e.g., coordinated motions (translations/expansions/rotations) or collective grasping/transporting.
### 3.2.4 Shared Control of Complex Robotics Systems
This final and main research axis will exploit the methods, algorithms and technologies developed in the previous axes for realizing applications involving complex semi-autonomous robots operating in complex environments together with human users. The leitmotiv is to realize advanced shared control paradigms, which essentially aim at blending robot autonomy and user's intervention in an optimal way for exploiting the best of both worlds (robot accuracy/sensing/mobility/strength and human's cognitive capabilities). A common theme will be the issue of where to “draw the line” between robot autonomy and human intervention: obviously, there is no general answer, and any design choice will depend on the particular task at hand and/or on the technological/algorithmic possibilities of the robotic system under consideration.
A prototypical envisaged application, exploiting and combining the previous three research axes, is as follows: a complex robot (e.g., a two-arm system, a humanoid robot, a multi-UAV group) needs to operate in an environment exploiting its onboard sensors (in general, vision as the main exteroceptive one) and deal with many constraints (limited actuation, limited sensing, complex kinematics/dynamics, obstacle avoidance, interaction with difficult-to-model entities such as surrounding people, and so on). The robot must then possess a quite large autonomy for interpreting and exploiting the sensed data in order to estimate its own state and the environment one (“Optimal and Uncertainty-Aware Sensing” axis), and for planning its motion in order to fulfil the task (e.g., navigation, manipulation) by coping with all the robot/environment constraints. Therefore, advanced control methods able to exploit the sensory data at its most, and able to cope online with constraints in an optimal way (by, e.g., continuously replanning and predicting over a future time horizon) will be needed (“Advanced Sensor-based Control” axis), with a possible (and interesting) coupling with the sensing part for optimizing, at the same time, the state estimation process. Finally, a human operator will typically be in charge of providing high-level commands (e.g., where to go, what to look at, what to grasp and where) that will then be autonomously executed by the robot, with possible local modifications because of the various (local) constraints. At the same time, the operator will also receive online visual-force cues informative of, in general, how well her/his commands are executed and if the robot would prefer or suggest other plans (because of the local constraints that are not of the operator's concern). This information will have to be visually and haptically rendered with an optimal combination of cues that will depend on the particular application (“Haptics for Robotics Applications” axis).
# 4 Application domains
The activities of Rainbow fall obviously within the scope of Robotics. Broadly speaking, our main interest is in devising novel/efficient algorithms (for estimation, planning, control, haptic cueing, human interfacing, etc.) that can be general and applicable to many different robotic systems of interest, depending on the particular application/case study. For instance, we plan to consider
• applications involving remote telemanipulation with one or two robot arms, where the arm(s) will need to coordinate their motion for approaching/grasping objects of interest under the guidance of a human operator;
• applications involving single and multiple mobile robots for spatial navigation tasks (e.g., exploration, surveillance, mapping). In the multi-robot case, the high redundancy of the multi-robot group will motivate research in autonomously exploiting this redundancy for facilitating the task (e.g., optimizing the self-localization of the environment mapping) while following the human commands, and vice-versa for informing the operator about the status of a multi-robot group. In the single robot case, the possible combination with some manipulation devices (e.g., arms on a wheeled robot) will motivate research into remote tele-navigation and tele-manipulation;
• applications involving medical robotics, in which the “manipulators” are replaced by the typical tools used in medical applications (ultrasound probes, needles, cutting scalpels, and so on) for semi-autonomous probing and intervention;
• applications involving a direct physical “coupling” between human users and robots (rather than a “remote” interfacing), such as the case of assistive devices used for easing the life of impaired people. Here, we will be primarily interested in, e.g., safety and usability issues, and also touch some aspects of user acceptability.
These directions are, in our opinion, very promising since nowadays and future robotics applications are expected to address more and more complex tasks: for instance, it is becoming mandatory to empower robots with the ability to predict the future (to some extent) by also explicitly dealing with uncertainties from sensing or actuation; to safely and effectively interact with human supervisors (or collaborators) for accomplishing shared tasks; to learn or adapt to the dynamic environments from small prior knowledge; to exploit the environment (e.g., obstacles) rather than avoiding it (a typical example is a humanoid robot in a multi-contact scenario for facilitating walking on rough terrains); to optimize the onboard resources for large-scale monitoring tasks; to cooperate with other robots either by direct sensing/communication, or via some shared database (the “cloud”).
While no single lab can reasonably address all these theoretical/algorithmic/technological challenges, we believe that our research agenda can give some concrete contributions to the next generation of robotics applications.
# 5 Highlights of the year
## 5.1 Awards
• Best Paper Award at the conference 2021 ICAT-EGVE (International Conference on Artificial Reality and Telexistence and Eurographics Symposium on Virtual Environments) 52
• Best Demonstration Awards at 2021 IEEE WHC (IEEE World Haptics) 61
## 5.2 Highlights
• P. Robuffo Giordano is part of the euRobotics “George Giralt” PhD Award panel for awarding the best PhD Thesis in robotics in Europe
• P. Robuffo Giordano has been elected member of the Section 07 of the Comité National de la Recherche Scientifique
• A. Krupa was promoted in 2021 to the grade of Inria Senior Research Scientist (Inria DR2)
• C. Pacchierotti has been proposed for the 2022 CNRS Bronze medal by the Section 7 of the CoNRS (The National Committee for Scientific Research).
# 6 New software and platforms
## 6.1 New software
### 6.1.1 HandiViz
• Name:
Driving assistance of a wheelchair
• Keywords:
Health, Persons attendant, Handicap
• Functional Description:
The HandiViz software proposes a semi-autonomous navigation framework of a wheelchair relying on visual servoing.
It has been registered to the APP (“Agence de Protection des Programmes”) as an INSA software (IDDN.FR.001.440021.000.S.P.2013.000.10000) and is under GPL license.
• Contact:
Marie Babel
• Participants:
Francois Pasteau, Marie Babel
• Partner:
INSA Rennes
### 6.1.2 UsTk
• Name:
Ultrasound toolkit for medical robotics applications guided from ultrasound images
• Keywords:
Echographic imagery, Image reconstruction, Medical robotics, Visual tracking, Visual servoing (VS), Needle insertion
• Functional Description:
UsTK, standing for Ultrasound Toolkit, is a cross-platform extension of ViSP software dedicated to 2D and 3D ultrasound image processing and visual servoing based on ultrasound images. Written in C++, UsTK architecture provides a core module that implements all the data structures at the heart of UsTK, a grabber module that allows acquiring ultrasound images from an Ultrasonix or a Sonosite device, a GUI module to display data, an IO module for providing functionalities to read/write data from a storage device, and a set of image processing modules to compute the confidence map of ultrasound images, generate elastography images, track a flexible needle in sequences of 2D and 3D ultrasound images and track a target image template in sequences of 2D ultrasound images. All these modules were implemented on several robotic demonstrators to control the motion of an ultrasound probe or a flexible needle by ultrasound visual servoing.
• URL:
• Contact:
Alexandre Krupa
• Participants:
Alexandre Krupa, Fabien Spindler
• Partners:
Inria, Université de Rennes 1
### 6.1.3 ViSP
• Name:
Visual servoing platform
• Keywords:
Augmented reality, Computer vision, Robotics, Visual servoing (VS), Visual tracking
• Scientific Description:
Since 2005, we develop and release ViSP [1], an open source library available from https://visp.inria.fr. ViSP standing for Visual Servoing Platform allows prototyping and developing applications using visual tracking and visual servoing techniques at the heart of the Rainbow research. ViSP was designed to be independent from the hardware, to be simple to use, expandable and cross-platform. ViSP allows designing vision-based tasks for eye-in-hand and eye-to-hand systems from the most classical visual features that are used in practice. It involves a large set of elementary positioning tasks with respect to various visual features (points, segments, straight lines, circles, spheres, cylinders, image moments, pose...) that can be combined together, and image processing algorithms that allow tracking of visual cues (dots, segments, ellipses...), or 3D model-based tracking of known objects or template tracking. Simulation capabilities are also available.
[1] E. Marchand, F. Spindler, F. Chaumette. ViSP for visual servoing: a generic software platform with a wide class of robot control skills. IEEE Robotics and Automation Magazine, Special Issue on "Software Packages for Vision-Based Control of Motion", P. Oh, D. Burschka (Eds.), 12(4):40-52, December 2005.
• Functional Description:
ViSP provides simple ways to integrate and validate new algorithms with already existing tools. It follows a module-based software engineering design where data types, algorithms, sensors, viewers and user interaction are made available. Written in C++, ViSP is based on open-source cross-platform libraries (such as OpenCV) and builds with CMake. Several platforms are supported, including OSX, iOS, Windows and Linux. ViSP online documentation allows to ease learning. More than 300 fully documented classes organized in 17 different modules, with more than 408 examples and 88 tutorials are proposed to the user. ViSP is released under a dual licensing model. It is open-source with a GNU GPLv2 or GPLv3 license. A professional edition license that replaces GNU GPL is also available.
• URL:
• Contact:
Fabien Spindler
• Participants:
Éric Marchand, Fabien Spindler, Francois Chaumette
• Partners:
Inria, Université de Rennes 1
### 6.1.4 DIARBENN
• Name:
Obstacle avoidance through sensor-based servoing
• Keywords:
• Functional Description:
DIARBENN's objective is to define an obstacle avoidance solution adapted to a mobile robot such as a powered wheelchair. Through a shared control system, the system corrects progressively and if necessary the trajectory when approaching an obstacle while respecting the user's intention.
• Contact:
Marie Babel
• Partner:
INSA Rennes
## 6.2 New platforms
### 6.2.1 Robot Vision Platform
Participant: François Chaumette [contact], Alexandre Krupa [contact], Eric Marchand [contact], Fabien Spindler [contact].
We exploit two industrial robotic systems built by Afma Robots in the nineties to validate our research in visual servoing and active vision. The first one is a 6 DoF Gantry robot, the other one is a 4 DoF cylindrical robot (see Fig. 2). These robots are equipped with monocular RGB cameras. The Gantry robot also allows mounting grippers on its end-effector. Attached to this platform, we can also find a collection of various RGB and RGB-D cameras used to validate vision-based real-time tracking algorithms.
In 2021, this platform has been used to validate experimental results in 2 accepted publications 4016.
### 6.2.2 Mobile Robots
Participants: Marie Babel [contact], Solenne Fortun [contact], François Pasteau [contact], Julien Pettré [contact], Quentin Delamare [contact], Fabien Spindler [contact].
To validate our research in personally assisted living topic (see Section 7.4.4), we have three electric wheelchairs, one from Permobil, one from Sunrise and the last from YouQ (see Fig. 3.a). The control of the wheelchair is performed using a plug and play system between the joystick and the low level control of the wheelchair. Such a system lets us acquire the user intention through the joystick position and control the wheelchair by applying corrections to its motion. The wheelchairs have been fitted with cameras, ultrasound and time of flight sensors to perform the required servoing for assisting handicapped people. A wheelchair haptic simulator completes this platform to develop new human interaction strategies in a virtual reality environment (see Fig. 3(b)).
Pepper, a human-shaped robot designed by SoftBank Robotics to be a genuine day-to-day companion (see Fig. 3.c) is also part of this platform. It has 17 DoF mounted on a wheeled holonomic base and a set of sensors (cameras, laser, ultrasound, inertial, microphone) that makes this platform interesting for robot-human interactions during locomotion and visual exploration strategies (Sect. 7.2.8).
Moreover, for fast prototyping of algorithms in perception, control and autonomous navigation, the team uses a Pioneer 3DX from Adept (see Fig. 3.d). This platform is equipped with various sensors needed for autonomous navigation and sensor-based control.
In 2021, these platforms was used to obtain experimental results presented in 3 papers 2, 7, 50.
### 6.2.3 Medical Robotic Platform
Participants: Alexandre Krupa [contact], Fabien Spindler [contact].
This platform is composed of two 6 DoF Adept Viper arms (see Figs. 4.a–b). Ultrasound probes connected either to a SonoSite 180 Plus or an Ultrasonix SonixTouch 2D and 3D imaging system can be mounted on a force torque sensor attached to each robot end-effector. The haptic Virtuose 6D or Omega 6 device (see Fig. 7.a) can also be used with this platform.
This platform was extended with a ATI Nano43 force/torque sensor attached to one of the Viper arm. It allows to perform experiments for needle insertion applications.
This testbed is of primary interest for researches and experiments concerning ultrasound visual servoing applied to probe positioning, soft tissue tracking, elastography or robotic needle insertion tasks (see Sect. 7.4.3). It can also be used to validate more classical tracking and visual servoing researches.
In 2021, this platform was used to obtain experimental results presented in 2 papers 8, 48.
Participants: Claudio Pacchierotti [contact], Paolo Robuffo Giordano [contact], Fabien Spindler [contact].
This platform is composed by 2 Panda lightweight arms from Franka Emika equipped with torque sensors in all seven axes. An electric gripper, a camera, a soft hand from qbrobotics or a Reflex TakkTile 2 gripper from RightHand Labs (see Fig. 5.b) can be mounted on the robot end-effector (see Fig. 5.a). A force/torque sensor from Alberobotics is also attached to one of the robots end-effector to get more precision during torque control. This setup is used to validate our researches in coupling force and vision for controlling robot manipulators (see Section 7.2.11) and in shared control for remote manipulation (see Section 7.4.1). Other haptic devices (see Section 6.2.6) can also be coupled to this platform.
2 new papers 19, 26 published this year include experimental results obtained with this platform.
### 6.2.5 Unmanned Aerial Vehicles (UAVs)
Participants: Joudy Nader [contact], Paolo Robuffo Giordano [contact], Claudio Pacchierotti [contact], Fabien Spindler [contact].
Rainbow is involved in several activities involving perception and control for single and multiple quadrotor UAVs. To this end, we exploit four quadrotors from Mikrokopter Gmbh, Germany (see Fig. 6.a), and one quadrotor from 3DRobotics, USA (see Fig. 6.b). The Mikrokopter quadrotors have been heavily customized by: $\left(i\right)$ reprogramming from scratch the low-level attitude controller onboard the microcontroller of the quadrotors, $\left(ii\right)$ equipping each quadrotor with a NVIDIA Jetson TX2 board running Linux Ubuntu and the TeleKyb-3 software based on genom3 framework developed at LAAS in Toulouse (the middleware used for managing the experiment flows and the communication among the UAVs and the base station), and $\left(iii\right)$ purchasing new Realsense RGB-D cameras for visual odometry and visual servoing. The quadrotor group is used as robotic platforms for testing a number of single and multiple flight control schemes with a special attention on the use of onboard vision as main sensory modality.
This year 3 papers 9, 7, 1 contain experimental results obtained with this platform.
### 6.2.6 Haptics and Shared Control Platform
Participants: Claudio Pacchierotti [contact], Paolo Robuffo Giordano [contact], Fabien Spindler [contact].
Various haptic devices are used to validate our research in shared control. We have a Virtuose 6D device from Haption (see Fig. 7.a). This device is used as master device in many of our shared control activities (see, e.g., Sections 7.4.1). It could also be coupled to the Haption haptic glove in loan from the University of Birmingham. An Omega 6 (see Fig. 7.b) from Force Dimension and devices in loan from Ultrahaptics complete this platform that could be coupled to the other robotic platforms.
This platform was used to obtain experimental results presented in 6 papers 19, 8, 7, 42, 45 published this year.
### 6.2.7 Portable immersive room
Participants: François Pasteau [contact], Fabien Grzeskowiak [contact], Marie Babel [contact].
To validate our research on assistive robotics and its applications in virtual conditions, we very recently acquire a portable immersive room that is planned to be easily deployed in different rehabilitation structures in order to conduct clinical trials. The system has been designed by Trinoma company and has been funded by Interreg ADAPT project.
# 7 New results
## 7.1 Optimal and Uncertainty-Aware Sensing
### 7.1.1 3D tracking of deformable objects from RGB-D data
Participants: Alexandre Krupa, Eric Marchand.
Within our research activities on deformable object tracking, this year we proposed a novel framework for tracking the deformation of soft objects using a RGB-D camera. It requires a coarse 3D mesh and physical model of the object based on FEM whose parameters (Young Modulus, Poisson's ratio, etc) do not need to be precise. The approach consists in minimizing both a geometric error and a direct photometric intensity error while relying on the co-rotational Finite Element Method as the underlying deformation model 48. The proposed method has been validated both on synthetic data with groundtruth and real data.
### 7.1.2 Trajectory Generation for Optimal State Estimation
Participants: Nicola De Carli, Gennaro Notomista, Claudio Pacchierotti, Paolo Robuffo Giordano.
This activity addresses the general problem of active sensing where the goal is to analyze and synthesize optimal trajectories for a robotic system that can maximize the amount of information gathered by (few) noisy outputs (i.e., sensor readings) while at the same time reducing the negative effects of the process/actuation noise. We have recently developed a general framework for solving online the active sensing problem by continuously replanning an optimal trajectory that maximizes a suitable norm of the Constructibility Gramian (CG) 64.
In 36, we have extended this framework for considering the problem of localization for a group of multiple robots that can obtain distance measurements and communicate only with local neighbors. We showed that, thanks to a proper change of coordinates, the CG for the multi-robot group can be computed in a decentralized way with only minor approximations. This allowed us to formulate an online and decentralized trajectory generation problem for optimal localization. We considered as case study the localization of a quadrotor group with noisy distance measurements and sensing constraints, and showed the effectiveness of the appraoch via a monte-carlo simulation. We are now considering the case of bearing measurments (obtained from onboard cameras) and the associated constraints of limited fov and possible occlusions. We are also working towards an experimental validation of the approach.
In 25, we have instead considered a different active sensing problem that involves a single robot but in an environmental monitoring task. The goal is to estimate some (possibly time-varying) parameters of a distributed scalar field representative of, e.g., a gas or other quantities in the atmosphere. The robot is equipped with a sensor able to locally measure the value of this field, and the estimation goal is to recover the (unknown) field parameters (e.g., location of the source, decaying rate) by suitably planning an optimal trajectory. To this end we have formulated a trajectory optimization problem that maximizes a norm of the CG and also takes into account the energy level of the robot by modeling a battery with a discharge dynamics that depends on the control effort. The results have been validated in simulation and are quite promising. We are now working on a multi-robot formulation of this problem.
### 7.1.3 Leveraging Multiple Environments for Learning and Decision Making
Participants: Maud Marchal, Thierry Gaugry, Antonin Bernardin.
Learning is usually performed by observing real robot executions. Physics-based simulators are a good alternative for providing highly valuable information while avoiding costly and potentially destructive robot executions. Within the Imagine project, we presented a novel approach for learning the probabilities of symbolic robot action outcomes. This is done by leveraging different environments, such as physics-based simulators, in execution time. To this end, we proposed MENID (Multiple Environment Noise Indeterministic Deictic) rules, a novel representation able to cope with the inherent uncertainties present in robotic tasks. MENID rules explicitly represent each possible outcomes of an action, keep memory of the source of the experience, and maintain the probability of success of each outcome. We also introduced an algorithm to distribute actions among environments, based on previous experiences and expected gain. Before using physics-based simulations, we proposed a methodology for evaluating different simulation settings and determining the least time-consuming model that could be used while still producing coherent results. We demonstrated the validity of the approach in a dismantling use case, using a simulation with reduced quality as simulated system, and a simulation with full resolution where we add noise to the trajectories and some physical parameters as a representation of the real system.
### 7.1.4 A Plane-based Approach for Indoor Point Clouds Registration
Participant: Eric Marchand.
Traditional 3D point clouds registration algorithms, based on Iterative Closest Point (ICP), rely on point matching of large point clouds. In well-structured environments, such as buildings, planes can be segmented and used for registration, similarly to the classical point-based ICP approach. Using planes tremendously reduces the number of inputs. An efficient plane-based registration algorithm has been proposed. The optimal transformation is estimated through a two-steps approach, successively performing robust plane-to- plane minimization and non-linear robust point-to-plane registration 39, 57, 38. This work was done in cooperation with IETR Lab.
### 7.1.5 Visual SLAM
Participant: Eric Marchand.
We proposed a novel visual SLAM method with dense planar reconstruction using a monocular camera: TT-SLAM. The method exploits planar template-based trackers (TT) to compute camera poses and reconstructs a multi-planar scene representation. Multiple homographies are estimated simultaneously by clustering a set of template trackers supported by superpixelized regions. Compared to ou previous work (RANSAC-based multiple homographies method), data association and keyframe selection issues are handled by the continuous nature of template trackers. A non-linear optimization process is applied to all the homographies to improve the precision in pose estimation. This work 53 was done in cooperation with the Mimetic team.
We also proposed a novel binary graph descriptor to improve loop detection for visual SLAM systems. Our contribution is twofold: i) a graph embedding technique for generating binary descriptors which conserve both spatial and histogram information extracted from images; ii) a generic mean of combining multiple layers of heterogeneous data into the proposed binary graph descriptor, coupled with a matching and geometric checking method. We also introduce an implementation of our descriptor into an incremental Bag-of-Words (iBoW) structure that improves efficiency and scalability, and propose a method to interpret Deep Neural Network (DNN) results. This work 31 was done in cooperation with the Mimetic team.
### 7.1.6 Learn Offsets for robust 6DoF object pose estimation
Participants: Mathieu Gonzalez, Eric Marchand.
Estimating the 3D translation and orientation of an object is a challenging task that can be considered within augmented reality or robotic applications. In 16 we propose a novel approach to perform 6 DoF object pose estimation from a single RGB-D image in cluttered scenes. We adopt an hybrid pipeline in two stages: data-driven and geometric respectively. The first data-driven step consists of a classification CNN to estimate the object 2D location in the image from local patches, followed by a regression CNN trained to predict the 3D location of a set of keypoints in the camera coordinate system. We robustly perform local voting to recover the location of each keypoint in the camera coordinate system. To extract the pose information, the geometric step consists in aligning the 3D points in the camera coordinate system with the corresponding 3D points in world coordinate system by minimizing a registration error, thus computing the pose.
### 7.2.1 Trajectory Generation for Minimum Closed-Loop State Sensitivity
Participants: Pascal Brault, Ali Srour, Quentin Delamare, Paolo Robuffo Giordano.
The goal of this research activity is to propose a new point of view in addressing the control of robots under parametric uncertainties: rather than striving to design a sophisticated controller with some robustness guarantees for a specific system, we propose to attain robustness (for any choice of the control action) by suitably shaping the reference motion trajectory so as to minimize the state sensitivity to parameter uncertainty of the resulting closed-loop system.
In 34 we have proposed to couple the previously introduced “state sensitivity’’ metric with an “input sensitivity” metric, which allows us to obtain trajectories that, when perturbed, require minimal change of the control inputs and in the final tracking error. We applied this machinery to the case of a planar quadrotor. An off-the-shelf nonlinear optimization scheme was also employed for allowing us to take into account (nonlinear) input constraints. A large statistical analysis was performed in simulation, showing the effectiveness of the approach in producing intrinsically-robust motion plans. We are now working towards an implementation on a real quadorotor by considering offsets in the center of mass (CoM) as one of the main sources of uncertainty. We are also working on the combination of sensitivity and observability metrics for taking into account both robustness and optimal state estimation when producing motion plans. Finally, we are studying how to formulate an optimization problem that can optimize both the trajectory and the control gains of a familty of controllers for further improving the robustness of the generated trajectory.
### 7.2.2 Comfortable path generation for wheelchair navigation
Participants: Guillaume Vailland, Marie Babel.
In the case of non-holonomic robot navigation, path planning algorithms such as Rapidly-exploring Random Tree (RRT) rarely provides feasible and smooth path without the need of additional processing. Furthermore, in a transport context like power wheelchair navigation, user comfort should be a priority and influence path planning strategy.
We then proposed a local path planner which guarantees curvature bounded value and continuous Cubic Bézier piecewise curves connections. To simulate and test this Cubic Bézier local path planner, we developed a new RRT version (CBB-RRT*) which generates on-the fly comfortable path adapted to non-holonomic constraints 51.
### 7.2.3 UWB beacon navigation of assisted power wheelchair
Participants: Vincent Drevelle, Marie Babel, Eric Marchand, François Pasteau, Merwane Bouri.
Typical problems in robots are those of perception of the environment and localization. Visual sensors are poorly adapted to the context of autonomous wheelchair navigation, both in terms of acceptability (intrusiveness) and in terms of adaptation to the wheelchair and of overall cost.
New sensors, based on Ultra Wide Band (UWB) radio technology, are emerging in particular for indoor localization and object tracking applications. This low-cost system allows for the measurement of distances between fixed beacons and a mobile sensor, in order to obtain localization at decimeter level accuracy in the best case. We seek to exploit these sensors for the navigation of a wheelchair, despite the low accuracy of the measurements they provide.
The problem here lies in the definition of an autonomous or shared sensor based control solution, which fully exploits the notion of measurement uncertainty related to UWB beacons. By modeling the measurements of uncertain distances by intervals, we will try to propagate these uncertainties to the calculation of the speeds to be applied to the wheelchair. This will be done by using the methods of set inversion and constraint propagation, which lead to the characterization of solutions in the form of sets.
### 7.2.4 Visual Servoing for Cable-Driven Parallel Robots
Participant: François Chaumette.
This study was done in collaboration with IRT Jules Verne (Zane Zake, Nicolo Pedemonte) and LS2N (Stéphane Caro) in Nantes (see Section 8.2). It was devoted to the analysis of the robustness of visual servoing to modeling and calibration errors for cable-driven parallel robots. Zane Zake defended her Phd in February and her previous works on pose estimation, control workspace, and tension management have been published this year 555456.
### 7.2.5 Singularities in visual servoing
Participant: François Chaumette.
This study is done in the scope of the ANR Sesame project (see Section 9.3).
We have performed a complete theoretical study about the singularities of image-based visual servoing and pose estimation (PnP problem) from the observation of four image points. Highly original results have been exhibited. In particular, it was shown that 2 to 6 camera positions correspond to singularities for a general configuration of 4 non-coplanar points, while it was wrongly believed before that no singularities occur for such configuration 4
### 7.2.6 Multi-sensor-based control for accurate and safe assembly
Participants: John Thomas, François Chaumette.
This study is done in the scope of the BPI Lichie project 9.3. Its goal is to design sensor-based control strategies coupling vision and proximetry data for ensuring precise positioning while avoiding obstacles in dense environements. The targeted application is the assembly of satellite parts.
### 7.2.7 Visual servo of a satellite constellation
Participants: Maxime Robic, Eric Marchand, François Chaumette.
This study is also done in the scope of the BPI Lichie project 9.3. Its goal is to control the orientation of a satellite constellation from a camera mounted on each of them to track particular objects on the ground. We study new control law compatible with the control of the satellites.
### 7.2.8 Visual Exploration of an Indoor Environment
Participants: Thibault Noël, Eric Marchand, François Chaumette.
This study is done in collaboration with the Creative company in Rennes (see Section 7.2.8) It is devoted to the exploration of indoor environments by a mobile robot, Pepper typically (see Section 6.2.2) for a complete and accurate reconstruction of the environment.
### 7.2.9 Model-Based Deformation Servoing of Soft Objects
Participants: Fouad Makiyeh, Alexandre Krupa, Maud Marchal, François Chaumette.
This study takes place in the context of the GentleMAN project (see Section 9.1.3). The objective is to elaborate a new visual servoing approach aiming to control the shape of an object towards a desired deformation. This year we developed a new control approach that relies on a coarse model of the soft object to be manipulated. This model is composed by a 3D mesh and we chose to represent the mechanical behavior of the object using a Mass-Spring model because it provides real-time capability. We derived an analytical expression of a new controller that allows us to indirectly move any feature point of the soft object to a desired 3D position by acting with the end-effector of a robot on a distant manipulated point. This controller was implemented in an eye-to-hand visual servoing scheme using a RGB-D camera and the approach was tested on several soft objects with different geometries and materials. The experimental results demonstrated that this approach can accurately position a feature point belonging to a soft object to a desired 3D location even if it is based on a model that approximates the physical behavior of the real object. This work has been recently submitted to the special isssue of IEEE RA-L devoted to robotic handling of deformable objects.
### 7.2.10 Manipulation of a deformable wire by two UAVs
Participants: Lev Smolentsev, Alexande Krupa, François Chaumette.
This study takes place in the context of the CominLabs MAMBO project (see Section 9.4). Its main objective is the development of a visual-based control framework for performing autonomous manipulation of a deformable wire attached between two UAVs using data provided by onboard RGB-D cameras. Toward this direction, we have developed a visual servoing approach that considers as visual features the coefficients of a parabolic curve representing the shape of the wire and we analytically derived the interaction matrix that relates the variations of this features to the RGB-D camera displacement. Preliminary results obtained from experiments using an eye-to-hand RGB-D camera observing a wire with one extremity attached to a 6-DOF robotic arm validated the modelling and the design of a control law that automatically positions the wire to a desired shape configuration.
### 7.2.11 Coupling Force and Vision for Controlling Robot Manipulators
Participants: Alexander Oliva, François Chaumette, Paolo Robuffo Giordano.
The goal of this activity is about coupling visual and force information for advanced manipulation tasks. To this end, we are exploiting the recently acquired Panda robot (see Sect. 6.2.4), a state-of-the-art 7-dof manipulator arm with torque sensing in the joints, and the possibility to command torques at the joints or forces at the end-effector. The use of vision in torque-controlled robot is limited because of many issues, among which the difficulty of fusing low-rate images (about 30 Hz) with high-rate torque commands (about 1 kHz), the delays caused by any image processing and tracking algorithms, and the unavoidable occlusions that arise when the end-effector needs to approach an object to be grasped.
In this context we have proposed a general framework for combining force and visual information directly in the visual feature space, by reformulating and unifying the classical admittance control law in the image space. The proposed visual/force control framework has been extensively evaluated via numerous experiments performed on the Panda robot in peg-in-hole tasks where both the pose and the exchanged forces could be regulated with high accuracy and good stability 26. We have recently considered the case of visual/force control for moving targets by exploiting a Kalman filter that can estimate the target state and provide this information to the control loop. In order to fast prototyping the developments on these activities, we have also developed a realistic dynamic simulator of the Franka robot called “FrankaSim’’ that has been publicly released.
### 7.2.12 End-to-end deep visual servoing
Participants: Eric Marchand, Samuel Felton.
We proposed a deep architecture and the associated learning strategy for end-to-end direct visual servoing. The considered approach allows to sequentially predict, in $se\left(3\right)$, the velocity of a camera mounted on the robot’s end-effector for positioning tasks. Positioning is achieved with high precision despite large initial errors in both cartesian and image spaces. Training is fully done in simulation, alleviating the burden of data collection. We demonstrate the efficiency of our method in experiments in both simulated and real-world environments. We also show that the proposed approach is able to handle multiple scenes. This work 40 is done in collaboration with the Lacodam team.
We also proposed a new framework to perform VS in the latent space learned by a convolutional autoencoder. We show that this latent space avoids explicit feature extraction and tracking issues and provides a good representation, smoothing the cost function of the VS process. Besides, our experiments show that this unsupervised learning approach allows us to obtain, without labelling cost, an accurate end-positioning, often on par with the best DVS methods in terms of accuracy but with a larger convergence area. This work 15 is done in collaboration with the Lacodam team.
## 7.3 Haptic Cueing for Robotic Applications
### 7.3.1 Wearable Haptics Systems Design
Participants: Claudio Pacchierotti, Maud Marchal, Thomas Howard, Xavier de Tinguy.
We have been working on wearable haptics since few years now, both from the hardware (design of interfaces) and software (rendering and interaction techniques) points of view.
In 3, we presents an approach for automatically adapting the hardware design of a wearable haptic interface for a given user. We analyze the performance of a 3-DoF fingertip cutaneous device as a function of its main geometrical dimensions. Then, starting from the user's fingertip characteristics, we define a numerical procedure that best adapts the dimension of the device to (i) maximize the range of renderable haptic stimuli, (ii) avoid unwanted contacts between the device and the skin, (iii) avoid singular configurations, and (iv) minimize the device encumbrance and weight. Together with the mechanical analysis and evaluation of the adapted design, we present a MATLAB script that calculates the device dimensions customized for a target fingertip as well as an online CAD utility for generating a ready-to-print STL file of the personalized design.
One of the main issues when designing haptic systems for the fingertip is their tracking, especially when interacting with tangible/real objects at the same time. In this respect, in 14, we combined tracking information from a tangible object instrumented with capacitive sensors and an optical tracking system, to improve contact rendering when interacting with tangibles in Virtual Reality. Combining capacitive sensing with optical tracking significantly improves the visuohaptic synchronization and immersion of the experience, which is promising for haptic-enabled interaction with tangible environments.
Finally, in the framework of H2020 project TACTILITY, we are working on designing interaction and rendering techniques for wearable electrotactile interfaces. In this respect, we proposed the use of electrotactile feedback to render the interpenetration distance between the user's finger and the virtual content that is touched 52. The approach consists of modulating the perceived intensity (frequency and pulse width modulation) of the electrotactile stimuli according to the registered interpenetration distance. We assessed the performance of four different interpenetration feedback approaches: electrotactile-only, visual-only, electrotactile and visual, and no interpenetration feedback. Results suggest that electrotactile feedback could be an efficient replacement of visual feedback for enhancing contact information in virtual reality avoiding the need of active visual focus and the rendering of additional visual artefacts.
### 7.3.2 Mid-Air Haptic Feedback
Participants: Claudio Pacchierotti, Thomas Howard, Guillaume Gicquel, Maud Marchal.
In the framework of H2020 projects H-Reality and E-TEXTURE, we are working to develop novel mid-air haptics paradigms that can convey the information spectrum of touch sensations in the real world, motivating the need to develop new, natural interaction techniques.
In 45, we developed an open-source framework to aid in designing mid-air stimuli, named DOLPHIN. It allows to the study of the impact of rendering parameters on perceived stimulus properties. This platform-agnostic framework standardizes stimulus descriptions as a step toward more replicability and easier communication in the field. It enables reproduction of stimuli between perceptual experiments and ensures that stimuli used in applications correspond to those evaluated in prior perceptual studies.
In 62, we kept studying the perceptual aspects of ultrasound haptic stimulation, investigating the influence of the rendering sampling strategies on a user's ability to differentiate arc curvatures.
### 7.3.3 Encountered-Type Haptic Devices
Participants: Maud Marchal, Thomas Howard.
Encountered-Type Haptic Displays (ETHDs) provide haptic feedback by positioning a tangible surface for the user to encounter. This allows users to freely elicit haptic feedback with a surface during a virtual simulation. ETHDs differ from most of current haptic devices which rely on an actuator always in contact with the user. In 23, we intend to describe and analyze the different research efforts carried out in this field. In addition, we analyze ETHD literature concerning definitions, history, hardware, haptic perception processes involved, interactions and applications. The paper proposes a formal definition of ETHDs, a taxonomy for classifying hardware types, and an analysis of haptic feedback used in literature. Taken together the overview of this survey intends to encourage future work in the ETHD field.
In 22, we propose an example of ETHD with an approach towards an infinite surface haptic display. Our approach, named ENcountered-Type ROtating Prop Approach (ENTROPiA) is based on a cylindrical spinning prop attached to a robot's end-effector serving as an ETHD. This type of haptic display allows the users to have an unconstrained, free-hand contact with a surface being provided by a robotic device for the users to encounter a surface to be touched. In our approach, the sensation of touching a virtual surface is given by an interaction technique that couples with the sliding movement of the prop under the users' finger by tracking their hand location and establishing a path to be explored. This approach enables large motion for a larger surface rendering, permits rendering multi-textured haptic feedback, and leverages the ETHD approach introducing large motion and sliding/friction sensations. As a part of our contribution, a proof of concept was designed for illustrating our approach. A user study was conducted to assess the perception of our approach showing a significant performance for rendering the sensation of touching a large flat surface. Our approach could be used to render large haptic surfaces in applications such as rapid prototyping for automobile design.
In 44, we propose a novel haptic paradigm for object manipulation in 3D immersive VR. It uses a robotic manipulator to move tangible objects in its workspace such that they match the pose of virtual objects to be interacted with. Users can then naturally touch, grasp and manipulate a virtual object while feeling congruent and realistic haptic feedback from the tangible proxy. The tangible proxies can detach from the robot, allowing natural and unconstrained manipulation in the 3D virtual environment. When a manipulated virtual object comes into contact with the virtual environment, the robotic manipulator acts as an encounter-type haptic display, positioning itself so as to render reaction forces of the environment onto the manipulated physical object.
### 7.3.4 Multimodal Cutaneous Haptics to Assist Navigation
Participants: Louise Devigne, Marco Aggravi, Inès Lacôte, Pierre-Antoine Cabaret, François Pasteau, Maud Marchal, Claudio Pacchierotti, Marie Babel.
Within the project Inria Challenge DORNELL, we got interested on using cutaneous haptics for aiding the navigation of people with sensory disabilities. In particular, we investigated the ability of vibrotactile sensations and tap stimulations in conveying haptic motion illusions 43 in a handle-like device. We also evaluated the capability of vibrotactile feedback for rendering spatialized impacts with external (virtual) objects.
## 7.4 Shared Control Architectures
### 7.4.1 Shared Control for Remote Manipulation
Participants: Paolo Robuffo Giordano, Claudio Pacchierotti, Rahaf Rahal, Raul Fernandez Fernandez.
As teleoperation systems become more sophisticated and flexible, the environments and applications where they can be employed become less structured and predictable. This desirable evolution toward more challenging robotic tasks requires an increasing degree of training, skills, and concentration from the human operator. In this respect, shared control algorithms have been investigated as one of the main tools to design complex but intuitive robotic teleoperation systems, helping operators in carrying out several increasingly difficult robotic applications such as assisted vehicle navigation, surgical robotics, brain-computer interface manipulation, rehabilitation. Indeed, this approach makes it possible to share the available degrees of freedom of the robotic system between the operator and an autonomous controller.
Along this general line of research, during this year we gave the following contributions:
• in 24 we presented an adaptive impedance control architecture for robotic teleoperation of contact tasks featuring continuous interaction with the environment. We used Learning from Demonstration (LfD) as a framework to learn variable stiffness control policies. Then, the learnt state-varying stiffness was used to command the remote manipulator, so as to adapt its interaction with the environment based on the sensed forces. The proposed system only relies on the on-board torque sensors of a commercial robotic manipulator and it does not require any additional hardware or user input for the estimation of the required stiffness. We also provide a passivity analysis of our system, where the concept of energy tanks is used to guarantee a stable behavior. Finally, the system was evaluated in a representative teleoperated cutting application. Results showed that the proposed variable-stiffness approach outperforms two standard constant-stiffness approaches in terms of safety and robot tracking performance.
• in 19 we focused on robotic manipulation of fragile, compliant objects, such as food items. In particular we developed a haptic-based, Learning from Demonstration (LfD) policy that enables pre-trained autonomous grasping of food items using an anthropomorphic robotic system. The policy combines data from teleoperation and direct human manipulation of objects, embodying human intent and interaction areas of significance. We evaluated the proposed solution against a recent state-of-the-art LfD policy as well as against two standard impedance controller techniques. The results showed that the proposed policy performs significantly better than the other considered techniques, leading to high grasping success rates while guaranteeing the integrity of the food at hand.
• in 5 we have proposed a shared control for robot manipulators transporting an object on a tray. Differently from many existing studies about remotely operated robots with firm grasping capabilities, we considered the case in which, in principle, the object can break its contact with the robot end-effector. The proposed shared-control approach automatically regulates the remote robot motion, commanded by the user, and the end-effector orientation to prevent the object from sliding over the tray. Furthermore, the human operator is provided with haptic cues informing about the discrepancy between the commanded and executed robot motion, assisting the operator throughout the task execution. We carried out several experiments and user's studies employing a 7-DOF torque-controlled manipulator. In all experiments, the results clearly show that our control approach outperforms the other solutions in terms of sliding prevention, robustness, commands tracking, and user's preference.
### 7.4.2 Shared Control for Multiple Robots
Participants: Marco Aggravi, Paolo Robuffo Giordano, Claudio Pacchierotti.
Following our previous works on flexible formation control of multiple robots with global requirements, in particular connectivity maintenance, in 9, 7, 1 we have instead presented a decentralized haptic-enabled connectivity-maintenance control framework for heterogeneous human-robot teams. The proposed framework controls the coordinated motion of a team consisting of mobile robots and one human, for collaboratively achieving various exploration and SAR tasks. The human user physically becomes part of the team, moving in the same environment than the robots, while receiving rich haptic feedback about the team connectivity and the direction toward a safe path. We carried out two human subjects studies, both in simulated and real environments. The results showed that the proposed approach is effective and viable in a wide range of SAR scenarios. Moreover, providing haptic feedback showed increased performance with respect to providing visual information only. Finally, conveying distinct feedback regarding the team connectivity and the path to follow performed better than providing the same information combined together.
### 7.4.3 Shared Control of Flexible Needles
Participant: Marco Aggravi, Claudio Pacchierotti, Alexandre Krupa.
We proposed a shared-control strategy where the user is only in charge of teleoperating directly and intuitively in the 3D ultrasound image the needle tip desired position via the use of a haptic interface. In this approach, an autonomous "low level" controller based on visual sevoing using 3D ultrasound images is in charge of handling the complexity of the 6-DOF motion that needs to be applied to the needle base in such a way to reach the desired needle tip position. We also proposed in this shared-control strategy to assist the user 3D navigation through kinesthetic stimulation by increasing the stiffness of the haptic device in the direction that is orthogonal to the one that points to the anatomical target and provide to the user a feedback on the needle tip cutting force. This force is obtained by subtracting an estimate of the friction force acting along the needle shaft from the total force that is measured at the base of the needle by a force sensor 8. In order to obtain a real-time estimate, we proposed a method that relies on the deformation of the needle shaft that is automatically tracked in 3D ultrasound. We then provided to the user both stimulation for 3D navigation assistance toward the target and the cutting force feedback by using the grounded haptic interface and a wearable cutaneous interface on the user forearm. We carried out a human subject study to validate the insertion system in a gelatine phantom and compare seven different feedback techniques. The best performance was registered when providing navigation cues through kinesthetic feedback and needle tip cutting force through cutaneous vibrotactile feedback.
### 7.4.4 Shared Control of a Wheelchair for Navigation Assistance
Participants: Louise Devigne, François Pasteau, Marie Babel.
Power wheelchairs allow people with motor disabilities to have more mobility and independence. In order to improve the access to mobility for people with disabilities, we previously designed a semi-autonomous assistive wheelchair system which progressively corrects the trajectory as the user manually drives the wheelchair and smoothly avoids obstacles.
Despite the COVID situation, INSA and the rehabilitation center of Pôle Saint Hélier managed to co-organize clinical trials in July 2021 at INSA and in September 2021 at Pôle Saint Hélier. Based on the previous trial results 2, the objective was to evaluate the clinical benefit of a driving assistance for people with disabilities experiencing high difficulties while steering a wheelchair. 18 people participated to the trials. We clearly confirmed the excellent ability of the system to assist users and the relevant usage of such an assistive technology.
In addition, in collaboration with MIS laboratory (Fabio Morbidi, Guillame Caron), we evaluated the use of additional visual sensors such as spherical vision to enhance the navigation experience and situation awareness by providing adequate feedback to the user 37. The idea is to generate an augmented view of the surrounding environment, presented to the user on a display. We conducted user trial at INSA in July 2021 with able-bodied subjects and older adults with mobility impairments. Our field results indicate that SpheriCol is effective in improving safety and situational awareness, and in supporting a driver's decision during challenging but prevalent maneuvers, such as reversing into an elevator or corridor centering.
Finally, driving safely such a vehicle is a daily challenge particularly in urban environments while navigating on sidewalks, negotiating curbs or dealing with uneven grounds. Indeed, differences of elevation have been reported to be one of the most challenging environmental barrier to negotiate, with tipping and falling being the most common accidents power wheelchair users encounter. To this aim, we proposed a shared-control algorithm which provides assistance while navigating with a wheelchair in an environment consisting of negative obstacles. We designed a dedicated sensor-based control law allowing trajectory correction while approaching negative obstacles e.g. steps, curbs, descending slopes. This shared control method takes into account the humanin-the loop factor. We are currently preparing clinical trials and ethics committee (Comité de Protection des Personnes) procedures to evaluate the clinical benefit of it.
### 7.4.5 Multisensory power wheelchair simulator
Participants: Guillaume Vailland, Louise Devigne, François Pasteau, Marie Babel.
Power wheelchairs are one of the main solutions for people with reduced mobility to maintain or regain autonomy and a comfortable and fulfilling life. However, driving a power wheelchair in a safe way is a difficult task that often requires training methods based on real-life situations. Although these methods are widely used in occupational therapy, they are often too complex to implement and unsuitable for some people with major difficulties.
In this context, we collaborated with clinicians to develop a Virtual Reality based power wheelchair simulator. This simulator is an innovative training tool adapted to any type of situations and impairments 50. It relies on a modular and versatile workflow enabling not only easy interfacing with any virtual display, but also with any user interface such as wheelchair controllers or feedback devices. A clinical trial has been conducted in May 2021 and October 2021 in which 26 power wheelchair regular users were asked to complete a clinically validated task designed by clinicians within four display conditions: using the HTC Vive Pro HMD, Immersia immersive room or a screen (with or without haptic and vestibular feedback). The objective of this study was to compare performances between the four conditions and to evaluate the Quality of Experience. First analyses clearly show that immersive conditions allow high driving performances to be achieved.
### 7.4.6 Integrating social interaction in a VR powered wheelchair driving simulator
Participants: Emilie Leblong, Antoine Cellier, Marie Babel.
Navigating in the city while driving a powered wheelchair, in a complex and dynamic environment made of various interactions with other humans, can be challenging for a person with disabilities. Learning how to drive a powered wheelchair remains then a major issue for the clinical teams prescribing these technical mobility aids. The work carried out as part of the Interreg ADAPT project has made it possible to design a powered wheelchair simulator in VR. This work was done in cooperation with Anne-Hélène Olivier (MimeTIC team) and Valérie Gouranton (Hybrid team).
To promote the transfer of skills from virtual to real, the use of such a platform requires the deployment of environmentally friendly interactive populated virtual environments. These are currently empty of any pedestrians, even though the question of social interaction in the framework of an inclusive urban mobility is fundamental.
The objective is to better understand how pedestrians and powered wheelchair users interact. Especially, this study will aim to characterize the personal space from both the perspective of the pedestrian and the powered wheelchair driver in a laboratory setting.
The second objective is to use these new models of interaction to improve dynamic virtual environments by including virtual humans that faithfully reproduce the behaviors modeled in terms of the simulator user's reaction in a handicap situation.
Finally, the third objective is to evaluate the fidelity of this new generation of wheelchair simulator by comparing the resulting interactions with the ones previously observed in real conditions. In particular, we will consider the perception of the risk of collision as well as the benefit of learning to drive on the simulator via clinical studies.
## 7.5 Crowd Simulation for Robotics
### 7.5.1 Perceptual evaluation of crowd simulations
Participants: Julien Pettré.
This topic has been addressed in collaboration with the MimeTIC team, and more especially with Ludovic Hoyet and Anne-Hélène Olivier who brought expertise on the perceptual evaluation of graphics content.
Simulating crowds requires controlling a very large number of trajectories and is usually performed using crowd motion algorithms for which appropriate parameter values need to be found. The study of the relation between parametric values for simulation techniques and the quality of the resulting trajectories has been studied either through perceptual experiments or by comparison with real crowd trajectories. In the work presented in 13, we integrate both strategies. A quality metric, QF, is proposed to abstract from reference data while capturing the most salient features that affect the perception of trajectory realism. QF weights and combines cost functions that are based on several individual, local and global properties of trajectories. These trajectory features are selected from the literature and from interviews with experts. To validate the capacity of QF to capture perceived trajectory quality, we conducted an online experiment that demonstrates the high agreement between the automatic quality score and non-expert users. To further demonstrate the usefulness of QF, we used it in a data-free parameter tuning application able to tune any parametric microscopic crowd simulation model that outputs independent trajectories for characters. The learnt parameters for the tuned crowd motion model maintain the influence of the reference data which was used to weight the terms of QF.
Our perceptual evaluation not only concerned the global trajctories as generated by simulators, but also the motion of characters resulting from an animation layer , as reported in 33. The more diverse the characters and their behaviors are, the more realistic the virtual crowd is expected to be perceived. Hence, creating virtual crowds is a trade-off between the cost associated with acquiring more diverse assets, namely more virtual characters with their animations, and achieving better realism. In this paper, our focus is on the perceived variety in virtual crowd character motions. We present an experiment exploring whether observers are able to identify virtual crowds including motion clones in the case of large-scale crowds (from 250 to 1000 characters). As it is not possible to acquire individual motions for such numbers of characters, we rely on a state-of-the-art motion variation approach to synthesize unique variations of existing examples for each character in the crowd. Participants then compared pairs of videos, where each character was animated either with a unique motion or using a subset of these motions. Our results show that virtual crowds with more than two motions (one per gender) were perceptually equivalent, regardless of their size. We believe these findings can help create efficient crowd applications, and are an additional step into a broader understanding of the perception of motion variety.
As we show that the motion of virtual characters can play a great role in immersive simulation, we started a research work on visually-driven animation techniques, the first results of which are presented in 47.
### 7.5.2 Influence of path curvature on collision avoidance behaviour between two walkers
Participants: Julien Pettré.
This topic has been addressed in collaboration with the MimeTIC team, and more especially with Richard Kulpa and Anne-Hélène Olivier who directed the thesis of S. Lynch, main contributor of this work.
Navigating crowded community spaces requires interactions with pedestrians that follow rectilinear and curvilinear trajectories. In the case of rectilinear trajectories, it has been shown that the perceived action opportunities of the walkers might be afforded based on a future distance of closest approach. However, little is known about collision avoidance behaviours when avoiding walkers that follow curvilinear trajectories. In this work, presented in 20 twenty-two participants were immersed in a virtual environment and avoided a virtual human (VH) that followed either a rectilinear path or a curvilinear path with a 5 m or 10 m radius curve at various distances of closest approach. Compared to a rectilinear path (control condition), the curvilinear path with a 5 m radius yielded more collisions when the VH approached from behind the participant and more inversions when the VH approached from in-front. During each trial, the evolution of the future distance of closest approach showed similarities between rectilinear paths and curvilinear paths with a 10 m radius curve. Overall, with few collisions and few inversions of crossing order, we can conclude that participants were capable of predicting future distance of closest approach of virtual walkers that followed curvilinear trajectories. The task was solved with similar avoidance adaptations to those observed for rectilinear interactions. These findings should inform future endeavors to further understand collision avoidance strategies and the role of—for example—non-constant velocities.
### 7.5.3 SPH crowds: Agent-based crowd simulation up to extreme densities using fluid dynamics
Participants: Julien Pettré, Wouter van Toll, Cedric Braga, Thomas Chatagnon.
In highly dense crowds of humans, collisions between people occur often. It is common to simulate such a crowd as one fluid-like entity (macroscopic), and not as a set of individuals (microscopic, agent-based). Agent-based simulations are preferred for lower densities because they preserve the properties of individual people. However, their collision handling is too simplistic for extreme-density crowds. Therefore, neither paradigm is ideal for all possible densities. In this paper 29, we combine agent-based crowd simulation with Smoothed Particle Hydrodynamics (SPH), a particle-based method that is popular for fluid simulation. We integrate SPH into the crowd simulation loop by treating each agent as a fluid particle. The forces of SPH (for pressure and viscosity) then augment the usual navigation behavior and contact forces per agent. We extend the standard SPH model with a dynamic rest density per particle, which intuitively controls the crowd density that an agent is willing to accept. We also present a simple way to let agents blend between individual navigation and fluid-like interactions depending on the SPH density. Experiments show that SPH improves agent-based simulation in several ways: better stability at high densities, more intuitive control over the crowd density, and easier replication of wave-propagation effects. Also, density-based blending between collision avoidance and SPH improves the simulation of mixed-density scenarios. Our implementation can simulate tens of thousands of agents in real-time. As such, this work successfully prepares the agent-based paradigm for crowd simulation at all densities.
This effort to design simulators better suited to dense crowd simulation has been accompanied by an experimental effort to better understand how humans behave under crowd pressure. Our study protocol is presented here 12. We also took advantage of working on the topic to write a state-of-the-art report presenting the recent evolutions of crowd simulation. 30
### 7.5.4 Studying factors influencing users' perception and action for Virtual Steering Navigation
Participant: Maud Marchal.
Virtual steering techniques enable users to navigate in larger Virtual Environments (VEs) than the physical workspace available. Even though these techniques do not require physical movement of the users (e.g. using a joystick and the head orientation to steer towards a virtual direction), recent work observed that users might unintentionally move in the physical workspace while navigating, resulting in Unintended Positional Drift (UPD). This phenomenon can be a safety issue since users may unintentionally reach the physical boundaries of the workspace while using a steering technique. In this context, as a necessary first step to improve the design of navigation techniques minimizing the UPD, we propose to analyse and model the UPD during a virtual navigation task in 11. In particular, we characterize and analyze the UPD for a dataset containing the positions and orientations of eighteen users performing a virtual slalom task using virtual steering techniques. We analyzed the performed motions and proposed two UPD models: the first based on a linear regression analysis and the second based on a Gaussian Mixture Model (GMM) analysis. Then, we assessed both models through a simulation-based evaluation where we reproduced the same navigation task using virtual agents. Our results indicate the feasibility of using simulation-based evaluations to study UPD.
Rotation gains in Virtual Reality (VR) enable the exploration of wider VEs compared to the workspace users have in VR setups. The perception of these gains has been consequently explored through multiple experimental conditions in order to improve redirected navigation techniques. While most of the studies consider rotations, in which participants can rotate at the pace they desire but without translational motion, we have no information about the potential impact of the translational and rotational motions on the perception of rotation gains. In 35, we estimated the influence of these motions and compared the perceptual thresholds of rotations gains through a user study (= 14), in which participants had to perform virtual rotation tasks at a constant rotation speed. The main results are that the rotation gains are less perceivable at lower rotation speeds and that translational motion makes detection more difficult at lower rotation speeds. Furthermore, the paper provides insights into the user's gaze and body motions behaviour when exposed to rotation gains. These results contribute to the understanding of the perception of rotation gains in VEs.
### 7.5.5 From HRI to CRI: Crowd Robot Interaction—Understanding the Effect of Robots on Crowd Motion
Participant: Julien Pettré, Javad Amirian, Fabien Grzeskowiak.
The results reported here have been obtained in close collaboration with partners of the CrowdBot project, including T. Carlson from UCL, R. Siegwart from ETHZ, A. Billard from EPFL. Around this topic, we first empirically studied the navigation of robots among crowds, and reported on simulation tools to evaluate robots' crowd navigation capabilities. We also explored simulation environments for broade kind of interactions 46. 2 thesis resulted from the project: Javad Amirian's 58 and Fabien Grzeskowiak's 59.
How does the presence of a robot affect pedestrians and crowd dynamics, and does this influence vary across robot types ? In this paper, we took the first step towards answering this question by performing a crowd-robot gate-crossing experiment presented in 32. The study involved 28 participants and two distinct robot representatives: A smart wheelchair and a Pepper humanoid robot. Collected data includes: video recordings; robot and participant trajectories; and participants’ responses to post-interaction questionnaires. Quantitative analysis on the trajectories suggests the robot affects crowd dynamics in terms of trajectory regularity and interaction complexity. Qualitative results indicate that pedestrians tend to be more conservative and follow “social rules” while passing a wheelchair compared to a humanoid robot. These insights can be used to design a social navigation strategy that allows more natural interaction by considering the robot effect on the crowd dynamics.
To further evaluate robots' crowd navigation capabilities, we designed the simulation-based challenge presented in 41. The evaluation of robot capabilities to navigate human crowds is essential to conceive new robots intended to operate in public spaces. This paper initiates the development of a benchmark tool to evaluate such capabilities; our long term vision is to provide the community with a simulation tool that generates virtual crowded environment to test robots, to establish standard scenarios and metrics to evaluate navigation techniques in terms of safety and efficiency, and thus, to install new methods to benchmarking robots' crowd navigation capabilities. This work explores the architecture of the simulation tools, introduces first scenarios and evaluation metrics, as well as early results to demonstrate that our solution is relevant to be used as a benchmark tool. B
### 7.5.6 Tracking Pedestrian Heads in Dense Crowd
Participant: Julien Pettré, Eric Marchand, Ramana Sundararaman.
# 8 Bilateral contracts and grants with industry
## 8.1 Bilateral contracts with industry
#### IRT JV Happy
Participant: François Chaumette.
No Inria Rennes 13521, duration: 36 months.
This project ended in June 2021. It was managed by IRT Jules Verne and achieved in collaboration with LS2N, the ACsSysteme company and Airbus. Its goal was to develop local sensor-based control methods for the assembly of large parts of aircrafts.
#### Airbus React
Participants: Julien Dufour, Fabien Spindler, François Chaumette.
No Inria Rennes 16165, duration: 12 months.
This project started in September 2021. It is in collaboration with Laas in Toulouse for Airbus. Its goal is to develop a vision-based localization system so that a robot arm is able to point accurately on an industrial piece.
## 8.2 Bilateral grants with industry
#### Creative
Participants: Thibault Noël, François Chaumette, Eric Marchand.
No Inria Rennes 15737, duration: 8 months.
This project funded by Creative started in February 2021. It supported Thibault Noël's salary before the agreement of a Cifre grant for his Ph.D. about visual exploration (see Section 7.2.8).
#### IRT JV Perform
Participant: François Chaumette.
No Inria Rennes 14049, duration: 38 months.
This project funded by IRT Jules Verne in Nantes ended in February 2021. It was achieved in cooperation with Stéphane Caro from LS2N in Nantes to support Zane Zake's Ph.D. about visual servoing of cable-driven parallel robots (see Section 7.2.4).
#### MX
Participants: François Pasteau, Marie Babel.
INSA Rennes, duration: July 2020 - December 2021.
This contract with MX (Acigné) aims to define an online load estimation in order to servo truck loaders.
#### Sopra-Steria
Participants: François Pasteau, Marie Babel.
INSA Rennes, duration: 12 months. This project funded by Sopra Steria aimed to design a smart rollator equiped with haptic feedback.
# 9 Partnerships and cooperations
## 9.1 International initiatives
### 9.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program
#### ISI4NAVE
Participants: Marie Babel, Claudio Pacchierotti, Louise Devigne, François Pasteau.
• Title:
• Duration:
2016 -> 2023
• Coordinator:
Marie Babel (marie.babel@irisa.fr)
• Partners:
• University College London
• Inria contact:
Marie Babel
• Summary:
This team aims at developing adapted interfaces that should improve the understanding of people who suffer from cognitive and/or visual impairments. It focuses on two main complementary objectives: (i) compensate both sensorimotor disabilities and cognitive impairments by designing innovative and adapted interfaces; (ii) enhance the driving experience and to bring a new tool for rehabilitation purposes by defining efficient physical human-robot interaction.
### 9.1.2 Inria associate team not involved in an IIL or an international program
#### FRANTIC
Participants: Claudio Pacchierotti, Paolo Robuffo Giordano, Nicola De Carli, Fabien Spindler.
• Title:
French-Russian Advanced and Novel TactIle Cyberworlds
• Duration:
2021 -> 2023
• Coordinator:
Claudio Pacchierotti (claudio.pacchierotti@irisa.fr)
• Partners:
• Skolkovo institute of science and technology (Skoltech)
• Inria contact:
Claudio Pacchierotti
• Summary:
Ubiquitous haptic interfaces enable interaction with a virtual or augmented reality system while freely exploring the environment, unimpaired and unconstrained. The objectives of the team are to study and develop an innovative set of perceptually-motivated haptic interfaces for immersive interaction in XR and telerobotics, collaborating with Skoltech Space in the fields of haptics and human-robot interaction.
### 9.1.3 Participation in other International Programs
#### GentleMAN
Participants: Fouad Makiyeh, Alexandre Krupa, François Chaumette, Fabien Spindler.
• Title:
Gentle and Advanced Robotic Manipulation of 3D Compliant Objects
• Duration:
August 2019 - December 2023
• Coordinator:
Sintef Ocean (Norway)
• Partners:
• Sintef Ocean (Norway)
• NTNU (Norway)
• NMBU (Norway)
• MIT (USA)
• QUT (Australia)
• Inria contact:
Alexandre Krupa
• Summary:
This project is granted by The Research Council of Norway. Its main objective is to develop a novel learning framework that uses visual, force and tactile sensing to develop new multi-modal learning models to enable robots to learn new and advanced skills for the manipulation of 3D compliant objects. The Rainbow group is involved in the elaboration of new approaches for visual tracking of deformable objects, active vision perception and visual servoing for deforming soft objects into desired shape (see Section 7.2.9).
#### BIFROST
Participants: Alexandre Krupa, François Chaumette, Fabien Spindler.
• Title:
A Visual-Tactile Perception and Control Framework for Advanced Manipulation of 3D Compliant Objects
• Duration:
July 2021 - December 2025
• Coordinator:
Sintef Ocean (Norway)
• Partners:
• Sintef Ocean (Norway)
• MIT (USA)
• Inria contact:
Alexandre Krupa
• Summary:
This project is granted by The Research Council of Norway. Its main objective is to develop a visual-tactile perception and Control framework for advanced manipulation of 3D compliant objects. The Rainbow group is in charge of elaborating novel visual servoing approaches fusing visual and tactile feedback for dexterous manipulation of soft objects.
#### Study on the distributed control of Heterogeneous Human-Robot Teams with tactile feedback for Collaborative Exploration and Patrolling
Participants: Claudio Pacchierotti, Paolo Robuffo Giordano.
• Title:
Study on the distributed control of Heterogeneous Human-Robot Teams with tactile feedback for Collaborative Exploration and Patrolling
• Duration:
January 2021 - December 2022
• Coordinator:
IRISA/Inria Rennes
• Partners:
• Skolkovo Institute of Science and Technology (Russia)
• Inria contact:
Claudio Pacchierotti
• Summary:
This project is jointly granted by the CNRS and the RFBR (Russian Foundation for Basic Research) as a CNRS International Emerging Action (IEA). It proposes a paradigm for the natural control of a heterogeneous team composed of humans and robots (grounded and aerial). One or more human users move in the same environment as the robotic team, directing their coordinated motion. Each unit in the team shares information with its neighbors, processes it in a distributed way, and carries out its given tasks (e.g., complex environment exploration). The project will advance research in the theory of distributed, shared, and multi-robot control, wearable haptics, and aerial field robotics.
## 9.2 European initiatives
### 9.2.1 FP7 & H2020 projects
#### PRESENT
Participants: Claudio Pacchierotti, Julien Pettré, Alberto Jovane, Adèle Colas.
• Title:
Photoreal REaltime Sentient ENTity
• Duration:
September 2019 - August 2022
• Coordinator:
• Partners:
• BRAINSTORMMULTIMEDIA SL (Spain)
• CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS (France)
• CREATIVEWORKERS-CREATIEVE WERKERS VZW (Belgium)
• ETUITUS SRL (Italy)
• INFOCERT SPA (Italy)
• THE FRAMESTORE LIMITED (U.K.)
• UNIVERSITAET AUGSBURG (Germany)
• UNIVERSITE RENNES II (France)
• Inria contact:
Julien Pettré
• Summary:
Our relationship with virtual entities is deepening. Already, we are using technologies like Siri, Alexa and Google Assistant to aid in day-to-day tasks. The EU-funded PRESENT project will develop a virtual digital companion, which will not only sound human but also look natural, demonstrate emotional sensitivity, and establish meaningful dialogue. Advances in photorealistic computer-generated characters, combined with emotion recognition and behaviour, and natural language technologies, will allow these virtual agents to not only look realistic but respond like a human. The project will demonstrate a set of practical tools, a pipeline and an application programming interface.
#### CLIPE
Participants: Julien Pettré, Tairan Yin, Vicenzo Abichequer.
• Title:
Creating Lively Interactive Populated Environments
• Duration:
March 2019 - February 2023
• Coordinator:
UNIVERSITY OF CYPRUS (Cyprus)
• Partners:
• ECOLE POLYTECHNIQUE (France)
• KUNGLIGA TEKNISKA HOEGSKOLAN (Sweden)
• MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DERWISSENSCHAFTEN EV (Germany)
• SILVERSKY3D VR TECHNOLOGIES LTD (Cyprus)
• THE PROVOST, FELLOWS, FOUNDATION SCHOLARS & THE OTHER MEMBERS OF BOARD OF THE COLLEGE OF THE HOLY UNDIVIDED TRINITY OF QUEEN ELIZABETH NEAR DUBLIN (Ireland)
• UNIVERSITAT POLITECNICA DE CATALUNYA (Spain)
• UNIVERSITY OF CYPRUS (Cyprus)
• Inria contact:
Julien Pettré
• Summary:
The project addresses the core challenges of designing new techniques to create and control interactive virtual characters, benefiting from opportunities open by the wide availability of emergent technologies in the domains of human digitization and displays, as well as recent progresses of artificial intelligence. CLIPE aspires to train the new generation of researchers in these techniques, looking at the area holistically. The training and research programme is based on a multi-disciplinary and cross-sectoral philosophy, bringing together industry and academia experts and focusing on both technical and transversal skills development.
#### CrowdDNA
Participants: Julien Pettré, Thomas Chatagnon.
• Title:
TECHNOLOGIES FOR COMPUTER-ASSISTED CROWDMANAGEMENT
• Duration:
November 2020 -March 2023
• Coordinator:
Inria (France)
• Partners:
• ECOLE NORMALE SUPERIEURE DE RENNES (France)
• FORSCHUNGSZENTRUM JULICH GMBH (Germany)
• ONHYS (France)
• UNIVERSIDAD REY JUAN CARLOS (Spain)
• UNIVERSITAET ULM(Germany)
• UNIVERSITE RENNES II (France)
• UNIVERSITY OF LEEDS (U.K.)
• Inria contact:
Julien Pettré
• Summary:
Crowd management is a difficult task. Large crowds gathering for an outdoor event and heavy pedestrian traffic are events of serious concern for officials tasked with managing public spaces. Existing methods rely on simulation technologies and require the measurement of simulation variables that are difficult to estimate. The EU-funded CrowdDNA project proposes a new technology based on innovative crowd simulation models. It facilitates predictions on the dynamics, behaviour and risk factors of high-density crowds, addressing the need for safe and comfortable mass events. The project suggests that the analysis of some specific macroscopic characteristics of a crowd such as its apparent motion can offer important information about its internal structure and allow the exact assessment of its state.
