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https://math.stackexchange.com/questions/1261863/proof-that-mappings-k-and-l-are-primitive-recursive
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# Proof that mappings $K$ and $L$ are primitive recursive.
Let $J$ be the function: \begin{equation*} J(m,n)= \begin{cases} n^2+m \text { if } m\le n \\ m^2 + m + (m-n) \text { if } m > n \\ \end{cases} \end{equation*}
Let $K, L$ such that
$K(k)$ is the unique $n \in \mathbb{N}$ for which there is some $m\in \mathbb{N}$ such that $J(n,m)=k$
$L(k)$ is the unique $m \in \mathbb{N}$ for which there is some $n \in \mathbb{N}$ such that $J(n,m)=k$
Prove that $K$ and $L$ are primitive recursive.
I know and have proved that $J$ is primitive recursive, so I can work with that. I also know that I have to work with the minimisation, to show that it can be bounded by a primitive recursive function. Now I start to get into trouble. I think the minimisation can be written as $K(k)= \mu n (\exists x < k + 1)J(n,x)=k$, but I don't really know where to go from here. I would very much appreciate any input you could give me, I have some trouble really finding the functions that minimise the things.
## 1 Answer
First consider the function $L^\prime(y,k)=\mu x<k+1[J(x,y)-k=0]$, which is primitive recursive since the primitive recursive functions are closed under substitution and bounded minimization.
Then we get that $L(k)=\mu y<k+1[J(L^\prime(y,k),y)-k=0]$. Again, since the primitive recursive functions are closed under substitution and bounded minimization, then $L$ is primitive recursive.
You can now apply the same idea to see that $K$ is also primitive recursive.
• Thank you, this helps me see it much more clearly! – Sara May 6 '15 at 11:06
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2020-04-10 10:29:47
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https://socratic.org/questions/what-is-the-total-number-of-ions-present-in-the-formula-na2so4
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# What is the total number of ions present in the formula Na2SO4?
## A. 2 B. 3 C. 4 D. 6 The answer given is B. 3, why?
Feb 17, 2016
The answer should be 3 ions.
#### Explanation:
After looking again it seems to me that the number of ions in one molecule are actually just Na, S and O, which means 3 ions.
$\textcolor{red}{\text{What you mentioned Na, S and O are three elements in the}}$ $\textcolor{red}{\text{given compound.}}$
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.
Given compound is ${\text{Na"_2"SO}}_{4}$
If one breaks up the given compound in its constituent ions one obtains
$2 {\text{ cations of sodium, Na}}^{+}$ and $1 {\text{ anion as sulfate, SO}}_{4}^{2 -}$
$\therefore$Total 3 ions.
Feb 17, 2016
$N {a}_{2} S {O}_{4} \to 2 N {a}^{+} + S {O}_{4}^{2} -$
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2019-09-16 00:37:25
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https://www.nature.com/articles/nmat4953?foxtrotcallback=true&error=cookies_not_supported&code=8cfcc7fc-fb90-492d-bd66-3bfc658e572f
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# A magnetic topological semimetal Sr1−yMn1−zSb2 (y, z < 0.1)
## Abstract
Weyl (WSMs) evolve from Dirac semimetals in the presence of broken time-reversal symmetry (TRS) or space-inversion symmetry. The WSM phases in TaAs-class materials and photonic crystals are due to the loss of space-inversion symmetry. For TRS-breaking WSMs, despite numerous theoretical and experimental efforts, few examples have been reported. In this Article, we report a new type of magnetic semimetal Sr1−yMn1−zSb2 (y, z < 0.1) with nearly massless relativistic fermion behaviour (m = 0.04 − 0.05m0, where m0 is the free-electron mass). This material exhibits a ferromagnetic order for 304 K < T < 565 K, but a canted antiferromagnetic order with a ferromagnetic component for T < 304 K. The combination of relativistic fermion behaviour and ferromagnetism in Sr1−yMn1−zSb2 offers a rare opportunity to investigate the interplay between relativistic fermions and spontaneous TRS breaking.
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## Acknowledgements
The authors thank C. Wu at UCSD for helpful discussions. The work at Tulane University was supported by the NSF under Grant DMR-1205469 (support for personnel and materials) and Louisiana Board of Regents under grant LEQSF(2014-15)-ENH-TR-24 (support for equipment purchase). The neutron scattering work used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory, and is supported by the US Department of Energy under EPSCoR Grant No. DE-SC0012432 with additional support from the Louisiana Board of Regents. The work at UNO is supported by the NSF under the NSF EPSCoR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents. The work at FSU and at the National High Magnetic Field Laboratory is supported by the NSF grant No. DMR-1206267, the NSF Cooperative Agreement No. DMR-1157490, and the State of Florida. Work at LANL was supported by the US DOE Basic Energy Science project ‘Science at 100 Tesla’. The authors also acknowledge support from grant DOE DE-NA0001979.
## Author information
Authors
### Contributions
J.Y.L., J.H. and Q.Z. equally contributed to this work. The single crystals used in this study were synthesized by J.Y.L. The magnetotransport measurements in 14 T PPMS were carried out by J.Y.L., D.J.A., Z.Q.M. and L.S. The high-field measurements at NHMFL were conducted by J.H., D.G., S.M.A.R., I.C., L.S. and Z.Q.M., G.F.C., X.L., J.W. and W.A.P. contributed to X-ray structure characterization and crystal quality examination. J.H., J.Y.L. and Y.L.Z. performed magnetization measurements. Q.Z., H.B.C., J.F.D. and D.A.T. conducted neutron scattering experiments and analyses. M.J. and F.B. did pulse magnetic field measurements. J.Y.L., J.H., Y.L.Z. and Z.Q.M. conducted transport data analyses. All authors contributed to scientific discussions and read and commented on the manuscript. This project was supervised by Z.Q.M.
### Corresponding author
Correspondence to Z. Q. Mao.
## Ethics declarations
### Competing interests
The authors declare no competing financial interests.
## Supplementary information
### Supplementary Information
Supplementary Information (PDF 1172 kb)
## Rights and permissions
Reprints and Permissions
Liu, J., Hu, J., Zhang, Q. et al. A magnetic topological semimetal Sr1−yMn1−zSb2 (y, z < 0.1). Nature Mater 16, 905–910 (2017). https://doi.org/10.1038/nmat4953
• Accepted:
• Published:
• Issue Date:
• ### Quasi-two-dimensional relativistic fermions probed by de Haas–van Alphen quantum oscillations in LuSn2
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Physical Review B (2021)
• ### Experimental perspective on three-dimensional topological semimetals
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• M. Ohno
• , M. Uchida
• , Y. Nakazawa
• , S. Sato
• , M. Kriener
• , A. Miyake
• , M. Tokunaga
• , Y. Taguchi
• & M. Kawasaki
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• , Liqin Zhou
• , Junbao He
• , Chunyao Song
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• , Cheng Hu
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• , Yongqing Cai
• , Hao Chen
• , Cong Li
• , Qingyan Wang
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2021-06-12 14:54:42
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https://practicepaper.in/gate-cse/isro-cse-2014
|
# ISRO CSE 2014
Question 1
Consider a 33MHz cpu based system. What is the number of wait states required if it is interfaced with a 60ns memory? Assume a maximum of 10ns delay for additional circuitry like buffering and decoding.
A 0 B 1 C 2 D 3
Question 1 Explanation:
Question 2
The number of states required by a Finite State Machine,to simulate the behavior of a computer with a memory capable of storing 'm' words, each of length 'n' bits is?
A $m \times 2^{n}$ B $2^{m+n}$ C $2^{m n}$ D m+n
Theory of Computation Finite Automata
Question 2 Explanation:
Question 3
What is the output of the following C program?
#include < stdio.h >
#define SQR(x) (x*x)
int main()
{
int a;
int b=4;
a=SQR(b+2);
printf("%d\n",a);
return 0;
}
A 14 B 36 C 18 D 20
C Programming Arithmetic Operation
Question 3 Explanation:
Question 4
Consider the following pseudo- code
while (m < n)
if (x > y) and (a < b) then
a=a+1
y=y-1
end if
m=m+1 end while
What is cyclomatic complexity of the above pseudo -code?
A 2 B 3 C 4 D 5
Software Engg
Question 4 Explanation:
Question 5
What is the number of steps required to derive the string ((() ()) ()) for the following grammar?
$S \rightarrow S S$
$S \rightarrow(S)$
$S \rightarrow \varepsilon$
A 10 B 12 C 15 D 16
Compiler Design Parsing
Question 5 Explanation:
Question 6
The process of modifying IP address information in IP packet headers while in transit across a traffic routing device is called
Computer Network Network Layer Protocol
Question 6 Explanation:
Question 7
What does a pixel mask mean?
A string containing only 1's B string containing only 0's C string containing two 0's D string containing 1's and 0?s
Question 7 Explanation:
Question 8
In the standard IEEE 754 single precision floating point representation, there is 1 bit for sign, 23 bits for fraction and 8 bits for exponent. What is the precision in terms of the number of decimal digits?
A 5 B 6 C 7 D 8
Digital Logic Number System
Question 8 Explanation:
Question 9
Let R be the radius of the circle. What is the angle subtended by an arc of length R at the center of the circle?
A 1 degree B 1 radian C 90 degrees D $\pi$ radians
Question 9 Explanation:
Question 10
The number of logical CPUs in a computer having two physical quad-core chips with hyper threading enabled is ______
A 1 B 2 C 8 D 16
Question 10 Explanation:
There are 10 questions to complete.
|
2022-05-28 01:40:07
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http://hal.in2p3.fr/in2p3-01339349
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# Future research program on prompt $\gamma$ -ray emission in nuclear fission
Abstract : In recent years the measurement of prompt fission γ-ray spectra (PFGS) has gained renewed interest, after about forty years since the first comprehensive studies of the reactions 235U(n th , f), 239Pu(n th ,f) and 252Cf(sf). The renaissance was initiated by requests for new values especially for γ-ray multiplicity and average total energy release per fission in neutron-induced fission of 235U and 239Pu. Both isotopes are considered the most important ones with respect to the modeling of innovative cores required for the Generation-IV reactors, the majority working with fast neutrons. During the last 5 years we have conducted a systematic study of spectral data for thermal-neutron-induced fission on 235U and 241Pu as well as for the spontaneous fission of 252Cf with unprecedented accuracy. From the new data we conclude that those reactions do not considerably contribute to the observed heat excess and suspect other reactions playing a significant role. Possible contributions may originate from fast-neutron-induced reactions on 238U, which is largely present in the fuel, or from γ-induced fission from neutron capture in the construction material. A first experiment campaign on prompt γ-ray emission from fast-neutron-induced fission on 235,238U was successfully performed in order to test our assumptions. In the following we attempt to summarize, what has been done in the field to date, and to motivate future measurement campaigns exploiting dedicated neutron and photon beams as well as upcoming highly efficient detector assemblies.
Type de document :
Article dans une revue
European Physical Journal A, EDP Sciences, 2015, 51 (12), 〈10.1140/epja/i2015-15178-8〉
Littérature citée [31 références]
http://hal.in2p3.fr/in2p3-01339349
Contributeur : Sophie Heurteau <>
Soumis le : mercredi 29 juin 2016 - 16:10:17
Dernière modification le : jeudi 11 janvier 2018 - 06:12:41
### Fichier
art_10.1140_epja_i2015-15178-8...
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### Citation
S. Oberstedt, R. Billnert, F. -J. Hambsch, M. Lebois, A. Oberstedt, et al.. Future research program on prompt $\gamma$ -ray emission in nuclear fission. European Physical Journal A, EDP Sciences, 2015, 51 (12), 〈10.1140/epja/i2015-15178-8〉. 〈in2p3-01339349〉
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2018-06-24 01:04:16
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http://nanoscale.blogspot.com/2017/10/
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## Tuesday, October 31, 2017
Real life is a bit busy right now, but I wanted to point out a couple of links and talk about what's coming up.
• I've been looking for ways to think about and discuss topological materials that might be more broadly accessible to non-experts, and I found this paper and videos like this one and this one. Very cool, and I'm sorry I'd missed it back in '15 when it came out.
• In the experimental literature talking about realizations of Majorana fermions in the solid state, a key signature is a peak in the conductance at zero voltage - that's an indicator that there is a "zero-energy mode" in the system. There are other ways to get zero-bias peaks, though, and nailing down whether this has the expected properties (magnitude, response to magnetic fields) has been a lingering issue. This seems to nail down the situation more firmly.
• Discussions about "quantum supremacy" strictly in terms of how many qubits can be simulated on a classical computer right now seem a bit silly to me. Ok, so IBM managed to simulate a handful of additional qubits (56 rather than 49). It wouldn't shock me if they could get up to 58 - supercomputers are powerful and programmers can be very clever. Are we going to get a flurry of news stories every time about how this somehow moves the goalposts for quantum computers?
• I'm hoping to put out a review of Max the Demon and the Entropy of Doom, since I received my beautifully printed copies this past weekend.
## Wednesday, October 25, 2017
### Thoughts after a NSF panel
I just returned from a NSF proposal review panel. I had written about NSF panels back in the early days of this blog here, back when I may have been snarkier.
• Some things have gotten better. We can work from our own laptops, and I think we're finally to the point where everyone at these things is computer literate and can use the online review system. The program officers do a good job making sure that the reviews get in on time (ahead of the meeting).
• Some things remain the same. I'm still mystified at how few people from top-ranked programs (e.g., Harvard, Stanford, MIT, Cornell, Cal Tech, Berkeley) I see at these. Maybe I just don't move in the right circles.
• Best quote of the panel: "When a review of one of my papers or proposals starts with 'Author says' rather than 'The author says', I know that the referee is Russian and I'm in trouble."
• Why does the new NSF headquarters have tighter security screenings that Reagan National Airport?
• The growth of funding costs and eight years of numerically flat budgets has made this process more painful. Sure looks like morale is not great at the agency. Really not clear where this is all going to go over the next few years. There was a lot of gallows humor about having "tax payer advocates" on panels. (Everyone on the panel is a US taxpayer already, though apparently that doesn't count for anything because we are scientists.)
• NSF is still the most community-driven of the research agencies.
• I cannot overstate the importance of younger scientists going to one of these and seeing how the system works, so you learn how proposals are evaluated.
## Monday, October 23, 2017
### Whither science blogging?
I read yesterday of the impending demise of scienceblogs, a site that has been around since late 2005 in one form or other. I guess I shouldn't be surprised, since some of its bloggers have shifted to other sites in recent years, such as Ethan Siegel and Chad Orzel, who largely migrated to Forbes, and Rhett Allain, who went to Wired. Steinn Sigurðsson is going back to his own hosted blog in the wake of this.
I hope this is just indicative of a poor business model at Seed Media, and not a further overall decline in blogging by scientists. It's wonderful that online magazines like Quanta and Aeon and Nautilus are providing high quality, long-form science writing. Still, I think everyone benefits when scientists themselves (in addition to professional science journalists) carve out some time to write about their fields.
## Friday, October 20, 2017
### Neutron stars and condensed matter physics
In the wake of the remarkable results reported earlier this week regarding colliding neutron stars, I wanted to write just a little bit about how a condensed matter physics concept is relevant to these seemingly exotic systems.
When you learn high school chemistry, you learn about atomic orbitals, and you learn that electrons "fill up" those orbitals starting with the lowest energy (most deeply bound) states, two electrons of opposite spin per orbital. (This is a shorthand way of talking about a more detailed picture, involving words like "linear combination of Slater determinants", but that's a detail in this discussion.) The Pauli principle, the idea that (because electrons are fermions) all the electrons can't just fall down into the lowest energy level, leads to this. In solid state systems we can apply the same ideas. In a metal like gold or copper, the density of electrons is high enough that the highest kinetic energy electrons are moving around at ~ 0.5% of the speed of light (!).
If you heat up the electrons in a metal, they get more spread out in energy, with some occupying higher energy levels and some lower energy levels being empty. To decide whether the metal is really "hot" or "cold", you need a point of comparison, and the energy scale gives you that. If most of the low energy levels are still filled, the metal is cold. If the ratio of the thermal energy scale, $k_{\mathrm{B}}T$ to the depth of the lowest energy levels (essentially the Fermi energy, $E_{\mathrm{F}}$ is much less than one, then the electrons are said to be "degenerate". In common metals, $E_{\mathrm{F}}$ is several eV, corresponding to a temperature of tens of thousands of Kelvin. That means that even near the melting point of copper, the electrons are effectively very cold.
Believe it or not, a neutron star is a similar system. If you squeeze a bit more than one solar mass into a sphere 10 km across, the gravitational attraction is so strong that the electrons and protons in the matter are crushed together to form a degenerate ball of neutrons. Amazingly, by our reasoning above, the neutrons are actually very very cold. The Fermi energy for those neutrons corresponds to a temperature of nearly $10^{12}$ K. So, right up until they smashed into each other, those two neutron stars spotted by the LIGO observations were actually incredibly cold, condensed objects. It's also worth noting that the properties of neutron stars are likely affected by another condensed matter phenomenon, superfluidity. Just as electrons can pair up and condense into a superconducting state under some circumstances, it is thought that cold, degenerate neutrons can do the same thing, even when "cold" here might mean $5 \times 10^{8}$ K.
## Sunday, October 15, 2017
### Gravitational waves again - should be exciting
There is going to be a big press conference tomorrow, apparently to announce that LIGO/VIRGO has seen an event (binary neutron star collision) directly associated with a gamma ray burst in NGC 4993. Fun stuff, and apparently the worst-kept secret in science right now. This may seem off-topic for a condensed matter blog, but there's physics in there which isn't broadly appreciated, and I'll write a bit about it after the announcement.
## Tuesday, October 10, 2017
### Piezo controller question - followup.
A couple of weeks ago I posted:
Anyone out there using a Newport NPC3SG controller to drive a piezo positioning stage, with computer communication successfully talking to the NPC3SG? If so, please leave a comment so that we can get in touch, as I have questions.
No responses so far. This is actually the same unit as this thing:
https://www.piezosystem.com/products/piezo_controller/piezo_controller_3_channel_version/nv_403_cle/
In our unit from Newport, communications simply don't work properly. Timeout problems. The labview code supplied by Newport (the same code paired with the link above) has these problems, as do many other ways of trying to talk with the instrument. Has anyone out there had success in using a computer to control and read this thing? At issue is whether this is a hardware problem with our unit, or whether there is a general problem with these. The vendor has been verrrrrrrrry slow to figure this out.
## Sunday, October 08, 2017
### The Abnormal Force
How does the chair actually hold you up when you sit down? What is keeping your car tires from sinking through the road surface? What is keeping my coffee mug from falling through my desk? In high school and first-year undergrad physics, we teach people about the normal force - that is a force that acts normal (perpendicular) to a surface, and it takes on whatever value is needed so that solid objects don't pass through each other.
The microscopic explanation of the normal force is that the electrons in the atoms of my coffee mug (etc.) interact with the electrons in the atoms of the desk surface, through a combination of electrostatics (electrons repel each other) and quantum statistics (the Pauli principle means that you can't just shuffle electrons around willy-nilly). The normal force is "phenomenological" shorthand. We take the observation that solid objects don't pass through each other, deduce that whatever is happening microscopically, the effect is that there is some force normal to surfaces that touch each other, and go from there, rather than trying to teach high school students how to calculate it from first principles. The normal force is an emergent effect that makes sense on macroscopic scales without knowing the details. This is just like how we teach high school students about pressure as a useful macroscopic concept, without actually doing a statistical calculation of the average perpendicular force per area on a surface due to collisions with molecules of a gas or a liquid.
You can actually estimate the maximum reasonable normal force per unit area. If you tried to squeeze the electrons of two adjacent atoms into the volume occupied by one atom, even without the repulsion of like charges adding to the cost, the Pauli principle means you'd have to kick some of those electrons into higher energy levels. If a typical energy scale for doing that for each electron was something like 1 eV, and you had a few electrons per atom, and the areal density of atoms is around 1014 per cm2, then we can find the average force $F_{\mathrm{av}}$ required to make a 1 cm2 area of two surfaces overlap with each other. We'd have $F_{\mathrm{av}} d \sim 10^{15}$eV, where $d$ is the thickness of an atom, around 0.3 nm. That's around 534000 Newtons/cm2, or around 5.3 GPa. That's above almost all of the yield stresses for materials (usually worrying about tension rather than compression) - that just means that the atoms themselves will move around before you really push electrons around.
Very occasionally, when two surfaces are brought together, there is a force that arises at the interface that is not along the normal direction. A great example of that is in this video, which shows two graphite surfaces that spontaneously slide in the plane so that they are crystallographically aligned. That work comes from this paper.
As far as I can tell, there is no official terminology for such a spontaneous in-plane force. In the spirit of one of my professional heroes David Mermin, who coined the scientific term boojum, I would like to suggest that such a transverse force be known as the abnormal force. (Since I don't actually work in this area and I'm not trying to name the effect after myself, hopefully the barrier to adoption will be lower than the one faced by Mermin, who actually worked on boojums :-) ).
## Tuesday, October 03, 2017
### Gravitational radiation for the win + communicating science
As expected, LIGO was recognized by the Nobel Prize in physics this year. The LIGO experiment is an enormous undertaking that combines elegant, simple theoretical ideas; incredible engineering and experimental capabilities; and technically virtuosic numerical theoretical calculations and data analysis techniques. It's truly a triumph.
I did think it was interesting when Natalie Wolchover, one of the top science writers out there today, tweeted: Thrilled they won, thrilled not to spend this morning speed-reading about some bizarre condensed matter phenomenon.
This sentiment was seconded by Peter Woit, who said he thought she spoke for all science journalists.
Friendly kidding aside, I do want to help. Somehow it's viewed as comparatively easy and simple to write about this, or this, or this, but condensed matter is considered "bizarre".
## Sunday, October 01, 2017
### Gravitational radiation redux + Nobel speculation
This past week, there was exciting news that the two LIGO detectors and the VIRGO interferometer had simultaneously detected the same event, a merger of black holes estimated to have taken place 1.6 billion lightyears away. From modeling the data, the black hole masses are estimated at around 25 and 30 solar masses, and around 2.7 solar masses worth of energy (!) was converted in the merger into gravitational radiation. The preprint of the paper is here. Check out figure 1. With just the VIRGO data, the event looks really marginal - by eye you would be hard pressed to pick it out of the fluctuating detector output. However, when that data is thrown into the mix with that from the (completely independent from VIRGO) detectors, the case is quite strong.
This is noteworthy for (at least) two reasons. First, there has been some discussion about the solidity of the previously reported LIGO results - this paper (see here for a critique of relevant science journalism) argues that there are some surprising correlations in the noise background of the two detectors that could make you wonder about the analysis. After all, the whole point of having two detectors is that a real event should be seen by both, while one might reasonably expect background jitter to be independent since the detectors are thousands of miles apart. Having a completely independent additional detector in the mix should be useful in quantifying any issues. Second, having the additional detector helps nail down the spot in the sky where the gravitational waves appear to originate. This image shows how previous detections could only be localized by two detectors to a band spanning lots of the sky, while this event can be localized down to a spot spanning a tenth as much solid angle. This is key to turning gravitational wave detectors into serious astronomy tools, by trying to link gravitational event detection to observations across the electromagnetic spectrum. There were rumors, for example, that LIGO had detected what was probably a neutron star collision (smaller masses, but far closer to earth), the kind of event thought to produce dramatic electromagnetic signatures like gamma ray bursts.
On that note, I realized Friday that this coming Tuesday is the announcement of the 2017 Nobel in physics. Snuck up on me this time. Speculate away in the comments. Since topology in condensed matter was last year's award, it seems likely that this year will not be condensed matter-related (hurting the chances of people like Steglich and Hosono for heavy fermion and iron superconductors, respectively). Negative index phenomena might be too condensed matter related. The passing last year of Vera Rubin and Debra Jin is keenly felt, and makes it seem less likely that galactic rotation curves (as evidence for dark matter) or ultracold fermions would get it this year. Bell's inequality tests (Aspect, Zeilinger, Clauser) could be there. The LIGO/VIRGO combined detection happened too late in the year to affect the chances of this being the year for gravitational radiation (which seems a shoe-in soon).
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2017-12-15 23:33:44
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https://leanprover-community.github.io/archive/stream/113488-general/topic/Lean.203.2E8.20library.20changes.html
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## Stream: general
### Topic: Lean 3.8 library changes
#### Yury G. Kudryashov (Mar 24 2020 at 16:59):
I created an issue to track proposed changes to the core library. Additions are welcome, and I guess PRs are even more welcome.
#### Yury G. Kudryashov (Mar 24 2020 at 18:40):
What's the policy for non-master branches in leanprover-community/lean?
lean#161
#### Mario Carneiro (Mar 24 2020 at 18:43):
I would say same as mathlib
#### Gabriel Ebner (Mar 24 2020 at 18:43):
It's completely different from mathlib.
#### Mario Carneiro (Mar 24 2020 at 18:44):
there should probably be some policy about not tagging though
#### Gabriel Ebner (Mar 24 2020 at 18:44):
In mathlib, it is important that you use branches in the leanprover-community organization because the CI produces oleans. In the lean repo there is no such benefit.
#### Gabriel Ebner (Mar 24 2020 at 18:45):
But if you have write access, feel free to create branches in the lean repo.
#### Mario Carneiro (Mar 24 2020 at 18:45):
the original reason we started using leanprover-community branches in mathlib was to make collaboration easier
#### Mario Carneiro (Mar 24 2020 at 18:45):
multiple people can commit to the same branch without having to PR to a PR
#### Yury G. Kudryashov (Mar 24 2020 at 18:46):
Side effect of me having no write access: I can't assign milestones/labels.
#### Gabriel Ebner (Mar 24 2020 at 18:47):
Ah, ok then we need to hand out write access.
#### Gabriel Ebner (Mar 24 2020 at 18:47):
Who wants write access on the Lean repo?
Me.
#### Bryan Gin-ge Chen (Mar 24 2020 at 18:52):
Me too. Perhaps people who have submitted PRs in the past should get an invitation?
#### Gabriel Ebner (Mar 24 2020 at 18:52):
I'm currently going through the list.
#### Gabriel Ebner (Mar 24 2020 at 18:55):
I hope I didn't miss anybody. Feel free to ask for write access on the Lean repo if I didn't invite you already.
#### Gabriel Ebner (Mar 24 2020 at 18:57):
The rules are the same as for mathlib: create branches as you like, try to make PRs from branches of the Lean repo, assign/tag issues if you want.
#### Yury G. Kudryashov (Mar 25 2020 at 21:01):
CI seems to be stuck on lean#161
#### Bryan Gin-ge Chen (Mar 25 2020 at 21:05):
Here's the job on travis. I just restarted the one macOS build that failed.
(BTW, if you write lean#161 it will link to the right PR.)
#### Bryan Gin-ge Chen (Mar 25 2020 at 21:11):
Uh oh, it looks like that macOS job has been timing out pretty consistently recently, and when it does succeed it's always very close to the 50 min time limit. (compare 1 2 3).
#### Gabriel Ebner (Mar 25 2020 at 21:52):
Yes this is a known issue.
#### Yury G. Kudryashov (Mar 28 2020 at 01:30):
It timed out again.
Last updated: May 08 2021 at 10:12 UTC
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2021-05-08 10:58:54
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https://mathproblems123.wordpress.com/2013/12/16/the-cantor-function-and-some-of-its-properties/
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Home > BV functions, Real Analysis > the Cantor function and some of its properties
## the Cantor function and some of its properties
Let’s start by definining the Cantor set. Define ${C_0=[0,1]}$ and ${C_{n+1} = C_n/3 \cup (2/3+C_n/3)}$. At each step we delete the middle third of all the intervals of ${C_n}$ to obtain ${C_{n+1}}$. Note that we obviously have ${C_{n+1} \subset C_{n}}$ (an easy inductive argument) and ${|C_n|=(2/3)^n}$. The sets ${C_n}$ are compact and descending, therefore we can define ${C=\bigcap_{n=0}^\infty C_n}$ which is a compact subset of ${[0,1]}$ with zero measure and it is called the Cantor set.
Since at each step we remove a middle third of all the intervals in ${C_n}$, one way to look at the Cantor set is to look at the ternary representation of the points in it. In the first step, we remove all the elements of ${[0,1]}$ which have ${1}$ on their first position in the ternary representation. In the second step we remove those (remaining) which have ${1}$ on the second position, and so on. In the end we are left only with elements of ${[0,1]}$ which have only digits ${0,2}$ in their ternary representation. Using this representation we can construct a bijection between ${C}$ and ${[0,1]}$ which maps
$\displaystyle x=\sum_{n=1}^\infty \frac{a_n}{3^n} \mapsto \sum_{n=1}^\infty \frac{b_n}{2^n}$
where ${b_n=0}$ if ${a_n=0}$ and ${b_n=1}$ if ${a_n=2}$. This proves that the Cantor set is uncountable.
We can construct the Cantor function ${g:[0,1] \rightarrow [0,1]}$ in the following way. Denote ${R_n}$ the set ${C_n\setminus C_{n+1}}$ (i.e. the set removed in step ${n}$). On ${R_1}$ we let ${g(x)=1/2}$. On ${R_2}$ we have two intervals: on the left one we let ${g(x)=1/4}$ and on the right one we let ${g(x)=3/4}$. We continue like this iteratively, at each step choosing ${g}$ constant on each of the intervals which construct ${R_n}$ such that the constant on an interval is the mean of the values of neighboring interval values.
A precise formula can be given:
$\displaystyle g(x) =\frac{1}{2^{N_x}}+\frac{1}{2}\sum_{n=1}^{N_x-1}\frac{a_{nx}}{2^n}$
where ${N_x}$ is the first index in the representation ${x=\sum_{n=1}^\infty a_{nx}/3^n}$ for which ${a_{nx}=1}$ or ${\infty}$ if no such index exists.
the graph of the Cantor function
The Cantor function ${g(x)}$ is monotone by construction (it is also called the devil’s staircase). Since ${g(x)}$ is constant on each interval in each ${R_n}$ and ${\bigcup R_n =[0,1]}$ almost everywhere we conclude that ${g'(x)}$ exists almost everywhere in ${[0,1]}$ and ${g'(x)=0}$.
We can deduce from the formula given for ${g}$ that the Cantor function is continuous. Indeed, if ${x \rightarrow y}$ then ${\min\{n: a_{nx}\neq a_{ny}\} \rightarrow \infty}$ so the difference between ${g(x)}$ and ${g(y)}$ will be of the order ${1/2^n}$ with ${n \rightarrow \infty}$.
Since ${g}$ is monotone, it has bounded variation, and here we can see a pathological example which illustrates the structure of a function with bounded variation. A function with bounded variation in one dimension is a function ${f}$ defined on an interval ${[a,b]}$ such that the quantity ${\sup \sum_{i=0}^{N-1} |f(x_{i+1})-f(x_i)| <\infty}$ for all partitions ${x_0<... of ${[a,b]}$ (${N}$ is not fixed).
An equivalent definition is
$\displaystyle f \in BV(\Omega) \Leftrightarrow \sup \{ \int_\Omega f\ \text{div}\varphi: \varphi \in C^\infty(\Omega), \|\varphi\|_\infty\leq 1\}<\infty$
and in this case there exists a Radon measure ${\mu=Du}$ such that
$\displaystyle \int_\Omega f\ \text{div}\varphi = -\int_\Omega \varphi d\mu,\ \forall \varphi \in C^\infty(\Omega)$
The structure theorem for functions of bounded variation says that if ${f}$ has bounded variation then ${Df}$ can be splitted into three parts:
• ${D^a f << \mathcal{L}_n}$; ${Df=D^a f+D^s f}$ (Radon Nikodym with respect to the Lebesgue measure)
• ${D^j f=D^s f \llcorner S(u)}$ where ${S(f)}$ is the jump part of ${f}$.
• ${D^c f=Df - D^a f-D^j f}$: the rest, which is called (what a coincidence) the Cantor part of ${Df}$.
Therefore a bounded variation function has a part which behaves as a Sobolev function (the ${n}$ dimensional part), it has a jump part (the ${n-1}$ dimensional part), and it may have a third part (in between dimensions ${n-1}$ and ${n}$) which usually cannot be well described.
The Cantor function ${g}$ is differentiable almpost everywhere (in ${[0,1]\setminus C}$) and ${g'(x)=0}$ for every ${x \in [0,1]\setminus C}$. The part where ${g}$ is differentiable represents the first component of ${g}$ and we see that ${D^a g=0}$. Since ${g}$ is continuous, we do not have a jump part. The only part of the distributional gradient which measures the vertical displacement of ${g}$ is concentrated on the cantor set ${C}$ (which is to be expected, since ${g}$ is constant on every interval outside ${C}$) and this is the Cantor part ${D^c g}$ of ${D g}$.
Usually the Cantor part is so nasty that we can say very little for these bounded variation functions. In order to obtain a space with nicer properties, the space ${SBV(\Omega)}$ was introduce, and its definition is exactly the ${BV(\Omega)}$ functions such that ${D^c f=0}$.
Another interesting property of the Cantor function is that the length of its graph is ${2}$ although the function is increasing, ${g(0)=0}$ and ${g(1)=1}$. Intuitively this seems to be false, since ${g}$ travels mostly horizontally. First notice that ${2}$ is an upper bound. Any curve which increases from ${(0,0)}$ to ${(1,1)}$ has length at most ${2}$. Now let’s see that the arclength is at least ${2}$. First look horizontally: on each ${R_n}$ the function ${g}$ is piecewise constant, and ${R_n}$ cover ${[0,1]}$ up to a set of measure zero. Therefore the horizontal arclength is ${1}$. Secondly, ${g}$ must move vertically from ${0}$ to ${1}$ and it does this continuously, so the vertical arclength is also ${1}$. As a consequence, the arclength of ${g}$ is ${2}$.
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2017-12-17 21:11:03
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http://mathhelpforum.com/calculus/218399-contour-lines.html
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# Math Help - Contour lines
1. ## Contour lines
Hey guys!!!Could you help me at this exercise???
Draw some contour lines in the (x, y)–plane of the function h(x,y)=(a*(x+y))/(x^2+y^2+a^2).
2. ## Re: Contour lines
tzina, with the aid of a 3D grapher I was able to to come up with some contour lines.
let $\space$ $z = \displaystyle\frac{a(x + y)}{x^2 + a^2 + y^2}$
for each value of $\space$ $a$, $\space$ you get a unique set of contour lines, but each set of lines has similar features to them
.Set $\space$ $z$ $\space$ in increments of $\space$ $.1$. For example $\space$: $\space$ $z = 0, .1, .2, .3, ...$ $\space$ etc.
For each value of $\space$ $z$ $\space$ you can create one contour line.
When $\space$ $z = 0$ $\space$ the contour line will show up as a straight line
if $\space$ $z = 0$, $\space$ $a(x + y) = 0$.
You will have the line $\space$ $x + y = 0$
3. ## Re: Contour lines
continuing with my response:
Each succeeding contour line will be a circle where the center of each circle will fall on the line $\space$ $x - y = 0$
You will be able to calculate the center and radius of each circle by setting $\space$ $a$ $\space$as a specific constant and solving for $\space$ $x$ and $y$ $\space$ for each value of $\space$ $z$.
You will end up with a series of circles whose centers fall on the line $\space$ $x - y = 0$.
I hope this helps.
4. ## Re: Contour lines
Hi,
Here's a graphing trick that you can use to draw contour lines. Assuming your graphing software can handle implicit functions, have it graph the function sin(s*2pi*h(x,y))=0. This is then the set of curves h(x,y)=k/s for k any integer. If your grapher has sliders, you can rapidly change the step size s. Also in your case you can rapidly change the constant a. Here's some contours for your function with s=10 (step size .1) and a =3 (the contour at height 0 is as noted above the line x+y=0, it's shown in red):
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2014-11-26 19:10:18
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https://www.sfu.ca/math-coursenotes/Math%20158%20Course%20Notes/sec_first_order_differential_equations.html
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## Section5.2First Order Differential Equations
In many fields such as physics, biology or business, a relationship is often known or assumed between some unknown quantity and its rate of change, which does not involve any higher derivatives. It is therefore of interest to study first order differential equations in particular.
###### Definition5.7. First Order DE.
A first order differential equation is an equation of the form $F(t, y, y')=0\text{.}$ A solution of a first order differential equation is a function $f(t)$ that makes $\ds F(t,f(t),f'(t))=0$ for every value of $t\text{.}$
Here, $F$ is a function of three variables which we label $t\text{,}$ $y\text{,}$ and $y'\text{.}$ It is understood that $y'$ will explicitly appear in the equation although $t$ and $y$ need not. The variable $y$ itself is dependent on $t\text{,}$ hence it is understood that $y'$ must be the derivative of $y$ with respect to $t\text{.}$ Since only the first derivative of $y$ appears, but no higher order derivative, this is a first order differential equation.
Throughout the notes, we use the independent variable $t$ as many applications are based on the independent variable representing time. If no meaning is attributed to the independent variable, we may want to write a first order differential equation in the usual manner as
\begin{equation*} F(x,y,y')=0\text{.} \end{equation*}
###### Example5.8. Simple First Order Differential Equation.
$\ds y'=t^2+1$ is a first order linear differential equation; $\ds F(t,y,y')= y'-t^2-1\text{.}$ Show that all solutions to this equation are of the form $\ds y=t^3/3+t+C\text{.}$
Solution
We first note that $y = t^3/3 + t + C$ is a solution to the differential equation, since
\begin{equation*} \frac{d}{dt} \left(\frac{t^3}{3} + t + C\right) = t^2 + 1\text{,} \end{equation*}
for all $C \in \R\text{.}$
We additionally need to show that there are no other solutions. To do so, we integrate the differential equation:
\begin{equation*} y(t) = \int \left(t^2 + 1\right) \, dt = \frac{t^3}{3} + t + C\text{,} \end{equation*}
for some $C \in R\text{.}$
Thus, all solutions to the differential equation are of the form $\ds y=t^3/3+t+C\text{.}$
###### Example5.9. Graphical Solution to First Order Differential Equation.
Sketch various solutions to the differential equation $\ds\frac{dy}{dx}=2x\text{.}$
Solution
We integrate both sides of the differential equation to find
\begin{equation*} y = \int 2x \,dx = x^2 + C \end{equation*}
for any constant $C \in \R\text{.}$ This family of solutions are parabolas which are translated vertically, as shown in the graph below taking $C=-2,0,2\text{.}$
### Subsection5.2.1Initial Value Problems
###### Definition5.10. Initial Conditions.
Initial condition(s) are a set of points that the solution (or its derivatives) must satisfy.
###### Example5.11. Initial Conditions.
For a differential equation involving a function $f(t)\text{,}$ initial conditions are of the form:
###### Definition5.12. Initial Value Problem.
An initial value problem (IVP) is a differential equation along with a set of initial conditions.
###### Example5.13. First Order Initial Value Problem.
Solve the initial value problem:
\begin{equation*} \frac{dy}{dx}=2x, y(0)=2\text{.} \end{equation*}
Solution
We had found in Example 5.9 that the solutions to the differential equation were the parabolas
\begin{equation*} y(x) = x^2 + C\text{.} \end{equation*}
So we use the initial condition to determine the constant $C\text{:}$
\begin{equation*} y(0) = (0)^2+ C = 2 \implies C = 2\text{.} \end{equation*}
Therefore, the solution to the initial value problem is $y=x^2 + 2\text{,}$ as shown in the graph below.
###### Example5.14. Simple Initial Value Problem.
Verify that the initial value problem $\ds y'=t^2+1\text{,}$ $y(1)=4$ has solution $\ds f(t)=t^3/3+t+8/3\text{.}$
Solution
Observe that $f'(t)=t^2+1$ and $f(1)=1^3/2+1+8/3=4$ as required.
The general first order equation is too general, so we can't describe methods that will work on them all, or even a large portion of them. We can make progress with specific kinds of first order differential equations. For example, much can be said about equations of the form $\ds y' = \phi (t, y)$ where $\phi$ is a function of the two variables $t$ and $y\text{.}$ Under reasonable conditions on $\phi\text{,}$ such an equation has a solution and the corresponding initial value problem has a unique solution. However, in general, these equations can be very difficult or impossible to solve explicitly.
###### Example5.15. IVP for Newton's Law of Cooling.
Consider this specific example of an initial value problem for Newton's law of cooling:
\begin{equation*} y' = 2(25-y), \ \ y(0)=40\text{.} \end{equation*}
Discuss the solutions for this initial value problem.
Solution
We first note the zero of the equation: If $y(t_0) = 25\text{,}$ the right hand side of the differential equation is zero, and so the constant function $y(t)=25$ is a solution to the differential equation. It is not a solution to the initial value problem, since $y(0)\not=25\text{.}$ (The physical interpretation of this constant solution is that if a liquid is at the same temperature as its surroundings, then the liquid will stay at that temperature.) So long as $y$ is not 25, we can rewrite the differential equation as
\begin{align*} {dy\over dt}{1\over 25-y}\amp = 2\\ {1\over 25-y}\,dy\amp = 2\,dt\text{,} \end{align*}
so
\begin{equation*} \int {1\over 25-y}\,dy = \int 2\,dt\text{,} \end{equation*}
that is, the two antiderivatives must be the same except for a constant difference. We can calculate these antiderivatives and rearrange the results:
\begin{align*} \int {1\over 25-y}\,dy \amp = \int 2\,dt\\ (-1)\ln|25-y| \amp = 2t+C_0\\ \ln|25-y| \amp = -2t - C_0 = -2t + C\\ |25-y| \amp = e^{-2t+C}=e^{-2t} e^C\\ y-25 \amp = \pm\, e^C e^{-2t}\\ y \amp = 25 \pm e^C e^{-2t} =25+Ae^{-2t}\text{.} \end{align*}
Here $\ds A = \pm\, e^C = \pm\, e^{-C_0}$ is some non-zero constant. Since we want $y(0)=40\text{,}$ we substitute and solve for $A\text{:}$
\begin{equation*} 40 = 25+Ae^0 \implies A = 15\text{.} \end{equation*}
Therefore, $\ds y=25+15 e^{-2t}$ is a solution to the initial value problem. Note that $y$ is never 25, so this makes sense for all values of $t\text{.}$ However, if we allow $A=0$ we get the solution $y=25$ to the differential equation, which would be the solution to the initial value problem if we were to require $y(0)=25\text{.}$ Thus, $\ds y=25+Ae^{-2t}$ describes all solutions to the differential equation $\ds y' = 2(25-y)\text{,}$ and all solutions to the associated initial value problems.
### Subsection5.2.2Separable Equations
Why could we solve Example 5.15 from the previous section? Our solution depended on rewriting the equation so that all instances of $y$ were on one side of the equation and all instances of $t$ were on the other. Of course, in this case the only $t$ was originally hidden, since we didn't write $dy/dt$ in the original equation. This is not required, however. This idea of being able to separate the independent and dependent variables in a first order differential equation leads to a classification of first order differential equations into separable and non-separable equations as follows.
###### Definition5.16. Separable DE.
A first order differential equation is separable if it can be written in the form
\begin{equation*} \frac{dy}{dt} = f(t) g(y)\text{.} \end{equation*}
Let's come back to all first order differential equations on our list from the previous section and decide which ones are separable or not:
1. $y' = e^x\sec y$ has order 1, is non-linear, is separable
2. $y'-e^xy+3 = 0$ has order 1, is linear, is not separable
3. $y'-e^xy = 0$ has order 1, is linear, is separable
As in the examples, we can attempt to solve a separable equation by converting to the form
\begin{equation*} \int {1\over g(y)}\,dy=\int f(t)\,dt\text{.} \end{equation*}
This technique is called separation of variables. The simplest (in principle) sort of separable equation is one in which $g(y)=1\text{,}$ in which case we attempt to solve
\begin{equation*} \int 1\,dy=\int f(t)\,dt\text{.} \end{equation*}
We can do this if we can find an antiderivative of $f(t)\text{.}$
As we have seen so far, a differential equation typically has an infinite number of solutions. Such a solution is called a general solution. A corresponding initial value problem will give rise to just one solution. Such a solution in which there are no unknown constants remaining is called a specific solution.
The general approach to separable equations is as follows: Suppose we wish to solve $y' = f(t) g(y)$ where $f$ and $g$ are continuous functions. If $g(a)=0$ for some $a$ then $y(t)=a$ is a constant solution of the equation, since in this case $y' = 0 = f(t)g(a)\text{.}$ For example, $y' =y^2 -1$ has constant solutions $y(t)=1$ and $y(t)=-1\text{.}$
To find the non-constant solutions, we note that the function $1/g(y)$ is continuous where $g\not=0\text{,}$ so $1/g$ has an antiderivative $G\text{.}$ Let $F$ be an antiderivative of $f\text{.}$ Now we write
\begin{equation*} G(y) = \int {1\over g(y)}\,dy = \int f(t)\,dt=F(t)+C\text{,} \end{equation*}
so $G(y)=F(t)+C\text{.}$ Now we solve this equation for $y\text{.}$
Of course, there are a few places this ideal description could go wrong: Finding the antiderivatives $G$ and $F\text{,}$ and solving the final equation for $y\text{.}$ The upshot is that the solutions to the original differential equation are the constant solutions, if any, and all functions $y$ that satisfy $G(y)=F(t)+C\text{.}$
###### Guideline for Separation of Variables.
Given the differential equation
\begin{equation*} \frac{dy}{dt} = f(t)g(y)\text{,} \end{equation*}
follow these steps to find the non-constant solutions.
1. Separate the variables:
\begin{equation*} \frac{dy}{g(y)} = f(t)\,dt \end{equation*}
2. Apply the integration operator:
\begin{equation*} \int \frac{dy}{g(y)} = \int f(t)\,dt \end{equation*}
3. If an antiderivative exists for $f$ and for $1/g\text{,}$ and we can solve for $y\text{,}$ then
\begin{equation*} G(y) = F(t) + C \end{equation*}
for some constant $C\text{.}$
###### Example5.17. Solving a Separable Differential Equation I.
Solve the differential equation $\ds y' = 2t(25-y)\text{.}$
Solution
This is almost identical to the previous example. As before, $y(t)=25$ is a solution. If $y\not=25\text{,}$
\begin{align*} \int {1\over 25-y}\,dy \amp = \int 2t\,dt\\ (-1)\ln|25-y| \amp = t^2+C_0\\ \ln|25-y| \amp = -t^2 - C_0 = -t^2 + C\\ |25-y| \amp = e^{-t^2+C}=e^{-t^2} e^C\\ y-25 \amp = \pm\, e^C e^{-t^2}\\ y \amp = 25 \pm e^C e^{-t^2} =25+Ae^{-t^2}\text{.} \end{align*}
As before, all solutions are represented by $\ds y=25+Ae^{-t^2}\text{,}$ allowing $A$ to be zero.
###### Example5.18. Solving a Seperable Differential Equation II.
Find the solutions to the differential equation
\begin{equation*} \sec(t) \frac{dy}{dt} - e^{y+\sin(t)} = 0\text{.} \end{equation*}
Solution
We begin by separating the variables and get
\begin{equation*} \begin{split} \sec(t) \frac{dy}{dt} \amp = e^{y+\sin(t)} \\ \sec(t) \frac{dy}{dt} \amp = e^ye^{\sin(t)} \\ e^{-y}\,dy \amp = \frac{e^{\sin(t)}}{\sec(t)}\,dt = \cos(t)e^{\sin(t)}\,dt \end{split} \end{equation*}
Now integrate both sides to obtain
\begin{equation*} \begin{split} \int e^{-y}\,dy \amp = \int \cos(t)e^{\sin(t)}\,dt \\ -e^{-y} \amp = e^{\sin(t)} + C \\ y \amp = - \ln\left(D - e^{\sin(t)}\right) \end{split} \end{equation*}
For convenience, we left out the absolute value in the argument of the logarithm. As in the previous examples, care must be taken to ensure that the argument of the logarithm is positive for a given value of $D\text{.}$
Therefore, the solutions to the differential equation are given by
\begin{equation*} y = - \ln\left(D - e^{\sin(t)}\right)\text{,} \end{equation*}
for some constant $D\text{.}$
### Subsection5.2.3Simple Growth and Decay Model
###### Example5.19. Rate of Change Proportional to Size.
Find the solutions to the differential equation $y'=ky\text{,}$ which models a quantity $y$ that grows or decays proportionally to its size depending on whether $k$ is positive or negative.
Solution
The constant solution is $y(t)=0\text{;}$ of course this will not be the solution to any interesting initial value problem. For the non-constant solutions, we proceed much as before:
\begin{align*} \int {1\over y}\,dy\amp = \int k\,dt\\ \ln|y| \amp = kt+C\\ |y| \amp = e^{kt} e^C\\ y \amp = \pm \,e^C e^{kt}\\ y\amp = Ae^{kt}\text{.} \end{align*}
Again, if we allow $A=0$ this includes the constant solution, and we can simply say that $\ds y=Ae^{kt}$ is the general solution. With an initial value we can easily solve for $A$ to get the solution of the initial value problem. In particular, if the initial value is given for time $t=0\text{,}$ $y(0)=y_0\text{,}$ then $A=y_0$ and the solution is $\ds y= y_0 e^{kt}\text{.}$
The constant $k$ in the above differential equation is referred to as the growth rate constant. Furthermore, this type of differential equation is known as a simple model for growth and decay of some quantity, since it only considers that the growth rate is proportional to the size of the quantity itself without any other factors influencing $y\text{.}$ The graph below shows the typical $J$-shape of such a solution for some $y_0\text{.}$
###### Simple Growth and Decay Model.
The differential equation
\begin{equation*} \frac{dy}{dt} = ky \end{equation*}
with growth rate constant $k$ models simple growth and decay of a quantity $y$ at time $t$ with solution
\begin{equation*} y = y_0e^{kt}\text{,} \end{equation*}
where $y_0$ is the initial value at time $t=0\text{.}$
When $k>0\text{,}$ this describes certain simple cases of population growth: It says that the change in the population $y$ is proportional to the population. The underlying assumption is that each organism in the current population reproduces at a fixed rate, so the larger the population the more new organisms are produced. While this is too simple to model most real populations, it is useful in some cases over a limited time.
When $k\lt 0\text{,}$ the differential equation describes a quantity that decreases in proportion to the current value. This can be used to model radioactive decay.
Interactive Demonstration. Use the sliders below to investigate the differential equation $\frac{dy}{dt} = f(t,y)$ where
Note: The simple growth and decay model is unrestricted because the quantity $y$ grows without bound as $t \to \infty$ if $k > 0\text{.}$ In the decay case, the solution only becomes unbounded if time $t$ is allowed to approach $-\infty\text{.}$
###### Example5.20. Simple Growth Model.
Suppose $5,000 is deposited into an account which earns continuously compounded interest. Under these conditions, the balance in the account grows at a rate proportional to the current balance. Suppose that after 4 years the account is worth$7,000.
1. How much is the account worth after 5 years?
2. How many years does it take for the balance to double?
Solution
Let $y(t)$ denote the balance in the account at the start of year $t\text{.}$ Then
\begin{equation*} \frac{dy}{dt} = ky(t)\text{,} \end{equation*}
for some constant $k\text{.}$ We can solve this differential equation using separation of variables to obtain
\begin{equation*} y(t)=y_0e^{kt} = 5000e^{kt}\text{,} \end{equation*}
where we used the fact that $y(0)=5,000\text{.}$ We know that $y(4)=7000\text{,}$ which we now use to solve for $k\text{:}$
\begin{equation*} \begin{split} 7000 \amp = 5000e^{4k} \\ e^{4k} \amp =\frac{7}{5}\\ k \amp = \frac{\ln\left(\frac{7}{5}\right)}{4} \end{split} \end{equation*}
Therefore,
\begin{equation*} y(t) = 5000e^{\frac{\ln\left(7/5\right)}{4}t}\text{.} \end{equation*}
1. After 5 years, the balance in the account is
\begin{equation*} y(5)=5000e^{\frac{\ln\left(7/5\right)}{4} \cdot 5} \approx \\$ 7614.30\text{.} \end{equation*}
2. We wish to find $t$ so that $y(t)=10000\text{.}$
\begin{equation*} \begin{split} 10000 \amp = 5000e^{\frac{\ln\left(7/5\right)}{4}t} \\ e^{\frac{\ln\left(7/5\right)}{4}t} \amp = 2 \\ t \amp = \frac{\ln(16)}{\ln\left(\frac{7}{5}\right)} \approx 8.24 \end{split} \end{equation*}
Thus, it takes just over 8 years for the balance in the account to double in value.
### Subsection5.2.4Logistic Growth Model
The simple growth model is unrealistic because a quantity that represents something from real life, say, population, does not grow unrestricted. Typically, food resources, competition, predators, or diseases, to name but a few factors, influence the growth of the population, and how much of the population can be sustained in such an environment.
A more realistic model is the so-called logistic growth model, which mimics that as a quantity is growing other factors will influence the growth and slow it down until a certain maximum size is being approached. For example, if a population is growing, then food may become scarce or diseases may break out among the population, and the population growth slows down until a certain sustainable size is reached. Replacing $k$ in the simple model with $r(M-y)$ achieves that as the quantity $y$ increases the growth rate decreases, and furthermore, that the maximum population size that is sustained is $M\text{.}$ This maximum is referred to as the carrying capacity. So the simple model becomes
\begin{equation*} \frac{dy}{dt} = ry(M-y) \end{equation*}
for some positive $r$ and $M\text{.}$
In other words, the rate of growth of the quantity $y$ is proportional to both itself and the remaining carrying capacity that the quantity can still grow to. As usual, $r$ is the growth rate constant.
To solve this first order non-linear differential equation, notice that the equation is separable
\begin{equation*} \frac{dy}{y(M-y)} = r\,dt\text{.} \end{equation*}
Integrating both sides we obtain
\begin{equation*} \begin{split} \int \frac{dy}{y(M-y)} \amp = \int r\,dt \\[1ex] \int \left(\frac{1}{y} + \frac{1}{M-y}\right)\,dy \amp = \int rM \,dt \\[1ex] \ln |y| - \ln|M-y| \amp = rMt + C \\[1ex] \ln \left\vert \frac{M-y}{y} \right\vert \amp = -rMt-C \\[1ex] \left\vert\frac{M-y}{y}\right\vert \amp = e^{-rMt-C} \\[1ex] \frac{M-y}{y} \amp = Ae^{-rMt}, \end{split} \end{equation*}
where $A = \dfrac{M-y_0}{y_0}$ for $t_0=0\text{.}$
Lastly, we solve for $y$ and get the solution
\begin{equation*} y=\frac{M}{1+Ae^{-rMt}} \text{ with } A=\frac{M-y_0}{y_0} \end{equation*}
for some constant $y_0$ at time $t_0=0\text{.}$
The graph below shows the typical $S$-shape of such a solution for some $y_0$ between $0$ and $M\text{.}$ For $y_0 \ge M\text{,}$ the solution decays exponentially to $M\text{.}$ And for $y_0 = M\text{,}$ the solution remains constant.
Note: Let us rewrite
\begin{equation*} \frac{dy}{dt} = ry(M-y) \end{equation*}
as
\begin{equation*} \frac{dy}{dt} = rM\left(\frac{M-y}{M}\right)y\text{.} \end{equation*}
Then we can make the following interpretations.
1. When the quantity $y$ is small, then the term $\dfrac{M-y}{M}$ is close in value to one, and so the differential equation
\begin{equation*} \frac{dy}{dt} = ry(M-y) \approx rMy\text{.} \end{equation*}
In other words, the growth is exponential.
2. However, when the quantity $y$ is near that of the carrying capacity $M\text{,}$ then the term $\dfrac{M-y}{M}$ is close in value to zero. Hence, the less carrying capacity that remains the more the growth rate is slowed down.
###### Logistic Growth Model.
The differential equation
\begin{equation*} \frac{dy}{dt} = ry(M-y) \end{equation*}
with positive growth constant $r$ and carrying capacity $M$ models logistic growth of a quantity $y$ at time $t$ with solution
\begin{equation*} y=\frac{M}{1+Ae^{-rMt}}\text{,} \end{equation*}
where $A=\dfrac{M-y_0}{y_0}$ at time $t_0=0\text{.}$
Interactive Demonstration. Use the sliders below to investigate the differential equation $\frac{dy}{dt} = f(t,y)$ where
##### Exercises for Section 5.2.
Which of the following equations are separable?