#### CrowdBot
Participants: Javad Amirian, Fabien Grzeskowiak, Solenne Fortun, Marie Babel, Julien Pettré, Fabien Spindler.
• Title:
• Duration:
January 2018 - December 2021
• Coordinator:
Inria (France)
• Partners:
• UCL (UK)
• SoftBank Robotics (France)
• Univ. Aachen (Germany)
• EPFL (Switzerland)
• ETHZ (Switzerland)
• Locomotec (Germany)
• Inria contact:
Julien Pettré
• Summary:
CROWDBOT will enable mobile robots to navigate autonomously and assist humans in crowded areas. Today’s robots are programmed to stop when a human, or any obstacle is too close, to avoid coming into contact while moving. This prevents robots from entering densely frequented areas and performing effectively in these high dynamic environments. CROWDBOT aims to fill in the gap in knowledge on close interactions between robots and humans during navigation tasks. The project considers three realistic scenarios: 1) a semi-autonomous wheelchair that must adapt its trajectory to unexpected movements of people in its vicinity to ensure neither its user nor the pedestrians around it are injured; 2) the commercially available Pepper robot that must navigate in a dense crowd while actively approaching people to assist them; 3) the under development robot cuyBot will adapt to compact crowd, being touched and pushed by people. These scenarios generate numerous ethical and safety concerns which this project addresses through a dedicated Ethical and Safety Advisory Board that will design guidelines for robots engaging in interaction in crowded environments. CROWDBOT gathers the required expertise to develop new robot capabilities to allow robots to move in a safe and socially acceptable manner. This requires achieving step changes in a) sensing abilities to estimate the crowd motion around the robot, b) cognitive abilities for the robot to predict the short term evolution of the crowd state and c) navigation abilities to perform safe motion at close range from people. Through demonstrators and open software components, CROWDBOT will show that safe navigation tasks can be achieved within crowds and will facilitate incorporating its results into mobile robots, with significant scientific and industrial impact. By extending the robot operation field toward crowded environments, we enable possibilities for new applications, such as robot-assisted crowd traffic management.
### 9.2.2 Other european programs/initiatives
#### Interreg VA France (Channel) England ADAPT
Participants: Marie Babel, François Pasteau, Fabien Grzeskowiak, Louise Devigne, Guillaume Vailland.
• Title:
Assistive Devices for empowering disAbled People through robotic Technologies
• Duration:
Jan 2017 - June 2022
• Coordinators:
ESIGELEC/IRSEEM Rouen
• Partners:
• INSA Rennes - IRISA, LGCGM, IETR (France)
• Université de Picardie Jules Verne -MIS (France),
• Pôle Saint Hélier (France), CHU Rouen (France),
• Réseau Breizh PC (France),
• Pôle TES (France),
• University College of London - Aspire CREATE (UK),
• University of Kent (UK),
• East Kent Hospitals Univ NHS Found. Trust (UK),
• Health and Europe Centre (UK),
• Plymouth Hospitals NHS Trust (UK),
• Canterbury Christ Church University (UK),
• Kent Surrey Sussex Academic Health Science Network (UK),
• CornwallMobility Center (UK)
• Inria contact:
Marie Babel
• Summary:
This project aims at developing innovative assistive technologies in order to support the autonomy and to enhance the mobility of power wheelchair users with severe physical/cognitive disabilities. In particular, the objective is to design and evaluate a power wheelchair simulator as well as to design a multi-layer driving assistance system.
## 9.3 National initiatives
#### Equipex Robotex
Participants: Fabien Spindler, François Chaumette.
no Inria Rennes 6388, duration: 10 years.
Rainbow was one of the 15 French academic partners involved in the Equipex Robotex network that started in February 2011. It was devoted to get and manage significant equipment in the main robotics labs in France. In the scope of this project, we have obtained the humanoid robot Romeo in 2015.
#### ANR Marsurg
Participants: Eric Marchand, François Chaumette, Fabiens Spindler.
no Inria 16162, duration: 48 months.
This project started in September 2021. It involves a consortium managed by ISIR (Paris) with Pixee Medical and Rainbow group. It aims at researching markerless augmented reality solution for orthopedic surgery
#### ANR Sesame
Participant: François Chaumette.
no Inria 13722, duration: 48 months.
This project started in January 2019. It involves a consortium managed by LS2N (Nantes) with LIP6 (Paris) and Rainbow group. It aims at analysing singularity and stability issues in visual servoing (see Section 7.2.5)
#### Inria Challenge DORNELL
Participants: Marie Babel, Maud Marchal, Claudio Pacchierotti, François Pasteau, Louise Devigne, Marco Aggravi, Inès Lacôte, Pierre-Antoine Cabaret, Lisheng Kuang.
• Title:
DORNELL: A multimodal, shapeable haptic handle for mobility assistance of people with disabilities
• Duration:
November 2020 - December 2024
• Coordinators:
Marie Babel, Claudio Pacchierotti
• Partners:
• Potioc Inria team
• MFX Inria team
• LGCGM (Rennes)
• Centre de rééducation Pôle Saint Hélier (Rennes)
• ISIR (Paris)
• Institut des jeunes aveugles (Yzeure)
• Inria contact:
Marie Babel, Claudio Pacchierotti
• Summary:
While technology helps people to compensate for a broad set of mobility impairments, visual perception and/or cognitive deficiencies still significantly affect their ability to move safely and easily. We propose an innovative multisensory, multimodal, smart haptic handle that can be easily plugged onto a wide range of mobility aids, including white canes, precanes, walkers, and power wheelchairs. Specifically fabricated to fit the needs of a person, it provides a wide set of ungrounded tactile sensations (e.g., pressure, skin stretch, vibrations) in a portable and plug-and-play format – bringing haptics in assistive technologies all at once. The project will address important scientific and technological challenges, including the study of multisensory perception, the use of new materials for multimodal haptic feedback, and the development of a haptic rendering API to adapt the feedback to different assistive scenarios and user’s wishes. We will co-design DORNELL with users and therapists, driving our development by their expectations and needs.
#### BPI Lichie
Participants: Maxime Robic, John Thomas, Julien Dufour, Eric Marchand, François Chaumette.
no Inria 14876, duration: 45 months.
This project started in March 2020. It involves a consortium managed by Airbus (Toulouse) with many companies, Onera and Inria. It aims at designing a new constellation of satellites with on-board imaging facilities. Robotics for the assembly of the satellites is also studied. As for Rainbow, this project funds Maxime Robic and John Thomas PhDs (see Sections 7.2.6 and 7.2.7, as well as the software developments achieved for ViSP by Julien Dufour.
#### ANR CAMP
Participants: P. Robuffo Giordano, Q. Delamare, F. Spindler.
• Title:
Intrinsically-Robust and Control-Aware Motion Planning for Robots in Real-World Conditions
• Duration:
October 2020 - September 2024
• Coordinator:
P. Robuffo Giordano
• Partners:
• LAAS (Toulouse)
• Univ. Twente (Netherlands)
• Inria contact:
P. Robuffo Giordano
• Summary:
An effective way of dealing with the complexity of robots operating in real (uncertain) environments is the paradigm of “feedforward/feedback” or “planning/control”: in a first step a suitable nominal trajectory (feedforward) for the robot states/controls is planned exploiting the available information (e.g., a model of the robot and of the environment). While there has been an effort in proposing “robust planners” or more “global controllers” (e.g., Model Predictive Control (MPC)), a truly unified approach that fully exploits the techniques of the motion planning and control/estimation communities is still missing and the existing state-of-the-art has several important limitations, namely (1) lack of generality, (2) lack of computational efficiency, and (3) poor robustness. In this respect, the ambition of CAMP is to (1) develop a general and unified “intrinsically-robust and control-aware motion planning framework” able to address all the above-mentioned issues, and to (2) demonstrate the applicability of this new framework to real robots in real-world challenging tasks. In particular we envisage two robotics demonstrators for showing at best the effectiveness and generality of our methodology: (1) an indoor pick- and-place/assembly task involving a 7-dof torque-controlled arm for a first validation in “controlled conditions” and (2) an outdoor cooperative mobile manipulation task involving an aerial manipulator (a quadrotor UAV equipped with an onboard arm) and a skid-steering mobile robot with an onboard arm for a final validation in much less favorable experimental conditions (see Sect. 7.2.1)
#### ANR MULTISHARED
Participants: P. Robuffo Giordano, C. Pacchierotti, V. Drevelle, J. Pettré, G. Notomista, J. Nader.
• Title:
Shared-Control Algorithms for Human/Multi-Robot Cooperation
• Duration:
September 2020 - August 2024
• Coordinator:
P. Robuffo Giordano
• Inria contact:
P. Robuffo Giordano
• Summary:
The goal of the Chaire AI MULTISHARED is to significantly advance the state-of-the-art in multi-robot autonomy and human-multi-robot interaction for allowing a human operator to intuitively control the coordinated motion of a multi-UAV group navigating in remote environments, with a strong emphasis on the division of roles between multi-robot autonomy (in controlling its motion/configuration and online decision-making) and human intervention/guidance for providing high-level commands to the group while being most aware of the group status via VR and haptics technology (see Sect. 7.1.2 and Sect. 7.4.2).
## 9.4 Regional initiatives
#### CominLabs MAMBO
Participants: Lev Smolentsev, Alexandre Krupa, François Chaumette, Paolo Robuffo Giordano, Fabien Spindler.
• Title:
Manipulation of Soft Bodies with Multiple Drones
• Duration:
October 2020 - September 2024
• Coordinator:
LS2N (Nantes)
• Partners:
• LS2N (Nantes)
• Inria contact:
Alexandre Krupa
• Summary:
This project is funded by the Labex CominLabs. It is led by the ARMEN team at LS2N (Nantes) and implies the collaboration of the Rainbow Project-Team. Its objective is to propose a scientific framework for allowing the manipulation of an object by the combined action of two drones equipped with onboard cameras and force sensors. The envisaged solution is to manipulate a deformable body (a slender beam) attached between the two drones in order to grasp an object on the floor and move it to another location. In the scope of this project, the Rainbow group is involved in the elaboration of new approaches for controlling the 2 drones by visual servoing using data provided by onboard RGB-D cameras (see Section 7.2.10).
#### Cartam
Participants: Julien Dufour, Fabien Spindler, François Chaumette.
No Inria Rennes 14041 and 13954, duration: 36 months.
• This project started in January 2019 . It is supported by Brittany region and FEDER program. It is managed by Unilet with Copeeks, Neotec Vision, Rainbow group, and our start-up Dilepix. It aims at designing a vision system able to detect and locate adventices in a field. We are in charge of tracking the adventices once they are detected and building a geo-localized cartography of the field locating them.
#### Silver Connect
Participants: Marie Babel.
• Title:
SilverConnect - Le digital au service des EHPAD
• Duration:
September 2018 - April 2021
• Coordinator:
Hoppen (Rennes)
• Partners:
• INSA Rennes
• Hoppen (Rennes)
• Centre de médecine physique et de réadaptation Pôle Saint Hélier (Rennes)
• Famileo (Rennes)
• Inria contact:
Marie Babel
• Summary:
This project started in November 2018 and is supported by Brittany region/BPI as well as FEDER. This project aims at designing a fall detection framework by means of vision-based algorithms coupled with deep learning solutions.
#### Ambrougerien
Participants: Marie Babel, Vincent Drevelle, François Pasteau, Merwane Bouri.
• Title:
Autonomie, MoBilité et fauteuil ROUlant robotisé : GEolocalisation indoor et Recharge IntelligENte
• Duration:
December 2020 - December 2024
• Coordinator:
DK Innovation (Plérin)
• Partners:
• INSA Rennes
• Hoppen (Rennes)
• Centre de médecine physique et de réadaptation Pôle Saint Hélier (Rennes)
• Inria contact:
Marie Babel
• Summary:
This project started in December 2020 and is supported by Brittany region and Rennes Métropole. AMBROUGERIEN aims at supporting the independence of people in electric wheelchairs. A dedicated interface allows the wheelchair to move autonomously to secure the transfer and to return to an intelligent induction recharging base. Information on the internal state of the wheelchairs facilitates fleet management.
Participants: Marie Babel, Vincent Drevelle, Maud Marchal, Claudio Pacchierotti, François Pasteau, Louise Devigne, Fabien Grzeskowiak, Guillaume Vailland, Anne-Hélène Olivier (MimeTIC), Bruno Arnaldi (Hybrid), Valérie Gouranton (Hybrid), Florian Nouviale (Hybrid), Alexandre Audinot (Hybrid).
• Title:
Academic Chair on Innovations, Handicap, Autonomy and Accessibility (IH2A)
• Duration:
September 2020 -
• Coordinator:
Marie Babel
• Partners:
• IETR Rennes
• LGCGM Rennes
• Centre de médecine physique et de réadaptation Pôle Saint Hélier (Rennes)
• Inria contact:
Marie Babel
• Summary:
This research chair (Innovations, Handicap, Autonomy and Accessibility - IH2A) is a continuation of the research work developed at INSA Rennes/Rainbow team on assistive robotics. The idea is to propose the most suitable technological solutions to compensate for sensory-motor handicaps limiting the mobility and autonomy of people in daily life tasks and leisure activities. The Chair thus aims at perpetuating these activities, both from a societal point of view and from a scientific and clinical point of view, and is intended to be an effective and innovative tool for the deployment of large-scale research in this area. The creation of a new type of multidisciplinary and innovative collaborative site of experimentations will allow the clinical and scientific validation of the technical assistance offered, while ensuring the accessibility of the solutions deployed.
# 10 Dissemination
Participant: François Chaumette, Paolo Robuffo Giordano, Claudio Pacchierotti, Marie Babel, Maud Marchal, Alexandre Krupa, Julien Pettré, Eric Marchand, Vincent Drevelle, Fabien Spindler.
## 10.1 Promoting scientific activities
### 10.1.1 Scientific events: organisation
#### General chair, scientific chair
• P. Robuffo Giordano was Program co-chair for the IEEE International Symposium On Multi-Robot And Multi-Agent Systems 2021 (IEEE MRS 2021)
• M. Marchal was Program co-chair for the Journal Papers Track of IEEE Virtual Reality and 3D User Interfaces 2021 (IEEE VR 2021). She was also Program co-chair for Conference Papers Track of IEEE Symposium on Mixed and Augmented Reality 2021 (IEEE ISMAR 2021).
### 10.1.2 Scientific events: selection
#### Member of the conference program committees
• P. Robuffo Giordano was Associate Editor of IEEE ICRA 2022
• E. Marchand has been Associate Editor of IEEE ICRA 2021.
• M. Babel was Associate Editor of IEEE ICRA 2022.
• C. Pacchierotti was Associate Editor of IEEE ICRA 2022, Editorial Committee Member of Eurohaptics 2022, and Chair of the Cross-Cutting Challenges for IEEE HAPTICS 2022.
• M. Marchal was a Member of the Technical Papers Committee of Siggraph 2021, a Program Committee Member of the ACM/SIGGRAPH conference on Motion, Interaction and Games 2021 (MIG 2021), a Program Committee Member of ACM/Eurographics Symposium on Computer Animation 2021 (SCA 2021).
• J. Pettré was member of program committee for CASA 2021 and ACM MIG 2021.
#### Reviewer
• P. Robuffo Giordano: IEEE MRS (1), IEEE/RSJ IROS (1), IEEE ICRA (2)
• F. Chaumette: IEEE ICRA (1)
• A. Krupa: IEEE ICRA (1)
• E. Marchand: IEEE ICRA (1), IEEE IROS (1), BMVC (3)
• F. Spindler: RobAgri (1)
• M. Babel: IEEE ICRA (1)
• C. Pacchierotti: IEEE HAPTICS (1)
• J. Pettré: ACM SIGGRAPH (1) ACM SIGGRAPH Asia (1) ACM SCA (1)
### 10.1.3 Journal
#### Member of the editorial boards
• P. Robuffo Giordano and F. Chaumette are Editors for the IEEE Transactions on Robotics
• A. Krupa is Associate Editor for the IEEE Transactions on Robotics
• E. Marchand is Senior Editor for the IEEE Robotics and Automation Letters
• C. Pacchierotti is Associate Editor for the IEEE Robotics and Automation Letters
• M. Marchal is Associate Editor of IEEE Transactions on Visualization and Computer Graphics, IEEE Computer Graphics and Applications, Computers & Graphics and IEEE Transactions on Haptics.
• J. Pettré is associate editor for Computer Graphics Forum and Computer Animation and Virtual Worlds.
#### Reviewer - reviewing activities
• P. Robuffo Giordano: IEEE T-RO (1), IEEE RA-L (2), IEEE L-CSS (1)
• F. Chaumette: IEEE RA-L (4)
• A. Krupa: IEEE RA-L (1)
• E. Marchand: IEEE RA-L (2), IEEE TVCG (1)
• M. Babel: Springer International Journal of Social Robotics (1), IEEE TRO (1), Disability and Rehabilitation: Assistive Technology (1)
• C. Pacchierotti: IEEE TOH (7), IEEE T-RO (1), Journal of Intelligent & Robotic Systems (1), ACM Computing Survey (1), Behavior & Information Technology (2), IEEE Transactions on Medical Robotics and Bionics (1), IEEE Transactions on Visualization and Computer Graphics (1), Frontiers in Virtual Reality (1), IEEE Computer Magazine (1)
• M. Marchal: ACM TOG (2), IEEE TOH (1)
• J. Pettré: Scientific Reports (1), Physica A (1), Computer and Graphics (1)
### 10.1.4 Invited talks
• P. Robuffo Giordano. “An Overview of Formation Control and Localization for Multiple Robot Systems”. ENS, Computer Science Department, Rennes, September 2021
• P. Robuffo Giordano “Recent Advances in Shared Control for Tele-manipulation and Tele-navigation”. DIAG, “La Sapienza” University of Rome, Italy, June 2021
• P. Robuffo Giordano. “Recent Advances in Shared Control for Tele-manipulation and Tele-navigation”. Rencontres mécatroniques 2021, ENS Rennes, April 2021
• M. Babel, “Robotique d'assistance et handicap : la mobilité pour tous ”60ème Congrès annuel du Club EEA, June 2021
• M. Babel, “ADAPT project : knowledge, technology ”, Interreg EDUCAT project closing event, June 2021
• C. Pacchierotti. “Less is more: the challenge of wearable haptics in the era of immersive technologies.” University of Aarhus, Herning (then held online), Denmark, 2021.
• C. Pacchierotti. “Touching virtual reality: Extending immersive experiences through haptics.” MatchPoints 2021, Aarhus (then held online), Denmark, 2021.
• M. Marchal. “Playing with tangibles in Virtual Reality”, Journées IHM IG RV, June 2021
• M. Marchal. “Multisensory feedback for 3D Hand Interaction with virtual environments”, Univ. Grenoble Alpes, September 2021.
• J. Pettré, “Virtual Population for Urban Digital Twins”, TICS4Ci (Réseau latino-americain sur les applications TIC pour les villes intelligentes).
### 10.1.5 Leadership within the scientific community
• F. Chaumette is a member of the "Haut Comité des Grandes Infrastructures de Recherche" of the French Ministery of Research. He also serves as a member of the Scientific Council of the Mathematics and Computer Science Department of INRAe. He is also a founding member of the Scientific Council of the GdR Robotique. Finally, he is in the Advisory Board of the H2020 ERA Chair AIFORS project.
• C. Pacchierotti is Senior Chair of the IEEE Technical Committee on Haptics and Secretary of the Eurohaptics Society.
• M. Marchal is a Board Member of Institut Universitaire de France and a junior member of the institute. She is a board member of GDR Informatique Graphique-Réalité Virtuelle.
### 10.1.6 Scientific expertise
• P. Robuffo Giordano has been elected member of the Section 07 of the Comité National de la Recherche Scientifique. He also served as expert/reviewer for the euRobotics “George Giralt” award for the best European PhD thesis in robotics, for evaluating research projects from the ANR, SNSF (Switzerland) and Advanced Grants from the ERC. He was reviewer for the H2020 projects AirBorne, HyFliers and PILOTING.
• F. Chaumette served as member of the IEEE RAS Fellow Evaluation Committee in 2021. He was also in the CNRS evaluation committe of the JRL lab (Tsukuba)
• A. Krupa served as expert/reviewer for the Best French PhD Thesis in robotics awarded by the GdR Robotique in 2021.
• Marie Babel serves since 2017 as an expert for the International Mission of the French Research Ministry (MEIRIES) - Campus France since 2017. In 2021, she also served as an expert for the Haute Autorité de Santé and for the Ligue du Cancer.
• M. Marchal served as a reviewer for Starting Grants from the ERC.
• F. Chaumette served as the president of the committee in charge of all the temporary recruitments (“Commission Personnel”) at Inria Rennes-Bretagne Atlantique and IRISA and was a member of the Head team of Inria Rennes-Bretagne Atlantique till August 2021. He serves for the Scientific Steering Committee (COSS) of IRISA. He is also a member of the Inria COERLE committee (in charge of the ethical aspects of all Inria research). This year, he evaluated the ethical part of the 30 projects selected by the Brittany Bienvenue Program.
• A. Krupa is a member of the CUMIR (“Commission des Utilisateurs des Moyens Informatiques pour la Recherche”) of Inria Rennes-Bretagne Atlantique.
• E. Marchand is the head of "Digital Signals and Images, Robotics" department at IRISA.
• M. Marchal is a council member of the INSA component of IRISA.cc
• J. Pettré serves as president of the CUMIR (“Commission des Utilisateurs des Moyens Informatiques pour la Recherche”) of Inria Rennes-Bretagne Atlantique.
## 10.2 Teaching - Supervision - Juries
### 10.2.1 Teaching
François Chaumette:
• Master SISEA: “Robot Vision”, 12 hours, M2, Université de Rennes 1
• Master ENS: “Visual servoing”, 6 hours, M1, Ecole Nationale Supérieure de Rennes;
• Master ESIR3: “Visual servoing”, 8 hours, M2, Ecole supérieure d'ingénieurs de Rennes.
Alexandre Krupa:
• Master FIP TIC-Santé: “Ultrasound visual servoing”, 6 hours, M2, Télécom Physique Strasbourg
• Master ESIR3: “Ultrasound visual servoing”, 9 hours, M2, Esir Rennes
• Master INSA1: “Computer programming”, 42 hours, L1, INSA Rennes
Eric Marchand:
• Master Esir2: “Colorimetry”, 12 hours, M1, Esir Rennes
• Master Esir2: “Computer vision: geometry”, 24 hours, M1, Esir Rennes
• Master Esir3: “Robotics Vision 1”, 24 hours, M2, Esir Rennes
• Master Esir3: “Robotics Vision 2”, 14 hours, M2, Esir Rennes
• Master MRI: “Computer vision”, 12 hours, M2, Université de Rennes 1
• Master ENS: “Computer vision”, 16 hours, M2, ENS Rennes
• Master MIA: “Augmented reality”, 4 hours, M2, Université de Rennes 1
• Licence ESIR 1: "System", 16h, L3, Université de Rennes 1
Marie Babel:
• Master INSA2: “Robotics”, 26 hours, M1, INSA Rennes
• Master INSA1: “Concepts de la logique à la programmation”, 20 hours, L3, INSA Rennes
• Master INSA1: “Langage C”, 12 hours, L3, INSA Rennes
• Master INSA2: “Computer science project”, 30 hours, M1, INSA Rennes
• Master INSA1: “Practical studies”, 16 hours, L3, INSA Rennes
• Master INSA2: “Image analysis”, 26 hours, M1, INSA Rennes
• Master INSA1: “Remedial math courses”, 50 hours, L3, INSA Rennes
• Master INSA 1: “Probability”, 14 hours, L3, INSA Rennes
• Master INSA: tutoring and support for students with disabilities, 30 hours, INSA Rennes
Claudio Pacchierotti:
• Master “Artificial Intelligence & Advanced Visual Computing”: INF644 – Virtual/Augmented Reality & 3D Interactions”, 6 hours, M2, École Polytechnique
• Master SIF: “Virtual Reality and Multi-Sensory Interaction”, 4 hours, M2, IRISA.
M. Marchal:
• Master SIF: “Computer Graphics”, 8 hours, M2, INSA/Univ. Rennes 1.
• Master INSA1: “Computer Graphics”, 20 hours, M1, INSA Rennes.
• Master INSA1: “Complexity and algorithms”, 26 hours, L3, INSA Rennes.
Vincent Drevelle:
• Master 2 ILA/CCNA: “Transverse project”, 30 hours, M2, Université de Rennes 1
• Master 1 Info: “Artificial intelligence”, 20 hours, M1, Université de Rennes 1
• Licence Info: “Computer systems architecture”, 60 hours, L1, Université de Rennes 1
• Portail Info-Elec: “Discovering programming and electronics”, 18 hours, L1, Université de Rennes 1
• Licence 3 Miage: “Computer programming”, 78 hours, L3, Université de Rennes 1
• Master 2 EEEA-SE: “Instrumentation, localization, GPS”, 4 hours, M2, Université de Rennes 1
• Master 2 EEEA-SE: “Multisensor data fusion”, 20 hours, M2, Université de Rennes 1
• Master 2 IL/CCN: “Mobile robotics”, 32 hours, M2, Université de Rennes 1
Julien Pettré:
• Master 2 SIF: "Motion for Animation and Robotics", 6 hours, Université de Rennes 1
• Master 2 Advanced 3D Graphics, "Crowd simulation", 3 hours, Ecole Polytechnique
Paolo Robuffo Giordano:
• Parcours Intelligence artificielle confirmés, SAFRAN, modules “Advanced Robotics”, 12 heures
### 10.2.2 Supervision
• Ph.D. in progress: Lisheng Kuang, “Design and development of novel wearable haptic interfaces for teleoperation of robots”, started in March 2020, supervised by Claudio Pacchierotti and Paolo Robuffo Giordano.
• Ph.D. in progress: Pascal Brault, “Planification et optimisation de trajectoires robustes aux incertitudes paramétriques pour des taches robotiques fondées sur l'usage de capteurs”, started in September 2019, supervised by Paolo Robuffo Giordano and Quentin Delamare
• Ph.D. in progress: Alexander Oliva, “Coupling Vision and Force for Robotic Manipulation”, started in October 2018, supervised by François Chaumette and Paolo Robuffo Giordano
• Ph.D. in progress: Ali Srour, “Robust and Control-Aware Motion Planning”,started in October 2021, supervised by Q. Delamare and Paolo Robuffo Giordano
• Ph.D. in progress: Maxime Bernard, “Shared Control for Multi-Robot Systems”,started in October 2021, supervised by Claudio Pacchierotti and Paolo Robuffo Giordano
• Ph.D. in progress: Nicola De Carli, “Reactive Trajectory Planning Methods for Formation Control and Localization of Multi-Robot System”,started in January 2021, supervised by P. Salaris (Univ. Pisa, Italy) and Paolo Robuffo Giordano
• Ph.D. in progress: Lev Smolentsev, “Manipulation of soft bodies with multiple drones”, started in November 2020, supervised by Alexandre Krupa, François Chaumette and Isabelle Fantoni (LS2N, Nantes)
• Ph.D. in progress: Fouad Makiyeh,“Shape servoing based on visual information using RGB-D sensor for dexterous manipulation of 3D compliant objects”, started in September 2020, supervised by Alexandre Krupa, Maud Marchal and François Chaumette
• Ph.D. in progress: Xi Wang “Robustness of Visual SLAM techniques to light changing conditions”, started in September 2018, supervised by Eric Marchand and Marc Christie (MimeTIC group)
• Ph.D. in progress: Samuel Felton “Deep Learning for visual servoing”, started in October 2019, supervised by Eric Marchand and Elisa Fromont (Lacodam group)
• Ph.D. in progress: Mathieu Gonzalez “SLAM in time varying environment”, started in October 2019, supervised by Eric Marchand and Jérome Royan (IRT B<>COM)
• Ph.D. in progress: Maxime Robic, “Visual servoing of a satellite constellation”, started in November 2020, supervised by Eric Marchand and François Chaumette.
• Ph.D. in progress: John Thomas, “Visual servoing of a satellite constellation”, started in December 2020, supervised by François Chaumette.
• Ph.D. in progress: Thibault Noël, “3D environment exploration”, started in October 2021, supervised by Eric Marchand and François Chaumette.
• Ph.D. in progress: Erwan Normand, “3D environment exploration”, started in October 2021, supervised by Eric Marchand, Maud Marchal and Claudio Pacchierotti.
• Ph.D. in progress: Emilie Leblong, “Taking into account social interactions in a virtual reality power wheelchair driving simulator: promoting learning for inclusive mobility”, started in October 2020, supervised by Marie Babel and Anne-Hélène Olivier (Mimetic team)
• Ph.D. in progress: Inès Lacôte, “Investigate haptic and multisensory illusions to design an assistive navigation”, started in January 2021, supervised by Maud Marchal, Claudio Pacchierotti, David Gueorguiev (ISIR, Paris)
• Ph.D. in progress: Pierre-Antoine Cabaret, “Design of navigation techniques for a multi-sensory handle for mobility assistance”, started in October 2021, supervised by Maud Marchal, Marie Babel and Claudio Pacchierotti
• Ph.D. in progress: Antoine Cellier, “Inclusive navigation in a power wheelchair for people with people with neurological pathologies: from virtual to real”, started in October 2021, supervised by Marie Babel and Valérie Gouranton (Hybrid team)
• Ph.D. in progress: Glenn Kerbiriou, “Semantic Modeling of the Face”, started in April 2021, supervised by Maud Marchal, Fabien Danieau and Quentin Avril (both from Interdigital)
• Ph.D. in progress: Alberto Jovane, “Modélisation de mouvements réactifs et comportements non verbaux pour la création d'acteurs digitaux pour la réalité virtuelle”, started in September 2019, supervised by Julien Pettré, Marc Christie, Ludovic Hoyet, Claudio Pacchierrotti.
• Ph.D. in progress: Thomas Chatagnon, “Micro-to-macro energy-based interaction models for dense crowds behavioral simulations ”, started in November 2020, supervised by Julien Pettré, as well as (from Mimetic team): Charles Pontonnier, Anne-Hélène Olivier and Ludovic Hoyet
• Ph.D. in progress: Adèle Colas, “Modélisation de comportements collectifs réactides et expressifs pour la réalité virtuelle”, started in December 2019, supervised by Julien Pettré, Claudio Pacchierrotti as well as (from Mimetic team): Anne-Hélène Olivier and Ludovic Hoyet
• Ph.D. in progress: Tairan Yin, “Création de scènes peuplées dynamiques pour la réalité virtuelle”, started in November 2020, supervised by Julien Pettré, and Marie-Paule Cani (Ecole Polytechnique) as well as (from Mimetic team): marc Christie and Ludovic Hoyet
• Ph.D. in progress: Vicenzo Abichequer, “Humains virtuels expressifs et réactifs pour la réalité virtuelle ”, started in November 2020, supervised by Julien Pettré, and Carol O'Sullivan (Trinity College Dublin) as well as (from Mimetic team): marc Christie and Ludovic Hoyet
• Ph.D. defended: Guillaume Vailland, “Power Wheelchair Navigation : From Virtual Reality Simulation Towards Real Smart Designs” defended in December 2021, supervised by Marie Babel and Valérie Gouranton (Hybrid team)
• Ph.D. defended: Ketty Favre, “Lidar-based localization”, defended in september 2021, supervised by Eric Marchand, Muriel Pressigout and Luce Morin (Vador group, IETR)
• Ph. D. defended: Fabien Grzeskowiak, “Crowd simulation and experiments for the evaluation of robot navigation in crowds”, defended in June 2021, supervized by Julien Pettré and Marie Babel
• Ph. D. defended: Javad Amirian, “Human motion trajectory prediction for robot navigation”, supervised by Julien Pettré and Jean-Bernard Hayet (CIMAT, Mexico) defended in July 2021
### 10.2.3 Juries
#### PhD and HDR juries
• P. Robuffo Giordano: Julian Erskine (Ph.D, President), LS2N, Nantes; Esteban Restrepo (Ph.D., reviewer) L2S, Paris; José de Jesús Castillo Zamora (Ph.D., reviewers) L2S, Paris; Amanhoud Walid (Ph.D., reviewer) EPFL, Switzerland; Mahmoud Hamandi (Ph.D., reviewer), LAAS-CNRS; Paolo Ferrari (Ph.D., reviewer) University of Rome “La Sapienza”, Italy; Luca Bigazzi (Ph.D., reviewer) University of Florence, Italy; Riccardo Mengacci (Ph.D., member), University of Pisa, Italy; Chiara Gabellieri (Ph.D., reviewer) University of Pisa, Italy
• F. Chaumette: Marianne Bakken (PhD, reviewer, NTNU, Norway), Ferran Argelaguet (HDR, president, Irisa)
• A. Krupa: Florent Nageotte (HDR, member), ICube, Université de Strasbourg, France
• E. Marchand: Kevin Chappellet (PhD, Lirmm), Tom François (PhD, Institut Pascal, president), Antoine André (PhD, Femto-ST, president), Simon Evain (PhD, Inria Rennes, president), Javad Amirian (PhD, Inria Rennes, president), Fabien Grzeskoviac (PhD, Irisa, president), Ific Goudé (PhD, Irisa Rennes, president), Florian Berton (PhD, Inria Rennes, president),
• M. Babel Franck Pouvrasseau (PhD, Université Paris Saclay, member), Hazem Khaled Mohamed Abdelkawy (PhD, Université Paris-Est Créteil, member), Tassut Tagnithammou (PhD, Université Paris-Saclay, member), Michael Gray (PhD, INSA Hauts de France, member), Yann Morere (HDR, Université de Lorraine, member), Viet Thuan Nguyen (Université Polytechnique Hauts de France, reviewer)
• M. Marchal: Thomas Buffet (PhD, Ecole Polytechnique, reviewer), Jocelyn Monnoyer (PhD, Aix Marseille Université, reviewer), Thibaud Delrieu, Université de Poitiers, reviewer), Daniel Lobo (University of Rey Juan Carlos, Madrid, Spain, reviewer), Mickaël Ly (Université Grenoble Alpes, reviewer), Camille Brunel (Université de Bordeaux, reviewer), David Lopez (Université de Lorraine, reviewer), Hector Bareiro (University of Rey Juan Carlos, Madrid, Spain, reviewer and president), Elodie Bouzbib (Sorbonne Université, reviewer and president), Camille Apamon (INSA Rennes, member)
• V. Drevelle: Krushna Shinde (PhD, member, Université de Technologie de Compiègne)
• J. Pettré: Manon Predhumeau (PhD, member, Université Grenoble-Alpes),
#### Other juries
• P. Robuffo Giordano: member of the selection committee for the recruitment of researchers (CRCN) at Inria Nancy-Grand Est; Member of the selection committee for a Professor position at the University of Montpellier
• F. Chaumette: member of the selection committee for the recruitment of researchers (CRCN) at Inria Rennes Bretagne Atlantique; Member of the selection committee for a Professor position at the University of Toulouse
• A. Krupa: member of the selection committee for the recruitment of an Assistant Professor at the Université de Strasbourg
• E. Marchand: president of the selection committee for the recruitment of an Assistant Professor at the Université de Rennes 1
• M. Babel: member of assistant professor committee selections at Centrale Nantes, Université Picardie Hauts-de-France, ENS Rennes
## 10.3 Popularization
### 10.3.1 Internal or external Inria responsibilities
E. Marchand is an Editor for Interstices )i(
### 10.3.3 Interventions
Due to the visibility of our experimental platforms, the team is often requested to present its research activities to students, researchers or industry. Our panel of demonstrations allows us to highlight recent results concerning the vision-based shared control using our haptic device for object manipulation, the control of a fleet of quadrotors, vision-based detection and tracking for space navigation in a rendezvous context, the semi-autonomous navigation of a wheelchair, the power wheelchair simulator and augmented reality applications. Some of these demonstrations are available as videos on VispTeam YouTube channel.
• Maud Marchal was invited by France culture for the radio program "La méthode scientifique" about haptics and virtual reality (podcast)
• Marie Babel was interviewed by Radio Laser on Nov. 2021 for the radio program "Voyages extraordinaires dans le monde des sciences" (podcast).
• Marie Babel gave a speech during a conference organized in Inria Rennes about "J'peux pas, j'ai informatique" on Nov. 2021 for secondary-school teachers.
• Claudio Pacchierotti and Maud Marchal have been featured in a video and article entitled “Le sens du toucher fait son entrée dans la réalité virtuelle” (The sense of touch makes its entrance in VR) on the French newspaper Le Monde.
• The CrowdDNA project has been the topic of several articles in press (Ouest France, Sciences Ouest, Actu IA), as well as a TV report in Télématin (France TV).
# 11 Scientific production
## 11.1 Major publications
• 1 articleM.Marco Aggravi, G.Giuseppe Sirignano, P.Paolo Robuffo Giordano and C.Claudio Pacchierotti. Decentralized control of a heterogeneous human-robot team for exploration and patrolling.IEEE Transactions on Automation Science and EngineeringAugust 2021, 1-17
• 2 articleE.Emilie Leblong, B.Bastien Fraudet, L.Louise Devigne, M.Marie Babel, F.François Pasteau, B.Benoit Nicolas and P.Philippe Gallien. SWADAPT1: assessment of an electric wheelchair-driving robotic module in standardized circuits: a prospective, controlled repeated measure design pilot study.Journal of NeuroEngineering and Rehabilitation181September 2021, 1-12
• 3 articleM.Monica Malvezzi, F.Francesco Chinello, D.Domenico Prattichizzo and C.Claudio Pacchierotti. Design of personalized wearable haptic interfaces to account for fingertip size and shape.IEEE Transactions on Haptics (ToH)142April 2021, 266 - 272
• 4 articleB.Beatriz Pascual-Escudero, A.Abhilash Nayak, S.Sébastien Briot, O.Olivier Kermorgant, P.Philippe Martinet, M.Mohab Safey El Din and F.François Chaumette. Complete Singularity Analysis for the Perspective-Four-Point Problem.International Journal of Computer Vision1294April 2021, 1217–1237
• 5 articleM.Mario Selvaggio, J.Jonathan Cacace, C.Claudio Pacchierotti, F.Fabio Ruggiero and P.Paolo Robuffo Giordano. A Shared-control Teleoperation Architecture for Nonprehensile Object Transportation.IEEE Transactions on RoboticsJune 2021, 1-15
• 6 inproceedingsR.Ramana Sundararaman, C.Cédric De Ameida Braga, E.Eric Marchand and J.Julien Pettré. Tracking Pedestrian Heads in Dense Crowd.CVPR 2021 - IEEE/CVF Conference on Computer Vision and Pattern RecognitionVirtual, United StatesIEEEJune 2021, 1-11
## 11.2 Publications of the year
### International journals
• 7 articleM.Marco Aggravi, A. A.Ahmed Alaaeldin Said Elsherif, P.Paolo Robuffo Giordano and C.Claudio Pacchierotti. Haptic-enabled decentralized control of a heterogeneous human-robot team for search and rescue in partially-known environments.IEEE Robotics and Automation Letters63March 2021, 4843-4850
• 8 articleM.Marco Aggravi, D. A.Daniel A L Estima, A.Alexandre Krupa, S.Sarthak Misra and C.Claudio Pacchierotti. Haptic teleoperation of flexible needles combining 3D ultrasound guidance and needle tip force feedback.IEEE Robotics and Automation Letters63March 2021, 4859-4866
• 9 articleM.Marco Aggravi, C.Claudio Pacchierotti and P.Paolo Robuffo Giordano. Connectivity-Maintenance Teleoperation of a UAV Fleet with Wearable Haptic Feedback.IEEE Transactions on Automation Science and Engineering183June 2021, 1243-1262
• 10 articleM.Marco Aggravi, G.Giuseppe Sirignano, P.Paolo Robuffo Giordano and C.Claudio Pacchierotti. Decentralized control of a heterogeneous human-robot team for exploration and patrolling.IEEE Transactions on Automation Science and EngineeringAugust 2021, 1-17
• 11 articleH.Hugo Brument, G.Gerd Bruder, M.Maud Marchal, A.-H.Anne-Hélène Olivier and F.Ferran Argelaguet Sanz. Understanding, Modeling and Simulating Unintended Positional Drift during Repetitive Steering Navigation Tasks in Virtual Reality.IEEE Transactions on Visualization and Computer Graphics2711November 2021, 4300-4310
• 12 articleT.Thomas Chatagnon, A.-H.Anne-Hélène Olivier, L.Ludovic Hoyet, J.Julien Pettré and C.Charles Pontonnier. Modeling physical interactions in human crowds: a pilot study of individual response to controlled external pushes.Computer Methods in Biomechanics and Biomedical Engineering2021, 1-2
• 13 articleB. C.Beatríz Cabrero Daniel, R.Ricardo Marques, L.Ludovic Hoyet, J.Julien Pettré and J.Josep Blat. A Perceptually-Validated Metric for Crowd Trajectory Quality Evaluation.Proceedings of the ACM on Computer Graphics and Interactive Techniques43September 2021, 1 - 18
• 14 articleX.Xavier De Tinguy, C.Claudio Pacchierotti, A.Anatole Lécuyer and M.Maud Marchal. Capacitive Sensing for Improving Contact Rendering with Tangible Objects in VR.IEEE Transactions on Visualization and Computer Graphics274April 2021, 2481-2487
• 15 articleS.Samuel Felton, P.Pascal Brault, E.Elisa Fromont and E.Eric Marchand. Visual Servoing in Autoencoder Latent Space.IEEE Robotics and Automation Letters2022
• 16 articleM.Mathieu Gonzalez, A.Amine Kacete, A.Albert Murienne and E.Eric Marchand. L6DNet: Light 6 DoF Network for Robust and Precise Object Pose Estimation with Small Datasets.IEEE Robotics and Automation Letters62April 2021, 2914-2921
• 17 articleS.Salma Jiddi, P.Philippe Robert and E.Eric Marchand. Detecting Specular Reflections and Cast Shadows to Estimate Reflectance and Illumination of Dynamic Indoor Scenes.IEEE Transactions on Visualization and Computer Graphics282February 2022, 1249 - 1260
• 18 articleE.Emilie Leblong, B.Bastien Fraudet, L.Louise Devigne, M.Marie Babel, F.François Pasteau, B.Benoit Nicolas and P.Philippe Gallien. SWADAPT1: assessment of an electric wheelchair-driving robotic module in standardized circuits: a prospective, controlled repeated measure design pilot study.Journal of NeuroEngineering and Rehabilitation181September 2021, 1-12
• 19 articleA.Aleksander Lillienskiold, R.Rahaf Rahal, P.Paolo Robuffo Giordano, C.Claudio Pacchierotti and E.Ekrem Misimi. Human-Inspired Haptic-Enabled Learning from Prehensile Move Demonstrations.IEEE Transactions on Systems, Man, and Cybernetics: SystemsJanuary 2021, 1-12
• 20 articleS.Sean Lynch, R.Richard Kulpa, L. A.Laurentius Antonius Meerhoff, A.Anthony Sorel, J.Julien Pettré and A.-H.Anne-Hélène Olivier. Influence of path curvature on collision avoidance behaviour between two walkers.Experimental Brain Research2391January 2021, 329-340
• 21 articleM.Monica Malvezzi, F.Francesco Chinello, D.Domenico Prattichizzo and C.Claudio Pacchierotti. Design of personalized wearable haptic interfaces to account for fingertip size and shape.IEEE Transactions on Haptics (ToH)142April 2021, 266 - 272
• 22 articleV.Victor Mercado, M.Maud Marchal and A.Anatole Lécuyer. ENTROPiA: Towards Infinite Surface Haptic Displays in Virtual Reality Using Encountered-Type Rotating Props.IEEE Transactions on Visualization and Computer Graphics273March 2021, 2237-2243
• 23 articleV. R.Victor Rodrigo Mercado, M.Maud Marchal and A.Anatole Lécuyer. “Haptics On-Demand”: A Survey on Encountered-Type Haptic Displays.IEEE Transactions on Haptics (ToH)143July 2021, 449-464
• 24 articleY.Youssef Michel, R.Rahaf Rahal, C.Claudio Pacchierotti, P.Paolo Robuffo Giordano and D.Dongheui Lee. Bilateral teleoperation with adaptive impedance control for contact tasks.IEEE Robotics and Automation LettersMarch 2021, 8
• 25 articleG.Gennaro Notomista, C.Claudio Pacchierotti and P.Paolo Robuffo Giordano. Online Robot Trajectory Optimization for Persistent Environmental Monitoring.IEEE Control Systems LettersSeptember 2021, 1-7
• 26 articleA.Alexander Oliva, P.Paolo Robuffo Giordano and F.François Chaumette. A General Visual-Impedance Framework for Effectively Combining Vision and Force Sensing in Feature Space.IEEE Robotics and Automation Letters63July 2021, 4441-4448
• 27 articleB.Beatriz Pascual-Escudero, A.Abhilash Nayak, S.Sébastien Briot, O.Olivier Kermorgant, P.Philippe Martinet, M.Mohab Safey El Din and F.François Chaumette. Complete Singularity Analysis for the Perspective-Four-Point Problem.International Journal of Computer Vision1294April 2021, 1217–1237
• 28 articleM.Mario Selvaggio, J.Jonathan Cacace, C.Claudio Pacchierotti, F.Fabio Ruggiero and P.Paolo Robuffo Giordano. A Shared-control Teleoperation Architecture for Nonprehensile Object Transportation.IEEE Transactions on RoboticsJune 2021, 1-15
• 29 articleW.Wouter Van Toll, T.Thomas Chatagnon, C.Cédric Braga, B.Barbara Solenthaler and J.Julien Pettré. SPH crowds: Agent-based crowd simulation up to extreme densities using fluid dynamics.Computers and Graphics98June 2021, 306-321
• 30 articleW.Wouter Van Toll and J.Julien Pettré. Algorithms for Microscopic Crowd Simulation: Advancements in the 2010s.Computer Graphics Forum4022021
• 31 articleX.Xi Wang, M.Marc Christie and E.Eric Marchand. Binary Graph Descriptor for Robust Relocalization on Heterogeneous Data.IEEE Robotics and Automation Letters2022
• 32 articleB.Bingqing Zhang, J.Javad Amirian, H.Harry Eberle, J.Julien Pettré, C.Catherine Holloway and T.Tom Carlson. From HRI to CRI: Crowd Robot Interaction—Understanding the Effect of Robots on Crowd Motion.International Journal of Social RoboticsJune 2021, 1-13
### International peer-reviewed conferences
• 33 inproceedingsR.Robin Adili, B.Benjamin Niay, K.Katja Zibrek, A.-H.Anne-Hélène Olivier, J.Julien Pettré and L.Ludovic Hoyet. Perception of Motion Variations in Large-Scale Virtual Human Crowds.MIG 2021 - 14th Annual ACM SIGGRAPH Conference on Motion, Interaction and GamesVirtual Event Switzerland, FranceACMNovember 2021, 1-7
• 34 inproceedingsP.Pascal Brault, Q.Quentin Delamare and P.Paolo Robuffo Giordano. Robust Trajectory Planning with Parametric Uncertainties.ICRA 2021 - IEEE International Conference on Robotics and AutomationXi'an, ChinaMay 2021, 11095-11101
• 35 inproceedingsStudying the Influence of Translational and Rotational Motion on the Perception of Rotation Gains in Virtual Environments.SUI 2021 - Symposium on Spatial User InteractionVirtual Event, United StatesNovember 2021, 1-12
• 36 inproceedingsN.Nicola De Carli, P.Paolo Salaris and P.Paolo Robuffo Giordano. Online Decentralized Perception-Aware Path Planning for Multi-Robot Systems.MRS 2021 - 3rd IEEE International Symposium on Multi-Robot and Multi-Agent SystemsCambridge, United KingdomIEEENovember 2021, 1-9
• 37 inproceedingsS.Sarah Delmas, F.Fabio Morbidi, G.Guillaume Caron, J.Julien Albrand, M.Meven Jeanne-Rose, L.Louise Devigne and M.Marie Babel. SpheriCol: A Driving Assistance System for Power Wheelchairs Based on Spherical Vision and Range Measurements.SII 2021 - 13th IEEE/SICE International Symposium on System IntegrationIwaki, JapanIEEEJanuary 2021, 505-510
• 38 inproceedingsA Plane-based Approach for Indoor Point Clouds Registration.ICPR 2020 - 25th International Conference on Pattern RecognitionMilan (Virtual), ItalyJanuary 2021, 7072-7079
• 39 inproceedingsK.Ketty Favre, M.Muriel Pressigout, E.Eric Marchand and L.Luce Morin. Plane-based Accurate Registration of Real-world Point Clouds.SMC 2021 - IEEE International Conference on Systems, Man, and CyberneticsMelbourne / Virtual, AustraliaIEEEOctober 2021, 1-6
• 40 inproceedingsS.Samuel Felton, E.Elisa Fromont and E.Eric Marchand. Siame-se(3): regression in se(3) for end-to-end visual servoing.ICRA 2021 - IEEE International Conference on Robotics and AutomationXi'an, ChinaIEEEMay 2021, 14454-14460
• 41 inproceedingsF.Fabien Grzeskowiak, D.David Gonon, D.Daniel Dugas, D.Diego Paez-Granados, J. J.Jen Jen Chung, J.Juan Nieto, R.Roland Siegwart, A.Aude Billard, M.Marie Babel and J.Julien Pettré. Crowd against the machine: A simulation-based benchmark tool to evaluate and compare robot capabilities to navigate a human crowd.ICRA 2021 - IEEE International Conference on Robotics and AutomationXian, ChinaMay 2021, 3879-3885
• 42 inproceedingsT.Thomas Howard, G.Guillaume Gicquel, M.Maud Marchal, A.Anatole Lécuyer and C.Claudio Pacchierotti. PUMAH : Pan-tilt Ultrasound Mid-Air Haptics.WHC 2021 - IEEE World Haptics ConferenceMontréal / Virtual, CanadaJuly 2021, 1
• 43 inproceedingsI.Inès Lacôte, C.Claudio Pacchierotti, M.Marie Babel, M.Maud Marchal and D.David Gueorguiev. Generating apparent haptic motion for assistive devices.WHC 2021 - IEEE World Haptics ConferenceMontréal / Virtual, CanadaIEEEJuly 2021, 1
• 44 inproceedingsV. R.Victor Rodrigo Mercado, T.Thomas Howard, H.Hakim Si-Mohammed, F.Ferran Argelaguet Sanz and A.Anatole Lécuyer. Alfred: the Haptic Butler On-Demand Tangibles for Object Manipulation in Virtual Reality using an ETHD.WHC 2021 IEEE World Haptics ConferenceMontreal, FranceIEEEJuly 2021, 373-378
• 45 inproceedingsL.Lendy Mulot, G.Guillaume Gicquel, Q.Quentin Zanini, W.William Frier, M.Maud Marchal, C.Claudio Pacchierotti and T.Thomas Howard. DOLPHIN: A Framework for the Design and Perceptual Evaluation of Ultrasound Mid-Air Haptic Stimuli.SAP 2021 - ACM Symposium on Applied PerceptionRennes, FranceSeptember 2021, 1-10
• 46 inproceedingsI.Iana Podkosova, K.Katja Zibrek, J.Julien Pettré, L.Ludovic Hoyet and A.-H.Anne-Hélène Olivier. Exploring behaviour towards avatars and agents in immersive virtual environments with mixed-agency interactions.VR 2021 - 28th IEEE Conference on Virtual Reality and 3D User InterfacesLisbon, PortugalIEEE2021
• 47 inproceedingsP.Pierre Raimbaud, A.Alberto Jovane, K.Katja Zibrek, C.Claudio Pacchierotti, M.Marc Christie, L.Ludovic Hoyet, J.Julien Pettré and A.-H.Anne-Hélène Olivier. Reactive Virtual Agents: A Viewpoint-Driven Approach for Bodily Nonverbal Communication.IVA 2021: ACM International Conference on Intelligent Virtual AgentsVirtual Event Japan, FranceACMSeptember 2021, 164-166
• 48 inproceedingsA.Agniva Sengupta, A.Alexandre Krupa and E.Eric Marchand. Visual Tracking of Deforming Objects Using Physics-based Models.ICRA 2021 - IEEE International Conference on Robotics and AutomationXi'an, ChinaIEEEMay 2021, 14178-14184
• 49 inproceedingsR.Ramana Sundararaman, C.Cédric De Ameida Braga, E.Eric Marchand and J.Julien Pettré. Tracking Pedestrian Heads in Dense Crowd.CVPR 2021 - IEEE/CVF Conference on Computer Vision and Pattern RecognitionVirtual, United StatesIEEEJune 2021, 1-11
• 50 inproceedingsG.Guillaume Vailland, L.Louise Devigne, F.François Pasteau, F.Florian Nouviale, B.Bastien Fraudet, E.Emilie Leblong, M.Marie Babel and V.Valérie Gouranton. VR based Power Wheelchair Simulator: Usability Evaluation through a Clinically Validated Task with Regular Users.VR 2021 - IEEE Conference on Virtual Reality and 3D User InterfacesLisbon, PortugalIEEEMarch 2021, 1-8
• 51 inproceedingsG.Guillaume Vailland, V.Valérie Gouranton and M.Marie Babel. Cubic Bézier Local Path Planner for Non-holonomic Feasible and Comfortable Path Generation.ICRA 2021 - IEEE International Conference on Robotics and AutomationXian, ChinaIEEEMay 2021, 7894-7900
• 52 inproceedingsS.Sebastian Vizcay, P.Panagiotis Kourtesis, F.Ferran Argelaguet Sanz, C.Claudio Pacchierotti and M.Maud Marchal. Electrotactile Feedback For Enhancing Contact Information in Virtual Reality.ICAT-EGVE 2021 - International Conference on Artificial Reality and Telexistence and Eurographics Symposium on Virtual EnvironmentsSankt Augustin, GermanySeptember 2021
• 53 inproceedingsX.Xi Wang, M.Marc Christie and E.Eric Marchand. TT-SLAM: Dense Monocular SLAM for Planar Environments.ICRA 2021 - IEEE International Conference on Robotics and AutomationXi'an, ChinaMay 2021, 11690-11696
• 54 inproceedingsZ.Zane Zake, F.François Chaumette, N.Nicolò Pedemonte and S.Stéphane Caro. Control Stability Workspace for a Cable-Driven Parallel Robot Controlled by Visual Servoing.Cable-Driven Parallel Robots, Proceedings of the 5th International Conference on Cable-Driven Parallel RobotsProceedings of the 5th International Conference on Cable-Driven Parallel Robots, CableCon 2021virtual, FranceJune 2021, 284-296
• 55 inproceedingsZ.Zane Zake, F.François Chaumette, N.Nicolò Pedemonte and S.Stéphane Caro. Moving-Platform Pose Estimation for Cable-Driven Parallel Robots.IROS 2021 - IEEE/RSJ International Conference on Intelligent Robots and SystemsPrague, Czech RepublicJune 2021, 1-8
• 56 inproceedingsZ.Zane Zake, F.François Chaumette, N.Nicolò Pedemonte and S.Stéphane Caro. Visual Servoing of Cable-Driven Parallel Robots with Tension Management.ICRA 2021 - IEEE International Conference on Robotics and AutomationXi'an, ChinaIEEEMay 2021, 6861-6867
### National peer-reviewed Conferences
• 57 inproceedingsRecalage basé plans de nuages de points pour la navigation dans des environnements structurés.ORASIS 2021 - 18ème édition des journées francophones des jeunes chercheurs en vision par ordinateurSaint Ferréol, FranceSeptember 2021, 1-8
### Doctoral dissertations and habilitation theses
• 58 thesisJ.Javad Amirian. Human motion trajectory prediction for robot navigation.Université de Rennes 1July 2021
• 59 thesisF.Fabien Grzeskowiak. Crowd simulation and experiments for the evaluation of robot navigation in public places.Université Rennes 1June 2021
### Other scientific publications
• 60 articleG.Guillaume Gravier, E.Elisa Fromont, N.Nicolas Courty, T.Teddy Furon, C.Christine Guillemot and P.Paolo Robuffo Giordano. Rennes - une IA souveraine au service de la vie publique.Bulletin de l'Association Française pour l'Intelligence Artificielle2021
• 61 inproceedingsT.Thomas Howard, X.Xavier De Tinguy, G.Guillaume Gicquel, M.Maud Marchal, A.Anatole Lécuyer and C.Claudio Pacchierotti. WeATaViX: Wearable Actuated Tangibles for Virtual Reality Experiences.WHC 2021 - IEEE World Haptics ConferenceMontréal / Virtual, CanadaJuly 2021, 1
• 62 inproceedingsL.Lendy Mulot, G.Guillaume Gicquel, W.William Frier, M.Maud Marchal, C.Claudio Pacchierotti and T.Thomas Howard. Curvature Discrimination for Dynamic Ultrasound Mid-Air Haptic Stimuli.WHC 2021 - IEEE World Haptics ConferenceMontréal / Virtual, CanadaJuly 2021, 1
## 11.3 Other
### Scientific popularization
• 63 miscEtude du comportement piéton en réalité virtuelle.July 2021
## 11.4 Cited publications
• 64 articleP.Paolo Salaris, M.Marco Cognetti, R.Riccardo Spica and P.Paolo Robuffo Giordano. Online Optimal Perception-Aware Trajectory Generation.IEEE Transactions on Robotics2019, 1-16 | 2022-07-01 20:21:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 26, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27048957347869873, "perplexity": 9955.708898562887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103945490.54/warc/CC-MAIN-20220701185955-20220701215955-00161.warc.gz"} |
https://www.snapsolve.com/solutions/Writethe-dimensional-formula-of-angular-momentum-Is-it-scale-or-vector--1672380080025601 | Home/Class 11/Physics/
Write the dimensional formula of angular momentum. Is it scale or vector ?
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## QuestionPhysicsClass 11
Write the dimensional formula of angular momentum. Is it scale or vector ?
Angular momentum = $${\left[{M}^{{{1}}}{L}^{{{2}}}{T}^{{-{1}}}\right]}$$. It is a vector.
4.6
4.6
Correct28
Incorrect0
Still Have Question? | 2021-12-08 06:26:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9227414727210999, "perplexity": 7019.457985982673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363445.41/warc/CC-MAIN-20211208053135-20211208083135-00061.warc.gz"} |
http://math.stackexchange.com/questions/258691/moduli-space-of-elliptic-curves-with-c-n-action | # Moduli space of elliptic curves with $C_n$ action
I would like to construct moduli space of elliptic curves with cyclic group $C_n$-action. In other words, I want to classify a pair $(C,\phi)$, where $C$ is an elliptic curve and $\phi:C_n\rightarrow Aut(C)$ is an action, up to $C_n$ equivariant isomorphisms. Is there any good description of this moduli space?
Since endowing $C$ with a $C_n$-action is the same as specifying an $n$-torsion, this may be related to level structure (which unfortunately I don't know anything about).
Note I mean by $Aut(C)$ simply isomorphism group of $C$, not respecting the group structure on $C$.
Note2 I forgot to say an important point; my $C_n$-action is always translation.
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It's not entirely clear what exactly you want. Adding a fixed $n$-torsion point $P$ does not give you an automorphism of the curve, since if $\phi$ is the map $Q\mapsto Q+P$, then $\phi(Q_1+Q_2)\neq \phi(Q_1)+\phi(Q_2)$. In fact, automorphism groups of elliptic curves over number fields are either $C_2$ (generic case - no CM, or CM by a ring with no exciting units), or $C_4$ or $C_6$ (if you have CM by $\mathbb{Q}(i)$, respectively by $\mathbb{Q}(\zeta_3)$). – Alex B. Dec 14 '12 at 15:00
I meant by $Aut(C)$ simply $isomorphism$ group of $C$, not respecting the group structure on $C$. (it is typical to write $Aut(X)$ for the automorphism group of a general variety $X$). Sorry for the confusion. – M. K. Dec 14 '12 at 20:38
Dear M.K., Why not read something about level structures; there are many sources? Also, do your elliptic curves have a fixed point on them? Regards, – Matt E Dec 14 '12 at 20:44
Dear Matt E, I briefly go through some lecture notes on level structure of elliptic curve, but it seems to me that it parametrizes a "pair" of $n$-torsion points and I am not convinced that it may be modified to help my case. As to your second question, elliptic curve always has an origin by definition. – M. K. Dec 14 '12 at 22:56
I forgot to say an important point; my $C_n$-action is always translation. Thank you all for comments clarifying ambiguous points. – M. K. Dec 14 '12 at 22:58 | 2016-05-28 18:37:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8827150464057922, "perplexity": 298.4395982551144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049278042.87/warc/CC-MAIN-20160524002118-00236-ip-10-185-217-139.ec2.internal.warc.gz"} |
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« Embed this message on: March 20, 2010, 12:58:49 pm »
This applet show reflection and refraction in 3D to illustrate
incident ray/reflected ray and refracted ray all lie on the same plane.
The reflected angle is the same as incident angle.
The refracted angle is determined by Snell's law $n_1 \sin\theta_1=n_2 \sin\theta_2$
The intensity for reflected wave and refracted wave were calculated for p-wave or s-wave.
The check box labeled "E" is used to toggle display of Electric field.
The electric field for p-wave and s-wave are perpendicular to each other.
At Browster angle (\theta=0.98 for default case), the p-wave dispappear. (Please DO try it !)
You can use mouse drag and drop to change the view point (viewing angle/rotate the 3D system).
Embed a running copy of this simulation
Embed a running copy link(show simulation in a popuped window)
Full screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list | 2021-04-18 20:58:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23682770133018494, "perplexity": 10147.870211750102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038860318.63/warc/CC-MAIN-20210418194009-20210418224009-00609.warc.gz"} |
https://tex.stackexchange.com/questions/227237/is-it-possible-to-use-the-width-of-one-tikz-node-to-calculate-the-width-of-anoth | # Question
Is it possible to store the variable width of a node and use that width to calculate the text width of another node in TikZ?
# Example
I have a list of similar nodes side-by-side (inline, not overlaid). Let's call the left column of nodes "A" and the right column of nodes "B". Column A nodes contain a number (counter variable, which means the node gets wider at each digit carry-over). Column B nodes should have some text width minus the text width of the corresponding node in column A.
• get width of node A = \fontsize{50}{60}\selectfont\countervalue{}
• text width of node B = (\textwidth) - (width of node A)
I think the answer I am looking for is hidden somewhere in this answer.
Did I understand correctly?
\documentclass[border=5mm,tikz]{standalone}
\usetikzlibrary{calc}
\usepackage{lipsum}
\begin{document}
\begin{tikzpicture}
\foreach\j in {1,10,100}{
\node[outer sep=0] (a-\j) at (0,{-7*log10(\j)}) {\j};
\path let \p1=($(a-\j.east)-(a-\j.west)$)
in
node[anchor=west,text width=\textwidth-\x1] (b-\j) at (a-\j.east)
{\lipsum[1]};
}
\end{tikzpicture}
\end{document}
• Hey yes sort of... I was hoping to learn from this how to derive \x in your example, but I am still unsure of what \x is and where it came from. I expected to see the width of the first node being set somewhere (like in a pgf variable). – Jonathan Komar Feb 8 '15 at 21:55
• @macmadness86: The width of node A is the distance between its west and east anchors. In the example above, the point \p1 is calculated by subtracting A.west from A.east (@percusse used the a-\j as node name instead of A, but the idea is the same). To refer to the x-coordinate of point \p1, you can use \x1, which is equal to the x-coordinate of A.east minus the x-coordinate of A.west. This difference is exactly the width of node A. – Herr K. Feb 8 '15 at 22:21 | 2019-08-23 14:33:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8098059296607971, "perplexity": 1300.5319457207456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318421.65/warc/CC-MAIN-20190823130046-20190823152046-00177.warc.gz"} |
https://proxies-free.com/linear-programming-solution-from-simplex-method-not-satisfying-the-given-constraint-function-and-thus-not-feasible/ | linear programming – Solution from simplex method not satisfying the given constraint function and thus not feasible?
–Attempted solution below–
I am given the following minimization linear programming problem:
$$minimize: 5x_{1} – 6x_{2} + 2x_{3}$$
$$subject to:$$
$$x_{1} + 3x_{2} + 7x_{3} = 8$$
$$-5x_{1} + 6x_{2} -3x_{3} = -8$$
$$x_{1}, x_{2}, x_{3} geq 0$$
I do not require slack variables since the constraint functions are equalities, so I applied the Simplex method as is which began with the following tableau:
$$begin{array}{|c|c|c|c|c|c|c|c|} hline x_{1} & x_{2} & x_{3} & textrm{b} \ hline 1 & 3 & 7 & 8 \ hline -5& 6 & -3 & -8 \ hline 5 & -6 & 2 & 0 \hlineend{array}$$
which made me choose the 3 in the $$x_{2}$$ column as the pivot element, which I divided that row by 3 to make the pivot element into a 1:
$$begin{array}{|c|c|c|c|c|c|c|c|} hline x_{1} & x_{2} & x_{3} & textrm{b} \ hline 1/3 & 1 & 7/3 & 8/3 \ hline -5& 6 & -3 & -8 \ hline 5 & -6 & 2 & 0 \hlineend{array}$$
then I subtracted row 2 by 6 times row 1, and added row 3 by 6 times row 1:
$$begin{array}{|c|c|c|c|c|c|c|c|} hline x_{1} & x_{2} & x_{3} & textrm{b} \ hline 1/3 & 1 & 7/3 & 8/3 \ hline -7 & 0 & -17 & -24 \ hline 7 & 0 & 16 & 16 \hlineend{array}$$
Which is where I stop since the bottom row no longer has any negative values remaining, so the resulting solution is $$x = (0, frac{8}{3}, 0)$$, but if I plug that into the second constraint function it does not equal $$-8$$, so I am confused why it does not work, I don’t think I mischose the pivot element nor did the row operations incorrectly.
Do I need to add slack variables despite the constraint function being equalities? Any hints will be appreciated! | 2020-12-01 08:07:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7598832845687866, "perplexity": 277.5737015633147}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141672314.55/warc/CC-MAIN-20201201074047-20201201104047-00537.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/college-physics-4th-edition/chapter-2-problems-page-70/94 | ## College Physics (4th Edition)
$T =\frac{W}{2~sin~\theta}$
We can consider the left part of the rope and the right part of the rope. The sum of the vertical component of the tension $T$ in each part of the rope is equal in magnitude to the crow's weight. We can find $T$: $T~sin~\theta + T~sin~\theta = W$ $2~T~sin~\theta = W$ $T =\frac{W}{2~sin~\theta}$ | 2020-04-07 05:04:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7616310119628906, "perplexity": 100.69937504921712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371665328.87/warc/CC-MAIN-20200407022841-20200407053341-00323.warc.gz"} |
http://tex.stackexchange.com/questions/197047/draw-parallel-lines-each-one-with-a-fixed-length | # Draw parallel lines each one with a fixed length
I am trying to create parallel lines in tikz. Although I can create those lines, I can't seem to be able to control the length of those lines, which I want them to stop in the red box's face.
My code is
\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.5]
%target
\draw[fill=gray!30,gray!30] (0,3)rectangle(2,-3);
%tracks
\draw[->,thick] (-2,2)--(0,2);
\draw[->,dashed] (0,2)--(-3,-3);
\draw[->,thick] (-2,1)--(2,1);
\draw[->,dashed] (2,1)-- +($(-3,-3)-(0,2)$);
\draw[->,thick] (-2,-1)--(2,-1);
\draw[->,dashed] (2,-1)-- +($(-3,-3)-(0,2)$);
\draw[->,thick] (-2,-2)--(0,-2);
\draw[->,dashed] (0,-2)-- +($(-3,-3)-(0,2)$);
%detector
\draw[red, rotate around={60:(-1,-3)}] (-3,0) rectangle (-2,-6);
\end{tikzpicture}
\end{document}
My output is
Any idea on how to control the length and end point of the parallel line?
-
Assuming the parallel lines are orthogonal to the "detector" then the projection modifiers can be used (see "The Syntax of Projection Modifiers" in the manual), which are drawn in blue below. I've left the original dashed lines in for comparison.
\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}[scale=0.5]
%target
\draw[fill=gray!30,gray!30] (0,3) rectangle(2,-3);
%detector
\draw[red, rotate around={60:(-1,-3)}]
(-3,0) rectangle (-2,-6)
(-2,0) coordinate (a)
(-2,-6) coordinate (b);
%tracks
\draw[->,thick] (-2,2)--(0,2);
\draw[->,dashed] (0,2)--(-3,-3);
\draw[->,thick] (-2,1)--(2,1);
\draw[->,dashed] (2,1)-- +($(-3,-3)-(0,2)$);
\draw[->,thick] (-2,-1)--(2,-1);
\draw[->,dashed] (2,-1)-- +($(-3,-3)-(0,2)$);
\draw[->,thick] (-2,-2)--(0,-2);
\draw[->,dashed] (0,-2)-- +($(-3,-3)-(0,2)$);
\draw [blue, solid, ->] (0,2) -- ($(a)!(0,2)!(b)$);
\draw [blue, solid, ->] (2,1) -- ($(a)!(2,1)!(b)$);
\draw [blue, solid, ->] (2,-1) -- ($(a)!(2,-1)!(b)$);
\draw [blue, solid, ->] (0,-2) -- ($(a)!(0,-2)!(b)$);
\end{tikzpicture}
\end{document}
-
Except the foreach part, I managed to write the identical code. But too slow :P – percusse Aug 19 '14 at 16:28
For a dirty and quick fix, I would suggest using polar coordinates to keep the lines parallel. Then you can use the second parameter to approximate the needed length to hit the detector:
\begin{tikzpicture}[scale=0.5]
%target
\draw[fill=gray!30,gray!30] (0,3)rectangle(2,-3);
%tracks
\draw[->,thick] (-2,2)--(0,2);
\draw[->,dashed] (0,2)--+(240:5.8);
\draw[->,thick] (-2,1)--(2,1);
\draw[->,dashed] (2,1)-- +(240:5.9);
\draw[->,thick] (-2,-1)--(2,-1);
\draw[->,dashed] (2,-1)-- +(240:4.2);
\draw[->,thick] (-2,-2)--(0,-2);
\draw[->,dashed] (0,-2)-- +(240:2.3);
%detector
\draw[red, rotate around={60:(-1,-3)}] (-3,0) rectangle (-2,-6);
\end{tikzpicture}
- | 2015-07-01 21:39:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8483191132545471, "perplexity": 6678.3103691648985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095270.70/warc/CC-MAIN-20150627031815-00044-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://onethingsimple.com/2017/10/sync-materialized-views-after-debounce-period/ | # Sync PSQL Materialized Views After Debounce Period
## Sync PSQL Materialized View After Updates and Debounce Period
Abstract:
You have a PSQL database utilizing a “MV” materialized view and want an automated process to determine if the MV should be refreshed because the underlying tables have changed. It should also wait a period of time between the last known update to a table and then execute the REFRESH MATERIAL VIEW command.
This article will show you how to set up a couple of functions to handle that process on as many MVs as you need. Materialized Views are like actual tables in that their data is stored in a table rather than queried each time it is used, and they are like views in that you can’t modify the data in the MV… and so in order for a materialized view to properly reflect the data in the tables it gets the original data from there has to be an explicit instruction to refresh. Refreshing takes both time and resources, refreshing a materialized view may take long seconds or minutes to complete. In older versions of Postgresql this would even cause data to become temporarily inaccessible (and still will without the use of the CONCURRENTLY parameter circa Postgresql >= 9.4).
Note on Postgresql versions: I am using PostgreSQL 9.5 while writing this article, I’ll assume you’re using Postgresql ≥9.5. The reason for this is that the REFRESH MATERIALIZED VIEW parameter CONCURRENTLY wasn’t added until approx 9.4 I believe and ON CONFLICT wasn’t available until 9.5 as I’ve come to understand. The ON CONFLICT can be worked around super simple-easy enough read my notes at end of article
Note See my notes at the bottom of the page for a JavaScript function that can build out the entire SQL for everything, including setting up triggers on all your tables. All it requires is a JavaScript array of your table names and the material views they effect. (this is optional, you can write it all by hand too)
### The Solution (step by step):
We’ll want to take into account:
• Does the materialized view need to be refreshed?
• How long since the last event occurred that might cause the materialized view to becoming out of sync with the data it represents?
• Is that amount of time sufficient to conclude that changes are not likely to be made again? Because more changes mean more refreshes, more resources measurable by units of time/CPU/money. (We’ll label the minimum amount of time to wait after the most recent change was made to initiate a refresh as the “debounce” time)
• Refresh the materialized view.
• Whenever changes are made to relevant tables, the timestamp will be recorded as an update to a table we’ll build with the sole purpose of reaching the above conclusions.
The project I used this technique for has two materialized views in our project, but one uses the other and so I don’t need to track them separately. I appreciate that is not the case for everyone and so I wrote this to account for the possibility of multiple independent materialized views.
##### Step 1. Create a Table To Record Latest Updates
We’ll create a table to keep track of when specific tables have been changed. This table “table_update_records” will be an audit of the latest changes to all tables that contain data our materialized view/s are meant to represent (as well as the update time of the materialized view). We’ll use these times later to determine if the debounce time has passed, which is our buffer to reduce the likelihood that we’ll need to make subsequent refreshes to an MV.
Columns in the table_update_records table:
id table_name rel udpated
Type: SERIAL VARCHAR(50) VARCHAR(50) TIMESTAMP
Purpose: Primary Key When was the update Table that was updated MV that would be effected by update
Notes: only required to track multiple MV refresh plans
If you’re not using Postgresql >= 9.5 then you will need to insert a row for each table that is a dependency of some materialized view (and the MV needs a row as well). Otherwise, you don’t need to do this manually as we’ll be using “upserts”. I feel having to do it manually is more prone to error, however, let’s insert some rows by hand to the table_update_records table in order to check out that it’s what we expect it to be and quickly review the difference between an update and an upsert before continuing. Imagine 2 tables and a materialized view respectively as (employees, profiles, and mv_profiles)
##### Step 2. Create a Function To Trigger Record Updates
The table table_update_records won’t update itself, let’s write a trigger function to do the updating for us. This function will trigger on any tables related to specific materialized views after tables perform some INSERT, UPDATE or DELETE.
I wrote two versions of this function. Both versions take a single argument (the materialized views table name), PSQL Trigger Functions can’t have explicit arguments so we’ll pass the argument when we set the trigger and access it on the TG_ARGV[] arguments array at “runtime”. Both also run after any INSERT, UPDATE, DELETE query and update the appropriate table_update_records.updated value to current_time.
Version A just assumes that if an UPDATE/INSERT/DELETE was made to a table then something was actually changed (no need to compare OLD and NEW row values for each row:
ALTERNATIVE
Version B will run after any INSERT, UPDATE, DELETE query, but it will check each row that was touched by the query and test that something is actually different.
If you want to make sure that a change was actually made, and that an update didn’t occur which simply left the table in the same state it was in before, then you can replace the above function with the following alternative version.
version B doesn’t assume that every UPDATE/INSERT/DELETE that is ran actually changes something, it checks each row involved for changes (I didn’t use this version because the likelihood of it being useful in my apps case is almost zero, if you feel that false positives “ie: refreshing materialized views for no reason” could be a problem then this version is for you):
#### Step 4. Create a function to compare update times and refresh materialized views if needed
Basically, we are going to write a Postgresql function that will check the table_update_records that audits most-recent updates and for some materialized view mat_view:
• Query the table_update_records table and see if the last updated table with rel = mat_view was a table, not the actual materialized view mat_view.
• If so… has it has been longer than the debounce time (ie: 1 minute, thirty-minutes, whatever) from the time of the last update and current time now?
• If so… Refresh the materialized view mat_view, and then update table_update_records.updated value where table_update_records.table_name is mat_view. So essentially after a refresh, we record an updated time of a change in the Materialized View row on table_update_records,
The way we know if the materialized view is out of sync is if the time of its last update is earlier than the update of a table with rel equal to the materialized view. So this means that we’ll need a row in the table_update_records row for the materialized view. I’ve made sure this will be added if not present with an upsert in the below function, you may enter it manually if you need to replace the upsert in the function with a normal insert.
The following function is not a trigger, it’s just a function, you will need to set up some mechanism to run it on a schedule. You may want to use a cron job, or if running on AWS a Lambda function.
GOTCHAS:
• Use SECURITY DEFINER in the declaration so that whichever user you use to
execute the function will have correct permissions to refresh tables. In
other words, you want a role/user on the cron job with low low priviledges,
but you need those priviledges to be able to run the commands inside the
function (duh’), so SECURITY DEFINER runs the function as the user who
defined it.
• Make sure to watch the order of the age() function, think of it like subtraction, the left value must be larger than the right or you’ll end up with a negative. Since the current_timestamp will always be greater than a moment from the past, age(current_timestamp, some_old_timestamp) should always be a positive time value. You can compare it directly to a string representation of some time unit, like “> ‘10 minutes’” or “< 1 hour”.
• Materialized Views cannot REFRESH with the CONCURRENTLY parameter if the materialized view hasn’t been populated yet or if it doesn’t have a unique index. The latter being the more likely point of failure. So if you didn’t include a unique index on your materialized view, then you can add one now. It can be a combination of columns, but it cannot include a WHERE clause. You might even want to alter the view, add a new column that stands as a unique index if the alternative is making a 5 column unique index.
So something like:
##### Step 5. Creating a cron job.
Now all that is left is to schedule the following line of SQL to run every 10 minutes, or 1 hour, or once a day. The two arguments to the function should be the ‘debounce_time’ and the ‘rel’ value (materialized view table name).
Remember, the debounce time isn’t the same as the cron job time. You might have this function run every 1 minutes with a debounce time of 15 minutes, which results in a 15 - 16-minute window between when the last change related to the data in our materialized view was made and the view refreshing.
I will leave the “how” up to you, basically, you need to create some script (bash, python, javascript) and a cron job or pgAdmin scheduled procedure. Since my app is on AWS I don’t have pgAdmin setup and so I chose AWS Lambda and a Python function. Easy peasy lemon squeezy.
###### Tracking multiple tables
You can set up multiple cron jobs or just make multiple SQL queries that change the arguments passed into the fn_cron_smart_mv_sync so it can track multiple tables.
###### The output from function return
Expect a call like select fn_cron_smart_mv_sync('10 minutes', 'mv_profiles'); will return results such as: | 2018-04-27 00:59:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2927781641483307, "perplexity": 1638.3742023428533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125948738.65/warc/CC-MAIN-20180427002118-20180427022118-00053.warc.gz"} |
http://mathhelpforum.com/advanced-applied-math/104665-truck-problem.html | # Math Help - truck problem
1. ## truck problem
A heavy truck of weight 400kN is shown in Fig. 4, parked on an extremely steep hill of slope theta with its hand
brake applied to the rear brakes only. Determine the reactions between the road and the truck’s wheels in
terms of theta. Find the steepest value of theta that the truck can manage without sliding if the coefficient of friction
between the tyres and the slippery road is m = 0.8. Would it be better if the truck were parked facing downhill?
Could someone explain how to solve this problem? Cheers.
2. Hello Haris
Originally Posted by Haris
A heavy truck of weight 400kN is shown in Fig. 4, parked on an extremely steep hill of slope theta with its hand
brake applied to the rear brakes only. Determine the reactions between the road and the truck’s wheels in
terms of theta. Find the steepest value of theta that the truck can manage without sliding if the coefficient of friction
between the tyres and the slippery road is m = 0.8. Would it be better if the truck were parked facing downhill?
Could someone explain how to solve this problem? Cheers.
Resolve horizontally, resolve vertically and take moments (for example, about the point of contact of the rear wheel with the road).
Don't forget to include the weight of the truck - apart from that, you have all the forces you need.
Then say that if the rear wheel is on the point of slipping, the friction force $= \mu \times$ normal contact force.
Provided you can deal OK with the sines and cosines of the angle $\theta$, it's very straightforward.
If you can't get it to work, post the equations that you've written down, and we'll take it from there.
3. I really am stuck. First of all how would you resolve the forces? Cheers.
4. Hello Haris
Call the points of contact with the road of the rear wheel and front wheel R and F respectively, and the respective normal contact forces $N_R$ and $N_F$. Call the friction force at R, $F_R$. All measured in kN.
Resolve vertically: $N_R\cos\theta+N_F\cos\theta+F_R\sin\theta = 400$ (1)
Resolve horizontally: $N_R\sin\theta+N_F\sin\theta=F_R\cos\theta$ (2)
Take moments about R: $400(3-1.3\tan\theta)\cos\theta=7N_F$ (3)
Equation (3) gives you $N_F$ directly. Substitute this into (1) and (2) and solve for the other two forces.
P.S.
It gives simpler equations if you resolve at right angles to the plane:
$N_R+N_F=400\cos\theta$ (1)
... and up the plane:
$F_R=400\sin\theta$ (2) | 2015-10-07 17:32:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24226197600364685, "perplexity": 1011.8100263994636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737875203.46/warc/CC-MAIN-20151001221755-00036-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://zabawki-smyka.pl/mine/755/Jan_1610290177/ | reactor for manufacture of nitrogen monoxide
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# reactor for manufacture of nitrogen monoxide
### reactor for manufacture of nitrogen monoxide
2020-2-27 Manufacture of Nitric Acid And Calcium Ammonium Nitrate in IFI, Arklow • The oxidation of nitrogen monoxide • The absorption of nitrogen dioxide in water. 1 . The heart of the nitric acid plant is the catalytic reactor where the mixture of ammonia and air reacts to form nitrogen monoxide
### 7.1: Catalytic Converters - Chemistry LibreTexts
Catalytic converters are used in exhaust systems to provide a site for the oxidation and reduction of toxic by-products (like nitrogen oxides, carbon monoxide, and hydrocarbons) of fuel into less hazardous substances such as carbon dioxide, water vapor, and nitrogen gas.
### The Haber Process - Chemistry LibreTexts
Composition. The proportions of nitrogen and hydrogen: The mixture of nitrogen and hydrogen going into the reactor is in the ratio of 1 volume of nitrogen to 3 volumes of hydrogen. Avogadro's Law says that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. That means that the gases are going into the reactor in the ratio of 1 molecule of nitrogen
### Technologies for Hydrogen, Carbon Monoxide, and
ATR (AutoThermal Reactor) POX (Partial Oxidation) Steam Methane Reformer (SMR) SMR is one of the predominant technologies for producing raw syngas (a gas mixture of hydrogen and carbon monoxide). Where only pure H2 is required as product, a water
### Nitric acid - Essential Chemical Industry
2021-3-3 · The nitrogen oxides are reduced to nitrogen. For example: The temperature at which the reaction is most effective depends on the fuel. For hydrogen, it is ca 450 K. For methane, it is much higher, ca 750 K. The acid from the absorption towers contains
### The Manufacture of Nitric Acid | Johnson Matthey ...
2021-3-5 · The manufacture of nitrogen fertilisers represents by far the largest proportion of the use of nitric acid. These fertilisers, generally with high nitrogen contents, provide the active nitrogen in the form of ammonium nitrate or as nitrophosphate formed by
### Ammonia Synthesis for Fertilizer Production
2015-8-19 · nitrogen to achieve a higher overall conversion (approximately 18% of reactants are converted to ... today becomes feedstock for the manufacture of urea, a more stable nitrate used for fertilizer. However, the modern syntheses of ammonia and urea require several necessary and costly
### Haber Process for the Production of Ammonia
2019-12-2 · All the carbon monoxide (CO) in the mixture is oxidised to CO 2 using steam and an iron oxide catalyst: CO(g) + H 2 O(g) iron oxide catalyst -----> H 2 (g) + CO 2 (g) The carbon dioxide (CO 2) is removed using a suitable base so that only the nitrogen gas (N 2) and hydrogen gas (H 2) remain and are used in the production of ammonia (NH 3).
### 13.1 Nitrogen Compounds | A* Chemistry
13.1 Nitrogen Compounds The lack of reactivity of nitrogen. Nitrogen, N2 exists as a diatomic molecule, two nitrogen atoms are bonded by a triple bond; Nitrogen is very unreactive because the bond energy is very high (about +944 kJ mol⁻¹) and reactions involving nitrogen tend to break the entire bond.; However, nitrogen still undergoes the following reactions:
### CHAPTER 13: Nitrogen and Sulfur - Mega Lecture
2020-4-24 · 4) Nitrogen and oxygen gas are fed into the reactor in a ratio of 1:3, which is the one demanded by the equation. Excess of reactants are not used because it wastes the space in the reactor and decrease the efficiency of the catalyst, since the excess reactants will have nothing to react with. 5) i.
### High Pressure Reactor- suflux
High Pressure Reactor Ilshin Autoclave’s High Pressure Reactor is divided into [Multi-purpose Reactor] for various tests and [Special Reactor] for a production and special tests. Many other compo..
### Technologies for Hydrogen, Carbon Monoxide, and
ATR (AutoThermal Reactor) POX (Partial Oxidation) Steam Methane Reformer (SMR) SMR is one of the predominant technologies for producing raw syngas (a gas mixture of hydrogen and carbon monoxide). Where only pure H2 is required as product, a water gas shift step is employed using steam as reactant to convert most of the CO in raw syngas to CO2 ...
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### The Haber Process for the manufacture of ammonia
2021-3-4 · This page describes the Haber Process for the manufacture of ammonia from nitrogen and hydrogen, and then goes on to explain the reasons for the conditions used in the process. It looks at the effect of temperature, pressure and catalyst on the composition of the equilibrium mixture, the rate of the reaction and the economics of the process ...
### The Manufacture of Nitric Acid | Johnson Matthey ...
2021-3-2 · The manufacture of nitrogen fertilisers represents by far the largest proportion of the use of nitric acid. These fertilisers, generally with high nitrogen contents, provide the active nitrogen in the form of ammonium nitrate or as nitrophosphate formed by the action of nitric acid on phosphate rock.
### Selection of carbon catalysts for the industrial ...
The reactor exit stream was combined with a nitrogen gas flow of 1 dm 3 min −1. The carbon monoxide and phosgene concentrations in the diluted reaction product were monitored continuously via IR spectroscopy – the mixture being passed through a 10 cm path length continuous flow IR gas cell equipped with NaCl windows.
### Dumas nitrogen determination - Lab Solutions
The nitrogen oxides are first reduced to elementary nitrogen in the reduction reactor (RF) while the secondary products, water and carbon dioxide, are separated in special trans (F1 to F3). A gas flow consisting of helium and nitrogen remains, from which the nitrogen can be measured using a thermal conductivity detector.
### Manufacturing of ammonia (Haber process
2017-4-13 · Raw materials • The raw material use for manufacture of ammonia are air water and hydrocarbons. • Coal can also be used in place of hydrocarbons 6. Haber process • Haber process for manufacture of ammonia from nitrogen and hydrogen this process also explain the conditions used in the process such as temperature pressure catalyst.
### 13.1 Nitrogen Compounds | A* Chemistry
13.1 Nitrogen Compounds The lack of reactivity of nitrogen. Nitrogen, N2 exists as a diatomic molecule, two nitrogen atoms are bonded by a triple bond; Nitrogen is very unreactive because the bond energy is very high (about +944 kJ mol⁻¹) and reactions involving nitrogen tend to break the entire bond.; However, nitrogen still undergoes the following reactions:
### Balances on Reactive Processes | Elementary Princ
"In a real coal gasification reactor, sulfur in the coal would form hydrogen sulfide in the product gas, nitrogen in the coal would form $\mathrm{N}_{2}$, some of the carbon monoxide formed in the first reaction would react with steam to form carbon dioxide and more hydrogen, and some of the carbon in the coal would react with hydrogen to form ...
### Selection of carbon catalysts for the industrial ...
2013-12-10 · • Phosgene used industrially in the manufacture of polyurethanes, polycarbonates, pharmaceuticals and agrochemicals. • Gas phase reaction of chlorine with an excess of carbon monoxide over an activated carbon catalyst • Highly exothermic process with peak temperatures reaching over 500oC • Issue with catalyst lives in different plants
### The Haber Process for the manufacture of ammonia
2021-3-4 · This page describes the Haber Process for the manufacture of ammonia from nitrogen and hydrogen, and then goes on to explain the reasons for the conditions used in the process. It looks at the effect of temperature, pressure and catalyst on the composition of the equilibrium mixture, the rate of the reaction and the economics of the process ...
### Ammonia - Essential Chemical Industry
2 天前 · (b) The manufacture of ammonia (The Haber Process) The heart of the process is the reaction between hydrogen and nitrogen in a fixed bed reactor. The gases, in stoichiometric proportions, are heated and passed under pressure over a catalyst (Figure 3). Figure 3 A diagram illustrating a conventional synthesis reactor (a converter).
### Dumas nitrogen determination - Lab Solutions
The nitrogen oxides are first reduced to elementary nitrogen in the reduction reactor (RF) while the secondary products, water and carbon dioxide, are separated in special trans (F1 to F3). A gas flow consisting of helium and nitrogen remains, from which the nitrogen can be measured using a thermal conductivity detector.
### NOx and CO removal | Nitrogen oxides | Carbon
2021-3-3 · manufacture custom catalysts for specific tasks. > See All; Equipment. We can provide a complete range of proprietary equipment, spare parts and consumables, designed and manufactured to work optimally. > See All; Process licensing. We are involved in shaping the solutions and new technologies that customers will base their business on in the ...
### Urea manufacturing process - SlideShare
2016-8-25 · The second reactor recieves the gas from the first reactor and recycle solution from the decomposition and concentration sections. Conversion of carbon dioxide to urea is approximately 60% at a pressure of 50 barg. The solution is then purified in the same process as was used for the liquid from the first reactor. 15. | 2021-10-21 14:34:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5291811227798462, "perplexity": 4177.07739578848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00371.warc.gz"} |
https://reverievisuals.com/2019/10/11/change-default-tmux-key-binding.html | # Changing tmux Default Key Binding
Bt default, command key bindings are prefixed by ctrl+b. For example, to vertically split a window type ctrl+b %.
After splitting a window into multiple panes, a pane can be resized by the hitting prefix key (e.g. ctrl+b) and, while continuing to hold ctrl, press left/right/up/down. Swapping panes is achived in the same manner, but by hitting o instead of a directional key.
Key bindings may be changed with the bind and unbind commands in tmux.conf. For example, the default prefix binding of ctrl+b can be changed to ctrl+a by adding the following commands in your configuration file:
unbind C-b
set -g prefix C-a
bind C-a send-prefix
Tip: Quote special characters to use them as prefix. You may also use alt (called Meta) instead of ctrl. For example: set -g prefix m-'\' | 2020-10-28 22:42:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2784240245819092, "perplexity": 11121.87444044273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107902038.86/warc/CC-MAIN-20201028221148-20201029011148-00125.warc.gz"} |
https://blog.liox.eu/2010/02/09/cartagen---maps-for-your-website/ | # Cartagen - maps for your website
· by Peter · Read in about 1 min · (68 words) ·
[iframe http://cartagen.org/find/brno,-czech?fullscreen=true 500 300]
Cartagen is web service which allows you to embed maps on your website using single line of code. Result is visible above and is quite cool, you can even zoom and navigate on this map. To achieve it on your page, simply put this in body of your page:
<iframe height='300' src='http://cartagen.org/find/brno,-czech?fullscreen=true' style='border:0;' width='500'/>
And here you have map of my city of residence:) | 2018-12-11 09:11:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38951918482780457, "perplexity": 5033.611141300305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823614.22/warc/CC-MAIN-20181211083052-20181211104552-00058.warc.gz"} |
https://zbmath.org/?q=an:0283.58010 | # zbMATH — the first resource for mathematics
The central limit theorem for geodesic flows on $$n$$-dimensional manifolds of negative curvature. (English) Zbl 0283.58010
##### MSC:
37D99 Dynamical systems with hyperbolic behavior 28D05 Measure-preserving transformations 53C20 Global Riemannian geometry, including pinching 60F05 Central limit and other weak theorems
Full Text:
##### References:
[1] S. N. Bernstein, Sur l’éxtension du théorème limite du calcul des probabilités aux sommes de quantitiés dépendantes, Math. Ann.97 (1926), 1–59. · JFM 52.0517.03 · doi:10.1007/BF01447859 [2] R. Bowen,Markov partitions for axiom Adiffeomorphisms, Amer. J. Math.92 (1970), 725–747. · Zbl 0208.25901 · doi:10.2307/2373370 [3] R. Bowen,Symbolic dynamics for hyperbolic flows (to appear). · Zbl 0336.58009 [4] W. Feller,An introduction to probability theory and its applications, Vol. 1, New York. · Zbl 0077.12201 [5] B. M. Gurevič,The structure of increasing decompositions for special flows, Theor. Probability Appl.10 (1965), 627–654, MR35 #3034. · doi:10.1137/1110077 [6] I. A. Ibragimov,Some limit theorems for stationary processes, Theor. Probablity Appl.7 (1962), 349–382. · Zbl 0119.14204 · doi:10.1137/1107036 [7] V. P. Leonov,On the dispersion of time-dependent means of a stationary stochastic process, Theor. Probability, Appl.6 (1961), 87–93. · Zbl 0128.12701 · doi:10.1137/1106007 [8] W. Parry,Intrinsic Markov chains, Trans. Amer. Math. Soc.112 (1964), 55–66. · Zbl 0127.35301 · doi:10.1090/S0002-9947-1964-0161372-1 [9] M. Ratner,Central limit theorem for Anosov flows on three-dimensional manifolds, Soviet Math. Dokl.10 (1969). · Zbl 0292.60052 [10] M. Ratner,Invariant measure with respect to an Anosov flows on a three-dimensional manifold, Soviet Math. Dokl.10 (1969). · Zbl 0188.26602 [11] M. Ratner,Markov partitions for Anosov flows on n-dimensional manifolds (to appear). · Zbl 0269.58010 [12] Y. G. Sinai,The central limit theorem for geodesic flows on manifolds of constant negative curvature, Soviet Math. Dokl.1 (1960), 938–987. · Zbl 0129.31103 [13] Y. G. Sinai,Markov partitions and C-diffeomorphisms, Functional. Anal. Appl.2 (1968), 64–89. · Zbl 0182.55003 · doi:10.1007/BF01075361 [14] Y. G. Sinai,Gibbs measures in ergodic theory, Uspehi Mat. Nauk (4)27 (1972), 21–63. · Zbl 0246.28008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-04-16 09:09:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7106156945228577, "perplexity": 6301.880469723025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038088731.42/warc/CC-MAIN-20210416065116-20210416095116-00374.warc.gz"} |
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## HOCHSCHILD COHOMOLOGY FOR ABSTRACT CONVEXITY AND SHANNON ENTROPY
Prelegent: TOMASZ MASZCZYK
2023-02-22 17:15
The Shannon entropy was introduced as a statistical measure of information loss but appeared in other fields of mathematics as well. We plan to sketch its relations with polylogarithms and motives after Cathelineau, Dupont, Bloch, Goncharov, Elbaz-Vincent, and Gangl, a cohomological interpretation by Kontsevich, and the information cohomology after Baudot and Bennequin. In the latter approach, Shannon entropy is a one-cocycle. Next, we survey the Faddeev algebraic-characterization theorem and the Fundamental Equation of Information Theory after Tverberg, Kendall, and Lee. Then we will sketch Gromov’s program and comment on the categorical interpretation by Baez, Fritz, and Leinster. Finally, we plan to present another cohomological derivation of Shannon entropy based on a new kind of Hochschild cohomology we construct for abstract convexity. The latter admits a cohomological interpretation of extensions of convex bodies by vector spaces, which are parallel to Hochschild extensions of associative algebras by square-zero ideals. Then, the Shannon entropy arises from a two-cocycle where the cocycle condition is the Fundamental Equation of Information Theory. | 2023-03-23 10:54:59 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8174246549606323, "perplexity": 3199.139093711489}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945144.17/warc/CC-MAIN-20230323100829-20230323130829-00758.warc.gz"} |
https://zbmath.org/serials/?q=se%3A4000 | ## The Kluwer International Series in Engineering and Computer Science
Short Title: Kluwer Int. Ser. Eng. Comput. Sci. Publisher: Kluwer Academic Publishers, Boston, MA ISSN: 0893-3405 Online: https://link.springer.com/bookseries/6524 Successor: Analog Circuits and Signal Processing Comments: Book series; No longer indexed
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5 Koob, Gary M. 4 Brayton, Robert K. 4 Sangiovanni-Vincentelli, Alberto L. 3 Gray, Robert Molten 3 Lau, Clifford G. Y. 2 Abramowicz, Witold 2 Benjamin, D. Paul 2 Bergé, Jean-Michel 2 Bestavros, Azer 2 Bhanu, Bir 2 Blahut, Richard E. 2 Bookman, Lawrence A. 2 Buttazzo, Giorgio C. 2 Camposano, Raul 2 Diniz, Paulo Sergio Ramirez 2 Gallager, Robert G. 2 Gao, GuangRong 2 Grizzaffi Maynard, Ann Marie 2 John, Lizy Kurian 2 Kacsuk, Peter 2 Kilov, Haim 2 Lavrač, Nada 2 Liu, Huan 2 Menezes, Alfred J. 2 Minker, Wolfgang M. 2 Motoda, Hiroshi 2 Pan, Yi 2 Pedrycz, Witold 2 Rutenbar, Rob A. 2 Sheu, Bing J. 2 Steyaert, Michiel S. J. 2 Sutherland, Stuart 2 Thrun, Sebastian 2 van Tilborg, André M. 2 van Tilborg, Henk C. A. 2 Vanstone, Scott Alexander 2 Venetsanopoulos, Anastasios N. 2 Walker, Robert A. 2 Zadeh, Lotfi Asker 1 Abd-El-Hafiz, Salwa K. 1 Abraham, Santosh G. 1 Agrawal, Vishwani D. 1 Airiau, Roland 1 Akansu, Ali N. 1 Akşit, Mehmet 1 Allstot, David J. 1 Anderson, John B. 1 Armstrong-Hélouvry, Brian 1 Augustin, Larry M. 1 Avizienis, Algirdas 1 Avresky, Dimiter R. 1 Baciu, George 1 Bajdechi, Ovidiu 1 Balafoutis, C. A. 1 Balarin, Felice 1 Bargiela, Andrzej 1 Bartlett, Marian Stewart 1 Basili, Victor R. 1 Bayoumi, Magdy A. 1 Bennacef, Samir 1 Beveridge, J. Ross 1 Bevinakoppa, Savitri 1 Bhattacharyya, Shuvra S. 1 Biemond, Jan 1 Bini, Gilberto 1 Blackburn, Robert L. 1 Blake, Ian F. 1 Blaum, Mario 1 Blinovsky, Vladimir Markovich 1 Boel, René K. 1 Bohanec, Marko 1 Bolsens, Ivo 1 Bramley, Randall B. 1 Brazdil, Pavel B. 1 Breuer, Peter T. 1 Brown, Ian D. 1 Brown, Stephen D. 1 Browne, James C. 1 Burger, Wilhelm 1 Bushnell, Michael L. 1 Cai, Kaiyuan 1 Canny, John F. 1 Cao, Xi-Ren 1 Carlberg, Ken 1 Carley, L. Richard 1 Castillo, Enrique F. 1 Catthor, Francky 1 Cauwenberghs, Gert 1 Cavin, Ralph K. III 1 Chakradhar, Srimat T. 1 Chan, Alvin 1 Chan, Stephen Chi-fai 1 Chen, Kefei 1 Chiodo, Massimiliano 1 Chipman, Susan F. 1 Chiprout, Eli 1 Choi, Joongho 1 Chomicki, Jan 1 Chow, Paul Kai-On 1 Cios, Krzysztof J. ...and 419 more Authors
all top 5
### Fields
241 Computer science (68-XX) 67 Information and communication theory, circuits (94-XX) 53 General and overarching topics; collections (00-XX) 31 Systems theory; control (93-XX) 10 Numerical analysis (65-XX) 9 Statistics (62-XX) 9 Biology and other natural sciences (92-XX) 8 Mathematical logic and foundations (03-XX) 8 Number theory (11-XX) 8 Operations research, mathematical programming (90-XX) 6 Probability theory and stochastic processes (60-XX) 6 Mechanics of particles and systems (70-XX) 3 Algebraic geometry (14-XX) 2 Combinatorics (05-XX) 2 Field theory and polynomials (12-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Commutative algebra (13-XX) 1 Ordinary differential equations (34-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Convex and discrete geometry (52-XX) 1 Fluid mechanics (76-XX) 1 Optics, electromagnetic theory (78-XX)
### Citations contained in zbMATH Open
162 Publications have been cited 1,832 times in 1,733 Documents Cited by Year
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Complexity of lattice problems. A cryptographic perspective. Zbl 1140.94010
Micciancio, Daniele; Goldwasser, Shafi
2002
Gradient estimation via perturbation analysis. Foreword by Yu-Chi Ho. Zbl 0746.90024
Glasserman, Paul
1991
Scale-space theory in computer vision. Zbl 0812.68040
Lindeberg, Tony
1993
Perturbation analysis of discrete event dynamic systems. Zbl 0744.90036
Ho, Yu-Chi; Cao, Xi-Ren
1991
Feature selection for knowledge discovery and data mining. Zbl 0908.68127
Liu, Huan; Motoda, Hiroshi
1998
Granular computing. An introduction. Zbl 1046.68052
Bargiela, Andrzej; Pedrycz, Witold
2003
Finite fields for computer scientists and engineers. Zbl 0662.94014
McEliece, Robert J.
1987
Uniform random numbers. Theory and practice. Zbl 0841.65004
Tezuka, Shu
1995
Applications of finite fields. Zbl 0779.11059
Blake, Ian F.; Gao, XuHong; Mullin, Ronald C.; Vanstone, Scott A.; Yaghoobian, Tomik
1993
Effective polynomial computation. Zbl 0794.11048
Zippel, Richard
1993
Optimization of stochastic models. The interface between simulation and optimization. Zbl 0909.90220
Pflug, Georg Ch.
1996
Logic minimization algorithms for VLSI synthesis. Zbl 0565.94020
Brayton, Robert K.; Hachtel, Gary D.; McMullen, Curtis T.; Sangiovanni- Vincentelli, Alberto L.
1984
Finite commutative rings and their applications. Zbl 1095.13032
Bini, Gilberto; Flamini, Flaminio
2002
Elliptic curve public key cryptosystems. Foreword by Neal Koblitz. Zbl 0806.94011
Menezes, Alfred J.
1993
Nonlinear digital filters: principles and applications. Zbl 0719.93080
Pitas, I.; Venetsanopoulos, A. N.
1990
Control of machines with friction. Zbl 0782.93003
Armstrong-Hélouvry, Brian
1991
Modeling and control of logical discrete event systems. Zbl 0875.68980
Kumar, Ratnesh; Garg, Vijay K.
1995
Synchronous programming of reactive systems. Zbl 0828.68038
Halbwachs, Nicolas
1993
Iterative identification and restoration of images. Zbl 0752.68093
Lagendijk, Reginald L.; Biemond, Jan
1991
Steady-state methods for simulating analog and microwave circuits. Zbl 0723.94009
Kundert, Kenneth S.; White, Jacob K.; Sangiovanni-Vincentelli, Alberto
1990
Conditional Monte Carlo: gradient estimation and optimization applications. Zbl 0874.62093
Fu, Michael; Hu, Jian-Qiang
1997
Feature extraction, construction and selection. A data mining perspective. Zbl 0912.00012
1998
Directed sonar sensing for mobile robot navigation. Zbl 0760.68008
Leonard, John J.; Durrant-Whyte, Hugh F.
1992
Hard real-time computing systems. Predictable scheduling algorithms and applications. Zbl 0890.68016
Buttazzo, Giorgio C.
1997
Deadline scheduling for real-time systems. EDF and related algorithms. Zbl 0931.68136
Stankovic, John A.; Spuri, Marco; Ramamritham, Krithi; Buttazzo, Giorgio C.
1998
Introduction to fuzzy reliability. Zbl 0877.68006
Cai, Kai-Yuan
1996
Real-time systems. Design principles for distributed embedded applications. Zbl 0930.68148
Kopetz, Hermann
1997
On optimal interconnections for VLSI. Zbl 0826.68067
Kahng, Andrew B.; Robins, Gabriel
1995
Fault-tolerant parallel computation. Zbl 0881.68050
Kanellakis, Paris Christos; Shvartsman, Alex Allister
1997
Finite fields: normal bases and completely free elements. Zbl 0864.11065
Hachenberger, Dirk
1997
An introduction to error correcting codes with applications. Zbl 0726.94006
Vanstone, Scott A.; van Oorschot, Paul C.
1989
Asymptotic waveform evaluation and moment matching for interconnect analysis. Zbl 0798.94002
Chiprout, Eli; Nakhla, Michel S.
1994
Managing uncertainty in expert systems. Zbl 0751.68069
Grzymala-Busse, Jerzy W.
1991
Stability of finite and infinite dimensional systems. Zbl 0916.93002
Gil’, Michael I.
1998
Data mining methods for knowledge discovery. Zbl 0912.68199
Cios, Krzysztof; Pedrycz, Witold; Swiniarski, Roman
1998
Machine learning. Discriminative and generative. Zbl 1030.68073
Jebara, Tony
2004
Fuzzy if-then rules in computational intelligence. Theory and applications. Zbl 1027.68118
2000
Data mining in finance. Advances in relational and hybrid methods. Zbl 0944.91027
Kovalerchuk, B. Ya.; Vityaev, E. E.
2000
Asymptotic combinatorial coding theory. Zbl 0879.94036
Blinovsky, V.
1997
Synchronization in real-time systems: a priority inheritance approach. Zbl 0753.68004
Rajkumar, Ragunathan
1991
Robot force control. Zbl 0940.93006
Siciliano, Bruno; Villani, Luigi
1999
Turbo codes. Principles and applications. Zbl 1011.94028
Vucetic, Branka; Yuan, Jinhong
2000
The TSQL2 temporal query language. Zbl 0859.68014
1995
Variation principle in informational macrodynamics. Zbl 1058.94007
2003
Adaptive filters: structures, algorithms, and applications. Zbl 0565.94001
Honig, Michael L.; Messerschmitt, David G.
1984
Switching and traffic theory for integrated broadband networks. Foreword by Robert G. Gallager. Zbl 0711.94023
Hui, Joseph Y.
1990
An introduction to fuzzy logic applications in intelligent systems. Zbl 0755.68018
1992
Perturbation techniques for flexible manipulators. Zbl 0794.93074
Fraser, Anthony R.; Daniel, Ron W.
1991
Nonholonomic motion planning. Zbl 0875.00053
1992
Loop tiling for parallelism. Zbl 0964.68025
Xue, Jingling
2000
Error detecting codes. General theory and their application in feedback communication systems. Zbl 0902.94029
Kløve, Torleiv; Korzhik, Valery I.
1995
Adaptive filtering. Algorithms and practical implementation. 2nd ed. Zbl 1145.93300
Diniz, Paulo Sergio Ramirez
2002
Discrete stochastic processes. Zbl 0925.60005
Gallager, Robert G.
1995
Artificial neural networks. Learning algorithms, performance evaluation, and applications. Zbl 0817.68121
Karayiannis, Nicolaos B.; Venetsanopoulos, A. N.
1992
Genetic programming and data structures: genetic programming + data structures = automatic programming! With a foreword by John R. Koza. Zbl 0899.68016
Langdon, William B.
1998
Fixed interval smoothing for state space models. Zbl 0972.62069
Weinert, Howard L.
2001
Knowledge discovery and measures of interest. Zbl 0992.68046
Hilderman, Robert J.; Hamilton, Howard J.
2001
Coordination of distributed problem solvers. Zbl 1081.68501
Durfee, Edmund
1988
Source coding theory. Zbl 0824.94008
Gray, Robert M.
1990
Neural network parallel computing. Zbl 0899.68095
Takefuji, Yoshiyasu
1992
Neural networks and speech processing. Zbl 0749.68079
Morgan, David P.; Scofield, Christopher L.
1991
Monte Carlo device simulation: Full band and beyond. Zbl 0753.00014
1991
Binary decision diagrams and applications for VLSI CAD. Zbl 0846.68022
Minato, Shin-ichi
1996
Fundamentals of cryptology. A professional reference and interactive tutorial. Incl. 1 CD-ROM. Zbl 0991.94002
van Tilborg, Henk C. A.
2000
Image segmentation and compression using hidden Markov models. Zbl 0984.68178
Li, Jia; Gray, Robert M.
2000
Parallel programming and compilers. Zbl 1081.68563
Polychronopoulos, Constantine D.
1988
Incremental version-space merging: a general framework for concept learning. Zbl 0744.68111
Hirsh, Haym
1990
The SECD microprocessor. A verification case study. Zbl 0757.68051
Graham, Brian T.
1992
Adaptive filtering. Algorithms and practical implementation. Zbl 1145.93301
Diniz, Paulo Sergio Ramirez
1997
Explanation-based neural network learning. A lifelong learning approach. Zbl 0861.68071
Thrun, Sebastian
1996
Compositional translation. Zbl 0900.68148
Rosetta, M. T.
1994
Load balancing in parallel computers. Theory and practice. Zbl 0879.68034
Xu, Chengzhong; Lau, Francis C. M.
1996
Bayesian modeling of uncertainty in low-level vision. Zbl 0716.68091
Szeliski, Richard
1989
Computer analysis of visual textures. Zbl 0814.68133
Tomita, Fumiaki; Tsuji, Saburo
1990
Dynamic analysis of robot manipulators: a Cartesian tensor approach. Zbl 0743.70008
Balafoutis, C. A.; Patel, R. V.
1991
Measurement of image velocity. Zbl 0758.68059
Fleet, David J.
1992
Explorations in automatic thesaurus discovery. Zbl 0818.68069
Grefenstette, Gregory
1994
Introduction to convolutional codes with applications. Zbl 0826.94001
Dholakia, Ajay
1994
Hardware-software co-design of embedded systems. The POLIS approach. Zbl 0878.68133
Balarin, Felice; Chiodo, Massimiliano; Giusto, Paolo; Hsieh, Harry; Jurecska, Attila; Lavagno, Luciano; Passerone, Claudio; Sangiovanni-Vincentelli, Alberto; Sentovich, Ellen; Suzuki, Kei; Tabbara, Bassam
1997
Turbo coding. Zbl 0996.94535
Heegard, Chris; Wicker, Stephen B.
1999
Generating abstraction hierarchies. An automated approach to reducing search in planning. Zbl 0822.68102
Knoblock, Craig A.
1993
Foundations of knowledge acquisition: machine learning. Zbl 0820.68098
1993
Perceptual metrics for image database navigation. Incl. 1 CD-ROM. Zbl 0973.68661
Rubner, Yossi; Tomasi, Carlo
2001
Coding approaches to fault tolerance in combinational and dynamic systems. Zbl 0988.68015
2002
Probability distributions involving Gaussian random variables. A handbook for engineers and scientists. Zbl 1135.60005
Simon, Marvin K.
2002
Automating knowledge acquisition for expert systems. Zbl 1081.68695
1988
Learning with nested generalized exemplars. Zbl 0699.68111
Salzberg, Steven L.
1990
Foundations of real-time computing: scheduling and resource management. Zbl 0756.90052
1991
Enabling technologies for computational science. Frameworks, middleware and environments. Zbl 0943.00027
2000
Floberg, Henrik
1997
Automatic learning techniques in power systems. Zbl 0891.68080
Wehenkel, Louis A.
1998
Genetic learning for adaptive image segmentation. Zbl 0819.68140
Bhanu, Bir; Lee, Sungkee
1994
Timed Boolean functions. A unified formalism for exact timing analysis. Zbl 0837.68046
Lam, William K. C.; Brayton, Robert K.
1994
CMOS integrated analog-to-digital and digital-to-analog converters. 2nd ed. Zbl 1042.94031
van de Plassche, Rudy
2003
Ontology learning for the semantic Web. Zbl 1005.68126
Maedche, Alexander
2002
Analog device-level layout automation. Zbl 0819.68016
Cohn, John M.; Garrod, David J.; Rutenbar, Rob A.; Carley, L. Richard
1994
Wireless personal communications: research developments. Zbl 0880.94002
1995
Software synthesis from dataflow graphs. Zbl 0923.68002
Battacharyya, Shuvra S.; Murthy, Praveen K.; Lee, Edward A.
1996
Petri nets in flexible and agile automation. Zbl 0851.00035
1995
QoS in packet networks. Zbl 1061.68014
Park, Kun I.
2005
Machine learning. Discriminative and generative. Zbl 1030.68073
Jebara, Tony
2004
Flexible neuro-fuzzy systems. Structures, learning and performance evaluation. With a foreword by Lotfi A. Zadeh. Zbl 1080.93014
Rutkowski, Leszek
2004
Web caching and its applications. Zbl 1054.68009
Nagaraj, S. V.
2004
Systematic design of sigma-delta analog-to-digital converters. Zbl 1072.94001
Bajdechi, Ovidiu; Huijsing, Johan H.
2004
Granular computing. An introduction. Zbl 1046.68052
Bargiela, Andrzej; Pedrycz, Witold
2003
Variation principle in informational macrodynamics. Zbl 1058.94007
2003
CMOS integrated analog-to-digital and digital-to-analog converters. 2nd ed. Zbl 1042.94031
van de Plassche, Rudy
2003
Computer architecture: a minimalist perspective. Zbl 1013.68051
Gilreath, William F.; Laplante, Phillip A.
2003
Complexity of lattice problems. A cryptographic perspective. Zbl 1140.94010
Micciancio, Daniele; Goldwasser, Shafi
2002
Finite commutative rings and their applications. Zbl 1095.13032
Bini, Gilberto; Flamini, Flaminio
2002
Adaptive filtering. Algorithms and practical implementation. 2nd ed. Zbl 1145.93300
Diniz, Paulo Sergio Ramirez
2002
Coding approaches to fault tolerance in combinational and dynamic systems. Zbl 0988.68015
2002
Probability distributions involving Gaussian random variables. A handbook for engineers and scientists. Zbl 1135.60005
Simon, Marvin K.
2002
Ontology learning for the semantic Web. Zbl 1005.68126
Maedche, Alexander
2002
Fixed interval smoothing for state space models. Zbl 0972.62069
Weinert, Howard L.
2001
Knowledge discovery and measures of interest. Zbl 0992.68046
Hilderman, Robert J.; Hamilton, Howard J.
2001
Perceptual metrics for image database navigation. Incl. 1 CD-ROM. Zbl 0973.68661
Rubner, Yossi; Tomasi, Carlo
2001
Face image analysis by unsupervised learning. Zbl 1005.68125
Bartlett, Marian Stewart
2001
Real-time database systems. Architecture and techniques. Zbl 1008.68016
2001
Error coding for engineers. Zbl 1005.94002
Houghton, A.
2001
Modeling from reality. Zbl 0976.68592
2001
Fuzzy if-then rules in computational intelligence. Theory and applications. Zbl 1027.68118
2000
Data mining in finance. Advances in relational and hybrid methods. Zbl 0944.91027
Kovalerchuk, B. Ya.; Vityaev, E. E.
2000
Turbo codes. Principles and applications. Zbl 1011.94028
Vucetic, Branka; Yuan, Jinhong
2000
Loop tiling for parallelism. Zbl 0964.68025
Xue, Jingling
2000
Fundamentals of cryptology. A professional reference and interactive tutorial. Incl. 1 CD-ROM. Zbl 0991.94002
van Tilborg, Henk C. A.
2000
Image segmentation and compression using hidden Markov models. Zbl 0984.68178
Li, Jia; Gray, Robert M.
2000
Enabling technologies for computational science. Frameworks, middleware and environments. Zbl 0943.00027
2000
Soft computing for knowledge discovery. Introducing cartesian granule features. Zbl 0973.68660
Shanahan, James G.
2000
Discrete event systems. Analysis and control. Papers from the 5th workshop, WODES 2000, Ghent, Belgium, August 21–23, 2000. Zbl 0996.00058
2000
Mathematical principles of fuzzy logic. Zbl 0940.03028
Novák, Vilém; Perfilieva, Irina; Močkoř, Jiří
1999
Robot force control. Zbl 0940.93006
Siciliano, Bruno; Villani, Luigi
1999
Turbo coding. Zbl 0996.94535
Heegard, Chris; Wicker, Stephen B.
1999
Wavelet, subband and block transforms in communications and multimedia. Zbl 0998.94502
1999
Parallel numerical computation with applications. Proceedings of the workshop on Frontiers of parallel numerical computations and applications, organized in the IEEE 7th symposium on the Frontiers on massively parallel computers (Frontiers ’99) at Annapolis, MD, USA, February 20–25, 1999. Zbl 0928.00049
1999
Computed synchronization for multimedia applications. Zbl 0948.68006
Owen, Charles B.; Makedon, Fillia
1999
Scheduling in parallel computing systems. Fuzzy and annealing techniques. Zbl 0998.68019
Salleh, Shaharuddin; Zomaya, Albert Y.
1999
Behavioral specifications of businesses and systems. Zbl 0948.68003
1999
Feature selection for knowledge discovery and data mining. Zbl 0908.68127
Liu, Huan; Motoda, Hiroshi
1998
Feature extraction, construction and selection. A data mining perspective. Zbl 0912.00012
1998
Deadline scheduling for real-time systems. EDF and related algorithms. Zbl 0931.68136
Stankovic, John A.; Spuri, Marco; Ramamritham, Krithi; Buttazzo, Giorgio C.
1998
Stability of finite and infinite dimensional systems. Zbl 0916.93002
Gil’, Michael I.
1998
Data mining methods for knowledge discovery. Zbl 0912.68199
Cios, Krzysztof; Pedrycz, Witold; Swiniarski, Roman
1998
Genetic programming and data structures: genetic programming + data structures = automatic programming! With a foreword by John R. Koza. Zbl 0899.68016
Langdon, William B.
1998
Automatic learning techniques in power systems. Zbl 0891.68080
Wehenkel, Louis A.
1998
Trellises and trellis-based decoding algorithms for linear block codes. Zbl 0990.94500
Lin, Shu; Kasami, Tadao; Fujiwara, Toru; Fossorier, Marc
1998
Fuzzy control of industrial systems. Theory and applications. Zbl 0910.93001
Shaw, Ian S.
1998
Wireless CMOS frequency synthesizer design. Zbl 0972.94505
Craninckx, J.; Steyaert, M.
1998
Information retrieval. Algorithms and heuristics. Zbl 0918.68020
Grossman, David A.; Frieder, Ophir
1998
Communication protocol specification and verification. Zbl 0914.68133
Lai, Richard; Jirachiefpattana, Ajin
1998
Parallel computing using optical interconnections. Zbl 0928.68046
1998
Conditional Monte Carlo: gradient estimation and optimization applications. Zbl 0874.62093
Fu, Michael; Hu, Jian-Qiang
1997
Hard real-time computing systems. Predictable scheduling algorithms and applications. Zbl 0890.68016
Buttazzo, Giorgio C.
1997
Real-time systems. Design principles for distributed embedded applications. Zbl 0930.68148
Kopetz, Hermann
1997
Fault-tolerant parallel computation. Zbl 0881.68050
Kanellakis, Paris Christos; Shvartsman, Alex Allister
1997
Finite fields: normal bases and completely free elements. Zbl 0864.11065
Hachenberger, Dirk
1997
Asymptotic combinatorial coding theory. Zbl 0879.94036
Blinovsky, V.
1997
Adaptive filtering. Algorithms and practical implementation. Zbl 1145.93301
Diniz, Paulo Sergio Ramirez
1997
Hardware-software co-design of embedded systems. The POLIS approach. Zbl 0878.68133
Balarin, Felice; Chiodo, Massimiliano; Giusto, Paolo; Hsieh, Harry; Jurecska, Attila; Lavagno, Luciano; Passerone, Claudio; Sangiovanni-Vincentelli, Alberto; Sentovich, Ellen; Suzuki, Kei; Tabbara, Bassam
1997
Floberg, Henrik
1997
TETROBOT. A modular approach to reconfigurable parallel robotics. Zbl 0912.70001
Hamlin, Gregory J.; Sanderson, Arthur C.
1997
Interleaving planning and execution for autonomous robots. Zbl 0865.93003
Nourbakhsh, Illah Reza
1997
Intelligent unmanned vehicles. Autonomous navigation research at Carnegie Mellon. Zbl 0865.93004
1997
Real-time search for learning autonomous agents. Zbl 0879.68090
Ishida, Toru
1997
Intelligent image databases. Towards advanced image retrieval. Zbl 0896.68044
Gong, Yihong
1997
Reasoning with complex cases. Zbl 0945.68147
Gebhardt, Friedrich; Voß, Angi; Gräther, Wolfgang; Schmidt-Belz, Barbara
1997
Optimization of stochastic models. The interface between simulation and optimization. Zbl 0909.90220
Pflug, Georg Ch.
1996
Introduction to fuzzy reliability. Zbl 0877.68006
Cai, Kai-Yuan
1996
Binary decision diagrams and applications for VLSI CAD. Zbl 0846.68022
Minato, Shin-ichi
1996
Explanation-based neural network learning. A lifelong learning approach. Zbl 0861.68071
Thrun, Sebastian
1996
Load balancing in parallel computers. Theory and practice. Zbl 0879.68034
Xu, Chengzhong; Lau, Francis C. M.
1996
Software synthesis from dataflow graphs. Zbl 0923.68002
Battacharyya, Shuvra S.; Murthy, Praveen K.; Lee, Edward A.
1996
Object-oriented behavioral specifications. Zbl 0857.68005
1996
Fault-tolerant real-time systems. The problem of replica determinism. Zbl 0864.68007
Poledna, Stefan
1996
Uniform random numbers. Theory and practice. Zbl 0841.65004
Tezuka, Shu
1995
Modeling and control of logical discrete event systems. Zbl 0875.68980
Kumar, Ratnesh; Garg, Vijay K.
1995
On optimal interconnections for VLSI. Zbl 0826.68067
Kahng, Andrew B.; Robins, Gabriel
1995
The TSQL2 temporal query language. Zbl 0859.68014
1995
Error detecting codes. General theory and their application in feedback communication systems. Zbl 0902.94029
Kløve, Torleiv; Korzhik, Valery I.
1995
Discrete stochastic processes. Zbl 0925.60005
Gallager, Robert G.
1995
Wireless personal communications: research developments. Zbl 0880.94002
1995
Petri nets in flexible and agile automation. Zbl 0851.00035
1995
Computational architectures integrating neural and symbolic processes. A perspective on the state of the art. Zbl 0827.68109
1995
Logic synthesis for field programmable gate arrays. Zbl 0839.68044
Murgai, Rajeev; Brayton, Robert K.; Sangiovanni-Vincentelli, Alberto
1995
Fourier transforms. An introduction for engineers. Zbl 0997.42500
Gray, Robert M.; Goodman, Joseph W.
1995
Co-synthesis of hardware and software for digital embedded systems. Zbl 0846.68050
Gupta, Rajesh Kumar
1995
Asymptotic waveform evaluation and moment matching for interconnect analysis. Zbl 0798.94002
Chiprout, Eli; Nakhla, Michel S.
1994
Compositional translation. Zbl 0900.68148
Rosetta, M. T.
1994
Explorations in automatic thesaurus discovery. Zbl 0818.68069
Grefenstette, Gregory
1994
Introduction to convolutional codes with applications. Zbl 0826.94001
Dholakia, Ajay
1994
Genetic learning for adaptive image segmentation. Zbl 0819.68140
Bhanu, Bir; Lee, Sungkee
1994
Timed Boolean functions. A unified formalism for exact timing analysis. Zbl 0837.68046
Lam, William K. C.; Brayton, Robert K.
1994
Analog device-level layout automation. Zbl 0819.68016
Cohn, John M.; Garrod, David J.; Rutenbar, Rob A.; Carley, L. Richard
1994
A formal approach to hardware design. Zbl 0827.68001
Staunstrup, Jørgen
1994
Physical design for multichip modules. Zbl 0820.68024
Sriram, M.; Kang, S. M.
1994
Mixed-mode simulation and analog multilevel simulation. Zbl 0826.68129
Saleh, Resve A.; Jou, Shyh-Jye; Newton, A. Richard
1994
Modeling with an analog hardware description language. Zbl 0820.68135
Mantooth, H. Alan; Fiegenbaum, Mike
1994
Coded-modulation techniques for fading channels. Zbl 0831.94001
Jamali, S. Hamidreza; Le-Ngoc, Tho
1994
Scale-space theory in computer vision. Zbl 0812.68040
Lindeberg, Tony
1993
...and 62 more Documents
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### Cited by 3,094 Authors
36 Novák, Vilém 15 Cassandras, Christos G. 14 Pedrycz, Witold 13 Futa, Yuichi 13 Heidergott, Bernd F. 13 Shidama, Yasunari 12 Holcapek, Michal 12 Kerre, Etienne E. 12 Kreinovich, Vladik Yakovlevich 11 Štěpnička, Martin 10 Kosheleva, Olga M. 10 Lindeberg, Tony 10 Murinová, Petra 10 Perfilieva, Irina G. 9 Bělohlávek, Radim 9 Dvořák, Antonín 9 Fu, Michael C. 9 Gil’, Michael Iosif 8 Blinovsky, Vladimir Markovich 8 Hachenberger, Dirk 8 Lerner, Vladimir S. 8 Okazaki, Hiroyuki 8 Schwarzmann, Alexander A. 8 Wang, Guojun 7 Cao, Xi-Ren 7 Daňková, Martina 7 De Cock, Martine 7 Rúa, Ignacio F. 7 Venetsanopoulos, Anastasios N. 7 Vychodil, Vilém 7 Xia, Li 6 Cornelis, Chris 6 Gaivoronski, Alexei A. 6 Godo, Lluís 6 Ruan, Da 6 Skowron, Andrzej 6 Vázquez-Abad, Felisa J. 6 Wardi, Yorai 6 Yager, Ronald R. 5 Deschrijver, Glad 5 Emiris, Ioannis Z. 5 Esteva, Francesc 5 Georgiou, Chryssis 5 Gerla, Giangiacomo 5 Gutierrez, Jaime 5 Kowalski, Dariusz R. 5 Kyuregyan, Melsik K. 5 Larrañaga, Pedro 5 Lecerf, Grégoire 5 Močkoř, Jiří 5 Nagy, James Gerard 5 Vetterlein, Thomas 5 Weickert, Joachim 5 Zhou, Hongjun 4 Chen, Hao 4 Costa, Sueli Irene Rodrigues 4 De Baets, Bernard 4 Ermoliev, Yuri M. 4 Faure, Henri 4 Gao, Shuhong 4 Ho, Yu-Chi 4 Hong, Liu Jeff 4 Inza, Iñaki 4 Kyureghyan, Gohar M. 4 Melamed, Benjamin 4 Micciancio, Daniele 4 Norkin, Vladimir I. 4 Panario, Daniel 4 Pawlak, Zdzisław 4 Pflug, Georg Ch. 4 Pitas, Ioannis 4 Schockaert, Steven 4 Sussner, Peter 4 Tadić, Vladislav B. 4 Van Gasse, Bart 4 Volk-Makarewicz, Warren M. 4 von zur Gathen, Joachim 4 Woźniakowski, Henryk 4 Xu, Yang 4 Zinov’ev, Viktor Aleksandrovich 3 Anderson, James H. 3 Blake, Ian F. 3 Bouhamidi, Abderrahman 3 Cai, Kaiyuan 3 Castellano, Giovanna 3 Charpin, Pascale 3 Chen, Gongliang 3 Climent, Joan-Josep 3 Cools, Ronald 3 Dai, Liyi 3 Doucet, Arnaud 3 Elia, Michele 3 Fanelli, Anna Maria 3 Ferrari, Agnaldo José 3 Garefalakis, Theodoulos 3 Gaujal, Bruno 3 Gokbayrak, Kagan 3 Gong, Guang 3 Gottwald, Siegfried 3 Hadjicostis, Christoforos N. ...and 2,994 more Authors
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### Cited in 372 Journals
115 Fuzzy Sets and Systems 53 Information Sciences 45 Theoretical Computer Science 41 Discrete Event Dynamic Systems 40 Automatica 37 Designs, Codes and Cryptography 36 International Journal of Approximate Reasoning 32 European Journal of Operational Research 31 Pattern Recognition 31 Finite Fields and their Applications 27 Journal of Mathematical Imaging and Vision 26 Real-Time Systems 23 Artificial Intelligence 22 Soft Computing 19 Journal of Symbolic Computation 19 Journal of Complexity 18 Discrete Applied Mathematics 18 Journal of Optimization Theory and Applications 18 Mathematics and Computers in Simulation 17 International Journal of Intelligent Systems 16 Information Processing Letters 16 Mathematics of Computation 15 Annals of Operations Research 15 Machine Learning 14 Journal of Computational and Applied Mathematics 13 Applied Mathematics and Computation 13 International Journal of Circuit Theory and Applications 13 Mathematical and Computer Modelling 13 Linear Algebra and its Applications 13 Applicable Algebra in Engineering, Communication and Computing 13 Formalized Mathematics 12 Journal of the Franklin Institute 11 Circuits, Systems, and Signal Processing 11 Mathematical Problems in Engineering 10 Computers & Mathematics with Applications 10 Information and Computation 9 Computer Methods in Applied Mechanics and Engineering 9 Journal of Computer and System Sciences 9 Journal of Automated Reasoning 9 Distributed Computing 8 Problems of Information Transmission 8 Kybernetika 8 International Journal of Parallel Programming 8 Queueing Systems 8 Mathematical Programming. Series A. Series B 8 International Journal of Computer Vision 8 International Journal of Applied Mathematics and Computer Science 7 Discrete Mathematics 7 Signal Processing 7 International Journal of Computer Mathematics 7 Journal of Computer and Systems Sciences International 7 Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology 7 SIAM Journal on Imaging Sciences 6 Journal of Mathematical Analysis and Applications 6 Computing 6 Systems & Control Letters 6 Journal of Computer Science and Technology 6 Computers & Operations Research 6 Formal Aspects of Computing 6 Multidimensional Systems and Signal Processing 6 International Journal of Adaptive Control and Signal Processing 6 The Annals of Applied Probability 6 Numerical Algorithms 6 Computational Statistics and Data Analysis 6 Annals of Mathematics and Artificial Intelligence 6 European Journal of Control 6 Nonlinear Dynamics 6 Journal of Discrete Mathematical Sciences & Cryptography 6 Journal of Mathematical Cryptology 6 The Annals of Applied Statistics 5 Advances in Applied Probability 5 International Journal of General Systems 5 Algorithmica 5 Journal of Intelligent & Robotic Systems 5 International Journal of Robust and Nonlinear Control 5 Journal of Algebra and its Applications 4 Communications in Algebra 4 Journal of Pure and Applied Algebra 4 Naval Research Logistics 4 Cybernetics and Systems 4 Annals of Pure and Applied Logic 4 Journal of Robotic Systems 4 Journal of Cryptology 4 Archive for Mathematical Logic 4 Archive of Applied Mechanics 4 Cybernetics and Systems Analysis 4 Computational Complexity 4 Formal Methods in System Design 4 Mechanism and Machine Theory 4 Probability in the Engineering and Informational Sciences 4 Quantitative Finance 4 Journal of Applied Mathematics 4 Quantum Information Processing 4 Journal of Applied Logic 4 Journal of Field Robotics 3 Acta Informatica 3 International Journal of Systems Science 3 Physica A 3 International Journal for Numerical Methods in Engineering 3 Journal of Algebra ...and 272 more Journals
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### Cited in 55 Fields
710 Computer science (68-XX) 315 Information and communication theory, circuits (94-XX) 209 Systems theory; control (93-XX) 199 Operations research, mathematical programming (90-XX) 191 Number theory (11-XX) 190 Mathematical logic and foundations (03-XX) 156 Numerical analysis (65-XX) 145 Statistics (62-XX) 94 Probability theory and stochastic processes (60-XX) 60 Combinatorics (05-XX) 58 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 53 Commutative algebra (13-XX) 47 Biology and other natural sciences (92-XX) 45 Field theory and polynomials (12-XX) 43 Mechanics of particles and systems (70-XX) 42 Order, lattices, ordered algebraic structures (06-XX) 34 Algebraic geometry (14-XX) 33 Linear and multilinear algebra; matrix theory (15-XX) 20 Ordinary differential equations (34-XX) 19 Mechanics of deformable solids (74-XX) 17 Calculus of variations and optimal control; optimization (49-XX) 17 Convex and discrete geometry (52-XX) 15 Dynamical systems and ergodic theory (37-XX) 13 Quantum theory (81-XX) 12 Optics, electromagnetic theory (78-XX) 11 Measure and integration (28-XX) 11 Partial differential equations (35-XX) 10 Category theory; homological algebra (18-XX) 10 Harmonic analysis on Euclidean spaces (42-XX) 9 Approximations and expansions (41-XX) 9 Statistical mechanics, structure of matter (82-XX) 8 Integral equations (45-XX) 8 Operator theory (47-XX) 7 Group theory and generalizations (20-XX) 6 General algebraic systems (08-XX) 6 Associative rings and algebras (16-XX) 6 Real functions (26-XX) 6 Difference and functional equations (39-XX) 5 History and biography (01-XX) 4 General and overarching topics; collections (00-XX) 4 Differential geometry (53-XX) 4 General topology (54-XX) 4 Fluid mechanics (76-XX) 3 Geometry (51-XX) 2 Nonassociative rings and algebras (17-XX) 2 Functions of a complex variable (30-XX) 2 Functional analysis (46-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Geophysics (86-XX) 1 $$K$$-theory (19-XX) 1 Potential theory (31-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Abstract harmonic analysis (43-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Relativity and gravitational theory (83-XX) | 2023-01-30 11:37:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5053279399871826, "perplexity": 11933.31136966731}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499816.79/warc/CC-MAIN-20230130101912-20230130131912-00043.warc.gz"} |
https://tex.stackexchange.com/questions/249764/answers-package-environ-package-equation-wont-compile | # answers package + environ package + equation won't compile
This MWE works perfectly when the I comment out equation in the teacher environment. With that equation in place I get this error:
(C:\Users\Ethan.Bolker\AppData\Roaming\MiKTeX\2.9\tex\latex\trimspaces\trimspac
es.sty))
! Missing \endcsname inserted.