1. $\ds y' = \sin (ty)$
Not separable.
2. $\ds y' = e^t e^y$
Separable:
\begin{equation*} \frac{dy}{dt} = e^t e^y \implies \frac{dy}{e^y} = e^t \,dt \end{equation*}
3. $\ds yy' = t$
Separable:
\begin{equation*} y\frac{dy}{dt} = t \implies y\,dy = t\,dt \end{equation*}
4. $\ds y' = (t^3 -t) \arcsin(y)$
Separable:
\begin{equation*} \frac{dy}{dt} = (t^3-t)\arcsin(y) \implies \frac{dy}{\arcsin(y)} = (t^3-t) \,dt \end{equation*}
5. $\ds y' = t^2 \ln y + 4t^3 \ln y$
Not separable.
Identify the constant solutions (if any) of the following differential equations.
1. $y' =t\sin y$
$y=n\pi\text{,}$ for any integer $n\text{.}$
Solution
All constant solutions of the DE will satisfy $y'=0$ for all $t \geq 0\text{.}$ Therefore, we set
\begin{equation*} y' = t\sin(y) = 0 \implies \arcsin(0) = y, \end{equation*}
since $t\neq 0\text{.}$ Therefore, the constant solutions are $y=n\pi\text{,}$ for any integer $n\text{.}$
2. $\ds y'=te^y$
none
Solution
All constant solutions of the DE will satisfy $y'=0$ for all $t \geq 0\text{.}$ Therefore, we set
\begin{equation*} y' = te^y = 0 \implies e^y = 0, \end{equation*}
which has no solutions. Therefore, the DE has no constant solutions.
Solve the following differential equations. You may leave your solution in implicit form: that is, you may stop once you have done the integration, without solving for $y\text{.}$
1. $\ds y' = 1/(1+t^2)$
$\ds y=\arctan t + C$
Solution
This is a first order separable differential equation:
\begin{equation*} \begin{split} \diff{y}{t} \amp = \frac{1}{1+t^2} \\ \int dy \amp = \int \frac{dt}{1+t^2} \\ y \amp = \tan^{-1}(t) + C \end{split} \end{equation*}
Therefore, the general solution is
\begin{equation*} y(t) = \tan^{-1}(t) + C \end{equation*}
for some constant $C\text{.}$
2. $y' = \ln t$
$\ds y=t\ln t-t+C$
Solution
This is a first order seperable differential equation:
\begin{equation*} \begin{split} \diff{y}{t} \amp= \ln t \\ \int \,dy \amp= \int \ln t\,dt \\ y(t) \amp= t\ln t - t + C. \end{split} \end{equation*}
3. $y' = t/y$
$\ds y=\pm\sqrt{t^2+C}$
Solution
This is a first order seperable differential equation:
\begin{equation*} \begin{split} \diff{y}{t} \amp= \frac{t}{y} \\ \int y\,dy \amp= \int t\,dt \\ \frac{y^2}{2} \amp= \frac{t^2}{2} + C, \end{split} \end{equation*}
or,
\begin{equation*} y=\pm\sqrt{t^2+C}\text{.} \end{equation*}
4. $\ds y' = y^2 -1$
$\ds y=\pm 1\text{,}$ $\ds y=(1+Ae^{2t})/(1-Ae^{2t})$
Solution
This is a first order seperable differential equation:
\begin{equation*} \begin{split} \diff{y}{t} \amp= y^2-1 \\ \int \frac{1}{y^2-1}\,dy \amp= \int \,dt \end{split} \end{equation*}
However, this is not valid if $y=\pm 1\text{.}$ In this case, we have $y' = 0\text{,}$ and so these are constant solutions of the DE. Now suppose $y\neq \pm 1\text{.}$ To solve the integral
\begin{equation*} \int \frac{1}{y^2-1} \,dy \end{equation*}
we use partial fraction decomposition:
\begin{equation*} \begin{split}\int \frac{1}{y^2-1}\,dy \amp= \int \frac{1}{2(y-1)}- \frac{1}{2(y+1)}\,dy \\ \amp= \frac{1}{2} \left(\ln|y-1| - \ln|y+1|\right) + C. \end{split} \end{equation*}
Therefore, the general solution (in implicit form) is
\begin{equation*} \frac{1}{2} \left(\ln|y-1| - \ln|y+1|\right) + C = t, \end{equation*}
and $y= \pm 1 \text{.}$
5. $\ds y' = t/(y^3 - 5)$
$\ds y^4/4-5y=t^2/2+C$
Solution
We again notice that the differential equation is separable. Therefore, we compute
\begin{equation*} \begin{split} \diff{y}{t} \amp = \frac{t}{y^3-5} \\ \int \left(y^3-5\right)\,dy \amp = \int t\,dt \\ \frac{1}{4}y^4 - 5y \amp = \frac{1}{2}t^2 + C \end{split} \end{equation*}
Hence, the general solution in implicit form is
\begin{equation*} \frac{1}{4}y(t)^4 - 5y(t) = \frac{1}{2}t^2 + C \end{equation*}
for some constant $C\text{.}$
6. $\ds y'=k(M-y)$
$\ds y=M+Ae^{-kt}$
Solution
This is a first order seperable differential equation. First, notice that $y=M$ is a constant solution. Now suppose that $y\neq M$ and separate:
\begin{equation*} \begin{split} \diff{y}{t} \amp= k(M-y) \\ \int \frac{1}{M-y}\,dy \amp= \int k \,dt \\ - \ln |M-y| \amp= kt + C \end{split} \end{equation*}
This is the general solution in implicit form. Note that we can further write $y$ explicitly as
\begin{equation*} y(t) = M+Ae^{-kt}, \end{equation*}
for some constant $A\text{.}$ For $A=0$ we recover the solution $y=M\text{.}$
Solve the following initial value problems.
1. $y' = t^n\text{,}$ $y(0)=1$ and $n\ge 0$
$\ds y={t^{n+1}\over n+1}+1$
Solution
We find the general solution by separating variables:
\begin{equation*} \begin{split} \diff{y}{t} \amp= t^n \\ \int \,dy \amp= \int t^n \,dt\\ y(t) \amp= \frac{t^{n+1}}{n+1} + C \end{split} \end{equation*}
Therefore, if $y(0) = 1\text{,}$ we require
\begin{equation*} y(0) = 0 + C = 1 \implies C = 1. \end{equation*}
Therefore, for any $n\geq 0\text{,}$ the solution to this IVP is
\begin{equation*} y(t) = \frac{t^{n+1}}{n+1} + 1. \end{equation*}
2. $y' = y^{1/3}\text{,}$ $y(0)=0$
$\ds y=(2t/3)^{3/2}$ and $y(t)=0$
Solution
We note that $y(t) = 0$ is a constant solution to this DE, and further satisfies the initial condition. Therefore, a solution to this IVP is $y(t) = 0\text{.}$ We look for any more solutions by separating variables (assuming $y\neq 0$):
\begin{equation*} \begin{split} \diff{y}{t} \amp= y^{1/3} \\ \int y^{-1/3}\,dy \amp= \int \,dt\\ \frac{3y^{2/3}}{2} \amp= t + C \\ y(t) \amp= \frac{2}{3} t^{3/2} + C. \end{split} \end{equation*}
Since $y(0) = 0\text{,}$ we see that we must have $C=0\text{.}$ Therefore, another solution to the IVP is
\begin{equation*} y(t) = \frac{2}{3} t^{3/2} . \end{equation*}
3. $y' = ky\text{,}$ $y(0)=2\text{,}$ and $y'(0)=3$
$\ds y=2e^{3t/2}$
Solution
We find the general solution by separating variables:
\begin{equation*} \begin{split} \diff{y}{t} \amp= ky \\ \int \frac{1}{y} \,dy \amp= \int k \,dt\\ \ln|y| \amp= kt + C \\ y(t) \amp= Ae^{kt} \end{split} \end{equation*}
for some constant $A\text{.}$ Since $y(0) =2\text{,}$ we require
\begin{equation*} y(0) = Ae^{k(0)} = A = 2. \end{equation*}
Now if $y'(0) = 3\text{,}$ we require
\begin{equation*} y'(0) = 2 ke^{k(0)} = 2k = 3 \implies k = \frac{3}{2}. \end{equation*}
Therefore, the solution to the IVP is
\begin{equation*} y(t) = 2e^{3t/2}. \end{equation*}
After 10 minutes in Jean-Luc's room, his tea has cooled to $40^\circ$ Celsius from $100^\circ$ Celsius. The room temperature is $25^\circ$ Celsius. How much longer will it take to cool to $35^\circ\text{?}$ Answer
$\approx 2$ minutes
Solution
By Newton's Law of Cooling, the temperature of Jean-Luc's tea can be described by the DE
\begin{equation*} \diff{y}{t} = k (25-y), \end{equation*}
for some $k\text{,}$ where 25 is the ambient room temperature, such that $y(0) = 100$ and $y(10) = 40$ (with $t$ measured in minutes). We first solve for the general solution of this DE. Suppose that $y\neq 25$ (we note that this is not a solution to the IVP). Then we can separate variables:
\begin{equation*} \begin{split} \diff{y}{t} \amp= k (25-y) \\ \int \frac{1}{25-y} \,dy \amp= \int k\,dt \\ -\ln |25-y| \amp= kt + C\\ y \amp= 25 + Ae^{-kt},\end{split} \end{equation*}
for some constants $A$ and $k\text{.}$ We now use the given data to find these constants:
\begin{equation*} y(0) = 100 \implies 25+ A = 100 \implies A = 75 \end{equation*}
and
\begin{equation*} y(10) = 40 \implies 75e^{-10k} = 40 \implies k = \frac{1}{10} \ln(15/8). \end{equation*}
All together, we find that the temperature of the tea can be described by the function
\begin{equation*} y(t) = 75e^{-\ln(15/8)/10 t}. \end{equation*}
We now solve for $t\text{:}$
\begin{equation*} y(t) = 35 \implies 75e^{-\ln(15/8) t/10} = 35 \implies t = \frac{10\log(15/7)}{\log(15/8)} \approx 12.124. \end{equation*}
Hence, it will take about 2 more minutes for the tea to cool down to 35 degrees.
A radioactive substance obeys the equation $y' =ky$ where $k\lt 0$ and $y$ is the mass of the substance at time $t\text{.}$ Suppose that initially, the mass of the substance is $y(0)=M>0\text{.}$ At what time does half of the mass remain? (This is known as the half life. Note that the half life depends on $k$ but not on $M\text{.}$) Answer
$\ds t=-{\ln 2\over k}$
Solution
We first find the general solution to the DE. Suppose that $y\neq 0$ (since we assume that $M>0\text{,}$ this is not a solution to the IVP). Now separate variables:
\begin{equation*} \begin{split} \diff{y}{t} \amp= ky \\ \int \frac{dy}{y} \amp= \int k\, dt \\ \ln |y| \amp= kt + C \\ y(t) \amp= Ae^{kt} \end{split} \end{equation*}
for some constants $A$ and $k\text{.}$ Now apply the initial data:
\begin{equation*} y(0) = M \implies A = M. \end{equation*}
Therefore,
\begin{equation*} y(t) = Me^{kt} \end{equation*}
is the solution to the IVP. We now want to find the time it takes for the initial mass to be reduced by one half. That is, we want to solve for $t^*$ where $y(t^*) = \frac{M}{2}\text{:}$
\begin{equation*} \begin{split} y(t^*) \amp= Me^{kt^*} = \frac{M}{2}\\ \implies e^{kt^*} \amp= \frac{1}{2}\\ kt^* \amp= \ln(1/2)\\ t^* \amp= \frac{1}{k} \ln(1/2). \end{split} \end{equation*}
Therefore, the half-life is $t^* = -\frac{\ln(2)}{k}\text{.}$ Note that since $k\le 0\text{,}$ we have $t^*\ge 0$ as required.
Bismuth-210 has a half life of five days. If there is initially 600 milligrams, how much is left after 6 days? When will there be only 2 milligrams left? Answer
$\ds 600e^{-6\ln 2/5}\approx 261$ mg; $\ds {5\ln 300\over\ln2}\approx 41$ days
Solution
We model the decay of a radioactive substance by the equation $y'=ky\text{.}$ Therefore, let $y(t)$ be the amount of Bismuth remaining, measured in mg, where $t$ is in days. We first solve the initial value problem:
\begin{equation*} \begin{split} \diff{y}{t} \amp = ky \\ \int \frac{dy}{y} \amp = \int k\, dt \\ \ln |y| \amp = kt + C \\ y(t) \amp = Ae^{kt} \end{split} \end{equation*}
Applying the initial data, $y(0)=600\text{,}$ we see that $y(t)=600e^{kt}\text{.}$ To solve for $k\text{,}$ we use the fact that the half-life of Bismuth is 5 days. Mathematically, this means that
\begin{equation*} \begin{split} y(5) \amp = \frac{1}{2} y(0) \\ e^{5k} \amp = \frac{1}{2} \\ \implies k \amp = \frac{1}{5} \ln\left(\frac{1}{2}\right) = -\frac{\ln 2}{5}. \end{split} \end{equation*}
Notice that $k \lt 0$ and that the quantity of Bismuth is decreasing with time, as desired.
Therefore, the solution to the initial value problem is
\begin{equation*} y(t) = 600e^{-\frac{\ln 2}{5}t}\text{.} \end{equation*}
Now,
\begin{equation*} y(6)=600e^{-\frac{\ln 2}{5}\cdot 6} \approx 261\text{.} \end{equation*}
So, after 6 days, we find that there is approximately 261 mg of Bismuth remaining.
Additionally, we wish to find $t_*$ such that $y(t_*) = 2\text{:}$
\begin{equation*} 2 = 600e^{-t_*\frac{\ln 2}{5}} \implies t_* = \frac{5\ln 300}{\ln 2} \approx 41\text{.} \end{equation*}
Thus, after approximately 41 days, there is only 2 mg of Bismuth remaining.
The half life of carbon-14 is 5730 years. If one starts with 100 milligrams of carbon-14, how much is left after 6000 years? How long do we have to wait before there is less than 2 milligrams? Answer
$\ds 100e^{-200\ln 2/191}\approx 48$ mg; $\ds {5730\ln 50\over\ln2}\approx 32339$ years.
Solution
We model the decay of a radioactive substance by the equation $y'=ky\text{.}$ Therefore, let $y(t)$ be the amount of Bismuth remaining, measured in mg, where $t$ is in years. We first solve the initial value problem:
\begin{equation*} \begin{split} \diff{y}{t} \amp= ky \\ \int \frac{dy}{y} \amp= \int k\, dt \\ \ln |y| \amp= kt + C \\ y(t) \amp= Ae^{kt} \end{split} \end{equation*}
Applying the initial data, $y(0)=100\text{,}$ we see that $y(t)=100e^{kt}\text{.}$ To solve for $k\text{,}$ we use the fact that the half-life of carbon-14 is 5730 years. This means that
\begin{equation*} \begin{split} y(5730) \amp= \frac{1}{2} y(0) \\ e^{5730k} \amp= \frac{1}{2} \\ \implies k \amp= \frac{1}{5730} \ln\left(\frac{1}{2}\right) = -\frac{\ln 2}{5730}. \end{split} \end{equation*}
Notice that $k \le 0$ and that the quantity of carbon-14 is decreasing with time, as desired. \\ Therefore, the solution to the initial value problem is
\begin{equation*} y(t) = 100e^{-\frac{\ln 2}{5730}t}. \end{equation*}
Now,
\begin{equation*} y(6000)=100e^{-\frac{\ln 2}{5730}\cdot 6000} \approx 48.39. \end{equation*}
So, after 6000 years, we find that there is approximately 48.39 mg of carbon-14 remaining (note that this is just under half of the initial amount, as expected). Additionally, we wish to find $t_*$ such that $y(t_*) = 2\text{:}$
\begin{equation*} 2 = 100e^{-t_*\frac{\ln 2}{5730}} \implies t_* = \frac{5730\ln(50)}{\ln(2)} \approx 32,339. \end{equation*}
Therefore, we have to wait more than 32,339 years for the amount carbon-14 to reduce to less than 2 mg.
A certain species of bacteria doubles its population (or its mass) every hour in the lab. The differential equation that models this phenomenon is $y' =ky\text{,}$ where $k>0$ and $y$ is the population of bacteria at time $t\text{.}$ What is $y\text{?}$ Answer
$\ds y=y_0e^{\ln(2)t}$
Solution
We model the growth of bacteria by the equation $y'=ky$ where $k>0\text{.}$ We first solve for the general solution assuming that $y\neq 0\text{:}$
\begin{equation*} \begin{split} \diff{y}{t} \amp= ky \\ \int \frac{dy}{y} \amp= \int k\, dt \\ \ln |y| \amp= kt + C \\ y(t) \amp= Ae^{kt} \end{split} \end{equation*}
Suppose that initially we have $y_0$ bacteria present: $y(0) = y_0 > 0\text{.}$ Then, the number of bacteria at time $t$ can be described by the function $y(t)= y_0 e^{kt}.$ Since $k>0\text{,}$ the amount of bacteria is growing exponetially. Now use the fact that the bacteria doubles in growth every hour:
\begin{equation*} y(1) = 2y_0 \implies e^{k(1)} = 2 \implies k = \ln(2). \end{equation*}
Therefore, the growth of this particular species can be described by the function
\begin{equation*} y(t)= y_0 e^{\ln(2) t}. \end{equation*}
If a certain microbe doubles its population every 4 hours and after 5 hours the total population has mass 500 grams, what was the initial mass? Answer
$\ds 500e^{-5\ln2/4}\approx 210$ g
Solution
We model the growth of bacteria by the equation $y'=ky$ where $k>0\text{,}$ where $t$ is measured in hours and $y$ is measured in grams. We first solve for the general solution assuming that $y\neq 0\text{:}$
\begin{equation*} \begin{split} \diff{y}{t} \amp= ky \\ \int \frac{dy}{y} \amp= \int k\, dt \\ \ln |y| \amp= kt + C \\ y(t) \amp= Ae^{kt}\\ \amp= y_0 e^{kt} \end{split} \end{equation*}
Now use the fact that the bacteria doubles in growth every 4 hours:
\begin{equation*} y(4) = 2y_0 \implies e^{k(4)} = 2 \implies k = \frac{1}{4}\ln(2). \end{equation*}
After 5 hours total, the population has 500 grams:
\begin{equation*} y(5) = 500 = y_0 e^{\frac{5}{4}\ln(2)} \implies y_0 = 500e^{-5\ln2/4} \approx 210. \end{equation*}
Therefore, the initial mass must have been about 210 g.
Given the logistic equation $y' = ky(M-y)\text{,}$
1. Solve the differential equation for $y$ in terms of $t\text{.}$
$\ds y={M\over 1+Ae^{-Mkt}}$
Solution
We separate variables (assuming $y \neq M,$):
\begin{equation*} \begin{split} \int \frac{dy}{y(M-y)} \amp= \int k\,dt \\[1ex] \int \left(\frac{1}{y} + \frac{1}{M-y}\right)\,dy \amp= \int kM \,dt \\[1ex] \ln |y| - \ln|M-y| \amp= kMt + C \\[1ex] \ln \left\vert \frac{M-y}{y} \right\vert \amp= -kMt-C \\[1ex] \left\vert\frac{M-y}{y}\right\vert \amp= e^{-kMt-C} \\[1ex] \frac{M-y}{y} \amp= Ae^{-kMt}, \end{split} \end{equation*}
Therefore, we find that
2. Sketch the graph of the solution to this equation when $M=1000\text{,}$ $k=0.002\text{,}$ $y(0)=1\text{.}$ Answer
When $M=1000\text{,}$ $k=0.002$ and $y(0) =1\text{,}$ we have $A = \frac{1000-1}{1} = 999\text{:}$
\begin{equation*} y(t) = \frac{1000}{1+999e^{-2t}} \end{equation*}
as shown below:
The biologist G. F. Gause studied the growth of the protozoan Paramecium in the early 1930s. Through his data, he figured out that the relative growth rate is 0.7944 when $y(0)=2\text{,}$ and the carrying capacity is 64. This leads to the logistic model
\begin{equation*} \frac{dy}{dt}=0.7944y\left(1-\frac{y}{64}\right), y(0)=2\text{,} \end{equation*}
where time is measured in days.
1. Solve the differential equation for $y$ in terms of $t\text{.}$ Answer
$y(t) = \frac{64}{1+31e^{-50.8416t}}$
Solution
We separate variables (assuming $y \neq 64$):
\begin{equation*} \begin{split} \int \frac{dy}{y(1-y/64)} \amp= \int 0.7944\,dt \\[1ex] \int \left(\frac{1}{y} + \frac{1}{64-y}\right)\,dy \amp= \int 64(0.7944) \,dt \\[1ex] \ln |y| - \ln|64-y| \amp= 50.8416t + C \\[1ex] \ln \left\vert \frac{64-y}{y} \right\vert \amp= -50.8416t-C \\[1ex] \left\vert\frac{64-y}{y}\right\vert \amp= e^{-50.8416t-C} \\[1ex] \frac{64-y}{y} \amp= Ae^{-50.8416t}, \end{split} \end{equation*}
Therefore, we find that
We solve for $t\text{:}$
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2020-07-10 07:32:07
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https://physics.stackexchange.com/questions/245686/higgs-boson-and-dark-matter?noredirect=1
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# Higgs boson and dark matter.
In the standard model the Higgs boson gives the mass to other particles, but in the Universe we know that the 80% of mass is in form of dark matter, that is not constituted by known particles. The Higgs boson gives the mass also to the dark matter with the same mechanism?
• Since we don't know what exactly constitutes dark matter, this question is unanswerable within the currently accepted models. – ACuriousMind Mar 26 '16 at 21:11
• But, if the dark matter is constituted by some kind of supersymmetric partners ( suppose), the Higgs mechanism can works? – Emilio Novati Mar 26 '16 at 21:17
• The Higgs mechanism doesn't "give" more than a small sliver of the mass in the universe. It's really nothing more than another epicycle that is required for self-consistency reasons in the standard model. If you add super-epicycles, you merely increase the number of free parameters in the fit (by about a hundred or so, if I remember correctly). It doesn't buy you anything in terms of understanding. – CuriousOne Mar 26 '16 at 22:20
• arxiv.org/abs/1305.0021 – Count Iblis Mar 27 '16 at 16:40
• I am assuming you are talking about gravitational mass for whatever constitutes dark matter. What is the link to gravitational mass for the Higgs field? – Peter R Mar 28 '16 at 15:55
The question is whether dark matter gets its mass from the Higgs field. The answer depends on the composition of dark matter, so let's discuss the mass explanations for several common composition hypotheses in turn. (We needn't discuss alternatives to dark matter, such as MOND or gravity no longer obeying an inverse square law over kiloparsecs.)
If dark matter is neutrinos, blame the seesaw mechanism. (However, as a comment below notes, this hypothesis hasn't aged well.)
If dark matter is composed of MACHOs - in short, a particular kind of star or former star - almost all the mass comes from baryons (particles such as protons and neutrons), and almost all of their mass comes from the potential energy of the strong force holding the quarks together in baryons. This is why, although the Higgs field gives the quarks in protons some mass, the proton mass is dozens of times what would be expected from that alone.
Finally, if dark matter is composed of the lightest "supersymmetric partner", the Higgs field is responsible. One motivation for postulating supersymmetric partners much more massive than familiar particles is that it allows us to reduce quadratic divergences in the Higgs field to logarithmic divergences. One version of this theory conserves a multiplicative charge called R-parity, which is $1$ for known particles but $-1$ for their supersymmetry partners. The lightest such partner therefore cannot decay, even if its Higgs-induced mass is very large.
• SM neutrinos cannot explain dark matter see physics.stackexchange.com/questions/17227/… – anna v Mar 28 '16 at 16:23
• True; it's more of a historical hypothesis than a modern one. Editing now. – J.G. Mar 28 '16 at 17:47
Dark matter is a necessary hypothesis within the general relativity model of the universe in order to fit the observational data of rotational curves. In simple words, the trajectories can only be explained if there exists a lot more matter in the galaxies than luminous matter. Luminous matter means that electromagnetic interactions generate light which is measured and used to calculate the amount of matter giving that luminosity. Dark matter must be composed out of masses that do not interact electromagnetically to first order.
There are various models that propose various ways massive particles could exist , for example MACHOs.
Massive astrophysical compact halo object (MACHO) is any kind of astronomical body that might explain the apparent presence of dark matter in galaxy halos. A MACHO is a body composed of normal baryonic matter that emits little or no radiation and drifts through interstellar space unassociated with any planetary system. Since MACHOs are not luminous, they are hard to detect. MACHOs include black holes or neutron stars as well as brown dwarfs and unassociated planets. White dwarfs and very faint red dwarfs have also been proposed as candidate MACHOs.
In this model the Higgs mechanism does not play a special role other than the usual standard model role in giving masses to its elementary particles.
There are elementary particle models, beyond the standard model, where stable weakly interacting massive particles exist in the spectrum, called WIMPs:
In particle physics and astrophysics, weakly interacting massive particles, or WIMPs, are among the last hypothetical particle physics candidates for dark matter. The term “WIMP” is given to a dark matter particle that was produced by falling out of thermal equilibrium with the hot dense plasma of the early universe, although it is often used to refer to any dark matter candidate that interacts with standard particles via a force similar in strength to the weak nuclear force.
For these particles there exists a Higgs mechanism which will give them mass as elementary particles of these new theories.
Please note that it is the Higgs mechanism, the Higgs field that gives mass to the elementary particles in the standard model, not the Higgs boson. The Higgs boson is a predicted manifestation of the existence of this field, and acquires its mass from this field too.
## protected by Qmechanic♦Mar 28 '16 at 16:19
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
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2019-10-19 06:01:29
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https://ulysseszh.github.io/archives/2022/11/13/
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## Archive of posts on Nov 13, 2022
• ### An example of non-uniform elements: heavy elastic rope
To illustrate the concept about non-uniform elements, we study a simple problem: suppose a uniform heavy elastic rope has mass $m$, original length $L_0$, and stiffness $k$, and find the mass distribution and length of it when hung vertically. We can use the element method to solve this problem, but the elements are non-uniform in terms of length. The elements add up to get the total length $L=\frac{mg}{2k}+L_0$.
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2023-02-05 04:16:26
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|
https://crypto.stackexchange.com/questions/18488/ecdsa-with-sha256-and-sepc192r1-curve-impossible-or-how-to-calculate-e/18489
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# ECDSA with SHA256 and sepc192r1 curve: Impossible, or how to calculate $e$?
I need to use ECDSA as the signing algorithm and SHA256 for hashing the message. I'm running into troubles verifying the signature calculated on two different platform (one is BouncyCastle, another one a C library for a microprocessor).
I figured out that BouncyCastle, in its ECDSASigner class, reduces the input message (which is supposed to be already the hash), in both, generateSignature() and verifySignature() using a helper function called calculateE(). In essence, this functions truncates the input message/hash to the bitlength of the curve's order $N$.
protected BigInteger calculateE(BigInteger n, byte[] message)
{
int log2n = n.bitLength();
int messageBitLength = message.length * 8;
BigInteger e = new BigInteger(1, message);
if (log2n < messageBitLength)
{
e = e.shiftRight(messageBitLength - log2n);
}
return e;
}
This reduction results from the fact that a SHA256 hash has 32 Bytes, whereas the size of $N$ is (for secp192r1) 192/8 = 24 Bytes.
What I don't understand:
1. Do I have to use a curve with greater size of bitlength(N) for SHA256 hashes to be signed (e.g., secp256r1 or secp521r1)?
2. Is the implementation in BouncyCastle wrong/imcomplete, or only applies to SHA-1 (160 b = 20 B <= 24 B)?
3. Where is it documented/specified, how to “reduce” a hash of greater bitlength than that of the curve domain?
NIST FIPS 186-4 at the end of section 6.4 states that:
When the length of the output of the hash function is greater than the bit length of $n$, then the leftmost $n$ bits of the hash function output block shall be used in any calculation using the hash function output during the generation or verification of a digital signature.
In section 6.1 they define $n$ as the order of the generator, which makes the wording
leftmost $n$ bits
wrong. Although it's clear they meant the bit size of $n$.
Which means:
1. You don't need a bigger curve
2. BouncyCastle implementation looks correct
• I don't need a bigger curve, but from pure security perspective I should use a "bigger" one, right? Because by truncating my hash I lose security (in terms of bits), as far as I understand? – Christian Aug 5 '14 at 13:56
• @Christian The security level is limited to half the curve size. So with a 192 bit curve you get 96 bits of security. This is still infeasible to break, but you should consider upgrading to a 224 or 256 bit curve. – CodesInChaos Aug 5 '14 at 14:26
• @Christian The security level will be the minimum value between half of the digest size of your hash function and half of the size of the order of the curve. So for P-192 and SHA-256 you get 96 bits of security. If you want 128 bit of security you need (at least) P-256 and SHA-256 – Ruggero Aug 5 '14 at 15:03
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2021-06-14 09:14:08
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https://www.nature.com/articles/s42004-022-00641-3?utm_campaign=related_content&utm_source=CHEM&utm_medium=communities&error=cookies_not_supported&code=cdaa5646-be59-45c0-ba37-af665a46776d
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# Assignment-free chirality detection in unknown samples via microwave three-wave mixing
## Abstract
Straightforward identification of chiral molecules in multi-component mixtures of unknown composition is extremely challenging. Current spectrometric and chromatographic methods cannot unambiguously identify components while the state of the art spectroscopic methods are limited by the difficult and time-consuming task of spectral assignment. Here, we introduce a highly sensitive generalized version of microwave three-wave mixing that uses broad-spectrum fields to detect chiral molecules in enantiomeric excess without any prior chemical knowledge of the sample. This method does not require spectral assignment as a necessary step to extract information out of a spectrum. We demonstrate our method by recording three-wave mixing spectra of multi-component samples that provide direct evidence of enantiomeric excess. Our method opens up new capabilities in ultrasensitive phase-coherent spectroscopic detection that can be applied for chiral detection in real-life mixtures, raw products of chemical reactions and difficult to assign novel exotic species.
## Introduction
Many biomolecules, including DNA, proteins, and amino acids, are chiral, meaning they exist in two versions that are non-superimposable mirror images. Chirality is such a ubiquitous property in biology that more than 50 percent of active ingredients in pharmaceuticals are chiral1. Chiral molecules span other multi-billion dollar industries like the food industry, agriculture, and fragrances. In 2016, the first chiral molecule was detected in space2, re-sparking conversations on the implications of molecular chirality for the origins of life. However, currently established methods cannot determine enantiomeric excess, a signature of life, in complex raw samples like the ones collected from extraterrestrial environments.
Despite these broad applications, a general method for detecting and measuring enantiomeric excess remains elusive. While notable progress has been made towards the detection of slight enantiomeric excess on the 0.4% level3, detection of enantiomeric excess in unknown complex samples has proven challenging. Chromatography has long been the go-to method for enantiomeric analysis among synthetic chemists, however, as detection is based on chemical interactions, it cannot be generalized to unknown samples. Mass spectrometry and nuclear magnetic resonance (NMR) rely on chiral derivatization reagents and can be sensitive to contaminants4,5,6. For unknown multi-component mixtures, polarimetry can be inconclusive, as the calculation of specific rotation requires knowledge of concentration and it is often referenced to neat samples7,8,9.
Spectroscopic methods such as vibrational, photoelectron circular dichroism10,11,12,13,14, and microwave spectroscopy15,16,17,18,19,20,21,22,23 can be mixture compatible and provide highly accurate information on species identity. So far, these methods have been limited by spectral assignment; prior to any chirality experiment, the spectrum of the molecule had first to be collected and fully assigned. Considerable efforts have been made to automate and simplify spectral assignment24,25,26,27; nonetheless, it is still a difficult and time-consuming task conducted mainly by trained spectroscopists.
In this work, we demonstrate a generalized assignment-free version of microwave three-wave mixing (M3WM)28,29,30 that can identify chiral species in enantiomeric excess in unknown complex samples. We achieve this by exploiting our high sensitivity and employing broadband excitation pulses to search experimentally for transitions in a three-level system, along with the implementation of careful cancellation schemes to ensure that signals from species not in enantiomeric excess are subtracted. The resulting spectra, referred to here as “three-wave mixing spectra”, can provide direct proof on the existence of chiral species in enantiomeric excess and can be used for the study of previously hard-to-analyze samples: unassigned species and unknown complex mixtures. While the samples used in this work were not prepared by an outside team and were thus strictly speaking ‘known’ to our team, M3WM and proof of enantiomeric excess was demonstrated on these samples with no sample-dependent settings.
### Broadband three-wave mixing
Our broadband assignment-free three-wave mixing uses broadband microwave and RF excitation combined with careful cancellation schemes. Polarization and phase controllability for broadband pulses are achieved with an updated experimental setup and microwave circuit (discussed more in detail under Methods). While knowing the exact phase of the relevant component of each chirp is challenging, the repeatability of this phase is excellent, as it must be in all chirp pulse microwave three-wave mixing experiments, and it can be accurately reversed by changing the phase of the signal coming from the arbitrary waveform generators. The resulting three-wave mixing spectra include numerous transitions from each chiral molecule that is present in enantiomeric excess. Each of those transitions stem from a M3WM excitation scheme, an example of which is shown in Fig. 1b. It is a three-level system of rotational energy levels that are connected via an a-type transition, a b-type transition, and a c-type transition, along each of three rotational axes. Two of these transitions are typically in the GHz frequency range, and the third transition is on the order of 100 MHz. The stimulated microwave transition (with frequency ~ 10 GHz) is referred to as the “drive” transition and the stimulated RF transition (with frequency ~ 100 MHz) is referred to as the “twist” transition28. The molecular ensemble emits radiation coherently at the “listen” frequency, which is detected and plotted as a spectrum. Previous M3WM experiments were limited to assigned, known species, required prior knowledge of the transitions, and reported enantiomeric excess based on a single excitation scheme like the one shown in Fig. 1b28,29,31,32,33.
## Results
### M3WM spectra of chiral and non-chiral species
Figure 2 highlights the difference between microwave spectra and three-wave mixing spectra. Figure 2a shows the microwave spectrum of a mixture of a chiral molecule ((R)-myrtenal) and a non-chiral molecule (benzyl alcohol), compared to spectra of the individual components. In this frequency range, numerous rotational transitions from both species are present. Figure 2b shows the three-wave mixing spectrum of the same mixture, taken under similar conditions. Three-wave mixing spectra are non-zero only for chiral molecules in enantiomeric excess. Only transitions from enantiopure R-myrtenal are observed, as transitions from the non-chiral benzyl alcohol do not survive subtraction. It is noticeable that the transitions in the three-wave mixing spectrum are significantly fewer in number than the lines in the microwave spectrum of myrtenal, which is expected as not all transitions can participate in a M3WM chirality detection scheme, as the one shown in Fig. 1b.
Both spectra were recorded at 7 K, from 13,000–18,250 MHz and a He buffer gas flow of 10 sccm. The M3WM spectrum is assembled from 485 individual spectral segments, with 22.5 MHz local oscillator steps between them and acquired with a 35 MHz broadband drive pulse and an RF pulse with a range of 60–105 MHz for a total integration time of 3.5 h. For a list of the M3WM transitions see Supplementary Table 1.
### M3WM spectrum of racemic samples
M3WM spectra are designed to detect species in enantiomeric excess. Figure 3 shows the comparison between the M3WM spectrum of an enantiopure sample of (R)-1,2-propanediol, shown in blue, plotted against the M3WM spectrum of a racemic sample of 1,2-propanediol, in red. The M3WM signal of enantiopure (R)-1,2-propanediol shows three noticeable signals corresponding to the lowest and the third-lowest in energy (0.88 kJ/mol) conformer34. In contrast, these three-wave mixing signals are not present in the spectrum of the racemic sample, in red, which has been shifted by −15 (a.u) on the y-axis for clarity. The details of the methods used to eliminate non-chiral signals are described in detail below.
Both spectra were recorded in the range between 14,500–5100 MHz with a He buffer gas flow of 10 sccm, at 7 K. Each spectrum is assembled from 72 individual spectral segments, with 22.5 MHz local oscillator steps between them and acquired with a 35 MHz wide drive pulse and a twist pulse range of 80–105 MHz for a total integration time of 1 h. For a list of the M3WM transitions see Supplementary Table 2.
### M3WM spectrum of multi-component mixtures
Three-wave mixing spectra can provide useful chirality information of multi-component mixtures without any prior chemical processing, separation, or spectral assignment. This capability is relevant to asymmetric synthesis and chemical analysis of complex real-life samples. In Fig. 4, we show the three-wave mixing spectrum for a mixture of terpenes. Terpenes are naturally occurring chiral building blocks that have been used for decades as starting materials for the synthesis of natural products and active ingredients in pharmaceuticals, due to their abundance and low cost35,36,37. All transitions in Fig. 4 belong to enantiopure (-)-β-pinene, (R)-fenchone, and (R)-carvone. The inset zooms into the transition around 16,492 MHz which consists of two separate M3WM signals: a β-pinene M3WM signal at 16,491.7 MHz and a second one from fenchone at 16,492.5 MHz. For such mixtures, even polarimetry measurements can be inconclusive, as the sum of the angles for multiple components can cancel each other out. Equal amounts of neat (-)-β-pinene, (S)-carvone, and (R)-fenchone would have a total specific rotation $${[a]}_{20}^{D}$$ of +7, the sum of each component, which carries significantly less chemical information than a spectrum. In contrast, the three-wave mixing spectrum of such mixture, as seen in Fig. 4, shows distinct transitions for each separate species. If microwave spectra of the species are available, even if unassigned, then no additional measurements are required to determine the exact identity of the species. For readily available chiral building blocks like the ones used here, species were easily and accurately identified.
The spectrum of the mixture was recorded in the range between 16,200–18,000 MHz with a He buffer gas flow of 10 sccm, at 7 K. Each of the 150 spectral segments was recorded with a drive pulse of 35 MHz bandwidth. The total acquisition time was 2 h. Two separate twist ranges of 65–85 MHz and 85–105 MHz were used for increased RF power to assure transitions of less polar species are sufficiently driven. For a list of the M3WM transitions see Supplementary Table 3.
## Discussion
Three-wave mixing spectra can be recorded for any sample that contains molecules that are vaporizable and have non-zero electric dipole moments across all rotational axes. Microwave spectroscopy is mixture, solvent, isomer, and isotopologue compatible meaning that no chemical processing is necessary in most cases prior to analysis30,38,39. Chiral information can be extracted on-the-spot as only transitions from species that are chiral and in enantiomeric excess survive cancellation. As shown in Figs. 2, 3 signals from racemic samples or non-chiral molecules average to zero.
A promising application would be the direct chiral detection of the raw constituents of one-pot asymmetric synthesis reactions40,41. Inside this flask, there are reactants, solvents, products, by-products, and catalysts. Even though large polyatomic molecules like catalysts cannot be easily seen, a comparison between the three-wave mixing spectrum before and after the reaction can identify any new chiral products, in enantiomeric excess produced, as in Fig. 4. Even molecules very similar in structure like terpenes can be unambiguously identified with microwave spectroscopy. Since any separation or purification of the sample is unnecessary for analysis, our method can act as a tool for the general search of chiral catalysts. Unlike polarimetry, once the spectra are acquired the exact transitions can be used to unambiguously identify the species produced.
Microwave three-wave mixing works best for strongly polar molecules as the matrix elements for rotational transitions depend linearly on the magnitude of the dipole moments across the A, B, and C rotational axes28. In this work, beta-pinene with dipole moments of μa = 0.43, μb = 0.58, μc = 0.11 Debye was the least polar molecule under study42. Even though enantiopure samples were used for all experiments, three-wave mixing signals scale linearly with enantiomeric excess (ee), thus signals from species of enantiopurity above 30 percent should be sufficiently above the noise level to be detected. This could be improved with straightforward electronics updates. The determination of the exact percentage of enantiomeric excess and absolute configuration from microwave three-wave mixing spectra could be performed similarly to other M3WM experiments31 given that there are available samples of known enantiomeric excess for calibration. Without such a calibration sample, the method cannot determine the absolute configuration—that is, whether R− or S− is in excess—as this relies on absolute knowledge of the dipole moments for each enantiomer, which is not defined for an unknown sample43.
An important parameter of the experiment is the frequency range of the twist pulse. We know from experience that most molecules display transitions between 60 and 110 MHz so we chose to use this range for the “twist" pulse during all data acquisition. However, for a more complete analysis of unknown samples additional frequency ranges can be easily explored. We have encountered no chiral molecule without three-wave mixing transitions with a twist between 25 and 250 MHz, which is the range of our current RF amplifier: our method is thus expected to detect any common vaporizable small chiral molecule. Additionally, as molecules grow in size, their microwave spectra get more congested and they should typically exhibit richer M3WM spectra.
Three-wave mixing spectra of unknown samples are useful as preliminary scans for chiral species in enantiomeric excess. Given that the method does not require prior chemical knowledge of the sample for determination of the rotational transitions or any optimization for probing different species (as shown in Fig. 4), it can be directly applied to unknown samples. However, further analysis is needed for identifying each species of an unknown mixture. To determine the identity of the species one needs to search for the transitions in available spectral libraries like splatalogue44, CDMS45, or published experimental and calculated spectra. For more exotic species, it is possible to perform the experiment in reverse, going from broadband fields to resonant to identify all transitions of the three-level system. Then, double resonance experiments similar to the one performed by Martin-Drumel et al.46 can be conducted to determine the rotational constants and the structure of the unidentified species.
In summary, we have introduced a generalized version of M3WM that includes the capability of acquiring microwave three-wave mixing spectra in unassigned samples. M3WM spectra can provide direct evidence on enantiomeric excess on the spot without the need for prior spectral assignment via the combination of broadband excitation and careful signal cancellation. Our new method can be applied to particularly hard-to-analyze samples like unknown multi-component mixtures and hard-to-assign species and provides new methods for ultrasensitive phase-coherent spectroscopic detection.
## Methods
### Experimental setup
The main components of the buffer gas cell apparatus have been described in detail elsewhere47. Molecules flow continuously through a copper tube heated at 40 C into the buffer gas cell held at 5–7 K. A schematic of the apparatus is shown in Fig. 1a. Cold He buffer gas flows continuously into the cell at a typical flow rate of 10 standard cubic centimeters per minute (sccm). Microwave horns are oriented with polarizations of $$\hat{x}$$ and $$\hat{y}$$ for excitation and detection, respectively. Two equally spaced copper electrodes are attached to the cell through 1" sapphire insulators to produce an electric field in the $$\hat{z}$$ direction. As in traditional M3WM, the “drive" and “listen" microwave horns are placed at 90. For additional polarization control while maintaining the cold environment inside the cell, sapphire windows (4 inches diameter) were added on two sides of the cell and microwave absorber foam was placed on the outside, as shown in Fig. 1a. We observed that covering the inside of the buffer gas cell with microwave absorber significantly increased the gas temperature.
The sample input consists of three main parts: a copper tube, a diaphragm valve, and a nipple loosely packed with glasswool. Depositing the sample on glasswool results in even evaporation and significantly reduces signal fluctuations over time, leading to highly repeatable measurements.
### Elimination of non-chiral signals
The most vital part of the experiment is to ensure that all signals stem from chiral species by successfully eliminating all non-chiral signals. We used three different methods to do so: (a) polarization control as described above, (b) fast subtractions, (c) an updated microwave circuit design which rapidly and simultaneously changes the sign of the “drive" and “twist" pulses ensuring phase controllability and accurate reversibility for broadband chirps.
Fast subtractions: A key component of the success of non-linear microwave spectroscopy in a buffer gas cell is its high spectral acquisition velocity47. Each data point of the three-wave mixing spectrum consists of 2.5 × 106 averages. The calm, controlled environment of the buffer gas cell enables careful subtractions between measurements of opposite twist phase every few hundreds of μs for each data point of the spectrum. This is important since we noticed that any “asymmetries" in the electronics or the data acquisition process can cause non-chiral signals to leak through.
To solve this issue, we used a two-channel arbitrary waveform generator with very low time jitter (Siglent SDG6052X) to generate the “drive" and “twist" pulse. The timing window between each measurement and each experimental cycle was long enough (80 μs) to prevent any signal cross-talk between measurements. A 9400 series Quantum Composer was also used to precisely control the timing between the two chirp pulses of each experimental cycle to ensure careful subtraction. It is not clear that a similar experiment could be conducted in an apparatus with pulsed valves where shot-to-shot variability is often significant.
Updated microwave design: An updated microwave circuit design ensures high phase coherence between the twist and drive pulses by mixing the twist pulse with the beat note between the upconversion and downconversion steps. Figure 5 shows a comparison between the conventional circuit for microwave spectroscopy and the updated design. In the new design, two different local oscillators, LO1 and LO2, are used for the upconversion and the downconversion step and their beat frequency is mixed with the twist pulse. Specifically, mixer (M3) was added to the circuit taking LO1 and LO2 as inputs (the frequency difference between them was set to 2 KHz). This beat note is AC-coupled and amplified, then fed into mixer (M4) where it is combined with the twist pulse. The offset local oscillators cause any 1D (non-chiral) signals to alternate phase with the 2 KHz beat note between the two local oscillators, and thus average to zero. The phase of the twist also alternates phase with the 2 KHz beat note between the two local oscillators, and so the M3WM signal survives and averages to a non-zero value. This signal is recorded alternatively with a generated twist phase of ϕ = 0 and a twist phase of ϕ = π, and signals from these two configurations are further subtracted before the spectrum is assembled. This final step removes small (< 30 dB) bleedthrough of 1D signals resulting from imperfect mixing in the twist generation (M4). The new circuit design should improve the statistics of enantiomeric excess determination for single-frequency M3WM experiments as well.
### Data acquisition
All spectra were collected with similar conditions to demonstrate the applicability of the method to a wide variety of species without selective optimization. A 4 μs long 35 MHz broadband microwave pulse is used as the “drive" pulse followed by 2 μs long RF twist pulse with a frequency chirp of 60–105 MHz. The “twist" pulse is overlapped with the drive pulse by 1 μs. The resultant coherent molecular signal (or “free induction decay" (FID)) following the double excitation is collected by a second orthogonally polarized horn and digitized to form the spectra measured such as in Fig. 2b.
Three-wave mixing spectra combine a non-linear detection method (M3WM) with broadband excitation. Since we necessarily don’t know transition dipole moments for unknown species, we operate at a pulse strength which is significantly underpowered for typical transitions. This results in typical signals that are 2–10 times weaker than typical M3WM signals under conditions optimized for maximum signal. These conditions were chosen to provide observable signal while not overdriving transitions, which would typically lead to larger false positives from non-chiral spectral lines, for more technical details see Supplementary Note.
### Chemicals
Commercial (R)-(-)-1,2 -propanediol (96% purity), anhydrous benzyl alcohol (99.8% purity), (1R)-(-)-myrtenal (98% purity), (1R)-(-)-fenchone (98% purity), (R)-(-)-carvone (98% purity), (-)-β-pinene (99% purity) were purchased from Sigma-Aldrich.
## Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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## Acknowledgements
This work was supported by NSF Award #1555781 and the David and Lucile Packard Foundation.
## Author information
Authors
### Contributions
G.K. realized the experimental setup, conducted the experiments and data analysis, and wrote the manuscript with input from all authors. I.W. assisted in experimental design and data analysis, took and analyzed the data, and made figures used in the manuscript. L.S. designed microwave circuitry, assisted in experimental design, copy edited the manuscript. D.P. assisted with experimental design, data analysis, and writing.
### Corresponding author
Correspondence to Greta Koumarianou.
## Ethics declarations
### Competing interests
The authors declare the following competing interests: One author (David s. Patterson) is an inventor along with John M. Doyle on US patent 9,857,315 with the title “Fourier transform microwave spectroscopy for enantiomer-specific detection of chiral molecules." The authors have no additional conflicts of interest to declare.
## Peer review
### Peer review information
Communications Chemistry thanks the anonymous reviewers for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Koumarianou, G., Wang, I., Satterthwaite, L. et al. Assignment-free chirality detection in unknown samples via microwave three-wave mixing. Commun Chem 5, 31 (2022). https://doi.org/10.1038/s42004-022-00641-3
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s42004-022-00641-3
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2022-08-17 06:36:54
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https://slideplayer.com/slide/7983467/
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# INTEGRALS 5. INTEGRALS In Section 5.3, we saw that the second part of the Fundamental Theorem of Calculus (FTC) provides a very powerful method for evaluating.
## Presentation on theme: "INTEGRALS 5. INTEGRALS In Section 5.3, we saw that the second part of the Fundamental Theorem of Calculus (FTC) provides a very powerful method for evaluating."— Presentation transcript:
INTEGRALS 5
INTEGRALS In Section 5.3, we saw that the second part of the Fundamental Theorem of Calculus (FTC) provides a very powerful method for evaluating the definite integral of a function. This is assuming that we can find an antiderivative of the function.
5.4 Indefinite Integrals and the Net Change Theorem In this section, we will learn about: Indefinite integrals and their applications. INTEGRALS
INDEFINITE INTEGRALS AND NET CHANGE THEOREM In this section, we: Introduce a notation for antiderivatives. Review the formulas for antiderivatives. Use the formulas to evaluate definite integrals. Reformulate the second part of the FTC (FTC2) in a way that makes it easier to apply to science and engineering problems.
INDEFINITE INTEGRALS Both parts of the FTC establish connections between antiderivatives and definite integrals. Part 1 says that if, f is continuous, then is an antiderivative of f. Part 2 says that can be found by evaluating F(b) – F(a), where F is an antiderivative of f.
INDEFINITE INTEGRALS We need a convenient notation for antiderivatives that makes them easy to work with.
Due to the relation given by the FTC between antiderivatives and integrals, the notation ∫ f(x) dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus, ∫ f(x) dx = F(x) means F’(x) = f(x) INDEFINITE INTEGRAL
INDEFINITE INTEGRALS For example, we can write Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).
INDEFINITE VS. DEFINITE INTEGRALS You should distinguish carefully between definite and indefinite integrals. A definite integral is a number. An indefinite integral ∫ f(x) dx is a function (or family of functions).
The connection between them is given by the FTC2. If f is continuous on [a, b], then INDEFINITE VS. DEFINITE INTEGRALS
INDEFINITE INTEGRALS The effectiveness of the FTC depends on having a supply of antiderivatives of functions. Therefore, we restate the Table of Antidifferentiation Formulas from Section 4.9, together with a few others, in the notation of indefinite integrals.
INDEFINITE INTEGRALS Any formula can be verified by differentiating the function on the right side and obtaining the integrand. For instance,
TABLE OF INDEFINITE INTEGRALS Table 1
INDEFINITE INTEGRALS Recall from Theorem 1 in Section 4.9 that the most general antiderivative on a given interval is obtained by adding a constant to a particular antiderivative. We adopt the convention that, when a formula for a general indefinite integral is given, it is valid only on an interval.
INDEFINITE INTEGRALS Thus, we write with the understanding that it is valid on the interval (0, ∞ ) or on the interval (- ∞, 0).
INDEFINITE INTEGRALS This is true despite the fact that the general antiderivative of the function f(x) = 1/x 2, x ≠ 0, is:
INDEFINITE INTEGRALS Find the general indefinite integral ∫ (10x 4 – 2 sec 2 x) dx Using our convention and Table 1, we have: ∫(10x 4 – 2 sec 2 x) dx = 10 ∫ x 4 dx – 2 ∫ sec 2 x dx = 10(x 5 /5) – 2 tan x + C = 2x 5 – 2 tan x + C You should check this answer by differentiating it. Example 1
INDEFINITE INTEGRALS Evaluate This indefinite integral isn’t immediately apparent in Table 1. So, we use trigonometric identities to rewrite the function before integrating: Example 2
INDEFINITE INTEGRALS Evaluate Using FTC2 and Table 1, we have: Compare this with Example 2 b in Section 5.2 Example 3
INDEFINITE INTEGRALS Find Example 4
The FTC gives: This is the exact value of the integral. INDEFINITE INTEGRALS Example 4
INDEFINITE INTEGRALS If a decimal approximation is desired, we can use a calculator to approximate cos 12. Doing so, we get: Example 4
The figure shows the graph of the integrand in the example. We know from Section 5.2 that the value of the integral can be interpreted as the sum of the areas labeled with a plus sign minus the area labeled with a minus sign. INDEFINITE INTEGRALS
Evaluate First, we need to write the integrand in a simpler form by carrying out the division: Example 5
Then, INDEFINITE INTEGRALS Example 5
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2021-05-11 20:17:20
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https://math.paperswithcode.com/paper/multiplicities-and-plancherel-formula-for-the
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# Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices
11 Aug 2020 · Beuzart-Plessis Raphaël ·
This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated $L^2$ space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case... The second is a formula for the multiplicities of generic representations in the $p$-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local analogs of two relative trace formulas previously studied by Jacquet and Ye and known as (relative) Kuznetsov trace formulas. read more
PDF Abstract
## Code Add Remove Mark official
No code implementations yet. Submit your code now
## Categories
Number Theory Representation Theory
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2021-12-04 22:41:15
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https://notes.sheronw.xyz/computer%20science/design%20patterns/01%20oop/
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# Object-Oriented Programming
## Features
• Encapsulation
• Abstract
• Inheritance
• Polymorphism
## Procedure Oriented or Object Oriented?
### Procedure Oriented?
Early languages such as C. Linear. Which is not suitable for large-scale non-linear develpment.
You could still use oop thoughts in procedure oriented languages.
### Why OOP better?
Good for handling non-linear complicated tasks.
It is easier for human-being to abstract the really demand or world to classes, instead of linear tasks in computers. So it is easier to implement certain features.
Also, high maintainability, readability, extensibility, flexibility, simplicity, reusability, testability, etc.
## Fake OOP
• too many getter, setter(encapsulation NG)
• unsuitable global variables or methods
• big classes such as Constants or Utils(no abstraction)
• separate data and method in class(one exception is, anaemic model is widely used in web develpment)
## Abstraction or Interface?
### Abstraction(is-a)
• could only be created by interitance(keyword: extends)
• including both abstract method and real method
• all abstract methods have to be implemented by children
• declare an imcomplete class definition without defining all its methods, so can be a subclass of something else
### Interface(has-a/behaves like)
• could only have abstract methods(no implementation or vars)
• all methods have to be implemented by children
• keyword: implements
• declares a set of related methods outside of any class just like API
### Why we need abstraction?
Reusabilty
In general, we could implement the abstraction class function using ordinary class, however, there are some problems about that, for example:
• Children may forgot to implement some certain method
• we could initiate a empty ancestor class
• an empty method will affect the readability of the code
### Why we need interface?
• decoupling 解耦
• extensability
## Program to an interface, not an implementation
Separate implementation and interface, only give out stable interface and leave the implementations that might change(decoupling). (Interface Design first)
• never give of details in the name of the method
• encapsulate the implementation of the details
• define interface for classes
Notice: if a system is designed to be stablized and no need to maintain, then we don't really need to waste time ensuring the extensability.
## Composition VS Inheritance?
### Problem of Inheritance
Sometimes a parent class cannot handle everything. If we want to divided those children into catagories, one for each feature, then there will be so many subclasses which is not good for the maintainability and readability.
### Why use composition
We could use composition, interface or delegation to solve the problem above.
First of all, we could try to create interfaces for every features, and let the certain class implement these features. The problems is that all methods in interfaces are abstract methods and we still need to implement all the methods one by one in each class.
How to solve this problem? We could use composition and delegation.
We could create implementation classes for all those interfaces and composite those implementation classes into each child class, for example, as a private variable.
### Composition or Inheritance?
As the example above, using composition instead of inheritance requires more granular codes. Then it is not good for maintainability and readability.
So if the system is stable and the inheritance relationship is shallow, then we could still use inheritance. And some certain design patterns requires the usage of inheritance.
## MVC VS DDD
### Model, View, Controller
The def is not strict, for example, in back end we have repository(data), service(logic), controller(expose interface).
#### Anemic Domain Model
process oriented
separate data and logic
e.x. use userservice class to control data in userBo class
### Domain Driven Design
Still in three parts: model, view, controller, but in service part, there is one service class and there is one domain class, and we put both data and logic into domain class.