\csname\endcsname
l.30 \end{teacher}
The code:
\documentclass{article}
\usepackage{amsmath}
\usepackage{environ}
\NewEnviron{teacher}{%
}
\begin{document}
An equation in the master file:
%
$$2+2 = 4$$
\begin{teacher}
In answersout: expand a macro: \LaTeX.
Try an equation:
% comment out the next three lines and the document compiles
$$2+2 = 4$$
\end{teacher}
\end{document}
Some history. A previous problem with the answers package led me to this question: Incomplete \iffalse error using answers package
In fact that was the first error I saw when I encountered this problem. That error message was replaced by the one above when I made my example minimal.
You can substitute the \protected@iwrite mechanism with a stronger one that has the defect that it expands nothing. So if you want to add titles to the teacher's notes you have to work in two steps.
\documentclass{article}
\usepackage{amsmath}
\usepackage{environ}
\newcounter{teacher}
\makeatletter
\NewEnviron{teacher}{%
\stepcounter{teacher}%
(See teacher note \theteacher)
\protect\subsection*{Teacher's note \theteacher}%
}
\begingroup
\def\protected@iwrite##1##2##3{\immediate\write##1{##3}}%
\endgroup
}
\makeatother
\begin{document}
An equation in the master file:
$$2+2 = 4$$
\begin{teacher}
In answersout: expand a macro: \LaTeX.
Try an equation:
$$2+2 = 4$$
\end{teacher}
\section*{Teacher's notes}
\end{document}
The problem here is that \BODY gets expanded before being written to the file. The result isn't plain text, and disaster follows. Protecting doesn't help, because the meaning of \BODY is lost at \end{teacher}. One way around this is to store the bodies of the environments using globally defined macros called \tchI, \tchII, \tchIII, etc. These can be protected, and written to the file.
\documentclass{article}
\usepackage{environ}
\newcounter{tchcount}
\setcounter{tchcount}{0}
\NewEnviron{teacher}{%
\stepcounter{tchcount}
\global\expandafter\let\csname tch\Roman{tchcount}\endcsname\BODY
}
\begin{document}
\begin{teacher}
Try an equation:
$$2+2 = 4$$
\end{teacher}
\begin{teacher}
And another
$$1+1 = 2$$
\end{teacher} | 2019-09-20 18:17:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 6, "x-ck12": 0, "texerror": 0, "math_score": 0.8726028800010681, "perplexity": 3592.955098244619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574058.75/warc/CC-MAIN-20190920175834-20190920201834-00160.warc.gz"} |
https://www.calculus-online.com/exercise/4469 | # Vectors – Calculate the length of diagonals of a parallelogram – Exercise 4469
Exercise
Calculate the lengths of the parallelogram diagonasl built on the vectors
$$\vec{a}=5\vec{p}+2\vec{q}$$
$$\vec{b}=\vec{p}-3\vec{q}$$
Given that
$$|\vec{p}|=2\sqrt{2}, |\vec{q}|=3$$
And the angle between them is equal to 45 degrees.
$$|\vec{a}+\vec{b}|=15$$
$$|\vec{b}-\vec{a}|=\sqrt{593}$$ | 2020-07-08 22:50:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 5, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7572703957557678, "perplexity": 965.6879289118203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897707.23/warc/CC-MAIN-20200708211828-20200709001828-00077.warc.gz"} |
https://existentialtype.wordpress.com/tag/polarization/ | ## Polarity in Type Theory
August 25, 2012
There has recently arisen some misguided claims about a supposed opposition between functional and object-oriented programming. The claims amount to a belated recognition of a fundamental structure in type theory first elucidated by Jean-Marc Andreoli, and developed in depth by Jean-Yves Girard in the context of logic, and by Paul Blain-Levy and Noam Zeilberger in the context of programming languages. In keeping with the general principle of computational trinitarianism, the concept of polarization has meaning in proof theory, category theory, and type theory, a sure sign of its fundamental importance.
Polarization is not an issue of language design, it is an issue of type structure. The main idea is that types may be classified as being positive or negative, with the positive being characterized by their structure and the negative being characterized by their behavior. In a sufficiently rich type system one may consider, and make effective use of, both positive and negative types. There is nothing remarkable or revolutionary about this, and, truly, there is nothing really new about it, other than the terminology. But through the efforts of the above-mentioned researchers, and others, we have learned quite a lot about the importance of polarization in logic, languages, and semantics. I find it particularly remarkable that Andreoli’s work on proof search turned out to also be of deep significance for programming languages. This connection was developed and extended by Zeilberger, on whose dissertation I am basing this post.
The simplest and most direct way to illustrate the ideas is to consider the product type, which corresponds to conjunction in logic. There are two possible ways that one can formulate the rules for the product type that from the point of view of inhabitation are completely equivalent, but from the point of view of computation are quite distinct. Let us first state them as rules of logic, then equip these rules with proof terms so that we may study their operational behavior. For the time being I will refer to these as Method 1 and Method 2, but after we examine them more carefully, we will find more descriptive names for them.
Method 1 of defining conjunction is perhaps the most familiar. It consists of this introduction rule
$\displaystyle\frac{\Gamma\vdash A\;\textsf{true}\quad\Gamma\vdash B\;\textsf{true}}{\Gamma\vdash A\wedge B\;\textsf{true}}$
and the following two elimination rules
$\displaystyle\frac{\Gamma\vdash A\wedge B\;\textsf{true}}{\Gamma\vdash A\;\textsf{true}}\qquad\frac{\Gamma\vdash A\wedge B\;\textsf{true}}{\Gamma\vdash B\;\textsf{true}}$.
Method 2 of defining conjunction is only slightly different. It consists of the same introduction
$\displaystyle \frac{\Gamma\vdash A\;\textsf{true}\quad\Gamma\vdash B\;\textsf{true}}{\Gamma\vdash A\wedge B\;\textsf{true}}$
and one elimination rule
$\displaystyle\frac{\Gamma\vdash A\wedge B\;\textsf{true} \quad \Gamma,A\;\textsf{true},B\;\textsf{true}\vdash C\;\textsf{true}}{\Gamma\vdash C\;\textsf{true}}$.
From a logical point of view the two formulations are interchangeable in that the rules of the one are admissible with respect to the rules of the other, given the usual structural properties of entailment, specifically reflexivity and transitivity. However, one can discern a difference in “attitude” in the two formulations that will turn out to be a manifestation of the concept of polarity.
Method 1 is a formulation of the idea that a proof of a conjunction is anything that behaves conjunctively, which means that it supports the two elimination rules given in the definition. There is no commitment to the internal structure of a proof, nor to the details of how projection operates; as long as there are projections, then we are satisfied that the connective is indeed conjunction. We may consider that the elimination rules define the connective, and that the introduction rule is derived from that requirement. Equivalently we may think of the proofs of conjunction as being coinductively defined to be as large as possible, as long as the projections are available. Zeilberger calls this the pragmatist interpretation, following Count Basie’s principle, “if it sounds good, it is good.”
Method 2 is a direct formulation of the idea that the proofs of a conjunction are inductively defined to be as small as possible, as long as the introduction rule is valid. Specifically, the single introduction rule may be understood as defining the structure of the sole form of proof of a conjunction, and the single elimination rule expresses the induction, or recursion, principle associated with that viewpoint. Specifically, to reason from the fact that $A\wedge B\;\textsf{true}$ to derive $C\;\textsf{true}$, it is enough to reason from the data that went into the proof of the conjunction to derive $C\;\textsf{true}$. We may consider that the introduction rule defines the connective, and that the elimination rule is derived from that definition. Zeilberger calls this the verificationist interpretation.
These two perspectives may be clarified by introducing proof terms, and the associated notions of reduction that give rise to a dynamics of proofs.
When reformulated with explicit proofs, the rules of Method 1 are the familiar rules for ordered pairs:
$\displaystyle\frac{\Gamma\vdash M:A\quad\Gamma\vdash N:B}{\Gamma\vdash \langle M, N\rangle:A\wedge B}$
$\displaystyle\frac{\Gamma\vdash M:A\wedge B}{\Gamma\vdash \textsf{fst}(M):A}\qquad\frac{\Gamma\vdash M:A\wedge B}{\Gamma\vdash \textsf{snd}(M):B}$.
The associated reduction rules specify that the elimination rules are post-inverse to the introduction rules:
$\displaystyle\textsf{fst}(\langle M,N\rangle)\mapsto M \qquad \textsf{snd}(\langle M,N\rangle)\mapsto N$.
In this formulation the proposition $A\wedge B$ is often written $A\times B$, since it behaves like a Cartesian product of proofs.
When formulated with explicit proofs, Method 2 looks like this:
$\displaystyle \frac{\Gamma\vdash M:A\quad\Gamma\vdash M:B}{\Gamma\vdash M\otimes N:A\wedge B}$
$\displaystyle\frac{\Gamma\vdash M:A\wedge B \quad \Gamma,x:A,y:B\vdash N:C}{\Gamma\vdash \textsf{split}(M;x,y.N):C}$
with the reduction rule
$\displaystyle\textsf{split}(M\otimes N;x,y.P)\mapsto [M,N/x,y]P$.
With this formulation it is natural to write $A\wedge B$ as $A\otimes B$, since it behaves like a tensor product of proofs.
Since the two formulations of “conjunction” have different internal structure, we may consider them as two different connectives. This may, at first, seem pointless, because it is easily seen that $x:A\times B\vdash M:A\otimes B$ for some $M$ and that $x:A\otimes B\vdash N:A\times B$ for some $N$, so that the two connectives are logically equivalent, and hence interchangeable in any proof. But there is nevertheless a reason to draw the distinction, namely that they have different dynamics.
It is easy to see why. From the pragmatic perspective, since the projections act independently of one another, there is no reason to insist that the components of a pair be evaluated before they are used. Quite possibly we may only ever project the first component, so why bother with the second? From the verificationist perspective, however, we are pattern matching against the proof of the conjunction, and are demanding both components at once, so it makes sense to evaluate both components of a pair in anticipation of future pattern matching. (Admittedly, in a structural type theory one may immediately drop one of the variables on the floor and never use it, but then why give it a name at all? In a substructural type theory such as linear type theory, this is not a possibility, and the interpretation is forced.) Thus, the verficationist formulation corresponds to eager evaluation of pairing, and the pragmatist formulation to lazy evaluation of pairing.
Having distinguished the two forms of conjunction by their operational behavior, it is immediately clear that both forms are useful, and are by no means opposed to one another. This is why, for example, the concept of a lazy language makes no sense, rather one should instead speak of lazy types, which are perfectly useful, but by no means the only types one should ever consider. Similarly, the concept of an object-oriented language seems misguided, because it emphasizes the pragmatist conception, to the exclusion of the verificationist, by insisting that everything be an object characterized by its methods.
More broadly, it is useful to classify types into two polarities, the positive and the negative, corresponding to the verificationist and pragmatist perspectives. Positive types are inductively defined by their introduction forms; they correspond to colimits, or direct limits, in category theory. Negative types are coinductively defined by their elimination forms; they correspond to limits, or inverse limits, in category theory. The concept of polarity is intimately related to the concept of focusing, which in logic sharpens the concept of a cut-free proof and elucidates the distinction between synchronous and asynchronous connectives, and which in programming languages provides an elegant account of pattern matching, continuations, and effects.
As ever, enduring principles emerge from the interplay between proof theory, category theory, and type theory. Such concepts are found in nature, and do not depend on cults of personality or the fads of the computer industry for their existence or importance.
Update: word-smithing.
Follow | 2015-05-24 02:51:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 21, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7503100633621216, "perplexity": 707.6738612197453}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207927824.81/warc/CC-MAIN-20150521113207-00214-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://twxb.org/twxb/article/abstract/20220501 | 1. 中国科学院紫金山天文台 南京 210023;2. 中国科学院行星科学重点实验室 南京 210023;3. 中国科学院比较行星学卓越创新中心 合肥 230026
P185;
Statistical Analysis of Discoveries and Discovery Scenarios of Near-Earth Asteroids
Author:
Affiliation:
1. Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023;2. Key Laboratory of Planetary Sciences, Chinese Academy of Sciences, Nanjing 210023;3. CAS Center for Excellence in Comparative Planetology, Hefei 230026;
Fund Project:
• 摘要
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• 图/表
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• 访问统计
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• 参考文献
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摘要:
近地小行星是一类可能对地球安全造成潜在威胁的太阳系小天体, 目前绝大部分的近地小行星是由地基望远镜发现的, 且数目仍在不断增加. 为了对我国未来开展近地小行星发现监测提供参考和借鉴, 利用国际小行星中心公开的数据库对所有近地小行星首次发现时刻的观测资料开展了多维度统计分析. 发现望远镜探测能力的限制会对近地小行星的发现造成选择效应, 导致不同轨道类型近地小行星发现的相对比例逐年变化且与直径有关. 另外, 结合数值模拟获得的轨道数据, 对近地小行星首次发现时的观测场景进行了还原, 获得了发现时刻近地小行星位置在不同天球坐标系的分布, 分析了其分布特征与季节、测站纬度和小行星直径的依赖关系. 最后, 通过分析数据定量考察了太阳、月球和银道面对近地小行星发现的影响, 发现地基望远镜一般难以发现来自太阳方向90$^\circ$范围内直径140m以下的近地小行星, 并且随着小行星直径的减小该限制范围也将变大; 月光污染对近地小行星发现的影响也非常显著, 望月前后几天的观测限制可导致约29%的目标无法被发现, 而且分析表明农历上半月发现的目标一般比下半月发现的更难以被跟踪观测; 银道面特别是银心方向会对近地小行星发现产生影响, 使得黄道面附近存在与季节相关的观测盲区''.
Abstract:
Near-Earth asteroids (NEAs) are a kind of small solar system bodies that may lead to potential hazard to the safety of the Earth. Currently, most of the NEAs are discovered with ground-based telescopes while the number is still growing. In order to provide references and experience to our future near-Earth asteroid discovery and monitoring, we perform a multi-dimensionally statistical analysis on the discovery data of NEAs with public database obtained from the website of Minor Planet Center (MPC). We find the constraint of observation ability can lead to selection effect on the discoveries, which causes a yearly dependence trend and a size-dependence characteristic of the relative proportion of different orbit types of discovered NEAs. Besides, combined with the orbits obtained from numerical simulations, we recover the discovery scenarios of these objects. The position distribution of the objects under different celestial coordinate systems are obtained, and the dependence on seasons, observatory latitudes, and the diameters are analyzed. Finally, we quantify the impact of the Sun, the Moon and the galactic plane on the discoveries by analyzing the observation data and find that ground-based telescopes generally have difficulty in discovering NEAs within $90^\circ$ from the Sun direction, and that this limitation generally has a greater impact on smaller-sized objects. The lunar position also has a significant effect on the discoveries, with the restriction on the nights before and after the full Moon resulting in 29% of NEAs being undiscovered, and analysis shows that objects found in the first half of the lunar calendar month are generally more difficult to be followed than those found in the second half. The galactic plane, especially the direction near the galactic center, also has an effect on the discoveries, resulting in a season-dependent blind spot'' for observations near the ecliptic.
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• 收稿日期:2021-11-15
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• 在线发布日期: 2022-09-30
• 出版日期: | 2022-11-26 15:40:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28169506788253784, "perplexity": 4698.007411169938}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446708010.98/warc/CC-MAIN-20221126144448-20221126174448-00065.warc.gz"} |
https://srdas.github.io/MLBook/DataScience.html | # Chapter 1 The Art of Data Science
All models are wrong, but some are useful.” — George E. P. Box and N.R. Draper in “Empirical Model Building and Response Surfaces,” John Wiley & Sons, New York, 1987.
So you want to be a “data scientist”? There is no widely accepted definition of who a data scientist is.1 Several books now attempt to define what data science is and who a data scientist may be, see Patil (2012), Patil (2011), and Loukides (2012). This book’s viewpoint is that a data scientist is someone who asks unique, interesting questions of data based on formal or informal theory, to generate rigorous and useful insights.2 It is likely to be an individual with multi-disciplinary training in computer science, business, economics, statistics, and armed with the necessary quantity of domain knowledge relevant to the question at hand. The potential of the field is enormous for just a few well-trained data scientists armed with big data have the potential to transform organizations and societies. In the narrower domain of business life, the role of the data scientist is to generate applicable business intelligence.
Among all the new buzzwords in business – and there are many – “Big Data” is one of the most often heard. The burgeoning social web, and the growing role of the internet as the primary information channel of business, has generated more data than we might imagine. Users upload an hour of video data to YouTube every second.3 87% of the U.S. population has heard of Twitter, and 7% use it.4 Forty-nine percent of Twitter users follow some brand or the other, hence the reach is enormous, and, as of 2014, there are more then 500 million tweets a day. But data is not information, and until we add analytics, it is just noise. And more, bigger, data may mean more noise and does not mean better data.
In many cases, less is more, and we need models as well. That is what this book is about, it’s about theories and models, with or without data, big or small. It’s about analytics and applications, and a scientific approach to using data based on well-founded theory and sound business judgment. This book is about the science and art of data analytics.
Data science is transforming business. Companies are using medical data and claims data to offer incentivized health programs to employees. Caesar’s Entertainment Corp. analyzed data for 65,000 employees and found substantial cost savings. Zynga Inc, famous for its game Farmville, accumulates 25 terabytes of data every day and analyzes it to make choices about new game features. UPS installed sensors to collect data on speed and location of its vans, which combined with GPS information, reduced fuel usage in 2011 by 8.4 million gallons, and shaved 85 million miles off its routes.5 McKinsey argues that a successful data analytics plan contains three elements: interlinked data inputs, analytics models, and decision-support tools.6 In a seminal paper, Halevy, Norvig, and Pereira (2009) argue that even simple theories and models, with big data, have the potential to do better than complex models with less data.
In a recent talk7 well-regarded data scientist Hilary Mason emphasized that the creation of “data products” requires three components: data (of course) plus technical expertise (machine-learning) plus people and process (talent). Google Maps is a great example of a data product that epitomizes all these three qualities. She mentioned three skills that good data scientists need to cultivate: (a) in math and stats, (b) coding, (c) communication. I would add that preceding all these is the ability to ask relevant questions, the answers to which unlock value for companies, consumers, and society. Everything in data analytics begins with a clear problem statement, and needs to be judged with clear metrics.
Being a data scientist is inherently interdisciplinary. Good questions come from many disciplines, and the best answers are likely to come from people who are interested in multiple fields, or at least from teams that co-mingle varied skill sets. Josh Wills of Cloudera stated it well - “A data scientist is a person who is better at statistics than any software engineer and better at software engineering than any statistician.” In contrast, complementing data scientists are business analytics people, who are more familiar with business models and paradigms and can ask good questions of the data.
## 1.1 Volume, Velocity, Variety
There are several “V”s of big data: three of these are volume, velocity, variety.8 Big data exceeds the storage capacity of conventional databases. This is it’s volume aspect. The scale of data generation is mind-boggling. Google’s Eric Schmidt pointed out that until 2003, all of human kind had generated just 5 exabytes of data (an exabyte is $$1000^6$$ bytes or a billion-billion bytes). Today we generate 5 exabytes of data every two days. The main reason for this is the explosion of “interaction” data, a new phenomenon in contrast to mere “transaction” data. Interaction data comes from recording activities in our day-to-day ever more digital lives, such as browser activity, geo-location data, RFID data, sensors, personal digital recorders such as the fitbit and phones, satellites, etc. We now live in the “internet of things” (or iOT), and it’s producing a wild quantity of data, all of which we seem to have an endless need to analyze. In some quarters it is better to speak of 4 Vs of big data, as shown in Figure 1.1.
A good data scientist will be adept at managing volume not just technically in a database sense, but by building algorithms to make intelligent use of the size of the data as efficiently as possible. Things change when you have gargantuan data because almost all correlations become significant, and one might be tempted to draw spurious conclusions about causality. For many modern business applications today extraction of correlation is sufficient, but good data science involves techniques that extract causality from these correlations as well.
In many cases, detecting correlations is useful as is. For example, consider the classic case of Google Flu Trends, see Figure 1.2. The figure shows the high correlation between flu incidence and searches about “flu” on Google, see Ginsberg et al. (2009); Culotta (2010). Obviously searches on the key word “flu” do not result in the flu itself! Of course, the incidence of searches on this key word is influenced by flu outbreaks. The interesting point here is that even though searches about flu do not cause flu, they correlate with it, and may at times even be predictive of it, simply because searches lead the actual reported levels of flu, as those may occur concurrently but take time to be reported. And whereas searches may be predictive, the cause of searches is the flu itself, one variable feeding on the other, in a repeat cycle.9 Hence, prediction is a major outcome of correlation, and has led to the recent buzz around the subfield of “predictive analytics.” There are entire conventions devoted to this facet of correlation, such as the wildly popular PAW (Predictive Analytics World).10 Pattern recognition is in, passe causality is out.
Data velocity is accelerating. Streams of tweets, Facebook entries, financial information, etc., are being generated by more users at an ever increasing pace. Whereas velocity increases data volume, often exponentially, it might shorten the window of data retention or application. For example, high-frequency trading relies on micro-second information and streams of data, but the relevance of the data rapidly decays.
Finally, data variety is much greater than ever before. Models that relied on just a handful of variables can now avail of hundreds of variables, as computing power has increased. The scale of change in volume, velocity, and variety of the data that is now available calls for new econometrics, and a range of tools for even single questions. This book aims to introduce the reader to a variety of modeling concepts and econometric techniques that are essential for a well-rounded data scientist.
Data science is more than the mere analysis of large data sets. It is also about the creation of data. The field of “text-mining” expands available data enormously, since there is so much more text being generated than numbers. The creation of data from varied sources, and its quantification into information is known as “datafication.”
## 1.2 Machine Learning
Data science is also more than “machine learning,” which is about how systems learn from data. Systems may be trained on data to make decisions, and training is a continuous process, where the system updates its learning and (hopefully) improves its decision-making ability with more data. A spam filter is a good example of machine learning. As we feed it more data it keeps changing its decision rules, using a Bayesian filter, thereby remaining ahead of the spammers. It is this ability to adaptively learn that prevents spammers from gaming the filter, as highlighted in Paul Graham’s interesting essay titled “A Plan for Spam.”11 Credit card approvals are also based on neural-nets, another popular machine learning technique. However, machine-learning techniques favor data over judgment, and good data science requires a healthy mix of both. Judgment is needed to accurately contextualize the setting for analysis and to construct effective models. A case in point is Vinny Bruzzese, known as the “mad scientist of Hollywood” who uses machine learning to predict movie revenues.12 He asserts that mere machine learning would be insufficient to generate accurate predictions. He complements machine learning with judgment generated from interviews with screenwriters, surveys, etc., “to hear and understand the creative vision, so our analysis can be contextualized.”
Machine intelligence is re-emerging as the new incarnation of AI (a field that many feel has not lived up to its promise). Machine learning promises and has delivered on many questions of interest, and is also proving to be quite a game-changer, as we will see later on in this chapter, and also as discussed in many preceding examples. What makes it so appealing? Hilary Mason suggests four characteristics of machine intelligence that make it interesting: (i) It is usually based on a theoretical breakthrough and is therefore well grounded in science. (ii) It changes the existing economic paradigm. (iii) The result is commoditization (e.g. Hadoop), and (iv) it makes available new data that leads to further data science.
Machine Learning (a.k.a. “ML”) has diverged and is now defined separate from traditional statistics. ML is more about learning and matching inputs with outputs, whereas statistics has always been interested more in analyzing data under a given problem statement or hypothesis. ML tends to be more heuristic, whereas econometrics and statistical analyses tend to be theory-driven, with tight assumptions. ML tends to focus more on prediction, econometrics on causality, which is a stronger outcome than prediction (or correlation). ML techniques work well with big data, whereas econometrics techniques tend toward too much significance with too much data. Hence, the latter is better served with dimension reduction, though ML may not in fact be implementable with small data. Under ML techniques, even when they work very well, it is hard to explain why, and also which variables in the feature set seem to work best. Under traditional econometrics and statistics, tracing the effects in the model is clear and feasible, making understanding of the model better. Deciding which approach fits a given problem best is a matter of taste, but experience often helps in deciding which one of the two methods applies better.
Let’s examine a definition of Machine Learning. Tom Mitchell, one of the founders of the field, stated a formal definition thus:
“A computer program is said to learn from experience $$E$$ with respect to some class of tasks $$T$$ and performance measure $$P$$ if its performance at tasks in $$T$$, as measured by $$P$$, improves with experience $$E$$.” – Mitchell (1997)
Domingos (2012) offers an excellent introductionmachine learning. He defines learning as the sum of Representation, Evaluation, and Optimization. Machine learning representation requires specifying the problem in a formal language that a computer can handle. These representations will differ for different machine learning techniques. For example, in a classification problem, there may be a choice of many classifiers, each of which will be formally represented. Next, a scoring function or a loss function is specified in order to complete the evaluation step. Finally, best evaluation is attained through optimization.
Once these steps have been undertaken and the best ML algorithm is chosen on the training data, we may validate the model on out-of-sample data, or the test data set. One may randomly choose a fraction of the data sample to hold out for validation. Repeating this process by holding out different parts of the data for testing, and training on the remainder, is a process known as cross-validation and is strongly recommended.
If it turns out that repeated cross-validation results in poor results, even though in-sample testing does very well, then it is possible evidence of over-fitting. Over-fitting usually occurs when the model is over-parameterized in-sample, so that it fits very well, but then it becomes less useful on new data. This is akin to driving by looking in the rear-view mirror, which does not work well when the road does not remain straight going forward. Therefore, many times, simpler and less parameterized models tend to work better in forecasting and prediction settings. If the model performs pretty much the same in-sample and out-of-sample, it is very unlikely to be overfit. The argument that simpler models overfit less is often made with Occam’s Razor in mind, but is not always an accurate underpinning, so simpler may not always be better.13
## 1.3 Supervised and Unsupervised Learning
Systems may learn in two broad ways, through “supervised” and “unsupervised” learning. In supervised learning, a system produces decisions (outputs) based on input data. Both spam filters and automated credit card approval systems are examples of this type of learning. So is linear discriminant analysis (LDA). The system is given a historical data sample of inputs and known outputs, and it “learns” the relationship between the two using machine learning techniques, of which there are several. Judgment is needed to decide which technique is most appropriate for the task at hand.
Unsupervised learning is a process of reorganizing and enhancing the inputs in order to place structure on unlabeled data. A good example is cluster analysis, which takes a collection of entities, each with a number of attributes, and partitions the entity space into sets or groups based on closeness of the attributes of all entities. What this does is reorganizes the data, but it also enhances the data through a process of labeling the data with additional tags (in this case a cluster number/name). Factor analysis is also an unsupervised learning technique. The origin of this terminology is unclear, but it presumably arises from the fact that there is no clear objective function that is maximized or minimized in unsupervised learning, so that no “supervision” to reach an optimal is called for. However, this is not necessarily true in general, and we will see examples of unsupervised learning (such as community detection in the social web) where the outcome depends on measurable objective criteria.
Supervised learning might include broad topics such as regression, classification, forecasting, and importance attribution. All these analyses are supported by the fact that the feature set ($$X$$ variables) are accompanied by tags ($$Y$$ variables). Unsupervised learning includes analyses such as clustering, and association models, e.g., recommendation engines, market baskets, etc.
## 1.4 Feature Selection
When faced with a machine learning problem, having the right data is paramount. Sometimes, especially in this age of Big Data, we may have too much data; abundance comes with a curse. Too much data also might mean featureless data, which is not useful to the data scientist. Hence, we might want to extract those data variables that are useful, through a process called “feature selection”. Dimension reduction is also a useful by product of feature selection, and pruning data might also mean that ML algorithms will run faster, and converge better.
Wikipedia defines feature selection as – “In machine learning and statistics, feature selection, also known as variable selection, attribute selection or variable subset selection, is the process of selecting a subset of relevant features (variables, predictors) for use in model construction.”14
Feature selection subsets the variable space. If there are $$p$$ columns of data, then we choose $$q \ll p$$ variates. Feature extraction on the other hand refers to transformation of the original variables to create new variables, i.e., functionals of $$p$$, such as $$g(p)$$. We will encounter these topics later on, as we work through various ML techniques.
## 1.5 Ensemble Learning
Ensemble models are simply combinations of many ML models. There are of course, many ways in which models may be combined to generate better ML models. It is astonishing how powerful this “model democracy” turns out to be where various models vote, for example, on a classification problem. In S. R. Das and Chen (2007), five different classifiers vote on classifying stock bulletin board messages into three categories of signals: Buy, Hold, Sell. In this early work, ensemble methods were able to improve the signal-to-noise ratio in classification.
Different classification models are not always necessary. One may instead calibrate the same model to different subsamples of the training data, delivering multiple similar, but different models. Each of these models is then used to classify out-of-sample, and the decision is made by voting across models. This method is known as bagging. One of the most popular examples of bagging algorithms is the random forest model, which we will encounter later when we examine classifiers in more detail.
In another technique, boosting, the loss function that is being optimized does not weight all examples in the training data set equally. After one pass of calibration, training examples are reweighted such that the cases where the ML algorithm made errors (as in a classification problem) are given higher weight in the loss function. By penalizing these observations, the algorithm learns to prevent those mistakes as they are more costly.
Another approach to ensemble learning is called stacking where models are chained to each other, so that the output of low-level models becomes the input of another higher-level model. Here models are vertically integrated in contrast to bagging, where models are horizontally integrated.
## 1.6 Predictions and Forecasts
Data science is about making predictions and forecasts. There is a difference between the two. The statistician-economist Paul Saffo has suggested that predictions aim to identify one outcome, whereas forecasts encompass a range of outcomes. To say that “it will rain tomorrow” is to make a prediction, but to say that “the chance of rain is 40%” (implying that the chance of no rain is 60%) is to make a forecast, as it lays out the range of possible outcomes with probabilities. We make weather forecasts, not predictions. Predictions are statements of great certainty, whereas forecasts exemplify the range of uncertainty. In the context of these definitions, the term predictive analytics is a misnomer for it’s goal is to make forecasts, not mere predictions.
## 1.7 Innovation and Experimentation
Data science is about new ideas and approaches. It merges new concepts with fresh algorithms. Take for example the A/B test, which is nothing but the online implementation of a real-time focus group. Different subsets of users are exposed to A and B stimuli respectively, and responses are measured and analyzed. It is widely used for web site design. This approach has been in place for more than a decade, and in 2011 Google ran more than 7,000 A/B tests. Facebook, Amazon, Netflix, and several others firms use A/B testing widely.15 The social web has become a teeming ecosystem for running social science experiments. The potential to learn about human behavior using innovative methods is much greater now than ever before.
## 1.8 The Dark Side
### 1.8.1 Big Errors
The good data scientist will take care to not over-reach in drawing conclusions from big data. Because there are so many variables available, and plentiful observations, correlations are often statistically significant, but devoid of basis. In the immortal words of the bard, empirical results from big data may be - “A tale told by an idiot, full of sound and fury, signifying nothing.”16 One must be careful not to read too much in the data. More data does not guarantee less noise, and signal extraction may be no easier than with less data.
Adding more columns (variables in the cross section) to the data set, but not more rows (time dimension) is also fraught with danger. As the number of variables increases, more characteristics are likely to be related statistically. Over fitting models in-sample is much more likely with big data, leading to poor performance out-of-sample.
Researchers have also to be careful to explore the data fully, and not terminate their research the moment a viable result, especially one that the researcher is looking for, is attained. With big data, the chances of stopping at a suboptimal, or worse, intuitively appealing albeit wrong result become very high. It is like asking a question to a class of students. In a very large college class, the chance that someone will provide a plausible yet off-base answer quickly is very high, which often short circuits the opportunity for others in class to think more deeply about the question and provide a much better answer.
Nassim Taleb17 describes these issues elegantly - “I am not saying there is no information in big data. There is plenty of information. The problem – the central issue – is that the needle comes in an increasingly larger haystack.” The fact is, one is not always looking for needles or Taleb’s black swans, and there are plenty of normal phenomena about which robust forecasts are made possible by the presence of big data.
### 1.8.2 Privacy
The emergence of big data coincides with a gigantic erosion of privacy. Human kind has always been torn between the need for social interaction, and the urge for solitude and privacy. One trades off against the other. Technology has simply sharpened the divide and made the slope of this trade off steeper. It has provided tools of social interaction that steal privacy much faster than in the days before the social web.
Rumors and gossip are now old world. They required bilateral transmission. The social web provides multilateral revelation, where privacy no longer capitulates a battle at a time, but the entire war is lost at one go. And data science is the tool that enables firms, governments, individuals, benefactors and predators, et al, en masse, to feed on privacy’s carcass. The cartoon in Figure 1.3 parodies the kind of information specialization that comes with the loss of privacy!
The loss of privacy is manifested in the practice of human profiling through data science. Our web presence increases entropically as we move more of our life’s interactions to the web, be they financial, emotional, organizational, or merely social. And as we live more and more of our lives in this new social melieu, data mining and analytics enables companies to construct very accurate profiles of who we are, often better than what we might do ourselves. We are moving from “know thyself” to knowing everything about almost everyone.
Humankind leaves an incredible trail of “digital exhaust” comprising phone calls, emails, tweets, GPS information, etc., that companies use for profiling. It is said that 1/3 of people have a digital identity before being born, initiated with the first sonogram from a routine hospital visit by an expectant mother. The half life of non-digital identity, or the average age of digital birth is six months, and within two years 92% of the US population has a digital identity.18 Those of us who claim to be safe from revealing their privacy by avoiding all forms of social media are simply profiled as agents with a “low digital presence.” It might be interesting to ask such people whether they would like to reside in a profile bucket that is more likely to attract government interest than a profile bucket with more average digital presence. In this age of profiling, the best way to remain inconspicuous is not to hide, but to remain as average as possible, so as to be mostly lost within a large herd.
Privacy is intricately and intrinsically connected to security and efficiency. The increase in transacting on the web, and the confluence of profiling, has led to massive identity theft. Just as in the old days, when a thief picked your lock and entered your home, most of your possessions were at risk. It is the same with electronic break ins, except that there are many more doors to break in from and so many more windows through which an intruder can unearth revealing information. And unlike a thief who breaks into your home, a hacker can reside in your electronic abode for quite some time without being detected, an invisible parasite slowly doing damage. While you are blind, you are being robbed blind. And unlike stealing your worldly possessions, stealing your very persona and identity is the cruelest cut of them all.
An increase in efficiency in the web ecosystem comes too at some retrenchment of privacy. Who does not shop on the internet? Each transaction resides in a separate web account. These add up at an astonishing pace. I have no idea of the exact number of web accounts in my name, but I am pretty sure it is over a hundred, many of them used maybe just once. I have unconsciously, yet quite willingly, marked my territory all over the e-commerce landscape. I rationalize away this loss of privacy in the name of efficiency, which undoubtedly exists. Every now and then I am reminded of this loss of privacy as my plane touches down in New York city, and like clockwork, within an hour or two, I receive a discount coupon in my email from Barnes & Noble bookstores. You see, whenever I am in Manhattan, I frequent the B&N store on the upper west side, and my credit card company and/or Google knows this, as well as my air travel schedule, since I buy both tickets and books on the same card and in the same browser. So when I want to buy books at a store discount, I fly to New York. That’s how rational I am, or how rational my profile says I am! Humor aside, such profiling seems scary, though the thought quickly passes. I like the dopamine rush I get from my discount coupon and I love buying books.19
Profiling implies a partitioning of the social space into targeted groups, so that focused attention may be paid to specific groups, or various groups may be treated differently through price discrimination. If my profile shows me to be an affluent person who likes fine wine (both facts untrue in my case, but hope springs eternal), then internet sales pitches (via Groupon, Living Social, etc.) will be priced higher to me by an online retailer than to someone whose profile indicates a low spend. Profiling enables retailers to maximize revenues by eating away the consumer’s surplus by better setting of prices to each buyer’s individual willingness to pay. This is depicted in Figure 1.5.
In Figure 1.5 the demand curve is represented by the line segment $$ABC$$ representing price-quantity combinations (more is demanded at lower prices). In a competitive market without price segmentation, let’s assume that the equilibrium price is $$P$$ and equilibrium quantity is $$Q$$ as shown by the point $$B$$ on the demand curve. (The upward sloping supply curve is not shown but it must intersect the demand curve at point $$B$$, of course.) Total revenue to the seller is the area $$OPBQ$$, i.e., $$P \times Q$$.
Now assume that the seller is able to profile buyers so that price discrimination is possible. Based on buyers’ profiles, the seller will offer each buyer the price he is willing to pay on the demand curve, thereby picking off each price in the segment $$AB$$. This enables the seller to capture the additional region $$ABP$$, which is the area of consumer’s surplus, i.e., the difference between the price that buyers pay versus the price they were actually willing to pay. The seller may also choose to offer some consumers lower prices in the region $$BC$$ of the demand curve so as to bring in additional buyers whose threshold price lies below the competitive market price $$P$$. Thus, profiling helps sellers capture consumer’s surplus and eat into the region of missed sales. Targeting brings benefits to sellers and they actively pursue it. The benefits outweigh the costs of profiling, and the practice is widespread as a result. Profiling also makes price segmentation fine-tuned, and rather than break buyers into a few segments, usually two, each profile becomes a separate segment, and the granularity of price segmentation is modulated by the number of profiling groups the seller chooses to model.
Of course, there is an insidious aspect to profiling, which has existed for quite some time, such as targeting conducted by tax authorities. I don’t believe we will take kindly to insurance companies profiling us any more than they already do. Profiling is also undertaken to snare terrorists. However, there is a danger in excessive profiling. A very specific profile for a terrorist makes it easier for their ilk to game detection as follows. Send several possible suicide bombers through airport security and see who is repeatedly pulled aside for screening and who is not. Repeating this exercise enables a terrorist cell to learn which candidates do not fall into the profile. They may then use them for the execution of a terror act, as they are unlikely to be picked up for special screening. The antidote? Randomization of people picked for special screening in searches at airports, which makes it hard for a terrorist to always assume no likelihood of detection through screening.20
Automated invasions of privacy naturally lead to a human response, not always rational or predictable. This is articulated in Campbell’s Law: “The more any quantitative social indicator (or even some qualitative indicator) is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor.”21 We are in for an interesting period of interaction between man and machine, where the battle for privacy will take center stage.
## 1.9 Theories, Models, Intuition, Causality, Prediction, Correlation
My view of data science is one where theories are implemented using data, some of it big data. This is embodied in an inference stack comprising (in sequence): theories, models, intuition, causality, prediction, and correlation. The first three constructs in this chain are from Emanuel Derman’s wonderful book on the pitfalls of models.^[“Models. Behaving. Badly.” Emanuel Derman, Free Press, New York, 2011.}
Theories are statements of how the world should be or is, and are derived from axioms that are assumptions about the world, or precedent theories. Models are implementations of theory, and in data science are often algorithms based on theories that are run on data. The results of running a model lead to intuition, i.e., a deeper understanding of the world based on theory, model, and data. Whereas there are schools of thought that suggest data is all we need, and theory is obsolete, this author disagrees. Still the unreasonable proven effectiveness of big data cannot be denied. Chris Anderson argues in his Wired magazine article thus:“22
Sensors everywhere. Infinite storage. Clouds of processors. Our ability to capture, warehouse, and understand massive amounts of data is changing science, medicine, business, and technology. As our collection of facts and figures grows, so will the opportunity to find answers to fundamental questions. Because in the era of big data, more isn’t just more. More is different.
In contrast, the academic Thomas Davenport writes in his foreword to Siegel (2013) that models are key, and should not be increasingly eschewed with increasing data:
But the point of predictive analytics is not the relative size or unruliness of your data, but what you do with it. I have found that “big data often means small math,” and many big data practitioners are content just to use their data to create some appealing visual analytics. That’s not nearly as valuable as creating a predictive model.
Once we have established intuition for the results of a model, it remains to be seen whether the relationships we observe are causal, predictive, or merely correlational. Theory may be causal and tested as such. Granger (1969) causality is often stated in mathematical form for two stationary23 time series of data as follows. $$X$$ is said to Granger cause $$Y$$ if in the following equation system,
$\begin{eqnarray*} Y(t) &=& a_1 + b_1 Y(t-1) + c_1 X(t-1) + e_1 \\ X(t) &=& a_2 + b_2 Y(t-1) + c_2 X(t-1) + e_2 \end{eqnarray*}$
the coefficient $$c_1$$ is significant and $$b_2$$ is not significant. Hence, $$X$$ causes $$Y$$, but not vice versa. Causality is a hard property to establish, even with theoretical foundation, as the causal effect has to be well-entrenched in the data.
We have to be careful to impose judgment as much as possible since statistical relationships may not always be what they seem. A variable may satisfy the Granger causality regressions above but may not be causal. For example, we earlier encountered the flu example in Google Trends. If we denote searches for flu as $$X$$, and the outbreak of flu as $$Y$$, we may see a Granger cause relation between flu and searches for it. This does not mean that searching for flu causes flu, yet searches are predictive of flu. This is the essential difference between prediction and causality.
And then there is correlation, at the end of the data science inference chain. Contemporaneous movement between two variables is quantified using correlation. In many cases, we uncover correlation, but no prediction or causality. Correlation has great value to firms attempting to tease out beneficial information from big data. And even though it is a linear relationship between variables, it lays the groundwork for uncovering nonlinear relationships, which are becoming easier to detect with more data. The surprising parable about Walmart finding that purchases of beer and diapers seem to be highly correlated resulted in these two somewhat oddly-paired items being displayed on the same aisle in supermarkets.24 Unearthing correlations of sales items across the population quickly lead to different business models aimed at exploiting these correlations, such as my book buying inducement from Barnes & Noble, where my “fly and buy” predilection is easily exploited. Correlation is often all we need, eschewing human cravings for causality. As Mayer-Schönberger and Cukier (2013) so aptly put it, we are satisfied “… not knowing why but only what.”
In the data scientist mode of thought, relationships are multifaceted correlations amongst people. Facebook, Twitter, and many other platforms are datafying human relationships using graph theory, exploiting the social web in an attempt to understand better how people relate to each other, with the goal of profiting from it. We use correlations on networks to mine the social graph, understanding better how different social structures may be exploited. We answer questions such as where to seed a new marketing campaign, which members of a network are more important than the others, how quickly will information spread on the network, i.e., how strong is the “network effect”?
A good data scientist learns how to marry models and data, and an important skill is the ability to define a problem well, and then break it down so that it may be solved in a facile manner. In a microcosm, this is what good programmers do, each component of the algorithm is assigned to a separate subroutine, that is generalized and optimized for one purpose. Mark Zuckerberg told a group of engineers the following – “The engineering mindset dictates thinking of every problem as a system, breaking down problems from the biggest stage down to smaller pieces. You get to the point where you are running a company, itself a complicated system segmented into groups of high-functioning people. Instead of managing individuals you are managing teams. And if you’ve built it well, then it’s not so different from writing code.” (Fortune, December 2016.)
Data science is about the quantization and understanding of human behavior, the holy grail of social science. In the following chapters we will explore a wide range of theories, techniques, data, and applications of a multi-faceted paradigm. We will also review the new technologies developed for big data and data science, such as distributed computing using the Dean and Ghemawat (2008) MapReduce paradigm developed at Google,25 and implemented as the open source project Hadoop at Yahoo!.26 When data gets super sized, it is better to move algorithms to the data than the other way around. Just as big data has inverted database paradigms, so is big data changing the nature of inference in the study of human behavior. Ultimately, data science is a way of thinking, for social scientists, using computer science.
### References
Patil, Dhanurjay. 2012. Data Jujitsu. Sebastopol, California: O’Reilly.
Patil, Dhanurjay. 2011. Building Data Science Teams. Sebastopol, California: O’Reilly.
Loukides, Michael. 2012. What Is Data Science. Sebastopol, California: O’Reilly.
Halevy, Alon Y., Peter Norvig, and Fernando Pereira. 2009. “The Unreasonable Effectiveness of Data.” IEEE Intelligent Systems 24 (2): 8–12. doi:10.1109/MIS.2009.36.
Ginsberg, Jeremy, Matthew Mohebbi, Rajan Patel, Lynnette Brammer, Mark Smolinski, and Larry Brilliant. 2009. “Detecting Influenza Epidemics Using Search Engine Query Data.” Nature 457: 1012–4. http://www.nature.com/nature/journal/v457/n7232/full/nature07634.html.
Culotta, Aron. 2010. “Towards Detecting Influenza Epidemics by Analyzing Twitter Messages.” In Proceedings of the First Workshop on Social Media Analytics, 115–22. SOMA ’10. New York, NY, USA: ACM. doi:10.1145/1964858.1964874.
Mitchell, Thomas M. 1997. Machine Learning. 1st ed. New York, NY, USA: McGraw-Hill, Inc.
Domingos, Pedro. 2012. “A Few Useful Things to Know About Machine Learning.” Commun. ACM 55 (10). New York, NY, USA: ACM: 78–87. doi:10.1145/2347736.2347755.
Das, Sanjiv R., and Mike Y. Chen. 2007. “Yahoo! For Amazon: Sentiment Extraction from Small Talk on the Web.” Manage. Sci. 53 (9). Institute for Operations Research; the Management Sciences (INFORMS), Linthicum, Maryland, USA: INFORMS: 1375–88. doi:10.1287/mnsc.1070.0704.
Siegel, Eric. 2013. Predictive Analytics. New Jersey: John-Wiley; Sons.
Granger, C W J. 1969. “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods.” Econometrica 37 (3): 424–38. https://ideas.repec.org/a/ecm/emetrp/v37y1969i3p424-38.html.
Mayer-Schönberger, Viktor, and Kenneth Cukier. 2013. Big Data: A Revolution That Will Transform How We Live, Work, and Think. Second. New York, NY: Houghton Mifflin Harcourt.
Dean, Jeffrey, and Sanjay Ghemawat. 2008. “MapReduce: Simplified Data Processing on Large Clusters.” Commun. ACM 51 (1): 107–13. doi:10.1145/1327452.1327492.
1. The term “data scientist” was coined by D.J. Patil. He was the Chief Scientist for LinkedIn. In 2011 Forbes placed him second in their Data Scientist List, just behind Larry Page of Google.
2. To quote Georg Cantor - “In mathematics the art of proposing a question must be held of higher value than solving it.”
3. Mayer-Schönberger and Cukier (2013), p8. They report that USC’s Martin Hilbert calculated that more than 300 exabytes of data storage was being used in 2007, an exabyte being one billion gigabytes, i.e., $$10^{18}$$ bytes, and $$2^{60}$$ of binary usage.
4. In contrast, 88% of the population has heard of Facebook, and 41% use it. See www.convinceandconvert.com/\7-surprising-statistics-about\-twitter-in-america/. Half of Twitter users are white, and of the remaining half, half are black.
5. “How Big Data is Changing the Whole Equation for Business,” Wall Street Journal March 8, 2013.
6. “Big Data: What’s Your Plan?” McKinsey Quarterly, March 2013.
7. At the h2o world conference in the Bay Area, on 11th November 2015.
8. This nomenclature was originated by the Gartner group in 2001, and has been in place more than a decade.
9. Interwoven time series such as these may be modeled using Vector Auto-Regressions, a technique we will encounter later in this book.
10. May be a futile collection of people, with non-working crystal balls, as William Gibson said - “The future is not google-able.”
11. “Solving Equation of a Hit Film Script, With Data,” New York Times, May 5, 2013.
12. See the excellent paper on this by Domingos (1999).
13. “The A/B Test: Inside the Technology that’s Changing the Rules of Business,” by Brian Christian, Wired, April 2012.
14. William Shakespeare in Macbeth, Act V, Scene V.
15. “Beware the Big Errors of Big Data” Wired, February 2013.
16. See “The Human Face of Big Data” by Rick Smolan and Jennifer Erwitt.
17. I also like writing books, but I am much better at buying them, and somewhat less better at reading them!
18. See http://acfnewsource.org.s60463.gridserver.com/science/random_security.html, also aired on KRON-TV, San Francisco, 2/3/2003.
19. “The End of Theory: The Data Deluge Makes the Scientific Method Obsolete.” Wired, v16(7), 23rd June, 2008.
20. A series is stationary if the probability distribution from which the observations are drawn is the same at all points in time. | 2022-07-01 03:46:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30697304010391235, "perplexity": 1756.6282382551506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00673.warc.gz"} |
https://www.lmfdb.org/EllipticCurve/Q/369600/ok/ | # Properties
Label 369600.ok Number of curves $2$ Conductor $369600$ CM no Rank $0$ Graph
# Related objects
Show commands: SageMath
sage: E = EllipticCurve("ok1")
sage: E.isogeny_class()
## Elliptic curves in class 369600.ok
sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.ok1 369600ok2 $$[0, 1, 0, -51633, -4525137]$$ $$59466754384/121275$$ $$31046400000000$$ $$[2]$$ $$1474560$$ $$1.4754$$
369600.ok2 369600ok1 $$[0, 1, 0, -2133, -119637]$$ $$-67108864/343035$$ $$-5488560000000$$ $$[2]$$ $$737280$$ $$1.1288$$ $$\Gamma_0(N)$$-optimal
## Rank
sage: E.rank()
The elliptic curves in class 369600.ok have rank $$0$$.
## Complex multiplication
The elliptic curves in class 369600.ok do not have complex multiplication.
## Modular form 369600.2.a.ok
sage: E.q_eigenform(10)
$$q + q^{3} - q^{7} + q^{9} + q^{11} - 6 q^{13} - 2 q^{17} + O(q^{20})$$
## Isogeny matrix
sage: E.isogeny_class().matrix()
The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.
$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$
## Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels. | 2022-08-13 22:21:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9330062866210938, "perplexity": 3724.686123782039}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571987.60/warc/CC-MAIN-20220813202507-20220813232507-00161.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-connecting-concepts-through-application/chapter-1-linear-functions-1-2-using-data-to-create-scatterplots-1-2-exercises-page-28/3 | ## Intermediate Algebra: Connecting Concepts through Application
$(A)$ and $(E)$
When putting a ruler up to the graph, the points $(A)$ and $(E)$ seem to line up the best. | 2018-07-20 02:48:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4833526909351349, "perplexity": 1233.8127504608258}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591481.75/warc/CC-MAIN-20180720022026-20180720042026-00434.warc.gz"} |
http://gauravtiwari.org/2011/02/page/2/ | # LAWS OF ENERGY DISTRIBUTION IN BLACK BODY RADIATION : First Part – Wein’s Laws
Image via Wikipedia
Various workers tried to explain the problem of energy distribution in black body radiation and finally the problem was successfully solved by German Physicist Max Planck. Before him, German Physicist Wilhelm Wein and British Physicist Lord Rayleigh & James Jean have tackled this problem and have given important laws. In fact, the work of there scientists paved the way for Planck to give his famous theory of radiation.
In this series of articles, I shall be discussing the various laws, special concentration on Planck’s law, concerning the black body in the brief.
# Wein’s Formula & Wein’s Laws
The problem of black body radiation was first theoretically tackled by Wein in 1893. Besides giving a general formula for the energy distribution in the blackbody radiation, he gave following important and useful laws. Continue reading
# Largest Prime Numbers
## What is a Prime Number?
An integer, say $p$ , [ $\ne {0}$ & $\ne { \pm{1}}$ ] is said to be a prime integer iff its only factors (or divisors) are $\pm{1}$ & $\pm{p}$ .
## As?
Few easy examples are:
$\pm{2}, \pm{3}, \pm{5}, \pm{7}, \pm{11}, \pm{13}$ …….etc. This list goes upto infinity & mathematicians are trying to find the larger one than the largest, because primes numbers has no distinct pattern (as any one cannot guess the next prime after one.) As of now the biggest prime number found is $M-47$ , called as Mersenne’s 47. This has an enormous value of $2^{43112609} -1$ . It is very hard to write it on paper because it consists of $12978189$ digits.
»M47 was Invented in 2008. Continue reading
# Albert Einstein
This name need not be explained. Albert Einstein is considered to be one of the best physicists in the human history.
The twentieth century has undoubtedly been the most significant for the advance of science, in general, and Physics, in particular. And Einstein is the most luminated star of the 20th century. He literally created cm upheaval by the publication, in quick succession, in the year 1905, two epoch-making papers, on the concept of the photon and on the Electrodynamics of moving bodies respectively, with yet another on the Mathematical analysis of Brownian Motion thrown in, in between.
The Electrodynamics of moving bodies was the biggest sensation and it demolished at one stroke some of the most cherished and supposedly infallable laws and concepts and gave the breath takingly new idea of the relativity of space and time.
# Derivative of x squared is 2x or x ? Where is the fallacy?
As we know that the derivative of $x^2$ , with respect to $x$ , is $2x$.
i.e., $\dfrac{d}{dx} x^2 = 2x$
However, suppose we write $x^2$ as the sum of $x$ ‘s written up $x$ times..
i.e.,
# Solving Ramanujan’s Puzzling Problem
Consider a sequence of functions as follows:-
$f_1 (x) = \sqrt {1+\sqrt {x} }$
$f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } }$
$f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } }$
……and so on to
$f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt {1+3 \sqrt {\ldots \sqrt {1+n \sqrt {x} } } } } }$
Evaluate this function as n tends to infinity.
Or logically:
Find
$\displaystyle{\lim_{n \to \infty}} f_n (x)$ .
### Solution
Ramanujan discovered
which gives the special cases
for and
Proof of Ramanujan’s nested radicals equation
Then keep replacing the the last part term by writing in the form of to give
Which is another form of our problem and is referred as Ramanujan’s Nested Radical. Comparing these two expressions & assuming
=$X$ , we can write the problem as:
$\displaystyle {\lim_{n \to \infty}} f_n (x)$
= $\sqrt {1+X}$
= $\sqrt {1+3}$
=$\sqrt {4}$
=$2$
-
For further info please refer the comments below. There is also a supportive article on Ramanujan Nested Radicals on this blog.
# L A S E R S
Light Amplified by Stimulated Emission of Radiation i.e. LASER is one of the most incredible discoveries of Physics. First produced in 1960s, it recently completed its 50th year.
Laser light is emitted when atoms make transition from one Quantum State to a lower one.
## Properties of Laser Light
1) Laser light is highly monochromatic.
2) Laser Light is highly coherent .
3) Laser light is highly directional.
4) Laser light can be sharply focused.
### Types of Lasers
There are many kinds of LASERs differing their operational wavelength & applications:
I. Gas LASERs:
The Helium-Neon Gas Laser, Carbon dioxide gas Laser
II. Chemical LASERs:
HF Laser
III. Solid LASERs: Ruby LASER (most popular), Hybrid Silicon LASER, Diode LASER
IV. Metal Vapour LASERs: Copper Vapour LASER, Gold Vapour LASER
V. Dye LASERs
VI. RAMAN LASERs
VII. Free Electron LASER
VIII. Gas Dynamic LASER
#### Uses
* The smallest Lasers are use in Fibre Optics– for -Voice & Data Transmission over Optical Fibres.
* The Largest lasers are used for nuclear fusion research, to measure astronomical distances and in military applications.
* Other uses are– Reading Bar Codes, Manufacturing and Reading CDs and DVDs, Performing Surgery, Surveying, Cutting hundred layes Cloth at a time in the garment industry, weldings and in generating holograms.
# Its a Mystery! – Simulacrum in Eagle Nebula
One of the Strangest photos that have Ever Been taken of space is that of the Eagle Nebula. The photo itself is supposed to show the birth of a star from the gaseous clouds.
When the photo was shown on CNN, [See Image1] hundreds of calls came in from people reporting — they could see a face in the cloud. When the color of the photo was adjusted, a large human form seemed to appear within the cloud. [See Image2]
### Image 2
So What did you see here?
This is what attracts us to mysteries. Mystreries are beautiful.
Scientists have not been able to explain this beautiful phenomenon.
# Raman Effect- Raman Spectroscopy- Raman Scattering
In constrast to other conventional brances of spectroscopy, Raman spectroscopy deals with the scattering of light & not with its absorption.
# Raman Effect
Raman Effect: An Overview
Chandrasekhar Venkat Raman discovered in 1928 that if light of a definite frequency is passed through any substance in gaseous, liquid or solid state, the light scattered at right angles contains radiations not only of the original frequency (Rayleigh Scattering) but also of some other frequencies which are generally lower but occasionally higher than the frequency of the incident light.
The phenomenon of scattering of light by a substance when the frequencies of radiations scattered at right angles are different (generally lower and only occasionally higher) from the frequency of the incident light, is known as Raman Scattering or Raman effect.
The lines of lower frequencies as known as Stokes lines while those of higher frequencies are called anti-stokes lines.
If f is the frequency of the incident light & f’ that of a particular line in the scattered spectrum, then the difference f-f’ is known as the Raman Frequency. This frequency is independent of the frequency of the incident light. It is constant and is characteristic of the substance exposed to the incident light.
A striking feature of Raman Scattering is that Raman Frequencies are identical, within the limits of experimental error, with those obtained from rotation-vibration (infrared) spectra of the substance.
Here is a home made video explaining the Raman Scattering of Yellow light:
And here is another video guide for Raman Scattering:
• Raman Spectroscopy can be used not only for gases but also for liquids & solids for which the infrared spectra are so diffuse as to be of little quantitative value.
• Raman Effect is exhibited not only by polar molecules but also by non-polar molecules such as O2, N2, Cl2 etc.
• The rotation-vibration changes in non-polar molecules can be observed only by Raman Spectroscopy.
• The most important advantage of Raman Spectra is that it involves measurement of frequencies of scattered radiations, which are only slightly different from the frequencies of incident radiations. Thus, by appropriate choice of the incident radiations, the scattered spectral lines are brought into a convenient region of the spectrum, generally in the visible region where they are easily observed. The measurement of the corresponding infrared spectra is much more difficult.
### Uses
• Investigation of biological systems such as the polypeptides and the proteins in aqueous solution.
• Determination of structures of molecules.
RAMAN was awarded the 1930 Physics Nobel Prize for this.
# Classical Theory of Raman Effect
The classical theory of Raman effect, also called the polarizability theory, was developed by G. Placzek in 1934. I shall discuss it briefly here. It is known from electrostatics that the electric field $E$ associated with the electromagnetic radiation induces a dipole moment $\mu$ in the molecule, given by
$\mu = \alpha E$ …….(1)
where $\alpha$ is the polarizability of the molecule. The electric field vector $E$ itself is given by
$E = E_0 \sin \omega t = E_0 \sin 2\pi \nu t$ ……(2)
where $E_0$ is the amplitude of the vibrating electric field vector and $\nu$ is the frequency of the incident light radiation.
Thus, from Eqs. (1) & (2),
$\mu= \alpha E_0 \sin 2\pi \nu t$ …..(3)
Such an oscillating dipole emits radiation of its own oscillation with a frequency $\nu$ , giving the Rayleigh scattered beam. If, however, the polarizability varies slightly with molecular vibration, we can write
$\alpha =\alpha_0 + \frac {d \alpha} {dq} q$ …..(4)
where the coordinate q describes the molecular vibration. We can also write q as:
$q=q_0 \sin 2\pi \nu_m t$ …..(5)
Where $q_0$ is the amplitude of the molecular vibration and $\nu_m$ is its (molecular) frequency. From Eqs. 4 & 5, we have
$\alpha =\alpha_0 + \frac {d\alpha} {dq} q_0 \sin 2\pi \nu_m t$ …..(6)
Substituting for $alpha$ in (3), we have
$\mu= \alpha_0 E_0 \sin 2\pi \nu t + \frac {d\alpha}{dq} q_0 E_0 \sin 2\pi \nu t \sin 2\pi \nu_m t$ …….(7)
Making use of the trigonometric relation $\sin x \sin y = \frac{1}{2} [\cos (x-y) -\cos (x+y) ]$ this equation reduces to:
$\mu= \alpha_0 E_0 \sin 2\pi \nu t + \frac {1}{2} \frac {d\alpha}{dq} q_0 E_0 [\cos 2\pi (\nu - \nu_m) t - \cos 2\pi (\nu+\nu_m) t]$ ……(8)
Thus, we find that the oscillating dipole has three distinct frequency components:
1• The exciting frequency $\nu$ with amplitude $\alpha_0 E_0$
2• $\nu – \nu_m$
3• $\nu + \nu_m$ (2 & 3 with very small amplitudes of $frac {1}{2} frac {d\alpha}{dq} q_0 E_0$ . Hence, the Raman spectrum of a vibrating molecule consists of a relatively intense band at the incident frequency and two very weak bands at frequencies slightly above and below that of the intense band.
If, however, the molecular vibration does not change the polarizability of the molecule then $(d\alpha / dq )=0$ so that the dipole oscillates only at the frequency of the incident (exciting) radiation. The same is true for the molecular rotation. We conclude that for a molecular vibration or rotation to be active in the Raman Spectrum, it must cause a change in the molecular polarizability, i.e., $d\alpha/dq \ne 0$ …….(9)
Homonuclear diatomic molecules such as $\mathbf {H_2 , N_2 , O_2}$ which do not show IR Spectra since they don’t possess a permanent dipole moment, do show Raman spectra since their vibration is accompanied by a change in polarizability of the molecule. As a consequence of the change in polarizability, there occurs a change in the induced dipole moment at the vibrational frequency.
REFERENCE:-
Principles in Physical Chemistry
[7th edition]
Puri, Sharma & Pathania | 2013-12-12 15:04:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 13, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6825315952301025, "perplexity": 1467.1533777122977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164641332/warc/CC-MAIN-20131204134401-00055-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://lensfun.github.io/manual/v0.3.1/structlfLensCalibRealFocal.html | lensfun 0.3.1.0
lfLensCalibRealFocal Struct Reference
Struct to save real focal length, which can depends on the (nominal) focal length. More...
#include <lensfun.h>
## Public Attributes
float Focal
float RealFocal
Real focal length. More...
## Detailed Description
Struct to save real focal length, which can depends on the (nominal) focal length.
## ◆ Focal
float lfLensCalibRealFocal::Focal
Nominal focal length in mm at which this calibration data was taken
## ◆ RealFocal
float lfLensCalibRealFocal::RealFocal
Real focal length.
When Lensfun speaks of “focal length”, the nominal focal length from the EXIF data or the gravure on the lens barrel is meant. However, especially for fisheye lenses, the real focal length generally differs from that nominal focal length. With “real focal length” I mean the focal length in the paraxial approximation, see http://en.wikipedia.org/wiki/Paraxial_approximation. Note that Hugin (as of 2014) implements the calculation of the real focal length wrongly, see http://article.gmane.org/gmane.comp.misc.ptx/34865.
The documentation for this struct was generated from the following file: | 2022-05-23 00:06:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.814667820930481, "perplexity": 5972.257445934616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662550298.31/warc/CC-MAIN-20220522220714-20220523010714-00013.warc.gz"} |
http://mathhelpforum.com/discrete-math/54451-recursion-problem.html | 1. ## Recursion problem
I need some direction to the recursion problem listed below : recursion is a topic which i recently studied and i have not get a hang of it.
Consider the following recursive recursive function A(m,n)
A(0,n) = n+1 n greater or equal to 0
A(m,0) = A(m-1,1) m > 0
A(m,n) = A(m-1,A(m,n-1)), m>0,n>0
Find A(2,n) for n greater or equal to 0
I am not sure whether this is the right approach but here is what i did.
A(2,n) = A(1,A(2,n-1))
= A(0,A(1,A(2,A(2,n-2)))
Then i got stuck
2. Originally Posted by tester85
I need some direction to the recursion problem listed below : recursion is a topic which i recently studied and i have not get a hang of it.
Consider the following recursive recursive function A(m,n)
A(0,n) = n+1 n greater or equal to 0
A(m,0) = A(m-1,1) m > 0
A(m,n) = A(m-1,A(m,n-1)), m>0,n>0
Find A(2,n) for n greater or equal to 0
I am not sure whether this is the right approach but here is what i did.
A(2,n) = A(1,A(2,n-1))
= A(0,A(1,A(2,A(2,n-2)))
Then i got stuck
Two hints:
1) this is the Ackermann function
2) try first to get a non-recursive expression for A(1, n), then go on to solve A(2, n).
3. Ok i will try now. I tired to solve for A(1,n)
Here is what i got.
A(1,n) = A(0,A(1,n-1))
= A(0,A(0,A(1,n-2)))
it seems like an endless loop is my approach wrong. I cannot go further even with the following link.
http://en.wikipedia.org/wiki/Ackermann_function
4. Originally Posted by tester85
Ok i will try now. I tired to solve for A(1,n)
Here is what i got.