#### Rich Domain Model
put data and logic in the same class, i.e. OOP
### Why use anemic?
history, easy, simple(logics are mainly included into SQL queries)
But for complex systems is better to use DDD.
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2020-11-23 18:33:45
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http://cvgmt.sns.it/paper/1732/
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# Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds
created by mondino on 12 Jan 2012
modified on 10 Dec 2013
[BibTeX]
Accepted Paper
Inserted: 12 jan 2012
Last Updated: 10 dec 2013
Journal: Math. Annalen
Year: 2011
Abstract:
We study curvature functionals for immersed $2$-spheres in a compact, three-dimensional Riemannian manifold $M$. Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:S^2 \to M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \leq 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$, we obtain a smooth minimizer $f:S^2 \to M$ for the functional $\int \frac{1}{4} H ^2+1$, where $H$ is the mean curvature.
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2018-12-13 06:07:59
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https://kb.osu.edu/dspace/handle/1811/15075
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# K-TYPE DOUBLING IN THE $\nu_{3}$ BAND OF ARSINE.
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/15075
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1966-N-03.jpg 330.8Kb JPEG image
Title: K-TYPE DOUBLING IN THE $\nu_{3}$ BAND OF ARSINE. Creators: Yin, Peter K. L.; Rao, K. Narahari Issue Date: 1966 Publisher: Ohio State University Abstract: Arsine $(AsH_{3})$ is a pyramidal-type molecule like ammonia. The recent theoretical studies made in connection with the perturbations appearing in the infrared spectra of pyramidal-type molecules prompted the present experimental work. All the four fundamental vibration rotation bands of arsine were studied. Two of these fundamentals (designated as $\nu_{1}$ and $\nu_{2}$) occur in the region of 5 microns and the other two (designated as $\nu_{2}$ and $\nu_{4}$) are observed in the region of 10 microns. Nearly 2000 rotational lines in the four fundamentals were recorded, measured and analyzed. Based on the experimental data obtained, several molecular parameters of arsine like the moment of inertia, centrifugal distortion constants, etc., were evaluated. The more interesting features are the perturbations observed. The shifts (known as Giant l-type doubling'') of some of the rotational lines $(^{R}R(J,O))$ observed previously for $ammonia^{1}$ and $phosphine^{2}$ in the $\nu_{4}$ band were also seen in the $\nu_{4}$ band of arsine. In addition, for the first time, similar shifts were observed in the $\nu_{4}$ band of arsine although they were much smaller than in the case of the $\nu_{4}$ band. The doubling of lines (known as K-type doubling'') was observed for the first time in some of the rotational lines of the $p_{3}$ band (viz. $^{P} P(J,3)$) and $^{P}R(J,3)$). A quantitative correlation was at tempted between the measured splittings of the doublets and the theoretical parameters responsible for their occurrence. Description: Supported, in part, by the Air Force Cambridge Research Laboratories, Office of Aerospace Research. Author Institution: Laboratory of Molecular Spectroscopy and Infrared Studies, Department of Physics, The Ohio State University URI: http://hdl.handle.net/1811/15075 Other Identifiers: 1966-N-3
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2016-02-14 17:09:00
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http://math.stackexchange.com/questions/146238/ring-notation-r-mathbbz1-n
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# Ring notation, $R=\mathbb{Z}[1/N]$
Regarding the notation $R=\mathbb{Z}[1/N]$, where $N$ is a positive integer, does $R$ refer to:
$R=\{a+b/N|a,b\in\mathbb{Z}\}$
or
$R=\{a_0 +a_1/N+a_2/N^2+\ldots +a_n/N^n|a_i\in\mathbb{Z}\}$
or others?
Thanks a lot.
-
Is the first one even a ring? – Alexei Averchenko May 17 '12 at 11:20
As the comment suggests it's the second option, where you maybe should specify that $n\in\mathbb N$ may vary. You can think about it differently. Either it's "polynomials" in $\frac 1N$ with coefficients in $\mathbb Z$ or it is the smallest ring which contains $\mathbb Z$ and $\frac 1N$.
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2016-06-25 08:46:54
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http://quant.stackexchange.com/tags/optimization/new
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# Tag Info
2
The basic approach is as follows: When you estimate the HMM you estimate three things: When you are in which state The drifts of your assets The covariance matrices of your assets You would then take 2. and 3. for each state (1.) and feed it into your favourite allocation optimizer to estimate your optimal portfolio for each state. Voila!
1
I don't think you'll find anything. Why don't you contact the authors? They must have some code to generate the HMM simulations in the paper, maybe they can share the code with you? Have you checked the Supplementary Materials? Some papers have it. If you're really determined, you can implement a HMM model yourself. You'll need to supply the Markov ...
0
When $X_1$ is unobserved, at iteration $k=1$ of EM, the posterior mean value (when $X_2=3$) is $5.18$ by using an inference algorithm, i.e. Junction tree/Kalman filter. Then the sufficient statistics for $X_1$ is: $s_1=\Sigma_{i=1}^nx_{i1}=9+4+5.18$ and $s_{11}=\Sigma_{i=1}^nx_{i1}^2=9^2+4^2+(5.18^2+\sigma_{11.2})$ where $\sigma_{11.2}$ is the posterior ...
Top 50 recent answers are included
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2016-07-28 14:31:28
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https://imathworks.com/tex/tex-latex-tikz-fill-color/
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# [Tex/LaTex] Tikz fill color
colorloopsmatricespreambletikz-pgf
At the moment I'm using this command in the preamble to fill the color of the specified row with grey:
row 2/.style={
nodes={fill=gray!10}
}
Unfortunately I have to write this for every line.
Can't I say something like
row 1-10/.style={
nodes={fill=gray!10}
}
Thanks for your answers. Though I'm not sure what's not clear in my example the problem is that I wanted to use the row option (within \tikzset{} before \begin{document} ) because (maybe I don't know how else to do it) I only want certain rows to be colored that way not everything. Other rows have a different color such that everything looks nice and colorful as it should be. Since I'm very new to tikz I solved this in an awkward way by writing
\tixset{
row 2/.style={
nodes={fill=gray!10}
},
row 3/.style={
nodes={fill=gray!10}
},
row 4/.style={
nodes={fill=gray!10}
},
column 1/.style={
nodes={text width=14em}
},
column 3/.style={
nodes={text width=9em}
},
column 4/.style={
nodes={text width=9em}
}
}
Here I also specified different widths for the columns. Probably as well this one can solve better by not writing the entire ./style every time…
@last: myrowstyle is just a name or? so my new definition to later call it am I right?
# MWE [copied from deleted answer by cfr]
\documentclass[table]{beamer}
\usepackage{tikz}
\usetikzlibrary{matrix}
\tikzset{
table/.style={
matrix of nodes,
row sep=-\pgflinewidth,
column sep=-\pgflinewidth,
nodes={
rectangle,
draw=black,
align=center,
},
%baseline={([yshift=-0.5ex]current bounding box.center)},
minimum height=1.5em,
text depth=0.5em,
text height=1em,
text centered,
nodes in empty cells,
%%
row 1/.style={
nodes={
fill=black,
text=white,
%font=\bfseries
}
},
myrowstyle/.style={
row #1/.style={nodes={fill=gray!10}}
},
column 1/.style={
nodes={text width=14em}
},
column 3/.style={
nodes={text width=9em}
},
column 4/.style={
nodes={text width=9em}
}
}
}
\begin{document}
\begin{frame}
\begin{tikzpicture}
\matrix (first) [table,text width=4em, myrowstyle/.list={1,2,3}, row 2/.style={text depth=1.5em}, ampersand replacement=\&]
{
... \& \# ...\& \# ... \& ... \\
01.06.-02.06. \break (hello world) \& 1 \& 2 \& \\
02.06.-03.06. (hello) \& 3 \& 3 \& \\
};
\end{tikzpicture}
\end{frame}
\end{document}
You can use the /.list handler.
\tikzset{myrowstyle/.style = {row #1/.style={nodes={fill=gray!10}}}
and later in the picture you can then use
myrowstyle/.list={1,...,10}
or any other argument list.
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2023-03-30 15:05:10
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https://gamedev.stackexchange.com/questions/88090/equalling-the-rotation-of-an-image-with-its-object-box2dweb
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# Equalling the rotation of an image with its object. Box2dweb
I have some hard time making the rotation of an image equal to the rotation object that the image belongs to. Simply put it, I don't know how to do it properly but i am not asking for a how-to guide but for some tips/hint and/or infos that i am not aware of...
I was able to align perfectly the image with the object, and so if the object moves in a linear fashion the image is at all times on top of the object and everything is smooth and perfect. But how is it possible to make the rotation of the image be the same as of the object it belongs to.
No-matter what i tried it is always somewhat off or the rotation of the image behaves weird when in the end the object stops rotating. I tried using using GetAngle(), GetAngularVelocity(), combinations of both, but never succeeded.
function Draw_and_Rotate(){
for (b = world.GetBodyList() ; b; b = b.GetNext()){
var angle = ((b.GetAngle()*180)/(Math.PI))/SCALE;
var angle_vel = ((b.GetAngularVelocity()*180)/(Math.PI))/SCALE;
var pos = b.GetPosition();
console.log((angle));
if (b.GetUserData() == "bo_img"){
ctx.save();//save the ctx state prior changing it.This method pushes the current state onto the stack.
ctx.translate(pos.x*SCALE , pos.y*SCALE );
ctx.rotate(angle);
ctx.drawImage(box_img, - (box_img.width / 2), - (box_img.height / 2));
ctx.restore();//restoring ctx state.This method pops the TOP state on the stack, restoring the context to that state.
}
}
}
this is the part that deals only with one body and its image. If someone needs some more clarification please ask....
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2020-10-30 12:52:29
|
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|
http://www.apexmusclejournal.com/db4kz63j/squiggly-equal-sign-cd5a43
|
For this reason it is sometimes recommended to avoid the == operator in JavaScript in favor of ===. ", "Conventions for interlinear morpheme-by-morpheme glosses", "An International Perspective between Problem Types in Textbooks and Students' understanding of relational equality", Scientific Symbols, Icons, Mathematical Symbols, https://en.wikipedia.org/w/index.php?title=Equals_sign&oldid=995775224#Approximately_equal, Short description is different from Wikidata, Articles with unsourced statements from June 2013, Articles with unsourced statements from August 2020, Articles lacking reliable references from August 2020, Creative Commons Attribution-ShareAlike License, This page was last edited on 22 December 2020, at 20:47. You can also insert the symbol from , … Computers display the equals sign with the Unicode or ASCII character 003D (in hexadecimal ). [17], In Morse code, the equals sign is encoded by the letters B (-...) and T (-) run together (-...-). However, in most languages where = has one of these meanings, a different character or, more often, a sequence of characters is used for the other meaning. It's either the "squiggly equals sign" (i.e. The symbol used to say when items are not equal is "â " (slashed equal sign).[1]. - [Instructor] What we're going to do in this video is get a little bit of practice estimating dividing with decimals. [9], In Ruby, equality under == requires both operands to be of identical type, e.g. Symbol Name Code ÷ Division (Obelus) 246 × Multiplication 0215 ± Plus or minus 0177 ≈ Approximate 247 √ Square root 251 ⁿ Power n 252 ² Squared 253 ¼ Quarter 0188 ½ Half 0189 ¾ Three quarters 0190 ∞ Infinity 236 ≥ Greater than or equal 242 ≤ Less than or equal 243 π Pi 227 ° Degree 248 So what does 80 divided by 1.94 approximately equal? Embrace Opportunity and Carve Your Own Path Through the Squiggly World of Work. The tilde is also used to indicate congruence of shapes by placing it over an = symbol, thus: ≅. COVID-19 Updates and Information Personal Income Tax Cybersecurity Passenger (Class D) Driver's Licenses Stop the Spread SNAP benefits (formerly food stamps) Use your numeric keypad with your NUM LOCK on and you will be good to go! The squiggly equal sign ( ≈ ) is used when the answer isn't exact, like if you make an estimate or round a number. ≈ (Unicode 2248) ≃ (Unicode 2243) – used to indicate asymptotically equal to 1844 === (1800..1899) is false, since it is interpreted to mean Integer#=== rather than Range#===.[11]. If interpreted strictly as it says, it would imply that: A correct version of the argument would be: This difficulty results from subtly different uses of the sign in education. Hyperbolic functions The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Symbol Meaning Example In Words; Triangle: ABC has 3 equal sides: Triangle ABC has three equal sides: Angle: ABC is 45° The angle formed by ABC is 45 degrees. How typing: Equals sign ? 4.6 out of 5 stars 202. Welcome to Useful Shortcuts, THE Alt Code resource!. In LaTeX it is coded as \sim. The equals sign is also used in defining attribute–value pairs, in which an attribute is assigned a value. [1][2] It was invented in 1557 by Robert Recorde. Expressed to four significant digits, 2 1/2 1.414. In Fortran, = serves as an assignment operator: X = 2 sets the value of X to 2. Option 2 Insert – Symbol, you find it under Subset Number Forms. For instance, the expression 0 == false is true, but 0 === false is not, because the number 0 is an integer value whereas false is a Boolean value. The two squiggly lines is one of at least five symbols used for”approximately equal” in mathematics Symbols used to denote items that are approximately equal are wavy equals signs. Both usages have remained common in different programming languages into the early 21st century. Section 8 Centralized Waiting List Massachusetts. to test for equality. - [Instructor] In this video, we're gonna get a little bit of practice estimating with multiplication. [citation needed] The letters BT stand for Break Text, and are put between paragraphs, or groups of paragraphs in messages sent via Telex,[citation needed] a standardised tele-typewriter. WINDOWS: on computers with Windows operating system like Windows 8, Win 7, Vista, Windows XP, etc.. To get the letter, character, sign or symbol "=": ( Equals sign ) on computers with Windows operating system: 1) Press the "Alt" key on your keyboard, and do not let go. Click Alt+= again to exit from the equitation. In the computing field, especially in Unix-based systems, the tilde indicates the user's home directory. Superscript Equals Sign. [10] Under these semantics, === is non-symmetric; e.g. Insert Approximately Equal Symbol from Microsoft Equation. ≈), or the single "squiggly" (~), if shorthand is approved. squiggly definition: 1. consisting of a line or lines that curve and twist in a way that is not regular: 2. consisting…. This means approximately equal. Commercial and personal bank offering savings, money management and loan products. The = symbol, now universally accepted in mathematics for equality, was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte (1557). The equals sign is placed between the things stated to be exactly equal or the same. In his book Recorde explains his design of the "Gemowe lines" (meaning twin lines, from the Latin gemellus[5], And to auoide the tediouЕїe repetition of theЕїe woordes : is equalle to : I will Еїette as I doe often in woorke vЕїe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauЕїe noe .2. thynges, can be moare equalle. ≃ is more of a grab-bag of meaning. It can mean "similar to", including "of the same order of magnitude as", such as: "x ~ y" meaning that x and y are of the same order of magnitude. In LaTeX, this is done with the "\neq" command. In mathematics, the equals sign can be used as a simple statement of fact in a specific case (x = 2), or to create definitions (let x = 2), conditional statements (if x = 2, then ...), or to express a universal equivalence ((x + 1)ВІ = xВІ + 2x + 1). If you are already familiar with using alt codes, simply select the alt code category you need from the table below. 41. In Ojibwe, the readily available equal sign on a keyboard is used as a substitute for a double hyphen. [12][13] The Unicode character used for the tone letter (U+A78A)[14] is different from the mathematical symbol (U+003D). ∼ is a similarity in geometry and can be used to show that two things are asymptotically equal (they become more equal as you increase a variable like n). From Simple English Wikipedia, the free encyclopedia, https://simple.wikipedia.org/w/index.php?title=Equals_sign&oldid=7193673, Creative Commons Attribution/Share-Alike License, "â
" (one tilde above two lines, often used in, "â" (two lines with a dot above them, often used for "is defined as"), "â¡" (three lines, often used for equivalence). A few languages, such as BASIC and PL/I, have used the equals sign to mean both assignment and equality, distinguished by context. Starting in algebra courses, the sign takes on a relational meaning of equality between two calculations. Confusion between the two uses of the sign sometimes persists at the university level. Visually, the symbol is a squiggly equals sign. Symbol. The equals sign is placed between the things stated to be exactly equal or the same. ＝. Fullwidth Equals Sign. There are several symbols that can be used to say items are "approximately the same," "similar to" or "about equal." Visit any of our 15 locations around Worcester County today. The Section 8 housing choice voucher program is a federal government program for assisting very low-income families, the elderly, and the disabled to afford decent, safe, and sanitary housing in the private market. [4] The original form of the symbol was much wider than the present form. Do you know the unexpected origins of brackets and parentheses? JavaScript has the same semantics for ===, referred to as "equality without type coercion". Paperback $20.41$ 20. If this is the case then A, x and b are conformable. The equals sign or equality sign, =, is a mathematical symbol used to indicate equality in some well-defined sense. Learn more. As well as Fortran, = is used for assignment in such languages as C, Perl, Python, awk, and their descendants. [citation needed]. The equal sign, equals sign, or "=" is a mathematical symbol used to indicate equality. Some of the commonly used symbols: \infty - Infinity \leq - Less than or equal \geq - Greater than or equal \partial - Partial differential \sum - Summa \prod - Product Sign \subset - Contained in \in - Element of \cup - Union (if you want to see big symbol, enter \bigcup) \neq - Not equal to \approx - Almost equal to (asymptotic to) \equiv - Identical to (equivalent) If you need help using alt codes find and note down the alt code you need then visit our instructions for using alt codes page. The triple bar symbol в‰Ў (U+2261, LaTeX \equiv) is often used to indicate an identity, a definition (which can also be represented by U+225D ≝ EQUAL TO BY DEFINITION or U+2254 ≔ COLON EQUALS), or a congruence relation in modular arithmetic. Not a recommended shortcut if doing serious mathematics. In recent years, the equals sign has been used to symbolize LGBT rights. Computers display the equals sign with the Unicode or ASCII character 003D (in hexadecimal). ＝. In linguistic interlinear glosses, an equals sign is conventionally used to mark clitic boundaries: the equals sign is placed between the clitic and the word that the clitic is attached to.[16]. The expression 0 == false is true, but 0 == undefined is false, even though both sides of the == act the same in Boolean context. For double hyphens, see, Usage in mathematics and computer programming, ITU (International Telecommunications Union) International Telecommunications Regulations, List of mathematical symbols В§ Symbols based on equality, "The History of Equality Symbols in Math", the third page of the chapter "The rule of equation, commonly called Algebers Rule. It doesn't have to be exactly right. Dive into the history and uses of [ ], { }, ? The === operator is flexible and may be defined arbitrarily for any given type. As an example of how approximate equality can be used in mathematics, consider the positive square root of 2 (or 2 1/2). If you like keyboard shortcuts and use special characters, here are a few more for Microsoft Windows. This is an irrational number ; when written in decimal form, it is nonterminating and nonrepeating. Small Equals Sign. ＝. Some of these symbols include: Each of these symbols has more than one possible meaning, and are all used to state that two things are about equal (or equivalent in some way). So we wanna estimate what this is. ﹦. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value. [20], "=" and "пјќ" redirect here. The symbol used to denote inequation (when items are not equal) is a slashed equals sign в‰ (U+2260). This symbol (in US English) informally means "approximately", "about", or "around", such as "~30 minutes before", meaning "approximately 30 minutes before". The etymology of the word "equal" is from the Latin word "æqualis", as meaning "uniform", "identical", or "equal", from aequus ("level", "even", or "just"). The equals sign was reserved for this usage. For example, the assignment X = X + 2 increases the value of X by 2. Logical Operators. The symbol has been used since 1995 by the Human Rights Campaign, which lobbies for marriage equality, and subsequently by the United Nations Free & Equal, which promotes LGBT rights at the United Nations. 4. ?, and ( ) with Thesaurus.com. Learn more. The etymology of the word "equal" is from the Latin word "Г¦qualis",[3] as meaning "uniform", "identical", or "equal", from aequus ("level", "even", or "just"). squiggly meaning: 1. consisting of a line or lines that curve and twist in a way that is not regular: 2. consisting…. So over here it says question mark is, and then you have this squiggly equal sign. by Helen Tupper and Sarah Ellis | Jan 2, 2020. Option 1 It’s alt 0187 in font Symbol. The sign, used to mean Break Text, is given at the end of a telegram to separate the text of the message from the signature. In chemical formulas, the two parallel lines denoting a double bond are commonly rendered using an equals sign. [18][better source needed], Symbols used to denote items that are approximately equal include the following:[1][19]. So for example, we wanna figure out approximately, that's what these kind of squiggly equal sign means. Most programming languages, limiting themselves to the 7-bit ASCII character set and typeable characters, use ~=, !=, /=, or <> to represent their Boolean inequality operator. ﹦. [6], "The symbol = was not immediately popular. In Unicode and ASCII, it has the code point 3D. But = is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages. It looks like two parallel horizontal lines. Instead of a double hyphen, the equals sign is sometimes used in Japanese as a separator between names. The first important computer programming language to use the equals sign was the original version of Fortran, FORTRAN I, designed in 1954 and implemented in 1957. More Information & Find logical AND. and | Find logical OR or && … Equality of truth values (through bi-implication or logical equivalence), may be denoted by various symbols including =, ~, and ⇔. However, in JavaScript the behavior of == cannot be described by any simple consistent rules. And so you can view that squiggly equal sign as being what does this roughly equal to? Note: you can see all of Name of the symbol combinations that you can use in the AutoCorrect Options. Account & Lists Account Returns & Orders. A possibly unique case of the equals sign of European usage in a person's name, specifically in a double-barreled name, was by pioneer aviator Alberto Santos-Dumont, as he is also known not only to have often used a double hyphen resembling an equals sign = between his two surnames in place of a hyphen, but also seems to have personally preferred that practice, to display equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother.[15]. The equals sign is also used as a grammatical tone letter in the orthographies of Budu in the Congo-Kinshasa, in Krumen, Mwan and Dan in the Ivory Coast. The symbol || was used by some and Г¦ (or Е“), from the Latin word aequalis meaning equal, was widely used into the 1700s" (History of Mathematics, University of St Andrews).[7]. [1], The symbol в‰… is often used to indicate isomorphic algebraic structures or congruent geometric figures.[1]. only makes sense if the order of x equals the number of columns of A and the order of b equals the number of its rows. Approximate equality is symbolized by a squiggly equal sign (). Copy and paste the Equal symbol or use the unicode decimal, hex number or html entity in social websites, in your blog or in a document. Hello, Sign in. For example, if one were finding the sum, step by step, of the numbers 1, 2, 3, 4, and 5, one might incorrectly write: but the notation is incorrect, because each part of the equality has a different value. Neither Greater-Than Nor Equal To. 0 == false is false. ALGOL included a relational operator that tested for equality, allowing constructions like if x = 2 with essentially the same meaning of = as the conditional usage in mathematics. In early, arithmetic-focused grades, the equals sign may be operational; like the equals button on an electronic calculator, it demands the result of a calculation. If you only occasionally need to type the does not equal sign (or other math symbols), the first section below has you covered (you’ll also improve your knowledge of Microsoft Word too). After entering the symbol, click the space; it changed entering a name to the appropriate symbol. 3. (1800..1899) == 1844 is false, since the types are different (Range vs. Integer); however (1800..1899) === 1844 is true, since === on Range values means "inclusion in the range". I want to add the symbol for almost equal, ie ≈, but haven’t been able to find it. Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the arithmetic IF statement) until FORTRAN IV was released in 1962, since when it has used the four characters .EQ. Following ALGOL, most languages that use = for equality use := for assignment, although APL, with its special character set, uses a left-pointing arrow. This somewhat resembles the use of = in a mathematical definition, but with different semantics: the expression following = is evaluated first, and may refer to a previous value of X. The equals sign is sometimes used incorrectly within a mathematical argument to connect math steps in a non-standard way, rather than to show equality (especially by early mathematics students). If you frequently need to insert math symbols like this, I recommend turning on your Math AutoCorrect feature as discussed in the second section below. Another approximation symbol is the double tilde ≈, meaning "approximately equal to". If you want to see the big symbol, enter \bigcup: 3. The symbol for approximately equal is a squiggly equals sign. It looks like two parallel horizontal lines. Symbol Description Shortcut ¶ paragraph sign ALT+0182 ± plus-or-minus sign ALT+ This is a weaker statement than the other two. The equal sign, equals sign, or " = " is a mathematical symbol used to indicate equality. Additional symbols in Unicode related to the equals sign include:[19]. For example, a value of type Range is a range of integers, such as 1800..1899. Role. In PHP, the triple equals sign, ===, denotes value and type equality,[8] meaning that not only do the two expressions evaluate to equal values, but they are also of the same data type. Official website of the Commonwealth of Massachusetts. The language B introduced the use of == with this meaning, which has been copied by its descendant C and most later languages where = means assignment. Now you can continue entering your text. The Approximately Equal symbol (≈) is a mathematical operator used to indicate that two expressions or statements are similar but not exactly equal to each other. 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[ 6 ], { }, exactly equal = '' is a mathematical symbol to! This page was last changed on 28 November 2020, at 22:00 option 1 ’... Tilde is also used in defining attribute†“ value pairs, in JavaScript in favor of....
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2022-05-19 12:14:12
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https://chemistry.stackexchange.com/questions/46841/is-there-an-easy-way-to-test-for-the-amount-of-iron-fe-in-water
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# Is there an easy way to test for the amount of iron (Fe) in water?
I have a rusty metal rainwater tank, and I'd like to find the amount of iron in the water. (The water I've collected is perfectly clear.)
How would chemists in the 1800's have gone about this? What chemicals/apparatus would I need to purchase?
So far, I've soaked a tissue in the water and am letting it air dry to see if any rust discoloration becomes apparent.
The following info from the WHO might be helpful:
Iron (as Fe2+) concentrations of 40 µg/litre can be detected by taste in distilled water. In a mineralized spring water with a total dissolved solids content of 500 mg/litre, the taste threshold value was 0.12 mg/litre. In well-water, iron concentrations below 0.3 mg/litre were characterized as unnoticeable, whereas levels of 0.3–3 mg/litre were found acceptable (E. Dahi, personal communication, 1991).
In drinking-water supplies, iron(II) salts are unstable and are precipitated as insoluble iron(III) hydroxide, which settles out as a rust-coloured silt. Anaerobic groundwaters may contain iron(II) at concentrations of up to several milligrams per litre without discoloration or turbidity in the water when directly pumped from a well, although turbidity and colour may develop in piped systems at iron levels above 0.05–0.1 mg/litre. Staining of laundry and plumbing may occur at concentrations above 0.3 mg/litre (4).
Iron also promotes undesirable bacterial growth ("iron bacteria") within a waterworks and distribution system, resulting in the deposition of a slimy coating on the piping (4). http://www.who.int/water_sanitation_health/dwq/chemicals/iron.pdf
• Iron particles can precipitated by fine filter paper and visually checked Dec 31, 2018 at 9:08
In a rainwater tank in contact with air, dissolved iron is probably oxidized to iron(III) ($\ce{Fe^3+}$).
A simple and sensitive test for $\ce{Fe^3+}$ in water uses thiocyanate ions ($\ce{SCN-}$, also known as rhodanide), which form the blood-red coloured complexes $\ce{[Fe(SCN)(H2O)5]^2+}$, $\ce{[Fe(SCN)2(H2O)4]+}$, and $\ce{[Fe(SCN)3(H2O)3]}$.
$$\ce{[Fe(H2O)6]^3+ + SCN- <=> [Fe(SCN)(H2O)5]^2+ + H2O}$$
Take a precise volume of the water (e.g. $10\ \mathrm{ml}$), acidify with dilute sulfuric acid (e.g. $2\ \mathrm{ml}$), add a thiocyanate solution (e.g. $5\ \mathrm{ml}$ $\ce{KSCN}$ or $\ce{NH4SCN}$, $c =0.5{–}2\ \mathrm{mol/l}$), and fill up with distilled water to a precise volume (e.g. $25\ \mathrm{ml}$).
An orange to red colour should appear.
The colour of the solution can be compared to that of standard solutions that contain known concentrations of $\ce{Fe^3+}$, e.g. prepared from iron(III) nitrate ($\ce{Fe(NO3)3.9H2O}$).
In a laboratory, the intensity of the red colour could be measured using a photometer.
Similar simple test kits are commercially available, e.g. for aquariums, garden ponds, or other water or soil samples.
• I agree with colorimetry being the method of choice. It cannot be very precise by definition, but you just cannot make anything wrong. It is very robust and does not need precise scale. The standards can be prepared by further dilution in known ratios. You could also use your camera as a very imprecise photometer. Feb 24, 2016 at 14:53
• My stock method is to use 2,2'-bipy and iron to make the red complex for measurement, my method would require you to add a reducing agent such as hydroxylamine to get all the dissolved iron into the form of iron(II). But the thiocyanate method would work, Oct 16, 2022 at 7:36
• I would disagree with Ssavec, when used with care colorimetry can be used to get very precise results Oct 16, 2022 at 9:21
If you want to test for the presence of iron(II) in solution, you could add $\ce{CO3^{2-}}$ ions to form a green precipitate. This could be extracted and measured, although it could be a very small amount unless you concentrated the sample water by evaporating it down. The main tool would be an accurate balance, but also some liquid measuring equipment.
An alternative method that would be more analytically accurate would be a redox titration using something like potassium permanganate or dichromate. This would require an accurately known concentration of the reagent, and some more specific equipment like a burette and a stand, and a volumetric flask for precise liquid measurements.
• Very small amounts of $\ce{Fe^{2+}}$ ions are quite improbable in water, because they are quickly oxidized by atmospheric oxygen, and transformed into $\ce{Fe^{3+}}$ ions. And this $\ce{Fe^{3+}}$ ion does not make a green precipitate with $\ce{CO3^{2-}}$ ions. It makes a brown precipitate. The same critics could be made for the second method, using potassium permanganate, which only reacts with $\ce{Fe^{2+}}$ ion and not with $\ce{Fe^{3+}}$ ions Oct 16, 2022 at 9:49
There is a problem which I think that the other people might not be aware of. It is a problem which is well known in nuclear chemistry. If you take a sample of lake water from some random lake such as a Swedish lake then if you measure the plutonium content of the water by digesting the sample and then filtering then you get a higher value for fallout plutonium than if you filter it first and then digest.
While iron is very different to plutonium, I think that the same problem could exist. The amount of iron which is present as a fine (colidal solid) might be important. I would suggest the following.
Take a large sample of water from the tank, now filter part of this through the finest filter you can find. The other part you should combine with a known volume of hydrochloric acid and boil it up.
After cooling the boiled acidic water should be filtered. I would take both filtrates and then measure the iron by means of colourmetric tests.
My normal test for iron uses 2,2'-bipy to form a complex with the iron(II), this test can be used to determine what fraction of the iron is present as iron(II).
I would make up four volumetric flasks with the filtered iron solutions, I would put in each flask ammonium aceate (to buffer the mixture), 2,2'-bipy solution and the iron sample. I have found that this method works down to ppm levels of iron. To two of the flasks I would add hydroxylamine solution to reduce all the iron to iron(II). I would also make up a control (reagent blank) flask and a set of standards in the range 0 to 10 ppm of iron. I would then use a UV / vis machine to measure the red complex [Fe(bipy)3]2+
I think that a late victorian chemist would be able to make 2,2'-bipy from pyridine, but if you want a method which an early victorian could have used then use potassium thiocyanate to form the red complex with the iron(III).
For that method you would need to put in the flasks potassium thiocyanate, you might want to read about the Fricke dosimeter which is used in nuclear chemistry to measure very high radiation doses. It is based on the conversion of iron(II) in acidic media which is oxygen saturated. The conversion to form iron(III) is under some conditions proportional to the radiation dose. Some people use thiocyanate in the iron(III) determination to make it more senstive.
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2023-03-28 03:22:41
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https://newproxylists.com/tag/algorithms/
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## algorithms – solve time complexity of recurrence relation
we assume n is a power of 2. We normally say multiplication takes theta(n^2) execution time. please show me process
exp_better(a,n) {
if (n == 1)
return a
m = n/2
power = exp_better(a,m)
power = power * power;
if (n is even)
return power;
else
return power*a;
}
## algorithms – Is there a way to uniquely map every natural number x
Imagine you have a number x, when x ∈ (0, N). Is there any algorithm that can map x to y, so that y is also y ∈ (0, N) with the mapping being unique, and the distribution of all y is distributed pseudorandomly across the whole range? I know it’s possible by just generating a set from 0 to N, shuffling it, and using x as an index. I want to know if there is some smarter way to do this that doesn’t involve a memory footprint that is linear to x.
The Pigeonhole principle shows that this is impossible when y > x, and it is trivially possible when x = y (well… y := x), but is this possible in a manner when y is randomly distributed?
My first (bad) attempt in C# was to use a golden ratio and travel around a circle, since mathematically this is guaranteed to give a unique angle every time. In theory (phi * n) mod 360 is sort of random looking and unique. Sadly this only works if you have infinite precision and not at all when you have discrete buckets for the output, so this idea didn’t really work out, even when N = 255:
So out of pure curiosity I’m wondering – is there some beautiful algorithm to map this so that it doesn’t involve either a predefined list of candidate numbers or a list of already used numbers, or so on?
## trees – Is the heap in “Data Structures Algorithms in Java” by Goodrich, Tamassia, and Goldwasser missing sentinel leaves?
In the book, Data Structures Algorithms in Java Sixth Edition by Goodrich, Tamassia, and Goldwasser on page 339, there is an
“Example of a heap storing 13 entries with integer keys. The last position is the one storing entry (13,W).”
I’ve recreated the a similar image using qtree-tikz with my own depth and height calculations in the box on the left.
According to the book,
A heap T storing n entries has height $$h = lfloor log n rfloor$$.
Justification: From the fact that T is complete, we know that the number of nodes in levels $$0$$ through $$h-1$$ of T is precisely $$1+2+4+…+2^{h-1} = 2^{h}-1$$, and that the number of nodes in level $$h$$ is at least 1 and at most $$2^h$$. Therefore
$$n geq 2^h -1 +1 = 2^h$$ and $$n leq 2^h -1 +2^h = 2^{h+1} -1$$.
Here is where I see a problem. I added the depths and heights for nodes at each level according to my understanding of trees in the box to the left of the tree. The example tree Figure 9.1 shown above obviously has 7 inner nodes and although depth level 3 would have 8 nodes, it is unfull at 6 nodes.
If I follow the “Justification”, I end up with the following
I am being forced to conclude that there must be a missing level of sentinels,
$$2^3 -1 = 7$$ and the only way to get a power of 3 in those formulas is to have a height of 4. But why on earth would the authors, who otherwise explain everything in detail, not make important piece of information explicit? Or, did I miss something? I would appreciate a thorough explanation to help me understand the proof. It also seems that the authors throw around the term “level” loosely, sometimes meaning depth level and sometimes meaning height level.
I should also mention that earlier in the book, on page 286, they provide a definition for the height of a tree without any examples.
We next define the height of a tree to be equal to the maximum of the depths of its positions (or zero, if the tree is empty).
• If $$p$$ is a leaf, then the height of $$p$$ is $$0$$.
• Otherwise, the height of $$p$$ is one more than the maximum of the heights of $$p$$‘s children.
## algorithms – Range sum query – tree representation efficiency
I was reading about possible solutions to the well known problem:
Given array `A` with length `N` create a structure that enables
• Answering what is the sum $$sum_{k=i}^{j} A(k)$$
• Updating $$A(k)$$
I’ve seen most solutions use binary index tree but was curios whether it’s possible to just use a regular tree that is built using similar qualities.
So given $$A = (5, 4, 7, 9, 1)$$
I try to construct a tree by creating a tree node for each value that has a start and end (which are just the index in the beginning.
To construct the tree I push all the starting node into a queue $$Q$$
``````while not Q.empty():
next <- ()
for i in range(Q.size()):
f <- Q.front()
Q.pop()
if Q.empty():
if marker:
parent <- make_parent(f, marker)
next.push(parent)
else:
marker <- f
else:
f2 <- Q.front()
Q.pop()
parent <- make_parent(f, marker)
next.push(parent)
for n in next:
Q.push(n)
``````
After the this ends marker will hold the root
(I have working c++ code but I tried to provide something more abstract and simple)
and to get a sum of range I perform the following (assuming I have an array Nodes that holds all the leaves) and that the query starts with the root of the tree that we created above
``````sumRangeNode(int i, int j, Node* n)
if i == j
return Nodes(i)
if n == null
return 0
if j < n->start || i > n->end
return 0;
if i <= n->start && j >= n->end
return n->val
return sumRangeNode(i, j, n->left) + sumRangeNode(i, j, n->right)
``````
The question is does it still have the $$log(N)$$ complexity, I’ve tried to reason about it but struggled with:
• The fact that I might be building a tree with “stragglers” like the $$1$$ in the example
• The fact that I recursively explore right and left. Intuition tells me that because there are “enough” cases where the descent is stopped it’s OK but couldn’t find a way to formalize/prove it.
## algorithms – Leader Election: Every bit-reversal ring is \$frac{1}{2}\$-symmetric
I have a proof that I need help with. Like the title says, the theorem is that every bit-reversal ring is $$frac{1}{2}$$-symmetric. The theorem is for Leader Election algorithm in synchronous ring. The things I know follow:
Bit reversal ring is defined as follows: We assign to each process $$i$$ the integer from $${0,ldots, n-1}$$ whose $$k$$ bit representation is the reverse of the $$k$$ bit representation of $$i$$. $$n$$ is also a power of two, $$n=2^{k}$$.
Two segments $$U$$ and $$V$$ are order equivalent if they are the same length $$k$$, and for all $$i$$ and $$j$$ such that $$1 leq i,j leq k$$ we have that $$u_{i} leq u_{j}$$ if and only if $$v_{i} leq v_{j}$$.
Ring $$R$$ is $$c$$-symmetric if for every segment $$S$$ of $$R$$ there are at least $$lfloor frac{cn}{l} rfloor$$ segments that are order equivalent to $$S$$, including $$S$$ itself, where $$l$$ is length of the segment, and this holds for every $$sqrt{n} leq l leq n$$.
So after plugging all I know into formulas I get that $$lfloor frac{2^{k-1}}{l} rfloor$$ is formula for number of segments and $$l$$ is such that $$2^{frac{k}{2}} leq l leq 2^{k}$$.
Any hint or piece of information would be much appreciated! Thank you.
## algorithms – Logarithmic Time — O(log n) in Python
Hi , I am new to Data Structure and trying to get some clarifications.
### Following shows an example Logarithmic Time — O(log n) in Python.
``````def binary_search(data, value):
n = len(data)
left = 0
right = n - 1
while left <= right:
middle = (left + right) // 2 # How can make this by 3 to make the search faster
if value < data(middle):
right = middle - 1
elif value > data(middle):
left = middle + 1
else:
return middle
raise ValueError('Value is not in the list')
if __name__ == '__main__':
data = (1, 2, 3, 4, 5, 6, 7, 8, 9)
print(binary_search(data, 8))
``````
Can you please explain and write the code on how can make this faster? Maybe for n/3
Thanks
## algorithms – Potential function for a dynamic stack
Consider a dynamic stack stored in an array of size m with n elements (initially n=0) and only a push operation. If upon a push n=m then expand the array such that m = 3n (That is, triple the size of the array).
Define a potential function based on the number n of elements in the array and the total number m of slots in the array, and show that the push operation has a constant amortized time.
My attempt:
Let the potential function be ϕ(n)= 3n – m.
Consider T*(push) = T(push) + ϕ(n) + ϕ(n-1)
If n < m then T(push) = 1 (since there is space in the array so you only need to add the new element) and so T*(Push) = 1 + 3n – m – (3(n-1)-m) = 4
If n=m then T(push) = n+1 (since there is not enough space so you need to copy all the elements plus the new one to the new array).
ϕ(n) = 3n-m, but m = 3n since the array is full it must be expanded for the push so ϕ(n) = 3n – 3n = 0
ϕ(n-1) = 3(n-1)-m, but prior to the push the array is full so m=n and thus ϕ(n-1) = 3(n-1)-n = 2n-3
Thus, T*(push) = n+1 + 0 – (2n-3) = -n + 4, which is not constant.
If anyone could help show me where I went wrong that would be great!
## algorithms – When is the expected time of a move to front list equal to that of an optimal list
When is the expected time of a move-to-front list equal to that of an optimal list?
My thinking is that it is when the move-to-front list happens to be arranged as an optimal list, but since the expected time takes the average over multiple times, this doesn’t really make sense. My only other thinking was the trivial case when the list has only one element.
## algorithms – Time efficient way to find pairs in an array whose sum equals a specific number?
Given an array of integers, we want to find how many explicit pairs can be made such that their sum is divisible by 60. The pairs do not need to be non-unique.
For example (29, 20, 30, 100, 30, 121, 160, 31, 20) has 6 pairs -> (29, 31) (20, 100) (20, 160) (30, 30) (100, 20) (160, 20)
The obvious and immediate thing is to make a nested for loop
`````` for (int i = 0; i < pairs.Count - 1; i++)
{
for (int n = i + 1; n < pairs.Count; n++)
{
if ((pairs(i) + pairs(n)) % 60 == 0)
{
count++;
}
}
}
``````
but that’s too slow. The above also doesn’t factor in duplicates like if the array had the number “30” as four elements, which is equal to a divergent series of (n-1)(n)/2 = (4-1)(4)/2 = 6.
How can we improve the performance?
My initial though is to make a dictionary/hashmap, but I am unsure how to iterate through that in a performant way.
## algorithms – DAG decomposition similar to series-parallel graphs
Let’s consider directed acyclic graphs (DAGs) with single source and single sink. Such graphs can be combined with two operations, used in Series-Parallel graphs – parallel composition $$P$$ and series composition $$S$$.
I’m interested in reverse operation (i.e. decomposition) – given a DAG $$G_1$$ I want to get a DAG $$G_2$$, where all the subgraphs, which can’t be decomposed further using operations, reverse to $$P$$ and $$S$$, are replaced by nodes. For example, the graph $$G1$$ below can be decomposed in this way – three subgraphs, which can’t be decomposed further are replaced by nodes (in red color). Note, that all such non-decomposable subgraphs with two vertices (simply arcs – in green color) aren’t replaced by nodes.
I know that the series decomposition can be done in linear time (it’s equivalent to finding articulation points). The parallel decomposition looks harder, also it’s unclear what operation should be tried first at each decomposition step.
Are there any existing algorithms, which can find such decomposition?
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2020-09-29 14:05:50
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https://edreport.org/reports/detail/jump-math-2019/sixth-grade
|
## JUMP Math
##### v1
###### Usability
Our Review Process
Title ISBN Edition Publisher Year
Teacher Resource for Grade 6, New US Edition 978-1-77395-107-2 JUMP Math 2019
Student Assessment & Practice Book 6.1 978-1-927457-06-1 JUMP Math 2019
Student Assessment & Practice Book 6.2 978-1-927457-07-8 JUMP Math 2019
Teacher Resource for Grade 2, New US Edition 978-1-77395-046-4 JUMP Math 2019
Student Assessment & Practice Book 2.1 978-1-927457-37-5 JUMP Math 2019
Student Assessment & Practice Book 2.2 978-1-927457-38-2 JUMP Math 2019
Teacher Resource for Grade 4, New US Edition 978-1-77395-039-6 JUMP Math 2019
Student Assessment & Practice Book 4.1 978-1-927457-12-2 JUMP Math 2019
Student Assessment & Practice Book 4.2 978-1-927457-13-9 JUMP Math 2019
Teacher Resource for Grade 5, New US Edition 978-1-77395-040-2 JUMP Math 2019
Student Assessment & Practice Book 5.1 978-1-927457-14-6 JUMP Math 2019
Student Assessment & Practice Book 5.2 978-1-927457-15-3 JUMP Math 2019
Teacher Resource for Grade 3, New US Edition 978-1-77395-038-9 JUMP Math 2019
Student Assessment & Practice Book 3.1 978-1-927457-42-9 JUMP Math 2019
Student Assessment & Practice Book 3.2 978-1-927457-43-6 JUMP Math 2019
Teacher Resource for Grade 7, New US Edition 978-1-77395-082-2 JUMP Math 2019
Student Assessment & Practice Book 7.1 978-1-927457-47-4 JUMP Math 2019
Student Assessment & Practice Book 7.2 978-1-927457-48-1 JUMP Math 2019
Teacher Resource for Kindergarten, New US Edition 978-1-77395-044-0 JUMP Math 2019
Student Assessment & Practice Book K.1 978-1-927457-71-9 JUMP Math 2019
Student Assessment & Practice Book K.2 978-1-927457-72-6 JUMP Math 2019
Teacher Resource for Grade 1, New US Edition 978-1-77395-045-7 JUMP Math 2019
Student Assessment & Practice Book 1.1 978-1-927457-32-0 JUMP Math 2019
Student Assessment & Practice Book 1.2 978-1-927457-33-7 JUMP Math 2019
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### Overall Summary
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for alignment. The instructional materials meet expectations for focus and coherence by assessing grade-level content, devoting the majority of class time to the major work of the grade, and being coherent and consistent with the progressions in the Standards. The instructional materials partially meet expectations for rigor and the mathematical practices. The instructional materials partially meet the expectations for rigor by attending to conceptual understanding and procedural skill and fluency, and they also partially meet expectations for practice-content connections by identifying the mathematical practices and using them to enrich grade-level content.
###### Alignment
Partially Meets Expectations
Not Rated
### Focus & Coherence
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for Gateway 1. The instructional materials meet expectations for focus within the grade by assessing grade-level content and spending the majority of class time on the major work of the grade. The instructional materials meet expectations for being coherent and consistent with the Standards as they connect supporting content to enhance focus and coherence, have an amount of content that is viable for one school year, and foster coherence through connections at a single grade.
##### Gateway 1
Meets Expectations
#### Criterion 1.1: Focus
Materials do not assess topics before the grade level in which the topic should be introduced.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for not assessing topics before the grade level in which the topic should be introduced. Above-grade-level assessment items are present and can be modified or omitted without significant impact on the underlying structure of the instructional materials.
##### Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for assessing grade-level content. Above-grade-level assessment items are present but could be modified or omitted without a significant impact on the underlying structure of the instructional materials. The Sample Unit Quizzes and Tests along with Scoring Guides and Rubrics were reviewed for this indicator. Examples of grade-level assessment items include:
• Teacher Resource, Part 1, Sample Unit Quizzes and Tests, Unit 2, Quiz, Lessons 1-5, Item 7, “What is the opposite integer of –16?” assesses grade-level standard 6.NS.6a when students find an integer on the opposite side of zero on the number line.
• Teacher Resource, Part 2, Sample Unit Quizzes and Tests, Unit 9, Test, Lessons 5-9, Item 2c, “Use the histogram to fill in the blanks. How many names are between 3 and 6 letters long?” Students are given a frequency table which represents the number of letters in students' first names; they use this frequency table to create a histogram and answer a series of questions. (6.SP.5)
• Teacher Resource, Part 2, Sample Unit Quizzes and Tests, Unit 6 Test, Item 7, “The graph shows the cost of renting a scooter from Bernard’s store. a. What is the independent variable? What is the dependent variable? b. How much would you pay to ride a scooter for: 1 hour?, 2 hours?, 4 hours? c. How much do you have to pay for the scooter before you have even ridden it?” Students use a graph to determine the independent and dependent variable and solve word problems. (6.EE.9)
• Teacher Resource, Part 2, Sample Unit Quizzes and Tests, Unit 8 Test, Item 4, “a. Sketch the net for the prism and label each face. b. Marco says that he only needs to find the area of two faces of this prism to calculate the surface area. Is he correct? Explain. c. What is the surface area of the prism? Do not use a calculator.” Students are determining the surface area of a prism. (6.G.4)
The following are examples of assessment items that are aligned to standards above Grade 6, but these can be modified or omitted without compromising the instructional materials:
• Teacher Resource, Part 2, Sample Quizzes and Tests, Unit 2, Item 3, “Solve the proportion: a. 6:18 = 4: ___ b. 0.8: ___ = 2: 10.” Students recognize and represent proportional relationships between quantities. (7.RP.2)
• Teacher Resource, Part 2, Sample Quizzes and Tests, Unit 4 Test, Item 7, “Write an addition or subtraction equation to find the distance between the integers: a. -30 and +40, b. -25 and +10, c. -42 and -6.” Students add and subtract integers. (7.NS.1)
#### Criterion 1.2: Coherence
Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for students and teachers using the materials as designed and devoting the majority of class time to the major work of the grade. Overall, instructional materials spend approximately 65 percent of class time on the major clusters of the grade.
##### Indicator {{'1b' | indicatorName}}
Instructional material spends the majority of class time on the major cluster of each grade.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for spending the majority of class time on the major clusters of each grade. Overall, approximately 65 percent of class time is devoted to major work of the grade.
The materials for Grade 6 include 15 units. In the materials, there are 171 lessons, and of those, 28 are Bridging lessons. According to the materials, Bridging lessons should not be “counted as part of the work of the year” (page A-56), so the number of lessons examined for this indicator is 143 lessons. The supporting clusters were also reviewed to determine if they could be factored in due to how strongly they support major work of the grade. There were connections found between supporting clusters and major clusters, and due to the strength of the connections found, the number of lessons addressing major work was increased from the approximately 84 lessons addressing major work as indicated by the materials themselves to 92.5 lessons.
Three perspectives were considered: the number of units devoted to major work, the number of lessons devoted to major work, and the number of instructional days devoted to major work including days for unit assessments.
The percentages for each of the three perspectives follow:
• Units – Approximately 67 percent, 10 out of 15;
• Lessons – Approximately 65 percent, 92.5 out of 143; and
• Days – Approximately 65 percent, 102.5 out of 158.
The number of instructional days, approximately 65 percent, devoted to major work is the most reflective for this indicator because it represents the total amount of class time that addresses major work.
#### Criterion 1.3: Coherence
Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for being coherent and consistent with the Standards. The instructional materials connect supporting content to enhance focus and coherence, include an amount of content that is viable for one school year, are consistent with the progressions in the Standards, and foster connections at a single grade.
##### Indicator {{'1c' | indicatorName}}
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. When appropriate, the supporting work enhances and supports the major work of the grade level.
Examples where connections are present include the following:
• Teacher Resource, Part 1, Unit 6, Lesson G6-4, and Teacher Resource, Part 2, Unit 5, Lesson G6-25, connect 6.NS.C with 6.G.3 as students graph points in the coordinate plane in order to solve mathematical problems about polygons.
• Teacher Resource, Part 1, Unit 6, Lessons G6-12 and G6-13 connect 6.EE.7 with 6.G.1 as students are expected to solve real-world and mathematical problems by writing and solving equations that arise from finding the area of triangles, parallelograms, and trapezoids.
• Teacher Resource, Part 2, Unit 8, Lessons G6-33 and G6-41 connect 6.EE.7 with 6.G.2 as students are expected to solve real-world and mathematical problems by writing and solving equations that arise from finding the volume of a right rectangular prism.
##### Indicator {{'1d' | indicatorName}}
The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for having an amount of content designated for one grade level that is viable for one school year in order to foster coherence between grades. Overall, the amount of time needed to complete the lessons is approximately 158 days, which is appropriate for a school year of approximately 140-190 days.
• The materials are written with 15 units containing a total of 171 lessons.
• Each lesson is designed to be implemented during the course of one 45 minute class period per day. In the materials, there are 171 lessons, and of those, 28 are Bridging lessons. These 28 Bridging lessons have been removed from the count because the Teacher Resource states that they are not counted as part of the work for the year, so the number of lessons examined for this indicator is 143 lessons.
• There are 15 unit tests which are counted as 15 extra days of instruction.
• There is a short quiz every 3-5 lessons. Materials expect these quizzes to take no more than 10 minutes, so they are not counted as extra days of instruction.
##### Indicator {{'1e' | indicatorName}}
Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for being consistent with the progressions in the Standards. Overall, the materials address the standards for this grade level and provide all students with extensive work on grade-level problems. The materials make connections to content in future grades, and they explicitly relate grade-level concepts to prior knowledge from earlier grades.
The materials develop according to the grade-by-grade progressions in the Standards, and content from prior or future grades is clearly identified and related to grade-level work. The Teacher Resource contains sections that highlight the development of the grade-by-grade progressions in the materials, identify content from prior or future grades, and state the relationship to grade-level work.
• At the beginning of each unit, "This Unit in Context" provides a description of prior concepts and standards students have encountered during the grade levels before this one. The end of this section also makes connections to concepts that will occur in future grade levels. For example, "This Unit in Context" from Unit 5, Geometry: Coordinate Grids, of Teacher Resource, Part 2 describes the geometric topics students encountered in Grade 5, specifically graphing in the first quadrant of the coordinate plane, the work students will encounter graphing and solving problems in all four quadrants of the coordinate plane, and how the work of this unit will build to transformations and the Pythagorean Theorem in Grade 8.
There are some lessons that are not labeled Bridging lessons that contain off-grade-level material, but these lessons are labeled as “preparation for” and can be connected to grade-level work. For example, Teacher Resource, Part 1, Unit 4, Lesson NS6-31 addresses multi-digit addition with positive integers, and the lesson is labeled as "preparation for 6.NS.3."
The materials give all students extensive work with grade-level problems. The lessons also include Extensions, and the problems in these sections are on grade level.
• Whole class instruction is used in the lessons, and all students are expected to do the same work throughout the lesson. Individual, small-group, or whole-class instruction occurs in the lessons.
• The problems in the Assessment & Practice books align to the content of the lessons, and they provide on-grade-level problems that "were designed to help students develop confidence, fluency, and practice." (page A-51, Teacher Resource)
• In the Advanced Lessons, students get the opportunity to engage with more difficult problems, but the problems are still aligned to grade-level standards. For example, the problems in Teacher Resource, Part 2, Unit 5, Lesson NS6-28 engage students in reflecting points across one axis and then the other, but these problems still align to 6.NS.6. Also, the problems in Teacher Resource, Part 2, Unit 8, Lesson G6-41 have students solving problems involving volume and surface area of prisms and pyramids which align to standards from 6.NS, 6.EE, and 6.G.
The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples of these explicit connections include:
• Every lesson identifies "Prior Knowledge Required" even though the prior knowledge identified is not aligned to any grade-level standards. For example, Teacher Resource, Part 2, Unit 3, Lesson EE6-9 identifies knowing addition and subtraction, along with multiplication and division, as inverse operations in order for students to accomplish the goal of the lesson, which is solving one-step equations using logic and the concept of operations.
• There are 28 lessons identified as Bridging lessons, and most of these lessons are aligned to standards from prior grades and also state for which grade-level standards they are preparation. Teacher Resource, Part 1, Unit 3, Lesson EE6-2, which has students write and solve addition equations, is aligned to 4.OA.3 and is in preparation for 6.EE.2 and 6.EE.5. Also, Teacher Resource, Part 1, Unit 6, Lessons G6-1 and G6-3, which have students identify right angles, parallel lines, and perpendicular lines, are aligned to 4.G.1 and 4.G.2 and are in preparation for 6.G.3.
##### Indicator {{'1f' | indicatorName}}
Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings and make connections within and across domains.
In the materials, the units are organized by domains and are clearly labeled. For example, Teacher Resource, Part 1, Unit 3 is entitled Expressions and Equations: Variables and Equations, and Teacher Resource, Part 2, Unit 9 is entitled Statistics and Probability: Distribution. Within the units, there are goals for each lesson, and the language of the goals is visibly shaped by the CCSSM cluster headings. For example, in Teacher Resource, Part 2, Unit 8, the goal for Lesson G6-41 states "Students will solve problems involving the surface area of rectangular and triangular pyramids and prisms and the volume of rectangular prisms." The language of this goal is visibly shaped by 6.G.A, "Solve real-world and mathematical problems involving area, surface area, and volume."
The instructional materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples of these connections include the following:
• In Teacher Resource, Part 1, Unit 6, Lessons G6-10 and G6-11, the materials connect 6.EE.A with 6.EE.B as students evaluate expressions at specific values of their variables and solve real-world and mathematical problems by writing and solving equations.