A(1,n) = A(0,A(1,n-1))
= A(0,A(0,A(1,n-2)))
it seems like an endless loop is my approach wrong. I cannot go further even with the following link.
First of all, A(0, A(1,n-1)) = A(1,n-1) + 1 from the definition of the function.
Let's write $x_n$ for A(1, n). Then you have a recurrence relation
$x_0 = 2$
$x_{n+1} = x_n + 1$
You can calculate a number of the terms, see what formula you likely get for $x_n$, and prove it by induction.
There are general theorems for solving such a recurrence relation, but they're too tedious to use for this exercise.
5. Ok noted with thanks. Now i know how to approach the problem.
6. Mistake spotted. Problem solved. | 2016-08-26 00:00:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7016555070877075, "perplexity": 1073.8268623828294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982294883.0/warc/CC-MAIN-20160823195814-00283-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://autoplot.org/ascii_data_source | # ascii data source
The ASCII table parser has enough controls and features that it should have its own page. This page describes in detail what it can do and how it is used.
# 1. What it can do
Autoplot's ascii parser looks in ascii files for tables of data. It breaks up the file into records and fields, and records with the correct number of fields are treated as data. Presently a record is always a newline-delimited line. (^M, ^J, and ^M^J are all treated as newline delimiters.) The record is generally broken up by tabs or whitespace into fields. When a line has the "right" number of fields, it is considered a data record. "Right" is in quotes because the user can explicitly control the number of fields, or the code can also guess the number of fields, essentially looking in the file for the data records. When a line contains the right number of fields, but the fields are not parsable, it tries to extract labels for the fields from the record. Non-records can contain metadata as well, and this is described later. Last, field parsers are "pluggable" so for example, times and nominal data can be parsed as well.
So in short, this:
• parses tables with the same number of fields in each record
• ignores lines that do not appear to be records
• grabs labels and units from a line that has the correct number of fields, but fields are not parseable.
• has pluggable field parsers to handle times and nominal data
• has pluggable record parsers to handle fixed columns and regular expressions as well.
## 1.1. What it cannot do
• parse records with variable field counts.
• parse records that span multiple lines. (Presently, this wouldn't be too difficult to support.
• parse multiple tables
# 2. Records and non-records
A record is a line that breaks up into the correct number of fields, and doesn't start with a comment character. Sometimes comments are treated as data fields in this case, but when they do not parse, the record is disposed of. There can be failures, for example:
data file created 1/2/2011
1 2 3 4
2 3 4 5
1 2 3 4
causes problems because the first line, which is a comment but isn't explicitly marked as such, is treated as a record of column headers. There is a control to skip a number of lines before parsing.
Generally there should be a space between implicit comments, or better yet prefix comments with hash marks when possible. Note too that the number of fields determines how if a line is a record or not. So if you have spaces in column headers and spaces are used to delimit files, then the column header will be missed.
Note this parser will not be able to parse all files. Hopefully, though, it's capable of parsing many files, and simple improvements can be made to it so that it handles more files.
## 2.1. Parsing Column Headers and Units
The parser tries to create useful labels and names for each column. The name is a URI-friendly string used to identify columns, which can also be used as a variable name. The label is a human-readable label for the column. Here are a few rules for them:
• density(nT) -> name="density" label="density" unit="nT"
• Ch1(34.4-68.3) -> column name="Ch1" label="Ch1(34.4-68.3)" unit=dimensionless
Note that if a label cannot be used, then "field<n>" is used for the column. Right now, there are limitations on what is used for a label, including the characters: a-z, A-Z, 0-9, and " _[]()". (TODO: I'm not sure why this restriction exists.)
# 3. Field parsing
The parser provides codes for automatically configuring parsing. These will break on field delimiters as it's the most inclusive, yet still has acceptable performance. (Note that the fixed columns parser has the best performance.) Fields will be broken up on delimiters found in records, first looking for commas, then tabs, and then whitespace when commas and tabs are not found. Note the ascii parser can be configured to use a particular delimiter when the guess fails.
Each field has it's own parser for converting text into a number. The default is just a number parser (Java's Double.parseDouble). Often though, the field is an ISO8601 formatted time, or time of some other format, and special field parsers are available for this. In Autoplot's ascii table data source, checking time in the GUI will use this instead. Note that an ISO8601 time in the first column is automatically handled.
To be more precise, the das2 Units are identified for each column, and the unit is responsible for interpreting each field. When time is detected, a das2 time unit is specified for the column, and the parser work. Note when the AsciiParser is used directly (not through the ascii data source), ordinal values can be handled as well.
# 4. Use in Autoplot
Note Autoplot's ASCII Table Data Source is a wrapper for QDataSet's ASCII parser, org.virbo.dsutil.AsciiParser. It abstracts the table read from the ascii file a bit further, pulling out individual records and also combining records to form times.
# 5. Source Code
The source code for further review is here:
AsciiParser.java, the code that parses ascii tables.
AsciiTableDataSource.java the Autoplot plugin data source that wraps AsciiParser.java and provides parsing of dates when dates span multiple fields. | 2017-04-27 03:25:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4568316638469696, "perplexity": 2006.427617168755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121865.67/warc/CC-MAIN-20170423031201-00063-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/tags/biophysics/new | # Tag Info
3
Replace "magnetically charged" by "magically charged" - I would rather agree to that :) (and that's actually how I read it by mistake! ;)) Writing of magnetic charges (=monopoles) actually doesn't increase the plausibility. I think it's just a hoax (I mean, that the university is involved, not the whole business) - though this word would be not ...
1
At the equator the effect on the local value of $g$ due to the rotation of the Earth and the non-spherical shape is only a few parts in a thousand. So even planting a tree at the top of a high mountain at the Equator probably will not have much of an effect on its ultimate height. I found this Scientific American article interesting although it did not ...
1
International Agency for Research on Cancer groups cell phone in group 2B, which is for possibly carcinogenic to humans. Not even probably. So, it is not that harmful . But, you can protect your by Electromagnetic shielding. You can figure out a way to connect conductive or magnetic materials either to your or the phone, but if you connect it to your ...
2
No they cannot. Any living system is an incredibly complicated, interacting quantum system. The countionous interaction (noise, heat bath, etc.) between particles makes it impossible for a coherent state to exist. However this is only true for the whole organism, on the level of single proteins, or other biomolecules quantum coherence may play an important ...
0
background radiation is very dependent on where you live it's different depending on altitude rocks around where you live ect. but generally comes to around 0.1 - 2 micro sieverts per hour the "ionizing radiation regulations 1999" states: Anyone under 18 should not be exposed to more than 6 milli sieverts per year above background radiation ~ 1 micro ...
1
It may well be that the cross-membrane voltage difference is largely homogeneous, but not because intracellular protons are very mobile. To the contrary, the cytosol mobility (rate of diffusion) of protons, and ions in general, is very different from the high mobility measured in water or dilute solutions of small molecular species. According to this ...
2
When you do a sit-up on a hard floor, your abdominal muscles are doing all the work to bring your upper body to a certain height above the floor. On a springy surface, as you start to sit up, the weight of your body becomes concentrated on the part of the floor beneath your hips, causing it to sink. This gives two advantages: The final height of your upper ...
2
There are obviously many biological factors contributing to this, but from a physics point of view one could talk about degrees of freedom: since a foamy floor has more of them it is easier for yourself to get in the optimal position to get up. I would also like to add a simple biological factor: it hurts more to move around when you are trying to get up on ...
2
It would still be here, it would just be turned inside out.
2
1) What can be say about the work done by the man to the weight? Unlike gravity, the force exerted by the man is in general not constant, does not depend only on position of the body (you may apply different force on the weight at the same height $h$ on the way up vs. on the way down), and is not conservative (does not arise from a potential). When the ...
4
1&2) Let's suppose there are only two forces in play: the one exerted by the man, and weight. The sum of works done on the body equals the variation of kinetic energy (this is a theorem of mechanics; I fail to find its English name). Since the body has no velocity at the start and at the end, the sum of works is zero. The net work done by the man on ...
1
The correct statement is - the total work done on the weight is zero i.e. the total energy of the weight before and after the experiment is same. However, when the man is lifting the weight he is obviously working against gravity. More importantly, when he is lowering the weight, he is still working against gravity, as gravity would rather lower the weight ...
0
VSD's work because a portion of the VSD molecule contains a chromophore, that is, a molecular structure that is responsible for absorbing light at one or more particular wavelengths. Absorption of light is entirely a function of the electronic structure of the chromophore. That is, light at a particular wavelength is absorbed by the molecule because the ...
Top 50 recent answers are included | 2016-04-29 06:20:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6302524209022522, "perplexity": 552.4832982552359}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860110764.59/warc/CC-MAIN-20160428161510-00025-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.giaascartomante.com/vernonia-cinerea-ylpjrsn/e18ba6-molar-mass-of-so4 | If the formula used in calculating molar mass is the molecular formula, the formula weight computed is the molecular weight. A) 65.2 g/mol B) 58.9 g/mol C) 11.4 g/mol D) 22.4 g/mol E) 43.9 g/mol. What mass of AgNO3 would you dissolve in water in order to get 1.00 g of silver? What is the density of the gas in #g*L^-1#? If 9.15 g of element X(s) is completely reacted with 4.00 L of fluorine gas at 250 C an... What is the mass in grams of 4.47 \times 10^{23} atoms of arsenic? … You dissolve .3g of an acid in a small amount of water. mass of one mole of that compound. Molar mass of EDTA is 372.244 g/mol. like 13k plus.. help please. Calculate the molecular or formula mass for Ca(NO3)2. a) 116 amu b) 102 amu c) 82 amu d) 164 amu. For instance, if you had a 80.0 g sample of a compound that was 20.0 g element X and 60.0 g element y then the percent composition of each element would be: B) 4 kg. 187 amu C. 0.00534 g D. 0.00534 amu. D)... Cytochromes are proteins that contain iron and are important in biological electron transport. Answers: 2. continue. How many kilograms of chalcopyrite must be mined to obtain 300 g of pure Cu? He isolated the compounds from a mold which had contaminated some of his experiments. A mole of Cd. Samples of helium, sulfur, copper, and mercury each contain 1 mole. The density of krypton gas at 1.2 atm and 50 ^oC is _____ g/L. 12 b. Calculate the mass, in grams, of the following sample. Which of the following samples has the greatest mass? As its formula indicates, this acid has an acid hydrogen ion. What is the molecular formula of this compound? The molar mass of an unknown gas was measured by an effusion experiment. 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Be sure your answers have the correct number of significant di... What mass of CuCl_2 + 6H_20 would be needed to make 500.00 mL of 0.1500 M CuCl_2 solution? Nicotine (C10H14N2), a member of the nightshade family of plants, is an oily liquid that makes up between 0.6 to 3% of a cigarette's mass. How many formula units are in 19.6 g of magnesium chloride? Find the volume, in liters, occupied by 0.220 mole of He. Use this an... How many moles of zinc are contained in 892 mg of zinc? How many grams are in 1.73 mol of dinitrogen pentoxide ( N 2 O 5 )? (a) What is its empirical formula? Calculate the number of moles of fluorine gas produced. The density of hexane is 0.6548 g/mL and its molar mass is 86.17 g/mol. Find the number of moles of Na_2CO_3 in 0.015 grams of Na_2CO_3. Calculations: Formula: Al2(so4)3.18H2o Molar Mass: 666.3984 g/mol 1g=1.50060384298642E-03 mol Percent composition (by mass): Element Count Atom Mass %(by mass) 2 Al + 6 HCl to 2 AlCl3 + 3 H2 a) 17.6 g H2 b) 8.79 g H2 c) 3.91 g H2 d) 2.93 g H2, What is the molar mass of (NH4)2O? By looking at a periodic table, we can conclude that the molar mass of lithium is $$6.94 \: \text{g}$$, the molar mass of zinc is $$65.38 \: \text{g}$$, and the molar mass of gold is $$196.97 \: \text{g}$$. According to the following reaction, how many grams of oxygen gas are required for the complete reaction of 25.8 grams of butane (C4H10)? Consider the following unbalanced chemical equation in which element X is unknown. d) 225 amu. 0.272 g of C c. 12.0 g of ammonia, NH 3 d. 7.30 g of propane, C3H8 e. 235 g of Fe2O3. (a) 87.0 g (b) 0.870 g (c) 1.94 g (d) 194 g (e) 0.194 g, How many moles of K2SO4 are in 15.0 g of K2SO4? Just add up the molar mass of every element in the hydrate and multiply by the amount of atoms it has in the compound. a. H2O b. NaF c. Na3PO4 d. K2Cr2O7 e. C6H12O6, Give the molar mass for the compound. This is how to calculate molar mass (average molecular weight), which is based on isotropically weighted averages. The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the total weight of the atom (or group of atoms) in the formula by the formula weight and multiplying by 100. 13. If 78.4 grams of aluminum metal react with excess hydrochloric acid, how many grams of hydrogen gas can be produced? How many grams of CO2 would burning a gallon of gasoline make? What massif NaCl (FW 58.8) would be required to react with 200 mL of 0.200M AgNO3 solution? a) 96.0 g/mol b) 92.0 g/mol c) 82.0 g/mol d) 78.0 g/mol. Calculate the mass of iodine solid that participates. What is the mass in grams in 4.25 moles of Na2CO3? These relative weights computed from the chemical equation are sometimes called equation weights. a. It is a crystalline ionic solid with a very bright, red-orange color. How many iron atoms are in 27.71 grams of iron? Each tablet contains 80.0 mg of propranolol. A)4.0 g Li B)4.0 g Na C)4.0 g Rb D)4.0 g K E)4.0 g Ca, What is the atomic mass for cadmium? The molar mass of an unknown gas was measured by an effusion experiment. Determine the molar mass of copper(II) phosphate. which is more than 50% oxygen on a molar basis and which is more than 50% oxygen by mass? Calculate the mass of each of the following amounts in order to determine which substance has the greater mass. If 52.4 mL of NaOH(aq) is required to titrate 0.643 g KHP to the equivalence point, what is the concentration of the NaOH... A) Allison R. Lang determines the mass of an unknown gas to be 10.21 g at 36 degrees Celsius and 780 mmHg in a 3.5 L container. What is its molar mass? The reason is that the molar mass of the substance affects the conversion. a. How many more grams of selenium than chromium does this represent? How many moles of iron atoms are contained in 3.49 g of iron? A 743.0 g sample of SO_2 contains how many moles of SO_2? Molar mass. The balanced equation will appear above. Convert 2.80 mol of chlorine gas, Cl2, to g. What is the mass, in g, of Avogadro's number of argon atoms? a) 0.10 g b) 0.20 g c) 0.30 g d) 0.40 g e) 0.50 g f) 0.90 g. In the reaction, 210 kJ of heart energy is used to form 3.0 moles of hydrogen. We use the most common isotopes. The chemical formula of this important biological molecule is C_27H46O. A.) Select one: a. 5.20 \times 10^3 mol \\3. What is the molar mass and name of this gas? How many carbon atoms are in 3.00 g of butane? What information is needed to calculate the mass of calcium oxide that can be produced from 4.7 kg of calcium carbonate? They are also inorganic chemistry usually. It may cause digestive tract irratation if swallowed. Calculate the formula mass of glucose, C6H12O6. How many moles in 1.35 g of each? a) Determine the number of grams in 0.750 moles of the compound Iron (II) phosphate, Fe3(PO4)2? Find the molar mass of the gas. モル質量と元素組成を計算するために化学式を入力してください: モル質量 of (SO4)2 is 192.1252 g/mol Convert between (SO4)2 weight … A.C_2H_4O_2 \\D.C_4H_{10}O \\B. Calculate the molecular weight of ibuprofen, C13H18O2. 187 g B. What is the molar mass of this compound? Consider a situation in which 112 g of P4 are exposed to 112 g of O2. 95.21 g/mol \\5. How many grams are in 1.36 x 10-3 moles of H2S? Determine the molar mass of propane: C3H8. (a) 159.62 g/mol (b) 249.70 g/mol (c) 177.64 g/mol (d) 185.72 g/mol (e) 446.48 g/mol. Perform the calculation: Molar mass of a gas if its density is 3.6 g/L at 15 degrees C and 825 torr. Find the mass of 1 molecule of C2H6. Calculate the mass, in grams, of each of the following: 1.4 \times 10^{23} leadatoms, Calculate the mass, in grams, of each of the following: 2.92 \times 10^{22} heliumatoms, Given the density of liquid bromine is 3.12 g/mL, what is the mass of 4.00 moles of bromine? When calculating molecular weight ), density 1.86 g/cm3 synthesize nitrogen triiodide 0.023... 0.323 g of calcium carbonate 25 } mol d. 4.88 \times 10^ { 23 formula... For bulk stoichiometric calculations, we are usually determining molar mass AgNO3 ( FW 58.8 ) would required... Of 1.48 moles of P2O5 that can theoretically be made from 112 g of ammonia is by. Copyrights are the same number of moles of nitrogen is present in 10.0 grams of solid KCl ( formula of... + 2 ( 32.1 ) + 12 ( 18.0 ) = 2.01588g/ mol contains., and the mass percent water in order to determine which substance has the formula weight of 58.1 grams 2.50. 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Gram sample of air measurements that 0.095 moles of P2O5 that can theoretically be made from g... | 2022-05-20 12:55:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7657082080841064, "perplexity": 4213.1505630997335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662532032.9/warc/CC-MAIN-20220520124557-20220520154557-00582.warc.gz"} |
http://libros.duhnnae.com/2017/jun8/14983312504-Enhanced-sensitivity-to-variation-of-m-e-m-p-in-molecular-spectra-Physics-Atomic-Physics.php | # Enhanced sensitivity to variation of $m {e}-m {p}$ in molecular spectra - Physics > Atomic Physics
Enhanced sensitivity to variation of $m {e}-m {p}$ in molecular spectra - Physics > Atomic Physics - Download this document for free, or read online. Document in PDF available to download.
Abstract: We propose new experiments with high sensitivity to a possible spatial ortemporal variation of the electron-to-proton mass ratio $\mu \equiv m e-m p$.We consider a nearly-degenerate pair of molecular vibrational levels, whereeach state is associated with a different electronic potential. The change inthe splitting between such levels, with respect to a change in $\mu$, can belarge both on an absolute scale and relative to the splitting. We demonstratethe existence of such pairs of levels in Cs$2$. The narrow spectral linesachievable with ultracold Cs$2$ in these long-lived levels make this systempromising for future searches for small variations in $\mu$.
Author: D. DeMille, S. Sainis, J. Sage, T. Bergeman, S. Kotochigova, E. Tiesinga
Source: https://arxiv.org/ | 2019-06-27 10:10:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7799721956253052, "perplexity": 3800.800559835845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628001089.83/warc/CC-MAIN-20190627095649-20190627121649-00446.warc.gz"} |
https://mathematica.stackexchange.com/questions/144433/naive-manual-implementation-of-dft-results-in-long-sum-which-is-very-inefficie | # (Naive) Manual implementation of DFT results in long sum which is very inefficient to plot, can my approach be saved?
I will do my best to keep this question as short as possible, I am afraid though that I wont be able to get my issue across if I don't elaborate on the background and provide my code as an example.
Issue: I enjoy using Mathematica for my Numerics homework. I have a hard time using Matlab because I always need to translate things (i.e. algorithms and closed formulas) first into Matrices and Vectors before I can turn to the actual problem.
Mathematica instead lets me dive right into the implementation, the price I pay for it is the performance. I have no doubt that this is my fault hence I am here to ask for advice.
My issue as described above has shown dramatically when I was implementing the Discrete Fourier Transformation (of a real valued function) for a class assignment (thus not allowed to use built in functions). I was given these two items:
Def: Let $f_r$, $0 \leq r \leq N-1$ be the values of a periodic function with length $b-a$. $$\hat{f_r}:= \frac{1}{N} \sum_{m=0}^{N-1} f_m e^{- \frac{ 2 \pi i}{N} m r}, \ 0 \leq r \leq N-1$$
Lemma: Let $f:[a,b] \to \mathbb{R}$ with data $f_r, 0 \leq r \leq N-1$, then the trigonometric Fourier Transformation is given by $$T(t) = \hat{A_0} + \sum_{m=1}^{\left\lfloor{\frac{N-1}{2}} \right\rfloor} \left( \hat{A_m} \cos \left( \frac{2 \pi}{b-a}mt \right) + \hat{B_m} \sin \left( \frac{2 \pi }{b-a}mt \right) \right) \\ + \begin{cases} \hat{A_{N/2}} \cos \left( \frac{ \pi N}{b-a}t\right) \text{, if N is even} \\ 0 \text{, if N is odd}\end{cases}$$ Where $\hat{A_0}=\hat{f_0}, \ \hat{A_{N/2}}=\hat{f_{N/2}}$ and
$\hat{A_m} = 2 \text{Re}( \hat{f_m}), \ \hat{B_m} = -2 \text{Im}(\hat{f_m})$
Since these are beautifully closed formulas I would indeed implement them using Mathematica, naively following what the definition/lemma told me to do.
The issue I encounter is when $N$ becomes large, I have already massive trouble when $N=2^{12}=4096$
My implementation:
fhatentry[r_, rvector_] :=
fhatentry[r,
rvector] = (1./Length[rvector]) Sum[
rvector[[m + 1]] Exp[-(2.*Pi*I*m*r)/Length[rvector]], {m, 0.,
Length[rvector] - 1}]
As given by the Definition, using rvector for $f_r$. Note that I have already tried improving my poor performance speed by forcing the function to recall its values (might be not a good idea) and using numerical values, i.e. instead of the symbol $2$, using it's numerical equivalent $2.$
As in the Lemma I have implemented
ahat[k_, rvector_] := 2 * Re[ fhatentry[k, rvector]]
bhat[k_, rvector_] := -2* Im[fhatentry[k, rvector]]
and
trigpol[t_, rvector_] :=
fhatentry[0, rvector] +
Sum[ ahat[m, rvector] Cos[ (2. Pi m t)/(b - a)] +
bhat[m, rvector] Sin[ (2. Pi m t)/(b - a)], {m, 1., Floor[ (Length[rvector] - 1)/2]}] +
If[ EvenQ[Length[rvector]],
fhatentry[ Length[rvector]/2,
rvector] Cos[ (Pi Length[rvector] t)/(b - a)], 0]
I have also given that $a=0,b=1$ and stored them in Mathematica accordingly using numerical values.
Question: I am given a vector, say data, with $4096$ entries which translate to the values of $f_r$ for some periodic function $f:[0,1] \to \mathbb{R}$ and I am interested in the plot in the region [0,1].
Plot[trigpol[t,data], {t,0,1}]
Does terminate and does its job, but it takes like 5 minutes to do so.
I understand if these kind of questions are tedious but I am curious on how to:
• Tune the performance of my code above to make it work faster, while still keeping it relatively legible and easy to understand what's going on.
Update
In the comments I was suggested to use a vectorized coding style to implement my functions, I have done that as follows
trigpol2[t_, rvector_] := With[{len = Length[rvector]},
fhatentry[0, rvector] +
Total[
ahat[#, rvector] & /@ Range[1, Floor[(len - 1)/2]] *
Cos[(2 Pi Range[1, Floor[(len - 1)/2]] t )/(b - a)]] +
Total[ bhat[#, rvector] & /@ Range[1, Floor[(len - 1)/2]]*
Sin[( 2 Pi Range[1, Floor[(len - 1)/2]] t)/(b - a)]]
+ If[ EvenQ[len],
fhatentry[ len/2, rvector] Cos[ (len t)/(b - a)], 0]
]
When plotting this function (as in the Question above) my Timing went from 198 to 251 (the output remains correct though)
• try Plot[Evaluate[trigpol[t, data]], {t, 0, 1}] or Plot[trigpol[t, data], {t, 0, 1}, Evaluated -> True] – kglr Apr 25 '17 at 2:05
• Vectorized operations on numerical data tends to be much faster than Summing element-wise, so we can try to implement this without losing too much readability. I would replace your fhatentry with fhatentry[r_, rvector_] := With[{len = Length[rvector]}, Total[rvector*Exp[-2 Pi I Range[0, len - 1]*r/len]]/len]. There is almost no point in memoizing this function because each value is only used twice: once for ahat and once for bhat. I'll let you think about how to "vectorize" the Sum in trigpol :) – Marius Ladegård Meyer Apr 25 '17 at 7:05
• Dear @MariusLadegårdMeyer - Thanks a lot for your advice, I tried my best to follow it and I implemented the vectorized version for trigpol, I will post it in the OP accordingly. I am sure the fault is entirely on my side, but it didn't help reduce the computation time, in fact it increased it. – Spaced Apr 25 '17 at 13:38
• Also, I think you will have more control over the time the plotting takes if you use ListPlot instead, where you choose the t points yourself. E.g. ListPlot[Table[{t,trigpol[t,data]},{t,0,1,1/99}]] will give you 100 plot points evenly spaced. Adjust accordingly. – Marius Ladegård Meyer Apr 25 '17 at 13:41 | 2019-09-15 13:04:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4863761365413666, "perplexity": 1866.9854512233505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514571360.41/warc/CC-MAIN-20190915114318-20190915140318-00186.warc.gz"} |
http://aas.org/archives/BAAS/v28n2/aas188/abs/S011006.html | Session 11 - Elliptical Galaxies.
Display session, Monday, June 10
Great Hall,
## [11.06] A Grid of Model Atmospheres and Synthetic Spectra for the Far Ultraviolet Analysis of Old Stellar Populations
T. M. Brown, A. F. Davidsen (JHU), H. C. Ferguson (STScI)
We present a grid of stellar synthetic spectra suitable for detailed comparison to far-ultraviolet (FUV) observations obtained with the Hopkins Ultraviolet Telescope, IUE, and HST. Our specific application is to study the hot stellar populations in elliptical galaxies, but we anticipate that the models will be useful for other purposes. The 1,497 spectra span a range of 10,000 K \leq T_eff \leq 250,000 K and 2 \leq log g \leq 8.5, with three metallicities: Z = Z_\sun, Z = 0.1 Z_\sun, and Z = 0.01 Z_\sun. A variety of simplifying assumptions have been made to reduce computer time and improve convergence, at the inevitable expense of some accuracy. Nevertheless, models in the grid reproduce the overall continuum shape and most of the absorption features seen in HUT spectra of four evolved stars at temperatures of 17000, 29900, 36100, and 55000 K. The most serious discrepancy is in the cores of the Lyman series lines, where the observed lines are not as deep as those obtained from the models. Until this problem is resolved, the Lyman series lines will not provide a reliable measure of T_eff or log g.
While the synthetic spectra in this grid may not be appropriate for detailed analysis of high S/N stellar spectra, they are sufficiently similar to the observed stars to provide a considerable advantage over existing models for the analysis of the FUV spectra of composite systems, such as elliptical galaxies and globular clusters, where the advantages of having a large, well-sampled grid of models tend to outweigh the known inadequacies of the individual grid points. | 2014-10-20 23:41:34 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9126537442207336, "perplexity": 1745.5928314233527}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507443451.12/warc/CC-MAIN-20141017005723-00260-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://cms.math.ca/10.4153/CJM-2003-015-7 | Canadian Mathematical Society www.cms.math.ca
location: Publications → journals → CJM
Abstract view
# The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$
Read article[PDF: 258KB]
Published:2003-04-01
Printed: Apr 2003
• David Savitt
Format: HTML LaTeX MathJax PDF PostScript
## Abstract
We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.
MSC Classifications: 11G20 - Curves over finite and local fields [See also 14H25] 14H25 - Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
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© Canadian Mathematical Society, 2015 : https://cms.math.ca/ | 2015-11-25 17:28:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28703224658966064, "perplexity": 1828.6829715101085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398445219.14/warc/CC-MAIN-20151124205405-00144-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/23253/reference-countable-models-of-non-euclidean-geometry | # Reference: Countable Models of (Non-)Euclidean Geometry
Has there been a survey written on the model theory of first-order (non-)Euclidean geometry in the spirit of Hilbert and Tarski? I'm especially interested in two aspects of the model theory:
1. Countable models
2. Distinguished/canonical models.
I'm interested in 2 because I don't see why the standard ${\mathbb{R}}^2$ model is any special.
-
Of course you can take the algebraic points in $\mathbb{R}^2$ to get a countable model. – Dylan Thurston May 2 '10 at 19:05
I sent you Greenberg's article an hour or two ago. I think you will like it. – Will Jagy May 3 '10 at 19:37
@Will/Colin: Can you send me the article, too, please? – Hans Stricker Dec 10 '10 at 16:45
Marvin Jay Greenberg got very interested in the foundations for the fourth edition(2007) of his book. This led to a survey article in the March 2010 M.A.A. Monthly. Table of contents:
It does not seem to say explicitly on the link, volume 117, number 3, March 2010, pages 198-219
I have a pdf of the article if you have no better way to look at it, just email me. Marvin sent me a copy because my results are in it, however I was not doing genuine foundations.
The other book is by the well known R. Hartshorne, called Geometry:Euclid and Beyond (2000)
That's a pretty good start, article and two books. You may also want to look up Victor Pambuccian on MathSciNet
To summarize the bits I expect you find most interesting, Hilbert gave a recipe for defining the hyperbolic plane first and finding the underlying field second, this is called the "field of ends" in English. I prefer the "Euclidean" fields, if there is a field element $a > 0$ then $\sqrt a$ is also in the field. The smallest field that does everything I find interesting is the "constructible field" which is all numbers arrived at by starting with the rationals and taking a finite number of square roots, mixed with other field operation of course. This is a subfield of the algebraic numbers. Hilbert himself concentrated on the milder Pythagorean fields, if $a \in F$ then $\sqrt{1 + a^2} \in F.$ So there may be positive field elements without square roots. As there is an intermediate step that usually requires interpreting the standard hyperbolic functions, Hartshorne wrote that entire section with a "multiplicative length" for segments, which corresponds to taking $e^x$ when $x$ is an ordinary length in a plane over the reals. Personally, I prefer $\sinh x$ because of the appearance of the (hyperbolic) Pythagorean Theorem and the integral triangles you get this way, but that is never going to be popular. | 2015-04-26 01:43:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7225496768951416, "perplexity": 595.1516117694166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246652114.13/warc/CC-MAIN-20150417045732-00280-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/tags/draw/hot | # Tag Info
41
You can load the image into inkscape, choose Path | Trace Bitmap to obtain an svg file (using only 8 colors), and then use svg2tikz to convert the svg file to a tex file using tikz. The result is the image below. The tex source is too big to post here; it looks like this. \documentclass{standalone} \usepackage[utf8]{inputenc} \usepackage{tikz} \begin{...
39
One rose...drawing with Mathcha.... \documentclass[a4paper,12pt]{article} \usepackage{tikz} \begin{document} \tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt \begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] %uncomment if require: \path (0,439); %set diagram left start at 0, and has height of 439 %Shape: ...
27
Using TikZ and its calc library you can use the ( $(<name1>)!<value>!(<name1>)$ ) syntax to find a point along the segment passing through (<name1>) and (<name2>). In the example below, ( $(a)!0.66666!(b)$ ) means a point that is two thirds away from (a) in the segment joining (a) and (b): \documentclass{article} \...
25
I think that the key here is to put more samples. This is your code, with 100 samples. I changed the domain too, from 0 to 2pi, because yours draws everything twice. I also added a view more alike to the plot in WA that you linked. \documentclass[border=2mm]{standalone} \usepackage {pgfplots} \pgfplotsset{compat=1.17} \begin{document} \begin{tikzpicture} \...
25
This is basic Asymptote code: // // trefoil.asy // // to get trefoil.png, run // asy -f png -render=4 trefoil.asy // // to view it in internal Asymptote 3d-viewer, run // asy -V trefoil.asy // // or, to get an interactive 3D vector WebGL graphics, // embedded within HTML file trefoil.html, run // asy -f html trefoil.asy // import graph3; import tube; ...
21
From the phase plot, your transfer function has: 1 integrator, 1 pole (at w=100), and 2 zeros (at w=0.1 and w=10). It can be written as follows: Here is my attempt using bodegraph package. You can also check this tutorial: The Easiest Way to Draw a BODE Plot in LaTeX!). \documentclass{standalone} \usepackage{bodegraph} \begin{document} \begin{...
20
\documentclass[border=5pt,tikz]{standalone} \begin{document} \begin{tikzpicture}[xscale=1,yscale=1] \fill[blue!20] (0,-2.5) --+ (0,2.5) arc (0:180:1.5) -- (-3,-2.5) arc (180:235:3) --+ (0,-.25) --+ (.45,-.25) --+ (.45,.015) (0,-2.5) arc (0:-55:3); \fill[white] (0,0) arc(0:180:1.5) --+ (0,-2) -| cycle; \fill[blue!10] (-1.5,-2) circle ({1....
18
You don't need to connect nodes. First draw background lines. All starting from (0,0). \foreach \angle in {0,1,...,359} \draw[cyan!50!black] (0,0)--++(\angle:4); Second, draw a circular node white filled: \node[circle, fill=white, text=cyan!50!black, text width=15mm, align=center]{Orion\\2000}; And third (although it's the first command), define the ...
17
\documentclass[a4paper]{article} \pdfpageheight\paperheight \pdfpagewidth\paperwidth \makeatletter \dimen4=.996264\paperheight \dimen6=.996264\paperwidth \pdfliteral page{% q n 0 0 m \strip@pt\dimen6 \space \strip@pt\dimen4 \space l s Q} \begin{document} zzzz \end{document}
17
Note sure if it is more efficient, but it is a lot easier to work with if you define the coordinates first, and then draw them. BTW: +1 for a nice usable MWE \documentclass[border=5pt,tikz]{standalone} \usetikzlibrary{backgrounds,calc} \definecolor{hellblau}{RGB}{18,158,181} \definecolor{dunkelblau}{RGB}{22,141,163} \newcommand{\changefont}[3]{\...
17
So I tried to reproduce the graphics you posted but removing the circle around the Earth and using it to make the equator. Yes, it's a shaded sphere but its position is accurate, trust me. If there is something unclear about the code, please feel free to ask so that it might help you if you decide to do more in TikZ. You can replace the shaded sphere in the ...
16
15
Welcome to TeX.SE! Here is a simple prototype of what you need. Hopefully, you may apply your desired customizations to it. \documentclass[border=1pt]{standalone} \usepackage{tikz} %TikZ central library is called. \usetikzlibrary{automata,positioning} % automata and positioning libraries are required to use nodes and coordinates in addition to placement ...
15
For complicated figures with graphics, I think there is some consensus that tikz is the way to go. It is probably worth learning. It is amazing, but can be intimidating for the beginner. For very simple pictures (lines, arrows, text, ovals) the picture environment has easy-to-learn tools. The \put command together with \line and \vector commands can recreate ...
15
Here is a solution that automates the process with the command \contour: Each circle must be given a name. For convenience, I also used the \coordinate command for each of the poles. Then you must choose, for each pole, a coordinate outside the main shape. The contours will be drawn parallel to the line from the external point to the corresponding pole. For ...
14
To connect the edges of the circle you can use the border anchors of nodes (A.120) and the to Operator to connect it to the outer square. \documentclass[tikz, border=5mm]{standalone} \begin{document} \begin{tikzpicture} \draw[clip] (-2,-2) rectangle +(3,3); \node[minimum size=5cm](A){}; \node[circle, draw, minimum size=1cm](B) at (A.center) {}; \foreach \...
14
\documentclass[border=3pt,tikz]{standalone} \usetikzlibrary{decorations.pathreplacing} \begin{document} \begin{tikzpicture} \draw[very thick,-{latex}] (0,0) -- (6,0) node[below]{$x$}; \foreach \x/\l in {1/-r,3/{},5/+r} \draw (\x,3pt) -- (\x,-3pt) node[below]{$x_0\l$}; \draw [decorate,decoration={brace,amplitude=5pt,mirror,raise=4ex}] (1,0) -- (5,0) ...
13
There is no every draw available. One possible way is to style every path. \documentclass{article} \usepackage{tikz} \begin{document} \begin{tikzpicture}[every path/.style={->,red,thick}] \draw(0,0)node[left]{$A$}--(5,0)node[right]{$B$}; \end{tikzpicture} \end{document} An alternative solution is to globally set draw for every picture. ...
13
Like this? \documentclass[border=10pt,multi,tikz]{standalone} \begin{document} \begin{tikzpicture} \foreach \m in {0,1,2}{ % draws shaded regions \draw [fill=gray!50!white, rounded corners] ({ 15+120*\m}:2.25) arc ({15+120*\m}:{129+120*\m}:2.25) -- ({129+120*\m}:1.75) arc ({129+120*\m}:{15+120*\m}:1.75) -- cycle; } % draws non shaded regions \...
13
\documentclass[border=2mm, tikz]{standalone} \usetikzlibrary{matrix, positioning} \tikzset{ trafficlight/.style = {% matrix of nodes, nodes in empty cells, rounded corners, draw = blue!70, fill = blue!30, nodes = {circle, minimum size=5mm, anchor=center, draw=black}, row 1/.style={nodes={fill=red}}...
13
Here's how to do it with pstricks. I also suggest a line with arms: \documentclass[svgnames]{article} \usepackage{pst-node} % \usepackage{auto-pst-pdf} to compile with pdflatex --enable-write18 (MiKTeX) % or pdflatex --shell-escape (TeXLive, MacTeX) \begin{document} This is a very long\Rnode{st}{ \underline{sentence that}} appears in this very \Rnode{ss}{\...
12
With tkz-euclide you can use a barycentric coordinate, which can be used to place points between other points. A point p lying on the line between the points A and B can be written as (a1 + a2)p = a1A + a2B, and (a1, a2) will be the barycentric coordinate of p. A point that is 2/3 along the way between A and B can be found by the vector sum A + (2/3)(B - A) ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2021-05-08 12:59:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.946893572807312, "perplexity": 5591.971561878629}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.7/warc/CC-MAIN-20210508121446-20210508151446-00528.warc.gz"} |
https://ahmetozer.org/ | # Finding one-way Latency on ISP
My service provider has a latency issue over three years.
# Network Prototyping with Linux Namespace
Linux namespaces are mostly use in container technologies. They very usefully, efficient and fast to deploy.
# Ripe Atlas USB Disk Error Problem
If you have a Ripe atlas probe version 3, you might be face this error with your probe.
It is happen to me on my probes in many times.
You can easily to solve this error. You just need a bit time and computer.
# Reinstall Server From Recovery Disk with RSYNC
Few weeks ago, my Amsterdam instance out of space. My all projects is in my workstation (also Github and Gitlab) and I make a sync with mirrorfy to server. I can purge or re install my server without losing (because everything has a backup) any data but I try to truncate log files to gain a bit space in server. I made a huge mistake while emptying log files 😁. Normally I have to write truncate command with */* arg while in /var/log folder to truncate files but lack of sleep cause to write a /*/* and most of the binaries and some important system files is gone 😂. | 2021-03-03 03:13:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22971849143505096, "perplexity": 6056.554548109296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178365186.46/warc/CC-MAIN-20210303012222-20210303042222-00188.warc.gz"} |
https://brilliant.org/problems/an-impossible-question/ | # An Impossible Question!
Level 1
A bear walks a great distance South. Then he takes a left, & walks a great distance. He again takes a left & walks a great distance. Amazingly, he finds itself where it started initially. Find the color of the bear!
Details & Assumptions : a great distance means the range between $1000$ - $3000 km$.
× | 2019-08-25 21:48:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3982703685760498, "perplexity": 2716.7647888089095}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330800.17/warc/CC-MAIN-20190825194252-20190825220252-00328.warc.gz"} |
https://andrescaicedo.wordpress.com/tag/erdos-rado-theorem/ | ## 580 -Partition calculus (6)
April 24, 2009
1. The ${\mbox{Erd\H os}}$-Rado theorem
Large homogeneous sets (of size ${\omega_1}$ or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by ${\mbox{Erd\H os}}$ and Rado using tree arguments (with ${\kappa+1}$ lowered to ${\kappa}$ in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and ${\mbox{Todor\v cevi\'c},}$ which proves the stronger version. For ${\kappa}$ a cardinal let ${2^{<\kappa}=\sup_{\lambda<\kappa}2^\lambda.}$
Theorem 1 (${\mbox{Erd\H os}}$-Rado) Let ${\kappa}$ be a regular cardinal and let ${\lambda=(2^{<\kappa})^+.}$ Then
$\displaystyle \lambda\rightarrow(\kappa+1)^2_\mu$
for all ${\mu<\kappa.}$ | 2020-05-25 13:48:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9475194215774536, "perplexity": 795.8676370045216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347388758.12/warc/CC-MAIN-20200525130036-20200525160036-00077.warc.gz"} |
https://chem.libretexts.org/Courses/Heartland_Community_College/CHEM_120%3A_Fundamentals_of_Chemistry/06%3A_Gases/6.06%3A_Gas_Law_Equations%3A__Relating_the_Pressure_and_Temperature_of_a_Gas | # 6.6: Gas Law Equations: Relating the Volume and Temperature of a Gas
Learning Objectives
• State whether the volume and temperature of a gas are directly or indirectly proportional.
• Write an equation for Charles's Law.
The experiments that investigated how changing the temperature of a constant amount of gas impacted its volume under isobarometric, or constant-pressure, conditions were performed by a French physicist named Jacques Charles. By increasing the temperature of a gas, its constituent particles move more quickly and, therefore, collide more often with the surfaces of the container in which they are held. If the container is flexible, increasing the frequency of these collisions causes the container to expand. Therefore, the volume and temperature of a gas are directly, or linearly, proportional to one another, and dividing these quantities yields a constant, k3, as shown below.
$$\dfrac{ \text{V}}{\text{T}}$$ = $$\rm{k_3}$$
Like k1 and k2, k3 is not a universal constant, because its value varies based on the identity of the gas that is being studied. Therefore, the most practical representation of Charles's Law directly relates the volume and temperature of a gas at the beginning of an experiment to their corresponding final values. This equation, which is shown below, can also be written using variables that have abbreviated or modified subscripts.
$$\dfrac{ \rm{V_{initial}}}{\rm{T_{initial}}}$$ = $$\dfrac{ \rm{V_{final}}}{\rm{T_{final}}}$$
$$\dfrac{ \rm{V_{i}}}{\rm{T_{i}}}$$ = $$\dfrac{ \rm{V_{f}}}{\rm{T_{f}}}$$
$$\dfrac{ \rm{V_{1}}}{\rm{T_{1}}}$$ = $$\dfrac{ \rm{V_{2}}}{\rm{T_{2}}}$$
6.6: Gas Law Equations: Relating the Volume and Temperature of a Gas is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. | 2022-12-07 16:32:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8905216455459595, "perplexity": 940.3359373525749}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711200.6/warc/CC-MAIN-20221207153419-20221207183419-00196.warc.gz"} |
http://www.mathwarehouse.com/geometry/polygon/ | Graphing Calculator Math Worksheets Algebara Solver Chart Maker
A+ A− B
# Polygons: Formula and Examples
## Exterior Angles and Interior Angles
Interior Angle Sum Theorem
## What is true about the sum of angles inside a polygon (ie interior angles) ?
Answer The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180
Examples:
• Triangle or ( '3-gon' )
• Quadrilateral which has four sides ( ' 4-gon')
• sum of interior angles: (4-2)180 = 360°
• Hexagon which has six sides ( '6-gon')
• sum of interior angles: (6-2)180 = 720°
Video Tutorial on Interior Angles of a Polygon
Definition of a Regular Polygon: A regular polygon is simply a polygon whose sides all have the same length and whose angles all have the same measure. The most well known example of a regular polygon is the equilateral triangle.
## What about when you just want 1 interior angle?
Measure of a Single Interior Angle
Answer: In order to find the measure of a single interior angle of a regular polygon (a polygon with sides of equal length and angles of equal measure) with n sides, we just divide the sum of the interior angles or (n-2) × 180 by the number of sides or n
The Formula
An interior angle of a regular polygon with n sides is $\frac{ (n -2) \cdot 180^{\circ} }{n}$
Example: To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows:
( (8-2) × 180) /8 = 135°
What is the total number degrees of all interior angles of a triangle?
What is the total number of degrees of all interior angles of the polygon on the left?
What is the sum measure of the interior angles of the polygon (a pentagon) on the left?
What is sum of the measures of the interior angles of the polygon (a hexagon) on the left?
Finding 1 interior angle of a regular Polygon
What is the measure of 1 interior angle of a regular octagon?
Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?
Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?
Challenge Problem)
What is the measure of 1 interior angle of a pentagon ?
## How about the measure of an exterior angle?
### Exterior Angles of a Polygon
Formula for sum of exterior angles:
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.
## Measure of a Single Exterior Angle
Formula to find 1 angle of a regular convex polygon of n sides =
1+2 +3 =360°
1+2 +3+ 4 =360°
1+2 +3+ 4 +5 =360°
Practice Problems
Calculate the measure of 1 exterior angle of a regular pentagon?
What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?
What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)? | 2014-04-18 18:16:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.260469526052475, "perplexity": 733.7973156202271}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00121-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2331700/is-fx-y-5x-4y-injective-or-surjective | # Is $f(x, y) = 5x - 4y$ injective or surjective?
Define $f : \Bbb Z\times\Bbb Z \to\Bbb Z$ by $f(x, y) = 5x - 4y$.
Is $f$ injective or surjective?
How would I go about proving this?
Thanks
• Do you know the definitions of injectivity and surjectivity? – Dave Jun 21 '17 at 21:19
• Yeah, I just have trouble formally proving if its surjective – ivince95 Jun 21 '17 at 21:59
• A good thing to do is just write down the definitions of injectivity and surjectivity, and see if the function satisfies these conditions. – Dave Jun 21 '17 at 22:28
HINT
• A function is injective if it maps different inputs for different values. Can you find $x,y,v,w,a$ such that $f(x,y) = a = f(v,w)$ with $(x,y) \ne (v,w)$?
• A function is surjective if it completely covers the set it maps to. Let $a \in \mathbb{Z}$. Can you find such $x,y \in \mathbb{Z}$ so that $f(x,y)=a$?
• @ivince95 that you picked such an example shows it is not injective, since different inputs lead to the same outputs. It should be surjective, can you prove that any integer can be written as a linear combination of $4$ and $5$? For example, you can write $1$ that way, and then any number, e.g. $29 = 29 \cdot 1$... – gt6989b Jun 21 '17 at 21:29
Surjective: Fix a $z \in \mathbb{Z}$ and try to find an $x=(x1,x2) \in \mathbb{Z}\times\mathbb{Z}$ with $f(x1,x2)=z$. For Injectivity, show that for $(x1,x2) \neq (y1,y2) \in \mathbb{Z}\times\mathbb{Z}\implies f(x1,x2) \neq f(y1,y2)$.
• Can you show me an example for the surjective? – ivince95 Jun 21 '17 at 22:07
Hints:
$$5x-4y=5a-4b\iff 5(x-a)=4(y-b)\implies\begin{cases}x=a\pmod 4\\y=b\pmod 5\end{cases}$$
For example: $\;5\cdot3-4\cdot7=5\cdot7-4\cdot12\;$
Also, since $\;gcd(4,5)=1\;$ , there exist $\;m,n\in\Bbb Z\;$ s.t. $\;5m+4n=1\;$ , so for any $\;t\in\Bbb Z\;$ ... | 2019-07-15 18:20:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9459032416343689, "perplexity": 218.14867807832726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195523840.34/warc/CC-MAIN-20190715175205-20190715201205-00486.warc.gz"} |
https://libwethair.wordpress.com/ | # Momentum Transfer & Dripping
Here’s the experiments on momentum transfer and dripping
Moving a hair from left to right with different momentum transfer on hair
A free fall of liquid with different momentum transfer on hair
Here’s the dripping experiment
and pouring water onto dry hairs
# Results Collection
I’ll put our main results here. This post will be kept updated.
Rubber band dipping/pulling
256 strands
4096 strands
Car wash
16,384 strands
Hair Flip
1024 strands w. sub-stepping
1024 strands w. CCD
4096 strands w. sub-stepping
# Latest Todo list
Here is an updated TODO list of the tasks currently on our radar, feel free to comment, prioritize, or add things.
Raymond:
• Compile code on CG cluster (stokes & others)
• Compare 4 vs 8 vs 16 vs 32 vs 64 cores on a scene
• Re-enable the Level Set Force
• Look into Newton-Raphson solver
• Look at faster fluids solver Christopher mentioned
Henrique:
• Unit test the Discrete Elastic Rod code
• merge codes together
• Update Elastic Rods code for new Compliant Solver
Christopher / Changxi / Eitan :
• Contribute to Abstract, Related Work, verify contributions
Other:
• Rendering / Houdini
• Set up scenes we want for the paper
# Linear Compliant Solver with Velocity Constraint
Here we briefly introduce the linear compliant solver with velocity constraint.
All the energies of constraint on positions may have the form
$E=\frac{1}{2}\phi^T\alpha\phi$ for the goal where $\phi(q)=0$.
All the energies of constraint on velocities may have the form
$E_v=\frac{1}{2}\dot{\phi}^T\beta\dot{\phi}$ for the goal where $latex \dot{\phi}(q,\dot{q})=0$.
We define $J=\frac{d\phi}{dq}, J_v=\frac{\partial\dot{\phi}}{\partial\dot{q}}$, and $J_{xv}=\frac{\partial\dot{\phi}}{\partial q}$.
By the KKT condition we have
where
is the geometric stiffness matrix with velocity change considered, and $\lambda$ is the Lagrange multiplier where $lambda=h^{-1}\mu=-K\phi$ and $lambda_v=h^{-1}\dot{\mu}=-K_v\phi$, where $\text{diag}K=\alpha$ and $\text{diag}K_v=\beta$.
By Schur complement the equations above can be re-formulated into an equation where the LHS is SPD, thus with even large system it’s possible to be solved with CG in parallel.
This is the extension to [Tournier et al. 2015] with velocity constraints added, where stiff drag forces may be also integrated implicitly.
Ref.
Tournier, Maxime, et al. “Stable constrained dynamics.” ACM Transactions on Graphics (TOG) 34.4 (2015): 132.
Here we show the results for our linear-variant quadrature and the inter-hair flow.
(theories come later)
2 Hairs scripted splitting after sticking
https://www.dropbox.com/s/f9823owiubuqnii/splitting2.mp4?dl=0
Inter-hair flow with hair fixed
2 hairs
https://www.dropbox.com/s/m2eevbmgsxrdgbl/liquid_share_basic.mp4?dl=0
7 hairs
https://www.dropbox.com/s/vmwxy47lkpvos4l/liquid_share_7.mp4?dl=0
Inter-hair flow with hair freely move
2 hairs
https://www.dropbox.com/s/7ioru00v3yuvc4s/liquid_share_basic_free.mp4?dl=0
7 hairs
https://www.dropbox.com/s/m2eevbmgsxrdgbl/liquid_share_basic.mp4?dl=0
Combined simple example
https://www.dropbox.com/s/o27zjxxlybamlkz/circle_new.mp4?dl=0
Raymond and I have recently made progress on the items for this week.
Raymond has coded up a mix of linear quadrature, with added samples, in order to smooth out the separation of strands. I’ll let him speak to that since he is more familiar with it.
Given our discussions last time, we have a new Adhesion-Collision model, with a modified profile on the Cohesion Table we use for lookups. Here are some views of the updates profiles:
We have modified the old table into essentially three regions, best seen from the Adhesion-Collision profile, the first picture above.
Region 1, from d = 0 to dmin (where dmin is the combined radii of two particles), denotes the penetration region. This accounts for everything from overlapping exactly all the way to just barely touching. This region is negative, linearly scaled, so that the two particles will push away from one another. Notably however, past the point of overlapping, when things begin tunneling, there is no special treatment, and particles would begin to push in the wrong direction. Ideally, there is never enough collision-contact drift to allow for that.
Region 2, from dmin to some threshold above dmin (for now 1/4 dmin), is the region when two particles are very close to one another. Normally adhesion would say to keep pulling together closer and closer, but instead now we scale this somehow (for now linearly), between the end of region 1 (contact or dmin) and the beginning of region 3, not worrying about tangent continuity.
Region 3, above the modified range or threshold near dmin, is where the plot remains exactly as it was before, in accordance with the principled profile Raymond derived for the Adhesion force. In order to determine the values for region 2, we look at the bounds of region 1 and region 3, and linearly scale between them.
Therefore there this model presents a few tunable parameters, those being:
• The scale constant, or form, at which region 1 changes
• The scale or profile for region 2
• The threshold or distance at which we transition from region 3 to region 2.
Implementing this new profile, we witness the following comparison videos:
OldTable
NewTable
NewTableCircleExample
From these videos we can see that the result is somewhat the same, but there is no need to do Collision response with the new Cohesion table.
Qualitative comparisons of different approaches below:
Original Original w/ No Collisions (Collisions ON, Old Cohesion Table) –Slow -converges -slight separation throughout (Collisions OFF, Old Cohesion Table) -fast –overlaps New New Deluxe (Collisions OFF, New Cohesion Table) -fast -no overlaps -very close proximity, touching (Collisions ON, New Cohesion Table) -may still want collisions elsewhere (think dry hairs in contact) -relatively fast, (few collisions if any) -no overlaps -very similar to New way
Looking at the number of collisions created from simply changing this table, we get the following below (X is time, Y is #of Collisions) from the vertical tests in the videos:
We are also sub-stepping the hairs in order to prevent penetrations/collision. This helps disperse the dynamics and collision handling of adhesion over several substeps, and effectively lower the timestep of the scene.
The next order of business, as we see it, is as follows:
1. Should we remove collisions altogether from the pipeline, and replace them with our Adhesion-Collision spring model, then we will need a way to handle underwater collisions as well, since the adhesion model does not apply for fully saturated hairs (they do not come together, but move freely, but still cannot penetrate)
2. Therefore we can simply continue to apply the non-penetration component of the Adhesion-Collision spring model, which acts as a spring/penalty force to push out collisions.
3. We aim to do this by simply applying the adhesion model when the two particles are overlapping, and ignoring its contribution elsewhere while under water
2. CCD (with thickness) replacing our instantaneous check
1. If we want to take large timesteps, or avoid sub-stepping, or in general be safer about collisions, we will need to switch our detection routine from instantaneous to continuos.
2. Additionally, since the adhesion forces begin to act when inside a neighborhood/threshold of the strands, we need a CCD routine with thickness involved.
3. In the future we can add additional spring forces to preserve volume based on these CCD tests if necessary, but for now we are assuming all collisions will be treated by the Adhesion-Collision force
3. Water sharing between neighboring strands
1. Nearby strands with overlapping water regions should share water freely between themselves, so that way they may come apart with some averaged share of the original sum of water
2. There are gravitational, frictional, and fluid-mechanical considerations which influence the behavior of sharing water, each of which Raymond is tackling individually to test
3. In the future, we may want to combine all of these considerations into one, so there will be no special case code
These are the items we are aware of, and actively working on, but any input or alternative directions will be graciously appreciated.
# Recent Progress
In this post we discuss the problems recently found and solved (or unsolved).
1. The preservation of volume for shallow water equation. Previously the water may disappear on hairs.
(Solution: see memo
https://www.dropbox.com/s/o4zzfqfegchfrf4/memo_swe.pdf?dl=0
for details)
https://www.dropbox.com/s/243jj2ktqbos5q9/circle.mp4?dl=0
2. The cohesion force makes the hairs coupled together, which cannot be integrated separately, and the nonlinearity makes the integrator iterates a lot. Trivially decoupling would result in trembling artifact.
Solution: we proposed a local-global solver. At first we integrate the hairs in parallel, using the linearized version of cohesion force (a.k.a. $\frac{\partial E}{\partial x}(x)=\frac{\partial E}{\partial x}(x_0)+(x-x_0)\frac{\partial^2 E}{\partial x^2}(x_0)$). After finished we use the result from the parallel solver as the initial value of a global implicit integrator which uses the nonlinear pair-wise cohesion force.
Result: the number of iterations/cost of time in total is generally the same as using a local solver only, but we have the nonlinear cohesion force applied.
https://www.dropbox.com/s/ylw8dr3oayj2u50/frame00001_1.mp4?dl=0
3. Effect of buoyancy.
Discussion: Previously only drag force is applied on the hairs. Nevertheless the drag force only models the viscous friction between water and hairs. Notice any object submerged in liquid would suffer from the pressures around, this should also been taken considered for hairs.
Solution: we decompose the velocity change after pressure solve into the difference along and perpendicular to hairs. We add the difference along hairs to the shallow water velocity, and add the one perpendicular to hairs as impulse on hairs before the going into next time step.https://www.dropbox.com/s/vi8mblex6zjuj6v/21hairs.mp4?dl=0
Remaining Problem
1. The cohesion force currently is applied to hair particles. However in reality the cohesion force is not a stair function. Only choosing applying or not will result in the popping artifact.
2. Collisions between hairs. The quadratic programming may not converge due to the singularity of the constraints. This is a known problem.
# Multiple Tests
Here’re multiple tests.
7 hairs with obstacles:
21 hairs naturally hanging
21 hairs with obstacles
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https://projecteuclid.org/euclid.gt/1513882905 | ## Geometry & Topology
### Intersections in hyperbolic manifolds
#### Abstract
We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any $n$ and any closed negatively curved manifold $M$ of dimension $≥3$, only finitely many hyperbolic $n$–manifolds are total spaces of orientable vector bundles over $M$.
#### Article information
Source
Geom. Topol., Volume 2, Number 1 (1998), 117-144.
Dates
Revised: 26 March 1998
Accepted: 17 July 1998
First available in Project Euclid: 21 December 2017
https://projecteuclid.org/euclid.gt/1513882905
Digital Object Identifier
doi:10.2140/gt.1998.2.117
Mathematical Reviews number (MathSciNet)
MR1633286
Zentralblatt MATH identifier
0931.57009
#### Citation
Belegradek, Igor. Intersections in hyperbolic manifolds. Geom. Topol. 2 (1998), no. 1, 117--144. doi:10.2140/gt.1998.2.117. https://projecteuclid.org/euclid.gt/1513882905
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• A G Reznikov, Harmonic maps, hypebolic cohomology and higher Milnor inequalities, Topology 32 (1993) 899–907 | 2020-01-25 09:22:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5449615716934204, "perplexity": 4811.93363190953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251671078.88/warc/CC-MAIN-20200125071430-20200125100430-00502.warc.gz"} |
https://electronics.stackexchange.com/questions/462293/mlx90632-eeprom-write-problem | # MLX90632 eeprom write problem
I am using an MLX90632 (MLX90632-S-LD-BCB-000) infrared temperature sensor to build an application which reads an object's temperature among other things. MLX90632 manual
The MLX90632 comes with a default refresh rate of 2 Hz which is unacceptably low for my application. Supposedly, the sensor has configurable refresh rate, which is configured by writing specific values at 2 specific memory addresses in its eeprom memory.
The datasheet says that the eeprom should be unlocked with a command, the specific memory positions should be erased by writing 0x0000 to them and then the eeprom should be unlocked again and write the actual values.
I had written the program below to do this task but it does not seem to work, when i read the sensor's refresh rate,after changing it, it has the default rate (eeprom contents not changed).
I have also seen Sparkfun's driver which has the same implementation as mine and they have put a comment there:
Requires unlocking the EEPROM, writing 0x0000, unlocking again, then writing value. The datasheet doesn't go a good job of explaining how writing to EEPROM works. This should work but doesn't. It seems the IC is very sensitive to I2C traffic while the sensor is recording the new EEPROM.
For this reason instead of polling the sensor to check its eeprom_busy bit i just put a delay, long enough for the memory operation to finish before trying anything else. Still does not work.
I wonder if anyone has managed to write new values to this sensor's eeprom memory and how they managed it. Is there anything wrong with the logic in my implementation?
Note: The program sends what it is supposed to send via i2c to the sensor - this has been checked with a logic analyzer - I am just looking for errors in the process
int main (void){
// initialize mcu code etc...
// ...
// ...
rate = mlx90632_get_refresh_rate(); //the rate we get here is 2Hz
mlx90632_set_refresh_rate_1Hz (); //write the new refresh rate to sensor's eeprom
// mlx90632_reset(); // a reset won't help here either
rate = mlx90632_get_refresh_rate(); //after changing the rate, still gets 2 Hz refresh rate
}
/*-------------------------------------------------------------------------*/
// function to set the refresh rate to 1 Hz
int32_t mlx90632_set_refresh_rate_1Hz ( void ){
uint32_t meas;
meas = 0x810D811D; //value supposed to be written to configure the 1Hz refresh rate
// put to sleep mode before writing to eeprom
mlx90632_mode_type previous_mode = mlx90632_get_mode();
if (previous_mode == continuous_mode)
mlx90632_set_mode(sleep_step_mode);
//EE_MEAS_1 and EE_MEAS_2 are two 16-bit registers, needed to be written for the configuration
// EE_MEAS_2 is right after EE_MEAS_1. We write the incrementally as one 32-bit register
if ( mlx90632_write_eeprom32( EE_MEAS_1, meas ) < 0 )
return -1;
mlx90632_set_mode(previous_mode);
}
/*-------------------------------------------------------------------------*/
int32_t mlx90632_write_eeprom32( uint16_t address, uint32_t value ){
while( mlx90632_eeprom_busy() ){;} //wait until sensor is free
if ( mlx90632_unlock_eeprom() < 0 ) // unlock eeprom for writing
return -1;
// erase eeprom register before writing
if ( mlx90632_write_reg32(address, 0x0000) < 0 ) return -1; //write a 32 bit value to the eeprom starting from "address"
nrf_delay_ms(500);
//while( mlx90632_eeprom_busy() ){;}
if ( mlx90632_unlock_eeprom() < 0 )
return -1;
// actually write the desired value
if ( mlx90632_write_reg32(address, value) < 0 ) return -1;
nrf_delay_ms(500);
//while( mlx90632_eeprom_busy() ){;} // to do: add timeout?
return 0;
}
/*-------------------------------------------------------------------------*/ | 2021-10-27 01:11:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23799817264080048, "perplexity": 6729.1763839132245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587963.12/warc/CC-MAIN-20211026231833-20211027021833-00051.warc.gz"} |
https://www.hackmath.net/en/math-problem/14763 | # Children's home
The children's home received a gift to Nicholas of 54 oranges, 81 chocolate figurines, and 135 apples. Every child received the same gift and nothing was left.
a) How many packages could be prepared?
b) what did the children find in the package?
n = 27
p = 2
c = 3
j = 5
### Step-by-step explanation:
$p=54\mathrm{/}n=54\mathrm{/}27=2$
$c=81\mathrm{/}n=81\mathrm{/}27=3$
$j=135\mathrm{/}n=135\mathrm{/}27=5$
Did you find an error or inaccuracy? Feel free to write us. Thank you!
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One crate will hold 50 oranges. If Bob needs to ship 932 oranges, how many crates will he need? | 2021-06-23 23:22:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25265222787857056, "perplexity": 2024.154006664163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488544264.91/warc/CC-MAIN-20210623225535-20210624015535-00114.warc.gz"} |
https://physics.stackexchange.com/questions/469052/energy-of-the-particles-in-the-particle-accelerator | # Energy of the particles in the particle accelerator
Recently I came across something and I was surprised. I always thought that huge amount of energy is required to accelerate particles in the accelerator in the particle physics.But looks like no. The peak energy of proton beams at the LHC now is around 7 trillion electron Volts (TeV), which is only like 0.00000121J. So energy involved in particles accelerators is not that much then or am I missing something.? May be since the mass of these partciles is so small, their velocity needs to really high to get this much energy and may be that is the big deal.?
• 7 TeVs are over 11 ergs! 7000 times more than the mass of a proton is not a lot? At the moment of impact, energywise, the protons are mostly kinetic energy. How do you define "that much"? – Cosmas Zachos Mar 28 at 0:14
• @CosmasZachos I think the OP means that LHC energy is not that high compared to other energy scales in nature, for instance in this list (which includes the LHC value too) here - en.wikipedia.org/wiki/Orders_of_magnitude_(energy) – Avantgarde Mar 28 at 2:06
• Similarly, energy of superlasers is not "that much" either. The key point is not the absolute amount of energy, but it's intensity, concentration in the small amount of matter, like in LHC, or in small volume and time window, like the laser power of the fusion projects. – Poutnik Mar 28 at 6:43
• Imagine energy needed to accelerate 1 g of protons. You would need energy equivalent to anihilation of 2x3.5 kg of matter and antimatter. Or fusion of about 1000 kg of hydrogen to helium, if I remember correctly . – Poutnik Mar 28 at 6:49
• In one of his books, Sean Carroll mentions that the total energy of all the 500 trillion protons is comparable to that of an "onrushing locomotive engine". – user191954 Mar 28 at 8:10
• $300\cdot10^{12}$ particles times $0.00000121J$ gives $363 MJ$... – cmaster Mar 28 at 6:58 | 2019-10-20 18:49:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6216902136802673, "perplexity": 898.3260905855661}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986718918.77/warc/CC-MAIN-20191020183709-20191020211209-00364.warc.gz"} |
http://mymathforum.com/complex-analysis/341781-fundamental-theorem-algebra-proof-print.html | My Math Forum (http://mymathforum.com/math-forums.php)
- Complex Analysis (http://mymathforum.com/complex-analysis/)
- - Fundamental Theorem of Algebra Proof (http://mymathforum.com/complex-analysis/341781-fundamental-theorem-algebra-proof.html)
zylo September 11th, 2017 02:34 PM
Fundamental Theorem of Algebra Proof
Fundamental Theorem of Algebra for Real or Complex Coefficients by Induction
.
$\displaystyle z^{n}+a^{n-1}z^{n-1}+....a^{1}z +a^{0}=0$
$\displaystyle z^{n}+a^{n-1}[(z-b_{1})(z-b_{2})...(z-b_{n-1})]=0$
$\displaystyle z^{n}=-a^{n-1}[(z-b_{1})(z-b_{2})...(z-b_{n-1})]$
which has exactly n roots
Ref
https://en.wikipedia.org/wiki/Fundam...rem_of_algebra
v8archie September 11th, 2017 03:01 PM
Quote:
Originally Posted by Your reference In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept.
I think your attempt has other problems too.