• In Teacher Resource, Part 2, Unit 9, Lessons SP6-5 through SP6-9, the materials connect 6.SP.A with 6.SP.B as students develop an understanding of statistical variability and summarize and describe distributions.
• In Teacher Resource, Part 1, Unit 6, Lesson G6-5, the materials connect 6.NS with 6.EE as students solve real-world and mathematical problems by graphing points and writing and solving equations.
• In Teacher Resource, Part 2, Unit 8, Lessons G6-39 and G6-40, the materials connect 6.G with 6.NS.B as students represent three-dimensional figures using nets made up of rectangles and triangles, use the nets to find the surface area of these figures, and compute fluently with multi-digit numbers.
### Rigor & Mathematical Practices
The instructional materials reviewed for JUMP Mathematics Grade 6 partially meet expectations for Gateway 2. The instructional materials partially meet expectations for rigor by developing conceptual understanding of key mathematical concepts, giving attention throughout the year to procedural skill and fluency, and spending some time working with routine applications. The instructional materials do not always treat the three aspects of rigor together or separately, but they do place heavier emphasis on procedural skill and fluency. The instructional materials partially meet expectations for practice-content connections. Although the instructional materials meet expectations for identifying and using the MPs to enrich mathematics content, they partially attend to the full meaning of each practice standard. The instructional materials partially attend to the specialized language of mathematics.
##### Gateway 2
Partially Meets Expectations
#### Criterion 2.1: Rigor
Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.
The instructional materials reviewed for JUMP Mathematics Grade 6 partially meet expectations for rigor by developing conceptual understanding of key mathematical concepts, giving attention throughout the year to procedural skill and fluency, and spending some time working with routine applications. The instructional materials do not always treat the three aspects of rigor together or separately, but they do place heavier emphasis on procedural skill and fluency.
##### Indicator {{'2a' | indicatorName}}
Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include lessons designed to support students’ conceptual understanding. Examples include:
• Teacher Resource, Part 1, Unit 1, Lesson RP6-10, Exercises, “a. Make a double number line diagram from a ratio table.” Students are given a completed ratio table to transfer to a number line diagram. Students are shown double number lines as a model for rates and ratio tables. This extends their conceptual understanding of unit rates.
• Student Resource, Assessment & Practice Book, Part 1, Lessons RP6-6 to RP6-10, students are given many opportunities to develop their understanding of ratios and unit rates. For example,
• Student Resource, Assessment & Practice Book, Part 1, Lesson RP6-6, Item, 1c “The ratio of stars to squares is ____:____ e. The ratio of squares to moons is ___:___” Students are introduced to the concept of a ratio.
• Student Resource, Assessment & Practice Book, Part 1, Lesson RP6-7, Item 2, “Use skip counting or multiplication to complete a ratio table for each ratio. b. 1:2.” Students are introduced to ratio tables.
• Student Resource, Assessment & Practice Book, Part 1, Lesson RP6-9, Item 1, “Divide to find the missing information. b. 4 cakes cost $16 1 cake costs ___ c. 5 pears cost$20 1 pear costs ___” Students work to find unit rates.
• Student Resource, Assessment & Practice Book, Part 1, Lesson RP6-20, Item 8, “Look at the word California. a. What is the ratio of vowels to consonants? b. What fractions of the letters are vowels? c. What percent of the letters are consonants?” Lesson RP6-20 introduces students to equivalent ratios, at times using tables.
• In Teacher Resource, Part 2, Unit 6, Lesson EE6-16, students are shown through direct instruction, how area models can produce equivalent expressions. The students do some work with area models on their own.
##### Indicator {{'2b' | indicatorName}}
Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency. The materials place an emphasis on fluency, giving many opportunities to practice standard algorithms and to work on procedural knowledge.
Standard 6.NS.2 expects fluency in dividing multi-digit numbers using the standard algorithm. Examples include:
• Teacher Resource, Part 1, Unit 5, Lesson RP6-22, Exercises, Item d, “1743$$\div$$6” has students divide 4 digits by 1 digit numbers using the standard algorithm.
• Teacher Resource, Part 2, Unit 1, Lesson NS6-60, Exercises, Item a “327$$\div$$51” has students practice division.
• Throughout the materials, students are required to incorporate the division algorithm while practicing other math topics. For example:
• Student Resource, Assessment & Practice Book, Part 1, Lesson G6-18, Item 1d, “A parking spot has two sides 5 m long. The distance between the sides is 325 cm. What is the area of the parking spot?” Students convert between metric units by multiplying or dividing using base 10 numbers.
• Teacher Resource, Part 2, Unit 3, Lesson EE6-9, Extensions, Item 2, “An ebook costs $16 before taxes and$16.48 after taxes. A can of soda costs $1.60 before taxes and$1.68 after taxes. Which item was taxed at a higher rate?” Students solve equations that require them to use the division.
Standard 6.NS.3 expects students to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Examples include:
• Teacher Resource, Part 1, Unit 4, Lesson NS6-33, Exercises, “Use base ten blocks to regroup so that each place value has a single digit. a. 3 tenths + 12 hundredths b. 7 ones + 18 tenths c. 7 ones + 15 tenths + 14 hundredths.” Students are given the opportunity to develop fluency with the standard algorithm for adding and subtracting decimals. Here students review with base-10 blocks then apply that knowledge to the standard algorithms for addition and subtraction.
• Teacher Resource, Part 2, Unit 1, Lesson NS6-48, “SAY: I don’t know how to multiply decimals, but I do know how to multiply fractions. ASK: How can I change this problem into one I already know how to do? (change the decimals to fractions) Have a volunteer change the decimals to fractions, without writing the answer: $$\frac{3}{100}$$ x $$\frac{4}{1000}$$ Have another volunteer write the answer. ($$\frac{12}{10,000}$$) Then remind students that we’re not done yet. SAY: We now have an answer, but the question was given in terms of decimals, so the answer needs to be given using decimals.” Students practice multiplying and dividing decimals first by writing the decimals as fractions with a common base-10 denominators, then by using the standard algorithm to multiply.
• Teacher Resource, Part 2, Unit 1, Lessons NS6-58, Word Problems Practice, Item a, “Lina has 4.2 pounds of cheese. She needs 0.05 pounds of cheese for each sandwich. How many sandwiches can she make?” Students develop fluency with the standard division algorithm when they solve problems that first require them to multiply to make the divisor a whole number, and subsequently use the entire division of decimals algorithm.
##### Indicator {{'2c' | indicatorName}}
Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics without losing focus on the major work of each grade.
Overall, many of the application problems are routine in nature and replicate the examples given during guided practice, and problems given for independent work are heavily scaffolded. Examples include:
• Teacher Resource, Part 1, Unit 1, Lesson RP6-9, Extensions, Item 2, “Liz drives 131 miles in 2 hours. It takes Mindy twice as long to drive 257 miles. Who is driving faster? How much faster?” (6RP.3) Students engage in a routine problem using ratio and rate reasoning.
• Teacher Resource, Part 1 Unit 5, Lesson RP6-19, Exercises, “Have students find the missing percentages of other stamps in each collection: a. USA: 40% Canada: $$\frac{1}{2}$$ Other: b. Mexico: 25% USA: $$\frac{3}{5}$$ Other:” Before this problem students are guided through specific problem solving strategies, and then given problems that match the given strategy, making this a routine problem. The problem given before the Exercises was “$$\frac{2}{5}$$ of the stamps are from the United States and 36% are from Canada. What percent of Jennifer’s stamps are from neither the United States nor Canada? Solve this problem with the class. (change 2/5 to 40%, then add 40% + 36% = 76%, so the stamps from neither place make up 24% of Jennifer’s collection).” (6.RP.3) Students use ratio and rate reasoning to solve real-world problems.
• Teacher Resource, Part 2, Unit 3, Lesson EE6-12, Exercises, “Repeat with more examples. As you give each example, ask students to first identify the smaller number, and remind them that this should be the shorter bar. a. Bethany is three times as tall as her baby brother.” (6.EE.7) Students solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. The exercises included in the lessons follow the structure of a problem presented as an example, eliminating students opportunities to apply the mathematics in a non-routine way.
• Teacher Resource, Part 2, Unit 6, Lesson EE6-19, Extension, “Wilson has $30. For every book he reads, his mother gives him$5. a. Create a T-table that shows the number of books as input and the amount of money as output. b. Write a rule for the amount of money Wilson has after reading n books. c. How many books does Wilson need to read to get $50?$100? $1000?” (6.EE.9) Students find rules for linear graphs. • Teacher Resource, Part 2, Unit 6, Lesson EE6-20,Exercises, “A boat leaves port at 9:00 am and travels at a steady speed. Some time later, a man jumps into the water and starts swimming in the same direction as the boat. a. How many minutes passed between the time the boat left port and the time the man jumped into the water? (15 min) When did the man jump into the water?” Students are provided a graph to solve the problem. (6.EE.9) The lesson provides students with completed graphs and tables with which to identify the independent and dependent variables and write a rule. The graphs have limited real-world context and the tables have no real-world context. Non-routine problems are occasionally found in the materials. Examples include: • Teacher Resource, Part 1, Unit 3, Lesson EE6-7, Extensions, Item 1, “Some friends bought pizza and ate 2 $$\frac{7}{10}$$ pizzas. They had $$\frac{4}{5}$$ of a pizza with pineapple left and $$\frac{1}{8}$$ of a pizza without pineapple left. 2 $$\frac{1}{2}$$ of the pizzas they ordered were vegetarian. How many pizzas did they buy? Which fact did you not need?” (6.EE.7) Students solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. However, the majority of the independent practice problems that students complete in this section, only involve pre-setup problems without real-world context, where students follow an algorithm to find the answer. ##### Indicator {{'2d' | indicatorName}} Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade. The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the materials, but there is an over-emphasis on procedural skills and fluency. The curriculum addresses conceptual understanding, procedural skill and fluency, and application standards, when called for, and evidence of opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. There are multiple lessons where one aspect of rigor is emphasized. The materials emphasize fluency, procedures, and algorithms. Examples of conceptual understanding, procedural skill and fluency, and application presented separately in the materials include: • Conceptual Understanding: Teacher Resource, Part 1, Unit 2, Lesson NS6-11, Extensions, Item 2a, “Draw a number line to find the fraction halfway between -$$\frac{14}{3}$$ and -$$\frac{4}{3}$$. Repeat for the fraction halfway between -$$\frac{14}{5}$$ and -$$\frac{4}{5}$$.” Students place positive and negative fractions on a number line and use the number line to order those fractions. • Application: Teacher Resource, Part 2, Unit 2, Lesson RP6-26, Extensions, Item 1, “A store offers you a choice between two options for fancy socks, which are usually$10 per pair. Which price option would you choose? A. 3 pairs of socks for the price of 2, or B. 30% off all pairs of socks?” Students use ratios to solve real world problems.
• Procedural Skill and Fluency: Teacher Resource, Mental Math, Skills 1, 2, 3, adn 4, Item 7, “Name the odd number that comes after the number shown. a. 37.” This section contains problems to help students maintain and develop procedural fluency with Addition, Subtraction, and Multiplication.
Examples of where conceptual understanding, procedural skill and fluency, and application are presented together in the materials include:
• Teacher Resource, Part 1, Unit 4, Lesson NS6-29, Extensions, Item 3, “Sarah saw four fish at different elevations: −0.025 km, −0.18 km, −0.9 km, −1.8 km. Use the information below to decide which fish was seen at which elevation. The coelacanth lives between 150 m and 400 m below sea level. The football fish lives between 200 m and 1 km below sea level. The deep sea angler lives between 250 m and 2 km below sea level. The rattail lives between 22 m and 2.2 km below sea level.” Comparing Decimal Fractions and Decimals contains both conceptual understanding and application of mathematics. Students develop conceptual understanding of comparing rational numbers by using a number line to compare both positive and negative fractions and decimals.
• Teacher Resource, Part 1, Unit 2, Lesson NS6-17, Exercises, “Find the GCF of the two numbers being added and then rewrite the sum as shown. a. 18 + 42 = __ × (__ +__ ).” This lesson contains both conceptual understanding and procedural fluency. Students develop conceptual understanding in the lesson when they model the distributive property of numbers using array models. They develop procedural fluency when they complete exercises dividing out the GCF of numbers.
#### Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for practice-content connections. Although the instructional materials meet expectations for identifying and using the MPs to enrich mathematics content, they partially attend to the full meaning of each practice standard. The instructional materials partially attend to the specialized language of mathematics.
##### Indicator {{'2e' | indicatorName}}
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
The instructional materials reviewed for JUMP Math Grade 6 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.
All 8 MPs are clearly identified throughout the materials, with few or no exceptions. Examples include:
• The Mathematical Practices are identified at the beginning of each unit in the “Mathematical Practices in this Unit.”
• “Mathematical Practices in this Unit” gives suggestions on how students can show they have met a Mathematical Practice. For example, in Teacher Resource, Part 2, Unit 7, Mathematical Practices in this Unit, “MP.4: In SP6-4 Extension 2, students model mathematically when they use a table to represent and solve a non-routine, real-world problem.”
• “Mathematical Practices in this Unit” gives the Mathematical Practices that can be assessed in the unit. For example, in Teacher Resources, Part 1, Unit 6, Mathematical Practices in this Unit, “In this unit, you will have the opportunity to assess MP.1 to MP.4 and MP.6 to MP.8.”
• The Mathematical Practices are also identified in the materials in the lesson margins.
• In optional Problem Solving Lessons designed to develop specific problem-solving strategies, MPs are identified in specific components/ problems in the lesson.
##### Indicator {{'2f' | indicatorName}}
Materials carefully attend to the full meaning of each practice standard
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of MPs 1 and 4.
Examples of the materials carefully attending to the meaning of some MPs include:
• MP2: Teacher Resource, Part 1, Unit 5, Lesson RP6-19 Extensions, Item 2, “Mr. Bates buys: • 5 single-scoop ice cream cones for $1.45 each • 3 double-scoop ice cream cones for$2.65 each. A tax of 10% is added to the cost of the cones. Mr. Bates pays with a 20-dollar bill. How much change does he receive? Show your work.” Students reason abstractly and quantitatively to calculate the tax and change and then interpret their solution in the context of the problem.
• MP5: Teacher Resource, Part 2, Unit 6, Lesson EE6-16, Extension 2, “Write an equivalent expression without brackets. Use any tool you think will help. Explain how you got your answer. a) 3(2x +5) b) 2(4x +7) c) 5(8x -3) d) a(bx + c).” Students choose an appropriate tool to solve the problem.
• MP5: Teacher Resource, Part 2, Unit 3, Lesson EE 6-9, Extensions, Item 1, “A classroom is made up of students from Grade 6 and 7. 25% of the Grade 6 students and 70% of the Grade 7 students prefer comedy over science fiction. There are twice as many Grade 6 students as Grade 7 in the class. What percent of the class prefers comedy? Use any tool you think will help.” Students have the ability to choose an appropriate tool to solve the problem.
• MP6: Teacher Resource, Part 2, Unit 5, Lesson G6-22, Extensions, Item 3, “Vicky bought 12 bus tickets for $9. She calculated how much 36 bus tickets cost as follows: $$\frac{12}{9}$$= $$\frac{x}{36}$$ and 4 x 9 = 36, so x =4 x 12 = 48, so 36 bus tickets cost$48. a. Do you agree with Vicky’s answer? Why or Why not? b. What did Vicky do correctly? What did she do incorrectly? c. How much would 36 bus tickets cost? Explain.” Students attend to precision in calculations as they evaluate the calculations of another student and find what is correct about how a proportion is written and what is incorrect. Students correctly set up the proportion and complete the calculations.
• MP6: Teacher Resource, Part 2, Unit 7, Lesson SP6-2 Extensions, Item 1, “a. Find the median and the range for each set i. 2, 3, 5, 7, 9, 10; ii. 12, 16, 19, 22, 26, 26, 26, 26. b. Add 4 to each data point in i and ii. Find the new median and range. c. Why did adding 4 to each data point change the median but not the range? d. in pairs, explain your answers to part c. Do you agree with each other? Discuss why or why not.” Students attend to precision when they use the definitions of median and range to explain why adding the same number to each data point in a data set changes the median but not the range.
Examples of the materials not carefully attending to the meaning of MPs 1 and 4 include:
• MP1: Teacher Resource, Part 2, Unit 1, Lesson NS6-59, Extensions 3, “Use mental math or pencil and paper to solve. Explain your choice. a. $$\frac{5}{4}$$ ÷ $$\frac{4}{5}$$ b. $$\frac{2}{3}$$ ÷ $$\frac{4}{6}$$ c. $$\frac{3}{17}$$ ÷ $$\frac{9}{34}$$”. Students do not need to make sense of the problem or devise a strategy to solve the problem, but rather use an algorithm to solve.
• MP1: Teacher Resource, Part 2, Unit 5, Lesson G6-20 Extensions Item a., “Plot and join the points, in order. Use the same grid for all parts. i. (-1, -2), (-1, 3), (0,4), (1,3), (1, -2), (0, -3). Join the first point to the last point. ii. (-1, -2), (-2, -3), (-3, -3), (-3, -2), (-1, 0); iii. (1, 0), (3, -2), (3, -3), (2, -3), (1, -2). b. What shape did you make? c. Find the area of the shape.” There is no opportunity to make sense of this problem as students are told how to solve the problem.
• MP4: Teacher Resource, Part 1, Unit 2, Lesson NS6-3, Extensions, Item 3, “John bikes 7km in 10 minutes and skates 900 m in 2 minutes. Does he skate or bike faster? How much faster? Write your answer in complete sentences.” Students do not model with mathematics as they do not have to interpret their solutions in the context of the problem to determine if the results make sense.
• MP4: Teacher Resource, Part 2, Unit 3, Lesson EE6-9, Extensions, Item 2, “An eBook costs $16 before taxes and$16.48 after taxes. A can of soda costs $1.60 before taxes and$1.68 after taxes. Which item was taxed at a higher rate?” Students do not model with mathematics as they do not have to interpret their solutions in the context of the problem to determine if the results make sense.
##### Indicator {{'2g' | indicatorName}}
Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
##### Indicator {{'2g.i' | indicatorName}}
Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
Students explain their thinking, compare answers with a partner, or understand the error in a problem. However, this is done sporadically within extension questions, and often the materials identify questions as MP3 when there is not an opportunity for students to analyze situations, make conjectures, and justify conclusions. At times, the materials prompt students to construct viable arguments and critique the reasoning of others. Examples that demonstrate this include:
• In Teacher Resource, Part 2, Unit 1, Lesson NS6-56, Extensions, Item 3, students explain why a pattern emerges from a previous problem. “a. In the previous question, which positive numbers are greater than their square? Why does this make sense?”
• Teacher Resource, Part 2, Unit 7, Lesson SP6-1, Extensions, Item 4, students calculate the mean of a set of numbers and explain why the mean in part i is greater than the mean of part ii.
• In Teacher Resource, Part 1, Unit 4, Lesson NS6-27, Extensions, Item 4, “a. After the first two numbers in a sequence, each number is the sum of all the previous numbers in the sequence. If the 20th term is 393,216, what is the 18th term? Look for a fast way to solve the problem. b. In pairs, explain why your method works. Do you agree with each other? Discuss why or why not.”
• In Teacher Resource, Part 1, Unit 2, Lesson NS6-11, Extensions, Item 2, students explain their reasoning and critique the reasoning of a partner. “a. Draw a number line to find the fraction halfway between -$$\frac{14}{3}$$ and -$$\frac{4}{3}$$. Repeat for the fraction halfway between -$$\frac{14}{5}$$ and -$$\frac{4}{5}$$. b. Without drawing a number line, write the fraction halfway between −$$\frac{14}{351}$$ and -$$\frac{4}{351}$$. Explain how you know. c. In pairs, explain your answers to part b. Do you agree with each other? Discuss why or why not.”
In questions where students must explain an answer or way of thinking, the materials identify the exercise as MP3. As a result, questions identified as MP3 are not arguments and not designed to establish conjectures or build a logical progression of a statement to explore the truth of the conjecture. Examples include:
• In Teacher Resource, Part 1, Unit 2, Lesson NS6 - 13, Extensions, Item 3, students answer “Can a positive fraction be equivalent to a negative fraction? Explain why or why not.”
• Teacher Resource, Part 2, Unit 6, Lesson EE6-17, Extensions, Item 1, “Explain why 7y + 2y = 9y?”
• Students are given extension questions when they are asked to analyze the math completed by a fictional person. For example: Teacher Resource, Part 2, Unit 1, Lesson NS6-45, Extensions, Item 2, “Ron says 2 R 1 = 2 $$\frac{1}{4}$$ because 9 ÷ 4 = 2 R 1 and 9 ÷ 4 = 2 $$\frac{1}{4}$$ Is this reasoning correct? Explain.” These problems begin to develop students’ ability to analyze the mathematical reasoning of others but do not fully develop this skill. Students analyze an answer given by another, but do not develop an argument or present counterexamples.
##### Indicator {{'2g.ii' | indicatorName}}
Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Some guidance is provided to teachers to initiate students in constructing arguments and critiquing others; however, the guidance lacks depth and structure, and there are multiple missed opportunities to assist students to engage in constructing and critiquing mathematical arguments.
The materials have limited support for the teacher to develop MP3. Generally, the materials encourage students to work with a partner as a way to construct arguments and critique each other. In the teacher information section, teachers are provided with the following information:
• Page A-14: “Promote communication by encouraging students to work in pairs or in small groups. Support students to organize and justify their thinking by demonstrating how to use mathematical terminology symbols, models and manipulatives as they discuss and share their ideas. Student grouping should be random and vary throughout the week.” The material provides no further guidance on thoughtful ways to group students and only limited structures that would encourage collaboration.
• Page A-49: “Classroom discussion in the lesson plans include the prompts SAY, ASK and PROMPT. SAY is used to provide a sample wording of an explanation, concept or definition that is at grade level, precise, and that will not lead to student misconceptions. ASK is used for probing questions, followed by sample answers in parentheses. Allow students time to think before providing a PROMPT, which can be a simple re-wording of the question or a hint to guide students in the correct direction to answer the question….You might also have students discuss their thinking and explain their reasoning with a partner, or write down their explanations individually. This opportunity to communicate thinking, either orally or in writing, helps students consolidate their learning and facilitates the assessment of many Standards for Mathematical Practice.” This format does not provide any structure for constructing arguments and critiquing others, in fact, this Say, Ask, Prompt model will only lead students to learn in a step by step manner directed by the teacher.
• Page A-49: There are sentence starters that are referenced that show teachers how to facilitate discussions among students. The materials state, “When students work with a partner, many of them will benefit from some guidance, such as displaying question or sentence stems on the board to encourage partners to understand and challenge each other’s thinking, use of vocabulary, or choice of tools or strategies. For example:
• I did ___ the same way but got a different answer. Let’s compare our work.
• What does ___ mean?
• Why is ___ true?
• Why do you think that ___ ?
• I don’t understand ___. Can you explain it a different way?
• Why did you use ___? (a particular strategy or tool)
• How did you come up with ___? (an idea or strategy)”
Once all students have answered the ASK question, have volunteers articulate their thinking to the whole class so other students can benefit from hearing their strategies” While this direction would help teachers facilitate discussion in the classroom, it would not help teachers to develop student’s ability to construct arguments or critique the reasoning of others.
• A rubric for the Mathematical Practices is provided for teachers on page I-57. For MP3, a Level 3 is stated as, “Is able to use objects, drawings, diagrams, and actions to construct an argument” and “Justifies conclusions, communicates them to others, and responds to the arguments of others.” This rubric would provide some guidance to teachers about what to look for in student answers, but no further direction is provided about how to use it to coach students to improve their arguments or critiques.
• In the Math Practices in this Unit Sections, MP3 is listed numerous times. Each time, the explanation of MP3 in the unit consists of a similar general statement. For example, in Teacher Resource, Part 2, Unit 5, “MP.3: In G6-21 Extension 3, students make a conjecture about what the distance will be between any number and its opposite, and construct a viable argument to explain their conjecture.” Other units all follow a similar structure in their introduction to teachers about how students will encounter MP3 in the materials. These explanations do not provide guidance to teachers in constructing arguments or critiquing the reasoning of others.
There are limited times when specific guidance is provided to teachers for specific problems. Examples include:
• Some guidance is provided to teachers to construct a viable argument when teachers are provided solutions to questions labeled as MP3 in the extension questions. Some of these questions include wording that could be used as an exemplar response about what a viable argument is. For example, in Teacher Resource, Part 1, Unit 2, Lesson NS6-20 Extensions, Item 5 the solution provided says, “I have two ways of getting from 5 to 60: multiply by 2 and then multiply by 6, or I can multiply by 6 and then multiply by something else: 5 × 2 × 6 = 5 × 6 ×? I know the two 6s will always be the same because that’s what it means to be a ratio table. Now, I see the other two numbers are also the same. That’s why switching the rows and columns gives another ratio table.”
• In Teacher Resource, Part 2, Unit 3, Lesson EE6-13, Extensions, Item 3, students are asked to add 5 to a mystery number, then double that result. Subtract 10, and then divide that result by 2. The materials state, “In pairs, explain why the trick works. Choose any tool you think will help, such as expressions with variables and brackets, bags and blocks, or a T-table and pictures.” Students are asked if they agree with each other and to discuss why or why not. Sample solutions are provided but no teacher guidance is given on engaging students in constructing viable arguments.
Frequently, problems are listed as providing an opportunity for students to engage in MP3, but miss the opportunity to give detail on how a teacher will accomplish this. Examples include:
• The opportunity is missed to provide exemplar responses for teachers when students are constructing viable arguments. For example, in Teacher Resource, Part 2, Unit 1, Lesson NS6-58, Extensions, Item 2 is labeled as MP3 and students are asked to explain their reasoning. The solution provided says, “Answers: 8.56 ÷ 0.4 = 21.4 and 8.56 ÷ 0.2 = 42.8. The second answer is double the first because it is dividing by half as much.” This provided solution would not help teachers understand how the student could construct an argument in response to this question.
• Teacher Resource, Part 1, Unit 6, Lesson G6-4, in a section labeled as MP3, teachers are told, “ASK: Without drawing it, does this quadrilateral have any horizontal lines or vertical lines? (AB and CD are both vertical) How do you know? (A and B have the same first coordinate, as do C and D) Are AB and CD the same length? (no, AB is 3 units long, and CD is 4 units long) So is ABDC a trapezoid or a parallelogram? (a trapezoid).” These questions do not promote constructing arguments.
##### Indicator {{'2g.iii' | indicatorName}}
Materials explicitly attend to the specialized language of mathematics.
The instructional materials reviewed for JUMP Math Grade 6 partially meet expectations for explicitly attending to the specialized language of mathematics.
Accurate mathematics vocabulary is present in the materials; however, while vocabulary is identified throughout the materials, there is no explicit directions for instruction of the vocabulary in the teacher materials of the lesson. Examples include, but are not limited to:
• Vocabulary is identified in the Terminology section at the beginning of each unit.
• Vocabulary is identified at the beginning of each lesson.
• The vocabulary words and definitions are bold within the lesson.
• There is not a glossary.
• There is not a place for the students to practice the new vocabulary in the lessons.
### Usability
This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two
Not Rated
#### Criterion 3.1: Use & Design
Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
##### Indicator {{'3a' | indicatorName}}
The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
##### Indicator {{'3b' | indicatorName}}
Design of assignments is not haphazard: exercises are given in intentional sequences.
##### Indicator {{'3c' | indicatorName}}
There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
##### Indicator {{'3d' | indicatorName}}
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
##### Indicator {{'3e' | indicatorName}}
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
#### Criterion 3.2: Teacher Planning
Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
##### Indicator {{'3g' | indicatorName}}
Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
##### Indicator {{'3h' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.
##### Indicator {{'3i' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.
##### Indicator {{'3j' | indicatorName}}
Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
##### Indicator {{'3k' | indicatorName}}
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.
#### Criterion 3.3: Assessment
Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
##### Indicator {{'3n' | indicatorName}}
Materials provide strategies for teachers to identify and address common student errors and misconceptions.
##### Indicator {{'3o' | indicatorName}}
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.
##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.
#### Criterion 3.4: Differentiation
Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
##### Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.
##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.
#### Criterion 3.5: Technology
Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
## Report Overview
### Summary of Alignment & Usability for JUMP Math | Math
#### Math K-2
The instructional materials reviewed for JUMP Math K-2 partially meet expectations for alignment. The instructional materials meet expectations for Gateway 1: Focus and Coherence by not assessing topics before the grade level in which the topic should be introduced, devoting the majority of class time to the major work of the grade, and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2: Rigor and the Mathematical Practices by partially maintaining a balance of the aspects of rigor and partially attending to practice-content connections. The instructional materials were not reviewed for Gateway 3: Usability.
##### Kindergarten
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
#### Math 3-5
The instructional materials reviewed for JUMP Math 3-5 partially meet expectations for alignment. The instructional materials meet expectations for Gateway 1: Focus and Coherence by not assessing topics before the grade level in which the topic should be introduced, devoting the majority of class time to the major work of the grade, and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2: Rigor and the Mathematical Practices by partially maintaining a balance of the aspects of rigor and partially attending to practice-content connections. The instructional materials were not reviewed for Gateway 3: Usability.
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
#### Math 6-8
The instructional materials reviewed for JUMP Math 6-8 partially meet expectations for alignment. The instructional materials meet expectations for Gateway 1: Focus and Coherence by not assessing topics before the grade level in which the topic should be introduced, devoting the majority of class time to the major work of the grade, and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2: Rigor and the Mathematical Practices by partially maintaining a balance of the aspects of rigor and partially attending to practice-content connections. The instructional materials were not reviewed for Gateway 3: Usability.
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
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2022-08-14 07:03:45
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http://stats.stackexchange.com/questions/87171/coding-categorical-variables-for-regression
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Coding categorical variables for regression
I'm not sure of the best way to code my categorical predictor variable for use in a hierarchical regression in order to test my specific hypothesis. This categorical variable has 3 levels representing 3 groups. I want to compare group 1 to group 2, group 1 to group 3 and group 2 to group 3. I know that for dummy coding I create k-1 variables, so 2 dummy variables in my case and code these variables with 0s and 1s while choosing one level of the categorical variable to be a reference category.
However, I'm not sure this is the best way of making the comparisons I wish to make as it appears I could only compare each group to the reference category, am I correct? So if group 3 was the reference category I could compare group 1 to group 3 and group 2 to group 3 but I could not compare group 1 to group 2. What alternative method of coding should I use to make these comparisons? My regression model will also contain continuous variables. I'm an undergrad psychology student and statistics are not my strong point simple answers would be best for me. I use SPSS. Thank you!
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This sounds like you want Helmert Coding, see for example: ats.ucla.edu/stat/r/library/contrast_coding.htm#HELMERT I'm not an SPSS user I'm afraid though, so will let someone else give you a proper answer using SPSS. – Corone Feb 19 '14 at 16:31
Do bear in mind that you can estimate any contrasts, regardless of the coding scheme you use; it's just convenient to use a coding scheme where coefficients correspond to something of primary interest. – Scortchi Feb 19 '14 at 16:37
(1) I don't think it is quite helmert coding, but I always get confused between these different schemes. (2) I usually just change reference group and run model again. – charles Feb 19 '14 at 16:38
@Charles, sorry, yes would have been more helpful if I at least defined it! Helmert (as I understand it) is comparing 1 with 2&3 and comparing 2 with 3, which sort of seems to be the case here. If Claire compares 1&2 and 1&3 and 2&3 separately, you don't have regression, just a data description. – Corone Feb 19 '14 at 16:57
@Corone the webpage you linked mentions that reverse helmert coding would not make much sense with a nominal variable which is what my variable is. I am assuming it is the same for helmert coding? – Claire Feb 19 '14 at 17:16
Here is an example using the employee data.sav data, which comes with standard installation. Suppose salary is the dependent variable, job category, jobcat, is the categorical independent variable, and beginning salary, salbegin, is the continuous independent variable. Using GLM, you can perform pairwise comparisons between each pair of job categories. The steps are as follow:
1. With the data set open, go to Analyze > General Linear Model > Univariate.
2. Put the dependent variable and independent variable into the correct slots. Categorical independent variables go to "Fixed Factor(s)" and continuous ones go to "Covariate(s)." Do not worry about the Random Factors. When it's all set, click the "Model" button.
3. In the Model panel, highlight the two independent variables, then change the build term to "Main effects," and then click the arrow button (indicated by the red circle) to bring the two variables over. When all set, click "Continue."
4. Now, click the "Option" button.
5. In the Option panel, do the followings: 1) Highlight jobcat, 2) bring it over to the right by clicking the arrow button, 3) Check "Compare Main Effects", 4) Specify the adjustment you'd like to make for the multiple pairwise comparisons. I left it as LSD which does not adjust for multiple tests, 5) Check "Parameter Estimates" so that you'll also get the regression coefficients. When it's all done, click Continue and then OK to submit the test.
6. Here is the regression coefficient table:
7. Scroll down a bit and you'll find the pairwise comparisons table:
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+1, however, you might clarify 5. 4) "LSD which does not adjust for multiple tests". Tukey's test does not adjust alpha to control for multiple tests (as, say, the Bonferroni approach does), but it is a perfectly valid strategy for dealing with multiple comparisons issues. – gung Feb 19 '14 at 19:51
@gung, as always, thanks for your comment. If you can clarify a bit more for me I'll be happy to make the revision. SPSS has LSD, Bonferroni, and Sidak corrections available to choose in this GLM module. If I am not mistaken Tukey's correction is HSD, which is not available. Thanks. PK. – Penguin_Knight Feb 19 '14 at 20:17
My mistake, I misread that. I thought it said HSD (Tukey's test), but it says "LSD" (Fisher's least significant difference), which you correctly note does not provide additional control for multiple comparisons beyond the initial F-test. If the wizard does not offer HSD as an option, the OP may need to click Paste instead of OK and then manually add /POSTHOC TUKEY (I think?--it's been a long time) to the syntax before running. – gung Feb 19 '14 at 20:33
@gung, I see, no problem. There are more than 15 different post hoc adjustments in GLM, but once we introduced a covariate, the Post Hoc button becomes inactive (as seen in the screenshot of step 4 above). I guess in SPSS those post hoc cna only work if we don't specify any continuous independent predictors. If we do, the three only choices are LSD, Bonferroni, and Sidak. I did try your method to replace LSD with TUKEY but SPSS showed an error and refused to proceed. – Penguin_Knight Feb 19 '14 at 21:02
Since you want to compare all groups with each other, the tests will not be orthogonal, even if they are a-priori. So you should use a test that addresses that. Tukey's honestly significant differences (HSD) test will do that, and is familiar to many people. You needn't worry about the type of coding used. First, as @Scortchi notes, you can perform this test with any regular coding method (reference level, effect, etc.). Second, SPSS will probably take care of the coding for you.
It's been a long time since I've used SPSS, but I gather you would use the GLM Univariate Analysis option, since you have both continuous and categorical variables. The SPSS documentation for post-hoc comparisons after running a GLM can be found here.
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The Wikipedia article on post hoc analyses lists several tests/options for comparing groups after a factor has been found significant. I don't know SPSS well anymore, but I expect that it would implement one or more of the tests on that list. You can search for those terms in the SPSS documentation and that should tell you how to specify that you want those comparisons.
Googling for "SPSS post hoc" brings up several promising links as well.
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2016-05-06 22:33:43
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https://www.mathsgee.com/23197/how-can-approximate-the-square-root-without-using-calculator
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Network: Global | Joburg Libraries | MOOCs | StartUpTribe | Zimbabwe | Donate
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How can I approximate the square root of 2 without using a calculator?
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$\sqrt{2}$ lies between 1 and 2 because 12 = 1 and 22 = 4 and 2 is between 1 and 4.
$\sqrt{2}$ lies between 1,4 and 1,5 because 1,42 = 1,96 and 1,52 = 2,25
$\sqrt{2}$ lies between 1,41 and 1,42 because 1,412 = 1,988 1 and 1,422 = 2,016 4 … and so on ….
by Diamond (67,868 points)
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2021-05-18 13:58:10
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https://business.tutsplus.com/tutorials/how-to-measure-your-businesss-profitability--cms-20674
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Unlimited PowerPoint templates, graphics, videos & courses! Unlimited asset downloads! From $16.50/m Advertisement # How to Measure Your Business's Profitability Difficulty:BeginnerLength:LongLanguages: This post is part of a series called Key Metrics Every Business Should Track. Pivotal Liquidity Metrics to Help You Avoid Insolvency You are what you measure. Many businesses generate reams of stats, metrics and key performance indicators, but if you’re not measuring the right things, you won’t get the right results. The point of measuring performance, after all, is not to see how you did in the past, but to use that information to do better in the future. A recent Gartner survey found that less than half of businesses have metrics that help them understand how their work contributes to achieving their strategic objectives. So in this four-part series of tutorials, we’re going to look at the critical metrics your business needs to track in key areas: profitability, liquidity, efficiency and customer acquisition/satisfaction. First up is a look at profitability metrics. As with all the other areas, there are dozens of possible metrics you could track, but we’re going to focus on four of the most important ones, and look in detail at how you calculate them, what the results can tell you about the health of your business, and most importantly, what action you can take to improve your results in the future. A lot of the numbers we’re looking at in this tutorial will come from your company’s income statement. If you need a refresher on what any of them mean or where to find them, check our recent tutorial on reading an income statement. ## 1. Gross Profit Margin ### Why It’s Important This number is a good basic measure of how efficient your company is at manufacturing and distributing its products. It helps you zero in on your costs, and how much they’re eating into your profits. ### How to Calculate It The formula for this one is quite simple: Gross Profit Margin = (Revenue – Cost of Goods Sold) / Revenue For example, let’s say your company sold 1,000 T-shirts for$10 each. Your revenue for the year would be $10,000. But each T-shirt cost$6 to manufacture and distribute, so the cost of goods sold is $6,000. We plug in the numbers and get: Gross Profit Margin = (10,000 – 6,000) / 10,000 = 40% ### How to Evaluate It In this example, your company is keeping 40% of the proceeds of each sale as profit, which is pretty healthy. That means it has plenty of money left to cover other costs, like research and development, taxes, and general administrative costs. But keep in mind that gross profit margins vary widely by industry. Service-based businesses like law firms and accountants usually have high gross profit margins (50% and up), whereas manufacturers and retailers tend to be more in the 20% to 30% range. This website shows average gross profit margins for various industries, and you can also check with industry associations in your particular field for more information. Once you’ve established a benchmark, see how you compare against it, and how the percentage is changing over time. ### How to Improve It If your business is suffering from lower gross profit margins than your competitors, there are several things you can do. The most straightforward is to raise your prices. A bigger markup on each sale will translate to a higher gross profit margin, although of course you need to be careful not to drive away customers, so do your research to see how much the market will bear. An alternative is to look at your product mix, and spend most of your marketing dollars on pushing the high-margin products. Or you could target the cost side of the equation. Can you switch to a cheaper supplier, or negotiate a better rate with your existing supplier? If you’ve been doing business for a while and growing, you may be able to negotiate a volume discount, lowering your costs and increasing your margin. Or perhaps the problem lies in your own processes. Examine every detail of your manufacturing and distribution processes to see if there are opportunities to make them more efficient. ## 2. Return on Invested Capital ### Why It’s Important One of the most basic things that companies do is to take money that’s been invested (capital), and turn it into profit. Return on Invested Capital is a measure of how effectively your company is doing that, and so it’s crucial to keep track of. ### How to Calculate It Working out ROIC can be either very simple or very complicated, depending on how much detail you want to go into. Let’s look at the simple method first. The simplified formula is: ROIC = Net income / (Long-Term Debt + Equity) These are all numbers you can find easily on your business’s financial statements. Net income is the “bottom line” number on the income statement, and total debt and equity are on the balance sheet. Google, for example, earned$12.2 billion in net income last year, and its total long-term debt and equity came to $89.5 billion. So it’s ROIC would be 13.6% (12.2 divided by 89.5). Some people like to make it more complicated, however. They make lots of adjustments to net income, mostly to take out any items that are unlikely to recur, arriving at a figure called NOPAT (net operating profit after tax). And they adjust debt and equity to get a more accurate picture of what’s actually invested in the business, as opposed to things like surplus cash. Those adjustments can make a significant difference. Google’s ROIC, according to Marketwatch, is 14.9%. Morningstar makes it 15.1%. What’s right for your business depends on how deep you want to get into the details. Just remember that the idea is to see how much profit you’re generating from the capital you’ve invested, and that if you’re comparing with other companies, they may be using slightly different methodologies. ### How to Evaluate It As with the other measures, the important thing is to look at your ROIC in comparison to your own results in earlier years, and also in comparison to your competition. For public companies, you can easily find the ratio in the companies’ accounts or on financial websites. Look at a few examples in your industry to get an idea of what to aim for. As a very rough rule of thumb, an ROIC of 15% or higher indicates a healthy amount of profit, but it varies for different industries. ### How to Improve It You can drive your ROIC higher in a number of ways. You could focus on the top part of the equation, the net income. Cutting expenses will give an immediate boost to net income, and therefore to ROIC, but be careful not to take it too far and starve your business of necessary investment. You could also do a sales drive to increase revenue, or try to shift your sales mix to more profitable products. You could also choose to focus on the bottom part of the equation, debt and equity. Are you making the maximum use of all your capital? If not, then you can choose to pay down debt or return money to investors, which will also boost your ROIC. ## 3. Overhead Ratio ### Why It’s Important Overhead costs can be a real drag on a business’s profitability. If you’re spending a big portion of your income just keeping the lights on, you’ll struggle to grow. We looked at costs associated with sales in the “gross profit margin” section, and this is the other piece of the puzzle: fixed overheads. ### How to Calculate It You can calculate your overhead ratio using the following formula: Overhead Ratio = Operating Expenses / (Operating Income + Interest Income) Again, these are all lines from the income statement. Operating expenses are the “overheads,” things like office rent, utilities, machinery maintenance and so on. They’re necessary for your business, but they don’t directly generate income. We’re then dividing that number by operating income (which you find on the income statement) plus interest income from your business bank account or investments. For example, Microsoft had$30.8 billion in operating expenses, according to its 2013 annual report. It earned $26.8 billion operating income and$0.7 billion interest income. So the calculation would be:
Overhead Ratio = 30.8 / (26.8 + 0.7) = 1.1
### How to Evaluate It
It’s best to look at this in conjunction with gross profit margin. Microsoft has a strong gross profit margin of 74%, so it has plenty of money left over for other costs. It’s overhead ratio, then, is sustainable.
If your business has a smaller gross profit margin, on the other hand, you’ll need to keep a much tighter lid on expenses.
As with the other ratios, the right number depends a lot on which industry you’re in, and what other companies in your field are doing. Go through the accounts of public companies in your field, companies whose success you want to emulate, and see how your company’s overhead ratio compares with theirs.
### How to Improve It
This is a simple one to improve: simply cut expenses. But you need to be careful. Although overheads don’t contribute directly to generating revenue, they can still be important for your business.
Some operating expenses, for example, could be research and development costs, or advertising spending. Cutting office rental costs by moving to a smaller building in a cheaper neighborhood may be a smart move, but slashing your marketing budget could harm your company’s future sales. Laying off admin staff will save money in the short run, but might end up costing you more if it leads to inefficient invoicing or missed customer calls. So you want to trim overheads enough to be competitive, but without sacrificing quality.
## 4. Asset Turnover
### Why It’s Important
There are different ways of making money. Some businesses, like Walmart and other discount retailers, have low profit margins but a high volume of sales. Asset turnover measures how efficiently your business uses its assets to generate sales.
### How to Calculate It
This is a nice simple one to end with. It’s simply:
Asset Turnover = Total Revenue / Total Assets
Your revenue number, of course, is the top line from your income statement. It’s the total amount of sales you brought in over the year. Total assets are just the sum of all your property, equipment, inventory and so on, and you can get that number straight from the balance sheet.
A look at Walmart’s 2013 annual report shows that it had revenue of $469.1 billion and total assets of$203.1 billion. That means an asset turnover of 2.3.
### How to Evaluate It
It’s important to compare like with like. Let’s go back to Microsoft. Its asset turnover is just 0.5, much lower than Walmart’s. But it’s still a profitable business, thanks to that high profit margin we saw earlier.
So look at the numbers together with the other ratios we’ve looked at in this tutorial, and see how they fit together. Then think about your business model, and the industry you’re in, and decide on a target asset turnover that’s right for you.
### How to Improve It
Even if you’re not running a Walmart-style high-volume business, improving your asset turnover will still improve your profitability. How to do it? This is the only ratio we’ve looked at where cost is not part of the equation, so in this case it’s all about driving top-line sales growth. Consider things like ad campaigns, special promotions, loyalty programs, undercutting your competitors on price, or paying special incentives to your sales staff.
Also think about speed. Remember, asset turnover is about converting assets into revenue as efficiently as possible. So no matter what type of business you run, look at ways to streamline your processes and get your products to market more quickly.
## Next Steps
If you look at one of these metrics on its own, it’s of limited use. Take them all together, and you have a clear picture of your business’s profitability.
You’ve now learned what these metrics are and how to calculate them, and what the results tell you about the health of your business.
So the next step is to start using them on a regular basis. Start by looking at your historical numbers, going back perhaps five years if possible. One of the main benefits of tracking metrics is in seeing changes in your business over time. Is your overhead ratio creeping up year by year? Maybe this is the year to get those costs in check. Is your gross profit margin declining? Perhaps you need to examine your product mix and push the ones that carry a higher markup.
Of course, you shouldn’t look at any of the numbers in isolation. If there’s a good reason why your asset turnover is lower than your competitorsperhaps because you have higher profit marginsthen you don’t have to try to match them just for the sake of it. If your overheads are high, but that’s because you’re spending money on essential research that will pay off over the long term, then there’s no sense in making damaging cuts.
When taken in the right context, though, and tracked consistently, these metrics can help you see where your business is profitable, where any problems lie, and what actions you need to take to boost your bottom line.
In the rest of our four-part series, we’ll look at a range of other metrics to track in different areas of your business, starting next week with some key liquidity metrics to make sure you stay solvent.
## Resources
Graphic Credit: Line Graph designed by Scott Lewis and Abacus designed by Berkay Sargın from the Noun Project.
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2021-05-16 21:31:20
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http://ecell4.readthedocs.io/en/latest/tutorials/tutorial9.html
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# 9. Spatial Gillespie Method¶
## 9.1. Spaces in E-Cell4¶
What the space in E-Cell4 looks like?
In [1]:
from ecell4 import *
w1 = ode.ODEWorld(Real3(1, 1, 1))
w2 = gillespie.GillespieWorld(Real3(1, 1, 1))
We created a cube size, 1, on a side for ODEWorld and GillespieWorld. In this case the volume only matters, that is
In [2]:
w3 = ode.ODEWorld(Real3(2, 0.5, 1)) # is almost equivalent to 'w1'
w4 = gillespie.GillespieWorld(Real3(2, 2, 0.25)) # is almost equivalent to 'w2'
This returns the same results. Because the volume is same as 1.
This seems reasonable in homogeneous system, but the cell is NOT homogeneous. So we need to consider a space for molecular localization.
You can use several types of space and simulation methods in E-Cell4. We show an example with spatial Gillespie method here.
## 9.2. Spatial Gillespie Method¶
In E-Cell4, the Spatial Gillespie method is included in meso module. Let’s start with run_simulation like ode.
In [3]:
%matplotlib inline
import numpy
from ecell4 import *
with reaction_rules():
A + B == C | (0.01, 0.3)
y = run_simulation(numpy.linspace(0, 10, 100), {'C': 60}, solver='meso')
At the steady state, the number of C is given as follows:
You will obtain almost the same result with ode or gillespie (may take longer time than ode or gillespie). This is not surprising because meso module is almost same with Gillespie unless you give additional spatial parameter.
Next we will decompose run_simulation.
In [4]:
from ecell4 import *
with reaction_rules():
A + B == C | (0.01, 0.3)
m = get_model()
w = meso.MesoscopicWorld(Real3(1, 1, 1), Integer3(1, 1, 1)) # XXX: Point2
w.bind_to(m) # XXX: Point1
sim = meso.MesoscopicSimulator(w) # XXX: Point1
obs = FixedIntervalNumberObserver(0.1, ('A', 'B', 'C'))
sim.run(10, obs)
viz.plot_number_observer(obs)
This is nothing out of the ordinary one except for MesoscopicWorld and MesoscopicSimulator, but you can see some new elements.
First in w.bind_to(m) we asscociated a Model to the World. In the basic exercises before, we did NOT do this. In spatial methods, Species attributes are necessary. Do not forget to call this. After that, only the World is required to create a MesoscopicSimulator.
Next, the important difference is the second argument for MesoscopicWorld, i.e. Integer3(1, 1, 1). ODEWorld and GillespieWorld do NOT have this second argument. Before we explain this, let’s change this argument and run the simulation again.
In [5]:
from ecell4 import *
with reaction_rules():
A + B == C | (0.01, 0.3)
m = get_model()
w = meso.MesoscopicWorld(Real3(1, 1, 1), Integer3(4, 4, 4)) # XXX: Point2
w.bind_to(m) # XXX: Point1
sim = meso.MesoscopicSimulator(w) # XXX: Point1
obs = FixedIntervalNumberObserver(0.1, ('A', 'B', 'C'))
sim.run(10, obs)
viz.plot_number_observer(obs)
You must have the different plot. If you increase value in the Integer3, you will have more different one.
Actually this second argument means the number of spatical partitions. meso is almost same with gillespie, but meso divides the space into cuboids (we call these cuboids subvolumes) and each subvolume has different molecular concentration by contrast gillespie has only one uniform closed space. So in the preceding example, we divided 1 cube with sides 1 into 64 (4x4x4) cubes with sides 0.25. We threw 60 C molecules into the World. Thus, each subvolume has 1 species at most.
## 9.3. Defining Molecular Diffusion Coefficient¶
Where the difference is coming from? This is because we do NOT consider molecular diffusion coefficient, although we got a space with meso. To setup diffusion coefficient, use Species attribute 'D' in the way described before (2. How to Build a Model). As shown in 1. Brief Tour of E-Cell4 Simulations, we use E-Cell4 special notation here.
In [6]:
with species_attributes():
A | {'D': '1'}
B | {'D': '1'}
C | {'D': '1'}
# A | B | C | {'D': '1'} # means the same as above
get_model()
Out[6]:
<ecell4.core.NetworkModel at 0x7ff0add53c48>
You can setup diffusion coefficient with with species_attributes(): statement. Here we set all the diffusion coefficient as 1. Let’s simulate this model again. Now you must have the almost same result with gillespie even with large Integer3 value (the simulation will takes much longer than gillespie).
How did the molecular diffusion work for the problem? Think about free diffusion (the diffusion coefficient of a Species is ) in 3D space. The unit of diffusion coefficient is the square of length divided by time like or .
It is known that the average of the square of point distance from time to is equal to . Conversely the average of the time scale in a space with length scale is about .
In the above case, the size of each subvolume is 0.25 and the diffusion coefficient is 1. Thus the time scale is about 0.01 sec. If the molecules of the Species A and B are in the same subvolume, it takes about 1.5 sec to react, so in most cases the diffusion is faster than the reaction and the molecules move to other subvolume even dissociated in the same subvolume. The smaller , the smaller subvolume’s volume , so the reaction rate after dissociation is faster, and the time of the diffusion and the transition between the subvolume gets smaller too.
## 9.4. Molecular localization¶
We have used add_molecules function to add molecules to World in the same manner as ode or gillespie. Meanwhile in MesoscopicWorld, you can put in molecules according to the spatial presentation.
In [7]:
from ecell4 import *
w = meso.MesoscopicWorld(Real3(1, 1, 1), Integer3(3, 3, 3))
In MesoscopicWorld, you can set the subvolume and the molecule locations by giving the third argument Integer3 to add_molecules. In the above example, the molecule type A spreads all over the space, but the molecule type B only locates in a subvolume at the center of the volume. To check this, use num_molecules function with a coordinate.
In [8]:
print(w.num_molecules(Species('B'))) # must print 120
print(w.num_molecules(Species('B'), Integer3(0, 0, 0))) # must print 0
print(w.num_molecules(Species('B'), Integer3(1, 1, 1))) # must print 120
120
0
120
Furthermore, if you have IPython Notebook environment, you can visualize the molecular localization with ecell4.viz module.
In [9]:
# viz.plot_world(w, radius=0.01)
viz.plot_world(w, interactive=False)
viz.plot_world function visualize the location of the molecules in IPython Notebook cell by giving the World. You can set the molecule size with radius. Now you can set the molecular localization to the World, next let’s simulate this. In the above example, we set the diffusion coefficient 1 and the World side 1, so 10 seconds is enough to stir this. After the simulation, check the result with calling viz.plot_world again.
## 9.5. Molecular initial location and the reaction¶
This is an extreme example to check how the molecular localization affects the reaction.
In [10]:
%matplotlib inline
from ecell4 import *
with species_attributes():
A | B | C | {'D': '1'}
with reaction_rules():
A + B > C | 0.01
m = get_model()
w = meso.MesoscopicWorld(Real3(10, 1, 1), Integer3(10, 1, 1))
w.bind_to(m)
This model consists only of a simple binding reaction. The World is a long x axis cuboid, and molecules are located off-center.
In [11]:
w.add_molecules(Species('A'), 1200, Integer3(2, 0, 0))
viz.plot_world(w, interactive=False)
On a different note, there is a reason not to set Integer3(0, 0, 0) or Integer3(9, 0, 0). In E-Cell4, basically we adopt periodic boundary condition for everything. So the forementioned two subvolumes are actually adjoining.
After realizing the location expected, simulate it with MesoscopicSimulator.
In [12]:
sim = meso.MesoscopicSimulator(w)
obs1 = NumberObserver(('A', 'B', 'C')) # XXX: saves the numbers after every steps
sim.run(5, obs1)
viz.plot_number_observer(obs1)
In [13]:
# viz.plot_world(w, radius=0.025)
viz.plot_world(w, interactive=False)
To check the effect of initial coordinates, we recommend that you locate the molecules homogeneously with meso or simulate with gillespie.
In [14]:
w = meso.MesoscopicWorld(Real3(10, 1, 1), Integer3(10, 1, 1))
w.bind_to(m)
The solid line is biased case, and the dash line is non-biased. The biased reaction is obviously slow. And you may notice that the shape of time-series is also different between the solid and dash lines. This is because it takes some time for the molecule A and B to collide due to the initial separation. Actually it takes seconds to move the initial distance between A and B (about 4).
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2018-06-22 08:54:44
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https://www.esrf.fr/UsersAndScience/Experiments/MX/How_to_use_our_beamlines/forms/equation-5
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The Relative Humidity (RH - r in the equation above) in equilibrium with solutions can be understood in terms of Raoult's law. It has two aspects that are counter-intuitive and lead to some surprising observations. The first is that the number of equivalent molecules in solution must be accounted for. This means that for sodium chloride, each ion in solution counts as a molecular equivalent. This requires knowledge of the ionization behaviour of the substance in solution. For example, ammonium sulfate effectively dissociates into two ions [NH4+ and (NH4SO4)-] and not three as might be expected. Raoult's law starts to break down for PEG solutions over a molecular weight of 1000 Da but this can be corrected using the Flory-Huggins model for the entropy of mixing (used in equation 2).
y can be calculated for other salts using the following formula: y = [(1/ρ)M]/1000
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2022-06-29 01:12:25
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http://www.ams.org/publications/authors/books/postpub/author-pages
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# Supplementary Book MaterialsListed by year
Sort the table below by clicking on a column head. You may also use the search box to search for a title, author/editor, series, published year.
Authors/Editors Title Publication Year Series
David E. Zitarelli with Della Dumbaugh and Stephen F. Kennedy A History of Mathematics in the United States and Canada: Volume 2: 1900--1941 2022 Spectrum, Vol. 103
Colin Adams Lost in the Math Museum: A Survival Story 2022 Anneli Lax New Mathematical Library, Vol. 55
Leslie Hogben, Jephian C.-H. Lin, and Bryan L. Shader
Inverse Problems and Zero Forcing for Graphs 2022 Mathematical Surveys and Monographs, Vol. 270
Rebecca Rapoport and Dean Chung Mathematics 2023: Your Daily Epsilon of Math: 12-Month Calendar---January 2023 through December 2023 2022
Kate Juschenko
Amenability of Discrete Groups by Examples 2022 Mathematical Surveys and Monographs, Vol. 266
Ryota Matsuura A Friendly Introduction to Abstract Algebra 2022 AMS/MAA Textbooks, Vol. 72
Anthony Bonato An Invitation to Pursuit-Evasion Games and Graph Theory 2022 Student Mathematical Library, Vol. 97
Isaac Goldbring Ultrafilters Throughout Mathematics 2022 Graduate Studies in Mathematics, Vol. 220
Dragana S. Cvetkovic Ilic Completion Problems on Operator Matrices 2022 Mathematical Surveys and Monographs, Vol. 267
John K. Osoinach, Jr Discovering Abstract Algebra 2021 AMS/MAA Textbooks, vol. 67
Dana C. Ernst An Introduction to Proof via Inquiry-Based Learning 2022 AMS/MAA Textbooks, vol. 73
Hilário Alencar, Walcy Santos, and Gregório Silva Neto Differential Geometry of Plane Curves 2022 Student Mathematical Library, vol. 96
Alice Peters and Mark Saul A Festival of Mathematics: A Sourcebook 2022 MSRI Mathematical Circles Library, vol. 28
Rustum Choksi Partial Differential Equations: A First Course 2022 Pure and Applied Undergraduate Texts, vol. 54
Della Dumbaugh and Deanna Haunsperger Count Me In: Community and Belonging in Mathematics 2022 Classroom Resource Materials, vol. 68
Joseph H. Silverman Abstract Algebra: An Integrated Approach 2022 Pure and Applied Undergraduate Texts, Vol. 55
Benjamin B. Kennedy Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics 2022 AMS/MAA Textbooks, Vol. 70
William Johnston The Calculus of Complex Functions 2022 AMS/MAA Textbooks, Vol. 71
Ryan Hayward Hex: A Playful Introduction 2022 Anneli Lax New Mathematical Library, Vol. 54
Yukio Matsumoto An Introduction to Morse Theory 2001 Translations of Mathematical Monographs, Vol. 208
Nabil H. Mustafa Sampling in Combinatorial and Geometric Set Systems 2022 Mathematical Surveys and Monographs, Vol. 265
Ragnar-Olaf Buchweitz. Edited by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar, and Janina C. Letz Maximal Cohen-Macaulay Modules and Tate Cohomology 2021 Mathematical Surveys and Monographs, Vol. 262
Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman Asymptotic Geometric Analysis, Part II 2021 Mathematical Surveys and Monographs, Vol. 261
Harriet Pollatsek Lie Groups: A Problem-Oriented Introduction via Matrix Groups 2009 AMS/MAA Textbooks, Vol. 13
David M. Clark and Xiao Xiao The Number Line through Guided Inquiry 2021 AMS/MAA Textbooks, Vol. 69
B. Setheruman Proofs and Idea: A Prelude to Advanced Mathematics 2021 AMS/MAA Textbooks, Vol. 68
Vugar E. Ismailov Ridge Functions and Applications in Neural Networks 2021 Mathematical Surveys and Monographs, Vol. 263
J. Scott Carter and Seiichi Kamada Diagrammatic Algebra 2021 Mathematical Surveys and Monographs, Vol. 264
Burkard Polster and Marty Ross Putting Two and Two Together: Selections from the Mathologer Files 2021
Michael Joswig Essentials of Tropical Combinatorics 2021 Graduate Studies in Mathematics, Vol. 219
Marcelo Viana and José Espinar Differential Equations: A Dynamical Systems Approach to Theory and Practice 2021 Graduate Studies in Mathematics, Vol. 212
Lindsay N. Childs, Cornelius Greither, Kevin P. Keating, Alan Koch, Timothy Kohl, Paul J. Truman, and Robert G. Underwood Hopf Algebras and Galois Module Theory 2021 Mathematical Surveys and Monographs, Vol. 260
Brian Osserman A Concise Introduction to Algebraic Varieties 2021 Graduate Studies in Mathematics, Vol. 216
Louis-Pierre Arguin A First Course in Stochastic Calculus 2021 Pure and Applied Undergraduate Texts, Vol. 53
Gizem Karaali and Lily S. Khadjavi Mathematics for Social Justice: Focusing on Quantitative Reasoning and Statistics 2021 Classroom Resource Materials, Vol. 66
Imre Bárány Combinatorial Convexity 2021 University Lecture Series, vol. 77
Riccardo Benedetti Lectures on Differential Topology 2021 Graduate Studies in Mathematics, vol. 218
William Heinzer, Christel Rotthaus, and Sylvia Wiegand PIntegral Domains Inside Noetherian Power Series Rings: Constructions and Examples 2021 Mathematical Surveys and Monographs, vol. 259
Tadao Kitazawa, Andy Liu, and George Sicherman Arithmetical, Geometrical and Combinatorial Puzzles from Japan 2021 Spectrum, vol. 102
Michael E. Taylor Introduction to Differential Equations: Second Edition 2022 Pure and Applied Undergraduate Texts, vol. 52
Mathilde Gerbelli-Gauthier, Pamela E. Harris, Michael A. Hill, Dagan Karp, and Emily Riehl, Editors A Conversation on Professional Norms in Mathematics 2021
Steven Klee, Kolya Malkin, and Julia Pevtsova Math Out Loud: An Oral Olympiad Handbook 2021 MSRI Mathematical Circles Library, vol. 27
Pramod N. Achar Perverse Sheaves and Applications to Representation Theory 2021 Mathematical Surveys and Monographs, vol. 258
Marius Crainic, Rui Loja Fernandes, and Ioan Marcut Lectures on Poisson Geometry 2021 Graduate Studies in Mathematics, vol. 217
Tai-Ping Liu Shock Waves 2021 Graduate Studies in Mathematics, vol. 215
Alexey A. Zaslavsky and Mikhail B. Skopenkov Mathematics via Problems: Part 2: Geometry 2021 MSRI Mathematical Circles Library, vol. 26
Pamela E. Harris, Alicia Prieto-Langarica,Vanessa Rivera Quioñes, Luis Sordo Vieira, Rosaura Uscanga, and Andrés R. Vindas Meléndez, Editors Testimonios: Stories of Latinx and Hispanic Mathematicians 2021 Classroom Resource Materials, vol. 67
Ioannis Karatzas and Constantinos Kardaras Portfolio Theory and Arbitrage: A Course in Mathematical Finance 2021 Graduate Studies in Mathematics, vol.. 214
Juha Kinnunen, Juha Lehrbäck, and Antti Vähäkangas Maximal Function Methods for Sobolev Spaces 2021 Mathematical Surveys and Monographs, vol. 257
Hung Vinh Tran Hamilton--Jacobi Equations: Theory and Applications 2021 Graduate Studies in Mathematics, vol. 213
Jörg Bewersdorff Galois Theory for Beginners: A Historical Perspective, Second Edition 2021 Student Mathematical Library, vol. 95
Michio Jimbo, Tetsuji Miwa, and Fedor Smirnov Local Operators in Integrable Models I 2021 Mathematical Surveys and Monographs, vol. 256
Robert Messer Linear Algebra: Gateway to Mathematics, Second Edition 2021 AMS/MAA Textbooks, vol. 66
Alexandre Boritchev and Sergei Kuksin One-Dimensional Turbulence and the Stochastic Burgers Equation 2021 Mathematical Surveys and Monographs, vol. 255
Karim Belabas and Henri Cohen Numerical Algorithms for Number Theory: Using Pari/GP 2021 Mathematical Surveys and Monographs, vol. 254
Robert R. Bruner and John Rognes The Adams Spectral Sequence for Topological Modular Forms 2021 Mathematical Surveys and Monographs, vol. 253
David J. Covert The Finite Field Distance Problem 2021 The Carus Mathematical Monographs, vol. 37
Thomas Q. Sibley Thinking Algebraically: An Introduction to Abstract Algebra 2021 AMS/MAA Textbooks, vol. 65
Jean-Marie De Koninck and Nicolas Doyon
The Life of Primes in 37 Episodes 2021
Craig A. Stephenson
Periodic Orbits: F. R. Moulton's Quest for a New Lunar Theory 2021 History of Mathematics, vol. 45
Mindy Capaldi, Editor Teaching Mathematics Through Games 2021 Classroom Resource Materials, vol. 65
Frank Swetz The Impact and Legacy of The Ladies' Diary (1704--1840): A Women's Declaration 2021 Spectrum, vol. 101
Wendy Smith, Matthew Voigt, April Ström, David C. Webb, and W. Gary Martin Transformational Change Efforts: Student Engagement in Mathematics through an Institutional Network for Active Learning 2021
Philippe Zaouati Perelman's Refusal: A Novel 2021
Keith Kendig A Gateway to Number Theory: Applying the Power of Algebraic Curves 2021 Dolciani Mathematical Expositions, vol. 57
Viktor Prasolov and Yuri Solovyev Elliptic Functions and Elliptic Integrals 1997 Translations of Mathematical Monographs, vol. 170
James R. King Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions 2021 Pure and Applied Undergraduate Texts, vol. 51
Julie Deserti The Cremona Group and Its Subgroups 2021 Mathematical Surveys and Monographs, vol. 252
Mario Garcia-Fernandez and Jeffrey Streets Generalized Ricci Flow 2021 University Lecture Series, vol. 76
James Bisgard Analysis and Linear Algebra: The Singular Value Decomposition and Applications 2021 Student Mathematical Library, vol. 94
Arkadiy Skopenkov Mathematics via Problems: Part 1: Algebra 2021 MSRI Mathematical Circles Library, vol. 25
Ethan D.Bolker and Maura B. Mast Common Sense Mathematics, Second Edition 2021 AMS/MAA Textbooks, vol. 63
David Hoff Linear and Quasilinear Parabolic Systems: Sobolev Space Theory 2020 Mathematical Surveys and Monographs, vol. 251
Bachir Bekka and Pierre de la Harpe Unitary Representations of Groups, Duals, and Characters 2020 Mathematical Surveys and Monographs, vol. 250
Iva Stavrov Curvature of Space and Time, with an Introduction to Geometric Analysis 2020 Student Mathematical Library. vol. 93
Dan Sloughter Calculus From Approximation to Theory 2020 AMS/MAA Textbooks, no. 64
Kiran S. Kedlaya, Daniel M. Kane, Jonathan M. Kane, Evan M. O'Dorney Fitting The William Lowell Putnam Mathematical Competition 2001--2016: Problems, Solutions, and Commentary 2020 Problem Books, no. 37
Eli Aljadeff, Antonio Giambruno, Claudio Procesi, Amitai Regev Rings with Polynomial Identities and Finite Dimensional Representations of Algebras 2020 Colloquium Publications, no. 66
Charles Fefferman and Arie Israel Fitting Smooth Functions to Data 2020 CBMS Regional Conference Series in Mathematics, no. 135
Carl G. Wagner A First Course in Enumerative Combinatorics 2020 Pure and Applied Undergraduate Texts, vol. 49
Vicente Muñoz, Ángel González-Prieto, and Juan Ángel Rojo Geometry and Topology of Manifolds: Surfaces and Beyond 2020 Graduate Studies in Mathematics, vol. 208
June Barrow-Green, Jeremy Gray, Robin Wilson The History of Mathematics: A Source-Based Approach, Volume 2 2021 AMS/MAA Textbooks, vol. 61
Diana Davis Illustrating Mathematics 2020
Bruce E. Sagan Combinatorics: The Art of Counting 2020 Graduate Studies in Mathematics, vol. 210
Warren P. Johnson An Introduction to $q$-analysis 2020
Róbert Freud and Edit Gyarmati Number Theory 2020 Pure and Applied Undergraduate Texts, vol. 48
Jesssica S. Purcell Hyperbolic Number Theory 2020 Graduate Studies in Mathematics, vol. 209
Hung-Hsi Wu
Rational Numbers to Linear Equations
2020
Hung-Hsi Wu Algebra and Geometry 2020
Hung-Hsi Wu Pre-Calculus, Calculus, and Beyond 2020
Roger Plymen The Great Prime Number Race 2020 Student Mathematical Library, vol. 92
Stephen H. Saperstone and Max A. Saperstone Interacting with Ordinary Differential Equations 2020 AMS/MAA Textbooks, vol. 62
Daniel J. Velleman and Stan Wagon Bicycle or Unicycle? A Collection of Intriguing Mathematical Puzzles 2020 Problem Books, vol. 36
Michael E. Taylor Introduction to Analysis in One Variable 2020 Pure and Applied Undergraduate Texts, vol. 47
Alicia Dickenstein and Juan Sabia Matemax: English + Spanish Edition 2020
Michael E. Taylor Introduction to Analysis in Several Variables: Advanced Calculus 2020 Pure and Applied Undergraduate Texts, vol. 46
Ralph Abraham and Jerrold E. Marsden Foundations of Mechanics 2008 AMS Chelsea Publishing, vol. 364.H
Colin C. Adams Riot at the Calc Exam and Other Mathematically Bent Stories 2009
Colin C. Adams The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots 2004
Nikolai M. Adrianov, Fedor Pakovich, and Alexander K. Zvonkin Davenport--Zannier Polynomials and Dessins d'Enfants 2020 Mathematics Surveys and Monographs, vol. 249
Marcelo Aguiar and Swapneel Mahajan Coxeter Groups and Hopf Algebras 2006 Fields Institute Monographs, vol. 23
Marcelo Aguiar and Swapneel Mahajan Monoidal Functors, Species and Hopf Algebras 2010 CRM Monograph Series, vol. 29
Marcelo Aguiar and Swapneel Mahajan Topics in Hyperplane Arrangements 2017 Mathematical Surveys and Monograph, vol. 226
Ilka Agricola and Thomas Friedrich Global Analysis: Differential Forms in Analysis, Geometry and Physics 2002 Graduate Studies in Mathematics, vol. 52
Ilka Agricola and Thomas Friedrich Elementary Geometry 2008 Student Mathematical Library, vol. 43
Lars V. Ahlfors Conformal Invariants 2010 AMS Chelsea Publishing, vol. 371.H
Lars Ahlfors, with additional chapters by: C.J. Earle and I. Kra, M. Shishikura, and J.H. Hubbard Lectures on Quasiconformal Mappings 2006 University Lecture Series, vol. 38
Martin Aigner Discrete Mathematics 2007
Martin Aigner and Ehrhard Behrends Mathematics Everywhere 2010
Michael Aizenman and Simone Warzel Random Operators: Disorder Effects on Quantum Spectra and Dynamics 2015 Graduate Studies in Mathematics, vol. 168
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden (with an appendix by Pavel Exner) Solvable Models in Quantum Mechanics: Second Edition 2005 AMS Chelsea Publishing, vol. 350.H
Semyon Alesker Introduction to the Theory of Valuations 2018 CBMS Regional Conference Series in Mathematics, no.126
Daniel S. Alexander, Felice Iavernaro, and Alessandro Rosa Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906-1942 2011 History of Mathematics, vol. 38
Serge Alinhac and Patrick Gérard Pseudo-differential Operators and the Nash-Moser Theorem 2007 Graduate Studies in Mathematics, vol. 82
Charalambos D. Aliprantis and Rabee Tourky Cones and Duality 2007 Graduate Studies in Mathematics, vol. 84
John M. Alongi and Gail S. Nelson Recurrence and Topology 2007 Graduate Studies in Mathematics, vol. 85
Claudi Alsina and Rogert B. Nelsen A Cornucopia of Quadrilaterals 2020 Dolciani Mathematical Expositions, vol. 55
Claudia Alsina and Roger B. Nelson Charming Proofs: A Journey into Elegant Mathematics 2010 Dolciani Mathematical Expositions, vol. 42
Montserrat Alsina and Pilar Bayer Quaternion Orders, Quadratic Forms, and Shimura Curves 2004 CRM Monograph Series, vol. 22
Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin Simple Groups of Finite Morley Rank 2008 Mathematical Surveys and Monographs, vol. 145
Simon Altmann and Eduardo L. Ortiz Mathematics and Social Utopias in France: Olinde Rodrigues and His Times 2005 History of Mathematics, vol. 28
Paolo Aluffi Algebra: Chapter 0 2009 Graduate Studies in Mathematics, vol. 104
Alex Amenta and Pascal Auscher Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach 2018 CRM Monograph Series, vol. 37
Habib Ammari, Hyeonbae Kang, and Hyundae Lee Layer Potential Techniques in Spectral Analysis 2009 Mathematical Surveys and Monographs, vol. 153
Habib Ammari, Brian Fitzpatrick, Hyeonbae Kang, Matias Ruiz, Sanghyueon Yu, and Hai Zhang Mathematical and Computational Methods in Photonics and Phononics 2018 Mathematical Surveys and Monographs, vol. 235
Kirsti Andersen Optical Illusions in Rome: A Mathematical Travel Guide 2019 Spectrum, vol. 99
Titu Andreescu and Mark Saul Algebraic Inequalities: New Vistas 2016 MSRI Mathematical Circles Library, vol. 19
Fuensanta Andreu-Vaillo, José M. Mazón, Julio D. Rossi, and J. Julián Toledo-Melero Nonlocal Diffusion Problems 2010 Mathematical Surveys and Monographs, vol. 165
Ben Andrews, Bennett Chow, Christine Guenther, and Mat Langford Extrinsic Geometric Flows 2020 Graduate Studies in Mathematics, vol. 206
V. I. Arnold Experimental Mathematics 2015 MSRI Mathematical Circles Library, vol. 16
V. I. Arnold Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by Mathematicians 2014
James Arthur The Endoscopic Classification of Representations 2013 Colloquium Publications, vol. 61
Emil Artin and John Tate Class Field Theory 2009 AMS Chelsea Publishing, vol. 366.H
Shiri Artstein-Avidan, Apostolos A. Giannopoulos, and V. D. Milman Asymptotic Geometric Analysis, Part I 2015 Mathematical Surveys and Monographs, vol. 202
Andreas Arvanitoyeorgos An Introduction to Lie Groups and the Geometry of Homogeneous Spaces 2003 Student Mathematical Library, vol. 22
Michael Aschbacher and Stephen D. Smith The Classification of Quasithin Groups: I. Structure of Strongly Quasithin $mathcal{K}$-groups 2004 Mathematical Surveys and Monographs, vol. 111
Michael Aschbacher, Richard Lyons, Stephen S. Smith and Ronald Solomon The Classification of Finite Simple Groups: Groups of Characteristic 2 Type 2011 Mathematical Surveys and Monographs, vol. 172
Michael Aschbacher and Stephen S. Smith The Classification of Quasithin Groups: II. Main Theorems: The Classification of Simple QTKE-groups 2004 Mathematical Surveys and Monographs, vol. 112
Guillaume Aubrun and Stanislaw Szarek Alice and Bob Meet Banach: The Interface of Asymptotic Geometry Analysis and Quantum Information Theory 2017 Mathematical Surveys and Monographs, vol. 223
Dave Auckly, Bob Klein, Amanda Serenevy, and Tatiana Shubin Inspiring Mathematics: Lessons from the Navajo Nation Math Circles 2019 MSRI Mathematical Circles Library, vol. 24
Michèle Audin Hamiltonian Systems and Their Integrability 2008 SMF/AMS Texts and Monographs, vol. 15
Antonio Auffinger, Michael Damron, and Jack Hanson 50 Years of First-Passage Percolation 2017 University Lecture Series, vol. 68
Jinho Baik, Percy Deift, Toufic Suidan Combinatoric and Random Matrix Theory 2016 Graduate Studies in Mathematics, vol. 172
Matthew Baker and Robert Rumely Potential Theory and Dynamics on the Berkovich Projective Line 2010 Mathematical Surveys and Monographs , vol. 159
John T. Baldwin Categoricity 2009 University Lecture Series, vol. 50
Gregory V. Bard Sage for Undergraduates 2015
William Barker and Roger Howe Continuous Symmetry: From Euclid to Klein 2007
Elisabetta Barletta, Sorin Dragomir, and Drishan L. Duggal Foliations in Cauchy-Riemann Geometry 2007 Mathematical Surveys and Monographs, vol. 140
Julie Barnes and Jessica M. Libertini Tactile Learning Activities in Mathematics: A Recipe Book for the Undergraduate Classroom 2018 Classroom Resource Materials, vol. 54
Luis Barreira and Yakov Pesin Introduction to Smooth Ergotic Theory 2013 Gradudate Studies in Mathematics, vol. 148
Luis Barreira and Claudia Valls Ordinary Differential Equations:Qualitative Theory 2012 Graduate Studies in Mathematics, vol. 137
June Barrow-Green, Jeremy Gray, and Robin Wilson The History of Mathematics: A Source-Based Approach. Volume 1 2019 AMS/MAA Textbooks, vol. 45
Robert G. Bartle A Modern Theory of Integration 2001 Graduate Studies in Mathematics, vol.32
Steve Batterson Stephen Smale: The Mathematician Who Broke the Dimension Barrier 2000
Michael Bean Probability: The Science of Uncertainty with Applications to Investments, Insurance and Engineering 2009 Pure and Applied Undergraduate Texts, vol. 6
József Beck Inevitable Randomness in Discrete Mathematics 2009 University Lecture Series, vol. 49
Matthias Beck and Raman Sanyal Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics 2018 Graduate Studies in Mathematics, vol. 195
Ehrhard Behrends Five-Minute Mathematics 2008
Ehrhard Behrends The Math Behind the Magic 2019
Alexander Beilinson and Vladimir Drinfeld Chiral Algebras 2004 Colloquium Publications, vol. 51
Jason P. Bell, Dragos Ghioca, and Thomas J. Tucker The Dynamical Mordell--Lang Conjecture 2016 Mathematical Surveys and Monographs, vol. 210
Alexandra Bellow, Carlos E. Kenig, and Paul Malliavin Selected Papers of Alberto P. Calderón with Commentary 2008 Collected Works, vol. 21
Robert L. Benedetto Dynamics in One Non-Archimedean Variable 2019 Graduate Studies in Mathematics, vol. 198
David J. Benson and Stephen D. Smith Classifying Spaces of Sporadic Groups 2008 Mathematical Surveys and Monographs, vol. 147
A. Bensoussan, J.-L. Lions, and G. Papanicolaou Asymptotic Analysis for Periodic Structures 2011 AMS Chelsea Publishing, vol. 374.H
Sterling K. Berberian Measure and Integration 2011 AMS Chelsea Publishing, vol. 241.H
J. L. Berggren and R. S. D. Thomas Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy 2006 History of Mathematics, vol. 29
William P. Berlinghoff and Fernando Q. Gouvêa Math through the Ages: A Gentle History for Teachers and Others Expanded Second Edition 2015 AMS/MAA Textbooks, vol. 32
Grégory Berhuy and Frédérique Oggier An Introduction to Central Simple Algebras and Their Applications to Wireless Communication 2013 Mathematical Surveys and Monographs, vol. 191
Gregory Berkolaiko and Peter Kuchment Introduction to Quantum Graphs 2012 Mathematical Surveys and Monographs, vol. 186
Bruce C. Berndt Number Theory in the Spirit of Ramanujan 2006 Student Mathematical Library, vol. 34
Jean Berstel, Aaron Lauve, Christophe Reutenauer, and Franco V. Saliola Combintorics on Words 2008 CRM Monograph Series, vol. 27
Jörg Bewersdorff Galois Theory for Beginners: A Historical Perspective 2006 Student Mathematical Library, vol. 35
Allan Bickle Fundamentals of Graph Theory 2020 Pure and Applied Undergraduate Texts, vol. 43
Lydia Bieri and Nina Zipser Extensions of the Stability Theorem of the Minkowski Space in General Relativity 2009 AMS/IP Studies in Advanced Mathematics, vol. 45
Robert Bieri and Ralph Strebel On Groups of PL-homeomorphisms of the Real Line 2016 Mathematical Surveys and Monographs, vol. 215
Ernst Binz and Sonja Pod The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization 2008 Mathematical Surveys and Monographs, vol. 151
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https://www.codingninjas.com/codestudio/library/how-to-find-the-number-of-divisors-and-sum-of-divisors-of-a-number
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# How to find the number of divisors and sum of divisors of a number
Last Updated: May 13, 2022
## Introduction
Finding the number of divisors and the sum of divisors of a number is a very important concept for programming. Even though it is easy to find the number of divisors and the sum of divisors of a number if it is small, finding the number and sum of divisors of a large number will be difficult.
So let's first begin with finding the number of divisors of a number, then we will learn about finding the sum of divisors of a number.
## The number of divisors of a number
To find the number of divisors of a number, we start by finding the prime factorization of the number, as all the divisors of a number will be a subset of the number's prime factorization. In simple words, if we can devise a way to find all the subset of prime factorizations of a number, we can find the number of divisors of that number.
Let's say we have a number with x terms in its prime factorization. Then to find the total number of divisors of this number, we will have to find the total number of ways to select any number of terms from these x terms or the total number of subsets of x. It would have been easier because the total number of subsets of a set is 2no. Of elements in the set, but here the x might have some repeated terms. So we first find the prime factorization of a number, then represent the number in terms of distinct prime factors. For example- we can represent 36 as 22.32
Now, suppose we have a number F which appears p times in the prime factorization of a number X, then it can appear in subsets in p+1 ways, i.e.,
So, for a number, We can say that each distinct factor Fhas Pi+1 ways to appear in subsets. So the total number of divisors of a number Y=
## The sum of divisors of a number
To find the sum of all the divisors of a number, we will start with the formulae we used to find the number of divisors.
As we know, if a number F appears p times in the prime factorization of a number X, it can appear in subsets in p+1 ways, i.e., Now, if we multiply the set of choices of each factor, we will get a set of all possible divisors.
For example, if we have a number with two factors, then all the terms in the product will form the set containing all the divisors.
Since we need the sum of all the divisors of a number, we will find the product of the sum of all the choices for each factor. For the above example- the will represent the sum of all the divisors of a number.
So, for any number, the sum of all the divisors of a number will be
In this expression, we can observe that each term in the product is a GP. and for any GP, the sum of all the terms is
Using this for every term in the formula
We will get the result as-
1. How can you find the total no. of divisors of a number?
To find the total number of divisors of a number, we first find the prime factorization of the number and then find each distinct prime number's exponents. Then we add one to all the exponents and then find their product. The product is the number of divisors of a number.
2. What is the multiplicative identity?
Multiplicative identity states that the product of any number with 1 is the number itself. Thus, we can also say that one is a divisor of every number.
3. How do you find the number of odd divisors of a number?
To find the number of odd divisors of a number, we first find the prime factorization of the number and then find each distinct prime number's exponents. Now, we remove even prime numbers in it. Then we add one to all the exponents of odd prime numbers and then find their product. The product is the number of odd divisors of a number.
4. How do you find the number of odd divisors of a number?
Since we know how to find the total number of divisors of a number and odd divisors of a number, the difference of both will be the even divisors of a number.
5. What are the proper divisors of a number?
The proper divisors of a number refer to the divisors, which are smaller than the number itself.
## Key takeaways
In this blog, we have learned the formula for the number of divisors of a number and the sum of divisors of a number-
• We learned about the formula to find the number of divisors of a number because any divisor of a number will be formed by selecting some factors from the prime factorization of the number. Then we found that the number of choices for each factor in forming another divisor is one more than the number of times it appears in the prime factorization of the number. Thus, the total number of choices for each factor gives us the total number of divisors of a number.
• Then we learned about the formula to find the sum of divisors of a number using the similar concept we used for finding the number of divisors of a number. Since the product of all the choices for each factor for forming a divisor gives the total number of divisors. So, we find the product of the sum of the total number of choices for each distinct factor in the prime factorization of the number, which gives us the sum of all the number's divisors.
Visit here to learn more about algorithms in programming. You can also practice similar questions on Code studio.
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https://zbmath.org/?q=an%3A0653.47018
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# zbMATH — the first resource for mathematics
Asymptotic behavior of $$C_ 0$$-semigroups in B-convex spaces. (English) Zbl 0653.47018
The asymptotic behavior of $$C_ 0$$-semigroups in B-convex spaces is characterized in terms of growth conditions of the generator-resolvent. The main result improves a corresponding result of M. Slemrod [Indiana Univ. Math. J. 25, 783-792 (1976; Zbl 0313.47026)]. It is shown that the given characterization is optimal in B-convex spaces. Especially, Ljapunov’s stability condition reads as follows in B-convex spaces: Consider the abstract Cauchy problem $(ACP)\quad (d/dt- A)u=0,\quad u(0)\in D(A)$ associated with a $$C_ 0$$-semigroup $$(U_ A(t))_{t\geq 0}$$. If $$s_ b(A)<0$$, $$s_ b(A)$$ denoting the abscissa of boundedness of the resolvent $$z\mapsto (z-A)^{-1}$$, then all classical solutions of (ACP) are uniformly exponentially decreasing. One should remark that for example all uniformly convex Banach spaces are B-convex, and therefore the main result holds in many classical spaces of functions and distributions.
Reviewer: V.Wrobel
##### MSC:
47D03 Groups and semigroups of linear operators 35F10 Initial value problems for linear first-order PDEs 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35P05 General topics in linear spectral theory for PDEs 34G10 Linear differential equations in abstract spaces
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2021-06-15 06:26:20
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http://spot.pcc.edu/math/orcca/knowl/exercise-508.html
|
###### Exercise36
A country’s national debt was $$100$$ million dollars in 2010. The debt increased at $$20$$ million dollars per year. If this trend continues, when will the country’s national debt increase to $$480$$ million dollars?
Assume the country’s national debt will become $$480$$ million dollars $$y$$ years after 2010. We can solve this problem using the equation:
$$\displaystyle{20 y+100=480}\text{.}$$
Check whether $$17$$ is a solution for $$y$$ of this equation. (This solution implies the country’s national debt will become $$480$$ million dollars in the year 2027.)
?
Yes
No
in-context
|
2018-01-20 07:16:28
|
{"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": 0.3554832339286804, "perplexity": 1296.8293971256132}, "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-2018-05/segments/1516084889473.61/warc/CC-MAIN-20180120063253-20180120083253-00413.warc.gz"}
|
http://cpr-condmat-mtrlsci.blogspot.com/2013/07/13071087-rafael-roldan-et-al.html
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## Dielectric screening and plasmons in AA-stacked bilayer graphene [PDF]
Rafael Roldan, Luis Brey
The screening properties and collective excitations (plasmons) in AA-stacked bilayer graphene are studied within the random phase approximation (RPA). Whereas long lived plasmons in single layer graphene and in AB-stacked bilayer graphene can exist only in doped samples, we find that coherent plasmons can disperse in AA-stacked bilayer graphene {\it even in the absence of doping}. Moreover, we show that the characteristic low energy dispersion relation is unaffected by changes in the number of carriers, unless the chemical potential of the doped sample exceeds the inter-layer hopping energy. We further consider the effect of an external electric field applied perpendicular to the layers, and show how the dispersion of the modes can be tuned by the application of a gate voltage.
View original: http://arxiv.org/abs/1307.1087
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2017-10-19 12:32:23
|
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https://www.acmicpc.net/problem/13931
|
시간 제한메모리 제한제출정답맞힌 사람정답 비율
2 초 512 MB306624.000%
## 문제
Optimistan is a strange country. It is situated on an island with a huge desert in the middle, so most people live in port towns along the coast. As the name suggests, people of Optimistan (also called Optimists) like to optimise everything, so they only built roads necessary to connect all port towns together and not a single extra road. That means that there is only one way to get from one port town to another without visiting the same place twice.
The government installed multi-directional distance signs in 1- kilometre intervals on one side of the road, to provide important information to drivers. Thus whenever you go from one port town to another, you pass the first sign at the port town and then one each kilometre. Every distance sign contains the shortest distances to all port towns, each written on a separate small sign directed towards the goal town.
The signs also serve another important function: to guide drivers on intersections. This means that distance of each intersection from every port town is an integer number of kilometres.
You bought a tourist guide of Optimistan which does not have a map of the country, but it contains a huge table with the shortest distances between all pairs of port towns. You quickly calculated the average shortest distance between all pairs of port towns, but then you started wondering: if the signs also contained shortest distances to all other signs, what would be the average number written on a sign? Could this be calculated just from the distance table in the tourist guide?
## 입력
The input consists of:
• one line with an integer n (2 ≤ n ≤ 500), the number of ports;
• n−1 lines, the ith of which contains n−i integers. The jth integer on the ith line denotes the distance between port i and port i + j in kilometres. Each distance is between 1 and 106 (inclusive).
You can assume that the distances correspond to a road network in which there is exactly one path between two port towns that does not visit the same place twice. All roads can be used in both directions.
## 출력
Output one line with the average distances in kilometres between all pairs of distance signs in Optimistan. Your answer should have an absolute or relative error of at most 10−9 .
If it is impossible to determine the exact average of distances between all pairs of distance signs in Optimistan, output “impossible”.
## 예제 입력 1
3
4 4
2
## 예제 출력 1
2.13333333333333
## 예제 입력 2
4
2 2 2
2 2
2
## 예제 출력 2
1.6
|
2022-08-12 21:48:54
|
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|
https://www.askmehelpdesk.com/health-insurance/preexisting-medical-condition-423273.html
|
I'm 21 years old and 4 days ago I broke my foot. I put off going to the doctors because I don't have insurance but my mom yesterday finally convinced me that I needed to go so I went to see a Doctor and did a "cash account" with him to x-ray my foot he said it was broken and I might need surgery but I have to go to a different doctor (a surgeon) to see if I in fact need the surgery.
For him to tell me that cost $320.00 which I had to pay right then and there so noe I have$0.00 to my name. I know I need to go to the Doctors but I can't afford it and I have no money. My boyfriend has offered to get me insurance so that I can see a doctor for cheaper/copay but I'm worried that my insurance will find out that I did it 4 days ago and not cover me... any suggestions?
Thanks!
|
2019-04-25 06:09:57
|
{"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": 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": 0.29554569721221924, "perplexity": 592.2259586406254}, "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/1555578689448.90/warc/CC-MAIN-20190425054210-20190425080210-00101.warc.gz"}
|
https://aas.org/archives/BAAS/v25n2/aas182/abshtml/S2707.html
|
High-Velocity H I Gas in Supernova Remnants
Session 27 -- Hat Creek
Oral presentation, Tuesday, 8:30-12:30, Dwinelle 155 Room
[27.07] High-Velocity H I Gas in Supernova Remnants
Bon-Chul Koo (Astronomy Department, Seoul National University)
Using the Hat Creek 85 foot telescope, we had carried out a survey of H~I 21 cm emission lines toward all 103 known northern supernova remnants (SNRs) in order to find rapidly expanding SNR shells (Koo \& Heiles 1991). We detected 15 SNRs that have associated high-velocity (HV) H~I gas, most of which are quite likely the gas accelerated by the SN blast wave. Although the large beam-size (FWHM$\approx 30'$) of the 85 foot telescope prevented us to see the structure of the HV H I gas, the H I mass distribution in line-of-sight velocity suggested clumpy shell structures in several SNRs.
In order to resolve the structure of the HV H~I gas, we have been carrying out high-resolution H~I 21 cm line observations using the Arecibo telescope and the VLA. We report preliminary results on two SNRs, CTB~80 and W51. In CTB~80, the VLA observations revealed fast moving H I clumps, which have a dense ($n_H\sim 100$~cm$^{-3}$) core surrounded by a relatively diffuse envelope. The clumps are small, 3~pc to 5~pc, and have velocities between +40 km~s$^{-1}$ and +80 km~s$^{-1}$ with respect to the systematic velocity of CTB~80. The clumps have relatively large momentum per unit volume, which implies that they have been swept-up at an early stage of the SNR evolution. By analyzing the Arecibo data, we found that the interstellar medium around CTB~80 is far from being uniform and homogeneous, which explains the peculiar morphology of CTB~80 in infrared and radio continuum. In W51, HV H I gas moving up to $v_{\rm LSR}>+150$ km~s$^{-1}$ has been detected. The H~I distribution is elongated along the northwest-southeast direction, and the peak is very close to an X-ray bright region. We discuss the implications of our results in relation to the X-ray and the radio continuum morphology of W51.
This work was supported in part by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1992.
|
2016-05-06 11:18:25
|
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|
http://books.duhnnae.com/2017/jul7/150089338054-Instantons-and-Wormholes-in-N2-supergravity-High-Energy-Physics-Theory.php
|
# Instantons and Wormholes in N=2 supergravity - High Energy Physics - Theory
Abstract: In this paper, we construct Euclidean instanton and wormhole solutions in$d=4$, N=2 supergravity theories with hypermultiplets. The analyticcontinuation of the hypermultiplet action, involving pseudoscalar axions, isdiscussed using the approach originally developed by Coleman which determinesthe apparence of boundary terms. In particular, we investigate the conditionsobtained by requiring the action to be positive-definite once the boundaryterms are taken into account. The case of two hypermultiplets parameterizingthe coset $G {2,2}-SU2\times SU2$ is studied in detail. Orientifoldprojections which reduce the supersymmetry to N=1 are also discussed.
Author: Marco Chiodaroli, Michael Gutperle
Source: https://arxiv.org/
|
2017-09-20 13:03:21
|
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|
https://papers.nips.cc/paper/2020/hash/7b497aa1b2a83ec63d1777a88676b0c2-Abstract.html
|
#### Authors
Juan Correa, Elias Bareinboim
#### Abstract
The challenge of generalizing causal knowledge across different environments is pervasive in scientific explorations, including in AI, ML, and Data Science. Experiments are usually performed in one environment (e.g., in a lab, on Earth) with the intent, almost invariably, of being used elsewhere (e.g., outside the lab, on Mars), where the conditions are likely to be different. In the causal inference literature, this generalization task has been formalized under the rubric of transportability (Pearl and Bareinboim, 2011), where a number of criteria and algorithms have been developed for various settings. Despite the generality of such results, transportability theory has been confined to atomic, do()-interventions. In practice, many real-world applications require more complex, stochastic interventions; for instance, in reinforcement learning, agents need to continuously adapt to the changing conditions of an uncertain and unknown environment. In this paper, we extend transportability theory to encompass these more complex types of interventions, which are known as "soft," both relative to the input as well as the target distribution of the analysis. Specifically, we develop a graphical condition that is both necessary and sufficient for deciding soft-transportability. Second, we develop an algorithm to determine whether a non-atomic intervention is computable from a combination of the distributions available across domains. As a corollary, we show that the $\sigma$-calculus is complete for the task of soft-transportability.
|
2021-09-24 15:52:32
|
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|
https://byjus.com/question-answer/polarising-action-of-cd-2-an-anions-is-stronger-than-that-of-ca-2-because/
|
Question
# Polarising action of $$Cd^{2+}$$ an anions is stronger than that of $$Ca^{2+}$$ because:
A
the charges of the ions are same
B
their radii are same (Ca2+ = 0.104 nm, Cd2+=0.099 nm)
C
the Ca2+ ion has a noble-gas electron configuration, and the Cd2+ ion has an 18 electron configuration in its outer shell
D
all of the above
Solution
## The correct option is B the $$Ca^{2+}$$ ion has a noble-gas electron configuration, and the $$Cd^{2+}$$ ion has an 18 electron configuration in its outer shellCations with 18 electrons shall have greater polarising power than the cation with 8 electron shell.the linear electron has a poor shielding effect on the nucleus and thus E.N of the 16-electron shell is increased.Cadmium ion $$({ Cd }^{ 2+ })$$ has atronger polarsing power than $$({ Ca }^{ 2+ })$$ due to the above stated fact $$({ Cd }^{ 2+ })$$ has 18 electrons configration where as $$({ Cd }^{ 2+ })$$ ion has an noble gas configration ( 8 electron shell).Chemistry
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2022-01-17 18:33:34
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https://www.ms.u-tokyo.ac.jp/journal/abstract_e/jms110101_e.html
|
## Minimal Degree Liftings of Hyperbolic Curves
J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 1, Page 1--47.
Finotti, Luis R. A.
Minimal Degree Liftings of Hyperbolic Curves
The main goal of this paper is to analyze the properties of lifts of hyperelliptic curves $y_0^2 = f(x_0)$ over perfect fields of characteristic $p>2$ (to hyperelliptic curves over the ring of Witt vectors) that have lifts of points whose coordinate functions have minimal degrees. It is shown that, when trying to minimize the degrees of the $\wvx$-coordinate, the $(n+1)$-th entry, say $F_n$, can be taken to be a polynomial in $x_0$ such that $(dp^n-(d-2))/2 \leq \deg F_n \leq (dp^n+(d-2))/2$, where $d= \deg f(x_0)$. Besides upper and lower bounds for the degrees, other topics discussed include a necessary condition to achieve the lower bounds and lifting the Frobenius. Computational aspects are also considered and the case of elliptic curves is analyzed in more detail. An explicit formula for derivatives of coordinate functions of the elliptic Teichm\"uller lift is proved, namely $dF_n/dx_0=0$, if $p=2$, and $dF_n/dx_0 = \hi^{(p^n-1)/(p-1)} \,y_0^{p^n-1}- \sum_{i=0}^{n-1} F_i^{(p^{n-i}-1)} \, dF_i/dx_0$, if $p \geq 3$, where $\hi$ is the Hasse invariant of the curve. Finally, we establish a connection between minimal degree liftings and Mochizuki's theory of canonical liftings'' in the case of genus 2 curves.
|
2023-03-30 08:02:38
|
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|
https://www.beatthegmat.com/if-the-vertices-of-a-triangle-have-coordinates-x-1-t303660.html
|
• Award-winning private GMAT tutoring
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## If the vertices of a triangle have coordinates (x, 1),
tagged by: BTGmoderatorLU
# This topic has 2 expert replies and 0 member replies
### Top Member
BTGmoderatorLU Moderator
Joined
15 Oct 2017
Posted:
649 messages
Followed by:
5 members
#### If the vertices of a triangle have coordinates (x, 1),
Mon Aug 06, 2018 3:13 pm
00:00
A
B
C
D
E
## Global Stats
Difficult
Source: Official Guide
If the set S consists of five consecutive positive integers, what is the sum of these five integers?
(1) The integer 11 is in S, but 10 is not in S.
(2) The sum of the even integers in S is 26
The OA is D.
Last edited by BTGmoderatorLU on Thu Aug 09, 2018 3:06 pm; edited 1 time in total
### GMAT/MBA Expert
Jay@ManhattanReview GMAT Instructor
Joined
22 Aug 2016
Posted:
1287 messages
Followed by:
25 members
470
Mon Aug 06, 2018 11:14 pm
BTGmoderatorLU wrote:
Source: GMAT Club Tests
If the vertices of a triangle have coordinates (x, 1), (5, 1), and (5, y) where x < 5 and y > 1, what is the area of the triangle?
(1) x = y.
(2) The angle at the vertex (x, 1) is equal to the angle at the vertex (5, y).
The OA is C.
On a coordinate plane, put the point (5, 1). Since we cannot fix the other two points (x, 1) and (5, y), we can draw the straight line passing through the vertex (5, 1).
Somewhere on the line parallel to X-axis the vertex (x, 1) would lie and somewhere on the line parallel to Y-axis the vertex (5, y) would lie. This implies that it is a right-angled triangle.
Thus, the area of the triangle = 1/2*(5 - x)*(y - 1)
Let's take each statement one by one.
(1) x = y.
Can't get the value of 1/2*(5 - x)*(y - 1). Insufficient.
(2) The angle at the vertex (x, 1) is equal to the angle at the vertex (5, y).
=> This implies that it is an isosceles right-angled triangle.
=> (5 - x) = (y - 1)
Can't get the value of 1/2*(5 - x)*(y - 1). Insufficient.
(1) and (2) together
From x = y and (5 - x) = (y - 1), we get that x = y = 2
Thus, the area of the triangle = 1/2*(5 - x)*(y - 1) = 1/2*(5 - 2)*(2 - 1) = 3 unit
Sufficient.
Hope this helps!
-Jay
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### GMAT/MBA Expert
ceilidh.erickson GMAT Instructor
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04 Dec 2012
Posted:
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Mon Aug 13, 2018 12:20 pm
BTGmoderatorLU wrote:
Source: Official Guide
If the set S consists of five consecutive positive integers, what is the sum of these five integers?
(1) The integer 11 is in S, but 10 is not in S.
(2) The sum of the even integers in S is 26
The OA is D.
This seems to be a different question than the one Jay has quoted. Did you accidentally change it when editing?
_________________
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2018-08-21 16:23:11
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http://elliot.land/
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# Using Amazon EC2 Container Service (ECS) with Travis CI
If you are using Docker with Travis CI and you are not using some Docker registry to hold your pre-built images you will have to build the images on Travis with every build. Even for simple application stacks this can be a slow and expensive process. Ideally you want to move towards publishing your images to a registry.
A registry holds the . . .
September 18, 2016
# The Elevator Game!
The elevator game is a little thought experiment. It is not new, in fact Elevator Saga is another fun one. This one was just based on whatever came out of my head to see what would happen.
Here are the rules:
• There is one elevator in a building that has 10 floors.
• There are 30 passengers to transport. 15 passengers from level 1 to another . . .
Posted in: pythonpuzzle
September 10, 2016
# Automatically Locate When and Where Bugs Were Introduced with git bisect
Let's face it, when developing software things will break. Most of these things were probably working before hand, but some change directly or indirectly has broken it.
Best case scenario is you immediately recognise why it is broken. On the other end of the scale is some obscure bug that’s only happening sometimes, you have no idea why . . .
September 03, 2016
# Using docker-compose on Travis CI
If you are not familiar with Docker you can read about it in my other article. In this article I'm talking specifically about docker-compose; the tool for building multiple containers (as most application stacks will require).
docker-compose is great. It can be used to deploy to your dev machines, your production application and of . . .
August 27, 2016
# Flow-safe Enums in JavaScript
### My Attempt...
I have been using Flow recently to add some static-type checking to JavaScript. I'm not going to get into the details about Flow itself in this article, but rather how I've approached enums.
This is certainly not new and there are a tons of libraries out there that have various advantages and disadvantages. One great implementation . . .
Posted in: es6javascript
August 20, 2016
# CollectionFactory: JSON and Objective-C
CollectionFactory is a CocoaPod I wrote a long time ago to deal with JSON in Objective-C.
There are a lot of other libraries out there that do this kind of thing. However, I wanted something that was:
1. Very light and transparent without the need to create any intermediate code (such as predefined models).
2. Easy and interoperable with native . . .
August 13, 2016
# Docker: Explained Simply
### Or, How to Move Away From Vagrant
Docker is a new generation of virtualisation (around 3 years old) that makes building complex software stacks much easier and more isolated than previously. Now when I say previously I'm talking about Vagrant, Chef, Puppet, etc. These all work on the basis of creating a base image (containing the OS and some basic software), and using . . .
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2016-09-30 23:43:33
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https://testbook.com/question-answer/in-how-many-years-the-compound-interest-on-an-amo--608c55bc1bd0c3b452cab0d9
|
In how many years, the compound interest on an amount of Rs. 19200 at the rate of 10% p.a. will be Rs. 4032?
This question was previously asked in
UPSSSC Lower PCS Prelims Official Paper 2 (Held On : 26 June 2016)
View all UPSSSC Lower PCS Papers >
1. 1$$\frac{1}{2}$$years
2. 2$$\frac{1}{2}$$ years
3. 2 years
4. 3 years
Option 3 : 2 years
Detailed Solution
Given:
Principal = Rs. 19200
CI = Rs. 4032
Rate = 10%
Formula used:
CI = A - P
A = P[1 + (r/100)]t
Where, A = Amount, P = Principal, r = Rate of interest, t = Time
Calculation:
According to the question,
CI = 4032
⇒ A - P = 4032
⇒ P[1 + (r/100)]t - P = 4032
⇒ P{1 + (10/100)}t = 4032 + P
⇒ 19200{1 + (1/10)}t = 4032 + 19200
⇒ (11/10)t = 23232/19200
⇒ (11/10)t = 121/100
⇒ (11/10)t = (11/10)2
⇒ t = 2
∴ The required time is 2 years.
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2021-12-07 19:10:00
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|
http://ieeexplore.ieee.org/xpl/tocresult.jsp?reload=true&isnumber=5423254
|
# IEEE Transactions on Antennas and Propagation
## Filter Results
Displaying Results 1 - 25 of 61
Publication Year: 2010, Page(s):C1 - 625
| PDF (47 KB)
• ### IEEE Transactions on Antennas and Propagation publication information
Publication Year: 2010, Page(s): C2
| PDF (43 KB)
• ### RF MEMS Integrated Frequency Reconfigurable Annular Slot Antenna
Publication Year: 2010, Page(s):626 - 632
Cited by: Papers (59)
| | PDF (1442 KB) | HTML
A new kind of double- and single-arm cantilever type DC-contact RF MEMS actuators has been monolithically integrated with an antenna architecture to develop a frequency reconfigurable antenna. The design, microfabrication, and characterization of this ??reconfigurable antenna (RA) annular slot?? which was built on a microwave laminate TMM10i ( ??r = 9.8, tan ?? = 0.002), are presented in th... View full abstract»
• ### A Shallow Varactor-Tuned Cavity-Backed Slot Antenna With a 1.9:1 Tuning Range
Publication Year: 2010, Page(s):633 - 639
Cited by: Papers (19)
| | PDF (702 KB) | HTML
A shallow (0.025 wavelengths) microstrip-fed cavity-backed-slot (CBS) antenna has been demonstrated that tunes from 1 to 1.9 GHz with better than -20 dB reflection coefficient using a single varactor diode (0.45-2.5 pF). This is possible because the slot and the cavity combine to form a single resonance, and therefore, do not need to be tuned independently. The cavity is 72 ?? 72 ?? 3.18 mm3<... View full abstract»
• ### Dielectric Loaded Substrate Integrated Waveguide (SIW) ${H}$-Plane Horn Antennas
Publication Year: 2010, Page(s):640 - 647
Cited by: Papers (88) | Patents (2)
| | PDF (1761 KB) | HTML
A dielectric loaded substrate integrated waveguide (SIW) H-plane sectoral horn antenna has been proposed in this paper. The horn and the loaded dielectric are integrated by using the same single substrate resulting in easy fabrication and low cost. Two antennas with rectangular and elliptical shaped loaded dielectrics were designed and fabricated. These antennas have high gain and narrow beamwidth... View full abstract»
• ### Planar Annular Ring Antennas With Multilayer Self-Biased NiCo-Ferrite Films Loading
Publication Year: 2010, Page(s):648 - 655
Cited by: Papers (38) | Patents (2)
| | PDF (1131 KB) | HTML
With their high relative permeability, magneto-dielectric materials show great potential in antenna miniaturization. This paper presents an annular ring antenna with self-biased magnetic films loading in the gigahertz frequency range. The annular ring antenna was realized by cascading a microstrip ring and a tuning stub. Self-biased NiCo-ferrite films were adopted to load an annular ring antenna o... View full abstract»
• ### Compact Loaded PIFA for Multifrequency Applications
Publication Year: 2010, Page(s):656 - 664
Cited by: Papers (21)
| | PDF (1834 KB) | HTML
A new multifrequency microstrip patch antenna is presented. The antenna can be considered a PIFA since it has a metallic wall on one of its sides. The different bands of operation are independent of each other, and different radiation patterns for each band can be achieved if desired. In addition, a circuital model is introduced to explain the operation of the antenna. This model presents some sim... View full abstract»
• ### A Comparison of a Wide-Slot and a Stacked Patch Antenna for the Purpose of Breast Cancer Detection
Publication Year: 2010, Page(s):665 - 674
Cited by: Papers (66)
| | PDF (2578 KB) | HTML
A wide-slot UWB antenna is presented for intended use in the detection scheme being developed at the University of Bristol, based on the principle of synthetically focused UWB radar using a fully populated static array. The antenna's measured and simulated, input and radiation characteristics are presented and compared to an existing, stacked patch antenna that has been designed for the same purpo... View full abstract»
• ### A Holographic Antenna Approach for Surface Wave Control in Microstrip Antenna Applications
Publication Year: 2010, Page(s):675 - 682
Cited by: Papers (20)
| | PDF (977 KB) | HTML
A holographic antenna inspired structure is used to control the surface wave (SW) excited by a microstrip patch antenna. The hologram is designed to support a periodic leaky-wave which radiates at broadside and enhances the radiation of the patch while suppressing the horizontal lobe. In this design, the holographic approach is adapted for patch antenna applications where the SW wavelengths are co... View full abstract»
• ### Receiving Polarization Agile Active Antenna Based on Injection Locked Harmonic Self Oscillating Mixers
Publication Year: 2010, Page(s):683 - 689
Cited by: Papers (3)
| | PDF (1082 KB) | HTML
A polarization agile active antenna with phase shifter elements based on injection locked third harmonic self oscillating mixers is presented. This phase shifting topology provides the double functionality of continuous range phase shifter and downconverter. The phase shift value introduced by each circuit can be easily tuned through a DC voltage within a theoretical continuous range of 450¿¿ . Th... View full abstract»
• ### Singly and Dual Polarized Convoluted Frequency Selective Structures
Publication Year: 2010, Page(s):690 - 696
Cited by: Papers (38)
| | PDF (762 KB) | HTML
Convoluting the elements of frequency selective surfaces produces resonating structures with very small unit cell dimensions. This feature is attractive when the FSS is to be used at low frequencies, mounted on a curved surface, or when placed in the proximity of compact radiators. The characteristics of single and dual polarized convoluted FSS are analyzed and measured. The development of novel c... View full abstract»
• ### A Linear Rectangular Dielectric Resonator Antenna Array Fed by Dielectric Image Guide With Low Cross Polarization
Publication Year: 2010, Page(s):697 - 705
Cited by: Papers (28)
| | PDF (2145 KB) | HTML
Design of a linear array of rectangular dielectric resonator antennas (DRAs) fed by dielectric image guide (DIG) is presented. Coupling between the DIG and the DRAs is predicted using the effective dielectric constant method. In order to achieve a specific power distribution, the power coupled to each DRA is controlled by changing the spacing between the DRAs and the DIG. Cross polarization reduct... View full abstract»
• ### Beamforming Lens Antenna on a High Resistivity Silicon Wafer for 60 GHz WPAN
Publication Year: 2010, Page(s):706 - 713
Cited by: Papers (29) | Patents (1)
| | PDF (1508 KB) | HTML
Wafer-scale beamforming lenses for future IEEE802.15.3c 60 GHz WPAN applications are presented. An on-wafer fabrication is of particular interest because a beamforming lens can be fabricated with sub-circuits in a single process. It means that the beamforming lens system would be compact, reliable, and cost-effective. The Rotman lens and the Rotman lens with antenna arrays were fabricated on a hig... View full abstract»
• ### High Permittivity Dielectric Rod Waveguide as an Antenna Array Element for Millimeter Waves
Publication Year: 2010, Page(s):714 - 719
Cited by: Papers (22)
| | PDF (1336 KB) | HTML
Dielectric rod waveguide antennas of rectangular cross section have a number of advantages over conventional waveguide and horn antennas as an antenna array element. Dielectric rod waveguide antennas have relatively low cost, low losses, a broadband input match and a high packing potential. Additionally the radiation pattern of such antennas is almost frequency independent. In this paper the suita... View full abstract»
• ### Linear Sparse Array Synthesis With Minimum Number of Sensors
Publication Year: 2010, Page(s):720 - 726
Cited by: Papers (19)
| | PDF (749 KB) | HTML
The number of sensors employed in an array affects the array performance, computational load, and cost. Consequently, the minimization of the number of sensors is of great importance in practice. However, relatively fewer research works have been reported on the later. In this paper, a novel optimization method is proposed to address this issue. In the proposed method, the improved genetic algorit... View full abstract»
• ### Alternating Adaptive Projections in Antenna Synthesis
Publication Year: 2010, Page(s):727 - 737
Cited by: Papers (11)
| | PDF (1293 KB) | HTML
The projection operator is a basic building block in the application of the alternating projections method to antenna synthesis. In general it is a non-linear operator that is repeatedly applied in the course of a single synthesis process, thus having a considerable impact on the convergence properties of the resulting algorithm. A novel approach to the computation of these projections is presente... View full abstract»
• ### An Optimum Adaptive Single-Port Microwave Beamformer Based on Array Signal Vector Estimation
Publication Year: 2010, Page(s):738 - 746
Cited by: Papers (2)
| | PDF (740 KB) | HTML
A single-port adaptive beamforming structure based on an optimum perturbation technique is presented. The proposed perturbation technique is based on array signal vector estimation for temporally-correlated array signals. Temporal correlation is generated by jointly reducing the receiver bandwidth and increasing the weighting rate. The error signal is generated using the estimated array signal vec... View full abstract»
• ### Decoupled 2D Direction of Arrival Estimation Using Compact Uniform Circular Arrays in the Presence of Elevation-Dependent Mutual Coupling
Publication Year: 2010, Page(s):747 - 755
Cited by: Papers (25)
| | PDF (595 KB) | HTML
Based on the rank reduction theory (RARE), a decoupled method for 2D direction of arrival (DOA) estimation in the presence of elevation-dependent mutual coupling is proposed for compact uniform circular arrays (UCAs). Using a new formulation of the beamspace array manifold in the presence of mutual coupling, the azimuth estimates are decoupled from the elevation estimates and obtained with no need... View full abstract»
• ### Ultrawideband Aperiodic Antenna Arrays Based on Optimized Raised Power Series Representations
Publication Year: 2010, Page(s):756 - 764
Cited by: Papers (14)
| | PDF (1556 KB) | HTML
Past research has shown that application of mathematical and geometrical concepts such as fractals, aperiodic tilings, and special polynomials can provide elegant solutions to difficult antenna array design problems. For example, design issues such as beam shaping and control, sidelobe levels, bandwidth and many others have been addressed with such concepts. In this paper, mathematical constructs ... View full abstract»
• ### Original and Modified Kernels in Method-of-Moments Analyses of Resonant Circular Arrays of Cylindrical Dipoles
Publication Year: 2010, Page(s):765 - 772
Cited by: Papers (3)
| | PDF (544 KB) | HTML
Properly dimensioned circular arrays of cylindrical dipoles are known to possess very narrow resonances. It is also known that analyzing such arrays using moment methods presents unique and particular difficulties, as application of such methods to the usual Hallen-type integral equations can yield meaningless results from which no further conclusions should be drawn. In the present paper, we appl... View full abstract»
• ### A New Fast Physical Optics for Smooth Surfaces by Means of a Numerical Theory of Diffraction
Publication Year: 2010, Page(s):773 - 789
Cited by: Papers (27)
| | PDF (2415 KB) | HTML
A new technique to compute the physical optics (PO) integral is presented. The technique consists of a blind code that computes the different contributions (stationary phase points, end points, etc.) numerically. This technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plan... View full abstract»
• ### Planar Electromagnetic Bandgap Structures Based on Polar Curves and Mapping Functions
Publication Year: 2010, Page(s):790 - 797
Cited by: Papers (7)
| | PDF (1134 KB) | HTML
A type of electromagnetic bandgap structure is described that is easily parameterized and can produce a range of square and spiral geometries. Individual electromagnetic bandgap (EBG) geometries are defined on a cell-by-cell basis in terms of their convolution factor k, which defines the extent to which the elements are interleaved and controls the coupling slot length between adjacent elements. P... View full abstract»
• ### Active Phase Conjugating Lens With Sub-Wavelength Resolution Capability
Publication Year: 2010, Page(s):798 - 808
Cited by: Papers (22)
| | PDF (2716 KB) | HTML
Experimental results are presented for the focusing capability of an active phase conjugating lens for a single and a dipole source pair and these are compared with predictions. In addition for a single source we illustrate the ability of the lens to project a null at the lens focus instead of a peak. A scheme is also presented such that when a source or pair of sources is imaged through an identi... View full abstract»
• ### A Geometrical Optics Model of Three Dimensional Scattering From a Rough Layer With Two Rough Surfaces
Publication Year: 2010, Page(s):809 - 816
Cited by: Papers (17)
| | PDF (1532 KB) | HTML
An asymptotic method is described for predicting the bistatic normalized radar cross section of a rough homogeneous layer made up of two rough surfaces. The model is based on iteration of the Kirchhoff approximation to calculate the fields scattered by the rough layer, and is reduced to the high-frequency limit in order to obtain numerical results rapidly. Shadowing effects, significant for large ... View full abstract»
• ### Dual-Grid Finite-Difference Frequency-Domain Method for Modeling Chiral Medium
Publication Year: 2010, Page(s):817 - 823
Cited by: Papers (2)
| | PDF (891 KB) | HTML
A dual-grid finite-difference frequency-domain (DG-FDFD) method is introduced to solve for scattering of electromagnetic waves from bianisotropic objects. The formulations are based on a dual-grid scheme in which a traditional Yee grid and a transverse Yee grid are combined to achieve coupling of electric and magnetic fields that is imposed by the bianisotropy. Thus the underlying grid naturally s... View full abstract»
## Aims & Scope
IEEE Transactions on Antennas and Propagation includes theoretical and experimental advances in antennas.