Maschke September 11th, 2017 05:18 PM
Quote:
Originally Posted by zylo (Post 580115) Fundamental Theorem of Algebra for Real or Complex Coefficients by Induction . $\displaystyle z^{n}+a^{n-1}z^{n-1}+....a^{1}z +a^{0}=0$ $\displaystyle z^{n}+a^{n-1}[(z-b_{1})(z-b_{2})...(z-b_{n-1})]=0$
How do you know $a^{n-1}$ isn't $0$? Like $z^{46} + z^2 + 1$, say. If $a^{n-1} = 0$ then you can't pull it out of the last $n-1$ terms like you did. And you can't fix this by trying $a^{n-2}$ because that might be $0$ as well. So at the very least you have to take the first nonzero coefficient you find and divide through by that. Your proof will get more complicated.
Maschke September 11th, 2017 05:55 PM
ps -- The FTA has been on my mind a bit lately ever since someone posted this thread about using the Intermediate Value theorem (IVT) to prove that some particular polynomial has a zero. In that particular case it was easy to find a couple of input values whose corresponding output had opposite signs. And it was an odd degree polynomial so you can just look at the limiting behavior as the input goes to infinity in both directions, which someone noted.
I noted that for very small (in absolute value) input values, a polynomial is dominated by its constant term; and for large values, it's dominated by its highest-order term. So that if the degree is odd, we have our proof.
That observation happens to be at the heart of one of the standard proofs of FTA. If $\sum_{i=0}^n a_n z^n = 0$ where $z \in \mathbb C$, then a tiny region around the origin is mapped into a tiny little region about $a_0$. And if $z$ is very large, a tiny region about the origin is mapped to a huge region of the plane that must include the origin, because the output is unbounded. You just take $z$ as large as you need it to be.
As $z$ goes from very small (in absolute value) to very large, there must be some particular $z$ such that the boundary of the output region necessarily crosses the origin. That's our zero!
As you can see, this argument depends crucially on the topological properties of continuous functions on the plane. If there's a hole at the origin where there should be a point, the proof fails. Every known proof of FTA depends on concepts from analysis.
It's interesting (as Wiki notes) that Gauss gets credit for the first rigorous proof (by the standards of the time) in 1799; but the last little detail wasn't addressed completely till 1920.
Maschke September 11th, 2017 07:55 PM
Now suppose we try to make your idea work for polynomials with all nonzero coefficients. Then you can indeed divide through by $a_{n-2}$. [Sorry I wrote those as superscripts above by mistake, I hope that's clear. Oh wait YOU made that mistake and I just copied it!] In your third line you have some other polynomial and you claim by induction that the right hand side has $n - 1$ roots. I agree.
But what's the $n$-th root? Are you saying it's $0$? But $0$ is not a root of the original poly you started with. What do you claim is the $n$-th root? How have you demonstrated that the ORIGINAL poly you started with has $n$ roots? I don't see how you are establishing your inductive step even for the special case of all nonzero coefficients.
JeffM1 September 12th, 2017 06:19 PM
Quote:
Originally Posted by Maschke (Post 580144) I don't see how you are establishing your inductive step even for the special case of all nonzero coefficients.
I think his reasoning is that there is one root of every polynomial of degree one when all coefficients (except possibly the constant term) are non-zero. That justifies assuming that there exists at least one integer k such that there are k roots of every polynomial of degree k when all coefficients (except possibly the constant term) are not zero. Then his n is k + 1 and k = n - 1. Consequently, he can indeed factor the sum of all of the terms of degree less than n into a product of n factors. He moves that product to the other side of the equation. So far so good.
He sees n factors on one side of the equation and concludes n roots by relying on (1) the zero product property, which does not apply except in the trivial case where z = 0, plus (2) the possibility that the coefficient a_(n-1) may be 0, which reduces to the trivial case and is excluded by hypothesis.
If my understanding is correct, his reasoning is completely bogus.
zylo September 14th, 2017 10:56 AM
OP attempt led nowhere. But sometimes you have to plunge in. I wasn't fishing, I thought I could do the geometry.
I next tried dividing $\displaystyle P_{n}(x)$ by $\displaystyle (x-d)$ thinking to select $\displaystyle d$ to make remainder 0. This led to a standard result of algebra
$\displaystyle P(x)=(x-d)Q(x)+R(d)$. BuT $\displaystyle R(d)=P(d)=0$ and you are right back where you started from.
But that suggested the next step which gave OP proof, illustrated by example.
Example 1
Divide $\displaystyle x^{4}+3x^{3}+5x^{2}+7x+9$ by $\displaystyle x^{3}+ax^{2}+bx+c$.
Think of numerical coefficients as symbols to make algebra (latex) easier, and illustrate patterns.
After 2 steps of standard long division process, $\displaystyle x+(3-a)$ is the quotient and the remainder is set to 0 by setting coefficients of $\displaystyle x^{2}, x,$ and constant term to 0:
1) $\displaystyle (5-b)-(3-a)a=0$
2) $\displaystyle (7-c)-(3-a)b=0$
3) $\displaystyle 9- (3-a)c=0$
From 3), $\displaystyle c=\frac{9}{3-a}$. Sub into 2) to get
$\displaystyle b(3-a)^{2}+7(3-a)+3=0$, which gives $\displaystyle (3-a)=\frac{k}{b}, b=\frac{k}{3-a}$. Sub into 1) to get a cubic in $\displaystyle a$ which is solvable.
Example 2
Divide $\displaystyle x^{5}+3x^{4}+5x^{3}+7x^{2}+9x+11$ by $\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d$.
Think of numerical coefficients as symbols to make algebra (latex) easier, and illustrate patterns.
After 2 steps of standard long division process, $\displaystyle x+(3-a)$ is the quotient and the remainder is set to 0 by setting coefficients of $\displaystyle x^{3}, x^{2}, x$, and constant term to 0:
1) $\displaystyle (5-b)-(3-a)a=0$
2) $\displaystyle (7-c)-(3-a)b=0$
3) $\displaystyle (9-d)- (3-a)c=0$
4) $\displaystyle 11-(3-a)d=0$
Work backwards from 4) following the same procedure as in Example 1 to get a cubic in "a," which is solvable.
The procedure and pattern is obvious, always leading to a cubic in "a," and holds for any complex polynomial because only the simple arithmetic of a field was used.
Conclusion:
1) Any polynomial is divisible by x-t for some t.
2) Then any polynomial $\displaystyle P_{n}$ is expressible by n factors $\displaystyle (x-a_{i})$. Standard proof by induction.
3) Any (not just to quartic) polynomial is regressively solvable for its n roots by the elementary (but complicated) procedure above.
Maschke September 14th, 2017 11:14 AM
Quote:
Originally Posted by zylo (Post 580368) 3) Any (not just to quartic) polynomial is regressively solvable for it's n roots by the elementary (but complicated) procedure above.
That contradicts a standard result known since Niels Henrik Abel's 1824 proof of the unsolvabiity of the quintic.
It's one thing to provide a false proof of something known to be true. The next level downward is to provide a false proof of something known to be false.
Or as Lloyd Bentsen might have said: I knew Niels Henrik Abel. Niels Henrik Abel was a friend of mine. And you Sir are no Niels Henrik Abel.
Maschke September 14th, 2017 12:54 PM
Can you please walk through your method to find a zero of
$f(x) = x^5 – x – 1$
This example of an unsolvable quintic was given by Van der Waerden.
https://en.wikipedia.org/wiki/Galois_theory
v8archie September 14th, 2017 03:15 PM
The method may work, but it does rely on one being able to find $t$.
All times are GMT -8. The time now is 09:20 PM. | 2019-10-24 05:20:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7526736855506897, "perplexity": 471.6162211899283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987841291.79/warc/CC-MAIN-20191024040131-20191024063631-00362.warc.gz"} |
https://socratic.org/questions/a-culture-started-with-2-000-bacteria-after-2-hours-it-grew-to-2-400-bacteria-pr | # A culture started with 2,000 bacteria. After 2 hours, it grew to 2,400 bacteria. Predict how many bacteria will be present after 10 hours?
Aug 29, 2016
4977 bacteria.
#### Explanation:
Growth with bacteria is exponential, so we can consider that we are working with a G.P.
$a = 2000$
$r = \frac{2400}{2000} = 1.2$
If we regard 2 hours as being one time unit, 10 hours is a further 4 units of time. (6th term)
Either 4 more starting with 2400 bacteria,
or 5 time units starting with 2000 bacteria.
After 10 hours: 2000 xx 1.2^5 = 4977 bacteria.
${T}_{6} = 2000 \times {1.2}^{5}$
Or
${T}_{5} = 2400 \times {1.2}^{4}$ | 2021-12-04 22:45:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4725373387336731, "perplexity": 2724.3070886494006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363125.46/warc/CC-MAIN-20211204215252-20211205005252-00143.warc.gz"} |
https://en.m.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_80.djvu/443 | # Page:Popular Science Monthly Volume 80.djvu/443
from this equation and these experiments and the value of Ne obtained from experiments on electrolysis; for Ne is merely the amount of electricity required to separate by electrolysis one gram-equivalent of any substance from a solution. The value of ${\displaystyle \scriptstyle {\sqrt {N}}e}$ obtained from the most accurate experiments on the electrolysis of silver is ${\displaystyle \scriptstyle 1.702\times 10^{7}}$ electrostatic units. The mean value of ${\displaystyle \scriptstyle {\sqrt {N}}e}$ obtained from 1,735 displacement measurements upon nine different drops was ${\displaystyle \scriptstyle 1,698\times 10^{7}}$ electrostatic units. | 2021-01-21 09:15:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29883691668510437, "perplexity": 1514.823302018377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703524270.28/warc/CC-MAIN-20210121070324-20210121100324-00465.warc.gz"} |
https://brilliant.org/problems/oh-save-me-from-gamma/ | # Another Isotope?
Algebra Level pending
The half life period of a certain radioactive material is one hour. If the initial sample weighed $$500\text{ g}$$, after how many hours will its mass be $$488.28125\text{ mg}$$?
×
Problem Loading...
Note Loading...
Set Loading... | 2017-01-21 02:04:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8759361505508423, "perplexity": 3477.980028294297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280899.42/warc/CC-MAIN-20170116095120-00317-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://eevibes.com/computing/object-oriented-programming/how-to-call-php-function-on-the-click-of-a-button/ | # How to call PHP function on the click of a Button?
How to call PHP function on the click of a Button? We will likewise acquaint one more strategy with execute a PHP work with the onclick() occasion utilizing the jQuery library. This technique calls a JavaScript work which will yield the substance of the PHP work in the site page.
We will likewise exhibit one more strategy to execute a PHP work with the onclick() occasion utilizing plain JavaScript to call the PHP work.
This article will acquaint a strategy with execute a PHP work utilizing the GET technique to send the information in the URL and the isset() capacity to really look at the GET information. This strategy calls the PHP work assuming that the information is set and executes the capacity.
## How to Call PHP function with the click of a Button?
We can utilize jQuery to execute the onclick() occasion by composing a capacity that will execute the PHP work. For instance, make a PHP record echo.php and compose a capacity php_func(). Compose a message Have an extraordinary day inside the capacity and call the capacity. In another PHP record, think of some jQuery inside the content tag. Remember to interface the website page with the jQuery source. In the HTML, compose a button tag with onclick() property. Compose the worth of the property as the test() work. Compose the text Click between the button labels. Make a void div tag underneath the button. Compose the capacity test() inside the content tag. Compose an AJAX technique with the URL as echo.php and compose a triumph() work with result as the boundary. Then, at that point, utilize the selector to choose the div tag and utilize the text() work with result as the boundary.
In the model beneath, we utilize the AJAX technique to play out a nonconcurrent HTTP demand. The URL species the URL to send the solicitation to, and the achievement() work runs when the solicitation is effective. The technique sends the solicitation to the echo.php document, which dwells in a similar area as the current PHP record. The solicitation becomes fruitful, and the achievement() work returns the outcome, and it gets printed.
Given an archive containing HTML and PHP code and the assignment is to call the PHP work subsequent to tapping on the button. There are different strategies to take care of this issue. Likewise, aside from doing this with the snap of a button, a PHP capacity can be called utilizing Ajax, JavaScript, and JQuery. However, this article for the most part centers around the button arranged methodology of calling the PHP Function.
Calling a PHP work utilizing the HTML button: Create a HTML structure report which contains the HTML button. Whenever the button is tapped the technique POST is called. The POST strategy depicts how to send information to the server. In the wake of tapping the button, the array_key_exists() work called.
Program 1:
<html> <head> <title> How to call PHP function on the click of a Button ? <body style="text-align:center;"> <h1 style="color:green;"> GeeksforGeeks <h4> How to call PHP function on the click of a Button ? <form method="post"> <input type="submit" name="button1" class="button" value="Button1" /> <input type="submit" name="button2" class="button" value="Button2" />
Output:
eevibes
Program 2: This program uses isset() function to call PHP function.
Output:
eevibes | 2022-07-07 10:15:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17056182026863098, "perplexity": 8527.514855174759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104690785.95/warc/CC-MAIN-20220707093848-20220707123848-00447.warc.gz"} |
https://mattermodeling.stackexchange.com/tags/td-dft/hot | # Tag Info
## Hot answers tagged td-dft
24
These are a few extra points to complement Andrew Rosen's comprehensive response: To be absolutely clear, typical DFT calculations are not performed at 0K, a better description of what happens is that they are performed "for a static crystal". Static crystal means that the atoms are fixed at their crystallographic positions (which is what a typical DFT ...
19
You are correct that KS-DFT, strictly speaking, involves calculations of a potential energy surface at 0 K. However, if you accept that the density functional approximation you are using is sufficiently accurate, it is not too difficult of a stretch to go from 0 K to finite temperature conditions for an application of interest. The key assumption is that the ...
16
CDFT: Current DFT Current DFT is defined via the generalized Hohenberg-Kohn theorem (HKT), which extends the traditional HKT to account for the effect of magnetic fields. The generalized HKT says that the scalar potential $\mathbf{V}$, the (nondegenerate) ground state wavefunction $\Psi$, and the vector potential $\mathbf{A}$ are uniquely determined by the ...
14
OF-DFT: Orbital-free density functional theory Hohenberg and Kohn established that the ground state energy, $E$, of interacting electrons in a potential, $v(\mathbf{r})$, is a functional of the electron density, $n(\mathbf{r})$: $$\tag{1} E[n] = F[n] + \int \mathrm{d}\mathbf{r} \, v(\mathbf{r}) n(\mathbf{r}) .$$ While this statement is formally true, we do ...
11
$\Delta$SCF This method generates excited states by changing the occupancy of a ground state determinant and then carrying out a new SCF with that initial guess, with some restriction throughout to prevent variational collapse back to the ground state [1]. The most common approach to stay out of the ground state is the Maximum Overlap Method (MOM), which ...
11
Basis set name versus number of total orbitals I would like to first address a part of the question that appears to be a misconception about the use of a 6-31+G(d,p) basis set, since you wrote: "In my understanding of such basis sets, it is difficult to do this." 6-31+G(d,p) is not a "big" or "small" basis set, unless we're ...
11
Kohn-Sham DFT may only be rigorous at zero temperature, but at nonzero temperature, Kohn-Sham-Mermin DFT is an equally rigorous replacement. There are two major differences Rather than deriving the orbital equations from a minimization of the energy, $E$, one minimizes the free energy $F = E - TS$, where $S$ is the entropy. A practical consequence is ...
10
Time-evolution of conceptual DFT quantities has been considered starting, I think, with Chattaraj ~2000. (I imagine there is some earlier work by Ghosh and/or Harbola, but I do not know a reference.) Example references: IJQC v91 633 (2003); J. Phys. Chem. A (Feature article) v21, 4513 (2019); Chapter 13 in "Theoretical Aspects of Chemical Reactivity&...
9
GW+BSE: Excited states in the framework of many-body Green's function comprise charged excitations, where the number of electrons in the system changes from $N$ to $N-1$ or $N + 1$, and natural excitations, where the number of electrons remains constant. In the $|N\rangle \rightarrow |N-1\rangle$ case, an electron in the valence band (occupied orbital) is ...
8
The expression you are describing is equation (6) from your first link: $$R_j=\frac{3\hbar c\ln(10)1000}{16\pi^2N_A}\int_\text{band j}\frac{\Delta\epsilon}{\omega}d\omega\tag{1}$$ which defines the rotatory strength $R_j$ of a band $j$ as the differential absorption coefficient integrated over that band, with the units changed via a prefactor containing the ...
8
I am not aware of any public codes that have a force implementation for excited state calculations using the Bethe-Salpeter equation (happy to be corrected on this front). However, the methodology to do this was published some time ago by Ismail-Beige and Louie in this paper, where they also have an in-house implementation that they use to validate the ...
8
"But since the DFT works not so well when dealing with semiconductors and excited-state calculations, wouldn't NAMD be too erroneous considering the nuclei vibrations?" First of all, DFT in this context might not be as bad as you think. For example, my answer to: What are some recent developments in density functional theory? shows that even 10 ...
8
This second dipole moment is almost surely the excited state. You can see the nuclear contributions are identical and the rotational constants are also the same. This means they are both calculations of the same geometry. You can also see the magnitude of the dipole moment increases in the second calculation. This is very common in excited states. The ...
7
In general, you should use the $S_1$ state by default, unless you have reasons to suspect that your molecule is anti-Kasha. IMHO satisfying one of the following criteria guarantees that your molecule obeys the Kasha rule (supposing that your calculation is accurate), and you basically need not suspect otherwise: The experimental fluorescence spectrum ...
7
The time scale is related to time derivative of the kinetic energy of electrons defined as: $$T(t) = \sum_{i} \int |\nabla \phi_{i}(\mathbf{r},t)|^{2} d^{3} \mathbf{r}$$ You have this for time-derivative of the kinetic energy: $$\frac{d T(t)}{d t} \simeq \frac{T(t=0)}{\tau}$$ Where $\tau$ is the relaxation time for kinetic energy in the order of period ...
7
The question cites a 2008 paper about fairly large molecules (e.g. bithiophene N-succinimidyl esters), in which TD-DFT gave significantly wrong results, so the authors recommended RI-CC2. The question then asks if there's any example where TD-DFT was insufficient for a smaller system. After searching the literature, I have found an example of a small ...
6
Density functional perturbation theory (DFPT) This method refers to the calculation of the linear response of the system under some external perturbation. Consider some set of parameters $\{\lambda_i\}$. The first and second derivatives of the total energy with respect to these parameters in DFT read: $$\frac{\partial E}{\partial\lambda_i}=\int\frac{\... 6 I think you should take care of all possible interactions to get close to the real picture. In periodic solids, there might be electron-hole interaction (solve BSE equation for it), el-phonon coupling, etc. Note that, QE epsilon.x is the lowest level of approximation for the solids (IPA) and it doesn't include any non-local part and local field effects. ... 5 You should consider the states (roots) with non-zero oscillator strength. It means that the transition from the ground to excited state is allowed by the transition dipole moment rule. So, in your case root number 3 is the first excited state with oscillator strength of 0.16, whilst root number 5 is the second excited state with oscillator strength of 0.24. ... 4 TD-DFT works best for valence-valence excitation energies, but does not very well for charge-transfer excitation (unless xc potentials are constructed to lower delocalization error). Core-excitation energies are also not described will with TD-DFT. 4 KS-DFT: Kohn-Sham DFT The KS-DFT is proposed to deal with the problems of orbital-free DFT (OFDFT), which has been explained by @wcw. OFDFT attempts to compute the energy of interacting electrons, as the functional of the density. While this brute force approach is in principle correct, in practice it is not very accurate. This is due to the lack of accurate ... 4 Real-time TDDFT (RT-TDDFT) This is the straightforward non-perturbative solution of the TDDFT equations by means of direct propagation in time. Pioneered by Theilhaber and Yabana & Bertsch it has since found its way into several molecular or solid-state codes. The TDDFT equations in the Kohn–Sham (KS) framework are$$ i \frac{\partial}{\partial t} \phi_i ...
4
The SIESTA code has a branch (rel-Max-2) developed by researchers from Max Plank institute that include the calculations of forces and real-time TDDFT. The TDDFT is merged into the main development branch and will be released in versions newer than 4.1 (i.e. 4.2 or 5.0). To download it, go to the Gitlab page: https://gitlab.com/siesta-project/siesta/-/tree/...
2
In the past I used this tool called Nancy_ex developed by my friend to extract transition density cube directly from the .fchk file. Taken from the description: An open source code for the analysis of electronic excited states: Natural Transition Orbitals; Detachment and Attachment Density Matrices; Quantum Chemical Charge-Transfer Descriptors. The code is ...
1
If I'm interpreting the manual correctly, it is not possible to define a complex external potential in CP2K. It specifies that the VALUES keyword to define the corresponding PARAMETERS of your potential has to be real. I don't know anything about the internal code of CP2K, so I don't know if this would be a simple modification of the code to accept complex ...
1
Someone else can probably answer this in more detail, but most software packages allow for simulated spectra to be calculated just as you have said (not just the excitations). You can probably take the spectra and treat it the same as your experimental spectra. At the very least, I was able to find a recent paper that appears to have done exactly just that. ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2021-10-17 10:49:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6314455270767212, "perplexity": 605.8922805463794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585171.16/warc/CC-MAIN-20211017082600-20211017112600-00132.warc.gz"} |
https://www.cs.bgu.ac.il/~algaf132/Main | Contents (hide)
# Welcome to Design of Algorithms for IAF homepage
## Introduction
This course is all about algorithms and this site is all about helping you to succeed in this course. It will be our main information channel so be sure to come back often (you can see the last updated pages by using the recent changes link on the left side tab.
The course syllabus and rules are all combined on the main page. Pay special attention to the academic integrity section.
Finally, make yourself a habit to check the announcements page for important messages.
## Instructors
Full name Web page E-mail Office Uri Stemmer www.uri.co.il stemmer at cs dot bgu dot ac dot il Building 37, Room 101 Ran Taig www.cs.bgu.ac.il/~taig/ taig at cs dot bgu dot ac dot il Building 37, Room 207
## Syllabus
Reductions. Greedy algorithms.
Loop invariants and correctness proofs.
Dynamic programming.
Minimum spanning trees. Cuts.
Shortest paths algorithms: Dijkstra, Bellman-Ford, Floyd-Warshall.
Depth First Search (DFS), strongly connected components, topological sort.
Maximum flow algorithms of Ford-Fulkerson, Dinitz. Applications to matching.
Introduction to complexity theory: complexity classes, Cook-Levin theorem, techniques for proving NP-completeness.
Randomized algorithms (if there will be enough time).
## Text Books
T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 2nd edition. An online student source is available at CLRS for students. A Hebrew version exists as well.
J. Kleinberg and E. Tardos, Algorithm Design.
You will have a final EXAM, a midterm and 6 homework assignments.
The course grade will be composed of at least 65% final exam, up to 20% midterm and 15% homework assignments.
The weight w of the midterm will decrease the higher the exam grade is, according to the following formula. Let e - final exam grade, m - midterm grade:
if (e > m) then:
if (e-m < 45) then:
w = 20-(e-m)/3
else
w = 5
else
w = 20
Cheating in university courses is regarded as a serious offense. To avoid any possible misunderstanding, please read the following carefully.
Academic dishonesty includes any act of obtaining, soliciting or making available to others, material related to homework assignments. If you commit any of the above, then you are guilty of academic dishonesty. If your partner commits any of the above and you submit the assignment jointly, then you are just as guilty of academic dishonesty. If you choose to work with a partner, then you are both personally responsible for what you submit together. Claiming that you were not aware of the fact that your partner copied the assignment from somebody else will not absolve you of any responsibility.
To eliminate any doubts, we make no distinction between the two (or more) sides of the cheating. If we suspect that Bob and Alice have copied an exercise one from the other, we see no way they could have done this without cooperation. It is your own responsibility to make sure that nobody can copy your assignment.
We will not tolerate academic dishonesty in this course. If you are suspected of academic dishonesty, then a complaint will be filed with the university disciplinary board (ועדת משמעת) and a detailed report placed in your academic records. The minimal penalty for this type of offense is a grade of zero in the course. You might also be expelled from the university.
Be aware that publishing the assignment solution in the internet before the deadline is a serious violation of the academic integrity. Please avoid that.
We reserve the right to check for academic dishonesty anytime after you have submitted an assignment.
# Assignments
You will have 6 homework assignments throughout this semester. You must submit all the assignment, submission is personal.
Please see here the assignments schedule.
You are welcome to discuss the assignments, but you are not allowd to publish your solution in the net, Cheating will not be tolerated - meaning we expect each of you to write the answers in his own words . | 2018-02-25 01:15:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4836297333240509, "perplexity": 2384.265280905063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816083.98/warc/CC-MAIN-20180225011315-20180225031315-00048.warc.gz"} |
https://aitopics.org/mlt?cdid=conferences%3A33534965&dimension=pagetext | to
### Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems
This paper is concerned with finding a solution x to a quadratic system of equations y_i = |< a_i, x >|^2, i = 1, 2, ..., m. We prove that it is possible to solve unstructured quadratic systems in n variables exactly from O(n) equations in linear time, that is, in time proportional to reading and evaluating the data. This is accomplished by a novel procedure, which starting from an initial guess given by a spectral initialization procedure, attempts to minimize a non-convex objective. The proposed algorithm distinguishes from prior approaches by regularizing the initialization and descent procedures in an adaptive fashion, which discard terms bearing too much influence on the initial estimate or search directions. These careful selection rules---which effectively serve as a variance reduction scheme---provide a tighter initial guess, more robust descent directions, and thus enhanced practical performance. Further, this procedure also achieves a near-optimal statistical accuracy in the presence of noise. Finally, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size.
### Gradient descent with momentum --- to accelerate or to super-accelerate?
We consider gradient descent with momentum', a widely used method for loss function minimization in machine learning. This method is often used with Nesterov acceleration', meaning that the gradient is evaluated not at the current position in parameter space, but at the estimated position after one step. In this work, we show that the algorithm can be improved by extending this acceleration' --- by using the gradient at an estimated position several steps ahead rather than just one step ahead. How far one looks ahead in this super-acceleration' algorithm is determined by a new hyperparameter. Considering a one-parameter quadratic loss function, the optimal value of the super-acceleration can be exactly calculated and analytically estimated. We show explicitly that super-accelerating the momentum algorithm is beneficial, not only for this idealized problem, but also for several synthetic loss landscapes and for the MNIST classification task with neural networks. Super-acceleration is also easy to incorporate into adaptive algorithms like RMSProp or Adam, and is shown to improve these algorithms.
### Solving Most Systems of Random Quadratic Equations
This paper deals with finding an $n$-dimensional solution $\bm{x}$ to a system of quadratic equations $y_i=|\langle\bm{a}_i,\bm{x}\rangle|^2$, $1\le i \le m$, which in general is known to be NP-hard. We put forth a novel procedure, that starts with a \emph{weighted maximal correlation initialization} obtainable with a few power iterations, followed by successive refinements based on \emph{iteratively reweighted gradient-type iterations}. The novel techniques distinguish themselves from prior works by the inclusion of a fresh (re)weighting regularization. For certain random measurement models, the proposed procedure returns the true solution $\bm{x}$ with high probability in time proportional to reading the data $\{(\bm{a}_i;y_i)\}_{1\le i \le m}$, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns, that is, $m\ge cn$. Empirically, the upshots of this contribution are: i) perfect signal recovery in the high-dimensional regime given only an \emph{information-theoretic limit number} of equations; and, ii) (near-)optimal statistical accuracy in the presence of additive noise. Extensive numerical tests using both synthetic data and real images corroborate its improved signal recovery performance and computational efficiency relative to state-of-the-art approaches.
### Fast, Sample-Efficient Algorithms for Structured Phase Retrieval
We consider the problem of recovering a signal x in R^n, from magnitude-only measurements, y_i = |a_i^T x| for i={1,2...m}. Also known as the phase retrieval problem, it is a fundamental challenge in nano-, bio- and astronomical imaging systems, astronomical imaging, and speech processing. The problem is ill-posed, and therefore additional assumptions on the signal and/or the measurements are necessary. In this paper, we first study the case where the underlying signal x is s-sparse. We develop a novel recovery algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple and can be obtained via a natural combination of the classical alternating minimization approach for phase retrieval, with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that our algorithm achieves a sample complexity of O(s^2 log n) with Gaussian samples, which matches the best known existing results. It also demonstrates linear convergence in theory and practice and requires no extra tuning parameters other than the signal sparsity level s. We then consider the case where the underlying signal x arises from to structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of b and block sparsity k=s/b. For this problem, we design a recovery algorithm that we call Block CoPRAM that further reduces the sample complexity to O(ks log n). For sufficiently large block lengths of b=Theta(s), this bound equates to O(s log n). To our knowledge, this constitutes the first end-to-end linearly convergent family of algorithms for phase retrieval where the Gaussian sample complexity has a sub-quadratic dependence on the sparsity level of the signal.
### Solving Random Systems of Quadratic Equations via Truncated Generalized Gradient Flow
We prove that as soon as the number of equations $m$ is on the order of the number of unknowns $n$, TGGF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with the time required to read the data $\left\{\left(\bm{a}_i;\,y_i\right)\right\}_{i 1} m$. Specifically, TGGF proceeds in two stages: s1) A novel \emph{orthogonality-promoting} initialization that is obtained with simple power iterations; and, s2) a refinement of the initial estimate by successive updates of scalable \emph{truncated generalized gradient iterations}. The former is in sharp contrast to the existing spectral initializations, while the latter handles the rather challenging nonconvex and nonsmooth \emph{amplitude-based} cost function. Numerical tests demonstrate that: i) The novel orthogonality-promoting initialization method returns more accurate and robust estimates relative to its spectral counterparts; and ii) even with the same initialization, our refinement/truncation outperforms Wirtinger-based alternatives, all corroborating the superior performance of TGGF over state-of-the-art algorithms. Papers published at the Neural Information Processing Systems Conference. | 2021-02-27 03:16:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8004862666130066, "perplexity": 559.1271773321016}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358064.34/warc/CC-MAIN-20210227024823-20210227054823-00513.warc.gz"} |
https://history.jes.su/s207987840001203-2-1/ | Body as an Object of the (non)Historical Research
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Body as an Object of the (non)Historical Research
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S207987840001203-2-1
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Using the example of the torture story, the author of the article tries to show how significant and indispensable the interdisciplinary approach is in the pastresearch. The torture issue directly affects the mind-body problem. Therefore, it has become an object of study for both the psychological and the philosophical and sociological science. In order to understand the causes of the perverted physicalpractices such as torture, the historian should/must consult the research results of these sciences.
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mind-body problem, torture practices, pain, autonomy of will, otherness, interdisciplinary approach in the past research
23.05.2015
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## References | 2020-08-15 15:57:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21098008751869202, "perplexity": 4690.96222558345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740929.65/warc/CC-MAIN-20200815154632-20200815184632-00075.warc.gz"} |
https://www.physicsforums.com/threads/relation-between-magnetic-and-electric-fields.870202/ | # Relation between magnetic and electric fields
tboyers
## Homework Statement
What is the electric field strength at the position of the proton in the figure?(Figure 1) Assume that B = 0.10 T and F = 3.4×10−13N .
Figure 1: https://session.masteringphysics.com/problemAsset/1385081/6/35.P29.jpg
## Homework Equations
Honestly I don't know. Since we have a velocity, b-field, and force i thought F=qvXb but that doesnt help with solving the electric field. Other than that I really don't know.
## The Attempt at a Solution
I really havent even been able to start. I have no idea how to approach the problem.
Mentor
## Homework Statement
What is the electric field strength at the position of the proton in the figure?(Figure 1) Assume that B = 0.10 T and F = 3.4×10−13N .
Figure 1: https://session.masteringphysics.com/problemAsset/1385081/6/35.P29.jpg
## Homework Equations
Honestly I don't know. Since we have a velocity, b-field, and force i thought F=qvXb but that doesnt help with solving the electric field. Other than that I really don't know.
## The Attempt at a Solution
I really havent even been able to start. I have no idea how to approach the problem.
Welcome to the PF.
The Relevant Equation is the Lorentz Force. Look that up in your study materials or Google it online, and then start filling in the known quantities to solve for the unknown E field strength and direction.
tboyers
Welcome to the PF.
The Relevant Equation is the Lorentz Force. Look that up in your study materials or Google it online, and then start filling in the known quantities to solve for the unknown E field strength and direction.
So I use F = qE + qvXB, they give me the v, B, F, and q since it is a proton. Then i just solve for E right? Where would the angle of 30degrees come in to play?
Mentor
So I use F = qE + qvXB, they give me the v, B, F, and q since it is a proton. Then i just solve for E right? Where would the angle of 30degrees come in to play?
Good! The Lorentz Force equation is a vector equation. Are you able to solve it using vectors?
tboyers
Good! The Lorentz Force equation is a vector equation. Are you able to solve it using vectors?
Ah ok, i got it, thank you.
blank_L14
I know this post was 4 years ago, but I'm working on a similar problem, and I can't seem to figure out how to do it with vectors. I have all the numbers needed, but I can't figure out where to use the 30 degrees
Mentor
I know this post was 4 years ago, but I'm working on a similar problem, and I can't seem to figure out how to do it with vectors. I have all the numbers needed, but I can't figure out where to use the 30 degrees
Welcome to PhysicsForums.
Define your x,y,z coordinate system (label the diagram), and write your vector Lorentz force equation. Can you show us that much? And then you just solve that equation -- you have the vector resultant force on the lefthand side (LHS) of the equation, and you have the electric force vector and magnetic force vector terms on the RHS...
blank_L14
blank_L14
Welcome to PhysicsForums.
Define your x,y,z coordinate system (label the diagram), and write your vector Lorentz force equation. Can you show us that much? And then you just solve that equation -- you have the vector resultant force on the lefthand side (LHS) of the equation, and you have the electric force vector and magnetic force vector terms on the RHS...
For the problem that I'm working on, F=2.4x10^-13 N and B=.17 T, the velocity is the same, and it's an electron. So the equation would be
F=qv x B + qE
I understand that much, but the angle between v and B is 90, so it just goes to qvB because sin(90)=1.
I know qE has to have some vector component attached to it, but I can't figure out what that is. And the force would just be the 2.4 x 10^-13, because if you take the x and y components and find the magnitude it's the same thing, right? So I'm just confused... because I can't get the right answer and I'm on my last attempt.
Homework Helper
The net force on the proton (which you are given) is the vector sum of the electric force (whose direction you know) and the magnetic force (whose magnitude and direction you know). Make a good drawing with these three forces and it should become clear how to solve this. There is more than enough info. supplied in the problem.
And the force would just be the 2.4 x 10^-13, because if you take the x and y components and find the magnitude it's the same thing, right?
Sorry I have no idea what this says. There are many correct ways to represent any vector. I think you only need the "vertical"(or y) components to get your answer, but your problem statement is a bit sketchy.
blank_L14 and berkeman
blank_L14
electric force (whose direction you know)
I'm sorry I guess I'm really dumb, but how do I know the direction of the electric force? I'm given the net force and because I'm given v and B, I can figure out the direction of the magnetic force (right hand rule), but I guess the direction of the electric force might be where I'm lost? As many times as I read your responses I can't figure out what I'm missing! I do not have a physics brain and vectors in these contexts have always confused me a lot. I appreciate all the help!
Mentor
but how do I know the direction of the electric force?
From the direction of the Electric field and the polarity of the charge, ##\vec{F} = q\vec{E}##
blank_L14
Mentor
BTW, it helps in posting if you learn to use LaTeX math symbols (it's pretty easy to learn). See the LaTeX Guide link at the lower left of the Edit window. You can also "Reply" to my post above to see the in-line LaTeX that I used to post that equation.
Mentor | 2022-08-13 18:14:51 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8229828476905823, "perplexity": 321.64402324131584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571982.99/warc/CC-MAIN-20220813172349-20220813202349-00590.warc.gz"} |
https://electronics.stackexchange.com/questions/432066/lithium-battery-performance-in-cold-temperatures-powering-ble-beacon | # Lithium battery performance in cold temperatures powering BLE Beacon
I am working on a project that requires the use of BLE Beacons in very low-temperature environments, on average -15°C but on occasion as low as -25°C.
We are currently trialling a brand of BLE Beacon that receives power from 4x AA 3.6V Lithium (Li-SOCl2) batteries in parallel. You may find the beacon product page here https://accent-systems.com/product/ibks-plus/.
We need the beacons to broadcast exclusively one Eddystone-TLM frame per second for a 1Hz advertising rate. At room temperatures, the beacon app estimates a useful life of 86 months. We will be satisfied if they last half that amount of time, but I am trying to determine a rational estimate for their true life in cold temperatures.
We are currently trialling them under these environmental conditions and we will monitor the battery voltage after 2 months, but I am concerned that battery voltage measure does not offer the full picture. My understanding of what causes battery performance to reduce in cold temperatures is not the absence of potential energy but the capacity for the battery to convert the chemical energy into electricity to supply current.
To determine how much current draw is needed to power the BLE beacon I used my multimeter and observed a current draw of 200μA (microAmps). The current draw was also not consistent with a baseline draw of 30μA with spikes of 100-400μA every second (Editor's Note: I was reading the multimeter wrong) 20μA (microAmps). The current draw was also consistent with a baseline draw of 3μA with spikes of 10-40μA every second. This seemed to make sense given the pulse-like behaviour of BLE beacon tech, additionally, at 86 months or roughly 62000h at approximately 0.2mA the 4 batteries supply 12400mAh which seems about right for 4 AA Lithium.
From this observation, my hypothesis is the amount of chemical conversion going on in the battery has to be extremely small. Perhaps small enough that even very cold temperatures would not demonstrably change the estimated life of the BLE beacon than that of room temperature.
I have attempted to research the subject, but most of the discussion is around cell phones and car batteries, the former requiring a pretty constant draw of current that is not sustainable in the cold, and the latter requiring a great burst of current upon ignition. Neither of these use-cases seems to be applicable.
The question is, with knowing how much current the circuit supplies, how do I make a practical estimate for the life of the BLE beacon at these temperatures?
## Update April 15, 2019
In performing greater research on the question of profiling the depth of discharge of Lithium batteries I soon learned the question is meaningfully dependent on the type of lithium-metal chemistry of the lithium battery. It was more productive to narrow my research specifically towards lithium-thionyl chloride cells. The exact battery provided by Accent Systems in their Lithium modified beacon is the SAFT LS 14500 (manufacturer information can be found here).
One of the serious problems associated with exploiting lithium–thionyl chloride cells (LTCC) is estimating their depth of discharge (DOD). Diagnostics of discharging cells of this type is very difficult. This is due to the distinctive feature that LTCC voltage changes very little over the whole duration of discharge to the catastrophically rapid drop in cell voltage at almost total discharge. In fact, LTCC discharge curves cited in most literature sources and advertising material (especially at currents that are not extremely large) are nearly parallel to the capacity axis.
Kanevskii, L. "Special Features of Discharge Characteristics of Different Types of Lithium-Thionyl Chloride Cells and the Problem of their Diagnostics." Russian Journal of Electrochemistry 45.8 (2009): 835-46. Web. 15 Apr. 2019
The following reference chart was pulled from the Tadiran Batteries Technical Brochure here with the LTCC battery chemistry highlighted in blue at the top showing a distinct steady voltage with and immediate drop-off.
The closest research I could get for the study of low-temperature discharge of LTCC batteries was a paper written in the late 80s written about LTCC batteries a memory back-up power source where the produced the following, very relevant result on an Li-SOCl2 battery of R6 size (ER6C) [ R6 is the IEC 60086 system nomenclature for a AA battery] :
When the cell was discharged at -40°C under a 360Ω load (corresponding to about 10 mA discharge), the working voltage reached 3.1 V and the discharge capacity was 900 mAh.
Uetani, Y., and T. Iwamaru. “Characteristics of a Lithium-Thionyl Chloride Battery as a Memory Back-up Power Source.” Journal of Power Sources, May 1987, pp. 47–52.
Based on the manufacturers stated nominal capacity of 2600 mAh under 2mA at 20°C with a 2.0V cut-off (found here) I am willing to assume a very similar battery capacity at a 10 mA discharge. Further, if I am to assume a linear proportional decrease in battery capacity from 20°C to -40°C I calculate a 28.33 mAh decrease per decrease in °C operational temperature. Put another way, I can solve for battery capacity in mAh (BC) with the following equation where T is the temperature in °C:
BC=28.33T+2033.2
So at -20°C I estimate 1466.6mAh per cell (about half the capacity at room temperature). So at 4x batteries, I should have 4x the battery capacity for a total of 5866.4mAh.
At an average approximate operational current consumption of 20µA (0.020mA) when programmed to advertise 1 Eddystone TLM frame at a 1000ms broadcast rate I should theoretically have a battery life of 263320 hours or 30 years! This strikes me as pretty dramatic and I feel perhaps I am underestimating the current draw of the circuit but even the Accent System IBKS Plus datasheet seems to attest to a 3.8µA Idle Current Consumption which is what I observed from my multimeter between advertisement broadcasts.
The question I have now is there anything particularly wrong with my calculations and stated assumptions. I know what battery capacity is variable on the current draw, but it seems from the literature that the battery capacity only increases as the current draw decreases which would only stand to increase my calculated estimate.
Perhaps a 2.0V cut-off is too liberal and if I were to consider a cut-off of something like 3.0V I would find that the total battery capacity would reduce. However, based on the documented discharge behaviour of LTCC batteries, it seems like the voltage drop from 3.4 to 0 is almost immediate and all at once so I'm not sure setting a higher voltage cut-off would decrease my estimate substantially either.
Ultimately I feel like 4-year battery life is almost guaranteed at the current advertisement settings, but if anyone could offer some corrections and/or validation to my estimates it would be greatly appreciated.
• "one Eddystone-TLM frame per second for a 60Hz advertising rate" seems to be a mixup. You probably want to look at a power with a scope or with something that integrates. For your chemistry, used-up capacity will show up more as an increase in impedance than a change in open circuit voltage; I'd probably do something like run a test with only single cell (or a faster transmit rate) for a month on a unit in a freezer vs one at room temperature and then warm it back up (or chill the warm one) and compare the impedance. Also, ask your vendor. – Chris Stratton Apr 11 at 21:19
• Have you considered replacing the built in Li-ions with ones designed for low temperature operation? According to Battery University, you can find ones rated down to -40C (batteryuniversity.com/learn/article/…) – Nate Strickland Apr 11 at 21:48
• This paper may also be useful to you: scirp.org/journal/PaperInformation.aspx?PaperID=80512 – Nate Strickland Apr 11 at 21:51
• @ChrisStratton, yes that was a mixup, thank you. I like the idea for your testing procedure, but perhaps you mean to say resistance as it is a DC circuit? All the same might this guide here for Measuring Internal Resistance be what you mean? That being said, would internal resistance not be inversely proportional to stored voltage, how does one measure not tell me the same thing as another? – Daniel Alksnis Apr 11 at 23:02
• @NateStrickland the manufacture state down to -40 ºC (w/ Lithium Ion batteres) as an operating temperature so I believe the batteries provided are -40 rated. That being said, it's not whether they operate but how long they can operate at low temperatures. Referencing your batteryuniversity link I find it reassuring when they say "Specialty Li-ion can operate to a temperature of –40°C but only at a reduced discharge rate; " as I believe a BLE Beacon could be considered a reduced discharge rate. I will dig into the script link sometime later. – Daniel Alksnis Apr 11 at 23:04 | 2019-09-21 20:05:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41476088762283325, "perplexity": 1889.322926644738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574662.80/warc/CC-MAIN-20190921190812-20190921212812-00363.warc.gz"} |
https://www.math.uni-potsdam.de/professuren/partielle-differentialgleichungen/publikationen/ansicht/null-mean-curvature-flow-and-outermost-mots | # Null mean curvature flow and outermost MOTS
#### Autoren: Theodora Bourni, Kristen Moore (2015)
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partial\Omega_0$, we show that there exists a weak solution to the null mean curvature flow, given as a limit of approximate solutions that are defined using the $\varepsilon$-regularization method. We show that the approximate solutions blow up on the outermost MOTS and the weak solution converges (as boundaries of finite perimeter sets) to a generalized MOTS.
zur Übersicht der Publikationen | 2021-09-18 07:58:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9462534785270691, "perplexity": 933.8558310482869}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056348.59/warc/CC-MAIN-20210918062845-20210918092845-00213.warc.gz"} |
https://plainmath.net/84716/integral-by-use-of-substitution-int-co | # Integral by use of substitution (int 3cos x dx)/(sqrt(1+3sin x))
Integral by use of substitution $\frac{3\mathrm{cos}xdx}{\sqrt{1+3\mathrm{sin}x}}$
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Eve Good
$\int \frac{3\mathrm{cos}\left(x\right)dx}{\sqrt{1+3\mathrm{sin}\left(x\right)}}$
Note that the derivative of $3\mathrm{sin}\left(x\right)+1$ is present $\left[3\mathrm{cos}\left(x\right)\right]$, so we can try to use u-substitution.
$u=1+3\mathrm{sin}\left(x\right)←$ what's inside the $\sqrt{}$
$du=3\mathrm{cos}\left(x\right)dx←$ precisely our numerator
After substitution, we have
$\int \frac{3\mathrm{cos}\left(x\right)dx}{\sqrt{1+3\mathrm{sin}\left(x\right)}}=\int \frac{1}{\sqrt{u}}du$
$\int \frac{3\mathrm{cos}\left(x\right)dx}{\sqrt{1+3\mathrm{sin}\left(x\right)}}=\int {u}^{-\frac{1}{2}}du$
Integrating
$\int \frac{3\mathrm{cos}\left(x\right)dx}{\sqrt{1+3\mathrm{sin}\left(x\right)}}=2{u}^{\frac{1}{2}}+C$
$\int \frac{3\mathrm{cos}\left(x\right)dx}{\sqrt{1+3\mathrm{sin}\left(x\right)}}=2\sqrt{u}+C$
Writing in terms of x
$\int \frac{3\mathrm{cos}\left(x\right)dx}{\sqrt{1+3\mathrm{sin}\left(x\right)}}=2\sqrt{1+3\mathrm{sin}\left(x\right)}+C$
###### Not exactly what you’re looking for?
agantisbz
$\int \frac{3\mathrm{cos}x}{\sqrt{1+3\mathrm{sin}x}}dx=\int \frac{1}{\sqrt{u}}du$
take $u=1+3\mathrm{sin}x$
$du=3\mathrm{cos}xdx=\int {u}^{-1/2}du={u}^{1/2}+c=\sqrt{1+3\mathrm{sin}x}+c$ | 2022-08-20 04:56:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 46, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8964090347290039, "perplexity": 2279.4627837256244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573908.30/warc/CC-MAIN-20220820043108-20220820073108-00268.warc.gz"} |
http://physics.stackexchange.com/tags/electric-circuits/hot?filter=month | # Tag Info
23
You can use a high vertical tube to store water in it (fill it from the bottom by pushing the water in) How much water can you store? It obviously depends on the pressure you apply to push it in. If you push harder, there will be more water stored. The tube is characterized not the amount of water, but by how easy it is to store the water. Its "capacity" ...
5
As drawn, the circuit, assuming ideal circuit elements, is problematic for the reason you've deduced (KVL gives a contradiction). One interpretation is that there is infinite large current for an infinitesimal time which instantaneously charges the capacitors to their final steady state voltages. To gain some insight, add a resistance $r$ in series with ...
5
A typical voltmeter contains an internal Ohmic resistor with known and very high resistance $R$ (called the "input resistance" or "input impedance"), and an extremely sensitive ammeter that measures the current through that resistor. When the voltmeter is connected in parallel across some circuit elements, then ideally the internal resistor has resistance ...
4
a. Immediately after the switch is closed, are either or both bulbs glowing? Explain. They will both glow as some current passes through them as the capacitor is charging. b. If both bulbs are glowing, which is brighter? Or are they equally bright? Explain. They are both equally bright, because an equal and opposite charge is flowing on to ...
4
Capacitance is "charge over voltage" – and one farad is "coulomb per volt" – because the capacity of capacitors (something that determines their "quality") is the ability to store a maximum charge on the plate ($+Q$ on one side, $-Q$ on the other side) given a fixed voltage. When you try to separate the charges, you unavoidably create electric fields ...
4
I think the question is quite interesting actually. When resistances are equal the voltage will divide itself equally because the resistances are coupled in series. Then we take each R to infinity and we assume that we do this to each resistance in the same manner. I think that the voltage across each resistance remains 5 Volts in this case when you take R ...
3
No, that circuit cannot exist in that regime. You are neglecting the internal resistance of the wires between the voltage source and the capacitors, and if the capacitors are discharged (in which case the voltage over them is zero) that's no longer a good approximation. You therefore need to insert a small resistance on either side of the voltage source, ...
3
We Use $C=Q/V$ because those were useful things to measure. It's often easy to forget, but many of the equations we use are chosen because the work, and because other equations didn't work. Never underestimate that part of the reality. We don't use "charge per unit volume" because that number is not constant. You can charge a capacitor up without ...
3
Remembering that resistance = $\frac V I$ work the resistance at $I=1$ and $I=2$. For the resistance to be constant the current-voltage characteristic must be a straight line and go through the origin. There is another parameter which is useful in some instances and that is called the incremental resistance $\frac {\Delta V} {\Delta I}$ which is related ...
3
I would try a more "dynamic" approach, where you actually have an RC circuit (https://en.wikipedia.org/wiki/RC_circuit ) - even a piece of wire has some non-zero capacitance. The typical time scale of transient processes in such circuits is $RC$. When $R\rightarrow \infty$, you have $RC\rightarrow \infty$, so you have to wait longer and longer for transients ...
3
Try redrawing it as so - should be easier to figure out. EDIT the top R2 in the second diagram should be R4.
3
From Kirchhoff's second law, the sum of all the voltages around a loop is equal to zero. That is, the sum of the voltages across the three elements of your circuit, R, L and C, must be equal to the time varying voltage from the source: $$V_R+V_L+V_C = V(t)$$ As $V_R=RI$, $V_L=L\frac{dI}{dt}$ and $V_C=\frac{Q}{C}$, we get your equation, which is correct: ...
3
So, we have series LCR circuit. $V$ is a constant voltage source. $L$, $C$, and $R$ represents the inductance, capacitance and resistance in the circuit respectively. A current $I$ flows through the circuit. Now, the current through each component is the same. So, the potential difference between each component added up together gives the emf $V$. ...
3
The fact that the cell has internal resistance & it acts as a source corrupts your argument. Your argument that connecting a resistance across one of the cells in parallel reduces the overall resistance of the circuit and hence expect the current to increase is wrong. I will derive an equation to obtain the current & potential drop across the bulb ...
2
A capacitor is used to store energy in form of electric fields. This electric field is created by charges on plates of capacitor. So, basically you are storing charge on capacitors. Let someone ask you how much charge you can store in your capacitor.What would you reply? Clearly , you reply " I may store 1mC or 100mC, depending on Potential difference ...
2
Georg Ohm's original experiments, 1825, established that for a set temperature, the current through a specific length of a conductor was proportional to the potential difference applied. Ohm's law is empirical; it cannot be derived directly from Maxwell's equations as it depends upon material properties. It is violated by many materials, and even then ...
2
Why are they considering a phase difference of $\phi$? Your calculations are not totally correct. The voltages across different impedants $V_C,V_R,V_L$ have a phase relationship between them and hence the different impedances $Z_C,Z_R,Z_L$ are not directly linearly related as you have done. Consider the following phasor diagram: I hope it is clear ...
2
I understand that capacitance is the ability of a body to store an electrical charge and the formula is $C = {Q \over V}$ Perhaps you just need to top thinking of capacitance as that. "Capacitance" sounds like "capacity", which leads to an intuitive trap like this: If I have a basket with a capacity of 2 apples, then a basket with more capacity can ...
2
Current only flows if there is a voltage across the resistors. If the resistors are ideally short circuited, no current can flow through them.
2
I took the liberty of redrawing this circuit for clarity. Can you take it from here?
2
If you send $R$ to infinity that's equivalent to an open circuit. Since an open circuit, as you can tell from the name, is not a closed loop, Kirchoff's law is not valid, hence your "paradox". Edit: Kirchoff's law is valid, but not in the form you wrote it. You cannot say that $V_R=5 V$ because when you say that $R \to \infty$ what you are saying is that it ...
2
But I wanted to know whether we can charge a capacitor while it is in use If, by "while it is in use", you mean while the capacitor is discharging, i.e., energy is flowing out of the capacitor to some load, then the answer is no since, by definition, if a capacitor is charging, energy is flowing into the capacitor. Put another way, a capacitor cannot ...
2
A lot of these problems are best done by first redrawing the circuit so that it is in a more accessible form. The correct answer is $\frac 8 3 \;\mu$F.
1
Is there an electric field around the poles of the battery before the circuit is attached, and is there still, after the circuit is connected? Yes. However adding the connecting wires is likely to change the distribution of the field. And why is the field equally strong everywhere in the wire, no matter the shape? Usually we design our circuits ...
1
in series the current is the same through each resistor. Not just the same, the current through each is identical. So that the lightbulbs will all have the same brightness but dimmer than if there was just one bulb on there. Well of course, series connected resistances add and so, the total resistance of two series connected bulbs is greater ...
1
Light bulbs, or any loads, in series will all have the same current. This is unrelated to Ohm's Law - it's Kirchhoff's Current Law and it applies if the loads are ohmic or not. Assuming your source voltage stays the same, adding bulbs in series will increase the total resistance which will decrease the total current and make all the bulbs dimmer. The ...
1
I can't understand at all everything after "As the resistance increases ...". Is that really how it was explained? Nonetheless there's an interesting point here. The analysis is not exactly the same as for an ohmic resistor, but not for the reason you suggest. The thing to consider is that the resistance of the light bulb depends on the temperature ...
1
You are correct that light bulbs are non-ohmic (they don't obey Ohm's Law). But that makes no difference. The same current flows through each, even if they have completely different resistances. Electrical current (charge per second) is like the flow of a river. If there are no leaks, and no tributaries joining the river, then the volume of water per ...
1
This is a typical electromagnet. Let's refer to your figure and say that $g$ is the width of the gap and $A_c$ the section of the core. If $g << \sqrt A_c$,the magnetic field $\vec B$ inside the core will be approximately the same as the magnetic field $\vec B_0$ in the air gap (this is because in this case the magnetic field lines will stay ...
1
Schrodinger's Cat explains well why only part of the impendance is taken. Why are they considering a phase difference of $\phi$ ... Also, why are they taking modulus of Z Here's an algebraic explanation: For the RLC circuit, we can write the total impedance in a general form, $$Z_T=R + j Z_r,$$ where $Z_r$ is the total reactive impedance of the $L$'s ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2016-05-26 06:54:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7488141655921936, "perplexity": 345.95537258241717}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049275764.85/warc/CC-MAIN-20160524002115-00225-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://sbseminar.wordpress.com/category/link-homology/ | ## The meaning of knot homologyOctober 1, 2009
Posted by Ben Webster in link homology, low-dimensional topology.
What I left out of my post on knot homology was, perhaps, the elevator pitch (if you’re in an elevator with a mathematician who already has some background). I’m giving a talk tomorrow which should include this stuff, and so one possibility is to just drop the slides for that on you, and let that speak for itself. Especially recommended are slides 12 and those past 18 (the rest is more standard quantum topology and categorification stuff).
But I’m not so sure that’s a wise plan. So let me try to say something more bloggy:
(more…)
## A hunka hunka burnin’ knot homologySeptember 24, 2009
Posted by Ben Webster in category O, Category Theory, combinatorics, homological algebra, link homology, low-dimensional topology, quantum groups, representation theory.
One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?
This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.
At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). (more…)
## Man and machine thinking about SPC4June 29, 2009
Posted by Scott Morrison in crazy ideas, link homology, low-dimensional topology, papers, Uncategorized.
I’ve just uploaded a paper to the arXiv, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, joint with Michael Freedman, Robert Gompf, and Kevin Walker.
The smooth 4-dimensional Poincaré conjecture (SPC4) is the “last man standing in geometric topology”: the last open problem immediately recognizable to a topologist from the 1950s. It says, of course:
A smooth four dimensional manifold $\Sigma$ homeomorphic to the 4-sphere $S^4$ is actually diffeomorphic to it, $\Sigma = S^4$.
We try to have it both ways in this paper, hoping to both prove and disprove the conjecture! Unsuprisingly we’re not particularly successful in either direction, but we think there are some interesting things to say regardless. When I say we “hope to prove the conjecture”, really I mean that we suggest a conjecture equivalent to SPC4, but perhaps friendlier looking to 3-manifold topologists. When I say we “hope to disprove the conjecture”, really I mean that we explain an potential computable obstruction, which might suffice to establish a counterexample. We also get to draw some amazingly complicated links:
## New Journal: Quantum TopologyJune 26, 2009
Posted by Noah Snyder in good journals, hopf algebras, link homology, planar algebras, subfactors, tqft.
The European Math Society Publishing House (a non-profit publishing company which also publishes the Journal of the EMS, CMH, and half a dozen other journals) just announced a new journal: Quantum Topology. I think this is very exciting as it fills a nice hole in the existing journal options. The list of main topics include knot polynomials, TQFT, fusion categories, categorification, and subfactors. So there should be lots of material of interest to people here.
## More slidesApril 27, 2009
Posted by Ben Webster in link homology, Soergel bimodules, talks.
1 comment so far
My tendency to write slideshows instead of actual posts continues. If you like to see oodles of subtle variations on the same talk, you can see my slides from speaking at ARTIN in Glasgow (which just happened to be coincidentally scheduled during the breaks of the categorification conference there), which is the 8th time I’ve given that talk this year (I’m giving a talk today which will be my 13th total talk of 2009. You can see why I’ve been spending more time with Beamer than on the blog).
However, if you’re looking for something newer, this time you have a chance to see the slideshow before the people coming to the talk. I’m speaking on my work with Geordie in about 45 minutes, and made a Beamer show to accompany part of the talk.
Notably, this is the first Beamer I’ve made with Tikz. I’m particularly proud of the picture on slide 17, which I’ve posted under the cut:
(more…)
## Interpreting the Hecke Algebra II: the sheafificationApril 9, 2009
Posted by Ben Webster in Algebraic Geometry, link homology, Soergel bimodules.
A while back, David wrote a post describing how to produce the Hecke algebra, and I described in comments (very tersely) how David’s description can be categorified. I thought I would expand on that a bit for selfish reasons that will soon become apparent. Not only is some beautiful geometry involved, there’s also a bonus connection to knot homology. (more…)
## Symplectic duality slidesNovember 24, 2008
Posted by Ben Webster in Algebraic Geometry, category O, crazy ideas, link homology, mathematical physics, QFT, talks.
I’ve been too lazy to write in detail about the progress in my research (well, I am writing six papers and applying to jobs, so it isn’t entirely due to laziness), but I did recently speak in the symplectic seminar at MIT, and have posted the slides on my webpage. Obviously, they’re less useful without someone to explain them, but given the current lack of an overarching paper on the subject (that’s no. 5 on the list, I promise), I thought it might be edifying. Executive summary below the cut. (more…)
## Hot’n'fresh from the arXiv: 2-block Springer fibersFebruary 15, 2008
Posted by Ben Webster in Algebraic Geometry, Algebraic Topology, link homology, the arXiv, things I don't understand.
So, in my first paper based on a blog post (well, loosely adapted), Catharina Stroppel and I have finally posted our paper on 2-block Springer fibers. It was longer in coming than I had expected, but I’m certainly glad that it’s up there now. (more…)
## Kronheimer on “Knot Groups and Lie Groups”November 11, 2007
Posted by Ben Webster in link homology, low-dimensional topology, representation theory, talks, topology.
So, I’m in lovely Edinburgh, Scotland (everyone I’ve told about this said “Scotland? In November?” but it’s not actually worse than New Jersey) in advance of the Maxwell Colloquium on Knot Homology.
By sheer luck, my trip here happened to overlap with the University of Edinburgh’s Whittaker Lecture which is a bit like the Bowen Lectures at Berkeley, except that there’s only one of them. By even more luck, the speaker with Prof. Peter Kronheimer (from Harvard) and the topic was “Knot Groups and Lie Groups.”
## Components of Springer fibers, category O, and Khovanov’s “functor valued invariant of tangles”September 3, 2007
Posted by Ben Webster in Algebraic Geometry, category O, Category Theory, crazy ideas, homological algebra, link homology.
As many of you know, my co-blogger Joel recently posted a preprint (with Sabin Cautis), which constructs a knot homology theory using the geometry of coherent s heaves and Fourier-Mukai transforms on convolutions of minuscule orbits in the affine Grasmannian of $SL_2$.
On the other hand, last year, Catharina Stroppel published a couple of papers on the relationship between Khovanov’s original construction of “a functor valued invariant of tangles” and various flavors of category O. From what I understand, underlying this is a philosophy that the $\mathfrak{sl}_n$ version of Khovanov-Rozansky will be related a block to category O that lies on a dimension $n-1$-dimensional wall of the Weyl chamber of $\mathfrak {sl}_d$ (where $d$ is a number relating to the number of strands in your tangle diagram). | 2013-05-23 06:27:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5471744537353516, "perplexity": 1442.4162337909436}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368702900179/warc/CC-MAIN-20130516111500-00075-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/65165/switching-fonts-in-section-titles-while-maintaining-a-serif-sans-serif-distincti | # Switching fonts in section titles while maintaining a serif/sans-serif distinction for ToC vs. title itself
The solution to this problem is probably something absurdly simple, but I'm too stupid to see it …
Consider a document written in blackletter. German blackletter typography rules require latin words and uppercase abbreviations to be set in antiqua. I have followed the advice of the fontspec package and created a new font family and a little \antiqua-macro. This works perfectly inside text paragraphs, because I know if I want a serif or a sans-serif typeface. However, I am not able to create something, that would render the antiqua part in sans-serif in the section title itself, while turning the same thing to the corresponding serif typeface in the table of contents … What's the simple solution?
\documentclass[12pt,a4paper,twoside]{scrreprt}
{numerous other packages}
\setmainfont[Mapping=tex-text]{UnifrakturMaguntia} % for blackletter
\newfontfamily\antiquafont[Mapping=tex-text]{LiberationSerif} %antiqua serif typeface
\newfontfamily\antiquasans[Mapping=tex-text]{LiberationSans} %antiqua sans-serif typeface
\newcommand\antiqua[1]{{\antiquafont #1}} %creates my antiqua-macro
\newcommand\antisans[1]{{\antiquasans #1}} %creates a similar macro for sans-serif antiqua typing
\begin{document}
\tableofcontents
So any appearance of the abbreviation \antiqua{NMR,} other abbreviations,
other paragraph text \antiqua{etc.} can be set perfectly here, but
it mustn't.
\end{document}
If I'm perfectly honest I wouldn't have expected this code to work without flaws, because there is no connection between the serif and the sans-serif part (save the fact that the name starts with the same word). The minimal example can of course be worked with any other combination of two fonts different enough to tell them apart.
-
Define an \antiqua command that checks whether the current font family is sans serif.
The example features a rather peculiar choice of fonts, which are just used for better showing the effect. So Termes is for normal text, Adventor for standard sans serif, Chorus is the replacement of Antiqua Sans and Cursor the replacement for Antiqua Serif. It's not meant to be an example of good typography. ;-)
\documentclass[12pt,a4paper,twoside]{scrreprt}
\usepackage{pdftexcmds}
\usepackage{fontspec}
\setmainfont{TeX Gyre Termes}
\newfontfamily\antiquaserif{TeX Gyre Cursor}
\newfontfamily\antiquasans{TeX Gyre Chorus}
\makeatletter
\DeclareRobustCommand{\antiqua}[1]{{%
\ifnum\pdf@strcmp{\f@family}{\sfdefault}=\z@
\antiquasans
\else
\antiquaserif
\fi
#1%
}}
\makeatother
\begin{document}
\tableofcontents
So any appearance of the abbreviation \antiqua{NMR,} other abbreviations,
other paragraph text \antiqua{etc.} can be set perfectly here; also
\end{document}
-
I think that what you need to do is to augment the instruction
\chapter{My Blackletter Text about \antisans{NMR}}
to
\chapter[My Blackletter Text about \antiqua{NMR}]{My
i.e., you need to provide an optional argument which tells (Xe)LaTeX how the chapter header should be typeset in the Table of Contents.
Addendum From some of the follow-up comments I take it that you would like to fully automate this process, i.e., you would rather not to have to write the optional argument(s) of the various sectioning commands. I don't have a clear idea how this might be done, but I suspect that it would require some serious re-coding of the sectioning commands.
-
Thanks for the simple answer. Although I would also like an automatic solution, if there is one; since I'm not too keen on copying a (few) two-line-title(s) again just to add the one difference of tag (which occurs a few times in the long IUPAC names of chemicals) … ;) – Jan Jul 29 '12 at 21:03
@Jan - Without access to the fonts to actually execute your MWE, I'm afraid I'm in a position to work out all the programming. Maybe somebody else will come up with an extended solution. Do note that I've managed to simplify the suggested solution (by no longer requiring a \protect instruction). Just how many sectioning headers does your document have that might require attention of the nature implied by your question? – Mico Jul 29 '12 at 21:21
UnifrakturMaguntia can be downloaded [here] (unifraktur.sourceforge.net/maguntia.html), UnifrakturCook is not far down the way. Liberation is standard on Linux environments, so it's probably downloadable somewhere only I don't know where. I don't see any reason though, why it should not work with any combination of two fonts? It's 17 headings, of which three contain quite a few \antisans-tags. – Jan Jul 29 '12 at 21:35
@Jan - thanks for providing the links; I've installed the fonts and was able to verify that my suggested solution indeed does the job. :-) However, as I note in an addendum to the answer, I can't see a way of making your objective of fully automating the process, at least not without some major rewriting of all sectioning commands. In the end, it may be less effort for you to provide the optional arguments of the sectioning commands by hand. This shouldn't be so hard to do anyway, right: You copy the argument of the sectioning command, paste it into [], and then do a search-and-replace. – Mico Jul 30 '12 at 0:49
A possible solution would be to define a switch that defines whether the command appears in the main text or inside a section heading, and that is called automatically by section headings
\documentclass[12pt,a4paper,twoside]{scrreprt}
\usepackage{etoolbox}
\setmainfont[Mapping=tex-text]{Times} % for blackletter
\newfontfamily\antiquafont[Mapping=tex-text]{Optima} %antiqua serif typeface
\newfontfamily\antiquasans[Mapping=tex-text]{Courier} %antiqua sans-serif typeface
\newcommand\antiqua[1]{{\antiquafont #1}} %creates my antiqua-macro
\newcommand\antisans[1]{{\iftoggle{insection}{\antiquasans #1}{\antiquafont #1}}} %creates a similar macro for sans-serif antiqua typing
\newtoggle{insection}
\makeatletter
\renewcommand\chapter{\if@openright\cleardoublepage\else\clearpage\fi
\thispagestyle{\chapterpagestyle}%
\global\@topnum\z@
\@afterindentfalse
\toggletrue{insection}
\secdef\@chapter\@schapter
}
\renewcommand\section{\@startsection{section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\toggletrue{insection}\ifnum \scr@compatibility>\@nameuse{scr@v@2.96}\relax
\setlength{\parfillskip}{\z@ plus 1fil}\fi
\raggedsection\normalfont\sectfont\nobreak\size@section}%
}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
{-3.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
{\toggletrue{insection}\ifnum \scr@compatibility>\@nameuse{scr@v@2.96}\relax
\setlength{\parfillskip}{\z@ plus 1fil}\fi
\raggedsection\normalfont\sectfont\nobreak\size@subsection
}%
}
\makeatother
\begin{document}
\tableofcontents
So any appearance of the abbreviation \antiqua{NMR,} \textsf{NMR} other
abbreviations,other paragraph text \antiqua{etc.} can be set perfectly here,
it mustn't.