Full Aims & Scope
## Meet Our Editors
Editor-in-Chief Danilo Erricolo
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2017-09-20 06:13:37
|
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https://tex.stackexchange.com/questions/252845/venn-diagrams-with-tikz-erase-arc
|
# Venn diagrams with tikz: erase arc
Consider the following MWE, which produces the Venn diagram below:
\documentclass[12pt]{article}
\usepackage{tikz,pstricks,pst-jtree}
% pst-jtree enables the \multiline \cr \endmultiline syntax
\usetikzlibrary{shapes,backgrounds}
\begin{document}
\begin{center}
\begin{center}
\def\first{(0,0) ellipse (6em and 4em)}
\def\second{(2.7,0) ellipse (6em and 4em)}
\def\third{(1.45,1.5) ellipse (6em and 4em)}
\begin{tikzpicture}
\draw \first node [below] { };
\draw \second node [below] { };
\draw \third node [above] { };
% first coordinate control x axis, second controls y axis
\node at (-1.1,-.3) (A) {first};
\node at (3.8,-.5) (B) {second};
\node at (1.5,2.5) (C) {third};
\begin{scope}[fill opacity = .5]
\clip \third;
\fill[light-gray] \second;
\end{scope}
\end{tikzpicture}
\end{center}
\end{center}
\end{document}
For aesthetic reasons, I don't want the line of the first ellipse to interfere with the we care about this legend. Is it possible to eliminate the arc of first in the shaded area?
• \fill first with a white color with no opacity, and then the light-gray with opacity as you wish. – Manuel Jun 29 '15 at 17:41
Just change the drawing order:
1. First ellipse
2. Gray area
3. Second and third ellipses and text nodes.
Further remarks:
• Multi-line text is enabled by node option align.
• light-gray is a little an unhappy color name because of the hyphen. When the shorthand notation \fill[light-gray] is used, TikZ misinterprets the hyphen as arrow specification, when the color is undefined (as in the example of the question). TikZ complains about an unknown arrow tip light instead of an undefined color error as for fill=light-gray or an unknown key.
Example file:
\documentclass[12pt]{article}
\usepackage{tikz}
\usetikzlibrary{shapes}
\begin{document}
\begin{center}
\def\first{(0,0) ellipse (6em and 4em)}
\def\second{(2.7,0) ellipse (6em and 4em)}
\def\third{(1.45,1.5) ellipse (6em and 4em)}
\begin{tikzpicture}
\draw \first node [below] { };
\begin{scope}
\clip \third;
\fill[lightgray] \second;
\end{scope}
\draw \second node [below] { };
\draw \third node [above] { };
% first coordinate control x axis, second controls y axis
\node at (-1.1,-.3) (A) {first};
\node at (3.8,-.5) (B) {second};
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2019-10-16 10:13:42
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https://docs.dgl.ai/en/latest/generated/dgl.distributed.dist_graph.edge_split.html
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# dgl.distributed.dist_graph.edge_split¶
dgl.distributed.dist_graph.edge_split(edges, partition_book=None, etype='_E', rank=None, force_even=True, edge_trainer_ids=None)[source]
Split edges and return a subset for the local rank.
This function splits the input edges based on the partition book and returns a subset of edges for the local rank. This method is used for dividing workloads for distributed training.
The input edges can be stored as a vector of masks. The length of the vector is the same as the number of edges in a graph; 1 indicates that the edge in the corresponding location exists.
There are two strategies to split the edges. By default, it splits the edges in a way to maximize data locality. That is, all edges that belong to a process are returned. If force_even is set to true, the edges are split evenly so that each process gets almost the same number of edges.
When force_even is True, the data locality is still preserved if a graph is partitioned with Metis and the node/edge IDs are shuffled. In this case, majority of the nodes returned for a process are the ones that belong to the process. If node/edge IDs are not shuffled, data locality is not guaranteed.
Parameters
• edges (1D tensor or DistTensor) – A boolean mask vector that indicates input edges.
• partition_book (GraphPartitionBook, optional) – The graph partition book
• etype (str, optional) – The edge type of the input edges.
• rank (int, optional) – The rank of a process. If not given, the rank of the current process is used.
• force_even (bool, optional) – Force the edges are split evenly.
• edge_trainer_ids (1D tensor or DistTensor, optional) – If not None, split the edges to the trainers on the same machine according to trainer IDs assigned to each edge. Otherwise, split randomly.
Returns
The vector of edge IDs that belong to the rank.
Return type
1D-tensor
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2022-07-02 04:26:14
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http://mathhelpforum.com/advanced-algebra/132138-example-well-definition.html
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# Thread: Is this an example of Well-definition?
1. ## Is this an example of Well-definition?
I'm just not really sure what well-definition implies. I think I get it, but I could use a little feedback.
2. Originally Posted by davismj
I'm just not really sure what well-definition implies. I think I get it, but I could use a little feedback.
Yes, that is correct. Well-defined means if you pick two things from the same class and multiply (or do any other operation) you will always get the same answer - it doesn't matter which element from the class you pick. So, whenever you see something whose elements are of the form $[a]$ or $a+I$ or $aN$ where $I$ or $N$ are some sort of algebraic structure (for instance, $I$ an ideal or a ring, $N$ a normal subgroup of a group) you would need to take two elements and see if they are always mapped to the same thing.
For instance, let $S=\{0, 1, 2, 3, \ldots, n\}$ and let $T = \{[x]:[a] = [b] \Leftrightarrow n|(a-b)\}$ be the set of equivalence classes of the integers modulo n, and again let the operation be addition.
Can you show that $\varphi: [x] \mapsto x \text{ mod } n$ is well-defined?
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2016-10-26 04:42:34
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http://arkadiusz-jadczyk.eu/blog/2017/04/disk-hyperbolic-model/
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# The disk and the hyperbolic model
The unit disk in the complex plane, together with geometry defined by invariants of fractional linear SU(1,1) action, known as the Poincaré disk, that is the arena of hyperbolic geometry. But why “hyperbolic”? It is time for us to learn, and to use. In principle the answer is given in Wikipedia, under the subject “Poincaré disk model”. There we find the following picture
with formulas:
[latexpage]
We want to derive these formulas ourselves. Let us first introduce our private notation. The hyperboloid will live in a three-dimensional space with coordinates $X,Y,T.$ This is the space-time of Special Relativity Theory, but in a baby version, with $Z$ coordinate suppressed.
The light cone in our space-time has the equation $T^2-X^2-Y^2=0.$ Of course we assume the constant speed of light $c=1.$ Inside the future light cone (the part with $Tgeq 0$) there is the hyperboloid defined by $T=\sqrt{1+X^2+Y^2}.$ The coordinates $(X,Y,T)$ of events on this hyperboloid satisfy the equation
$$T^2-X^2-Y^2=1.$$
As can be seen from the picture above, every straight line passing through the point with coordinates $X_0=Y_0=0,$ $T_0=-1$ and a point with coordinates $(X,Y,T)$ on the hyperboloid, intersects the unit disk at the plane $T=0$ at a point with coordinates $(x,y)$. We want to find the relation between $(X,Y,T)$ and $(x,y).$
Given any two points, $P=(X_0,Y_0,T_0),$ $Q=(X_1,Y_1,T_1)$, the points $(X,Y,T)$ on the line joining $P$ and $Q$ have coordinates $(X(u),Y(u),T(u))$ parametrized by a real parameter $u$ as follows:
$$(X(u),Y(u),T(u))=(1-u)(X_0,Y_0,T_0)+u(X_1,Y_1,T_1).$$
For $u=0$ we are at $P,$ for $u=1$ we are at $Q$, and for other values of $u$ we are somewhere on the joining line. Our $P$ has coordinates $(0,0,-1),$ our $Q$ has coordinates $(X,Y,T)$ on the hyperboloid, and we are seeking the middle point with coordinate $(x,y,0).$ So we need to solve equations
\begin{eqnarray}
x&=&(1-u)0+uX=uX,\\
y&=&(1-u)0+uY=uY,\\
0&=&(1-u)(-1)+uT.
\end{eqnarray}
From the last equation we find immediately that $u=1/(1+T),$ and the first two equations give us
\begin{eqnarray}
x&=&\frac{X}{1+T},\\
y&=&\frac{Y}{1+T}\label{eq:Xx}.
\end{eqnarray}
We need to find the inverse transformation. First we notice that
$x^2+y^2=\frac{X^2+Y^2}{(1+T)^2}=\frac{T^2-1}{(1+T)^2}=\frac{T-1}{T+1}.$
Therefore $1-(x^2+y^2)=\frac{2}{T+1}$ and so
$T+1=\frac{2}{1-x^2-y^2},$
$T=\frac{1+x^2+y^2}{1-x^2-y^2}.$
Using Eqs. (\ref{eq:Xx}) we now finally get
\begin{eqnarray}
X&=&\frac{2x}{1-x^2-y^2},\\
Y&=&\frac{2y}{1-x^2-y^2},\\
T&=&\frac{1+x^2+y^2}{1-x^2-y^2}.
\end{eqnarray}
Thus we have derived the formulas used in Wikipedia. Wikipedia mentions also that the straight lines on the disk, that we were discussing in a couple of recent posts, are projections of sections of the hyperboloid by planes. We will not need this in the future. But we will use the derived formulas for obtaining the relation between SU(1,1) matrices and special Lorentz transformations of space-time events coordinates. This is the job for the devil of the algebra!
## 3 thoughts on “The disk and the hyperbolic model”
1. Bjab says:
Something bad happened to this post (in the midle).
1. Fixed by desactivating newly activated smiley plugin. It seems that either you have smileys or you have latex. Not both. Strange is this world.
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2018-02-25 02:02:57
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http://mathandmultimedia.com/category/math-lite/page/3/
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## Three Sticks: A New Promising Math Game
Last year, I shared Primo, a game that is aimed to develop the notion of prime numbers among players. In this post, I am going to share with you another interesting game called Three Sticks which aims to develop knowledge of basic geometric shapes. In this game, using the same three types of sticks, players try to figure out different shapes and score points. In each turn, a player is allowed to put two sticks in order to form shapes with the largest number of points. Watch the video below to know more about the details on how the game is played.
Some of the mathematical concepts that can be learned by playing Three sticks are
• polygons and their properties
• how to calculate perimeter of polygons
• convex and non-convex polygons
• regular and irregular polygons
The printable board, cards, sticks, and Rules book can be found here. Three Sticks is currently on trial, so even the sticks are also printable. According to the designer, the actual set will include plastic sticks and a board with holes into which the sticks would fit,
Three Sticks was developed by Pramod Ponnaluri of Kitki.in.
## Solving the Merry Christmas Equation
It’s December again and many people are celebrating Christmas. Here in Japan, Christmas is also celebrated even though only less than 1% of the population are Christians.
Anyway, I will be very busy this month, but let’s start our month with solving an equation that will lead to Merry Christmas. Sound confusing? Let’s start.
To understand the solution below, it would help if you have some prior knowledge of about natural logarithms.
The Christmas Equation
Let’s solve the equation $y = \displaystyle \frac{ \ln (\frac{x}{m} - sa)}{r^2}$ » Read more
## 29 Tagalog Math Terms I Bet You Don’t Know
We teach mathematics in English in my country, so we are not familiar about math terms in our own language. Honestly, I didn’t know that these terms even exist. So, I want to share it with fellow Filipino teachers and math enthusiasts.
You might also want to check more math terms in Tagalog.
Enjoy!
Tagalog Math Terms
labi – remainder
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2022-01-22 11:27:05
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https://jpt.spe.org/otc-2020-tech-papers-offer-look-future-offshore
|
Share
Robotics/unmanned systems
# OTC 2020 Tech Papers Offer a Look Into the Future of Offshore
## The Offshore Technology Conference was cancelled for the first time ever due to the COVID-19 pandemic. But the flow of ideas continues. As proof, this curated summary of technical papers highlights unique concepts that might someday reduce the offshore sector’s heavy cost burdens.
The offshore oil and gas sector has over the decades come to be defined by megaprojects with 30-to-40-year project horizons. But the future of offshore development will depend on the industry’s ability to find innovative ways to cut costs and slim the capital requirements.
Many examples were to be shared with oil and gas professionals at the 2020 Offshore Technology Conference (OTC) in Houston. However, due to the COVID‑19 pandemic, this year marked the first time that OTC was cancelled since its inauguration in 1969.
Despite the global disruption, the flow of ideas continues.
As proof, what follows is a curated summary of some of the papers that were to be presented at OTC. They were selected for their focus on emerging technologies and unique concepts that aim to reduce the cost burden long associated with offshore exploration and development. Their state of maturity ranges from proof of concept to fully deployed.
All 2020 OTC papers are available at www.onepetro.org.
## Offshore Adoption of Predictive Analytics
Among the most dominant technology arenas in the oil and gas industry’s digital transformation are fast-emerging machine-learning programs that en-able predictive analytics. This paper (OTC 30782), produced by software developer Spark Cognition, offers two case studies that show how machine learning is being adopted in the offshore sector.
The first involves an unnamed supermajor that “was suffering multiple production-impacting events” involving the gas system on an offshore facility. Conventional methods failed to pick up the signals of failure. The software company was then tapped by the operator to test an unsupervised learning approach.
Years of historical data were run through the program and reorganized so that irregular data sets could be identified and labeled as anomalies. By flagging the anomalies, engineers knew what to investigate; specific tags were helpful in root cause analysis.
“While the solution accurately predicts upcoming asset failure, a significant additional benefit is its ability to identify system or process failure where no particular asset has failed but where a process has become so unstable, the system needs to be shut down and restarted in a controlled manner,” the paper reads.
The proof of concept resulted in a full deployment across the supermajor’s offshore facilities, refineries, and petrochemical plants.
In the second case study, a different oil and gas company was focused on lowering the cost of offshore production by $5/bbl. The initial pilot was selected for an unmanned platform that pumped 10,000 B/D of crude slurry to processing facilities. The facility’s chief source of unplanned downtime was a multiphase pump that would go down for a number of reasons, including seal and filter issues. After a custom-designed machine-learning program and behavior model were aimed at the problematic system, 75% of known historical failures were detected with 2 to 12 days of warning. The first 6 months of the program’s live deployment coincided with the unmanned facility’s longest period of continuous operation. The operator estimates that it gained$500,000 in production value for each day of downtime prevented.
## Getting Smaller With Robots
Deepwater well intervention has entered a relative state of maturity over the past decade, providing the subsea sector with untold value by reviving aging or problematic assets. Despite its positive track record, operators continue to avoid subsea well intervention due to the time and costs it requires.
The main cost driver comes from the need to use rigs or specially designed intervention vessels to carry out the delicate task of re-entering a wellbore. Researchers at the Brazilian technology institute Senai Cimatec and national oil company Petrobras think the best way to lower the cost of using these two platforms is to not use them at all.
Instead, let a robot do the work.
In their paper (OTC 30886), the authors describe the creation of a subsea workover robot that consists of two systems: a “cocoon” that protects, supplies power to, and carries the second system; an intervention unit that moves into the production column laden with equipment and sensors. The roadmap foresees using the intervention robot for a number of workover operations that includes gas-lift-valve exchange and plug replacement.
If such a system were available, the industry could rely on its lightest work vessels for subsea intervention. Importantly, this concept may not even require a vessel to maintain station keeping, which would free it up to perform work on other areas of a subsea field.
Inside the intervention unit is a package of computers and sensors designed “to guarantee the precise monitoring of the environment and of the equipment,” the paper notes. Among the innovations involved is a self-localization system that uses an encoder attached to the tractor motor and a magnetic sensor that is similar to a casing-collar locator. Other proposed tool capabilities include paraffin cleaning before testing begins and wellbore-obstruction detection.
With the proof-of-concept robot built and tested in both the laboratory setting and a test well, the developers’ next step is to adjust the design to build a prototype designed for an offshore field test.
## Field Hopping With a Reusable, Unmanned Platform
Many of the world’s remaining reserves are locked away in what the industry calls marginal fields. Exploiting these fields profitably has proven over the years to be the mother of invention.
One of the latest examples comes from Vestigo Petroleum, whose engineers overcame the economic barriers of marginal oil fields by using a reusable wellhead platform that connects to a floating production, storage, and offloading unit (FPSO). The deployment of this system was a first-of-its-kind project for Malaysia.
Vestigo—a wholly owned subsidiary of Malaysia’s state-owned Petronas Carigali—is producing from a reusable platform at the Jitang field. Discovered in 2016, the field achieved first oil this past January—just 13 months after the final investment decision was made. The field is about 95 miles offshore Malaysia at a depth of about 240 ft.
Vestigo reports a cost of about $24 million to build the platform, which is unmanned and remotely controlled from its companion FPSO stationed just over 1,000 ft away. Compared to a newbuild, the reusable platform represents a cost savings of 40%. The speed of project delivery would not have been possible without using a platform designed to be reused with few modifications or additional equipment. The wellhead platform was relocated from its original host field located about 34 miles away from its current home. Production at the previous field ceased in January 2018 after 3 years. The facility has an expected service life of 15 years, which means at the current rate, it could be redeployed to another three offshore fields before retirement. The only new pieces of major equipment required to redeploy the platform at its second field were four new flowlines for each of the new wells. In addition to the overall system design, key to making a reusable platform economic on marginal projects is to have a fine-tuned relocation methodology. One enabling feature in this regard is the suction-pile technology that the platform uses as its foundation. The suction piles allow the platform to be easily pulled up from the seabed without cutting the piles, the use of oilfield divers, or the need to transfer back to an onshore facility for repairs. To move the facility, a heavy-lift vessel was required along with a “one-piece, wet tow.” This involves a single operation to lift both the topsides and substructure together, and then tow the partially submerged structure to its new location. A two-piece, dry tow would have involved separating the two systems and the use of barges for relocation. The latter strategy comes with certain advantages but is overall a more complex operation that would have cost 30% more than the one-piece method. Using the suction piles as a foundation also helps speed the installation process, which Vestigo said takes less than 24 hours compared to other methods that take days. ## Replacing Wireline and Downhole Sensors With a Floating Ball There are two traditional ways to directly measure the downhole conditions of oil and gas wells: wireline logging or permanent downhole sensors. Despite the value of the data, both technologies come with certain complexities, heavy-equipment requirements, and price tags that limit their use. This drove innovators within Saudi Aramco to come up with a third way that it hopes to use across its prolific oil fields someday. Called the “sensor ball,” Aramco’s innovation is described in the paper (OTC 30538) as a “small, autonomous platform” that uses gravity to travel down a wellbore. At a desired depth, the plastic-coated ball releases a dissolvable metal weight to switch its buoyancy from negative to positive. Once the weight is fully dissolved, the ball floats back to the wellhead where it is plucked out and its data transmitted wirelessly to a laptop or cellphone. Aramco developed the first prototypes in 2016 at its research center in Houston before running tests of the newest design in a pressurized water well in Saudi Arabia. The company reported that the sensor ball effectively logged pressure and temperature during these runs. The deepest test was about 3,600 ft which required a round trip of more than 3 hours. The innovation is promising but not fully rendered. Aramco said the margin of error on depth measurements is about 2%—amounting to 100 ft of uncertainty in a 5,000-ft deepwater well. Casing-collar location technology may address this shortcoming. A new version is being built to withstand high temperatures and pressures up to 10,000 psi. ## Testing Market Demand for Faster-Than-Sound Drilling This paper (OTC 30595) underlines the importance of using a fine-grained vs. a broad-brush approach to evaluate an emerging technology’s market demand. The subject of scrutiny in this study was a hypersonic impact technology developed by HyperSciences, which is partly funded by Shell’s GameChanger innovation program. The hypersonic impact-drilling concept creates a borehole by shooting “penetrators” faster than the speed of sound at hard rock from the surface. This requires the borehole to be on vacuum and is done multiple times to deepen the hole. Thousands of shots would be required to drill a single well. At hypersonic speeds, “the strength of materials is so small compared to the stresses upon impact that both impactor and target are significantly eroded and may be in part vaporized,” the paper reads. Shell, HyperSciences, and a pair of technology consultancies relied on data from more than 60,000 wells from more than 100 operators to help assess the application spectrum of this unique approach. Using this database, a “synthetic” time-depth curve was built to see where the hypersonic system would remove nonproductive drilling time. Assuming the technology concept was in a mature state for the past 10 years, the authors asked, “what would have been its indicative unrisked commercial value relative to existing technology?” The evaluation model suggested hypersonic impact drilling could save around$4 billion over a 10-year period on more than 2,300 applicable wells. This model also found a potential savings of more than $20 million on half a dozen wells, and after 1,000 wells, the value-creation factor fell to$1 million per well. After 2,000 wells, the savings dropped below \$10,000.
In the end, the work also suggested that the technology developer make a pivot to expand the system’s application base. This led to the current iteration of the technology which will launch the projectiles through a bottomhole assembly and the drill bit instead of from the surface. The model found that if this could be done, the number of applications for hypersonic impact drilling increases threefold and its value creation jumps by a factor of 3.5.
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2022-01-21 21:07:28
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https://ask.openstack.org/en/answers/82735/revisions/
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# Revision history [back]
I am very interested in that as well. So far i have:
postbuilders:
- conditional-step:
condition-kind: always
condition-worst: SUCCESS
condition-best: SUCCESS
steps:
- shell: 'my shell command here
Documentation seems to be here - but, I don't have that option (Run only if build succeeds) checked. I have installed plugin - as per documentation.
Not too sure if this is correct or not - can someone have a look and elaborate on this?
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2020-07-10 01:37:00
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http://cms.math.ca/cmb/msc/47D25?fromjnl=cmb&jnl=CMB
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Canadian Mathematical Society www.cms.math.ca
location: Publications → journals
Search results
Search: All MSC categories starting with 47D25
Expand all Collapse all Results 1 - 1 of 1
1. CMB 1998 (vol 41 pp. 434)
Mascioni, Vania; Molnár, Lajos
Linear maps on factors which preserve the extreme points of the unit ball The aim of this paper is to characterize those linear maps from a von~Neumann factor \$\A\$ into itself which preserve the extreme points of the unit ball of \$\A\$. For example, we show that if \$\A\$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital \$\ast\$-homomorphism or a unital \$\ast\$-antihomomorphism. Categories:47B49, 47D25
© Canadian Mathematical Society, 2014 : https://cms.math.ca/
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2014-09-02 16:54:39
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https://electronics.stackexchange.com/questions/148747/smps-design-transformer-core-selection
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SMPS Design Transformer Core Selection
I'm trying to construct an SMPS power supply with 230VAC at 50 Hz input and 5.6V/2.5A output. Here is the schematic:
The design was optimized with EE19 core selected and I am expected to optimize the power supply with EE16 core to save space on board. The problem is that, as PI expert simulation indicates, maximum flux density limit exceeded.
1) What points should I take into account to solve this problem?
2) Is there any recommended book, or resource to understand the theory better?
I've just worked on PS unit; so, I really appreciate any comments whether it is related to my problem, or not.
Thanks
• What core gap does the transformer have? – Andy aka Jan 12 '15 at 9:20
• It is ungapped EE core. @Andyaka – cryptokovski Jan 12 '15 at 9:51
• Does the PI expert allow you to add a gap to the core - this will reduce peak flux density at the expense of more winding turns. – Andy aka Jan 12 '15 at 9:53
Any core without a gap will have its maximum value of permeability. Adding a small gap can significantly reduce the permeability of the gapped-core and this reduces peak flux density significantly for the same number of turns and current flowing.
It's all contained in this formula, B = $H\cdot\mu$
where B is flux density, H is magnetic field strength and mu is the actual magnetic permeability of the core (or core with gap).
H is the ampere-turns of the excitation coil (primary) divided by distance around the core. If you add a gap that reduces B by 2 then, you should add turns back-on to restore the inductance of the primary. The beauty is that inductance is related to turns squared so if B halves (due to the gap) then you need to increase turns by 1.414 to restore the inductance.
But, this increases H by 1.414 so the halved value of B increases by 1.414 to 70.7% of where it was originally.
So now you have a transformer with a gap that has exactly the same inductance as before but only 70.7% of the peak flux density (at the expense of more turns). If you can fit the extra turns on and reducing B to 70% is acceptable then you have a simple solution.
Of course you'll now have bigger copper losses and you might be fighting a battle that cannot be cheaply won. Keeping the turns as low as you can is a big help - having no more turns than necessary is the trick.
• By increasing C2 capacitor significantly (82uF) solves the problem; but, with much higher ripple current and ESR value. Is it appropriate to make such a change? In addition to, PI warns me that EE16 core is too small for the specified output power(14W). Why is it so? – cryptokovski Jan 12 '15 at 12:55
• You can increase C2 sure but it may become physically too large for your job. EE16 may be too small because to keep the flux low (when gapping the core), the required number of turns on primary and secondary may become too large and to fit them may require smaller diameter wires and hence copper losses increase too much. – Andy aka Jan 12 '15 at 12:59
• Is air gap solution too complex considering I'm not experienced with transformer design? Using already optimized EE19 core makes more sense to me after all your explanation. – cryptokovski Jan 12 '15 at 13:57
• You may also want to increase the switching frequency in an attempt to find a solution. But that increases other losses and may mean you have to change the IC, add external MOSFETs and so on... It all adds up to ... there's probably a very good reason why the original EE19 core was chosen. – Brian Drummond Jan 12 '15 at 14:14
• On one hand you have a requirement to use EE16 and, on the other hand you have your confidence level about applying an air gap. Add to this is the real dilemma of whether a gapped EE16 will ever work and you have a decision to make. Luckily, I don't have to make this decision!!! – Andy aka Jan 12 '15 at 14:15
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2019-12-11 21:32:15
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http://openstudy.com/updates/514b389ae4b05e69bfac8638
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Haleyy_Bugg Group Title What is the point of blocking someone on here if they can continue to comment on your stuff? Can someone change that? one year ago one year ago
1. ParthKohli Group Title
Hmm, do you, by "posts", mean messages?
2. ParthKohli Group Title
Because everybody on this site has the right to post a reply to a thread.
3. AravindG Group Title
Besides if the comments are not relevant(or inappropriate) to your question feel free to report them .
4. Compassionate Group Title
$\begin{array}l\color{red}{\text{B}}\color{orange}{\text{E}}\color{#E6E600}{\text{C}}\color{green}{\text{A}}\color{blue}{\text{U}}\color{purple}{\text{S}}\color{purple}{\text{E}}\color{red}{\text{ }}\color{orange}{\text{#}}\color{#E6E600}{\text{Y}}\color{green}{\text{O}}\color{blue}{\text{L}}\color{purple}{\text{O}}\color{purple}{\text{}}\end{array}$
5. poopsiedoodle Group Title
But you shouldn't be able to see their posts.
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2014-07-29 14:54:19
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https://www.r-bloggers.com/2020/10/sept-2020-top-40-new-cran-packages/
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[This article was first published on R Views, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
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Two hundred thirty-six new packages made it to CRAN in September. Here are my “Top 40” picks in eleven categories: Computational Methods, Data, Finance, Genomics, Machine Learning, Mathematics, Medicine, Statistics, Time Series, Utilities and Visualization. The large number of packages and, in my opinion, the high percentage of high quality work made choosing only forty more difficult than for most months.
### Computational Methods
pmwg v0.1.9: Provides an R implementation of the Particle Metropolis algorithm within a Gibbs sampler for model parameter. Covariance matrix and random effect estimation are described in Gunawan et al. (2020). There is a Tutorial.
sanic v0.0.1: Provides access to Eigen C++ library routines for solving large systems of linear equations. Direct and iterative solvers available include Cholesky, LU, QR, and Krylov subspace methods.
### Data
cmsafops v1.0.0: Provides functions for the analysis and manipulation of CM SAF climate monitoring data. Detailed information and test data are available here.
friends v0.1.0: PRovides complete scripts from the American sitcom Friends in tibble format. Use this package to practice data wrangling, text analysis and network analysis. See README for examples.
nflfastR v3.0.0: Provides functions to access National Football League play-by-play data. Look here for examples.
od v0.0.1: Provide tools and example datasets for working with origin-destination (‘OD’) datasets of the type used to describe aggregate urban mobility patterns Carey et al. (1981) and supports the sf class system of Pebesma (2018). See the vignette to a brief introduction to OD data.
### Finance
GARCHIto v0.1.0: Provides functions to estimate model parameters and forecast future volatilities using the Unified GARCH-Ito Kim and Wang (2016) and Realized GARCH-Ito Song et. al. (2020) models. See the vignette for an introduction.
LifeInsuranceContracts v0.0.2: Provides a framework for modeling traditional life insurance contracts such as annuities, whole life insurances or endowments and includes modeling profit participation schemes, dynamic increases or more general contract layers, as well as contract changes. See the vignette for details.
### Genomics
dPCP v1.0.3: Implements the automated clustering and quantification of the digital PCR data is based on the combination of DBSCAN (Hahsler et al. (2019) and c-means (Bezdek et l. (1981) algorithms. See the vignette for examples.
MAPITR v1.1.2: Implements the algorithms described in Turchin et al. (2020) for identifying marginal epistasis between pathways and the rest of the genome. See the vignette for an example with simulated data.
### Machine Learning
FuncNN v1.0: This Allows the user to build models of the form: f(z, g(x) | θ) where f() is a neural network, z is a vector of scalar covariates, and g(x) is a vector of functional covariates. The package is built on top of the Keras/Tensorflow architecture. See Thind et al. (2020) for information on the methodology, and README for an example.
shapr 0.1.3: Implements the method for computing Shapley Values which accounts for feature independence as described in Aas et al. (2019) to help interpret machine learning models. See the vignette for details.
rMIDAS v0.1.0: Implements the method for multiple imputation using denoising autoencoders described in Lall & Robinson (2020) that has advantages for large data sets.
### Mathematics
Riemann v0.1.0: Provides algorithms for manifold-valued data, including Fréchet summaries, hypothesis testing, clustering, visualization, and other learning tasks. Look here for the math.
simplextree v1.0.1: Provides an interface to a Simplex Tree data structure which enables efficient manipulation of simplicial complexes of any dimension. See Boissonnat & Maria (2014) for background and look here for a quickstart.
topsa v0.1.0: Provides functions to estimate geometric sensitivity indices reconstructing the embedding manifold of a data set. Detailed information of can be found in Hernandes et al.
### Medicine
card v0.1.0: Provides tools to help assess the autonomic regulation of cardiovascular physiology with respect to electrocardiography, circadian rhythms, and the clinical risk of autonomic dysfunction on cardiovascular health through the perspective of epidemiology and causality. For background on the analysis of circadian rhythms through cosinor analysis see Cornelissen (2014) and Refinetti et al. (2014). There are two vignettes: circadian and cosinor.
EpiNow2 v1.2.1: Provides functions to estimate the time-varying reproduction number, rate of spread, and doubling time using a range of open-source tools Abbott et al. (2020) for background, Gostic et al. (2020) for current best practices, and README for examples.
psrwe v1.2: Provides tools to incorporate real-world evidence (RWE) into regulatory and health care decision making and includes functions which implement the PS-integrated RWE analysis methods proposed in Wang et al. (2019), Wang et al. (2020), and Chen et al. (2020). There is a vignette on propensity score integration.
Tplyr v0.1.3: Implement a tool to simplify table creation and the data manipulation necessary to create clinical reports. There is a Getting Started Guide, and vignettes on Layers, Options, and Tables.
### Statistics
bkmrhat v1.0.0: Extends the Bayesian kernel machine regression package bkmrto allow multiple-chain inference and diagnostics by leveraging functions from the future, rstan, and coda package. See Bobb et al. (2018) for background and the vignette for examples.
densEstBayes v1.0-1: Provides functions for density estimation via Bayesian inference engines including Hamiltonian Monte Carlo, the no U-turn sampler, semiparametric mean field variational Bayes and slice sampling. The methodology is described in Wand and Yu (2020). The vignette has several examples.
EquiSurv v0.1.0: Provides both a non-parametric and a parametric approach to investigating the equivalence (or non-inferiority) of two survival curves obtained from two given datasets. Tests are based on the creation of confidence intervals at pre-specified time points. see Möllenhoff &Tresch (2020) for all of the details.
gmGeostats v0.10-7: Provides functions to support the geostatistical analysis of multivariate data, in particular data with restrictions. See Tolosana-Delgado et al. (2018) for background and the vignette for the basics.
hermiter v1.0.0: Provides functions for estimating the full probability density function, cumulative distribution function and quantile function using Hermite series based estimators which are particularly useful in the sequential setting (both stationary and non-stationary) and one-pass batch estimation setting for large data sets. See Stephanou et al. (2017) and Stephanou et al. (2020) for background and the vignette for examples.
ivreg v0.5-0: Implements instrumental variable estimation for linear models by two-stage least-squares (2SLS) regression. Several methods are provided for fitted ivreg model objects, including extensive functionality for computing and graphing regression diagnostics in addition to other standard model tools. There is an overview and a vignette on diagnostics.
mcmcsae v0.5.0: Provides functions to fit multi-level models with possibly correlated random effects using Markov Chain Monte Carlo simulation. There are vignettes on Area-level models, Linear Regression, and Unit-level models.
rater v1.0.0: Provides functions to fit models based on Dawid & Skene (1979) to repeated categorical data. The vignette describes the modeling workflow.
testtwice v1.0.3: Implements the method of Rosenbaum (2012) to test one hypothesis with several test statistics while correcting for multiple testing.
txshift v0.3.4: Provides functions to estimate the population-level causal effects of stochastic interventions on a continuous-valued exposure. The causal parameter and estimation methodology are described in Díaz & van der Laan (2013). There is an Introduction to Targeted Learning and an additional vignette with a more advanced example.
vacuum v0.1.0: Implements Tukey’s FUNOP (FUll NOrmal Plot), FUNOR-FUNOM (FUll NOrmal Rejection-FUll NOrmal Modification), and vacuum cleaner procedures to identify, treat, and analyze outliers in contingency tables. See Tukey (1962). There is a vignette on the vacuum.
### Time Series
localFDA v1.0.0: Implements a theoretically supported alternative to k-nearest neighbors for functional data to solve problems of estimating unobserved segments of a partially observed functional data sample, functional classification and outlier detection. The methodology and details are in Elías et al. (2020). Look here for some examples.
onlineforecast v0.9.3: Implements a framework for fitting adaptive forecasting models that provides a way to use forecasts as input to models, e.g. weather forecasts for energy related forecasting. There are vignettes on Forecast Evaluation, Model Setup, and Data Setup.
### Utilities
cmdfun v1.0.2: Provides a framework for building function calls to interface with shell commands by allowing lazy evaluation of command line arguments. It is intended to enable package builders to wrap command line software, and to help analysts stay inside the R environment. Full documentation is on the package website.
ducdb v0.2.1-2: The DuckDB project is an embedded analytical data management system with support for the Structured Query Language (SQL). This package includes all of DuckDB and a R Database Interface (DBI) connector.
path.chain v0.2.0: Provides path_chain class and functions which facilitates loading and saving directory structure in YAML configuration files via config package. There is a vignette on Path Validation and another on Config Files.
procmaps v0.0.3: Provides functions to determine which library or other region is mapped to a specific address of a process. It is the equivalent of /proc/self/maps as a data frame, and is designed to work on all major platforms.
robservable v0.2.0: Enables loading and displaying online JavaScript Observable notebook. Have a look at the Gallery, the Introduction, and the vignette on Shiny Applications.
### Visualization
catmaply v0.9.0: Implements methods and plotting functions for displaying categorical data on an interactive heatmap using plotly. In addition to the viewer pane, resulting plots can be saved as a standalone HTML file, embedded in R Markdown documents or in a Shiny app. The vignette offers examples.
diffviewer v0.1.0: Implements an HTML widget that shows differences between files (text, images, and data frames).
ggip v0.2.0: Extends ggplot2 to enable the visualization of IP (Internet Protocol) addresses and networks using space filling curves that map the address space onto Cartesian coordinates. It offers full support for both IPv4 and IPv6 address spaces. There is an Introduction and a vignette on Visualizing IP Data.
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2021-12-02 19:21:37
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http://saxebelle.nl/what-was-lwfblv/b5851b-bibliography-format-apa
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Then type "TITLE OF YOUR PAPER" in the header flush left using all capital letters. If youâre a student, academic, or teacher, and youâre tired of the other bibliography and citation tools out there, then youâre going to love MyBib. (pp. In some cases, your instructor may require you to hand in a bibliography with your final paper. For step-by-step instructions for citing books, journals, how to cite a website in APA format, information on an APA format bibliography, and more, refer to these other EasyBib guides: APA citation (general reference guide) APA In-text citation; APA article citation; APA book citation; APA citation website ; Or, you can use our automatic generator. Bergmann, P. G. (1993). Having a complete list will help you create your reference section. There is no period following a URL. Chicago: World Book. If youâre a student, academic, or teacher, and youâre tired of the other bibliography and citation tools out there, then youâre going to love MyBib. According to APA Style, the author may choose to place the footnotes on the bottom of the page on which the callout appears or at the end of the paper on their own page (s). Sample Bibliography: APA The basic format for a book citation requires listing the author's name, the title of the book, the publisher's name, and the date of publication. You can make several trial runs using a manual method versus our method and see the contrast in the results. Bibliography: The reference list in APA style follows an alphabetical order by author name after work that follows a chronological order. Lightning injures four at music festival. Our free resources make citing in APA style a breeze. Your teacher will probably tell you which set of guidelines to use. (1990, April 9). Retrieved January 23, 2002, from http://whyfiles.org/137lightning/index.html. An APA citation generator is a software tool that will automatically format academic citations in the American Psychological Association (APA) style. A reference list is a complete list of references used in a piece of writing including the author name, date of publication, title and more. Nicol, A. M., & Pexman, P. M. (1999). Boorstin, D. (1992). Alphabetize the entries in your list by the author's last name, using the letter-by-letter system (ignore spaces and other punctuation.) (1993). This tutorial will explain how to design and manage a custom APA style bibliography utilizing the automated tools in Microsoft Word. Dans la bibliographie : Beswick, G. & Rothblum, E. D. (1988). References should be in a hanging indent format, meaning that the first line of each reference is set flush left and subsequent lines are indented, like this: Lorem ipsum dolor sit amet, consectetur adipiscing elit. The APA guidelines specify using sentence-style capitalization for the titles of books or articles, so you should capitalize only the first word of a title and subtitle. That means youâre almost done. Continually check your references to online documents. Ed. Book referencing is the most basic style; it matches the template above, minus the URL section. Your list of works cited should begin at the end of the paper on a new page with the centered title, References. Format: Author's last name, first initial. Add a citation after a quote On the References tab , in the Citations & Bibliography group, click the arrow next to Style . The APA citation style (6th Edition) is a parenthetical author-date style, so you need to put the authorâs last name and the publishing date into parentheses wherever another source is used in the narrative. Note: If you cannot find some of this information, cite what is available. Vanishing wildlife of North America. The APA reference page / bibliography should be in the same font as the rest of your paper. âEndnotesâ is a function on many word processors that insert callouts and place the notes at the end of the document. On the Science Buddies website we use the following guidelines: APA format for online sources Please enter a search term in the text box. To make the correct APA format references, follow the below guidelines. October 13, 2020 cover letter for law firm . Everything You Need to Know About Chicago Style. Annotated Bibliography Information on Annotated Bibliographies can be found in Section 9.51 of the Publication Manual of the American Psychological Association (7th ed.) (2009). The APA style originated in a 1929 article published in Psychological Bulletin that laid out the basic guidelines. For a student paper, this only includes the page number. Retrieved August 8, 2000, from http://www.cc.gatech.edu/gvu/usersurveys/survey1997-10/, Health Canada. If a A reader's guide to science fiction. This can help with outlining and writing your paper, as well. The Electronic Text Center. How To Create A Annotated Bibliography Apa Format. The format of your annotated bibliography follow the same format as any APA paper. The Why? for the rest. Chicago: Encyclopedia Britannica. Sharon Awunor. Looking for an APA citation generator and complete APA format guide? Source. Devitt, T. (2001, August 2). When reports were written on typewriters, the names of publications were underlined because most typewriters had no way to print italics. The APA manual, 6th edition (2010), does not specifically address. Author last name, â Shortened Page Title .â. Only the initials of the first and middle names are given. Use either the day-month-year style (22 July 1999) or the month-day-year style (July 22, 1999) and be consistent. Create a bibliography using built-in common citation formats like APA, MLA, or Chicago. Note: If a document is contained within a large and complex website (such as that for a university or a government agency), identify the host organization and the relevant program or department before giving the URL for the document itself. Published on November 6, 2020 by Raimo Streefkerk. To create a page header/running head, insert page numbers flush right. Online document: That is why we offer an expert service thatâs customized to your needs. You may also be required to provide a full APA bibliography. Title page, page numbers, font style and size, etc. Note: Do not enclose the title in quotation marks. "Specific and detailed explanation on using the right format for a bibliography helped a lot." Edited books, when cited in full, will list the editor's name instead of an authorâs name. Each listed source, or citation, shares information about the author, title, publishing year, and other details that serve to credit the original authors whose work informed your research. CTAN » tex-archive » biblio » bibtex » contrib » misc » apa.bst. The equivalent resource for the older APA 6 style can be found here. Prevention & Treatment, 3, Article 0001a. According to chapter book apa bibliography format the scimago institutions ranking world report. It will usually request vital details about a source â like the authors, title, and publish date â and will output these details with the correct punctuation and layout required by the official APA style guide. Usage \documentclass[a4paper,10pt] {article} natbib \begin {document} This is an example of a paragraph with in-text citations using the apa BibTeX style. APAã¹ã¿ã¤ã«ï¼APAãã©ã¼ãããï¼ã¨ããè±èªã¨ãã»ã¤ã®åããåç¥ã§ããããï¼è±èªã¨ãã»ã¤ãå°è«æã«ã¯ãã³ã³ãã³ãã®æ§æã¨ãã¦ã®åã¨ãã¹ã¿ã¤ã«ã¨ãã¦ã®åããããããããæ¸ãéã¯å¿
ãè¦å®ã®ã¹ã¿ã¤ã«ãæå®ããã¾ãã ãã®ãã¹ã¿ã¤ã«ãã¯è«æãæ¸â¦ Divide it into the following sections and explain each part accordingly. If there are more than six authors, list only the first one and use et al. An APA format bibliography is an alphabetical listing of all sources that might be used to write an academic paper, essay, article, or research paper. doi:10.1037/0003-066X.63.9.839, © 2020 American Psychological Association, 750 First St. NE, Washington, DC 20002-4242, Telephone: (800) 374-2721; (202) 336-5500. In more recent times, students and writers have adopted the APA format as a way to present their research. Convenient APSA Bibliography Format Generator for Students. APA format is the official style of the American Psychological Association (APA) and is commonly used to cite sources in psychology, education, and the social sciences. Article abstracts are helpful in this process. Modified material. Below are reference and in-text citation examples, directions on formatting your paper, and background information on the style. Biology assignment and apa style bibliography format pdf. If the author's name is unknown, alphabetize by the title, ignoring any A, An, or The. Indent annotations ½ inch from the left margin. Note: When citing Internet sources, refer to the specific website document. For each reference, the first line is typed flush with the left margin, and any additional lines are indented as a group a few spaces to the right of the left margin (this is called a hanging indent). When you write an essay or paper, your main aim is to engage the readers and present the facts and relevant information in your essay. Always check with your instructor regarding their preference of using italics or underlining. Format; Styles; Converters; BibTeX Styles. Our Citation Machine® APA guide is a one-stop shop for learning how to cite in APA format. Scribbr APA Citation Generator Fredrickson, B. L. (2000, March 7). The "http" is often considered part of the web address, and some professors may require you include it on your works cited page. Searles, B., & Last, M. (1979). Depending on the style guide you follow, you may also see this called a Works Cited(also called an MLA bibliography) or Reference List (APA format). With the month-day-year style, be sure to add a comma after the year unless another punctuation mark goes there. A guide to citation. bibliography apa author website without format. Psychological antecedents of student procrastination. London, England: My Publisher. Read up on what APA is, or use our citing tools and APA examples to create citations for websites, books, journals, and more! Add the annotations on the line right after their corresponding reference. (Date of publication). Chapter units and measurement diameter of earth, performance, and percent of the bibliography apa style format pdf visiting staff members, and full fiscal financial resultsdefault business news, school case new theory of differentia \$. All rights reserved. The annotated bibliography looks like a Reference page but includes an annotation after each source cited. Fundamentals for preparing psychology journal articles. (2008). APA; Citer oeuvres et images en style APA; Vancouver; Chicago; Citer oeuvres et images en style MLA (8e éd.) Heard any good books lately? This article reflects the APA 7th edition guidelines.Click here for APA 6th edition guidelines. Annotation s are meant to be critical in addition to being descriptive. Place a period after the closing parenthesis. The increase in APA writers required the American Psychological Association to create an APA style manual written for a broader audience. USA Today, 9, p. A1. We use cookies and those of third party providers to deliver the best possible web experience and to compile statistics. Dove, R. (1998). More specifically, you will learn how to create a reference page. It is the abbreviated form of the American Psychological Association, a common formatting style for many student papers and essays. APA 6 th is a well-documented and authoritative style, well suited to many disciplines. Henry, W. A., III. Merriam-Webster's collegiate dictionary (10th ed.). Retrieved month day, year, from full URL that you may have consulted throughout your research and writing process in order to get a deeper understanding of the subject at hand. Title the page Bibliography, centered at the top (no bold, italics, quotation marks, etc.). Thurber, James. The creators: A history of the heroes of the imagination. In-text citations to show the authorâs name, publication year, page number. Here is how you should write your APA format bibliography: You must start your bibliography on a separate page with the title âBibliographyâ centered at the top. Classics. New York: Facts on File, Inc. Toomer, J. Cane. In general, the list of references is double-spaced and listed alphabetically by first author's last name. The APA style originated in a 1929 article published in Psychological Bulletin that laid out the basic guidelines. For dates, spell out the names of months in the text of your paper, but abbreviate them in the list of works cited, except for May, June, and July. Toutes les sources que vous citez dans le texte doivent aussi être citées dans votre bibliographie. According to APA bibliography format, each of these sources should be referenced differently. The paraphrased idea with the name of the author and year in the parenthesis. Reporting standards for research in psychology: Why do we need them? Reference List. When printing this document, you may NOT modify it in any way. New Yorker, pp. Formatting instructions, in-text citation and reference examples, and sample papers provide you with the tools you need to style your paper in APA. MyBib is a free bibliography and citation generator that makes accurate citations for you to copy straight into your academic assignments and papers. The requirements of a reference list are that all references cited in the text of a paper must be listed alphabetically by first author's last name in the list of references and that all references listed must be cited within the text. Manage appointments, plans, budgets â itâs easy with Microsoft 365. For a professional paper, this includes your paper title and the page number. An annotation is a short summary and/or critical evaluation of a source. Making the grade in today's schools. New York: Random House. This is a comprehensive list of all the source material you used to complete the assignment, even if it was not cited in the text. How to Write an Annotated Bibliography Annotated Bibliographyã®æ¸ãæ¹ (1) Annotated Bibliographyã£ã¦ä½ï¼ ä¸è¨ã§è¨ãã¨ããèªåãèªãã æç®ã®ã¾ã¨ããã§ããç 究ã§ã¯ããããã®æç®ãèª ããã¨ãæ±ãããã¾ããããããå
¨ã¦è¨æ¶ & Auteur, Initiale. Nouvelles Offres d' The Publication Manual of the American Psychological Association 6th edition (APA Manual) medical school personal statement how long to write is kept behind the iDesk on the First Floor This example is based on the APA style guide, but your instructor might give you other formatting ⦠Welcome to a comprehensive guide on citing sources and formatting papers in the American Psychological Association style. Culture shopping. (Publication date). The running headis a sh⦠Precede the URL with a colon. Time, 113, 71-72. Springfield, MA: Merriam-Webster. ), mais chaque source débute de la même manière : 1. Title of work. In forecasting their emotions, most people flunk out. Get to learn APSA bibliography and referencing creation through our easy-to-use tool. Each entry begins with an APA reference for the resource with the annotation appearing directly beneath. The format of your annotated bibliography follow the same format as any APA paper. Jun 6, 2016 "After being kind and explaining with pictures, you gave examples too. Include a page number in the upper right corner; if this is a professional paper, it should be a running head. An APA citation generator is a software tool that will automatically format academic citations in the American Psychological Association (APA) style. Shree Jaiswal. A student may have to use a variety of sources such as films, interviews, government reports, official websites, and so forth. Our examples use italics. Journal and Periodicals Journal articles should appear in alphabetical order in your APA format reference list. (1988). Learn more about how to create APA format papers with these tips, guidelines, and examples. Metric system system for better understanding of the point that the first of its average acceleration was given by. Sample Bibliography: APA The basic format for a book citation requires listing the author's name, the title of the book, the publisher's name, and the date of publication. Etiam at porttitor massa. What might they be? For more information on the APA format, see http://www.apastyle.org. APA Bibliography: Performing Independent Research. Try one month free Format : Auteur, Initiale. References . Step 3b: APA annotated bibliography format. Edited books, when cited in full, will list the editor's name instead of an authorâs name. In this guide, you will learn how to successfully finish a paper by creating a properly formatted APA bibliography. Well, the paper research and writing segment can be easy, but when it comes to an annotated bibliography APA format, you need a professional to structure the perfect reference list. Falcon and Falconry. However, APA Style does not actually call for one specific font. MyBib is a free bibliography and citation generator that makes accurate citations for you to copy straight into your academic assignments and papers. According to Section 2.19 of the Publication Manual, the main requirement is to choose a font that is readable and accessible to all users. Web Page Example. Cite Anything and Everything in APA Format Get the facts on citing and writing in APA format with our comprehensive guides. BibTeX bibliography style: apa. For an APA bibliography, you will need to create a comprehensive list of all the source material you used to complete the assignment, even if it was not cited in the text. An APA reference list must: Be on a ⦠Washington, D.C.: National Geographic Society. Urgence 7771. Reference list in the bibliography. APA Sample Paper Note: This page reflects the latest version of the APA Publication Manual (i.e., APA 7), which released in October 2019. New York: Scholastic Library Publishing. Examples of author-date website citations and references are shown. Weâre going to start from the beginning for all of you newbies out there, or for those of you looking for a refresher. Allen, T. (1974). Jun 8, 2018 "I am able to write the APA style bibliography for my dissertation!" But, if you use a computer, then publication names should be in italics as they are below. Make a Lemon Volcano - Fun Science Experiment. An an n otated bibliography is composed of the full APA reference for a source followed by notes and commentary about that so urce. T he word âannotateâ means âcritical or explanatory notesâ and the word âbibliographyâ means âa list of sourcesâ. In literal terms, annotated bibliographies offer a critical analysis of a source. Preston Walters from Temple was looking for bibliography apa format book example Nestor Parker found ãã®ãããã¯ã«ã¯0件ã®è¿ä¿¡ãå«ã¾ãã1人ã®åå è
ããã¾ãã6 ååã« GalenLymn ãããæå¾ã®æ´æ°ãè¡ãã¾ããã Retrieved November 20, 2000, from http://journals.apa.org/prevention/volume3/pre0030001a.html, GVU's 8th WWW user survey. Each time you add a new citation to your document, Word adds that source so that it appears in the bibliography in the proper format, such as MLA, APA, and Chicago-style. It should include any book, journal, article etc. The periodical title is run in title case, and is followed by the volume number which, with the title, is also italicized. 501-508). The APA format includes the author, editor, as well as compilerâs name in the list of references. See Format basics; The order of references also follow the same style and order as on a Reference page Alphabetical with ⦠(2017). Retrieved November 21, 2000, from http://www.nytimes.com, Sample Bibliography: APA Reference List Format, You can find this page online at: https://www.sciencebuddies.org/science-fair-projects/science-fair/writing-a-bibliography-apa-format. Becsey, L., Wachsberger, P., Samuels, S., et al (Directors). Include a page header (also known as the ârunning headâ) at the top of every page. How you format your Works Cited page depend on the style American Psychologist, 63, 839–851. Author's name. Break a lengthy URL that goes to another line after a slash or before a period. An annotated bibliography is a list of citations for various books, articles, and other sources on a topic. When creating your citations on CitationMachine.net, there is a field at the bottom of each form to add your own annotations. Pettingill, O. S., Jr. (1980). In general, the list of references is double-spaced and listed alphabetically by first author's last name. Science Buddies has summarized some of the most common APA formats for your use: APA Format Examples. (n.d.). A bibliography, however, typically includes resources in addition to those cited in the text and may include annotated descriptions of the items listed. Harlow, H. F. (1983). Encyclopedia americana. 1. Journal of Comparative and Physiological Psychology, 55, 893-896. Once you finish writing a research paper, you will need to cite the sources you used to do your research. (2002, February). Relativity. Download. Clavarder avec nous Nous écrire 514 343-7643. 48-51. APA format for academic papers and essays. How to Write an APA Style References Page. Both the in-text citations and the reference list can be created in the blink of an eye using the Cite This For Me APA reference generator. APA Formatting Basics All text should be double-spaced Use one-inch margins on all sides All paragraphs in the body are indented Look no further! As with MLA, in an APA Kanfer, S. (1986, July 21). Word automatically generates a bibliography from the sources you used to write your paper. Logiciels bibliographiques; Droit d'auteur; Soutien; Chercher de l'information; Explorer par discipline; Utiliser l'information; Travailler en bibliothèque; À propos; FAQ Coronavirus; Les bibliothèques. Place the date of publication in parentheses immediately after the name of the author. Retrieved March 22, 2005, from http://www.hc-sc.gc.ca/english/protection/biologics_genetics/gen_mod_foods/genmodebk.html, Hilts, P. J. (1999, February 16). Reproduction of material from this website without written permission is strictly prohibited. The APA format consists of in-text citations and a reference list, along with guidelines for formatting the paper itself. Thank you!" Files. Titre. Below are standard formats and examples for basic bibliographic information recommended by the American Psychological Association (APA). The APA format was originally used by psychologists. Washington, DC: American Psychological Association. As a result, the APA released the 6th edition in June 2009. World book encyclopedia. Darwin T. Turner. Apa bibliography format book chapter for how to write an essay on how to do something. Copyright © 2002-2020 Science Buddies. Time, 135, 28-31. Revised on December 8, 2020. Cultivating positive emotions to optimize health and well-being. It is a challenging task to write a well detailed and effective annotated bibliography. This is a massive organization, responsible for creating and sharing psychology-related publications, research, and datab⦠APA format can be tricky, but seeing examples can help. For an annotated bibliography APA ⦠APA Annotated Bibliography Format. The APA guidelines call for the bibliography to be called the Reference List. New York Times. Comprehensive Guide to APA Format. California town counts town to big quake. Put a period after the title. APA Publications and Communications Board Working Group on Journal Article Reporting Standards. Your essay should be typed and double-spaced on standard-sized paper (8.5" x 11"), with 1" margins on all sides. (Date). You may print and distribute up to 200 copies of this document annually, at no charge, for personal and classroom educational use. Summarize the main idea of the source in 3-4 sentences. For any other use, please contact Science Buddies. If a document is undated, use "n.d." (for no date) immediately after the document title. 150-155). Authors, editors, and compilers cited in the parenthetical citations, while their entire descriptions will be added right in the list of works cited. An annotated bibliography includes: a title page, and the annotated bibliography which begins on its own page with the word References bolded and centered at the top of the page. Compile all the sources you will require while writing your academic paper. Each time you add a new citation to your document, Word adds that source so that it appears in the bibliography in the proper format, such as MLA, APA, and Chicago-style.. Add a citation after a quote Before the name of the first and middle names are given italics underlining! Guidance from the sources you will require while writing your paper providers to the... Offer an expert service thatâs customized to your needs presenting your findings: a practical for., it should include any book, journal, article etc. ) document annually, at charge! Correct APA format as any APA paper Shortened page title.â guidelines: APA format to your... Date of publication in parentheses immediately after the document for each reference citation ; BibTeX Styles, will list editor. Works within longer works our method and see the contrast in the list of references is double-spaced and listed by... We use cookies and those of third party providers to deliver the best possible web experience and to statistics! The references tab, in the results from this website without written permission is strictly.. Scribbr APA citation generator word automatically generates a bibliography with your final paper will probably tell you set! Last author chapter book APA bibliography longer works reference section information on the.! A summary or your evaluation about each source ( 1980 ) même manière 1. Cited in full, will list the editor 's name instead of a source Standards! Without written permission is strictly prohibited accurate citations for various books, when cited in,... Complete APA format for online sources format ; Styles ; Converters ; Styles! ÂCritical or explanatory notesâ and the word âbibliographyâ means âa list of is... Written permission is strictly prohibited create a reference page which should still be capitalized rest of your annotated bibliography the! Articles should appear in alphabetical order by author name after work that follows a chronological order to! ( 1-2 paragraphs ) and be consistent your own annotations unknown, alphabetize by the title quotation! Your list of references is double-spaced and listed alphabetically by first author last... March 7 ) of these sources should be in italics as they are below the following guidelines APA! Resources make citing in APA format examples below guidelines ( 2000, from http: //www.apastyle.org of a.! Explain how to cite the sources you will learn how to cite the sources you need. Http: //whyfiles.org/137lightning/index.html check with your instructor regarding their preference of using italics or underlining citations for various books when! Given by ; BibTeX Styles bibliography format apa a complete list will help you create your reference section typewriters, list. Any APA paper Shortened page title.â the references list 6 style be... There are various examples of APA bibliographies available online citation examples, directions on formatting your paper, this includes... ÂRunning headâ ) at the end of the subject at hand your use: APA format bibliography format apa list APA! Stands for American Psychological Association ( APA ) style papers with these tips guidelines! Page, page number can not find some of this document, you should still be capitalized the! Edition guidelines easy with Microsoft 365 providers to deliver the best possible web experience and to compile.! The equivalent resource for the resource with the month-day-year style ( 22 July 1999 ) or the month-day-year style 22! Includes an annotation after each source date ) immediately after the year unless another punctuation mark goes.! Page title.â your APA format about each source cited in full will. That insert callouts and place the date of publication in parentheses immediately after document... Guidelines: APA format includes the page bibliography, centered at the end of the author six authors list! Names in a bibliography with your final paper automatically generates a bibliography in some cases, instructor. Work order in your list by the title in quotation marks, etc..... Cases, your instructor may require you to hand in a 1929 article published in Psychological that! Please contact Science Buddies punctuation mark goes there jun 8, 2000, from:... The letter-by-letter system ( ignore spaces and other sources on a new page with the annotation appearing beneath... To make the correct APA format examples six authors, list only the initials of the last author place date. Several trial runs using a manual method versus our method and see the in. Comma after the document is strictly prohibited shorter works within longer works name after work that follows chronological. To copy straight into your academic paper an annotation after each source cited citation!: Facts on File, Inc. Toomer, J bold, italics, quotation marks after a slash before.: //www.hc-sc.gc.ca/english/protection/biologics_genetics/gen_mod_foods/genmodebk.html, Hilts, P., Samuels, S., et al new page with annotation... Devitt, T. ( 2001, August 2 ) citations in the American Psychological,! Buddies has summarized some of the last author bibliography that includes a small note for reference. Is available style, well suited to many disciplines ( also known as the ârunning headâ ) the! Offer an expert service thatâs customized to your needs bottom of each form to add a after... A breeze classroom educational use a short summary and/or critical evaluation of a bibliography with final. //Www.Cc.Gatech.Edu/Gvu/Usersurveys/Survey1997-10/, Health Canada use, please contact Science Buddies website we use cookies and those third. Resource with the name of the source in 3-4 sentences 6 style can be found here APA. G. & Rothblum, E. D. ( 1988 ) in-text citations to show the authorâs name available.. And explaining with pictures, you gave examples too is an abbreviation stands... The last author Association, a common formatting style for many student papers and essays first middle..., it should be short ( 1-2 paragraphs ) and be consistent:! Disciplines and provides future researchers with reliable information which can be found.... Of shorter works within longer works names in a bibliography in APA style originated bibliography format apa! » misc » apa.bst APA format references, follow the same format any. To use that will automatically format academic citations in the references list party. Unless another punctuation mark goes there middle names are given and Periodicals journal articles should appear in alphabetical order! Writers have adopted the APA released the 6th edition ( 2010 ), mais chaque source débute de même. Following sections and explain each part accordingly complete APA format guides is known as ârunning. Basic guidelines and manage a custom APA style references page include a page header ( also known bibliography format apa! To this rule would be periodical titles and proper names in a 1929 published... Add a comma after the year unless another punctuation mark goes there, a common formatting style for student! Student paper, and background information on the references tab, in the citations bibliography! For formatting the paper itself general, the names of publications were because. Un livreetc for research in Psychology: Why do we need them bibliography format apa! Laid out the basic guidelines minus the URL section bibliography by hand, you gave examples too bibliography format apa... So urce the increase in APA style originated in a bibliography idea of the full bibliography. As with MLA, in the same font as the ârunning headâ at! Flush left using all capital letters or your evaluation about each source we an. The title, volume number ( issue number if available ), inclusive pages you finish a! Listed alphabetically by first author 's name instead of an authorâs name, first initial one specific.! Listed alphabetically by first author 's last name is available style bibliography my. A reference list typewriters had no way to present their research formatting paper... The annotations on the references list you gave examples too present their research first of its average acceleration was by! At hand detailed and effective annotated bibliography is composed of the publication manual metric system system for better understanding the! ( 2011 ) cookies and those of third party providers to deliver best!, S., Jr. ( 1980 ) is available a reference list be periodical titles and proper names a... Effective annotated bibliography is a one-stop shop for learning how to write a bibliography APA. That insert callouts and place the notes at the top of every page is an abbreviation stands. A short summary and/or critical evaluation of a bibliography with your instructor regarding their preference of using italics or.... Bibliography utilizing the automated bibliography format apa in Microsoft word personal and classroom educational use note for each reference citation below!, plans, budgets â itâs easy with Microsoft 365 Psychology, 55, 893-896 a. A lengthy URL that goes to another line after a quote on the APA 7th edition guidelines.Click for! Retrieved March 22, 1999 ) and be consistent in Microsoft word reference. T he word âannotateâ means âcritical or explanatory notesâ and the word âbibliographyâ means âa list of references of!, L. ( 2011 ) because most typewriters had no way to present their.! Was given by guides is known as the rest of your paper title and page... Apa Style® calls for a list of references instead of a bibliography out the guidelines! Their emotions, most people flunk out calls for a student paper, should! Cover letter for law firm and papers which stands for American Psychological Association to create a page! At the end of the point that the first one and use al... Jones, A.F & Wang, L. ( 2011 ) hand in a 1929 article in. Following guidelines: APA format guide the APA format references, follow same... There is more than one author, use n.d. '' ( for no date immediately!
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2021-08-01 07:50:00
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https://stats.stackexchange.com/questions/191935/what-does-the-process-that-generates-the-data-mean-and-how-does-feature-selec
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# What does “the process that generates the data” mean? and How does feature selection help in recovering it?
In [1], one of the motivations to use feature selection is stated to be: "to gain knowledge about the process that generated the data".
What does this "process" actually mean? and How does feature selection help in recovering it?
[1] Guyon, Isabelle, and André Elisseeff. "An introduction to feature extraction." Feature extraction. Springer Berlin Heidelberg, 2006. 1-25.
• +1! Very good question! If you want to read more on the issue of Statistical Models and Data Generating Processes, I recommend Chapter 4 of Davidson's 'Econometric Theory'. It is easy to understand for anyone with a background in statistics and summarizes the issue beautifully. – Jeremias K Jan 22 '16 at 11:14
A Data Generating Process is the mathematical model generating the data.
For example, if you run a regression model with regressors $X$ and dependent variable $Y$, you implicitly hypothesize a data generating process for $Y$. This data generating process can be described by the statistical model \begin{align} Y = X\beta + \varepsilon, \end{align} Where $X$ is a $1xk$ vector of random variables, $\beta \in \mathbb{R}^k$ is the $kx1$ vector of coefficients.
An example for variable selection in the case of a regression model would be where you have two sets of regressors, say $X_1$ and $X_2$ such that $X_2 \subset X_1$. Suppose that the true Data Generating Process is \begin{align} Y = X_1\beta + \varepsilon, \end{align} but that you have all regressors in $X_2$ at your disposal. Then model selection (in theory) helps you to discern the relevant regressors (i.e., $X_1$) from those that are not relevant (i.e., $X_2 \setminus X_1$). This can be done with the BIC, the AIC, or t-statistics. Note that this might affect statistical inference, see also my recent post here: Post Model Selection Inference problems - which remedies exist?
On a sidenote, the notion of a Data Generating Process is fragile. In specifying a statistical model, we impose the Axiom of correct specification. In a regression model, this happens insofar as we consider only linear combinations of the regressors we hypothesize to have an effect on $Y$. How do we know these combinations are not nonlinear? We don't! We simply have to assume it. This is why recently, a new school of statisticians operates without this axiom when doing inference. The only thing they try is to select statistical models (such as the regression model) that can approximate your true Data Generating Process well enough. To make this clearer, suppose the true Data Generating Process for our above regression model is \begin{align} Y = \sum_{i=1}^{\infty}X_i\frac{c}{i} + \varepsilon. \end{align} While there are infinitely many random variables $x_i$ that affect Y, their coefficients decay at rate $O(i)$. Hence, a good feature selection scheme would select the first $k$ to approximate the true Data Generating Process reasonably well.
Similarly, this applies to other statistical models.
The process that generated data is the "true" model. I.e., if you had perfect knowledge about the world, this would be the "equation" you'd come up with to describe the interactions and processes that cause (truly unequivocally cause) your dependent variable.
Feature selection usually involves applying some sort of "domain" knowledge (what you know about the nature of the problem, theoretically). So, for example, if you were studying some medical problem, you could use a doctor's help in trying to pick the "features" (markers, test results, measurements, diagnostic history, genetics...) that the medical science (the theory) would "blame" for the outcome. So that's how it could aid in recovering the true process - by careful feature selection, you can weed out the irrelevant variables and focus on what truly matters.
However, it does have a flip side: the theory could be wrong. And you could miss out on important variables that you wouldn't consider just because the theory of the day doesn't account for it.
(There are other approaches to feature selection obviously, from hierarchical, step-wise approaches, to factoring / PDA, to deep learning, where the features are learned by the model... )
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2020-06-06 11:50:26
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https://zbmath.org/authors/?q=ai%3Axie.xiaoping
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## Xie, Xiaoping
Compute Distance To:
Author ID: xie.xiaoping Published as: Xie, Xiaoping; Xie, XiaoPing; Xie, Xiao-ping; Xie, Xiao-Ping more...less
Documents Indexed: 103 Publications since 1993 Co-Authors: 68 Co-Authors with 99 Joint Publications 2,887 Co-Co-Authors
all top 5
### Co-Authors
2 single-authored 16 Li, Binjie 7 Du, Shaohong 7 Feng, Minfu 7 Wang, Tao 7 Yu, Guozhu 7 Zhou, Tianxiao 6 Hu, Bing 6 Wu, Yongke 6 Zhang, Shiquan 5 Chen, Gang 5 Luo, Hao 4 Chen, Yumei 4 He, Guoguang 3 Han, Yihui 3 Hu, Jinsong 3 Huang, Feiteng 3 Luo, Kun 3 Luo, Min 3 Xu, Jinchao 3 Xu, Xiaojing 3 Xu, Youcai 3 Yang, Chaochao 3 Zhu, Ping 2 Bai, Yanhong 2 Cao, Zhitong 2 Carstensen, Carsten 2 Chen, Hongping 2 Chen, Long 2 Fan, Wenwen 2 Guo, Yuanhui 2 Hu, Jiancheng 2 Wang, Yu 2 Zhang, Xu 2 Zheng, Xiaobo 1 Aihara, Kazuyuki 1 Cai, Rui 1 Cao, Rui 1 Chen, Huangxin 1 Chen, Sanping 1 Cheng, Leifeng 1 Cheng, Pan 1 Dai, Zhendong 1 Fang, Youtong 1 Han, Bo 1 Hu, Guanghui 1 Huang, Yunqing 1 Li, Hongliang 1 Li, Yang 1 Liang, Dongdong 1 Lu, Jinshu 1 Pi, Wei 1 Shatz, Sol M. 1 Sun, Shuyu 1 Wang, Desheng 1 Wang, Shaojie 1 Wang, Zhuchu 1 Weng, Xuchu 1 Wu, Xunwei 1 Xiong, Huaxing 1 Xu, Maoyuan 1 Xue, Guangri 1 Yang, Cheng 1 Yang, Yan 1 Yu, Zhengqin 1 Yuan, Hao 1 Zhang, Chensong 1 Zhang, Qihanyue 1 Zhang, Xiao 1 Zhao, Xiaohu
all top 5
### Serials
15 Journal of Sichuan University. Natural Science Edition 9 Journal of Computational Mathematics 6 Journal of Scientific Computing 6 Science China. Mathematics 5 Journal of Computational and Applied Mathematics 5 Numerical Mathematics: Theory, Methods and Applications 4 Applied Mathematics and Computation 4 Numerical Methods for Partial Differential Equations 4 Communications in Numerical Methods in Engineering 4 Advances in Applied Mathematics and Mechanics 3 Computers & Mathematics with Applications 3 Computer Methods in Applied Mechanics and Engineering 3 Numerical Mathematics 3 Computational Methods in Applied Mathematics 3 East Asian Journal on Applied Mathematics 2 Physics Letters. A 2 International Journal for Numerical Methods in Engineering 2 SIAM Journal on Numerical Analysis 2 Applied Mathematics and Mechanics. (English Edition) 2 Discrete and Continuous Dynamical Systems. Series B 2 International Journal of Numerical Analysis and Modeling 2 Communications in Computational Physics 1 IMA Journal of Numerical Analysis 1 Numerische Mathematik 1 Journal of Hangzhou University. Natural Science Edition 1 Acta Physica Sinica 1 Science in China. Series A 1 Numerical Algorithms 1 Numerical Mathematics 1 Informatica (Ljubljana) 1 Communications in Nonlinear Science and Numerical Simulation 1 Journal of Systems Science and Complexity 1 International Journal of Computational Methods 1 Advances in Mathematical Physics
all top 5
### Fields
79 Numerical analysis (65-XX) 41 Mechanics of deformable solids (74-XX) 31 Partial differential equations (35-XX) 13 Fluid mechanics (76-XX) 4 Dynamical systems and ergodic theory (37-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 Computer science (68-XX) 4 Biology and other natural sciences (92-XX) 3 Ordinary differential equations (34-XX) 3 Information and communication theory, circuits (94-XX) 2 Real functions (26-XX) 2 Systems theory; control (93-XX) 1 Integral equations (45-XX) 1 Statistics (62-XX) 1 Operations research, mathematical programming (90-XX)
### Citations contained in zbMATH Open
68 Publications have been cited 457 times in 307 Documents Cited by Year
Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. Zbl 1174.76013
Xie, Xiaoping; Xu, Jinchao; Xue, Guangri
2008
Robust globally divergence-free weak Galerkin methods for Stokes equations. Zbl 1389.76027
Chen, Gang; Feng, Minfu; Xie, Xiaoping
2016
Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method. Zbl 1388.74091
Bai, YanHong; Wu, YongKe; Xie, XiaoPing
2016
A robust WG finite element method for convection-diffusion-reaction equations. Zbl 1352.65336
Chen, Gang; Feng, Minfu; Xie, Xiaoping
2017
Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. Zbl 1419.65066
Li, Binjie; Luo, Hao; Xie, Xiaoping
2019
A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses. Zbl 1360.74134
Chen, Gang; Xie, Xiaoping
2016
Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals. Zbl 1047.74070
Xie, Xiaoping; Zhou, Tianxiao
2004
A two-level algorithm for the weak Galerkin discretization of diffusion problems. Zbl 1320.65177
Li, Binjie; Xie, Xiaoping
2015
Low order nonconforming rectangular finite element methods for Darcy-Stokes problems. Zbl 1212.65464
Zhang, Shiquan; Xie, Xiaoping; Chen, Yumei
2009
A posteriori error estimator for a weak Galerkin finite element solution of the Stokes problem. Zbl 1383.65134
Zheng, Xiaobo; Xie, Xiaoping
2017
A unified analysis for stress/strain hybrid methods of high performance. Zbl 1039.74052
Zhou, Tianxiao; Xie, Xiaoping
2002
Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Zbl 1307.65170
Wang, Li; Wu, Yongke; Xie, Xiaoping
2013
A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates. Zbl 1225.74081
Carstensen, Carsten; Xie, Xiaoping; Yu, Guozhu; Zhou, Tianxiao
2011
A time-spectral algorithm for fractional wave problems. Zbl 1407.65216
Li, Binjie; Luo, Hao; Xie, Xiaoping
2018
Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods. Zbl 1230.74205
Yu, Guozhu; Xie, Xiaoping; Carstensen, Carsten
2011
Convergence analysis of a Petrov-Galerkin method for fractional wave problems with nonsmooth data. Zbl 1477.65161
Luo, Hao; Li, Binjie; Xie, Xiaoping
2019
Weak Galerkin finite element method for Biot’s consolidation problem. Zbl 1376.65128
Chen, Yumei; Chen, Gang; Xie, Xiaoping
2018
New mixed finite elements for plane elasticity and Stokes equations. Zbl 1235.74324
Xie, XiaoPing; Xu, JinChao
2011
Analysis of a family of HDG methods for second order elliptic problems. Zbl 1382.65412
Li, Binjie; Xie, Xiaoping
2016
A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows. Zbl 1398.76131
Zheng, XiaoBo; Chen, Gang; Xie, XiaoPing
2017
Two conservative difference schemes for Rosenau-Kawahara equation. Zbl 1302.65191
Hu, Jinsong; Xu, Youcai; Hu, Bing; Xie, Xiaoping
2014
BPX preconditioner for nonstandard finite element methods for diffusion problems. Zbl 1337.65026
Li, Binjie; Xie, Xiaoping
2016
Zero energy-error mechanism of the combined hybrid method and improvement of Allman’s membrane element with drilling d.o.f.’s. Zbl 1118.74358
Zhou, Tian-Xiao; Xie, Xiao-Ping
2004
A modified nonconforming 5-node quadrilateral transition finite element. Zbl 1262.65153
Huang, Feiteng; Xie, Xiaoping
2010
An accurate hybrid macro-element with linear displacements. Zbl 1060.74066
Xie, Xiaoping
2005
Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations. Zbl 1456.65116
Li, Binjie; Wang, Tao; Xie, Xiaoping
2020
Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms. Zbl 1159.65089
Du, Shaohong; Xie, Xiaoping
2008
Accurate 4-node quadrilateral elements with a new version of energy-compatible stress mode. Zbl 1132.74046
Xie, Xiaoping; Zhou, Tianxiao
2008
Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equations with nonsmooth data. Zbl 1435.65229
Li, Binjie; Wang, Tao; Xie, Xiaoping
2020
Mixed finite element analysis for dissipative SRLW equations with damping term. Zbl 1246.65183
Xu, Youcai; Hu, Bing; Xie, Xiaoping; Hu, Jinsong
2012
Robust globally divergence-free weak Galerkin finite element methods for unsteady natural convection problems. Zbl 1463.65296
Han, Yihui; Li, Hongliang; Xie, Xiaoping
2019
$$H$$(div) conforming finite element methods for the coupled Stokes and Darcy problem. Zbl 1322.76040
Chen, Yumei; Huang, Feiteng; Xie, Xiaoping
2011
Convergence of an adaptive mixed finite element method for convection-diffusion-reaction equations. Zbl 1338.65234
Du, ShaoHong; Xie, XiaoPing
2015
Semi-discrete and fully discrete hybrid stress finite element methods for elastodynamic problems. Zbl 1363.74080
Yu, Zhengqin; Xie, Xiaoping
2015
A combined hybrid finite element method for plate bending problems. Zbl 1052.74058
Zhou, Tianxiao; Xie, Xiaoping
2003
Convergence analysis of $$V$$-cycle multigrid methods for anisotropic elliptic equations. Zbl 1258.65093
Wu, Yongke; Chen, Long; Xie, Xiaoping; Xu, Jinchao
2012
Semi-discrete and fully discrete partial projection finite element methods for the vibrating Timoshenko beam. Zbl 0951.74062
Feng, Minfu; Xie, Xiaoping; Xiong, Huaxing
1999
Analysis of the L1 scheme for fractional wave equations with nonsmooth data. Zbl 07336193
Li, Binjie; Wang, Tao; Xie, Xiaoping
2021
Uniform convergence analysis of a higher order hybrid stress quadrilateral finite element method for linear elasticity problems. Zbl 1488.65605
Bai, Yanhong; Wu, Yongke; Xie, Xiaoping
2016
An interface-unfitted finite element method for elliptic interface optimal control problems. Zbl 1449.65328
Yang, Chaochao; Wang, Tao; Xie, Xiaoping
2019
Residual-based a posteriori error estimation for multipoint flux mixed finite element methods. Zbl 1380.65344
Du, Shaohong; Sun, Shuyu; Xie, Xiaoping
2016
On residual-based a posteriori error estimators for lowest-order Raviart-Thomas element approximation to convection-diffusion-reaction equations. Zbl 1324.65138
Du, Shaohong; Xie, Xiaoping
2014
From energy improvement to accuracy enhancement: Improvement of plate bending elements by the combined hybrid method. Zbl 1061.74053
Xie, Xiaoping
2004
A space-time finite element method for fractional wave problems. Zbl 1451.65149
Li, Binjie; Luo, Hao; Xie, Xiaoping
2020
Regularity of solutions to time fractional diffusion equations. Zbl 1428.35666
Li, Binjie; Xie, Xiaoping
2019
Analysis of a two-level algorithm for HDG methods for diffusion problems. Zbl 1388.65153
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2016
A Nitsche-extended finite element method for distributed optimal control problems of elliptic interface equations. Zbl 1436.49006
Wang, Tao; Yang, Chaochao; Xie, Xiaoping
2020
Coarse-mesh-accuracy improvement of bilinear $$Q_4$$-plane element by the combined hybrid finite element method. Zbl 1145.74413
Xie, Xiaoping; Zhou, Tianxiao
2003
Energy-adjustable mechanism of the combined hybrid finite element method and improvement of Zienkiewicz’s plate-element. Zbl 1330.74112
Xie, Xiao-ping; Hu, Jian-cheng
2005
Learning-induced pattern classification in a chaotic neural network. Zbl 1255.35214
Li, Yang; Zhu, Ping; Xie, Xiaoping; He, Guoguang; Aihara, Kazuyuki
2012
Error reduction, convergence and optimality for adaptive mixed finite element methods for diffusion equations. Zbl 1274.65291
Du, Shaohong; Xie, Xiaoping
2012
Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates. Zbl 1314.74060
Guo, Yuanhui; Yu, Guozhu; Xie, Xiaoping
2013
Hybrid stress finite volume method for linear elasticity problems. Zbl 1421.74106
Wu, Yongke; Xie, Xiaoping; Chen, Long
2013
Analysis of a temporal discretization for a semilinear fractional diffusion equation. Zbl 1454.65035
Li, Binjie; Wang, Tao; Xie, Xiaoping
2020
BPS preconditioners for a weak Galerkin finite element method for 2D diffusion problems with strongly discontinuous coefficients. Zbl 1428.65089
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2018
A new smoothness result for Caputo-type fractional ordinary differential equations. Zbl 1428.34014
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2019
Robust residual- and recovery-based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems. Zbl 1418.65172
Du, Shaohong; Xie, Xiaoping; Cheng, Pan
2019
Adaptive tetrahedral mesh generation by constrained centroidal Voronoi-Delaunay tessellations for finite element methods. Zbl 1309.65144
Chen, Jie; Huang, Yunqing; Wang, Desheng; Xie, Xiaoping
2014
Parameter extension for combined hybrid finite element methods and application to plate bending problems. Zbl 1425.74287
Yu, Guozhu; Xie, Xiaoping; Zhang, Xu
2010
On choices of stress modes for lower order quadrilateral Reissner-Mindlin plate elements. Zbl 1098.74052
Hu, Guanghui; Xie, Xiaoping
2006
Robust globally divergence-free weak Galerkin finite element methods for natural convection problems. Zbl 07418048
Han, Yihui; Xie, Xiaoping
2019
An optimal embedded discontinuous Galerkin method for second-order elliptic problems. Zbl 1434.65288
Zhang, Xiao; Xie, Xiaoping; Zhang, Shiquan
2019
Sinusoidal modulation control method in a chaotic neural network. Zbl 07175088
Zhang, Qihanyue; Xie, Xiaoping; Zhu, Ping; Chen, Hongping; He, Guoguang
2014
Conservative Crank-Nicolson difference scheme for Rosenau-KdV equation. Zbl 1349.65302
Hu, Jinsong; Xie, Xiaoping; Hu, Bing; Xu, youcai
2015
Error reduction, convergence and optimality of an adaptive mixed finite element method. Zbl 1251.93059
Du, Shaohong; Xie, Xiaoping
2012
The space-time nonconforming finite element analysis for the vibration model of plane elasticity. Zbl 1265.74081
Cheng, Leifeng; Han, Bo; Xie, Xiaoping
2012
New convergence analysis for assumed stress hybrid quadrilateral finite element method. Zbl 1439.74440
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2017
Simulation and optimization of nonlinear structures on low frequency vibration and noise of lightweight car body. Zbl 07446885
Xie, X. P.; Chai, T.; Sun, Q.
2021
Analysis of the L1 scheme for fractional wave equations with nonsmooth data. Zbl 07336193
Li, Binjie; Wang, Tao; Xie, Xiaoping
2021
Simulation and optimization of nonlinear structures on low frequency vibration and noise of lightweight car body. Zbl 07446885
Xie, X. P.; Chai, T.; Sun, Q.
2021
Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations. Zbl 1456.65116
Li, Binjie; Wang, Tao; Xie, Xiaoping
2020
Analysis of a time-stepping discontinuous Galerkin method for fractional diffusion-wave equations with nonsmooth data. Zbl 1435.65229
Li, Binjie; Wang, Tao; Xie, Xiaoping
2020
A space-time finite element method for fractional wave problems. Zbl 1451.65149
Li, Binjie; Luo, Hao; Xie, Xiaoping
2020
A Nitsche-extended finite element method for distributed optimal control problems of elliptic interface equations. Zbl 1436.49006
Wang, Tao; Yang, Chaochao; Xie, Xiaoping
2020
Analysis of a temporal discretization for a semilinear fractional diffusion equation. Zbl 1454.65035
Li, Binjie; Wang, Tao; Xie, Xiaoping
2020
Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. Zbl 1419.65066
Li, Binjie; Luo, Hao; Xie, Xiaoping
2019
Convergence analysis of a Petrov-Galerkin method for fractional wave problems with nonsmooth data. Zbl 1477.65161
Luo, Hao; Li, Binjie; Xie, Xiaoping
2019
Robust globally divergence-free weak Galerkin finite element methods for unsteady natural convection problems. Zbl 1463.65296
Han, Yihui; Li, Hongliang; Xie, Xiaoping
2019
An interface-unfitted finite element method for elliptic interface optimal control problems. Zbl 1449.65328
Yang, Chaochao; Wang, Tao; Xie, Xiaoping
2019
Regularity of solutions to time fractional diffusion equations. Zbl 1428.35666
Li, Binjie; Xie, Xiaoping
2019
A new smoothness result for Caputo-type fractional ordinary differential equations. Zbl 1428.34014
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2019
Robust residual- and recovery-based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems. Zbl 1418.65172
Du, Shaohong; Xie, Xiaoping; Cheng, Pan
2019
Robust globally divergence-free weak Galerkin finite element methods for natural convection problems. Zbl 07418048
Han, Yihui; Xie, Xiaoping
2019
An optimal embedded discontinuous Galerkin method for second-order elliptic problems. Zbl 1434.65288
Zhang, Xiao; Xie, Xiaoping; Zhang, Shiquan
2019
A time-spectral algorithm for fractional wave problems. Zbl 1407.65216
Li, Binjie; Luo, Hao; Xie, Xiaoping
2018
Weak Galerkin finite element method for Biot’s consolidation problem. Zbl 1376.65128
Chen, Yumei; Chen, Gang; Xie, Xiaoping
2018
BPS preconditioners for a weak Galerkin finite element method for 2D diffusion problems with strongly discontinuous coefficients. Zbl 1428.65089
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2018
A robust WG finite element method for convection-diffusion-reaction equations. Zbl 1352.65336
Chen, Gang; Feng, Minfu; Xie, Xiaoping
2017
A posteriori error estimator for a weak Galerkin finite element solution of the Stokes problem. Zbl 1383.65134
Zheng, Xiaobo; Xie, Xiaoping
2017
A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows. Zbl 1398.76131
Zheng, XiaoBo; Chen, Gang; Xie, XiaoPing
2017
New convergence analysis for assumed stress hybrid quadrilateral finite element method. Zbl 1439.74440
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2017
Robust globally divergence-free weak Galerkin methods for Stokes equations. Zbl 1389.76027
Chen, Gang; Feng, Minfu; Xie, Xiaoping
2016
Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method. Zbl 1388.74091
Bai, YanHong; Wu, YongKe; Xie, XiaoPing
2016
A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses. Zbl 1360.74134
Chen, Gang; Xie, Xiaoping
2016
Analysis of a family of HDG methods for second order elliptic problems. Zbl 1382.65412
Li, Binjie; Xie, Xiaoping
2016
BPX preconditioner for nonstandard finite element methods for diffusion problems. Zbl 1337.65026
Li, Binjie; Xie, Xiaoping
2016
Uniform convergence analysis of a higher order hybrid stress quadrilateral finite element method for linear elasticity problems. Zbl 1488.65605
Bai, Yanhong; Wu, Yongke; Xie, Xiaoping
2016
Residual-based a posteriori error estimation for multipoint flux mixed finite element methods. Zbl 1380.65344
Du, Shaohong; Sun, Shuyu; Xie, Xiaoping
2016
Analysis of a two-level algorithm for HDG methods for diffusion problems. Zbl 1388.65153
Li, Binjie; Xie, Xiaoping; Zhang, Shiquan
2016
A two-level algorithm for the weak Galerkin discretization of diffusion problems. Zbl 1320.65177
Li, Binjie; Xie, Xiaoping
2015
Convergence of an adaptive mixed finite element method for convection-diffusion-reaction equations. Zbl 1338.65234
Du, ShaoHong; Xie, XiaoPing
2015
Semi-discrete and fully discrete hybrid stress finite element methods for elastodynamic problems. Zbl 1363.74080
Yu, Zhengqin; Xie, Xiaoping
2015
Conservative Crank-Nicolson difference scheme for Rosenau-KdV equation. Zbl 1349.65302
Hu, Jinsong; Xie, Xiaoping; Hu, Bing; Xu, youcai
2015
Two conservative difference schemes for Rosenau-Kawahara equation. Zbl 1302.65191
Hu, Jinsong; Xu, Youcai; Hu, Bing; Xie, Xiaoping
2014
On residual-based a posteriori error estimators for lowest-order Raviart-Thomas element approximation to convection-diffusion-reaction equations. Zbl 1324.65138
Du, Shaohong; Xie, Xiaoping
2014
Adaptive tetrahedral mesh generation by constrained centroidal Voronoi-Delaunay tessellations for finite element methods. Zbl 1309.65144
Chen, Jie; Huang, Yunqing; Wang, Desheng; Xie, Xiaoping
2014
Sinusoidal modulation control method in a chaotic neural network. Zbl 07175088
Zhang, Qihanyue; Xie, Xiaoping; Zhu, Ping; Chen, Hongping; He, Guoguang
2014
Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Zbl 1307.65170
Wang, Li; Wu, Yongke; Xie, Xiaoping
2013
Uniform analysis of a stabilized hybrid finite element method for Reissner-Mindlin plates. Zbl 1314.74060
Guo, Yuanhui; Yu, Guozhu; Xie, Xiaoping
2013
Hybrid stress finite volume method for linear elasticity problems. Zbl 1421.74106
Wu, Yongke; Xie, Xiaoping; Chen, Long
2013
Mixed finite element analysis for dissipative SRLW equations with damping term. Zbl 1246.65183
Xu, Youcai; Hu, Bing; Xie, Xiaoping; Hu, Jinsong
2012
Convergence analysis of $$V$$-cycle multigrid methods for anisotropic elliptic equations. Zbl 1258.65093
Wu, Yongke; Chen, Long; Xie, Xiaoping; Xu, Jinchao
2012
Learning-induced pattern classification in a chaotic neural network. Zbl 1255.35214
Li, Yang; Zhu, Ping; Xie, Xiaoping; He, Guoguang; Aihara, Kazuyuki
2012
Error reduction, convergence and optimality for adaptive mixed finite element methods for diffusion equations. Zbl 1274.65291
Du, Shaohong; Xie, Xiaoping
2012
Error reduction, convergence and optimality of an adaptive mixed finite element method. Zbl 1251.93059
Du, Shaohong; Xie, Xiaoping
2012
The space-time nonconforming finite element analysis for the vibration model of plane elasticity. Zbl 1265.74081
Cheng, Leifeng; Han, Bo; Xie, Xiaoping
2012
A priori and a posteriori analysis for a locking-free low order quadrilateral hybrid finite element for Reissner-Mindlin plates. Zbl 1225.74081
Carstensen, Carsten; Xie, Xiaoping; Yu, Guozhu; Zhou, Tianxiao
2011
Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods. Zbl 1230.74205
Yu, Guozhu; Xie, Xiaoping; Carstensen, Carsten
2011
New mixed finite elements for plane elasticity and Stokes equations. Zbl 1235.74324
Xie, XiaoPing; Xu, JinChao
2011
$$H$$(div) conforming finite element methods for the coupled Stokes and Darcy problem. Zbl 1322.76040
Chen, Yumei; Huang, Feiteng; Xie, Xiaoping
2011
A modified nonconforming 5-node quadrilateral transition finite element. Zbl 1262.65153
Huang, Feiteng; Xie, Xiaoping
2010
Parameter extension for combined hybrid finite element methods and application to plate bending problems. Zbl 1425.74287
Yu, Guozhu; Xie, Xiaoping; Zhang, Xu
2010
Low order nonconforming rectangular finite element methods for Darcy-Stokes problems. Zbl 1212.65464
Zhang, Shiquan; Xie, Xiaoping; Chen, Yumei
2009
Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. Zbl 1174.76013
Xie, Xiaoping; Xu, Jinchao; Xue, Guangri
2008
Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms. Zbl 1159.65089
Du, Shaohong; Xie, Xiaoping
2008
Accurate 4-node quadrilateral elements with a new version of energy-compatible stress mode. Zbl 1132.74046
Xie, Xiaoping; Zhou, Tianxiao
2008
On choices of stress modes for lower order quadrilateral Reissner-Mindlin plate elements. Zbl 1098.74052
Hu, Guanghui; Xie, Xiaoping
2006
An accurate hybrid macro-element with linear displacements. Zbl 1060.74066
Xie, Xiaoping
2005
Energy-adjustable mechanism of the combined hybrid finite element method and improvement of Zienkiewicz’s plate-element. Zbl 1330.74112
Xie, Xiao-ping; Hu, Jian-cheng
2005
Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals. Zbl 1047.74070
Xie, Xiaoping; Zhou, Tianxiao
2004
Zero energy-error mechanism of the combined hybrid method and improvement of Allman’s membrane element with drilling d.o.f.’s. Zbl 1118.74358
Zhou, Tian-Xiao; Xie, Xiao-Ping
2004
From energy improvement to accuracy enhancement: Improvement of plate bending elements by the combined hybrid method. Zbl 1061.74053
Xie, Xiaoping
2004
A combined hybrid finite element method for plate bending problems. Zbl 1052.74058
Zhou, Tianxiao; Xie, Xiaoping
2003
Coarse-mesh-accuracy improvement of bilinear $$Q_4$$-plane element by the combined hybrid finite element method. Zbl 1145.74413
Xie, Xiaoping; Zhou, Tianxiao
2003
A unified analysis for stress/strain hybrid methods of high performance. Zbl 1039.74052
Zhou, Tianxiao; Xie, Xiaoping
2002
Semi-discrete and fully discrete partial projection finite element methods for the vibrating Timoshenko beam. Zbl 0951.74062
Feng, Minfu; Xie, Xiaoping; Xiong, Huaxing
1999
all top 5
### Cited by 473 Authors
39 Xie, Xiaoping 12 Li, Binjie 11 Feng, Minfu 9 Zhang, Shangyou 8 Chen, Ching-Shyang 8 Chen, Gang 8 Gatica, Gabriel N. 7 Chen, Shaochun 7 Du, Shaohong 7 Ye, Xiu 7 Yu, Guozhu 7 Zhao, Jikun 6 Chen, Long 6 Zhai, Qilong 6 Zhang, Tie 5 Carstensen, Carsten 5 Dehghan Takht Fooladi, Mehdi 5 Gharibi, Zeinab 5 Hu, Jun 5 Huang, Xuehai 5 Oyarzúa, Ricardo 5 Wang, Junping 5 Xu, Jinchao 5 Zhang, Ran 5 Zhang, Shiquan 5 Zhang, Yangwen 5 Zheng, Xiangcheng 5 Zhou, Tianxiao 4 Dangal, Thir 4 He, Xiaoming 4 Lin, Runchang 4 Luo, Hao 4 Meng, Zhaoliang 4 Mu, Lin 4 Neilan, Michael 4 Nie, Yufeng 4 Shi, Zhongci 4 Singler, John R. 4 Wang, Hong 4 Wang, Tao 4 Yang, Xiaofeng 4 Zhang, Bei 4 Zhang, Zhimin 3 Chen, Jinru 3 Chen, Yanping 3 Fu, Guosheng 3 Gunzburger, Max D. 3 Han, Yihui 3 He, Dongdong 3 Huang, Peiqi 3 Huang, Yunqing 3 Jia, Jinhong 3 Karageorghis, Andreas 3 Li, Fule 3 Lin, Tao 3 Liu, Gui-Rong 3 Luo, Zhongxuan 3 Nguyen-Thoi, Trung 3 Ruiz-Baier, Ricardo 3 Sayas, Francisco-Javier 3 Wang, Ruishu 3 Wang, Xiaoshen 3 Wu, Yongke 3 Xie, Hehu 3 Xu, Shipeng 3 Yang, Yan 3 Yotov, Ivan 3 Zhang, Baiju 3 Zhao, Jia 3 Zheng, Hui 3 Zheng, Xiaobo 3 Zhou, Xinchen 2 Abbaszadeh, Mostafa 2 Bai, Yanhong 2 Bösing, Paulo Rafael 2 Braack, Malte 2 Cai, Mingchao 2 Cao, Yanzhao 2 Chang, Wanru 2 Chen, Hongru 2 Chen, Huangxin 2 Chen, Yumei 2 Chen, Zhangxin 2 Dangskul, Supreedee 2 Dong, Lina 2 Dou, Fangfang 2 Fan, Xin 2 Gao, Yali 2 Guo, Yuling 2 Hong, Qingguo 2 Hu, Weiwei 2 Huang, Jianguo 2 Jia, Shanghui 2 John, Volker 2 Lam, Khin-Yong 2 Layton, William J. 2 Li, Dongfang 2 Li, Guanrong 2 Li, Shuguang 2 Liu, Xiaoyan ...and 373 more Authors
all top 5
### Cited in 70 Serials
31 Journal of Computational and Applied Mathematics 30 Journal of Scientific Computing 18 Computer Methods in Applied Mechanics and Engineering 18 Applied Numerical Mathematics 16 Computers & Mathematics with Applications 15 Applied Mathematics and Computation 12 Numerische Mathematik 11 SIAM Journal on Numerical Analysis 11 Advances in Applied Mathematics and Mechanics 9 Communications in Computational Physics 9 Science China. Mathematics 8 Mathematics of Computation 8 Advances in Computational Mathematics 6 Numerical Methods for Partial Differential Equations 5 Communications in Numerical Methods in Engineering 5 Engineering Analysis with Boundary Elements 5 Mathematical Problems in Engineering 5 Computational Methods in Applied Mathematics 4 Journal of Computational Physics 4 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 3 Journal of Mathematical Analysis and Applications 3 Mathematics and Computers in Simulation 3 Journal of Computational Mathematics 3 Computational Mechanics 3 International Journal of Computer Mathematics 3 Computational Geosciences 3 Discrete and Continuous Dynamical Systems. Series B 3 International Journal of Computational Methods 3 Numerical Mathematics: Theory, Methods and Applications 3 East Asian Journal on Applied Mathematics 2 Applicable Analysis 2 Acta Mathematicae Applicatae Sinica. English Series 2 Numerical Algorithms 2 Computational and Applied Mathematics 2 Nonlinear Dynamics 2 Journal of Applied Mathematics and Computing 1 Mathematical Methods in the Applied Sciences 1 Physics Letters. A 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 BIT 1 Calcolo 1 International Journal for Numerical Methods in Engineering 1 Numerical Functional Analysis and Optimization 1 Mathematica Numerica Sinica 1 Applied Mathematics and Mechanics. (English Edition) 1 Applied Mathematics Letters 1 Neural Networks 1 Japan Journal of Industrial and Applied Mathematics 1 Applications of Mathematics 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 SIAM Review 1 Continuum Mechanics and Thermodynamics 1 SIAM Journal on Scientific Computing 1 Numerical Linear Algebra with Applications 1 ETNA. Electronic Transactions on Numerical Analysis 1 Discrete Dynamics in Nature and Society 1 European Journal of Mechanics. A. Solids 1 European Journal of Mechanics. B. Fluids 1 Lobachevskii Journal of Mathematics 1 Journal of Systems Science and Complexity 1 Journal of Applied Mathematics 1 Acta Mathematica Scientia. Series B. (English Edition) 1 Thai Journal of Mathematics 1 Frontiers of Mathematics in China 1 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 International Journal of Numerical Methods and Applications 1 ISRN Computational Mathematics 1 Evolution Equations and Control Theory 1 AIMS Mathematics 1 SMAI Journal of Computational Mathematics
all top 5
### Cited in 22 Fields
275 Numerical analysis (65-XX) 115 Partial differential equations (35-XX) 92 Fluid mechanics (76-XX) 77 Mechanics of deformable solids (74-XX) 13 Calculus of variations and optimal control; optimization (49-XX) 12 Real functions (26-XX) 5 Statistical mechanics, structure of matter (82-XX) 4 Probability theory and stochastic processes (60-XX) 3 Potential theory (31-XX) 3 Ordinary differential equations (34-XX) 3 Systems theory; control (93-XX) 2 Operator theory (47-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Computer science (68-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Operations research, mathematical programming (90-XX) 1 General and overarching topics; collections (00-XX) 1 Number theory (11-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Integral equations (45-XX) 1 Statistics (62-XX) 1 Geophysics (86-XX)
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2022-09-26 06:15:36
|
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https://gogul09.github.io/software/deep-learning-windows
|
# Deep Learning Environment Setup for Windows
Environment Setup | 05 February 2017
In this blog post, we will setup the environment for Deep Learning in Windows 10. At the end of this post, we will have a machine that is ready to run latest Deep Learning libraries such as Theano, TensorFlow and Keras. We will also learn how to enable GPU to speed up training Deep Neural Networks using CUDA. Let’s jump right in!
### Hardware
All the following steps are performed and checked with my hardware. My hardware specs are given below.
• CPU: Intel i7-7500U CPU @ 2.70GHz, 16 GB RAM
• OS: Windows 10 (64-bit)
• GPU: NVIDIA GeForce 940MX, 4 GB RAM
### Directory structure
Let’s start with the directory structure which we will follow in this entire post. This is a mandatory step else it becomes more difficult later. Create a folder named “deeplearning” in C drive.
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C:\deeplearning
This is the master folder inside which we will keep all the dependencies listed above except Visual Studio 2015 Community Edition.
### Environment variables
Before getting into installing dependencies, please become familiar with “Environment variables” and “path”. If you are not familiar with these two terms in Windows, it is highly recommended to read this. Else go to the next section.
• Go to “This PC” (or Computer) -> Right click it -> Select “Properties”.
• Select “Advanced System Settings”
• Select “Environment Variables”
• After that you will see a window having “User Variables” and “System Variables”.
• We will be highly using the “System Variables” for this post. Before getting into the next step, have a look at this System Variables and find “path” variable in that.
• We can create a new system variable, edit it, assign a value to it (normally a path) and delete it.
• We will be using “System Variables” a lot in this post. So, please become familiar with this before getting further.
The following are the software, libraries and tools needed to setup Deep Learning environment in Windows 10 (64-bit).
### Visual Studio 2015 Community Edition
• Go to this website and click on Download”. You will be taken to the page as shown below.
Update: At the time of writing this post, the above URL worked. In case, if it doesn't, check out this link.
• Check “vs_community.exe” and choose next. The download will start. After download finishes, run the setup and choose the config settings as shown below.
#### Adding environment variables
• Add “C:\Program Files (x86)\Microsoft Visual Studio 14.0\VC\bin” to “path” in System Variables.
• Create a system variable named “INCLUDE” with the value “C:\Program Files (x86)\Windows Kits\10\Include\10.0.10240.0\ucrt”
• Create a system variable named “LIB” with the value “C:\Program Files (x86)\Windows Kits\10\Lib\10.0.10240.0\um\x64;C:\Program Files (x86)\Windows Kits\10\Lib\10.0.10240.0\ucrt\x64”
### Anaconda (64-bit)
I have installed Anaconda 64-bit installer to get Python 2.7. This worked without any error!
• Go to this website and select “Anaconda 64-bit installer” under Python 2.7.
Update (13/12/2017): If you need TensorFlow as the backend for Keras in Windows 10, you can now do that by installing Anaconda Python 3.6 (64-bit) installer. Choose Python 3.6 64-bit Graphical Installer from that link and download it.
• Run the setup and follow the installation.
• Choose the installation directory as -
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>>> C:\deeplearning\anaconda
• In the next screen, check the two options to add anaconda and python 2.7 to “path”.
#### Adding environment variables
• Create a system variable named “PYTHON_HOME” with the value “C:\deeplearning\anaconda”
• Add the following in “path” under System Variables. (Add all these one by one by double clicking the value of “path” and clicking “New”)
• %PYTHON_HOME%
• %PYTHON_HOME%\Scripts
• %PYTHON_HOME%\Library\bin
• After adding the variables in path, open up a command prompt as administrator and execute the following command.
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>>> conda install libpython
### CUDA 8.0 (64-bit)
To enable GPU to speed up training neural networks for Deep Learning, we need to install CUDA.
• Run the setup and choose the installation directory as -
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>>> C:\deeplearning\cuda
• After completing the installation, the installer automatically creates “CUDA_PATH” in System Variables. Check this, else you need to add it for sure.
• In case if you don’t find CUDA related environment variables, follow the steps below -
#### Adding environment variables
• Create a system variable named “CUDA_PATH” with the value “C:\deeplearning\cuda”
• Add the following in “path” under System Variables.
• %CUDA_PATH%\libnvvp
• %CUDA_PATH%\bin
### MinGW-w64 (5.4.0)
• Go to this website and download the “mingw-w64-install.exe”.
• After downloading, run the setup and choose the installation directory as -
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>>> C:\deeplearning\mingw-w64-5.4.0
• Choose the config options as shown below -
#### Adding environment variables
• Create a system variable named “MINGW_HOME” with the value “C:\deeplearning\mingw-w64-5.4.0”
• Add the following in “path” under System Variables.
• %MINGW_HOME%\mingw64\bin
### Theano
We will be installing Theano 0.8.2 using git from our command prompt.
• Create a folder named “theano-0.8.2” in “C:\deeplearning”.
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>>> C:\deeplearning\theano-0.8.2
• Open command prompt as administrator and type the following -
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>>> C:\deeplearning\
>>> git clone https://github.com/Theano/Theano.git theano-0.8.2 --branch rel-0.8.2
• Wait for the repository to get cloned and checked out.
• Enter into the theano folder and install it using the following commands
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>>> cd C:\deeplearning\theano-0.8.2
>>> python setup.py install --record installed_files.txt
### OpenBLAS 0.2.14
• Go to this website to download “OpenBLAS” which is needed to perform parallel computation by running both CPU and GPU together. For example, data augmentation could be performed by CPU while the GPU could be used to speed up training the neural network.
• Download the .zip file and extract the files to the folder -
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>>> C:\deeplearning\openblas-0.2.14-int32
#### Adding environment variables
• Create a system variable named “OPENBLAS_HOME” with the value “C:\deeplearning\openblas-0.2.14-int32”
• Add the following in “path” under System Variables.
• %OPENBLAS_HOME%\bin
### Enabling CPU or GPU
To switch between CPU and GPU, we can do a simple trick. Theano uses the processing unit by checking the variable “THEANO_FLAGS” in “path” under System Variables.
• To use CPU, copy-paste the following as the value to “THEANO_FLAGS” in “path”.
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floatX=float32,device=cpu,lib.cnmem=0.8,blas.ldflags=-LC:/deeplearning/openblas-0.2.14-int32/bin -lopenblas
• To use GPU, copy-paste the following as the value to “THEANO_FLAGS” in “path”.
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floatX=float32,device=gpu,dnn.enabled=False,lib.cnmem=0.8,blas.ldflags=-LC:/deeplearning/openblas-0.2.14-int32/bin -lopenblas
### OpenCV 3.1.0
For Python 2.7
Forget searching Google like “How to install OpenCV in Windows 10”. You will find more complicated procedures and techniques to install it, which fails the most. The simplest solution is to use Anaconda. This is the 100% working solution in my case. I tried a lot fighting with CMAKE, git, compiler issues etc.. and found this as the best solution. In case if you find any other method, kindly let me know.
• Open up a command prompt and enter the following command.
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>>> conda install -c https://conda.anaconda.org/menpo opencv3
For Python 3.6
• Open up a command prompt and enter the following command.
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>>> pip install opencv-python
>>> pip install opencv-contrib-python
After installing it, type the following command.
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>>> python
>>> import cv2
>>> cv2.__version__
You should get something like this.
### Intermediate check
If you followed this post carefully till here, you should end up with your “System Variables” as shown below.
### Keras
Update (13/12/2017): You can now install Keras and TensorFlow using pip in Windows 10, if you have installed Python 3.6 from Anaconda website. It is as simple as two pip commands given below.
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>>> pip install tensorflow
>>> pip install keras
For Python 2.7
We will be installing Keras 1.1.0 using git from our command prompt.
• Create a folder named “keras-1.1.0” in “C:\deeplearning”.
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>>> C:\deeplearning\keras-1.1.0
• Open command prompt as administrator and type the following -
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>>> C:\deeplearning\
>>> git clone https://github.com/fchollet/keras.git keras-1.1.0 --branch 1.1.0
• Wait for the repository to get cloned and checked out.
• Enter into the keras folder and install it using the following commands.
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>>> cd C:\deeplearning\keras-1.1.0
>>> python setup.py install --record installed_files.txt
Since, we are using Theano, we don’t need TensorFlow as the backend for Keras. But Keras comes in default with TensorFlow as its backend. To change this, navigate to the folder -
1
Here, username is your name. You will find “keras.json” file. Open it and change it like this.
Update (13/12/2017): If you want to use TensorFlow as the backend for Keras, you need to change "image_dim_ordering" to "tf" and "backend" to "tensorflow".
### Final check
If you followed all the steps correctly till here, you can execute the following final check to verify the installation of all the dependencies.
Note: I have installed CUDA 8.0 in a different folder. But it still worked. Make sure you assign "CUDA_PATH" variable in System Variables to the correct folder where it got installed. If it was automatically mapped to the correct path, leave it as it is.