\chapter{Test \antisans{Test}}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod
tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam,
quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo
consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse
cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non
proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
\end{document}
Caveat: One has to redefine all section commands (e.g., part, section, subsections ...) and insert the switch in the right place.
-
Thanks for this answer. I thought I might add that there are no chapters in my actual document that would require this, only sections and subsections. Just copying the chapter-commands and replacing 'section' where it says 'chapter' doesn't do the job unfortunately. – Jan Jul 30 '12 at 8:23
Almost three years later, revisiting the same problem I had before in a new document (PhD thesis as opposed to master's thesis, for those interested) and having a lot more time on my hands, I came up with a much nicer hack. I created a new command \antisanssubsection with the following definition:
\makeatletter
\def\antisans#1{{\antiquafont #1}}
\def\antisanssubsection#1{%
\section[#1]{%
\def\antisans##1{{\antiquasans ##1}}#1}}%
\makeatother
This included redefining \antisans{} as if it were \antiqua{}.
Typing:
\antisanssection{This is my Blackletter and \antisans{Antiqua} Section Title
Containing a Pointless \antisans{Lorem Ipsum}}
will now compile to:
\section[This is my Blackletter and {\antiquafont Antiqua} Section Title
Containing a Pointless {\antiquafont Lorem Ipsum}]{This is my Blackletter
and {\antiquafont Antiqua} Section Title Containing a Pointless {\antiquafont
Lorem Ipsum}}
(Line breaks included for clarity, not present in the original).
For some reason I did not manage to code the opposite (having \antisans{} defined the way it is, and switching its definition within the \section[ ] stuff, but that is part of a follow-up question.
- | 2015-11-30 13:36:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8400231599807739, "perplexity": 2049.0423607599437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398462665.97/warc/CC-MAIN-20151124205422-00056-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.math.princeton.edu/events/hermitian-k-theory-and-cobordism-2012-03-08t200003 | # Hermitian K-theory and Cobordism
-
Po Hu, Wayne State University
Fine Hall 214
I will discuss Z/2-equivariant motivic spectra. As an example, I will talk about a Z/2-equivariant motivic spectrum representing Karoubi's Hermitian K-theory, and my joint solution with Kriz and Ormsby of Thomason's homotopy limit problem. As another example, I will talk about motivic Hermitian cobordism, and its topological realization, topological Hermitian cobordism. This is an RO(G)-graded (Z/2 \times Z/2)-equivariant spectrum, whose RO(G)-graded homotopy groups we have computed. | 2018-03-17 22:17:29 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9468178749084473, "perplexity": 6895.740097419203}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645362.1/warc/CC-MAIN-20180317214032-20180317234032-00084.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/cpaa.2015.14.383 | American Institute of Mathematical Sciences
March 2015, 14(2): 383-396. doi: 10.3934/cpaa.2015.14.383
Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function
1 Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130
Received November 2013 Revised September 2014 Published December 2014
We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided that the ratio of the chemotactic coefficient to the motility of cells is not too large.
Citation: Qi Wang. Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function. Communications on Pure and Applied Analysis, 2015, 14 (2) : 383-396. doi: 10.3934/cpaa.2015.14.383
References:
[1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Stuttgart, Leipzig, (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1. [2] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Advances in Mathematical Sciences and Applications, 9 (1999), 347-359. [3] M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22. [4] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1983), 217-237. doi: 10.1016/0025-5564(81)90055-9. [5] D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. [7] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. [8] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2003), 51-69. [9] T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [10] T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst-Series B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125. [11] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxisi system, J. Diff. Equation, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [12] M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 583-623. doi: 10.1007/s002850050049. [13] E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415. [14] E. F. Keller and L. A. Segel, Model for Chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234. [15] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248. [16] T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. doi: 10.1137/09075161X. [17] R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, Journal of Mathematical Biology, 61 (2010), 739-761. doi: 10.1007/s00285-009-0317-0. [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. [19] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Soc. Appl., 8 (1997), 145-156. [20] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [21] T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28. [22] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol., 42 (1973), 63-105. [23] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469. [24] C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740. doi: 10.1016/j.nonrwa.2011.07.006. [25] Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [27] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190. doi: 10.1002/mma.1346. [28] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr, 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. [29] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Jap, 45 (1997), 241-265.
show all references
References:
[1] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Stuttgart, Leipzig, (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1. [2] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Advances in Mathematical Sciences and Applications, 9 (1999), 347-359. [3] M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22. [4] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1983), 217-237. doi: 10.1016/0025-5564(81)90055-9. [5] D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag-Berlin-New York, 1981. [7] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165. [8] D. Horstmann, From 1970 until now: the Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2003), 51-69. [9] T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [10] T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst-Series B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125. [11] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxisi system, J. Diff. Equation, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [12] M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 583-623. doi: 10.1007/s002850050049. [13] E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415. [14] E. F. Keller and L. A. Segel, Model for Chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234. [15] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248. [16] T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. doi: 10.1137/09075161X. [17] R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, Journal of Mathematical Biology, 61 (2010), 739-761. doi: 10.1007/s00285-009-0317-0. [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. [19] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Soc. Appl., 8 (1997), 145-156. [20] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [21] T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28. [22] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol., 42 (1973), 63-105. [23] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469. [24] C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740. doi: 10.1016/j.nonrwa.2011.07.006. [25] Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. [26] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [27] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190. doi: 10.1002/mma.1346. [28] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr, 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. [29] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Jap, 45 (1997), 241-265.
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2020 Impact Factor: 1.916 | 2022-05-16 12:05:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7030260562896729, "perplexity": 4507.437330283055}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510117.12/warc/CC-MAIN-20220516104933-20220516134933-00210.warc.gz"} |
https://byjus.com/question-answer/how-many-principles-were-there-in-panchsheel-6485/ | Question
# How many principles were there in Panchsheel?6485
Solution
## The correct option is D 5 Panchsheel agreement was based on 5 principles in total. The five principles of Panchsheel are: mutual respect for each other’s territorial integrity and sovereignty, mutual non-aggression, mutual non-interference, equality and cooperation for mutual benefit and peaceful co-existence.
Suggest corrections | 2021-11-27 02:18:16 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8966501355171204, "perplexity": 13344.822610051848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358078.2/warc/CC-MAIN-20211127013935-20211127043935-00057.warc.gz"} |
https://sttwiki.org/index.php?title=Voyages&oldid=316439 | # Voyages
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Voyages are a feature introduced in Update 3.0. They allow sending a manned ship on a long-away mission to collect resources and crew[1]. During a Voyage your crew can face Hazards and Dilemmas around the universe.
A voyage starts after a selection of crew and a ship, resulting in a combined value for each skill and a starting amount of antimatter. As the voyage progresses, activities will occur. A player will see a quick (20s) countdown timer at the bottom of the voyage log per activity and a long (2h) timer at the top of the screen for Dilemmas. A player has no influence on which activities occur, and only with Dilemmas can the player influence the outcome or reward.
Every expiration of the "quick" timer results in a voyage activity of type normal, hazard, or dilemma. These activities can reduce or increase available antimatter and may add a reward which can be claimed when the voyage returns successfully. The voyage will progress through normal and hazard activities, but every two hours the dilemma will halt voyage progress until the player makes a choice. This means the voyage will progress at most 2 hours at a time without player interaction to continue the voyage.
As the voyage progresses, the antimatter will eventually reduce to zero, resulting in a failed voyage. Before the antimatter has expired, the player may recall the voyage, collecting the voyage rewards after a recall time. After the antimatter has expired, the player may continue the voyage by spending a Voyage Revival Token or Dilithium, or the player may abandon the voyage causing the crew to immediately become available with the loss of any gained voyage rewards.
## Starting a Voyage
Voyages are found in the bottom left corner of the Galaxy map. To start a voyage you have to select a starship and twelve crew members. During the Voyage, the crew can not be used in away missions, shuttle missions, and can't be dismissed or fused. However they can be used in the Gauntlet or for skirmish events.
Crew selection occurs by filling the 12 voyage slots, 2 for each skill, with a crew member that has the named skill. The end result is a sum for each skill composed from all selected crew. A slot may only be filled by a crew member that has the named skill, but any other skills the crew member has also contributes to the total for that skill value.
The amount per skill contributed by selected crew equals their average roll (base skill plus average proficiency), so for example a crew with 20+((5+15)/2) and 10+((20+40)/2) will contribute 30 and 40. Note that such a crew may only occupy one of the two slots or one of the two slots.
Because crew with three skills generally have a higher amount when combining the averages of all three skills than crew with two skills or one skill, crew with fewer than three skills should be avoided when attempting to maximize skill totals, thus helping to maximizing voyage duration. Also, crew with three skills are more versatile since they can occupy one of six slots.
### Featured skills
Each voyage has two featured skills: one primary marked with a gold star, and one secondary with a silver star. These skills are tested more often: the primary skill will be tested 35% of the time, the secondary skill 25%, and each of the other skills 10%.[2]
Data collected from the community suggests that the skill selected as the primary when the voyage begins is slightly more likely to be Command or Science (22.5% chance of each) and slightly less likely to be Engineering (10% chance), with the other three skills having a 15% chance of being selected. Secondary skills are equally distributed among all six.
It is important to prioritize the featured skills as they will, cumulatively, be tested around 60% of the time. Completely neglecting the other four, however, will cause your voyage to perform poorly, so some balance is required to maximize voyage duration from your active crew.
### Antimatter
A voyage starts with a certain amount of Antimatter (AM), which is required to power the starship on the voyage, depending on the ship and crew selected.
Each ship carries a certain maximum amount of Antimatter (AM). The amount of ship AM depends on the ship rarity and level (gaining 50 AM per level), as follows:
Rarity AM at level 1 AM at max level
Common 1050 1300
Uncommon 1250 1550
Rare 1550 1900
Super Rare 1850 2250
Legendary 2050 2500
Starting Antimatter can also be increased during crew selection. Each of the crew slots along with the ship slot give you the opportunity to match traits. A matched ship traits adds 150 AM, while each matched crew trait adds 25 AM.
There is some balance required between selecting a crew that matches a trait, which would add 25 starting AM, and crew that has higher voyage skill values. A single missed hazard costs more 30 AM, so a trait match crew may save your voyage a single hazard. However, increased voyage skill values result in a better chance to pass the hazard. In addition, each time the voyage is revived, it will restart with the starting AM, so if you plan to revive multiple times, starting AM is worth much more than a chance to pass a hazard in a voyage over 12 hours, where chances to pass hazard are very low.
### Crew Traits
Following is a table of crew traits used by voyage slot type over 100+ voyages (Contributed by Greybeard), which has been confirmed by further data collected from the community:
Trait
Astrophysicist x x x x
Bajoran x x x
Borg x x x
Brutal x x x x x
Cardassian x x x
Civilian x x x x x x
Communicator x x x
Costumed x x x x x
Crafty x x x x
Cultural Figure x x x
Cyberneticist x x
Desperate x x x x x
Diplomat x x x
Duelist x x x
Exobiology x
Explorer x x x
Federation x x x x x x
Ferengi x
Gambler x x x
Hero x x x
Hologram x x x x
Human x x x x x x
Hunter x x
Innovator x x x
Inspiring x x x
Jury Rigger x x x
Klingon x x x
Marksman x
Maverick x x x
Physician x x x
Pilot x x x
Prodigy x x
Resourceful x x x x x
Romantic x x x x x
Romulan x x
Saboteur x x
Scoundrel x x x
Starfleet x x x x x x
Survivalist x x x
Tactician x x x x x
Telepath x x x x
Undercover Operative x x x x
Veteran x x x
Villain x x x
Vulcan x x x x
This indicates that certain traits are favored in certain slots ( only has 6!) and there are only four traits which can appear in all possible slots: Civilian, Federation, Human, and Starfleet.
## During the Voyage
Every expiration of the "quick" timer results in a voyage activity of type normal, hazard, or dilemma.
A new normal activity (consuming 1 AM) will occur every 20s with the expiration and reset of the quick timer. This means there are three activites per minute or 30 in a 10 minute period.
Every 140 seconds, a normal activity will grant a reward (every seventh activity).
Every 80 seconds, a hazard activity occurs instead of a normal activity, resulting in an increase of 5 AM or a loss of 30 AM, unless a reward-granting normal activity is due to take place, in which case it occurs instead.
Every 2 hours a dilemma activity occurs instead of a hazard activity (with the expiration of the long timer). When a dilemma is available, the voyage progress will halt until the player makes a choice.
As the voyage progresses, the antimatter will eventually reduce to zero, resulting in a failed voyage. Before the antimatter has expired, the player may recall the voyage, collecting the voyage rewards after a recall time. After the antimatter has expired, the player may continue the voyage by spending a Voyage Revival Token or dilithium, or the player may abandon the voyage causing the crew to immediately become available with the loss of any gained voyage rewards.
### Normal activity
Normal activities range from poetry competitions to exploring alien planets. Each of these activities deducts 1 AM. Every 7th normal activity (every 140s) will produce a Reward.
### Rewards
Rewards occur during normal activities, and in the process your crew will unearth resources (credits, honor, chronitons, components, crew experience training, replicator rations, items or crew) which your crew takes back to the ship. Activities which give rewards also deduct 1 AM.
### Hazards
Hazards test one skill indicated by the hazard marker. If your crew avoids the hazard, you will gain 5 AM. If they fail you will lose 30 AM.
Hazards test your crew's total capability in a skill set, not the individual skill of a single crew member. The total is computed as the sum of a single skill type over all crew for the base plus average proficiency (base + ((max + min) / 2)).
Hazards do not give rewards. Hazards are what will burn through your AM reserves very quickly if your crew isn't up to the challenge.
#### Difficulty of Hazards
The hazards increase in chance of failure with voyage duration, meaning that in the beginning the chance to win all hazards is very high until a certain point when the chance of winning a hazard is 0%. At that point, your crew will fail every hazard and your voyage will quickly lose its reserves of AM.
The hazard failure chance threshold increases steadily and is compared against your crew's total skill value, so the larger the skill values are the longer it will take for a hazard to reach a high chance of failure.
Below is a chart of hazard first failure times, which is the first voyage time (in minutes) at which a skill fails a hazard. The Voyage Skill Value is the composite skill value for all crew on the voyage for a single skill.
### Dilemmas
If your crew has survived enough hazards lasting two hours, your crew will be presented with a Dilemma - a stop point during your voyage that presents a situation that requires interaction. You, the player, must choose a course of action, all of which award some loot, possibly including Honor (30 to 100), Chronitons (30 to 75), Ship Schematics, Crew or Items.
During a Dilemmas, the voyage is paused. AM is not consumed and the timer does not advance. Until a solution is chosen by the player, the voyage will not continue. This can be used as a fail-safe by players -- for instance during an overnight voyage. However, if your skills are not strong enough, you can deplete your AM before reaching the next Dilemma.
A helpful method to use for determining whether your voyage will meet the next dilemma is to divide your current AM by 21 to get a worst-case estimate of the number of minutes your voyage has left if it fails all hazards. This is the result of the following calculation:
Worst-case voyage AM decay rate (if all hazards fail)
• One hazard every 80 seconds, skip every sixth (every 480s is a reward instead);
• 30 AM loss per hazard, +3 AM loss for three ticks at 20s, 40s, and 60s
• = -33AM/80s
• = -.4125 AM/s
• Non-30 AM loss every 480s
• = 30AM/480s = .0625 AM/s
• Add 1 loss every 480s
• = 1/480 = .002083
• = -.4125 + .0625 + .002083 = -.347916 (*60 s/m) = 20.875 AM/m
So a voyage with 630 AM remaining will continue another 30 minutes if all hazards fail, and longer if any pass. This also means having 2400 AM guarantees another 2h of voyage time, which is long enough to make it to a dilemma.
#### List of Dilemmas
Dilemma Name Rarity Choices Part Required
A Higher Duty, Part 1 3 1 of 3
A Higher Duty, Part 2 2 2 of 3 A Higher Duty, Part 1
A Higher Duty, Part 3 2 3 of 3 A Higher Duty, Part 2
A Life Alone, Part 1 2 1 of 2
A Life Alone, Part 2 2 2 of 2 A Life Alone, Part 1
Blood-Red Tide 3
Blow by Blow 2 1 of 2
Burden of Proof 3 2 of 2 Class Act
By Our Own Hands 2
Champion of the People, Part 1 2 1 of 3
Champion of the People, Part 2 2 2 of 3 Champion of the People, Part 1
Champion of the People, Part 3 2 3 of 3 Champion of the People, Part 2
Class Act 2 1 of 2
Cracks in the Wall 3 1 of 2
First Author 3 2 of 2 Cracks in the Wall
Friends in Need 3** 2 of 2 Blow by Blow
Interference, Part 1 3 1 of 2
Interference, Part 2 2 2 of 2 Interference, Part 1
Ladders and Chains 2
Next Stop, Absolution 3
Off the Books, Part 1 2 1 of 2
Off the Books, Part 2 2 2 of 2 Off the Books, Part 1
Outstretched Talons 2
Profit in the Wind 2
Retrofit 2 1 of 2
Sticks and Stones 3
The Beginning of the End of the World 3 1 of 2
The Buried Years 2
The Cost of Living 3
The Greater Need 2
The Tools at Hand 3** 2 of 2 Retrofit
The Voice of the Prophets 3 2 of 2 The Beginning of the End of the World
True to the Uniform 3
What Smiles May Hide 2
Where Earth Meets the Sky 2
Winter's Price 2
** Choices shown are based on a prior Dilemma.
Uncommon choice reward is generally 30 / 30 / 2* Crew
Rare choice reward is generally 60 / 50 / 3* Crew
Super Rare choice reward is generally 100 / 75 / 4* Crew
## Rewards over time
As time progresses rewards improve, making longer trips more valuable. For instance, it is possible for a player to obtain only 50 chronitons after a 2-hour Voyage, and 300 after a 6-hour Voyage.
Dilemmas, one every two hours, have a unique reward table. It is possible, for instance, to obtain a batch of 600 schematics, even for otherwise unobtainable ships (such as the NX-01 Enterprise), making Voyages the only currently available way to obtain elusive ships.
### Before the first dilemma
In the first two hours players will be mostly awarded with common crew, basic components, and batches of 3. The first dilemma threshold crew reward is usually an uncommon and in very few cases a rare, as well as 30 Honor and 30.
### Between the first and the third dilemma
Between 2 and 6 hours of voyage time players can be awarded rare crew, common components, batches of 8 and rare Replicator Rations. Most dilemmas still award an uncommon crew, 30 Honor and 30, but from the fourth hour (second dilemma) onwards, dilemmas which award a rare (or very occasionally a super rare) crew, 60 Honor and 50 become available, and from the sixth hour (third dilemma) those which give a super rare crew member, 100 Honor and 75.
Note that dilemmas which award one of the voyage-exclusive super rare crew do not award any honor or chronitons.
### After the third dilemma
When your trip lasts more than 6 hours, mid-voyage rewards increase remarkably, including uncommon components, super rare crew and batches of 15 . The same loot table and percentages apply for voyages of any longer duration.
### Reward Drop Percentages
This table lists approximate drop rates (over 300 voyages from Greybeard) for items by type between dilemmas.
Item 0-2h 2-6h 6+h
Chrons 15% 15% 15%
Credits 100% 100% 100%
Crew* 0.75%
Crew**
Crew*** 0.75%
Crew**** 0.75%
Honor 0.75% 2% 2%
Item 27%
Item* 27%
Item** 30%
Replicator Fuel** 0.8%
Replicator Fuel*** 2%
Replicator Fuel**** 2%
Replicator Fuel*****
Schematic
Training*** 40%
Training**** 38%
Training***** 31%
This table lists approximate drop rates (over 300 voyages from Greybeard) for items by type at dilemmas. (Note that the 97% rate for chrons/honor is due to those not rewarded when selecting voyage-exclusive crew from a dilemma sequence.)
Item 2h 4h 6h+
Chrons 100% 100% 97%
Credits
Crew*
Crew** 98% 80% 65%
Crew*** 2% 19% 15%
Crew**** 1% 20%
Honor 100% 100% 97%
Item
Item*
Item**
Replicator Fuel**
Replicator Fuel***
Replicator Fuel****
Replicator Fuel***** 1%
Schematic 0.5%
Training***
Training****
Training*****
## Recalling a Voyage
As long as you have AM remaining, you can recall the starship. The voyage home does not consume any AM. The ship will take a certain time to return to you, and you will receive the rewards and be able to use the crew otherwise.
The trip back requires 40% of the Voyage time (e.g. if your trip was four hours long it will take one hour, 36 minutes for the ship to come back). You can speed up the return with dilithium.
After the return you will have a chance to see the complete log of your Voyage.
## Running out of Antimatter
A player can run out of antimatter only because of normal activity (one AM per action) or of hazards (30 AM per failure). When reaching 0 AM, the ship is stranded in space: the player has the choice of aborting the mission (sacrificing all the rewards) or refilling the AM by spending a Voyage Revival Token or dilithium.
If any voyage revival tokens are in your inventory, they must all be spent before you are able to spend dilithim to revive the voyage. They can be earned in some events as threshold rewards and are sometimes offered as compensation for bugs and server crashes in the game.
The amount of dilithium needed to restore the antimatter depends on how long the voyage has lasted thus far with the equation: $\frac{Time_\left(Min.\right)}{5}=$ .
This works out to:
Hours Cost per Step Total Cost
2 24 24
4 48 72
6 72 144
8 96 240
10 120 360
12 144 504
14 168 672
16 192 864
18 216 1080
20 240 1320
22 264 1584
24 288 1872
If you choose to abandon the mission, then your crew will be returned to active duty immediately without a waiting period, but you will not receive any of the rewards you've collected.
It is not possible to recall a ship and obtain the rewards without at least 1 AM left.
## Strategies
### The WereDragon Strategy
Further tests has shown that the gold skill will occur nearly 36% of the time while silver skill 24% of the time (Sample size: 3,165 skill checks in 10 voyages). The other skills will fall between 7% to 12% (with one occurrence of 15%).
The silver and gold skills will occur 60% of the time. These skills should have your primary focus. If you are intending to do a 6 hour voyage, they should be minimally at 6,000. The higher, the better. 7,000 would be recommended.
Choose a skill to sacrifice. You want that skill as low as possible to boost your other skills. As an example, lets sacrifice engineering. A fully equipped lvl 100 Gangster Spock has the stats of 276 eng, 992 sec, and 671 sci for voyages. You can use Gangster Spock as your engineer to boost up your Security and Science. Engineering skill checks will fail most of the time and eventually all the time as the voyage progress, BUT you are playing the odds. In this example, Engineering will only come up 7% to 12% of the time. That's it. That is a very small percentage. And by sacrificing engineering, you can boost the other skills and those skills will take longer before they start failing skill checks.
A modified strategy (called the NATE mod), is to sacrifice 2 skills but not to the same degree. With a pure WereDragon strategy, it is quite possible to have the skill beneath 1,000. In the NATE mod, the 2 skills will fall between 1,000 and 3,000. Again, the thought is the same. You are playing the odds. You sacrifice 2 skills to boost the other 4 skills.
## Player suggestions
• A player-made tool to calculate an approximate duration of your Voyage using your crew's skills can be found here.
• Due to the fact that all the skill points of a crew contribute to the ship's total, Voyages have a slight bias in favour of crew members with points in three skills.
• As the difficulty increases at a certain point you will start failing all hazards. At that point you can roughly calculate how long it will take you to run out of AM based on your current reserve. The pattern of hazards every four activities and rewards every seven means that, once all hazards are being failed, antimatter depletes at an average rate of slightly less than 22 AM per minute. There is no point in recalling the ship when it has hundreds, or thousands, of AM in reserve, but waiting too much might result in accidentally running out. Running a voyage is safe until you have 31 AM or less in reserve - using the pattern can enable you to work out the optimal time to recall (immediately after the last reward you can collect before you reach a hazard that would bring your AM below 1).
• Voyages offer a great amount of common and uncommon crew, and are one of the fastest and cheaper ways for newer players to obtain all copies of such crew. | 2022-12-08 19:10:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.336921751499176, "perplexity": 3361.5523119803797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711360.27/warc/CC-MAIN-20221208183130-20221208213130-00397.warc.gz"} |
https://chemistry.meta.stackexchange.com/questions/3030/lets-do-the-bounty-dance/3037 | # Let's do the bounty dance!
I have had this on my mind for quite some time already and I thought that the time before the year passes might be good to introduce this idea.
## Let's get some Bounties rolling...
Bounties are a good way to draw attention to questions. There are various reasons, why one would want to award bounties. I like it when there are a few questions in the featured tab of the main site. It sure feels great getting one, but it also is awesome to see that the offered bounty pays out in a very good answer.
Now sometimes you come across a good question that did not get the necessary attention to draw some good answers. You might be interested in the answer, but not as interested as giving away your hard earned reputation.
I have passed the magic 10k threshold some time ago and as a moderator I do not need any more reputation points any way. So I decided to bounty off some. The general aim is to gain some top quality answers and to give back a token of gratitude.
(Also it might push some people higher on the privileges chain, so that they can more effectively help me moderate the page \(^O^)/)
## What are the rules for this?
It's quite simple. There are only a few rules.
### Questions:
• A question that should be awarded a bounty to draw attention should be at least about a month old. (Preferably the asker is still active.)
• Should have a score of five or more, but not more than 15.
• Should not have an accepted answer. (In cases that the answer is outdated this point can be voided.)
• The last bounty on this question may not be younger than half a year.
• No questions.
• An answer that is excellent (score > 15), but not the accepted answer, can get an additional bounty to mark it as great.
• The question must be of our minimal quality standards, proper formatting, etc.
• The answer should not yet have a bounty awarded.
• The answerer must not be a 10k reputation user or yourself.
Conditions are subject to change.
## How to enter?
Post an answer here with the current title and link to the question and answer. Make sure the post follows the above guidelines. Please also state why the linked post should be bountied. Wait.
You can enter as many questions as you find interesting. Obviously this is an experiment and we will have to evaluate the experience eventually, but I think this is not coming too soon.
I will probably go though the list Fridays before leaving into the weekend and feature the question I liked most. It's also most likely that I will award a bounty around that time.
Have fun and please don't be shy.
## Can I become a sponsor?
First of all: Thank you for considering that option. Obviously it would be very selfish of me to be the only one handing out the bounties, so of course this is allowed.
If you think that you have gained sufficient reputation and want to get rid of some yourself, find a post from the list that you like and feature it. When you do, add an editorial note above the body stating when you applied (or later awarded) the bounty. That's it, nothing more. Have fun with it.
If you have any questions, please use the comment section to ask them. Posts that are no entries might get deleted as NAA.
• This was $\ldots$ unexpected. – M.A.R. Dec 8 '15 at 18:17
• I like this. Is there a way that I can use the search to identify questions asked within a certain range of dates? – Ben Norris Dec 9 '15 at 0:03
• @Ben yes, IIRC it must be in the search tips. – M.A.R. Dec 9 '15 at 20:40
• One search that may be useful is "score:4 last active:1y.. hasaccepted:no" – Geoff Hutchison Dec 14 '15 at 19:24
• Why no homework questions as a rule? A good quality homework question should be allowed to jostle for position alongside all other categories. – long Mar 10 '16 at 2:45
• @long I included that, because at the time of the write up, I was already thinking about rewriting the homework policy from scratch and I did not want to include anything related to that. I agree that there are good homework questions, that certainly would be a good candidate for a bounty and a general answer. However, in these cases it usually does not take much effort to rewrite it in a more general sense, so that it can have a wider audience and maybe serve as an example for future homework questions. – Martin - マーチン Mar 10 '16 at 4:27
• And what about duplicate questions which qualifi all rules? For example this – Fawad Mar 4 '17 at 11:19
• Thanks for handing out bounties to questions, this would make the site better. I think this post itself could fund alot of bounties from the rep it gets. Here, have 5 points! :) – Pritt says Reinstate Monica May 22 '17 at 9:11
• @Pritt there is no reputation gained or lost from meta ;) – Martin - マーチン May 22 '17 at 9:58
• @Martin What?!? I lost about 30 rep on "disagreement downvotes" on the real meta.stackexchange. – Pritt says Reinstate Monica May 22 '17 at 11:43
• @pritt Mother-meta, as I tend to call it, is another story. On the per-site metas there is no rep loss or gain. – Martin - マーチン May 22 '17 at 12:20
• @Martin That's great! I'm quite afraid to post on meta nowadays. I guess the per site meta is morr friendlier. – Pritt says Reinstate Monica May 22 '17 at 12:49
• Why do you want a score of five or more?? – Bitthal Maheshwari May 22 at 21:43
• @Bitthal I wanted the question to be pre-approved by a couple of members, showing that it has general interest. Most voting thresholds on the site are about 5 members, so I think it is fair to say that when this score is reached, the question has already demonstrated it could reach out more. – Martin - マーチン May 22 at 22:07
# Priorly Bountied Questions:
(note numerical order of question numbers)
# Currently Nominated Questions:
• @Martin-マーチン Make sure to remove the question from the Currently Nominated Questions too I modified it for clarity. – A.K. Mar 26 '19 at 14:19
• oh well... I did not see that... that would have made things easier ;) – Martin - マーチン Mar 26 '19 at 14:22
• @Martin-マーチン Might it make sense to delete some of the answers to uncrowd this post? or start over? – A.K. Mar 30 '19 at 2:31
Editorial Notice: the bounty has been applied 2015-12-12.
If I am allowed to nominate another question:
Do vinyl cations adopt a classical or non-classical structure?
Rationale:
• Good question
• Poster is a frequent and active user who is 65% of the way to accessing the moderator tools (I can shamelessly appeal to Martin's shameless ulterior motive).
• Comments refer to a source with the answer and provide a brief summary of the answer.
• I want a definitive answer to this question - vinyl cations are described in the reactions of alkynes in undergraduate texts and I want to have something better so say about the mechanism of alkyne addition reactions. Since this is a topic that impacts introductory organic chemistry, a lot of other folks might benefit from the answer.
• You are allowed to enter as many questions as you like, I am not intending to make this a one time thing. As for the question itself, it is a good one and I remember doing some research on it that I was not able to go through with, so I am very intrigued any way :D – Martin - マーチン Dec 9 '15 at 2:42
• Since you chose my other nomination was chosen, I am giving this post a bounty. – Ben Norris Dec 12 '15 at 14:16
• I did not do this one because Jan wanted to join the dance... if you choose to give the bounty, mark it as I did, I am going to add about sponsoring on Monday – Martin - マーチン Dec 12 '15 at 14:39
• Haha, there goes my altruist badge xD Bu~t I’ll find another post I with to bounty one day ^^ – Jan Dec 12 '15 at 19:22
A question I would like for a bounty is:
d-orbital splittings in WS2 monolayer
• Almost a month old question
• very, very interesting question
• 3 favourites
• waiting patiently/dying hard for an answer.
• question has not received enough attention.
Editorial notice: Bounty applied 2015-12-21 and awarded 2015-12-29
How is the pKa of extremely weak acids determined?
Good candidate for bountying because
• it's a good question.
• it's somehow fundamental.
• there seems to be a definitive answer, glancing at Ron's comments.
• it's from Dissenter – our Socratic, who's likely to return.
• as of now, it has 7 votes and has had no bounties. Seriously, how many of you knew timelines existed?
Editorial Notice: the bounty has been applied 2015-12-14.
I'd like to suggest Why is Gold So Popular in Nanotechnology
• Only one answer (which doesn't mention thiols)
• No discussion of the properties of gold on the nanoscale
• Asker is a fairly frequent participant
Editorial Notice: the bounty has been applied 2015-12-11 and awarded 2015-12-18.
My nomination: Symmetrize nearly symmetric molecule
Rationale:
• This is a strong well-researched question.
• The comments to the question seem to suggest that a solution is out there, and the OP was getting close to it.
• The OP is still around (checked in 11 hours ago).
• There are no answers and no active bounties.
• This question is favorited by two people, so others are interested in the answer.
• We could easily nominate many of Dissenter's questions: chemistry.stackexchange.com/questions/40138/… – Ben Norris Dec 9 '15 at 0:19
• ^^ Yes, we could. He asked really good questions in the past and played a substantial part in evolving the community. You can enter this as a new entry, so that it does not get lost in the process. (However, this one had a bounty in November O.o) – Martin - マーチン Dec 9 '15 at 2:47
• @Martin-マーチン Are you sure "he" is the right pronoun? We don't know who Dissenter is.. :-) – Geoff Hutchison Dec 9 '15 at 16:01
• @BenNorris Thanks for the reminder - I favorited this question to come back and answer. I guess if you want to give me a bounty, I won't complain! – Geoff Hutchison Dec 9 '15 at 16:02
• I will state, for the record, that as a 10+k user, I'm going to turn any bounties around to other questions. :-) – Geoff Hutchison Dec 14 '15 at 19:06
Editorial Notice: The bounty has been applied 2015-12-31.
I’m going to suggest Alkylation of conjugated nitriles - regioselectivity.
• An interesting question, probably orbitals but I don’t know the answer.
• Posted in May.
• No answers, no edit history (thus no bounty so far).
• Score of 11 at the time of posting.
• OP still around as far as I know.
Editorial notice: the bounty has been applied 2016-09-23.
This question is not yet old enough to be eligible by my own rules, but I would like to revive this thread so I am nominating it anyway:
Why is an S-S bond stronger than an O-O bond?
I think this question is very intriguing and one might come by quite tempting easy explanations. In order to further understanding of chemical bonding a canonical answer would be really helpful. I am almost certain, that there are already reports about such investigations out there. However, I do not have the time to find it myself.
As for the user argument; Ina is very active on the site and has proven a good sport and he also asks quite interesting questions. While an answer is rewarding for everyone, I think he has supplied us with a solid foundation and hence really deserves some attention.
If you comment, I'll take it back.
• Oh wait, what's this? Someone saw my question worthy! \o/ – M.A.R. Mar 23 '16 at 10:53
• @IͶΔ (read the small print) – Martin - マーチン Mar 23 '16 at 11:02
• I'm not "you", so it doesn't count – M.A.R. Mar 23 '16 at 11:17
• I know this has a great answer on it, but I’ll take that as an opportunity to give said answer it’s due reward. And I need to get rid of some rep ;p – Jan Sep 23 '16 at 17:46
Editorial notice: Bounty applied on 2017/04/12, awarded 2017/04/17.
And here I am, suggesting another one:
How were old style stereographic structures produced?
• score of 18 as of now, no downvotes
• rather interesting from the historical point of view, and still relatively relevant to chemistry
• OP was last seen 27th April, although the last post seems a bit older
• no bounties.
• I added the bounty to reward the now existing answer. – Martin - マーチン Apr 12 '17 at 5:54
Editorial notice: Bounty applied on 2016-06-05. It could not be awarded due to lack of answers.
Is hybridization used in ab initio valence bond calculation?
• A two year old question
• asker is possibly not active any more, but has registered
• I think (but to be honest I am far away from the topic, so I require the compchem experts’ expertise) that it is an interesting question.
• +13 score as of now. (+19 after the bounty dance)
• no answer yet. (save one deleted NAA and another deleted one after the dance.)
• This entry is a sad, sad story. So many bounties, no answer... – Martin - マーチン Apr 12 '17 at 5:42
• @Martin-マーチン Jan, Update: this question does now have an answer. Thoughts? – A.K. Nov 15 '18 at 4:38
• @A.K. Unfortunately I think the current answer completely misses the point. – Martin - マーチン Nov 15 '18 at 12:41
Editorial notice: Bounty applied 2016-09-29.
I'd like to "revive" this thread with this question: Why can't Pd/C and H2 reduce both the alkene and carbonyl portions of α,β-unsaturated carbonyls?
It seems to have been mentioned in the comments of one earlier answer, but really deserves an answer of its own.
Rationale:
• Good conceptual question about hydrogenation procedures in organic synthesis, a very common reaction
• Question score of +14/0
• OP is Dissenter - enough said!
• Last bounty applied is almost a year old.
I recognise that hydrogenation mechanisms are not actually particularly well-elucidated, but still think that this is worth another shot. Surely there is something in the literature about them.
I'd like to enter one myself:
Chemistry of Walter Mitty's negative developer/stop bath/fixer?
I think it is a very interesting topic, that currently only has a link-only answer. From a general point of view the process is interesting for a wide audience.
• FWIW: I added an answer thoughts? – A.K. Mar 26 '19 at 20:50
I'd like to nominate the recently active
What chemical properties make LSD so psychoactive?
I am currently unable to give out a bounty, since I'm busy on a business trip. I find the question quite compelling. The current answers, however, offer little to no insight into which properties of the compound are actually responsible for the effect. For a better understanding of this I think an answer must go deeper than only the chemical structure of a few functional groups and investigate the mechanism of binding to the receptors.
Editorial note: Bounty applied 7 Sept 2018. No Bounty Awarded.
I would like to nominate the followng and maybe revive this thread:
Reaction of glucose with hydroiodic acid
I believe it is either of good teaching value or should have included phosphorous in the equation.
Editorial Note: This question now has an accepted answer as of May 13 '17
I'd like to add the following:
Functional difference of Benedict's solution and Fehling's solution
It has several upvotes and stars but has been left unanswered for six months. According to data explorer, it is one of the highest weighted unanswered questions on Chemistry.SE.
Editorial Note: answered 2019-01-27, bounty (retroactively) awarded 2019-03-26
I think a small bounty on the following question would be good. Until recently it was closed as homework, but I think it has a larger applicability to some chemical concepts. It could use a general answer: Quadrupole moment of a molecule.
Another nomination:
Melting and boiling point trend in Group II
This question has good stats and probes at an interesting anomaly.
Score: 13 $$\quad$$ Views:7600+ $$\quad$$ stars: 2 | 2020-09-27 01:26:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4746253192424774, "perplexity": 1700.0606159772747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400249545.55/warc/CC-MAIN-20200926231818-20200927021818-00067.warc.gz"} |
http://mathoverflow.net/questions/107441/does-every-nonempty-definable-finite-set-have-a-definable-member | # Does every nonempty definable finite set have a definable member?
I asked this on MSE yesterday ( http://math.stackexchange.com/q/197873/39378 ) but no one has answered it yet. I hope it's not too soon to post it here.
Here are a few ways to formalize the question, so you can pick your favorite and answer it. Assume whatever large cardinals you like.
(1) Is it consistent with ZFC that there is an inaccessible cardinal $\delta$ and a nonempty finite set that is first-order definable without parameters over $(V_\delta,\in)$ but has no elements that are first-order definable without parameters over $(V_\delta,\in)$?
(2) Is there any model of ZFC that has a finite nonempty set, first-order definable without parameters over the model, with no element that is first-order definable without parameters over the model?
(3) Is it consistent with ZFC that there is an ordinal-definable finite nonempty set with no ordinal-definable member? (I am aware of the question A question about ordinal definable real numbers, but that question asks about sets of real numbers and I already know the answer to my question for sets of real numbers, or indeed for sets of subsets of any ordinal, because they are definably linearly ordered.)
(4) Any of the above formulations with ZFC replaced by ZF.
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Here is a related question on pointwise algebraic models of set theory, in which every set is an element of a finite definable set: mathoverflow.net/questions/71537/…;. – Joel David Hamkins Sep 18 '12 at 11:02
Also this question: mathoverflow.net/questions/10413/… – François G. Dorais Sep 18 '12 at 14:18
@Joel That is interesting. It looks like what that says in general is that given a nonempty definable finite set $S$ one may assume by shrinking $S$ that every element $A \in S$ is definable from some element $x \in A$. In the present situation we probably don't know that $x$ is definable, so there is no obvious induction on rank and indeed the problem seems to be nontrivial already at very low rank. For example, what if $S$ is a finite set of countable sets of reals, e.g., a finite set of Turing degrees or constructibility degrees or $E_0$-classes? – Trevor Wilson Sep 18 '12 at 15:04
Suppose our finite set is the set of the square roots of -1. This two-element set is certainly definable in ZF. But does there exist a definition in the (primitive) language of ZF that is satisfied by i alone or by -i alone? (Note that the symbols i and -i do not belong to the (primitive) language of ZF but must be defined in that language before they can appear in definitions). – Garabed Gulbenkian Sep 18 '12 at 19:59
@Garabed I assume you are talking about the set of square roots of $-1$ in $\mathbb{C}$. Whether this pair has a definable member depends on how you define $\mathbb{C}$. In the usual construction of $\mathbb{C}$ by pairs of real numbers, both imaginary units are definable. However, using François's answer below, one can show that it is consistent that there is a definable field isomorphic to $\mathbb{C}$ with no definable imaginary unit. See my answer to math.stackexchange.com/q/181464/39378 for a proof. – Trevor Wilson Sep 18 '12 at 21:54
I believe the answers to these questions are all positive. This kind of problem was discussed by Groszek and Laver in Finite groups of OD-conjugates [Period. Math. Hungar. 18 (1987), 87-97, MR0895774]. Answering a question of Mycielski, they show that there can be two sets of reals $x,y$ such that $\lbrace x,y\rbrace$ is ordinal definable but neither $x$ nor $y$ is ordinal definable. They also prove a lot of other interesting things about OD conjugates.
Here is the brief argument from the intro to that paper. Suppose $u, v$ are two mutually Sacks generic reals over $L$. Both $u$ and $v$ have minimal degree over $L$. Let $x$ and $y$ be the $L$-degrees of $u$ and $v$ respectively. Then $x$ and $y$ satisfy the same formulas with ordinal parameters because Sacks forcing is homogeneous. However, $\lbrace x, y \rbrace$ is definable (without parameters) since these are the only two minimal $L$-degrees in $L[u,v]$.
I don't think $u$ and $v$ (or any two distinct reals) can satisfy the same formulas with ordinal parameters. What about formulas like "my 17th component is 0"? Nevertheless, the argument is probably OK since the $L$-degrees $x$ and $y$ do seem to satisfy the same formulas with ordinal parameters. – Andreas Blass Sep 18 '12 at 14:56
Absolutely, Andreas, I just corrected. (Of course, the proof in the paper did not claim that $u$ and $v$ satisfy the same formulas with ordinal parameters.) – François G. Dorais Sep 18 '12 at 15:20
@Trevor: $x$ and $y$ are sets of reals. – François G. Dorais Sep 18 '12 at 15:28 | 2016-07-24 01:16:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.948344349861145, "perplexity": 112.74778856573762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823805.20/warc/CC-MAIN-20160723071023-00301-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://www.gap-system.org/Manuals/pkg/CAP-2017.09.25/doc/chap3_mj.html | Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 Ind
### 3 Morphisms
Any GAP object satisfying IsCapCategoryMorphism can be added to a category and then becomes a morphism in this category. Any morphism can belong to one or no category. After a GAP object is added to the category, it knows which things can be computed in its category and to which category it belongs. It knows categorical properties and attributes, and the functions for existential quantifiers can be applied to the morphism.
#### 3.1 Attributes for the Type of Morphisms
##### 3.1-1 CapCategory
‣ CapCategory( alpha ) ( attribute )
Returns: a category
The argument is a morphism $$\alpha$$. The output is the category $$\mathbf{C}$$ to which $$\alpha$$ was added.
##### 3.1-2 Source
‣ Source( alpha ) ( attribute )
Returns: an object
The argument is a morphism $$\alpha: a \rightarrow b$$. The output is its source $$a$$.
##### 3.1-3 Range
‣ Range( alpha ) ( attribute )
Returns: an object
The argument is a morphism $$\alpha: a \rightarrow b$$. The output is its range $$b$$.
#### 3.2 Categorical Properties of Morphisms
‣ AddIsMonomorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsMonomorphism. $$F: \alpha \mapsto \mathtt{IsMonomorphism}(\alpha)$$.
‣ AddIsEpimorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsEpimorphism. $$F: \alpha \mapsto \mathtt{IsEpimorphism}(\alpha)$$.
‣ AddIsIsomorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsIsomorphism. $$F: \alpha \mapsto \mathtt{IsIsomorphism}(\alpha)$$.
‣ AddIsSplitMonomorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsSplitMonomorphism. $$F: \alpha \mapsto \mathtt{IsSplitMonomorphism}(\alpha)$$.
‣ AddIsSplitEpimorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsSplitEpimorphism. $$F: \alpha \mapsto \mathtt{IsSplitEpimorphism}(\alpha)$$.
‣ AddIsOne( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsOne. $$F: \alpha \mapsto \mathtt{IsOne}(\alpha)$$.
‣ AddIsIdempotent( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsIdempotent. $$F: \alpha \mapsto \mathtt{IsIdempotent}(\alpha)$$.
#### 3.3 Non-Categorical Properties of Morphisms
Non-categorical properties are not stable under equivalences of categories.
##### 3.3-1 IsIdenticalToIdentityMorphism
‣ IsIdenticalToIdentityMorphism( alpha ) ( property )
Returns: a boolean
The argument is a morphism $$\alpha: a \rightarrow b$$. The output is true if $$\alpha = \mathrm{id}_a$$, otherwise the output is false.
‣ AddIsIdenticalToIdentityMorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsIdenticalToIdentityMorphism. $$F: \alpha \mapsto \mathtt{IsIdenticalToIdentityMorphism}(\alpha)$$.
##### 3.3-3 IsIdenticalToZeroMorphism
‣ IsIdenticalToZeroMorphism( alpha ) ( property )
Returns: a boolean
The argument is a morphism $$\alpha: a \rightarrow b$$. The output is true if $$\alpha = 0$$, otherwise the output is false.
‣ AddIsIdenticalToZeroMorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsIdenticalToZeroMorphism. $$F: \alpha \mapsto \mathtt{IsIdenticalToZeroMorphism }(\alpha)$$.
‣ AddIsEndomorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsEndomorphism. $$F: \alpha \mapsto \mathtt{IsEndomorphism}(\alpha)$$.
‣ AddIsAutomorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsAutomorphism. $$F: \alpha \mapsto \mathtt{IsAutomorphism}(\alpha)$$.
#### 3.4 Equality and Congruence for Morphisms
##### 3.4-1 IsCongruentForMorphisms
‣ IsCongruentForMorphisms( alpha, beta ) ( operation )
Returns: a boolean
The arguments are two morphisms $$\alpha, \beta: a \rightarrow b$$. The output is true if $$\alpha \sim_{a,b} \beta$$, otherwise the output is false.
‣ AddIsCongruentForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsCongruentForMorphisms. $$F: (\alpha, \beta) \mapsto \mathtt{IsCongruentForMorphisms}(\alpha, \beta)$$.
##### 3.4-3 IsEqualForMorphisms
‣ IsEqualForMorphisms( alpha, beta ) ( operation )
Returns: a boolean
The arguments are two morphisms $$\alpha, \beta: a \rightarrow b$$. The output is true if $$\alpha = \beta$$, otherwise the output is false.
‣ AddIsEqualForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsEqualForMorphisms. $$F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphisms}(\alpha, \beta)$$.
##### 3.4-5 IsEqualForMorphismsOnMor
‣ IsEqualForMorphismsOnMor( alpha, beta ) ( operation )
Returns: a boolean
The arguments are two morphisms $$\alpha: a \rightarrow b, \beta: c \rightarrow d$$. The output is true if $$\alpha = \beta$$, otherwise the output is false.
‣ AddIsEqualForMorphismsOnMor( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsEqualForMorphismsOnMor. $$F: (\alpha, \beta) \mapsto \mathtt{IsEqualForMorphismsOnMor}(\alpha, \beta)$$.
#### 3.5 Basic Operations for Morphisms in Ab-Categories
##### 3.5-1 IsZeroForMorphisms
‣ IsZeroForMorphisms( alpha ) ( operation )
Returns: a boolean
The argument is a morphism $$\alpha: a \rightarrow b$$. The output is true if $$\alpha \sim_{a,b} 0$$, otherwise the output is false.
‣ AddIsZeroForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsZeroForMorphisms. $$F: \alpha \mapsto \mathtt{IsZeroForMorphisms}(\alpha)$$.
‣ AdditionForMorphisms( alpha, beta ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a,b)$$
The arguments are two morphisms $$\alpha, \beta: a \rightarrow b$$. The output is the addition $$\alpha + \beta$$.
‣ AddAdditionForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation AdditionForMorphisms. $$F: (\alpha, \beta) \mapsto \alpha + \beta$$.
##### 3.5-5 SubtractionForMorphisms
‣ SubtractionForMorphisms( alpha, beta ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a,b)$$
The arguments are two morphisms $$\alpha, \beta: a \rightarrow b$$. The output is the addition $$\alpha - \beta$$.
‣ AddSubtractionForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation SubtractionForMorphisms. $$F: (\alpha, \beta) \mapsto \alpha - \beta$$.
‣ AdditiveInverseForMorphisms( alpha ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a,b)$$
The argument is a morphism $$\alpha: a \rightarrow b$$. The output is its additive inverse $$-\alpha$$.
‣ AddAdditiveInverseForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation AdditiveInverseForMorphisms. $$F: \alpha \mapsto -\alpha$$.
##### 3.5-9 ZeroMorphism
‣ ZeroMorphism( a, b ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a,b)$$
The arguments are two objects $$a$$ and $$b$$. The output is the zero morphism $$0: a \rightarrow b$$.
‣ AddZeroMorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation ZeroMorphism. $$F: (a,b) \mapsto (0: a \rightarrow b)$$.
#### 3.6 Subobject and Factorobject Operations
Subobjects of an object $$c$$ are monomorphisms with range $$c$$ and a special function for comparision. Similarly, factorobjects of an object $$c$$ are epimorphisms with source $$c$$ and a special function for comparision.
##### 3.6-1 IsEqualAsSubobjects
‣ IsEqualAsSubobjects( alpha, beta ) ( operation )
Returns: a boolean
The arguments are two subobjects $$\alpha: a \rightarrow c$$, $$\beta: b \rightarrow c$$. The output is true if there exists an isomorphism $$\iota: a \rightarrow b$$ such that $$\beta \circ \iota \sim_{a,c} \alpha$$, otherwise the output is false.
‣ AddIsEqualAsSubobjects( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsEqualAsSubobjects. $$F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsSubobjects}(\alpha,\beta)$$.
##### 3.6-3 IsEqualAsFactorobjects
‣ IsEqualAsFactorobjects( alpha, beta ) ( operation )
Returns: a boolean
The arguments are two factorobjects $$\alpha: c \rightarrow a$$, $$\beta: c \rightarrow b$$. The output is true if there exists an isomorphism $$\iota: b \rightarrow a$$ such that $$\iota \circ \beta \sim_{c,a} \alpha$$, otherwise the output is false.
‣ AddIsEqualAsFactorobjects( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsEqualAsFactorobjects. $$F: (\alpha, \beta) \mapsto \mathtt{IsEqualAsFactorobjects}(\alpha,\beta)$$.
##### 3.6-5 IsDominating
‣ IsDominating( alpha, beta ) ( operation )
Returns: a boolean
In short: Returns true iff $$\alpha$$ is smaller than $$\beta$$. $$\\$$ Full description: The arguments are two subobjects $$\alpha: a \rightarrow c$$, $$\beta: b \rightarrow c$$. The output is true if there exists a morphism $$\iota: a \rightarrow b$$ such that $$\beta \circ \iota \sim_{a,c} \alpha$$, otherwise the output is false.
‣ AddIsDominating( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsDominating. $$F: (\alpha, \beta) \mapsto \mathtt{IsDominating}(\alpha,\beta)$$.
##### 3.6-7 IsCodominating
‣ IsCodominating( alpha, beta ) ( operation )
Returns: a boolean
In short: Returns true iff $$\alpha$$ is smaller than $$\beta$$. $$\\$$ Full description: The arguments are two factorobjects $$\alpha: c \rightarrow a$$, $$\beta: c \rightarrow b$$. The output is true if there exists a morphism $$\iota: b \rightarrow a$$ such that $$\iota \circ \beta \sim_{c,a} \alpha$$, otherwise the output is false.
‣ AddIsCodominating( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsCodominating. $$F: (\alpha, \beta) \mapsto \mathtt{IsCodominating}(\alpha,\beta)$$.
#### 3.7 Identity Morphism and Composition of Morphisms
##### 3.7-1 IdentityMorphism
‣ IdentityMorphism( a ) ( attribute )
Returns: a morphism in $$\mathrm{Hom}(a,a)$$
The argument is an object $$a$$. The output is its identity morphism $$\mathrm{id}_a$$.
‣ AddIdentityMorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IdentityMorphism. $$F: a \mapsto \mathrm{id}_a$$.
##### 3.7-3 PreCompose
‣ PreCompose( alpha, beta ) ( operation )
Returns: a morphism in $$\mathrm{Hom}( a, c )$$
The arguments are two morphisms $$\alpha: a \rightarrow b$$, $$\beta: b \rightarrow c$$. The output is the composition $$\beta \circ \alpha: a \rightarrow c$$.
##### 3.7-4 PreCompose
‣ PreCompose( L ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a_1, a_{n+1})$$
This is a convenience method. The argument is a list of morphisms $$L = ( \alpha_1: a_1 \rightarrow a_2, \alpha_2: a_2 \rightarrow a_3, \dots, \alpha_n: a_n \rightarrow a_{n+1} )$$. The output is the composition $$\alpha_{n} \circ ( \alpha_{n-1} \circ ( \dots ( \alpha_2 \circ \alpha_1 ) ) )$$.
‣ AddPreCompose( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation PreCompose. $$F: (\alpha, \beta) \mapsto \beta \circ \alpha$$.
##### 3.7-6 PostCompose
‣ PostCompose( beta, alpha ) ( operation )
Returns: a morphism in $$\mathrm{Hom}( a, c )$$
The arguments are two morphisms $$\beta: b \rightarrow c$$, $$\alpha: a \rightarrow b$$. The output is the composition $$\beta \circ \alpha: a \rightarrow c$$.
##### 3.7-7 PostCompose
‣ PostCompose( L ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a_1, a_{n+1})$$
This is a convenience method. The argument is a list of morphisms $$L = ( \alpha_n: a_n \rightarrow a_{n+1}, \alpha_{n-1}: a_{n-1} \rightarrow a_n, \dots, \alpha_1: a_1 \rightarrow a_2 )$$. The output is the composition $$((\alpha_{n} \circ \alpha_{n-1}) \circ \dots \alpha_2) \circ \alpha_1$$.
‣ AddPostCompose( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation PostCompose. $$F: (\alpha, \beta) \mapsto \alpha \circ \beta$$.
#### 3.8 Well-Definedness of Morphisms
##### 3.8-1 IsWellDefinedForMorphisms
‣ IsWellDefinedForMorphisms( alpha ) ( operation )
Returns: a boolean
The argument is a morphism $$\alpha$$. The output is true if $$\alpha$$ is well-defined, otherwise the output is false.
‣ AddIsWellDefinedForMorphisms( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation IsWellDefinedForMorphisms. $$F: \alpha \mapsto \mathtt{IsWellDefinedForMorphisms}( \alpha )$$.
#### 3.9 Basic Operations for Morphisms in Abelian Categories
##### 3.9-1 LiftAlongMonomorphism
‣ LiftAlongMonomorphism( iota, tau ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(t,k)$$
The arguments are a monomorphism $$\iota: k \hookrightarrow a$$ and a morphism $$\tau: t \rightarrow a$$ such that there is a morphism $$u: t \rightarrow k$$ with $$\iota \circ u \sim_{t,a} \tau$$. The output is such a $$u$$.
‣ AddLiftAlongMonomorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation LiftAlongMonomorphism. The function $$F$$ maps a pair $$(\iota, \tau)$$ to a lift $$u$$ if it exists, and to fail otherwise.
##### 3.9-3 ColiftAlongEpimorphism
‣ ColiftAlongEpimorphism( epsilon, tau ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(c,t)$$
The arguments are an epimorphism $$\epsilon: a \rightarrow c$$ and a morphism $$\tau: a \rightarrow t$$ such that there is a morphism $$u: c \rightarrow t$$ with $$u \circ \epsilon \sim_{a,t} \tau$$. The output is such a $$u$$.
‣ AddColiftAlongEpimorphism( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation ColiftAlongEpimorphism. The function $$F$$ maps a pair $$(\epsilon, \tau)$$ to a lift $$u$$ if it exists, and to fail otherwise.
#### 3.10 Lift/ Colift
• For any pair of morphisms $$\alpha: a \rightarrow c$$, $$\beta: b \rightarrow c$$, we call each morphism $$\alpha / \beta: a \rightarrow b$$ such that $$\beta \circ (\alpha / \beta) \sim_{a,c} \alpha$$ a lift of $$\alpha$$ along $$\beta$$.
• For any pair of morphisms $$\alpha: a \rightarrow c$$, $$\beta: a \rightarrow b$$, we call each morphism $$\alpha \backslash \beta: c \rightarrow b$$ such that $$(\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta$$ a colift of $$\beta$$ along $$\alpha$$.
Note that such lifts (or colifts) do not have to be unique. So in general, we do not expect that algorithms computing lifts (or colifts) do this in a functorial way. Thus the operations $$\mathtt{Lift}$$ and $$\mathtt{Colift}$$ are not regarded as categorical operations, but only as set-theoretic operations.
##### 3.10-1 Lift
‣ Lift( alpha, beta ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(a,b) + \{ \mathtt{fail} \}$$
The arguments are two morphisms $$\alpha: a \rightarrow c$$, $$\beta: b \rightarrow c$$ such that there is a lift $$\alpha / \beta: a \rightarrow b$$ of $$\alpha$$ along $$\beta$$, i.e., a morphism such that $$\beta \circ (\alpha / \beta) \sim_{a,c} \alpha$$. The output is such a lift or $$\mathtt{fail}$$ if it doesn't exist.
‣ AddLift( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation Lift. The function $$F$$ maps a pair $$(\alpha, \beta)$$ to a lift $$\alpha / \beta$$ if it exists, and to fail otherwise.
##### 3.10-3 Colift
‣ Colift( alpha, beta ) ( operation )
Returns: a morphism in $$\mathrm{Hom}(c,b) + \{ \mathtt{fail} \}$$
The arguments are two morphisms $$\alpha: a \rightarrow c$$, $$\beta: a \rightarrow b$$ such that there is a colift $$\alpha \backslash \beta: c \rightarrow b$$ of $$\beta$$ along $$\alpha$$., i.e., a morphism such that $$(\alpha \backslash \beta) \circ \alpha \sim_{a,b} \beta$$. The output is such a colift or $$\mathtt{fail}$$ if it doesn't exist.
‣ AddColift( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation Colift. The function $$F$$ maps a pair $$(\alpha, \beta)$$ to a colift $$\alpha \backslash \beta$$ if it exists, and to fail otherwise.
#### 3.11 Inverses
Let $$\alpha: a \rightarrow b$$ be a morphism. An inverse of $$\alpha$$ is a morphism $$\alpha^{-1}: b \rightarrow a$$ such that $$\alpha \circ \alpha^{-1} \sim_{b,b} \mathrm{id}_b$$ and $$\alpha^{-1} \circ \alpha \sim_{a,a} \mathrm{id}_a$$.
‣ AddInverse( C, F ) ( operation )
Returns: nothing
The arguments are a category $$C$$ and a function $$F$$. This operations adds the given function $$F$$ to the category for the basic operation Inverse. $$F: \alpha \mapsto \alpha^{-1}$$.
#### 3.12 Tool functions for caches
##### 3.12-1 IsEqualForCacheForMorphisms
‣ IsEqualForCacheForMorphisms( phi, psi ) ( operation )
Returns: true or false
Compares two objects in the cache
‣ AddIsEqualForCacheForMorphisms( c, F ) ( operation )
By default, CAP uses caches to store the values of Categorical operations. To get a value out of the cache, one needs to compare the input of a basic operation with its previous input. To compare morphisms in the category, IsEqualForCacheForMorphism is used. By default this is an alias for IsEqualForMorphismsOnMor, where fail is substituted by false. If you add a function, this function used instead. A function $$F: a,b \mapsto bool$$ is expected here. The output has to be true or false. Fail is not allowed in this context. | 2018-02-25 23:50:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8547921776771545, "perplexity": 1601.1156369501937}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891817523.0/warc/CC-MAIN-20180225225657-20180226005657-00543.warc.gz"} |
http://sky296blog106fc2com.somee.com/write-my-case-study/page-281-2018-09-03.html | # ⚡ Mit university fees in rupees
Monday, September 10, 2018 11:52:50 AM
Fuzzy neural network A fuzzy neural network or neuro-fuzzy system is a se marier au present machine that finds the parameters of a fuzzy system (i.e., fuzzy sets, fuzzy rules) by exploiting approximation techniques from neural networks. Both neural networks and fuzzy systems have some things in common. They can be used for solving a problem (e.g. pattern recognition, regression or density estimation) if there does not exist any mathematical model of the given problem. They solely do have certain disadvantages and advantages which almost completely disappear by combining both concepts. Neural networks can only come into play if the problem is expressed by a sufficient amount of observed examples. These observations funko pop steven universe connie used to train the black box. On the one mit university fees in rupees no prior knowledge about the problem needs to be given. On the other hand, however, it is not straightforward to extract mit university fees in rupees rules from the neural network's structure. On the contrary, a fuzzy university of notre dame australia demands linguistic rules instead of learning examples as prior knowledge. Furthermore the input and output variables have to be described linguistically. If the knowledge is incomplete, wrong or contradictory, then the fuzzy system must be tuned. Since there is not any formal approach for it, the tuning is performed in a heuristic way. This is usually very time consuming and error-prone. It is desirable for fuzzy systems to have an automatic adaption procedure which is comparable to neural networks. As it can be seen in Table 1, combining both approaches should unite advantages and exclude disadvantages. Compared to a common neural network, destino educação escolas inovadoras pdf weights and propagation and activation functions of fuzzy neural networks differ a lot. Although there are many different approaches to model a fuzzy neural network (Buckley and Hayashi, 1994, 1995; Nauck and Kruse, 1996), most of them agree on certain characteristics such as the following: A neuro-fuzzy system based on an underlying fuzzy system is trained by means of a data-driven learning method derived from neural network theory. This heuristic only takes into account local information to cause local changes in the fundamental fuzzy system. It can be represented as a set of fuzzy rules at any time of the learning universe sandbox 2 download free 2019, i.e., before, during and after. Thus the system might be initialized with or without prior knowledge in terms of fuzzy rules. The learning procedure is constrained to ensure the semantic properties universities with rolling admissions in canada the underlying fuzzy system. A neuro-fuzzy system approximates a n-dimensional unknown function which is partly represented by training examples. Fuzzy rules can thus be interpreted as vague prototypes of the training data. Bachelor of planning nirma university neuro-fuzzy mit university fees in rupees is represented as special three-layer feedforward neural network as it is shown in Figure 1. The first layer mit university fees in rupees to the input variables. Research paper example high school second layer symbolizes the fuzzy rules. The third layer represents the output variables. The fuzzy sets are converted as (fuzzy) connection weights. Some approaches also use five layers where the fuzzy sets are encoded in what are the best art colleges units of the second and fourth layer, respectively. However, these models can tom callahan educational consultant transformed into a three-layer architecture. One can basically distinguish between three different kinds my childhood memories essay fuzzy neural networks, i.e., mit university fees in rupees, concurrent and hybrid FNNs (Nauck et al., 1997). In the case mit university fees in rupees cooperative neural fuzzy systems, both artificial neural network and fuzzy system work independently from each other. The ANN tries to learn the parameters from the fuzzy system. This can be either performed offline what does nc stand for in education online while the fuzzy system is applied. Figure 2 depicts four different kinds of cooperative fuzzy neural networks. The upper left fuzzy neural network learns pleasure of childhood essay in english set from given training data. This is usually performed by fitting membership functions with a neural network. The fuzzy sets are then determined offline. They are then utilized to form the fuzzy system by fuzzy rules that are given (not learned) as well. The upper right neuro-fuzzy system determines fuzzy rules from training data by a neural network. Here as well, the neural networks learns offline before the fuzzy system is initialized. The rule learning usually done by clustering on self-organizing feature maps (Bezdek et al., 1992; Vuorimaa, 1994). It is also possible to apply fuzzy clustering methods to obtain rules. In the lower left neuro-fuzzy model, the system learns all membership function parameters online, i.e., while the fuzzy system is applied. Thus initially fuzzy rules and membership functions must be defined beforehand. Moreover, the error has to be measured in order to improve and guide the learning step. The lower right one determines rule weights for all fuzzy rules by a neural network. This can be done online and offline. A rule weight is interpreted as the mit university fees in rupees of a rule (Kosko, 1992). They are multiplied with the rule output. In (Nauck et al., 1997) the authors argue that the semantics of rule weights are not clearly defined. They could be replaced by modified membership functions. However, this could destroy the interpretation of fuzzy sets. Moreover, identical linguistic values might be represented differently in dissimilar rules. Hybrid neuro-fuzzy systems are homogeneous and usually resemble mit university fees in rupees networks. Here, the fuzzy system sandvine internet phenomena report interpreted as special kind of neural network. The advantage of such hybrid NFS is its architecture since both fuzzy system and neural network do not have to communicate any more with each other. They are one fully fused entity. These mit university fees in rupees can learn online how do you say research paper in spanish offline. Figure 3 shows such a hybrid FNN. The rule base of a fuzzy system is interpreted as a neural network. Fuzzy sets can be regarded as weights whereas the input and output variables and the rules are modeled as neurons. Neurons can be included or deleted in the learning step. Finally, the neurons of the network represent the fuzzy knowledge base. Obviously, the major drawbacks of both underlying systems are thus overcome. In order to build a fuzzy controller, membership functions which express the linguistic terms of the inference rules have to be defined. In fuzzy set theory, there does not exist any formal approach to define these functions. Any shape (e.g., triangular, Gaussian) can be considered as membership function with an arbitrary set of parameters. Thus the optimization of these functions in terms of generalizing the data is very important for fuzzy systems. Neural networks can be used to solve this problem. By fixing a distinct shape of the membership mit university fees in rupees, say triangular, the neural network must optimize their parameters by gradient descent (Nomura et al., 1992). Thus, aside information about the shape of the membership functions, training data must be available mit university fees in rupees well. Another approach (Hayashi et al., 1992) is to group the training data $$\$$ into $$M$$ clusters. Every cluster represents a rule $$R_m$$ where $$m = 1, 2, \ldots, M\ .$$ Hence these rules are not defined linguistically but rather by crisp data points $$\mathbf x = (x_1, x_2, \ldots, x_n)\ .$$ Thus a neural network with $$n$$ input units, hidden layers and $$M$$ output units might be applied to train on the pre-defined clusters. For testing, an arbitrary pattern $$x$$ is presented to the trained neural network. Every output unit $$m$$ will return a degree to which extend $$x$$ may fit to the antecedent of rule $$R_m\ .$$ To guarantee the characteristics of a fuzzy system, the learning algorithm must enforce the following mandatory constraints: Fuzzy sets must stay normal and convex. Fuzzy sets mit university fees in rupees not exchange their relative positions (they must not pass each other). Fuzzy sets must always overlap. Additionally there do exist some optional constraints like the following: Fuzzy sets must stay symmetric. The membership degrees must sum up to 1. An important hybrid fuzzy neural network has been introduced in (Berenji, 1992). The ARIC (approximate reasoning-based intelligent control) is presented as a neural network mit university fees in rupees a prior defined rule base is tuned by updating the network's prediction. Thus the advantages of fuzzy systems and neural networks are easily combined as presented in Table 1. The ARIC is represented by two feed-forward neural networks, the action-state evaluation network (AEN) and the action selection network (ASN). The ASN is a multilayer neural network representation of fh aachen university of applied sciences biomedical engineering fuzzy system. It then again consists of two separate. The first one represents the fuzzy inference and the second one computes a confidence measure based on the current and next system state. Both parts are eventually combined to the ASN's output. As it is shown in Figure 1, the first layer represents the rule antecedents, whereas the second layer corresponds to the implemented fuzzy rules and the third layer symbolized the system action. The network flow is at follows. In the first layer the system variables are fuzzified. In pre thesis presentation ppt next step these membership values are multiplied by the attached weights of the connections between the first and second layer. In the latter layer, every rule's input corresponds to the minimum of its input connections. A rule's conclusion is installed as membership function. This function maps the inverse rule input value. Its output values is then multiplied by the weights of the connections between second and third layer. The final output mit university fees in rupees is eventually computed by the weighted average of all rules' conclusions. The AEN (which is as three-layer feed-forward neural network as well) aims to forecast the system behavior. The hidden layer obtains as input both the system state and an error signal from the underlying system. The output of the networks shall represent the prediction of the next reinforcement which depends on the weights mit university fees in rupees the system state. The weights are changed by a reinforcement procedure which takes into consideration the outputs of both networks ASN and AEN, respectively. ARIC was successfully applied to the cart-pole balancing problem. Whereas the ARIC model can be easily interpret as a set of fuzzy-if-then rules, the ASN network to mit university fees in rupees the weights is rather difficult to understand. It is a working neural network architecture that utilizes aspects of mit university fees in rupees systems. However, a semantic interpretation of some learning universal naati and pte institute is not possible. Berenji and Khedkar (1992) introduced an improvement of the their former approach named GARIC (generalized ARIC). This idea does not suffer philippine railway institute careers different interpretations of the linguistic values anymore by refraining from weighted connections in the ASN. Instead the fuzzy sets are represented as nodes in the network. Moreover the learning procedure changes parameters of these nodes and thus the shape of the membership functions. GARIC is also able to use any kind of membership functions in the conclusion since a different defuzzifier and a differentiable soft-minimum function are used. Note that the ANFIS model (Jang, 1993) also implements a Sugeno-like fuzzy system in a network structure. Here a mixture of plain backpropagation and least mean squares best deal on universal orlando tickets is used to train the system. Both the ANFIS and the GARIC model are not so easy to interpret as, e.g., Mamdani-type fuzzy systems. Therefore models like NEFCON (Nauck, 1994), NEFCLASS (Nauck and Kruse, 1996) and NEFPROX (Nauck and Kruse, 1997) have been developed for neuro-fuzzy control, classification and regression, respectively. They all mit university fees in rupees Mamdani-type fuzzy systems and thus use special learning algorithms . Berenji, H.R. (1992). A Universal baby monitor app Learning Based Architecture for Fuzzy Logic Control. International Journal of Approximate Reasoning 6267-292. Berenji, H. R. and Khedkar, P. (1992). Learning and Tuning Fuzzy Logic Controllers Through Reinforcements, IEEE Trans. Neural Networks3pp. 724-740. Bezdek, J. C., Tsao, E. C.-K. and Pal, N. R. (1992). Fuzzy Kohonen Clustering Networks, in Proc. IEEE Int. Conf. on Fuzzy Systems 1992 (San Diego), pp. 1035-1043. Buckley, J. J. and Hayashi, Y. (1994). Fuzzy neural networks: A survey, Fuzzy Sets marijuana essay titles Systems 66pp. 1-13. Buckley, J. J. and Hayashi, Y. (1995). Neural networks for fuzzy systems, Fuzzy Sets and Systems 71pp. 265-276. Hayashi, I., Nomura, H., Yamasaki, H. and Wakami, N. (1992). Construction of Fuzzy Inference Rules by NFD and Mit university fees in rupees. International Journal of Approximate Reasoning6pp. 241-266. Jang, J.-S. R. (1993). ANFIS: Adaptive-Network-Based Fuzzy Inference Systems, IEEE Transaction on Systems, Man, and Cybernetics 23pp. 665-685. Kosko, B. (1992). Neural Networks and Fuzzy Systems. A Dynamical Systems Approach to Machine Intelligence (Prentice-Hall, Englewood Cliffs). Nauck, D. (1994). A Fuzzy Perceptron as a Generic Model for Neuro-Fuzzy Approaches, in Proc. Fuzzy-Systeme 94 (Munich). Nauck, D. and Kruse, R. (1996). Neuro-Fuzzy Classification with NEFCLASS, in P. Kleinschmidt, A. Bachem, U. Derigs, D. Fischer, U. Leopold-Wildburger and R. Möhring (eds.), Operations Research Proceedings 1995(Berlin), pp. 294-299. Nauck, D. child observation case study examples Kruse, R. (1997). Function Approximation by NEFPROX, in Proc. Second European Workshop on Fuzzy Decision Analysis and Neural Networks for Management, Planning, and Optimization (EFDAN'97)(Dortmund), pp. 160-169. Nomura, H., Hayashi, I. graduate medical education directory Wakami, N. (1992). A Learning Method of Fuzzy Inference Rules by Descent Method, in Proc. IEEE Int. Conf. on Fuzzy Systems 1992 (San Diego), pp. 203-210. Vuorimaa, P. (1994). Fuzzy Self-Organizing Map, Fuzzy Sets and Systems 66pp. 223-231. Tony J. Prescott (2008) Action selection. Scholarpedia, 3(2):2705. Robert Babuska and Ebrahim Mamdani (2008) Fuzzy control. Scholarpedia, 3(2):2103. Milan Mares (2006) Fuzzy sets. Scholarpedia, 1(10):2031. Teuvo Kohonen and Timo Honkela (2007) Kohonen network. Scholarpedia, 2(1):1568. Rodolfo Llinas (2008) Neuron. Scholarpedia, 3(8):1490.
Web hosting by Somee.com | 2020-04-02 04:31:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6174293160438538, "perplexity": 1947.753731135253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506580.20/warc/CC-MAIN-20200402014600-20200402044600-00337.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/187707-cyclic-subgroups-finite-group-g-print.html | # Cyclic Subgroups of a finite group G
All elements of a finite cyclic group are of the form $a^m$, where $a^n=e$. Imagine that there were two elements $a^m$ and $a^k$ such that k is different from m, but $a^m=a^k$. Let $m=na_1+r_1$ and $k=na_2+r_2$ where the r's are the rest of the division by n and thus less than n. Lets assume, without loss of generality, that $r_1 >r_2$. Then assume $a^m=a^{r_1}$ is equal to $a^{r_2}=a^k$ and we'll get a contradiction. $a^{r_1}=a^{r_2}$ implies $a^{r_1-r_2}=e$ and thus there would be a number $r_1-r_2 such that $a^{r_1-r_2}=e$, which is a contradiction since n is the smallest natural number for which $a^n=e$. | 2017-07-21 12:59:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9820824265480042, "perplexity": 73.9790790593403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423774.37/warc/CC-MAIN-20170721122327-20170721142327-00131.warc.gz"} |
https://inductivestep.github.io/R-notes/visualising-data-in-the-tidyverse.html | # Chapter 3 Visualising data in the tidyverse
By the end of this chapter you will:
• Have explored a key example visualisation in depth, using packages in the tidyverse.
• Understand what happens when you tweak this example in various ways.
• Know where to look for ideas and code for other visualisations.
## 3.1 Getting setup
You will need to create a new R or Markdown (Rmd) file (depending on your preference – I recommend Markdown) and save it somewhere sensible where you can find it again in a few months time.
We will be using the Gapminder dataset dataset from last time.
gap <- read.csv("gapminder.csv")
Previously we used the “base” plot function:
plot(lifeExp ~ year,
data = gap,
xlab = "Year",
ylab = "Life expectancy at birth")
We can do better than this.
A collection of packages called the tidyverse has become an industry standard in R (though see also an alternate view).
This command will include tidyverse and make a bit of noise as it arrives…
library(tidyverse)
If that didn’t work, there are two things you can do. You could try saving your R/Rmd file. This may prompt RStudio to notice that the package isn’t installed and ask you if you want to install it.
Alternatively, just use the install.packages command as per last chapter:
install.packages("tidyverse")
To see if that worked, run this:
ggplot(data = gap,
mapping = aes(x = year, y = lifeExp)) +
geom_point() +
labs(x = "Year",
y = "Life expectancy at birth")
Ta-da: a graph! This used ggplot, which is part of the ggplot2 package which, in turn, is part of tidyverse.
The rest of this tutorial will explore how to develop this into a more useful visualisation.
## 3.2 An interlude on functions
Previously, I described R functions as magical computational machines which take inputs and transform them in some way, giving an output.
Above, we have seen that the output of a function can be a picture. It can also be a vibration (that’s how sounds are made) or anything else that can be plugged into a computer. It might be a humble number, like a mean.
Sometimes I’ll call functions “commands” and sometimes I’ll call the inputs “options” or “parameters” or “arguments.” Hopefully it will be clear from the context what I mean. If not, even after scratching your head, then do ask!
## 3.3 A scatterplot in ggplot
Let’s build the previous example step-by-step.
ggplot(data = gap,
mapping = aes(x = year, y = lifeExp))
This first part tells ggplot what data to use and an aesthetic mapping. Aesthetics in tidyverse are properties of the objects in your plot and the mapping tells ggplot how those objects relate to your data. Two basic properties are x and y locations on a plot. Here they have been mapped to year and life expectancy, respectively.
When you run that code, you will see that nothing was actually done with the mappings. The next stage is to add a geom – a geometric object – for each row of data. That’s where the point geom, geom_point, comes in:
ggplot(data = gap,
mapping = aes(x = year, y = lifeExp)) +
geom_point()
Note how the + symbol is used here to mean adding elements to a plot. The meaning of + depends on context.
I could also have made this plot by giving a name to the first part:
the_basic_plot <- ggplot(data = gap,
mapping = aes(x = year, y = lifeExp))
Then added this to the geom:
plot_with_stuff_on <- the_basic_plot + geom_point()
The plot hasn’t displayed yet, though.
### 3.3.1 Warm-up activity
1. What do you need to do to get the plot_with_stuff_on plot to display?
2. How could you change the axis labels on plot_with_stuff_on? (Look up for a clue!)
a. What do you need to do to get the plot_with_stuff_on plot to display?
plot_with_stuff_on
b. How could you change the axis labels on plot_with_stuff_on?
Either:
plot_with_stuff_on +
labs(x = "Year",
y = "Life expectancy at birth")
Or:
final_plot <- plot_with_stuff_on +
labs(x = "Year",
y = "Life expectancy at birth")
final_plot
## 3.4 Another aesthetic: colour
This is a simple change, but begins to highlight patterns in the data. Here I have just copied and pasted a chunk from above and added the mapping colour = continent.
ggplot(data = gap,
mapping = aes(x = year,
y = lifeExp,
colour = continent)) +
geom_point() +
labs(x = "Year",
y = "Life expectancy at birth")
Can you spot any patterns in the graph?
A legend has appeared at the right hand side explaining what the colours represent.
By default the legend title is the same as the variable name. In this case it’s “continent” which is clear, but sometimes it will be something like “group_2_id” which is less pleasing on the eye (and I cringe when I see something like this in a journal article).
The legend title is easy to change by adding another option to labs:
ggplot(data = gap,
mapping = aes(x = year,
y = lifeExp,
colour = continent)) +
geom_point() +
labs(x = "Year",
y = "Life expectancy at birth",
colour = "Continent")
Now the legend has an uppercase “C.”
## 3.5 Another geom: jitter
Making graphs often involves playing around with different ways of showing the information. Here’s the jitter geom, which is the same as the point geom but with “a small amount of random variation to the location of each point” (see ?geom_jitter).
ggplot(data = gap,
mapping = aes(x = year,
y = lifeExp,
colour = continent)) +
geom_jitter() +
labs(x = "Year",
y = "Life expectancy at birth",
colour = "Continent")
### 3.5.1 Activity to develop your help-searching skill!
How can you vary the amount of jitter?
Tip: you might find the help useful:
?geom_jitter
If that doesn’t deliver anything useful, try this reference link.
There are two options, width and height, which specify how wide the jitteriness is. Set these to zero, and the plot is indistinguishable from the point geom:
ggplot(data = gap,
mapping = aes(x = year,
y = lifeExp,
colour = continent)) +
geom_jitter(width = 0, height = 0) +
labs(x = "Year",
y = "Life expectancy at birth",
color = "Continent")
Here’s a little jitter added only to the width:
ggplot(data = gap,
mapping = aes(x = year, y = lifeExp, colour = continent)) +
geom_jitter(width = 1, height = 0) +
labs(x = "Year",
y = "Life expectancy at birth",
color = "Continent")
## 3.6 Aggregating/summarising data by group
Last time, we saw how to calculate the mean of a variable. Here’s the mean of life expectancy, across all countries and years:
mean(gap$lifeExp) ## [1] 59.47444 I don’t know what to make of that! Typically we want to calculate means by group rather than for a whole variable. This is known as aggregating or summarising by group. For instance, looking at the plots above it seems that there will be a mean difference in life expectancy between continents, and it would be interesting to see that. For this, we will use dplyr (pronounced “DEE-ply-er”). It’s part of tidyverse so already included, but it’s useful to know the name of this specific part for when you are searching for help. I’m going to work through an example in excruciating detail, but it will be worth it I promise. The punchline is that to calculate mean life expectancy by year and continent, you do this: mean_life_exp_gap <- gap %>% group_by(year, continent) %>% summarise(mean_life_exp = mean(lifeExp)) ## summarise() regrouping output by 'year' (override with .groups argument) (Have a look and see.) Here’s a longer worked example. Step 1. Use group_by to tell R what variables you want to group the data by. The first parameter of group_by is the dataset you want to group. The remaining parameters are the variables in that dataset to group by: grouped_gap <- group_by(gap, year, continent) So this says, group the gap data frame by year and continent. This new variable, grouped_gap is a grouped data frame. It has all the same information as before, plus a little note (semi-hidden) to say that analyses on this should be grouped. Here’s how to peek at this note: group_vars(grouped_gap) ## [1] "year" "continent" Step 2. Use summarise on this grouped data frame to calculate what you want. The first argument of summarise is the data frame (grouped or otherwise) followed by new variable names and what you want them to contain. summarised_grouped_gap <- summarise(grouped_gap, mean_life_exp = mean(lifeExp)) ## summarise() regrouping output by 'year' (override with .groups argument) Let’s have a look at the top 10 rows: head(summarised_grouped_gap, 10) ## # A tibble: 10 x 3 ## # Groups: year [2] ## year continent mean_life_exp ## <int> <chr> <dbl> ## 1 1952 Africa 39.1 ## 2 1952 Americas 53.3 ## 3 1952 Asia 46.3 ## 4 1952 Europe 64.4 ## 5 1952 Oceania 69.3 ## 6 1957 Africa 41.3 ## 7 1957 Americas 56.0 ## 8 1957 Asia 49.3 ## 9 1957 Europe 66.7 ## 10 1957 Oceania 70.3 It worked! We could now use this in ggplot (and shall do so below). ### 3.6.1 Activity Do the same again but this time calculate means only by year, averaging across continents. ### 3.6.2 Answer grouped_gap_year <- group_by(gap, year) summarised_grouped_year <- summarise(grouped_gap_year, mean_life_exp = mean(lifeExp)) ## summarise() ungrouping output (override with .groups argument) summarised_grouped_year ## # A tibble: 12 x 2 ## year mean_life_exp ## <int> <dbl> ## 1 1952 49.1 ## 2 1957 51.5 ## 3 1962 53.6 ## 4 1967 55.7 ## 5 1972 57.6 ## 6 1977 59.6 ## 7 1982 61.5 ## 8 1987 63.2 ## 9 1992 64.2 ## 10 1997 65.0 ## 11 2002 65.7 ## 12 2007 67.0 ## 3.7 Pipes R analyses often feel like making information flow along a pipe, transforming it in various ways as it goes. Maybe reshaping it, selecting some variables, filtering, grouping, calculating. Finally, out flows an answer. This leads to another member of the tidyverse family, magrittr, named after René Magritte because of his 1929 painting showing a pipe and a caption “Ceci n’est pas une pipe” (“This is not a pipe”). You may have noticed that both group_by and summarise had a data frame as their first argument. They also both outputted a data frame. The forward pipe operator, %>%, allows you to pass the data frame along your information flow, without having to save results in interim variables. You start with the name of the input data frame and then pipe it into the first function. For example, here is how to group the data: gap %>% group_by(year, continent) As before you can then save the result: grouped <- gap %>% group_by(year, continent) To flow this onto summarise, just add another pipe like so: grouped <- gap %>% group_by(year, continent) %>% summarise(mean_life_exp = mean(lifeExp)) ## summarise() regrouping output by 'year' (override with .groups argument) The %>% is purely designed to make the flow of information easier to see and hopefully also easier to design. ## 3.8 Plot the mean life expectancy by continent By here you hopefully get the gist of how to use pipes to group data frames and summarise them. There will be further opportunities to practice this skill. Here’s an aggregated data frame with mean life expectancy by year and continent: mean_life_exp_gap <- gap %>% group_by(year, continent) %>% summarise(mean_life_exp = mean(lifeExp)) ## summarise() regrouping output by 'year' (override with .groups argument) You can view this to check the information is as you expect: View(mean_life_exp_gap) Here are the variable names, for ease of reference. names(mean_life_exp_gap) ## [1] "year" "continent" "mean_life_exp" ### 3.8.1 Actvity Now your challenge is to plot the mean life expectancy by year, with colour showing the continent. You could try adapting an example from above to help you. ### 3.8.2 Answer ggplot(mean_life_exp_gap, aes(x = year, y = mean_life_exp, colour = continent)) + geom_point() ## 3.9 Yet another geom: line Instead of plotting points for each year, you may wish to join the data with lines. Here’s how – just use geom_line: ggplot(mean_life_exp_gap, aes(x = year, y = mean_life_exp, colour = continent)) + geom_line() + labs(x = "Year", y = "Life expectancy at birth", colour = "Continent") ### 3.9.1 Activity How could you add points back to the lines? ### 3.9.2 Answer Simply use + again: ggplot(mean_life_exp_gap, aes(x = year, y = mean_life_exp, colour = continent)) + geom_point() + geom_line() I’ve been a bit lazy here and haven’t bothered changing the axis labels and legend title. That is fine when playing around with different visualisations and learning. Just remember to tidy it all up before adding to a written report! ## 3.10 Filtering data along the pipeline Analysing by continent clearly doesn’t do the data justice: in the jittered points we saw there was loads of variation within continent. The mean plots highlighted that improvement in life expectancy in Africa stalled around 1990. I wonder if this was the same for all countries therein? The next tidyverse function we will explore to help us is called filter. (See the help for lots of examples using a Star Wars dataset.) Here is how to filter the data so we only have rows for Africa: gap %>% filter(continent == "Africa") %>% head(10) ## country continent year lifeExp pop gdpPercap ## 1 Algeria Africa 1952 43.077 9279525 2449.008 ## 2 Algeria Africa 1957 45.685 10270856 3013.976 ## 3 Algeria Africa 1962 48.303 11000948 2550.817 ## 4 Algeria Africa 1967 51.407 12760499 3246.992 ## 5 Algeria Africa 1972 54.518 14760787 4182.664 ## 6 Algeria Africa 1977 58.014 17152804 4910.417 ## 7 Algeria Africa 1982 61.368 20033753 5745.160 ## 8 Algeria Africa 1987 65.799 23254956 5681.359 ## 9 Algeria Africa 1992 67.744 26298373 5023.217 ## 10 Algeria Africa 1997 69.152 29072015 4797.295 Note the double equals, ==, not to be confused with = which is used to set inputs (also known as arguments). To see how == works, compare: 11 + 3 == 14 ## [1] TRUE And: 11 + 3 == 2 ## [1] FALSE Now I’m going to try piping this filtered data frame directly into ggplot, without saving it. This should work because ggplot’s first argument is the data frame. gap %>% filter(continent == "Africa") %>% ggplot(aes(x = year, y = lifeExp, colour = country)) + geom_point() + geom_line() Well… it did… but the plot is very busy and I’m not sure I could distinguish between all those colours! Let’s try again without the legend to see what’s going on. At this point you may wonder, “How on earth will I be able to remember all these commands?” I will share a trick. Attempt 2: gap %>% filter(continent == "Africa") %>% ggplot(aes(x = year, y = lifeExp, colour = country)) + geom_point() + geom_line() + theme(legend.position = "none") ### 3.10.1 Activity One of the countries’ life expectancies dropped below 25. Can you work out which one it was by using filter? Tip: == was equals. You can use < for less than. 2 < 3 ## [1] TRUE ### 3.10.2 Answer gap %>% filter(lifeExp < 25) ## country continent year lifeExp pop gdpPercap ## 1 Rwanda Africa 1992 23.599 7290203 737.0686 So the answer is Rwanda. ## 3.11 Other handy tools: select, slice, bind, and arrange Often you will have datasets with a huge number of variables and will want to select a few of those to make the tables easier to read. The command for that is select; give it the names of the variables you want. Another useful function is arrange which sorts a data frames by the variable(s) you provide. Here is an example illustrating both. I have also added the operator & for “and.” gap %>% filter(year == 2007 & continent == "Africa") %>% arrange(lifeExp) %>% select(country, lifeExp) ## country lifeExp ## 1 Swaziland 39.613 ## 2 Mozambique 42.082 ## 3 Zambia 42.384 ## 4 Sierra Leone 42.568 ## 5 Lesotho 42.592 ## 6 Angola 42.731 ## 7 Zimbabwe 43.487 ## 8 Central African Republic 44.741 ## 9 Liberia 45.678 ## 10 Rwanda 46.242 ## 11 Guinea-Bissau 46.388 ## 12 Congo, Dem. Rep. 46.462 ## 13 Nigeria 46.859 ## 14 Somalia 48.159 ## 15 Malawi 48.303 ## 16 Cote d'Ivoire 48.328 ## 17 South Africa 49.339 ## 18 Burundi 49.580 ## 19 Cameroon 50.430 ## 20 Chad 50.651 ## 21 Botswana 50.728 ## 22 Uganda 51.542 ## 23 Equatorial Guinea 51.579 ## 24 Burkina Faso 52.295 ## 25 Tanzania 52.517 ## 26 Namibia 52.906 ## 27 Ethiopia 52.947 ## 28 Kenya 54.110 ## 29 Mali 54.467 ## 30 Djibouti 54.791 ## 31 Congo, Rep. 55.322 ## 32 Guinea 56.007 ## 33 Benin 56.728 ## 34 Gabon 56.735 ## 35 Niger 56.867 ## 36 Eritrea 58.040 ## 37 Togo 58.420 ## 38 Sudan 58.556 ## 39 Madagascar 59.443 ## 40 Gambia 59.448 ## 41 Ghana 60.022 ## 42 Senegal 63.062 ## 43 Mauritania 64.164 ## 44 Comoros 65.152 ## 45 Sao Tome and Principe 65.528 ## 46 Morocco 71.164 ## 47 Egypt 71.338 ## 48 Algeria 72.301 ## 49 Mauritius 72.801 ## 50 Tunisia 73.923 ## 51 Libya 73.952 ## 52 Reunion 76.442 This filters gap to data from 2007 and Africa, sorts it by life expectancy, and then selects the country and life expectancy variables. The slice family of functions can be used to zoom into the top or bottom slices of rows, particular rows, or a random sample. Here’s an example. First save the previous chunk results above in africa2007: africa2007 <- gap %>% filter(year == 2007 & continent == "Africa") %>% arrange(lifeExp) %>% select(country, lifeExp) The following R code saves the “head” of the dataset, which has the lowest life expectancies. The n is 3, so three rows are returned. Note the single = here: it’s an parameter setting n to 3 rather than an equality == checking whether n is 3. africa2007min <- africa2007 %>% slice_head(n = 3) africa2007min ## country lifeExp ## 1 Swaziland 39.613 ## 2 Mozambique 42.082 ## 3 Zambia 42.384 Do this again for the tail, i.e., the bottom of the dataset, which has the highest values for life expectancy. africa2007max <- africa2007 %>% slice_tail(n = 3) africa2007max ## country lifeExp ## 1 Tunisia 73.923 ## 2 Libya 73.952 ## 3 Reunion 76.442 We can bind the two data frames together again using bind_rows: top_and_bottom <- bind_rows(africa2007min, africa2007max) top_and_bottom ## country lifeExp ## 1 Swaziland 39.613 ## 2 Mozambique 42.082 ## 3 Zambia 42.384 ## 4 Tunisia 73.923 ## 5 Libya 73.952 ## 6 Reunion 76.442 ## 3.12 Filtering for members of a vector The top_and_bottom data frame has the names of countries with the top and bottom three life expectancies. top_and_bottom$country
## [1] "Swaziland" "Mozambique" "Zambia" "Tunisia" "Libya"
## [6] "Reunion"
Next we are going to filter the data set to only these countries, using the %in% operator which returns TRUE if a value is in the vector you provide and FALSE otherwise.
Here are two examples:
"Libya" %in% top_and_bottom$country ## [1] TRUE "Uganda" %in% top_and_bottom$country
## [1] FALSE
gap %>%
filter(country %in% top_and_bottom$country) %>% ggplot(aes(x = year, y = lifeExp, colour = country)) + geom_line() We can add Rwanda back in by using the c operator (“c” for “combine”). Here’s an example to show how c works: some_numbers <- c(1,2,3) c(some_numbers,4) ## [1] 1 2 3 4 Back to the graph. Below I have also enlarged the size of the lines to make the colours easier to distinguish. gap %>% filter(country %in% c(top_and_bottom$country, "Rwanda")) %>%
ggplot(aes(x = year,
y = lifeExp,
colour = country)) +
geom_line(size = 1) +
labs(x = "Year",
y = "Mean life expectancy (years)",
colour = "Country")
You might now consider a qualitative analysis of these countries (or lookup Wikipedia, for the purposes of a weekly R exercise) to conjecture why there are these differences.
## 3.13 Final challenge
### 3.13.1 Activity
Plot life expectancy against GDP per capita for all countries in the dataset at the most recent time point. Colour the points by continent.
Here’s how I did it.
First, check the variable names:
names(gap)
## [1] "country" "continent" "year" "lifeExp" "pop" "gdpPercap"
So we want lifeExp and gdpPercap.
The most recent year is:
max(gap$year) ## [1] 2007 (You could also find that by looking at the data frame using View.) Now make filter and make the graph in one go: gap %>% filter(year == 2007) %>% ggplot(aes(x = gdpPercap, y = lifeExp, colour = continent)) + geom_point() + labs(y = "Mean life expectancy (years)", x = "GDP per capita (US$, inflation-adjusted)",
colour = "Continent",
title = "Life expectancy and GDP per capita in 2007")
## 3.14 More ideas for visualisations
Check out these references, all available for free online: | 2021-01-27 05:00:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31759515404701233, "perplexity": 3749.625183106112}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704820894.84/warc/CC-MAIN-20210127024104-20210127054104-00300.warc.gz"} |
http://psiepsilon.wikia.com/wiki/T-Duality | ## FANDOM
118 Pages
String Theory
Prior to the First Superstring Revolution
Early History S-Matrix Theory
Regge Trajectory
Bosonic String Theory Worldsheet
String
Bosonic String Theory
String Perturbation Theory
Tachyon Condensation
Supersymmetric Revolution Supersymmetry
RNS Formalism
GS Formalism
BPS
Superstring Revolutions
First Superstring Revolution GSO Projection
Type II String Theory
Type IIB String Theory
Type IIA String Theory
Type I String Theory
Type H String Theory
Type HO String Theory
Type HE String Theory
Second Superstring Revolution T-Duality
D-Brane
S-Duality
Horava-Witten String Theory
M-Theory
Holographic Principle
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
(2,0) Theory
Twistor String Theory
F-Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
Phenomenology String Theory Landscape
Minimal Supersymmetric Standard Model
String Phenomenology
T-Duality is a duality, or equivalence between two String Theoryies. In contrast to S-Duality, T-Duality is a purely stringy concept. T-Duality equates a String Theory compactified around a distance $R$ with another compactified around a distance of $\frac{\alpha'}{R}=\frac{\ell_s^2}{R}$.
## T-Duality for Bosonic StringsEdit this section
Beginning with a simple toy model, Bosonic String Theory, we compactify a spatial dimension, say $x^9$ (9 is randomly chosen as Bosonic String Theory is 26-dimensional, not 10-dimensional), such that: .
$x^9\sim x^9+2\pi R$
The ground Wavefunction is $e^{ip_0^9x^9/\hbar}$. It is clear that this is single-valued only when
$p_0^9=\hbar \frac nR$
There fore, the momenta is quantised by the above equation. Then,
$\alpha_0^\mu=\tilde\alpha_0^\mu= \frac{\ell_s}{\hbar}\frac nR$
$\alpha_0^\mu+\tilde\alpha_0^\mu= \frac{2\ell_s}{\hbar} \frac nR$ =
However, a Closed String can actually wrap as many times around a circle as it wishes, not necessarily once. This "as many times" is called the winding number $w$ Then,
$x^9\sim x^9+2\pi R$
$\alpha_0^\mu-\tilde\alpha_0^\mu= \frac{wR}{\ell_s}$
When $w=0$ (uncompactified), the RHS becomes 0.
If we consider the momentum, it is still $$p=\frac nR$$, but the left- and right- moving momenta are:
$p_-= \hbar\left(\frac nR - \frac1{\ell_s^2} wR \right)$
$p_+= \hbar\left(\frac nR + \frac1{\ell_s^2} wR \right)$
The mass spectrum is also intuitively modified as (which is clear from the relation between the mass and the momenta); :
$$m=\frac{2\pi T\ell_s}{c_0^2} \sqrt{N+\tilde N-a-\tilde a + \ell_s^2 \frac{n^2}{R^2} \frac{1}{\ell_s^2}w^2R^2 }$$
If we talke the limit as $R\to\infty$, $w\to0$ and if we take the limit as $R\to0$, then $n\to0$, and the compactified dimension reappers. This is explained by the following transformations between the winding number and the momentum quantisation number; and between the compactification radii.
$w\leftrightarrow n$
$R\leftrightarrow \frac{ell_s^2}{R}$
These transformations are called T-Duality. This is also equivalent to negating the right-moving mode of oscillation; and therefore, the field $X^\mu$. Considering T-Duality for Open Strings, we see that a normal derivative becomes a tangential derivative, therefore exchanging Dirchilet and Newmann Boundary conditions.
This immediately makes the existence of D-Branes necessary.
## T-Duality for Type II StringsEdit this section
Most of the ideas of the previous section carry over to Type II (Type IIA and Type IIB) Strings; however there is an additional result. In the previous section, we learnt that T-Duality negates the bosonic field $X^\mu$. By the manifest Worldsheet Supersymmetry of RNS String Theory, this also implies that the fermionic field $\psi ^\mu$ is negated.
This immediately implies that the GSO Projection also flips sign:
$\operatorname{T}: \mathcal P^-_\operatorname{GSO}\leftrightarrow \mathcal P^+_\operatorname{GSO}$
As the Type IIA String Theory and Type IIB String Theory differ only by the GSO Projection, this means that T-Duality exchanges Type IIA String Theory and Type IIB String Theory.
## T-Duality for Type H StringsEdit this section
Type H String Theory, or Heterotic String Theory, is also affected by T-Duality. The weight lattice of $\frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}_2}$ (the gauge group of the Type HO String Theory is given by $\Gamma^{16}$ while the weight lattice of $E(8)\times E(8)$ is given by $\Gamma^8\oplus\Gamma^8$. Since $\Gamma^{8}\oplus \Gamma^8\oplus\Gamma^{1,1}= \Gamma^{16} \oplus\Gamma^{1,1}$, it follows that Type HO String Theory is T-Dual to Type HE String Theory. | 2018-08-20 03:23:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7929980158805847, "perplexity": 2036.663455750205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215523.56/warc/CC-MAIN-20180820023141-20180820043141-00510.warc.gz"} |
https://talkstats.com/threads/share-your-functions-code.18603/page-11#post-97845 | # Share your functions & code
#### Lazar
##### Phineas Packard
Ok this is weird. It just works for me. I am going to claim that I knew about Dason's seeInternet2 thing but honestly I am not sure why it worked for me.
P.S. Want to try the log log scale now. Hurrying away to try this.
#### trinker
##### ggplot2orBust
Nicely done Dason and nice post Lazar. Brian Diggs is definitely an R guru.
#### trinker
##### ggplot2orBust
@jimmy that's actually a really good way to:
a) become a little dorkier
When I made an R hangman game I learned a ton though I haven't really played it since but the learning remains.
#### trinker
##### ggplot2orBust
May I suggest R in Action. I really like this a lot as a new to intermediate book in R. The same author (Kabacoff; great guy) also does the Quick R web site (LINK). We also have a thread on here on R =resources that may be helpful (LINK).
#### Dason
Ever wished R was 0 indexed instead of 1 indexed?
Code:
"[" <- function(x, y){base::[(x, y+1)}
"[<-" <- function(x, i, value){base::[<-(x, i+1, value)}
Which results in
Code:
> x <- rnorm(5)
> x
[1] 1.0696787 0.9464659 -0.3197787 -0.4608032 1.0657153
> x[0]
[1] 1.069679
> x[0] <- 1
> x
[1] 1.0000000 0.9464659 -0.3197787 -0.4608032 1.0657153
Note: For the love of God don't ever do this. Plus it breaks really easily. For instance you can't use double indexing on matrices or dataframes - I haven'te quite worked out how to write over the double indexing function...
#### derksheng
##### New Member
Function for efficiently creating headings when you're working on a big project.
Code:
h <- function(char,type=c(1,2,3)){
if(type==1){
writeClipboard(paste("#%-------------#",toupper(char),"#-------------%#"))
}
if(type==2){
writeClipboard(paste("#------#",char,"#------#"))
}
if(type==3){
writeClipboard(paste("#---",char,"---#"))
}
}
E.G.
1) Type h("main heading",1) into R
2) Press CTRL+V
Last edited:
#### derksheng
##### New Member
If you use the above for your headings and subheadings in a project, you can use this function to automatically create an index to go into the preamble of your code (to assist co-authors in browsing code, and helping you navigate massive projects). Filename is the name of your code file, e.g. "quantileproject.r".
Code:
indexcreation <- function(filename){
con <- file(filename,"r",blocking=FALSE)
loc1 <- grep("#%-------------#",file,fixed=TRUE)
loc2 <- grep("#------#",file,fixed=TRUE)
loc3 <- grep("#--- ",file,fixed=TRUE)
names1 <- file[loc1]
names2 <- file[loc2]
names3 <- file[loc3]
names1 <- sapply(seq(length(loc1)),function(i) { strsplit(names1," ")[[i]][2] })
names2 <- sapply(seq(length(loc2)),function(i) { strsplit(names2," ")[[i]][2] })
names3 <- sapply(seq(length(loc3)),function(i) { strsplit(names3," ")[[i]][2] })
names <- list(names1,names2,names3)
linenumber <- list(loc1,loc2,loc3)
numberdots <- unlist(sapply(1:length(linenumber),function(i) { rep(length(linenumber)+1-i,length(linenumber[[i]]))*3 }))
numberdots <- sapply(seq(unlist(linenumber)),function(i) { numberdots[i] - length(strsplit(as.character(unlist(linenumber)[i]),"")[[1]]) })
dots <- sapply(1:length(numberdots),function(i) { paste(rep(".",numberdots[i]),collapse="") })
index <- sapply(seq(length(unlist(linenumber))),function(i) { paste(unlist(linenumber)[i],dots[i],unlist(names)[i],sep="") })
locations <- unlist(linenumber)
locations <- match(sort(locations),locations)
index <- unlist(index)[locations]
writeClipboard(matrix(index))
}
Just run the function then CTRL+V into your txt file's preamble.
#### trinker
##### ggplot2orBust
When I want to provide a reproducible example I like to provide the data.frame I'm using in structure format as it minimizes the typing of those helping. It can be annoying to use dput to do this in that you have to cut and paste from the console. Here's a dput2 that copies the text to the users clipboard (if mac or windows user) and places the dataframe name and arrow operator in front as well as indenting rows 2:n.
Code:
dput2 <- function(x, indents = 4) {
z <- capture.output(dput(x))
y <- as.character(substitute(x))
y <- y[length(y)]
z[1] <- paste(y, "<-", z[1])
z[-1] <- paste0(paste(rep(" ", indents), collapse=""), z[-1])
zz <- as.matrix(as.data.frame(z))
dimnames(zz) <- list(c(rep("", nrow(zz))), c(""))
if (Sys.info()["sysname"] == "Windows") {
writeClipboard(z, format = 1)
}
if (Sys.info()["sysname"] == "Darwin") {
j <- pipe("pbcopy", "w")
writeLines(z, con = j)
close(j)
}
noquote(zz)
}
#try it:
dput2(mtcars)
dput2(head(mtcars))
#### Dason
I think you meant to put dput2 and not repex.
And this isn't working for me!
Code:
dput2(mtcars + 2)
Of course I'm just being facetious but I had to find a way to break your function
#### trinker
##### ggplot2orBust
Yeah dason it was supposed to be dput2, originally it was repex for reproducible example but I couldn't remember the name for the function. And yes I knew it was fairly simple to break as I used:
Code:
dput2(head(head(mtcars, 20)))
but generally I want to dput a dataframe or the head of a dataframe. Anything else it works but you'll have to rename the object it's being assigned to. I don't really have any work around for all the different scenerios that could break it but if anyone has an easy solution I'm all eyes.
#### Dason
I doubt there is a direct solution. Consider
Code:
val <- 10
dput2(head(mtcars, val))
How should it know which variable you want the name to be? You aren't dputting either object directly so there is no clear way to tell.
#### trinker
##### ggplot2orBust
an alternative would be to just use dat as the object we assign to since you're likely making a reproducible example anyway. What are your thoughts on that?
#### Dason
No I think the way you're doing it is fine. In 99% of the cases you'll correctly identify the name.
#### trinker
##### ggplot2orBust
Thought I'd share this one cause it's simple and has the possibility for lots of applications. outer is pretty nice but it doesn't take vectors. I once asked at SO how to make outer work with vectors and got two responses (LINK). I recieved two great responses but have stuck with the Vectorize solution though it's slower because it is more readable to me. Anyway this v.outer is a vectorized version of outer that you can supply a function to and that functions arguments. Acts on a matrix or data.frame.
Code:
v.outer <-
function(x, FUN, digits = 3, ...){
FUN <- match.fun(FUN)
if (is.matrix(x)) {
x <- as.data.frame(x)
}
if (is.list(x) & !is.data.frame(x)){
if (is.null(names(x))) {
names(x) <- paste0("X", seq_along(x))
}
nms <- names(x)
} else {
nms <- colnames(x)
}
z <- outer(
nms,
nms,
Vectorize(function(i,j) FUN(unlist(x[[i]]), unlist(x[[j]]), ...))
)
dimnames(z) <- list(nms, nms)
if (is.numeric(z)) {
z <- round(z, digits = digits)
}
z
}
v.outer(mtcars, cor)
v.outer(mtcars, cor, method="kendall")
#### trinker
##### ggplot2orBust
ps I realize cor is a lttle silly in that it already gives you this output but it was the first function that came to mind that takes an x and y vector and so I used it. here's an example with a function for pooled sd (again this is a bit silly in that you'd do pooled for all and the function for pooled may be incorrect) but it can help people understand more:
Code:
pooled.sd <- function(x, y) {
n1 <- length(x)
n2 <- length(y)
s1 <- sd(x)
s2 <- sd(y)
sqrt(((n1-1)*s1 + (n2-1)*s2)/((n1-1) + (n2-1)))
}
v.outer(mtcars, pooled.sd)
PS can anyone think of functions that take an x and y vectors and return a single value?
#### Lazar
##### Phineas Packard
Code:
> euc.dist <- function(x,y) sqrt(sum((x - y) ^ 2))
> v.outer(mtcars, euc.dist)
#### Nathan G
##### New Member
I'm new to R, but I'm learning fast... With \Huge{ thanks! } to everyone adding to this thread.. Just finished going through all the code.. Simply amazing stuff. I couldn't get everything to work: The unwrap() function was especially frustrating. I tried for a couple hours to no avail; I did get wrap() to work fairly quickly with some substitutions to make it functional on my mac. I realize now the functions I was writing were very raw. But here they are:
1) a function that renames the column names to latex math format for my lab reports - yes I'm an undergrad
2) a search function that returns the first occurrences of unique values in a data.frame column
Code:
#################################################
# 1)
# puts $dollar signs in front and behind all column names col_{sub} ->$col_{sub}$# ################################################### amscols <- function(x){ colnames(x) <- paste( "$" , colnames(x) , "\$" , sep = "" ) x }
#################################################
# 2)
# Returns a data.frame of the first occurances of all unique values of the "search" column
#
###############################################
getfirsts <- function(data, searchcol){
# Receives a data.frame and a "search" column
# Returns a data.frame of the first occurances of all unique values of the "search" column
rows <- as.data.frame(match(unique(data[[searchcol]]), data[[searchcol]]))
firsts = data[rows[[1]],]
return(firsts)
} | 2022-08-11 00:20:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3527389168739319, "perplexity": 6481.707238522288}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571222.74/warc/CC-MAIN-20220810222056-20220811012056-00274.warc.gz"} |
https://www.easy2boot.com/discussion/?tid=200000014&mid=200001907 | ## Topic: Easy2Boot
Date 13/06/2020
By kike
### Subject Re: Re: iso windows 7 x64 bits pro efi
Thank you very much for your answer, I have tried and it has not worked, I have copied the file and mounted with ultraiso ..., and it keeps giving me the same errors
I have even added boot.wim and install.wim of that iso in another iso that yes is efi booteable (the one that fails me is udf booteable), to see if it succeeded but when booting it gives me error file: \ winedows \ system32 \ boot \ winload.efi
status: 0xc000000f
missing or corrupt
The problem is that I have an iso w7 that does boot with efi, but when it repairs it tells me that it is not the installed version of windows
And the iso that if you let me repair let me repair in legacy, but in efi it gives me those errors ...
Can you convert a bootable iso udf to bootable efi ??
Or some other possibility?
Where would be the tools to create an imgptn ???
Thank you so much for your help
Back | 2020-07-13 22:28:26 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8184277415275574, "perplexity": 5088.37244706154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657146845.98/warc/CC-MAIN-20200713194203-20200713224203-00052.warc.gz"} |
https://controls.ame.nd.edu/mediawiki/index.php/P._559,_line_-4 | # P. 559, line -4
"the frequency of motion is ${\displaystyle (\alpha n)/(2L)=\left(n{\sqrt {\tau }}\right)/\left(2L{\sqrt {\rho }}\right)}$ Hz, where ${\displaystyle \tau }$ is the tension in" | 2022-06-28 05:27:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5888485908508301, "perplexity": 1900.2180219522934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103355949.26/warc/CC-MAIN-20220628050721-20220628080721-00549.warc.gz"} |
https://rforanalytics.com/11.1-value-at-risk.html | ## 11.1 Value at Risk
• Value at Risk (VaR) is the most widely used market risk measure in financial risk management and it is also used by practitioners such as portfolio managers to account for future market risk. VaR can be defined as loss in market value of an asset over a given time period that is exceeded with a probability $$\theta$$. For a time series of returns $$r_{t}$$ , $$VaR_{t}$$ would be such that
$$$P[r_{t}<-VaR_{t}[I_{t-1}]=\theta \tag{11.1}$$$
where $$I_{t-1}$$ represents the information set at time t-1.
• Despite the appealing simplicity of VaR in its offering of a simple summary of the downside risk of an asset portfolio, there is no single way to calculate it (see ) for an overview on VaR methods in finance).
1% VaR
• Convert the prices to returns
library(ggplot2)
# calculate normal density for the returns
# prices to returns
bhp2 = BHP\$BHP.AX.Close
# covert to returns
bhp_ret = dailyReturn(bhp2, type = "log")
den1_r = coredata(bhp_ret)
den1_bhp = dnorm(x = den1_r, mean = mean(den1_r), sd = sd(den1_r))
data_rd = data.frame(den1_r, den1_bhp)
# change column names
colnames(data_rd) = c("x", "y")
# normal quantile
var1 = quantile(den1_r, 0.01)
p3 = ggplot(data_rd, aes(x = x, y = y)) + geom_line(size = 2) + geom_vline(xintercept = var1,
lty = 2, col = "red", size = 2) + theme_bw() + labs(title = "Normal Distribution and 1% (Empirical) VaR")
p3
• In distribution terms, for a distribution F, VaR can be defined as its p-th quantile given by
$$$VaR_{p}(V_{p})=F^{-1}(1-p) \tag{11.2}$$$
where $$F^{-1}$$ is the inverse of the distribution function also called as the quantile function. Hence VaR is easy to calculate once a distribution for the return series can be defined.
• VaR is the q-th quantile of the distribution of over a time horizon t, which is a well accepted measure of risk in financial management.
### References
Manganelli, Simone, and Robert F Engle. 2001. “Value at Risk Models in Finance.” | 2022-08-16 00:23:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.6136389374732971, "perplexity": 2441.5483571726745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572215.27/warc/CC-MAIN-20220815235954-20220816025954-00441.warc.gz"} |
http://chelseacourt.net/thread-53754-1-1.html | # 新東爵人報到, 各位大大好!
剛收東爵樓, 由於我從外區搬入, 假如我有些蠢問題, 煩請各位大大幫忙解答. 謝謝. 請問由爵悅庭去灣仔上班, 是否最方便是巴士930? 因為早上我知有兩班930A, 7:55開出, 預計多久才會到達灣仔?
歡迎~~~~~ 交通可參考呢個 POST~~ http://www.chelseacourt.net/view ... mp;extra=#pid628426
Depends on which one u take. There are two buses in the morning, of u take the first one, u can arrive WC at around 8:40, but if u take the second one, normally u can arrive at around 8:55. Largely depends I. The traffic condition! P.S. I take this bus every morning, and I'm working at Revenue Tower! | 2018-12-10 09:10:02 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9420453310012817, "perplexity": 14745.905174877425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823320.11/warc/CC-MAIN-20181210080704-20181210102204-00437.warc.gz"} |
https://dsp.stackexchange.com/questions/33567/is-there-any-audio-dsp-without-biquad-constraint | # Is there any audio DSP without biquad constraint?
I found a codec with audio DSP, ADAU1772. It supports 32 biquads, that is $32\cdot5=160$ parameters.
I need to calculate filter coefficients on-line. The result is in transfer function form, i.e.,
$$[a_1, a_2, \cdots, a_{m-1}, b_0, b_1, \cdots, b_{n-1}]$$
If I want to use an audio DSP, I must transform the result into SOS matrix.
In MATLAB, here is a function tf2sos does the job. It needs to solve high-order equations to find poles and zeros. Maybe it's difficult for MCUs.
• Assume number of coefficients $m=n=32$, is there any method I can write all the 64 coefficients directly into a chip and make it work?
• Or is there any audio DSP chips without biquad-structure constraint? | 2020-02-21 18:22:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3501707911491394, "perplexity": 2300.4041622678164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145534.11/warc/CC-MAIN-20200221172509-20200221202509-00288.warc.gz"} |
https://groupprops.subwiki.org/wiki/Supercharacter_theories | # Supercharacter theory
(Redirected from Supercharacter theories)
## Definition
Let $G$ be a finite group and $F$ be a splitting field for $G$. A supercharacter theory of $G$ over $F$ is a partition $\kappa$ of the conjugacy classes of $G$ and a partition $\kappa^\vee$ of the irreducible characters of $G$ over a splitting field such that:
1. The identity element is its own block in $\kappa$
2. $|\kappa| = |\kappa^\vee|$
3. For each block $K \in \kappa^\vee$, there exists a corresponding character $\chi^K$ which is a positive integer combination of the characters in $K$ such that $\chi^K$ is constant on the blocks of $\kappa$.
### Terminology
Each block of conjugacy classes is termed a superconjugacy class, and the term may also be used for the union of all the conjugacy classes (i.e., the set of group elements).
Any positive integer combination that works for (3) is termed a supercharacter for this supercharacter theory. All supercharacters for a given block are scalar multiples of each other. We may distinguish the supercharacter that uses the smallest positive integer combination from the others.
## Examples
### Extreme cases
For every finite group, there are two extreme supercharacter theories. Further, unless the group is trivial or cyclic of order two, these are distinct from each other:
• The supercharacter theory where all the blocks are singleton subsets, i.e., each conjugacy class is its own block in $\kappa$ and each irreducible representation is its own block in $\kappa^\vee$.
• The supercharacter theory where the identity element is one block and all other conjugacy classes are the other block. On the representation side, the trivial representation forms one block and all other representations form another block.
## References
• Supercharacter formulas for pattern groups by Persi Diaconis and Nathaniel Thiem | 2020-02-17 13:10:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 18, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8415204882621765, "perplexity": 275.9474696103003}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875142323.84/warc/CC-MAIN-20200217115308-20200217145308-00459.warc.gz"} |
http://strategywiki.org/wiki/Mount%26Blade/Skills | Skills are divided into three categories: personal, party, and leadership skills. All skills range from level 0-10 (although for party skills, the effective level can exceed this with bonuses). Skills can be improved by spending skill points. Some skill points are given during character creation, one skill point is earned per level up, and also one extra skill point is rewarded per attribute point spent on Intelligence. Each skill has a base attribute that limits the skill to 1/3 of the attribute's value (rounded down). For example, with a STR of 14 Ironflesh may not exceed level 4. Starting at STR 15, Ironflesh may be increased to 5. These attribute limits can be surpassed using bonuses from books, but cannot exceed level 10.
## Personal Skills
These are skills you or a companion can possess. They don't affect the party.
### Ironflesh
• Base Attribute: Strength
• Effect: +2 HP per level
At early levels it isn't very important, but as you fight tougher enemies every bit of protection helps. This skill has a very large effect on determining whether an NPC survives autoresolved combat.
### Power Strike
• Base Attribute: Strength
• Effect: +8% melee damage per level
### Power Throw
• Base Attribute: Strength
• Effect: +10% thrown weapon damage per level
More powerful throwing weapons cannot be used until this skill is sufficiently high.
### Power Draw
• Base Attribute: Strength
• Effect: +14% bow damage for each point (up to 4 plus Power Draw requirement of the bow). Also increases how long the character can aim for before losing accuracy.
This is required for the more powerful bows, and is overall a valuable skill for making archery less difficult. You will benefit from up to four points in this beyond the power draw requirement of the bow (if any).
E.g. if you have a bow that requires Power Draw 2, you may have up to 6 points in this skill in order to add damage to your arrow. If you add a seventh point to this skill, you will not receive an additional +14% damage bonus to your arrow above the maximum 6 points:
${\displaystyle MAXPowerDrawBonus=BowPowerDrawReq.+4}$
### Weapon Master
• Base Attribute: Agility
• Effect: Each point increases by 40 the limit at which weapon proficiencies can be raised with weapon proficiency points. Also increases the rate at which proficiencies are improved by use in combat.
This skill allows you to continue using weapon proficiency points after you have passed the normal cap.
Weapon Master level maximum weapon proficiency 0 1 2 3 4 5 6 7 8 9 10 60 100 140 180 220 260 300 340 380 420 460
This cap is only for the purpose of spending additional proficiency points; using weapons in combat will continue to raise your skill regardless of what the proficiency cap is set at, but it will do so faster at higher levels of Weapon Master.
### Shield
• Base Attribute: Agility
• Effect: Reduces damage to shields by 8% per level and improves shield speed and coverage.
Due to a bug, raising this skill also raises the Shield skill of enemies, and is therefore a poor choice for characters who make use of ranged weapons. This bug is fixed in Warband.
### Athletics
If you need help with wiki markup, see the wiki markup page. If you want to try out wiki markup without damaging a page, why not use the sandbox?
Cleanup required: Don't need the analysis here, we need the guide
• Base Attribute: Agility
• Effect: Increases running speed.
This isn't very obvious early on, but with a few levels it will start to really take effect. This skill is made less important by having a horse, but it's still a useful backup if you lose your horse or if you want to fight on foot. It is also useful during sieges and tournaments, where a horse might not be available. Note that the combined weight of your armor and equipment will reduce the speed bonus your Athletics skill grants.
#### Analysis
The goal of the following analysis is to provide an accurate account of the effects of the athletic skill.
To acquire this data, a contributor performed a series of timing measurements from a specific point, A, to a different specific point, B, in the game world. Different encumbrances and athletic skill points were used. Therefore, running speed is not known in units. Speed is only known relative to another based on Athletic skill and encumbrance differences. This contributor did not test for speed increase of more than base running speed. All calculations are an estimate. Agility attribute during the measurements was 15 or 18.
Assumptions
1. Running speed increase per Athletic skill level is linear.
2. Running speed change (+/-) per unit encumbrance is linear, and a decrease in running speed from encumbrance occurs at greater than 0. Base running speed = running speed at 0 encumbrance and 0 Athletic skill.
Conclusions
1. 50 units of encumbrance decreases running speed by about 22%, or .44% speed reduction per 1 unit.
2. 1 Athletic skill point is equivalent to about 4.16 units less encumbrance, or 1.83% increase in running speed.
Examples
1. Encumbrance 50 with 0 Athletic skill points reduces base running speed by 22%.
2. Encumbrance 50 with 2 Athletic skill points reduces base running speed by 18.3%.
3. Encumbrance 50 with 4 Athletic skill points reduces base running speed by ??% (estimated 14.64%).
4. Encumbrance 35 with 0 Athletic skill points reduces base running speed by 15.4%.
5. Encumbrance 35 with 4 Athletic skill points reduces base running speed by 8.1%.
### Riding
• Base Attribute: Agility
• Effect: Increases riding speed and maneuver, allows riding of more difficult horses.
### Horse Archery
• Base Attribute: Agility
• Effect: Reduces penalties for using ranged weapons on a moving horse by 10% per level.
Reduces the aiming reticule spread while mounted, allowing for much greater accuracy. All ranged weapons: bows, crossbows, throwing weapons and firearms are affected by this skill.
Note that the negative effects of being mounted and using ranged weapons are only applied while you are actually moving, thus this skill has no effect if you are mounted but stationary.
### Trainer
• Base Attribute: Intelligence
• Effect: Every day at midnight lower-level party members gain experience
Experience points are not given to party members that are fully upgraded. If multiple companions have this skill they will each help train party members.
Experience gains per level are as follows:
Trainer level EXP to each party member per day 1 2 3 4 5 6 7 8 9 10 4 10 16 23 30 38 46 55 65 80
This skill also helps you train peasants against bandits faster.
The total experience points given to a soldier is the amount from the training skill multiplied by the number of levels the player or companions are over the soldier to be trained.
For instance, if you are level 30 and have 1 point in training and are training a level 1 recruit you will give 116 experience points to the recruit.
That's 30-1= 29. Training level of 1 gives 4 points of experience.
29 X 4 = 116.
Stack this with companions and higher skill levels and you will be able to train even the highest tiered troops relatively quickly.
## Party Skills
Points Bonus
0-1 pts (+0)
2-4 (+1)
5-7 (+2)
8-9 (+3)
10+ (+4)
These are skills that each character individually possesses, but they are applied to the party as a whole. If the party leader (you) has these skills, a bonus to the skill is applied as per the table to the right. The leader's bonus is applied to the party even if a companion has a higher skill level than you. An additional (+1) can be earned from certain Books. If characters are listed as Wounded their Party skills will be disabled until they regain some health.
### Looting
• Base Attribute: Agility
Increases the amount of loot obtained by 10% per skill level.
Looting also increases the quality of the loot. For example, at level 1 looting you will find mostly rusty, broken and cracked prefixes, negative ones. If your looting is 10, you will find more items with positive prefixes such as balanced, reinforced, or tempered. This skill works in village looting, and in any other battle or siege. It also increases the number of cattle obtained when stealing them from a village.
### Tracking
• Base Attribute: Intelligence
• Effect: Makes onscreen tracks appear and gradually become more informative. Also increases the distance at which tracks can be seen.
This skill makes tracks appear on the world map, gradually becoming more accurate and providing additional information. At level 1 the tracks indicate party movement, but little else. As levels progress the accuracy of party size predictions and hours the tracks remain visible for increases. Tracks will also gain colors, allowing you to easily follow a particular path regardless of what other paths it crosses.
### Tactics
• Base Attribute: Intelligence
• Effect: Every 2 points to this raises your battle advantage by 1.
Before each battle your Tactics score is compared to that of the enemy leader, and this determines how many men each side has at the beginning of the battle.
Additionally, if you choose to send your men in while you stay back or are knocked out during battle this skill will have an effect on the aftermath losses. This skill also reduces how many men you need to leave behind if you retreat.
### Path-finding
• Base Attribute: Intelligence
• Effect: Increases party map speed by 3% per level.
### Spotting
• Base Attribute: Intelligence
• Effect: Increases party sight range by 10% per level.
### Wound Treatment
• Base Attribute: Intelligence
• Effect: Increases party healing speed 20% per level. Also applies to the healing of crippled horses in the inventory.
### Surgery
• Base Attribute: Intelligence
• Effect: Each point adds 4% to the chance that struck down troops will be knocked out rather than killed. This is added to a base chance of 25%.
This skill is quite useful for giving your troops the best chance you can to reach higher levels and become effective fighters. It also lessens the annoyance factor of spending top dollar on some new, high tier troops only to have a significant fraction of them be killed in their first battle.
### First Aid
• Base Attribute: Intelligence
• Effect: Heroes regain 5% per level of hit-points lost during a particular battle. This is added to a base rate of 10%.
This skill takes effect both between battle waves and after combat is over. This only heals up to the level of health before you entered battle, so if you go into battle with low health you can only heal back up to that level and no further.
### Engineer
This skill also affects the cost of building village improvements.
• Base Attribute: Charisma
• Effect: Reduces trading penalty by 5%
The Trade skill reduces the cost penalty applied to buying and selling. See Trade for more details. Trade also reduces the time it takes to collect taxes for tax collection quests.
These skills affect the party, but only if the person with these skills is the party leader.
### Inventory Management
• Base Attribute: Intelligence
• Effect: Increases inventory size by 6 squares
Very important when pillaging villages or attacking caravans.
### Persuasion
• Base Attribute: Intelligence
• Effect: Helps you make other people accept your point of view. There is a random factor involved in determining how successful you are.
Useful for persuading Lords to pay debts or support peace. Also used for persuading lords to rebel and reduce chances that recruited prisoners will run. Perhaps most importantly, it can be used to keep your companions from leaving your group due to low morale.
### Prisoner Management
• Base Attribute: Charisma
• Effect: Increases prisoner capacity by 5
Needed if you want to recruit or sell prisoners. Also needed for the Bring Prisoners quest. This skill has no effect for NPC lords, who have no limit to the number of prisoners they can take. | 2016-06-29 18:08:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33059653639793396, "perplexity": 4167.486417571463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397795.31/warc/CC-MAIN-20160624154957-00043-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1993882/intuition-behind-unit-hypersphere-n-ball-volume | # Intuition behind unit hypersphere/n-ball volume [duplicate]
Obviously the volume of a unit hypercube is always $1$.
I know that the volume of a unit hypersphere is given by: $$\dfrac{\pi^\frac{d}{2}}{\Gamma\Big(\dfrac{d}{2} + 1\Big)}$$ which rises sharply until about $d=5$, then falls off a cliff.
Why should it be this way? I can't get my head around why volume should increase with dimension only to a point.
• Tag edits or suggestions very much appreciated. – OJFord Nov 1 '16 at 1:00
• It is more meaningful to look at the volume of the hypersphere inscribed in the unit hypercube. – dxiv Nov 1 '16 at 1:24
This question has a thorough answer here
The ultimate reason is, of course, that the typical coordinate of a point in the unit ball is of size $$\frac{1}{\sqrt{n}}\ll 1$$. This can be turned into a simple geometric argument (as suggested by fedja) using the fact that an $$n$$-element set has $$2^n$$ subsets:
At least $$n/2$$ of the coordinates of a point in the unit ball are at most $$\sqrt{\frac{2}{n}}$$ in absolute value, and the rest are at most $$1$$ in absolute value. Thus, the unit ball can be covered by at most $$2^n$$ bricks (right-angled parallelepipeds) of volume $$\left(2\sqrt{\frac{2}{n}}\right)^{n/2},$$ each corresponding to a subset for the small coordinates. Hence, the volume of the unit ball is at most $$2^n \cdot \left(2\sqrt{\frac{2}{n}}\right)^{n/2} = \left(\frac{128}{n}\right)^{n/4}\rightarrow0.$$ In fact, the argument shows that the volume of the unit ball decreases faster than any exponential, so the volume of the ball of any fixed radius also goes to $$0$$.
Another one says,
The reason is because the length of the diagonal cube goes to infinity.
The cube in some sense does exactly what we expect. If it's side lengths are $$1$$, it will have the same volume in any dimension. So lets take a cube centered at the origin with side lengths $$r$$. Then what is the smallest sphere which contains this cube? It would need to have radius $$r\sqrt{d}$$, so that radius of the sphere required goes to infinity.
Perhaps this gives some intuition.
Hope this helps.
• Thanks for that, the second quote in particular was exactly what I needed. :) – OJFord Nov 1 '16 at 15:25 | 2021-01-26 09:57:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8636548519134521, "perplexity": 210.10242593959384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704799711.94/warc/CC-MAIN-20210126073722-20210126103722-00193.warc.gz"} |
https://tex.stackexchange.com/questions/161354/how-to-make-subsections-not-appear-in-the-text | # How to make subsections not appear in the text? [duplicate]
I have the following preamble and I want the subsection not to be visible in the text. Instead of Chapter 1 "Blah Blah" I only want "Blah Blah to be visible. With this code I did it bu the subsections are still visible in the text. e.g. 2.5 "blah blah". How can I also remove 2.5 ??
\documentclass[a4paper,12pt]{book}
\usepackage{titlesec}
\titleformat{\chapter}[display]{\normalfont\huge\bfseries}{}{20pt}{\Huge}
\let\cleardoublepage\clearpage
\usepackage[english,greek]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{blindtext}
\usepackage[pdftex]{graphicx}
\usepackage[skip=2pt, font=small, labelformat = empty]{caption}
\newcommand{\gr}{\selectlanguage{greek}}
\newcommand{\en}{\selectlanguage{english}}
\usepackage{wrapfig}
\usepackage[paperwidth=17cm, paperheight=24cm]{geometry}
\usepackage[onehalfspacing]{setspace}
\usepackage{textcomp}
\useshorthands{;}
\defineshorthand{;}{?}
\usepackage{fancyhdr}
\fancyhf{}
\fancyfoot[LE,RO]{\thepage}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{ \markboth{#1}{} }
\renewcommand{\sectionmark}[1]{ \markright{#1}{} }
• Put \renewcommand\thesubsection{} in your preamble. Feb 19 '14 at 22:08
No need to use titlesec for this. The easiest way to get all sectional units unnumbered is to set secnumdepth to -2 (or to -1 if parts should be numbered):
\documentclass{book}
\setcounter{secnumdepth}{-2}
\begin{document}
\chapter{Test chapter}
\section{Test section}
\subsection{Test subsection}
\subsubsection{Test subsubsection}
\end{document}
• Niiiice...!!! It worked perfectly..!! Thanks a lot..!! :-) Feb 19 '14 at 22:19
• @Stefanos You're welcome! Don't forget that you can accept answers that best solved your problems by clicking the checkmark to their left; in case of doubt, please see How do you accept an answer?. Feb 19 '14 at 22:23
• One more thing ...!! With the same preamble, each section is defined to start at almost the 2/3 of the page, meaning there is a small space and then goes the "Section 1: blah blah". Can I start each section at the beginning of the page like the other pages? Feb 19 '14 at 22:24
• @Stefanos As a general rule, one question per post. Please consider opening a new question with this new requirement. Feb 19 '14 at 22:25 | 2022-01-26 12:04:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7670848965644836, "perplexity": 2412.128225673823}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304947.93/warc/CC-MAIN-20220126101419-20220126131419-00507.warc.gz"} |