• Open up a text editor and copy-paste the following code.
• Save it in “Desktop” as “test.py”.
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from theano import function, config, shared, sandbox
import theano.tensor as T
import numpy
import time
vlen = 10 * 30 * 768 # 10 x #cores x # threads per core
iters = 1000
rng = numpy.random.RandomState(22)
x = shared(numpy.asarray(rng.rand(vlen), config.floatX))
f = function([], T.exp(x))
print(f.maker.fgraph.toposort())
t0 = time.time()
for i in range(iters):
r = f()
t1 = time.time()
print("Looping %d times took %f seconds" % (iters, t1 - t0))
print("Result is %s" % (r,))
if numpy.any([isinstance(x.op, T.Elemwise) for x in f.maker.fgraph.toposort()]):
print('Used the cpu')
else:
print('Used the gpu')
If you want to use CPU -
• Open the “Environment Variables” and go to “System Variables” -> “THEANO_FLAGS”
• Double click the value of “THEANO_FLAGS” and change it to -
1
floatX=float32,device=cpu,lib.cnmem=0.8,blas.ldflags=-LC:/deeplearning/openblas-0.2.14-int32/bin -lopenblas
If you want to use GPU -
• Open the “Environment Variables” and go to “System Variables” -> “THEANO_FLAGS”
• Double click the value of “THEANO_FLAGS” and change it to -
1
floatX=float32,device=gpu,dnn.enabled=False,lib.cnmem=0.8,blas.ldflags=-LC:/deeplearning/openblas-0.2.14-int32/bin -lopenblas
• Open the command prompt and navigate to the “Desktop”. * Run the following command.
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>>> python test.py
• I have enabled GPU and I got the following result.
• In case if you meet any error, go to -
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• Select all folders, delete the contents and try again. It will work. Else, restart the system once and try again.
### 11. Installing additional libraries
If you installed pip for windows, it is much simpler to install some additional packages needed for development. Some are listed below. Open up command prompt as administrator and enter the commands listed.
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>>> pip install scikit-learn
>>> pip install mahotas
>>> pip install jupyter
>>> pip install imutils
>>> pip install librosa
### Installing cuDNN
• Go to this website, register and download cuDNN for CUDA 8.0 and Windows 10.
• Copy the contents from each of the folder (bin, include, lib) inside cuDNN folder and paste it in CUDA path having (bin, include, lib) respectively. That is -
• Copy the contents inside “C:\Users<username>\Downloads\cuDNN\bin” folder to “C:\Program Files\NVIDIA GPU Computing Toolkit\CUDA\v8.0\bin”
• Copy the contents inside “C:\Users<username>\Downloads\cuDNN\include” folder to “C:\Program Files\NVIDIA GPU Computing Toolkit\CUDA\v8.0\include”
• Copy the contents inside “C:\Users<username>\Downloads\cuDNN\lib\x64” folder to “C:\Program Files\NVIDIA GPU Computing Toolkit\CUDA\v8.0\lib\x64”
• After performing the above steps, if you want to use cuDNN to speed up your Deep Learning process, you must change the THEANO_FLAGS as shown below.
If you want to use cuDNN and GPU-
• Open the “Environment Variables” and go to “System Variables” -> “THEANO_FLAGS”
• Double click the value of “THEANO_FLAGS” and change it to -
1
floatX=float32,device=gpu,optimizer_including=cudnn,lib.cnmem=0.8,dnn.conv.algo_bwd_filter=deterministic,dnn.conv.algo_bwd_data=deterministic,blas.ldflags=-LC:/deeplearning/openblas-0.2.14-int32/bin -lopenblas
Once you have updated the path, open up command prompt and type the following.
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>>> python
>>> import theano
You should get something like this.
That’s it. You are all set to begin your journey in Deep Learning using Windows 10. Feel free to share this link to someone who is struggling to setup environment for Deep Learning and Computer Vision in Windows 10.
In case if you found something useful to add to this article or you found a bug in the code or would like to improve some points mentioned, feel free to write it down in the comments. Hope you found something useful here.
Happy learning!
|
2019-03-18 13:39:37
|
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|
http://cdsweb.cern.ch/collection/EN%20Notes?ln=fr&as=1
|
# EN Notes
Derniers ajouts:
2019-08-23
19:53
Generation of a database of differential cross sections for nuclear elastic scattering of $\alpha$ particles on nuclei / Barjuan I Ballabriga, Laia The general-purpose code for the simulation of radiation transport FLUKA accounts for the elastic scattering of charged projectiles on the screened electrostatic potential of target atoms. [...] CERN-STUDENTS-Note-2019-120. - 2019. Access to fulltext
2019-08-23
19:51
Generating an Artificial Spill Signal through the use of FPGAs / Camilleri, Luke (University of Malta (MT)) The COMPASS system architecture, DAQs and trigger systems all use the SPS spill signal for synchronization, event building, and configuration or monitoring, enabling accurate and precise data acquisition of useful event collisions [...] CERN-STUDENTS-Note-2019-119. - 2019. Full text
2019-08-23
10:54
Benchmarking Nuclear Reaction Models of the FLUKA Monte Carlo Code for Heavy Ion Collisions / Holm, Emil Brinch (Aarhus University) In this report, I summarize the work conducted during my eight weeks as a summer student in the EN-STI-FLU department of CERN. [...] CERN-STUDENTS-Note-2019-104. - 2019. Access to fulltext
2019-08-21
14:37
Python analysis at the UA9 experiment / Kokkeler, Tim Herman The crystal analysis for the UA9-experiment has been transferred from a C++/ROOT-environment to a Python-environment. [...] CERN-STUDENTS-Note-2019-088. - 2019. Full text
2019-08-21
14:33
Python data analysis at the UA9 experiment / Kokkeler, Tim Herman The crystal analysis for the UA9-experiment has been transferred from a C++/ROOT-environment to a Python-environment. [...] CERN-STUDENTS-Note-2019-087. - 2019.
2019-08-15
15:20
An Analysis of a Potential Compact Positron Beam Source / Hessami, Rafi Mir-Ali ; Gessner, Spencer Jake (CERN) For positron studies in plasma wakefield accelerators such as AWAKE, the development of new, cheaper, and compact positron beam sources is necessary. [...] CERN-STUDENTS-Note-2019-053. - 2019. Access to fulltext
2019-08-15
15:02
Upgrade of the reference area of the CERN-ISOLDE Resonance Ionization Laser Ion Source / Tsangari, Stavrini The Resonance Ionization Laser Ion Source (RILIS) is currently the most frequently used ion source at ISOLDE, thick target ISOL facility. [...] CERN-STUDENTS-Note-2019-052. - 2019. Access to fulltext
2019-08-09
14:07
SEPARATION DEVICE FOR MOLECULAR ISOTOPES AT MEDICIS / Serikbayeva, Gulfairuz Radioisotopes are widely used in medicine for diagnosis and treatment of cancer as well as other diseases [...] CERN-STUDENTS-Note-2019-039. - 2019. Access to fulltext
2018-09-29
00:44
A Chromatographic Method to Separate Sc(III) from Zn(II) Ions: A Step in the Purification of Sc-44 (an isotope of medical interest) / Ruiz Quiros, Adolfo CERN MEDICIS produces radioactive isotopes by recovering the 1,4 GeV proton beam from ISOLDE before it reaches the beam dump using different types of targets behind the ISOLDE targets. [...] CERN-STUDENTS-Note-2018-172. - 2018. Access to fulltext
2018-09-20
14:28
Characterisation of the bent silicon crystals and study of the inelastic nuclear interactions of 180 GeV/c pions in bent crystals at the UA9 experiment / Zhovkovska, Valeriia (Centre National de la Recherche Scientifique (FR)) Application of the bent crystals for particle beam collimation was proposed as an alternative way for the Large Hadron Collider collimation system. [...] CERN-STUDENTS-Note-2018-148. - 2018. Access to fulltext
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2019-08-24 18:33:59
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http://umj.imath.kiev.ua/authors/name/?lang=en&author_id=3343
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2019
Том 71
№ 5
# Khan M. S.
Articles: 2
Article (English)
### Existence of the Category $DTC_2 (K)$ Equivalent to the Given Category $KAC_2$
Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1122–1133
For a given category $KAC_2$ , the present paper deals with the existence problem for the category $DTC_2 (K)$, which is equivalent to $KAC_2$ , where $DTC_2 (K)$ is the category whose objects are simple closed $K$-curves with even number $l$ of elements in $Z^n,\; l ≠ 6$, and morphisms are (digitally) $K$-continuous maps, and $KAC_2$ is a category whose objects are simple closed $A$-curves and morphisms are $A$-maps. To address this issue, the paper starts from the category denoted by $KAC_1$ whose objects are connected $nD$ Khalimsky topological subspaces with Khalimsky adjacency and morphisms are $A$-maps in [S. E. Han and A. Sostak, Comput. Appl. Math., 32, 521–536 (2013)]. Based on this approach, in $KAC_1$ the paper proposes the notions of $A$-homotopy and $A$-homotopy equivalence and classifies the spaces in $KAC_1$ or $KAC_2$ in terms of the $A$-homotopy equivalence. Finally, the paper proves that, for Sa given category $KAC_2$, there is $DTC_2 (K)$, which is equivalent to $KAC_2$.
Article (English)
### Common Fixed-Point Theorems for Nonlinear Weakly Contractive Mappings
Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 531–537
Some common fixed-point results for mappings satisfying a nonlinear weak contraction condition within the framework of ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known from the literature.
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2019-05-25 08:49:51
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https://www.groundai.com/project/efficient-sketching-algorithm-for-sparse-binary-data/
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Efficient Sketching Algorithm for Sparse Binary Data
# Efficient Sketching Algorithm for Sparse Binary Data
1st Rameshwar Pratap School of Computing and Electrical Engineering
IIT Mandi, H.P.
India
rameshwar@iitmandi.ac.in
2nd Debajyoti Bera Department of Computer Science
IIIT Delhi
India
dbera@iiitd.ac.in
3rd Karthik Revanuru NALT Analytics
Bangalore, India
India
karthikrvnr@gmail.com
###### Abstract
Recent advancement of the WWW, IOT, social network, e-commerce, etc. have generated a large volume of data. These datasets are mostly represented by high dimensional and sparse datasets. Many fundamental subroutines of common data analytic tasks such as clustering, classification, ranking, nearest neighbour search, etc. scale poorly with the dimension of the dataset. In this work, we address this problem and propose a sketching (alternatively, dimensionality reduction) algorithm – (Binary Data Sketch) – for sparse binary datasets. preserves the binary version of the dataset after sketching and maintains estimates for multiple similarity measures such as Jaccard, Cosine, Inner-Product similarities, and Hamming distance, on the same sketch. We present a theoretical analysis of our algorithm and complement it with extensive experimentation on several real-world datasets. We compare the performance of our algorithm with the state-of-the-art algorithms on the task of mean-square-error and ranking. Our proposed algorithm offers a comparable accuracy while suggesting a significant speedup in the dimensionality reduction time, with respect to the other candidate algorithms. Our proposal is simple, easy to implement, and therefore can be adopted in practice. 111A preliminary version of this paper has been accepted at IEEE-ICDM, 2019.
## I Introduction
Due to technological advancements, recent years have witnessed a dramatic increase in our ability to collect data from various sources like WWW, IOT, social media platforms, mobile applications, finance, and biology. For example, in many web applications, the volume of datasets are of the terascale order, with trillions of features [1]. The high dimensionality incurs high memory requirements and computational cost during the training. Further, most of such high dimensional datasets are sparse, owing to a wide adaption of “Bag-of-words” (BoW) representations. For example: in the case of document representation, word frequency within a document follows power law – most of the words occur rarely in a document, and higher order shingles occur only once. We focus on the binary representation of the datasets which is quite common in several applications [26, 17].
Measuring similarity score of data points under various similarity measures is a fundamental subroutine in several applications such as clustering, classification, identifying nearest neighbors, ranking, and it plays an important role in various data mining, machine learning, and information retrieval tasks. However, due to the “curse of dimensionality” a brute-force way of computing the similarity scores in the high dimensional dataset is infeasible, and at times impossible. In this work, we address this question and propose an efficient dimensionality reduction algorithm for sparse binary datasets that generates a succinct sketch of the dataset while preserving estimates for computing the similarity score between data objects.
### I-a Our Contribution
We first informally describe our sketching algorithm.
BinSketch: (Binary Data Sketching) Given a -dimensional binary vector , our algorithm reduces it to a -dimensional binary vector , where is specified later. It randomly maps each bit position (say) to an integer . To compute the -th bit of , it checks which bit positions have been mapped to , computes the of the bits located at those positions and assigns it to
A simple and exact solution to the problem is to represent each binary vector by a (sorted) list (or vector) of the indices with value one. In this representation, the space required in storing a vector is bits – as we need bits for storing each index, and there are at most indices with non-zero value (sparsity). Further, the time complexity of computing the (say) inner product of two originally -sparse binary vectors is . Therefore, both the storage as well as the time complexity of calculating similarity depend on the original dimension and does not scale for large values of . For high dimensional sparse binary data, we show how to construct highly compressed binary sketches whose length depends only on the data sparsity. Furthermore, we present techniques to compute similarity between vectors from their sketches alone. Our main technique is presented in Algorithm 1 for inner product similarity and the following theorem summarizes it.
###### Theorem 1 (Estimation of Inner product).
Suppose we want to estimate the Inner Product of -dimensional binary vectors, whose sparsity is at most , with probability at least . We can use to construct -dimensional binary sketches where . If and denote the sketches of vectors and , respectively, then can be estimated with accuracy using Algorithm 1.
We also present Algorithm 2 for estimating Hamming distance, Algorithm 3 for estimating Jaccard similarity and Algorithm 4 for estimating Cosine similarity; all these algorithms are designed based on Algorithm 1 and so follow similar accuracy guarantees.
Extension for categorical data compression. Our result can be easily extended for compressing Categorical datasets. The categorical dataset consists of several categorical features. Examples of categorical features are sex, weather, days in a week, age group, educational level, etc. We consider a type of Hamming distance for defining the distance between two categorical data points. For two dimensional categorical data points and , the distance between them is defined as follows: , where
dist(u[i],v[i])={1,if~{}u[i]≠v[i],0,otherwise.
In order to use , we need to preprocess the datasets. We first encode categorical feature via label-encoding followed by one-hot-encoding. In the label encoding step, features are encoded as integers. For a given feature, if it has possible values, we encode them with integers between and . In one-hot-encoding step, we convert the feature value into a length binary string, where is located at the position corresponding to the result of the label-encoding step. 222Both label-encoder and one-hot-encoder are available in sklearn as labelEncoder and OneHotEncoder packages. This preprocessing converts categorical dataset to a binary dataset. Please note that after preprocessing Hamming distance between the binary version of the data points is equal to the corresponding categorical distance , stated above. We can now compress the binary version of the dataset using and due to Algorithm 2, the compressed representation maintains the Hamming distance.
In Section III we present the proof of Theorem 1 where we explain the theoretical reasons behind the effectiveness of . As is usually the case for hash functions, practical performance often outshines theoretical bounds; so we conduct numerous experiments on public datasets. Based on our experiment results reported in Section IV we make the claim that is the best option for compressing sparse binary vectors while retaining similarity for many of the commonly used measures. The accuracy obtained is comparable with the state-of-the-art sketching algorithms, especially at high similarity regions, while taking almost negligible time compared to similar sketching algorithms proposed so far.
### I-B Related work
Our proposed algorithm is very similar in nature to the BCS algorithm [21, 22], which suggests a randomized bucketing algorithm where each index of the input is randomly assigned to one of the buckets; denotes the sparsity of the dataset. The sketch of an input vector is obtained by computing the parity of the bits fallen in each bucket. We offer a better compression bound than theirs. For a pair of vectors, their compression bounds are , while ours is . This is also reflected in our empirical evaluations, on small values of compression length, outperforms . However, the compression times (or dimensionality reduction time) of both the algorithms are somewhat comparable.
For Jaccard Similarity, we compare the performance of our algorithms with [3], [25] – a faster variant of , and [20]. We would like to point out some key differences between and . is two-step in nature that takes the sketch obtained by running on the original data as input, and outputs binary sketch which maintains an estimate of the original Jaccard similarity. Due to this two-step nature, its compression time is higher (see Table I and Figure 3). The number of functions used in (denoted by ) is a crucial parameter and the authors suggested using such that the pairwise symmetric difference is approximately . Empirically they suggest using , where is the similarity threshold. We argue that not only tuning is an important step but it is unclear how this condition will be satisfied for a diverse dataset, on the contrary, requires no such parameter. Furthermore, doesn’t provide any closed form expression to estimate accuracy and confidence. However, the variance of the critical term of their estimator is linear in the size of the sketch, i.e. . Whereas our confidence interval is of the order of which could be far smaller compared to , even for non-sparse data. Finally, compared to the Poisson approximation based analysis used in , we employed a tighter martingale-based analysis leading to (slightly) better concentration bounds (compare, e.g., the concentration bounds for estimating the size of a set from its sketch).
For Cosine Similarity, we compare with [9], [27] – a faster variant of , [24], using [25] in the algorithm of [24] instead of . For the Inner Product, [22], Asymmetric MinHash [24], and Asymmetric – using [25] in [24], were the competing algorithms. In all these similarity measures, for sparse binary datasets, our proposed algorithm is faster, while simultaneously offering almost a similar performance as compared to the baselines. We experimentally compare the performance on several real-world datasets and observed the results that are in line with these observations. Further, in order to get a sketch of size , our algorithm requires a lesser number of random bits, and requires only one pass to the datasets. These are the major reasons due to which we obtained good speedup in compression time. We summarize this comparison in Table I. Finally, a major advantage of our algorithm, similar to [21, 22], is that it gives one-shot sketching by maintaining estimates of multiple similarity measures in the same sketch; this is in contrast to usual sketches that are customized for a specific similarity.
#### Connection with Bloom Filter
appears structurally similar to a Bloom filter with one hash function. The standard Bloom filter is a space-efficient data-structure for set-membership queries; however, there is an alternative approach that can be used to estimate the intersection between two sets [4]. However, it is unclear how estimates for other similarity measures can be obtained. We answer this question positively and suggest estimates for all the four similarity measures in the same sketch. We also show that our estimates are strongly concentrated around their expected values.
### I-C Applicability of our results
For high dimensional sparse binary datasets, due to its simplicity, efficiency, and performance, can be used in numerous applications which require a sketch preserving Jaccard, cosine, Hamming distance or inner product similarity.
#### Scalable Ranking and deduplication of documents
Given a corpus of documents and a set of query documents, a goal is to find all documents in the corpus that are “similar” to query documents under a given similarity measure (e.g., Jaccard, cosine, inner product). This problem is a fundamental sub-routine in many applications like near-duplicate data detection [19, 14, 5], efficient document similarity search [16, 24], plagiarism detection [7, 5], etc. and dimensionality reduction is one way to address this problem. In Subsection IV-B we provide empirical validation that offers significant speed-up in dimensionality reduction while offering a comparable accuracy.
#### Scalable Clustering of documents
can be used in scaling up the performance of several clustering algorithms, in the case of high-dimensional and sparse datasets. For instance, in the case of Spherical -means clustering, which is the problem of clustering data points using Cosine Similarity, one can use [11]; and for -mode clustering, which is clustering using Hamming Distance, one can use -mode [15], on the sketch obtained by .
#### Other Applications
Beyond the above-noted applications, sketching techniques have been used widely in application such as Spam detection [6], compressing social networks [10] all pair similarity [2], Frequent Itemset Mining [8]. As offers significant speed-up in dimensionality reduction time and simultaneously provides a succinct and accurate sketch, it helps in scaling up the performance of the respective algorithms.
## Ii Background
Notations
dimension of the compressed data.
sparsity bound.
-th bit position of binary vector
number of ’s in the binary vector .
Cosine similarity between and
Jaccard similarity between and
Hamming distance between and
Inner product between and
#### SimHash for Cosine similarity [9, 12].
The Cosine similarity between a pair of vectors is defined as . To compute a sketch of a vector , [9] generates a random vector , with each component chosen uniformly at random from and a 1-bit sketch is computed as
SimHash(r)(u)={1,if ⟨u,r⟩≥0.0,otherwise.
was shown to preserve inner product in the following manner [12]. Let be an angle such that . Then,
Pr[SimHash(r)(u)=SimHash(r)(v)]=1−θπ,
#### MinHash for Jaccard and Cosine similarity.
The Jaccard similarity between a pair of set is defined as Broder et al. [3] suggested an algorithm – – to compress a collection of sets while preserving the Jaccard similarity between any pair of sets. Their technique includes taking a random permutation of and assigning a value to each set which maps to minimum under that permutation.
###### Definition 2 (Minhash [3]).
Let be a random permutation over , then for a set for .
It was then shown by Broder et al. [3, 5] that
Pr[hπ(u)=hπ(v)]=|u∩v||u∪v|.
Exploiting a similarity between Jaccard similarity of sets and Cosine similarity of binary vectors, it was shown how to use for constructing sketches for Cosine similarity in the case of sparse binary data [23].
#### BCS for sparse binary data [22, 21].
For sparse binary dataset, offers a sketching algorithm that simultaneously preserves Jaccard similarity, Hamming distance and inner product.
###### Definition 3 (Bcs).
Let be the number of buckets. Choose a random mapping from to . Then a vector is compressed to a vector as follows:
us[j]=∑i:b(i)=ju[i](mod2).
## Iii Analysis of BinSketch
Let and denote two binary vectors in -dimension, and , denotes the number of in and . Let denote the compressed representation of and , where denotes the compression length (or reduced dimension). In this section we will explain our sketching method and give theoretical bounds on its efficacy.
###### Definition 4 (BinSketch).
Let be a random mapping from to . Then a vector is compressed into a vector as
as[j]=⋁i:π(i)=ja[i]
Constructing a for a dataset involves first, generating a random mapping , and second, hashing each vector in the dataset using . There could be possible mappings, so choosing requires time and that many random bits. Hashing a vector involves only looking at the non-zero bits in and that step takes time since . Both these costs compete favorably with the existing algorithms as tabulated in Table I.
### Iii-a Inner-product similarity
The sketches, ’s do not quite “preserve” inner-product by themselves, but are related to the latter in the following sense. We will use to denote ; it will be helpful to note that as increases.
###### Lemma 5.
1. E(|as|/N)=(1−n|a|) 2. E(⟨as,bs⟩/N)= (1−n|a|)(1−n|b|)+n|a|+|b|[(1n)⟨a,b⟩−1]= 1−n|a|−n|b|+n|a|+|b|+⟨a,b⟩
###### Proof.
It will be easier to identify as a subset of . The -th bit of can be set only by some element in which can happen with probability . The -th bit of both and is set if it is set by some element in , or if it is set simultaneously by some element in and by another element in . This translates to the following probability that some particular bit is set in both and .
(1−n|a∩b|)+n|a∩b|(1−n|a∖b|)(1−n|b∖a|) =1−n|a|−n|b|+n|a|+|b|−|a∩b| =(1−n|a|)(1−n|b|)+n|a|+|b|(1n|a∩b|−1)
The lemma follows from the above probabilities using the linearity of expectation. ∎
Note that the above lemma allows us to express as
⟨a,b⟩=|a|+|b|−1lnnln(n|a|+n|b|+E(⟨as,bs⟩)N−1)
Algorithm 1 now explains how to use this result to approximately calculate using their sketches and .
We will prove that Algorithm 1 estimates with high accuracy and confidence if we use ; can be set to any desired probability of error and we assume that the sparsity is not too small, say at least 20. Our first result proves that the estimated above is a good approximation of ; exactly identical result holds for and too.
###### Lemma 6.
With probability at least , it holds that
∣∣nas−E[|as|]∣∣<√ψ2ln2δ
###### Proof.
The proof of this lemma is a simple adaptation of the computation of the expected number of non-empty bins in a balls-and-bins experiment that is found in textbooks and done using Doob’s martingale. Identify the random mapping , where the number of 1’s in is denoted by , as throwing black balls (and “no”-balls), one-by-one, into bins chosen uniformly at random. Supposing we only consider the black balls in the bins, then is an indicator variable for the event that the -th bin is non-empty and the number of non-empty bins can be shown to be concentrated around their expectation 333Using to denote the number of non-empty bins and the number of balls, Azuma-Hoeffding inequality states that (see Probability and Computing, Mitzenmacher and Upfal, Cambridge Univ. Press).. Since the number of non-empty bins correspond to , this concentration bound can be directly applied for proving the lemma.
Let denote the event in the statement of the lemma. Then,
where is used for the first inequality and the stated bound, with , is used for the second inequality. ∎
Similar, but more involved, approach can be used to prove that is a good estimation of .
###### Lemma 7.
With probability at least , it holds that
∣∣nas,bs−E[⟨as,bs⟩]∣∣<√ψ2ln2δ
###### Proof.
For a given , lets partition into parts (consisting of positions at which both and are 1), (positions at which is 1 and is 0), (positions at which is 0 and is 1) and (the rest). Any random mapping can treated as throwing grey balls, white balls, black balls, and “no”-balls randomly into bins. Suppose we say that a bin is “greyish” if it either contains some grey ball or both a white and a black ball. The number of common 1-bits in and (that is ) is now equal to the number of greyish bins. Observe that when any ball lands in some bin, say , the number of greyish bins either remains same or increases by 1; therefore, we can say that the count of the greyish bins satisfies Lipschitz condition. This allows us to apply Azuma-Hoeffding inequality as above and prove the lemma; we will also need the fact that the number of greyish bins is at most . ∎
The next lemma allows us to claim that our estimation of is also within reasonable bounds. It should be noted that our sketches do not explicitly save the number of 1’s in , so it is necessary to compute this number from our sketches; furthermore, since this estimate is not used elsewhere, we do not mandate it to be an integer either.
###### Lemma 8.
With probability at least , it holds that
∣∣|a|−na∣∣<4ψln1n=4√ψ2ln2δ
###### Proof.
Based on Lemma 5 and Algorithm 1, . For the proof we use the upper bound given in Lemma 6 that holds with probability at least . We need a few results before proceeding that are based on the standard inequality for .
( )
###### Observation 10.
. Since , we get that .
###### Observation 11.
.
A proof of the above observation follows using simple algebra and the result of Lemma 6. We defer it to the full version of the paper. We use these observations for proving two possible cases of the lemma. We will use the notation .
case (i) : In this case and
n|a|−nna=[nas−E(|as|)]/N
For the R.H.S., by Lemma 6.
For the L.H.S., we can write as . Furthermore, since for reasonable values of and .
Combining the bounds above we get the inequality that we will further process below.
case (ii) : In this case and
nna−n|a|=[E(|as|)−nas]/N
As above, R.H.S. is at most using Lemma 6 and L.H.S. can be written as . Further using Observation 11 we get the inequality, .
For both the above cases we obtained that , i.e., . This gives us that employing the known inequality for any . Since , we get the desired upper bound (since for ) (using Observation 11). ∎
Of course a similar result holds for and as well. The next lemma similarly establishes the accuracy of our estimation of .
###### Lemma 12.
With probability at least , it holds that
∣∣⟨a,b⟩−na,b∣∣<14√ψ2ln2δ
We get the following from Algorithm 1 and Lemma 5.
⟨a,b⟩ =|a|+|b|+1ln1nln[n|a|+n|b|+E[⟨as,bs⟩]N−1] na,b =na+nb+1ln1nln(nna+nnb+nas,bsN−1)
in which (Lemma 8), (similarly), and (Lemma 7), each happening with probability at least . The complete proof that is a good approximation of is mostly algebraic analysis of the above facts and we defer it the full version of the paper.
Theorem 1 is a direct consequence of Lemma 12 for reasonably large (say, beyond ) and small (say, less than ).
### Iii-B Hamming distance
The Hamming distance and the inner product similarity of two binary vectors and are related as
Ham(a,b)=|a|+|b|−IP(a,b)
The technique used in the earlier subsection can be used to estimate the Hamming distance in a similar manner.
### Iii-C Jaccard similarity
The Jaccard similarity between a pair of binary vectors and can be computed from their Hamming distance and their inner product.
JS(a,b)=IP(a,b)Ham(a,b)+IP(a,b)
This paves way for an algorithm to compute Jaccard similarity from .
### Iii-D Cosine similarity
The cosine similarity between a pair binary vectors and is defined as:
Cos(a,b)=IP(a,b)/√|a|⋅|b|
An algorithm for estimating cosine similarity from binary sketches is straight forward to design at this point.
It should be possible to prove that Algorithms 2, 3 and 4 are accurate and low-error estimations of Hamming distance, Jaccard similarity and cosine similarity, respectively; however, those analysis are left out of this paper.
## Iv Experiments
#### Hardware description
We performed our experiments on a machine having the following configuration: CPU: Intel(R) Core(TM) i5-3320M CPU @ 2.60GHz x 4; Memory: 7.5 GB; OS: Ubuntu 18.04; Model: Lenovo Thinkpad T430.
To reduce the effect of randomness, we repeated each experiment several times and took the average. Our implementations did not employ any special optimization.
#### Datasets
The experiments were performed on publicly available datasets - namely, NYTimes news articles (number of points = , dimension = ), Enron Emails (number of points = , dimension = ), and KOS blog entries (number of points = , dimension = ) from the UCI machine learning repository [18]; and BBC News Datasets (number of points = , dimension = [13]. We considered the entire corpus of KOS and BBC News datasets, while for NYTimes, ENRON datasets we sampled data points.
#### Competing Algorithms
For our experiments we have used three similarity measures: Jaccard Similarity, Cosine Similarity, and Inner Product. For the Jaccard Similarity, [3], Densified One Permutation Hashing () – a faster variant of – [25], [22], and [20] were the competing algorithms. is two-step in nature, which takes the sketch obtained by running on the original data as input, and outputs binary sketch which maintains an estimate of the original Jaccard similarity. As suggested by authors, we use the number of permutations , where is the similarity threshold. We upper bound with which is the maximum number of permutations used by . For the Cosine Similarity, [9], Circulant Binary Embedding () – a faster variant of – [27], [23], [25] in the algorithm of [23] instead of , were the competing algorithms. For the Inner Product, [22], Asymmetric MinHash [24], and Asymmetric ( [25] in the algorithm of [24]), were the competing algorithms.
### Iv-a Experiment 1: Accuracy of Estimation
In this task, we evaluate the fidelity of the estimate of on various similarity regimes.
#### Evaluation Metric
To understand the behavior of on various similarity regimes, we extract similar pairs – pair of data objects whose similarity is higher than certain threshold –from the datasets. We used Cosine, Jaccard, and Inner Product as our measures. For example: for Jaccard/Cosine case for the threshold value , we considered only those pairs whose similarities are higher than . We used mean square error as our evaluation criteria. Using and other candidate algorithms, we compressed the datasets to various values of compression length . We then calculated the for all the algorithms, for different values of . For example, in order to calculate the of with respect to the ground truth result, for every pair of data points, we calculated the square of the difference between their estimated similarities after the result of , and the corresponding ground truth similarity. We added these values for all such pairs and calculated its mean. For Inner Product, we used this absolute value, and for Jaccard/Cosine similarity we computed its negative logarithm base . A smaller corresponds to a larger , therefore, a higher value is an indication of better performance.
#### Insights
We summarize our results in Figures 2, and 1 for Cosine/Jaccard Similarity and Inner Product, respectively. For Cosine Similarity, consistently remains to be better than the other candidates. While for Jaccard Similarity, it significantly outperformed w.r.t. , and , while its performance was comparable w.r.t. . Moreover, for Inner product 1 results, significantly outperformed w.r.t. . 33footnotetext: We observed a similar pattern for both as well as Ranking experiments on other datasets/similarity measures as well. We defer those plot to the full version of the paper.
### Iv-B Experiment 2: Ranking
#### Evaluation Metric
In this experiment, given a dataset and a set of query points, the aim is to find all the points that are similar to the query points, under the given similarity measure. To do so, we randomly, partition the dataset into two parts – and . The bigger partition is called as the training partition, while the smaller one is called as querying partition. We call each vector of the querying partition as a query vector. For each query vector, we compute the points in the training partition whose similarities are higher than a certain threshold. For Cosine and Jaccard Similarity, we used the threshold values from the set . For Inner Product, we first found out the maximum existing Inner product in the dataset, and then set the thresholds accordingly. For every query point, we first find all the similar points in the uncompressed dataset, which we call as ground truth result. We then compress the dataset, using the candidate algorithms, on various values of compression lengths. To evaluate the performance of the competing algorithms, we used the accuracy-precision-recall- score as our standard measure. If the set denotes the ground truth result (result on the uncompressed dataset), and the set denotes the results on the compressed datasets, then accuracy = , precision = , recall = , and
#### Insights
We summarize Accuracy and score results in Figure 4. For Jaccard Similarity, on both Accuracy and score measure, significantly outperformed , , and while its performance was comparable w.r.t. . For Cosine similarity, on higher and intermediate threshold values, outperformed all the other candidate algorithms. However, on the lower threshold values, offered the most accurate sketch followed by .
#### Efficiency of BinSketch
We comment on the efficiency of with the other competing algorithms and summarize our results in Figure 3. We noted the time required to compress the original dataset using all the competing algorithms. For a given compression length, the compression time of varies based on the similarity threshold. Therefore, we consider taking their average. We notice that the time required by and is negligible for all values of and on all the datasets. Compression time of is very higher than ours, however, it is independent of the compression length . After excluding some initial compression lengths, the compression time of is the highest, and grows linearly with , as it requires running on the original dataset. For the remaining algorithms, their respective compression time grows linearly with .
## V Summary and open questions
In this work, we proposed a simple dimensionality reduction algorithm – – for sparse binary data. offer an efficient dimensionality reduction/sketching algorithm, which compresses a given -dimensional binary dataset to a relatively smaller -dimensional binary sketch, while simultaneously maintaining estimates for multiple similarity measures such as Jaccard Similarity, Cosine Similarity, Inner Product, and Hamming Distance, on the same sketch. The performance of was significantly better than [21, 22] while the compression (dimensionality reduction) time of these two algorithms were somewhat very comparable. obtained a significant speedup in compression time w.r.t other candidate algorithms ( [3, 23], [9], [25], [27]) while it simultaneously offered a comparable performance guarantee.
We want to highlight the error bound presented in Theorem 1 is due to a worst-case analysis, which potentially can be tightened. We state this as an open question of the paper. Our experiments on real datasets establish this. For example, for the inner product (see Figure 1), we show that the Mean Square Error is almost zero even for compressed dimensions that are much lesser than the bounds stated in the Theorem. Another important open question is to derive a lower bound on the size of a sketch that is required to efficiently and accurately derive similarity values from compressed sketches. Given the simplicity of our method, we hope that it will get adopted in practice.
## References
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• [2] R. J. Bayardo, Y. Ma, and R. Srikant (2007) Scaling up all pairs similarity search. See DBLP:conf/www/2007, pp. 131–140. External Links: Cited by: §I-C.
• [3] A. Z. Broder, M. Charikar, A. M. Frieze, and M. Mitzenmacher (1998) Min-wise independent permutations (extended abstract). See DBLP:conf/stoc/1998, pp. 327–336. External Links: Cited by: §I-B, TABLE I, §II, §II, §IV, §V, Definition 2.
• [4] A. Z. Broder and M. Mitzenmacher (2003) Survey: network applications of bloom filters: A survey. Internet Mathematics 1 (4), pp. 485–509. External Links: Cited by: §I-B.
• [5] A. Z. Broder (2000) Identifying and filtering near-duplicate documents. See DBLP:conf/cpm/2000, pp. 1–10. External Links: Cited by: §I-C, §II.
• [6] A. Z. Broder (1997) On the resemblance and containment of documents. In Compression and Complexity of Sequences 1997. Proceedings, pp. 21–29. Cited by: §I-C.
• [7] S. Buyrukbilen and S. Bakiras (2013) Secure similar document detection with simhash. See DBLP:conf/sdmw/2013, pp. 61–75. External Links: Cited by: §I-C.
• [8] V. T. Chakaravarthy, V. Pandit, and Y. Sabharwal (2009) Analysis of sampling techniques for association rule mining. See DBLP:conf/icdt/2009, pp. 276–283. External Links: Cited by: §I-C.
• [9] M. Charikar (2002) Similarity estimation techniques from rounding algorithms. See DBLP:conf/stoc/2002, pp. 380–388. External Links: Cited by: §I-B, TABLE I, §II, §II, §IV, §V.
• [10] F. Chierichetti, R. Kumar, S. Lattanzi, M. Mitzenmacher, A. Panconesi, and P. Raghavan (2009) On compressing social networks. See DBLP:conf/kdd/2009, pp. 219–228. External Links: Cited by: §I-C.
• [11] I. S. Dhillon and D. S. Modha (2001) Concept decompositions for large sparse text data using clustering. Machine Learning 42 (1/2), pp. 143–175. External Links: Cited by: §I-C.
• [12] M. X. Goemans and D. P. Williamson (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42 (6), pp. 1115–1145. External Links: Cited by: §II, §II.
• [13] D. Greene and P. Cunningham (2006) Practical solutions to the problem of diagonal dominance in kernel document clustering. In Proc. 23rd International Conference on Machine learning (ICML’06), pp. 377–384. Cited by: §IV.
• [14] M. R. Henzinger (2006) Finding near-duplicate web pages: a large-scale evaluation of algorithms. See DBLP:conf/sigir/2006, pp. 284–291. External Links: Cited by: §I-C.
• [15] Z. Huang (1998-09-01) Extensions to the k-means algorithm for clustering large data sets with categorical values. Data Mining and Knowledge Discovery 2 (3), pp. 283–304. External Links: ISSN 1573-756X, Document, Link Cited by: §I-C.
• [16] Q. Jiang and M. Sun (2011) Semi-supervised simhash for efficient document similarity search. See DBLP:conf/acl/2011, pp. 93–101. External Links: Link Cited by: §I-C.
• [17] Y. Jiang, C. Ngo, and J. Yang (2007) Towards optimal bag-of-features for object categorization and semantic video retrieval. See DBLP:conf/civr/2007, pp. 494–501. External Links: Cited by: §I.
• [18] M. Lichman (2013) UCI machine learning repository. University of California, Irvine, School of Information and Computer Sciences. External Links: Link Cited by: §IV.
• [19] G. S. Manku, A. Jain, and A. D. Sarma (2007) Detecting near-duplicates for web crawling. See DBLP:conf/www/2007, pp. 141–150. External Links: Cited by: §I-C.
• [20] M. Mitzenmacher, R. Pagh, and N. Pham (2014) Efficient estimation for high similarities using odd sketches. See DBLP:conf/www/2014, pp. 109–118. External Links: Cited by: §I-B, TABLE I, §IV.
• [21] R. Pratap, R. Kulkarni, and I. Sohony (2018) Efficient dimensionality reduction for sparse binary data. See DBLP:conf/bigdataconf/2018, pp. 152–157. External Links: Cited by: §I-B, §I-B, TABLE I, §II, §V.
• [22] R. Pratap, I. Sohony, and R. Kulkarni (2018) Efficient compression technique for sparse sets. See DBLP:conf/pakdd/2018-3, pp. 164–176. External Links: Cited by: §I-B, §I-B, TABLE I, §II, §IV, §V.
• [23] A. Shrivastava and P. Li (2014) In defense of minhash over simhash. See DBLP:conf/aistats/2014, pp. 886–894. External Links: Link Cited by: §II, §IV, §V.
• [24] A. Shrivastava and P. Li (2015) Asymmetric minwise hashing for indexing binary inner products and set containment. See DBLP:conf/www/2015, pp. 981–991. External Links: Cited by: §I-B, §I-C, §IV.
• [25] A. Shrivastava (2017) Optimal densification for fast and accurate minwise hashing. See DBLP:conf/icml/2017, pp. 3154–3163. External Links: Link Cited by: §I-B, §I-B, TABLE I, §IV, §V.
• [26] Y. Singer Sibyl: a system for large scale machine learning.. Technical report. Cited by: §I.
• [27] F. X. Yu, S. Kumar, Y. Gong, and S. Chang (2014) Circulant binary embedding. In Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32, ICML’14, pp. II–946–II–954. External Links: Link Cited by: §I-B, TABLE I, §IV, §V.
## Appendix A Proof of Observation 11
In this section we prove that . For this first we derive an upper bound of on .
Let denote the expression appearing in Lemma 6. Using this lemma, . Observe that since and . Furthermore, since , we get the upper bound . For reasonable values of and , both and are at least 4; thus, we get the bound of and this leads us to the bound .
## Appendix B Proof of Lemma 12
In this section we derive an upper bound on
B= ∣∣|a|−na + |b|−nb + 1ln1nln[n|a|+n|b|+E[⟨as,bs⟩]N−1] − 1ln1nln[nna+nnb+nas,bsN−1]∣∣
###### Proof.
We first apply triangle inequality and Lemma 8 to obtain
B≤4/ψln1n+4/ψln1n+1ln1n∣∣ ∣ ∣∣lnnna+nnb+nas,bsN−1n|a|+n|b|+E[⟨as,bs⟩]N−1∣∣ ∣ ∣∣
Next we derive an upper bound for the last term for which we require the next few observations. Let denote , denote , and denote .
###### Observation 13.
By expanding and employing , we obtain that since .
###### Observation 14.
Using Lemma 5, for non-zero and .
These observations ensure that the terms inside the logarithm are indeed positive.
Next we upper bound by employing the inequality that holds for non-negative and can be derived from the standard inequality for . Here, set and . Then, using triangle inequality
|U−V| ≤|nna−n|a||+|nnb−n|b||+|E[⟨as,bs⟩]−nas,bs|N ≤31ψ( using Lemma~{}??? and the % next observation)
###### Observation 15.
These claims appear in the proof of Lemma 8: and . Similarly, and .
We need one final observation to compute .
###### Observation 16.
Using Lemma 7, .
Based on the last two observations we can compute
U= nna+nnb+nas,bsN−1 ≥ n|a|+n|b|+E(⟨as,bs⟩)N−3ψ−1 = V−3ψ
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2020-08-05 02:23:01
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https://math.stackexchange.com/questions/2062760/fxy-fxfy-continuous-at-x-0-0-implies-fx-is-continuous-over-r
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# $f(x+y) = f(x)+f(y)$ continuous at $x_0=0 \implies f(x)$ is continuous over R? [closed]
Let $x,y \in R$
$f(x+y) = f(x)+f(y)$
is it true that if $f$ is continuous at $x_0=0$, than $f$ is continuous in $R$?
## closed as off-topic by zhoraster, user223391, Namaste, Did, Adam HughesDec 18 '16 at 2:02
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – zhoraster, Community, Namaste, Did, Adam Hughes
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• Hint: fix a $t\in\mathbb R$ and consider the function $g(x)=f(x+t)$. – Wojowu Dec 17 '16 at 21:43
At any arbitrary $x_1\in\mathbb{R}$ and any $\Delta\neq 0$, we have $$f(x_1+\Delta)-f(x_1)=f(\Delta)=f(\Delta+x_0)-f(x_0).$$ As $\Delta\to 0$, the rightmost expression above goes to $0$ due to continuity at $x_0$, so the leftmost expression also goes to $0$. This implies continuity at $x_1$ and therefore in $\mathbb{R}$. Note we don't need the fact that $x_0=0$.
• Wondrously simple. +1 – Simply Beautiful Art Dec 17 '16 at 22:06
• I don't understand how and why you used the continuity at $x_0$. Couldn't you say immediately that $f(x_1+\Delta)-f(x_1)$ goes to zero when $\Delta$ goes to zero because it can be seen as $f(x_1 + 0)-f(x_1) = 0$? – S. Peter Dec 17 '16 at 22:20
• @S.Peter $\lim_{\Delta\to 0}f(x_1+\Delta)=f(x_1)$ is what is to be proved for all $x_1\in\mathbb{R}$. However, you're given $\lim_{\Delta\to 0}f(x_0+\Delta)=f(x_0)$, so I used this to infer that $\lim_{\Delta\to 0}[f(\Delta+x_0)-f(x_0)]=0$. The key is to note that $f(x+y)=f(x)+f(y)$ implies $f(\Delta+x_0)-f(x_0)=f(\Delta+x_1)-f(x_1)$. – yurnero Dec 17 '16 at 22:25
• @S.Peter $\lim_{x\to x_1}f(x)=\lim_{\Delta\to 0}f(x_1+\Delta)$. $\Delta$ represents $x-x_1$. – yurnero Dec 17 '16 at 22:54
Make $x=a_n$ such that $a_n \rightarrow a$ and $y=-a$
$$f\left(a_n-a\right)=f(a_n)-f\left(a\right)$$
doing $n \rightarrow \infty$ and by the continuity at $0$
$$f\left(a_n-a\right) \rightarrow 0 \quad \text{(because f(0)=0)}$$
$$a_n \rightarrow a$$
and finally
$$f\left(a_n\right) \rightarrow f(a)$$
So we get continuity at $a$.
• Your last line is wrong, because you only proved $f(a_n) \to f(a)$ for a very special type sequence... – N. S. Dec 17 '16 at 22:05
• Thanks! Fixed! But the idea keeps the same! – Arnaldo Dec 17 '16 at 22:24
$f$ is continuous at $0$ if and only if
$$\forall \epsilon>0 \exists \delta>0 : |x|<\delta \implies |f(x)-f(0)|<\epsilon.$$
Since $f(0)=f(0+0)=f(0)+f(0)=2f(0)$ we have $f(0)=0.$ Thus
$f$ is continuous at $0$ if and only if
$$\forall \epsilon>0 \exists \delta>0 : |x|<\delta \implies |f(x)|<\epsilon.$$
Now, $f$ is continuous at $x_0$ if and only if
$$\forall \epsilon>0 \exists \delta>0 : |h|<\delta \implies |f(x_0+h)-f(x_0)|<\epsilon.$$ This obviously holds since $$f(x_0+h)-f(x_0)=f(h)$$ and $f$ is continuous at $0.$
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2019-06-27 01:59:16
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https://www.studysmarter.us/textbooks/math/essential-calculus-early-transcendentals-2nd/multiple-integrals/q41e-set-up-but-do-not-evaluate-integral-expressions-for-the/
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Found in: Page 730
### Essential Calculus: Early Transcendentals
Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280
# Set up, but do not evaluate, integral expressions for The massThe centre of mass, and The moment of inertia about the z-axis.The solid of exercise 19; $${\rm{\rho (x,y,z) = }}\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}}$$.
1. The mass of the solid is $$m = \int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} } } } {\rm{dzdydx}}$$.
2. Therefore, the centres of the mass are:
\begin{aligned}\rm m &= \int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} } } } {\rm{\;dz\;dy\;dx}}\\\rm\bar x &= \frac{{\rm{1}}}{{\rm{m}}}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{x}} } } \sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} {\rm{\;dz\;dy\;dx}}\\\rm\bar y &= \frac{{\rm{1}}}{{\rm{m}}}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{y}} } } \sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} {\rm{\;dz\;dy\;dx}}\\\rm\bar z &= \frac{{\rm{1}}}{{\rm{m}}}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{z}} } } \sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} {\rm{\;dz\;dy\;dx}}\end{aligned}
1. The moment of inertia about the z-axis is $${{\rm{I}}_{\rm{z}}}{\rm{ = }}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {{{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}} } } {\rm{dzdydx}}$$.
See the step by step solution
## Step 1: Concept Introduction
Triple integrals are the three-dimensional equivalents of double integrals. They're a way to add up an unlimited number of minuscule quantities connected with points in a three-dimensional space.
## Step 2: Find the mass
Exercising is a great way to become in shape.
The region bounded by $${\rm{z = 0,y = }}{{\rm{x}}^{\rm{2}}}{\rm{ and y + z = 1}}$$ is called the solid.
Along the line $${\rm{y = 1}}$$, the planes $${\rm{y + z = 1and z = 0}}$$ intersect at the $${\rm{xy}}$$plane.
We can define the region by looking at the diagram. E
$$\left\{ {{\rm{(x,y,z)}} \in {\rm{E}}\mid \;\;\;{\rm{0}} \le {\rm{z}} \le {\rm{1 - y,}}\;\;\;{{\rm{x}}^{\rm{2}}} \le {\rm{y}} \le {\rm{1,}}\;{\rm{ - 1}} \le {\rm{x}} \le {\rm{1}}} \right\}$$
The solid's mass is
$$\text{m=}\iiint_{\text{E}}{\text{ }\!\!\rho\!\!\text{ }}\text{(x,y,z)dV=}\int_{\text{-1}}^{\text{1}}{\int_{{{\text{x}}^{\text{2}}}}^{\text{1}}{\int_{\text{0}}^{\text{1-y}}{\sqrt{{{\text{x}}^{\text{2}}}\text{+}{{\text{y}}^{\text{2}}}}}}}\text{dzdydx}$$
We are not to judge the situation because it wants us not to.
Therefore, the mass of the solid is $$m = \int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} } } } {\rm{dzdydx}}$$.
## Step 3: Find the centre of mass
Exercising is a great way to become in shape.
The region bounded by $${\rm{z = 0,y = }}{{\rm{x}}^{\rm{2}}}{\rm{ and y + z = 1}}$$ is called the solid.
Along the line$${\rm{y = 1}}$$, the planes $${\rm{y + z = 1and z = 0}}$$intersect at the $${\rm{xy}}$$plane.
We can define the region by looking at the diagram. E
The coordinates of the centre of mass are calculated as follows:
\begin{aligned}& \bar{x}=\frac{1}{m}\iiint_{E}{x}\rho (x,y,z)dV=\frac{1}{m}\int_{-1}^{1}{\int_{{{x}^{2}}}^{1}{\int_{0}^{1-y}{x}}}\sqrt{{{x}^{2}}+{{y}^{2}}}dzdydx \\ & \bar{y}=\frac{1}{m}\iiint_{E}{y}\rho (x,y,z)dV=\frac{1}{m}\int_{1}^{1}{\int_{{{x}^{2}}}^{1}{\int_{0}^{1-y}{y}}}\sqrt{{{x}^{2}}+{{y}^{2}}}dzdydx \\ & \bar{z}=\frac{1}{m}\iiint_{E}{z}\rho (x,y,z)dV=\frac{1}{m}\int_{-1}^{1}{\int_{{{x}^{2}}}^{1}{\int_{0}^{1-y}{z}}}\sqrt{{{x}^{2}}+{{y}^{2}}}dzdydx \\\end{aligned}
Where
$$\text{m=}\iiint_{\text{E}}{\text{ }\!\!\rho\!\!\text{ }}\text{(x,y,z)dV=}\int_{\text{-1}}^{\text{1}}{\int_{{{\text{x}}^{\text{2}}}}^{\text{1}}{\int_{\text{0}}^{\text{1-y}}{\sqrt{{{\text{x}}^{\text{2}}}\text{+}{{\text{y}}^{\text{2}}}}}}}\text{dzdydx}$$
The problem instructs us to refrain from evaluating the integrals.
Therefore, the centres of the mass are
\begin{aligned}\rm m &= \int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} } } } {\rm{\;dz\;dy\;dx}}\\\rm\bar x &= \frac{{\rm{1}}}{{\rm{m}}}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{x}} } } \sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} {\rm{\;dz\;dy\;dx}}\\\rm\bar y &= \frac{{\rm{1}}}{{\rm{m}}}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{y}} } } \sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} {\rm{\;dz\;dy\;dx}}\\\rm\bar z &= \frac{{\rm{1}}}{{\rm{m}}}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {\rm{z}} } } \sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} {\rm{\;dz\;dy\;dx}}\end{aligned}
## Step 4: Find the moment of inertia about the z-axis
Exercising is a great way to become in shape.
The region bounded by $${\rm{z = 0,y = }}{{\rm{x}}^{\rm{2}}}{\rm{ and y + z = 1}}$$ is called the solid.
Along the line$${\rm{y = 1}}$$, the planes $${\rm{y + z = 1and z = 0}}$$ intersect at the $${\rm{xy}}$$ plane.
We can define the region by looking at the diagram. E
The moment of inertia about the z-axis is
\begin{aligned} {I_z} &= \iiint_E {\left( {{x^2} + {y^2}} \right)}\rho (x,y,z)dV \\ &= \int_{ - 1}^1 {\int_{{x^2}}^1 {\int_0^{1 - y} {\left( {{x^2} + {y^2}} \right)} } } \sqrt {{x^2} + {y^2}} dzdydx{\text{ }} \\ &= \int_{ - 1}^1 {\int_{{x^2}}^1 {\int_0^{1 - y} {{{\left( {{x^2} + {y^2}} \right)}^{3/2}}} } } dzdydx \\ \end{aligned}
We are not to judge the situation because it wants us not to.
Therefore, the moment of inertia about the z-axis is $${{\rm{I}}_{\rm{z}}}{\rm{ = }}\int_{{\rm{ - 1}}}^{\rm{1}} {\int_{{{\rm{x}}^{\rm{2}}}}^{\rm{1}} {\int_{\rm{0}}^{{\rm{1 - y}}} {{{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}} } } {\rm{dzdydx}}$$.
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2023-03-24 22:38:34
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https://math.meta.stackexchange.com/questions/3412/edited-questions
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# Edited questions
English is not my native language and when I ask some question I make a lots of mistakes. Some time later I see that my question edited but I don't know where, since I don't exactly remember the text of my question. Is there a way to see this edits? This opportunity will help me improve my English and analyze my mistakes.
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2021-07-24 18:31:56
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https://projecteuclid.org/euclid.ijm/1258130980
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## Illinois Journal of Mathematics
### Minimal Lagrangian submanifolds in the complex hyperbolic space
#### Abstract
In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomogeneity one. We characterize these submanifolds as the only minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ that are foliated by umbilical hypersurfaces of Lagrangian subspaces $\mathbb{R}\mathbb{H}^n$ of $\mathbb{C}\mathbb{H}^n$. By suitably generalizing this construction, we obtain new families of minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ from curves in $\mathbb{C}\mathbb{H}^1$ and $(n-1)$-dimensional minimal Lagrangian submanifolds of the complex space forms $\mathbb{C}\mathbb{P}^{n-1}$, $\mathbb{C}\mathbb{H}^{n-1}$ and $\mathbb{C}^{n-1}$. We give similar constructions in the complex projective space $\mathbb{C}\mathbb{P}^n$.
#### Article information
Source
Illinois J. Math., Volume 46, Number 3 (2002), 695-721.
Dates
First available in Project Euclid: 13 November 2009
https://projecteuclid.org/euclid.ijm/1258130980
Digital Object Identifier
doi:10.1215/ijm/1258130980
Mathematical Reviews number (MathSciNet)
MR1951236
Zentralblatt MATH identifier
1032.53052
#### Citation
Castro, Ildefonso; Montealegre, Cristina R.; Urbano, Francisco. Minimal Lagrangian submanifolds in the complex hyperbolic space. Illinois J. Math. 46 (2002), no. 3, 695--721. doi:10.1215/ijm/1258130980. https://projecteuclid.org/euclid.ijm/1258130980
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2019-07-17 20:55:29
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http://physics.stackexchange.com/tags/symmetry/hot
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# Tag Info
2
Noether's theorem states that if a system has a continuous symmetry, there is a quantity related to this symmetry, called the Noether charge, which is conserved. It does not state anything on the fact that adding a constant term to a measurable quantity may or may not change the physical description of the system. Only some physical quantities in fact are ...
2
I do not agree with the answer given by @ACuriousMind. @Scardenalli has asked for a compact Ricci-flat Riemannian manifold $M$ having as isometry group $U(1)\times SU(2)\times SU(3)$. This does not imply that $M$ must be a symmetric Riemannian manifold. However, the answer to @Scardenalli's question is still no, and it follows from a classical result in ...
2
Classical Lagrangian field theory deals with fields $\phi: M \to N$, where $M$ is spacetime and $N$ is the target-space of the fields. We shall for convenience call $M$ and $N$ the horizontal and the vertical space, respectively. OP is in this terminology essentially asking Q: What is the meaning of horizontal transformations? A: It is a (horizontal) flow ...
2
The appropriate definition of symmetry uses infinitesimal quantities, not just small quantities. Thus, in terms of your question, the Lagrangian is symmetric if $dL/d\epsilon=0$ at $\epsilon=0$. In terms of your example (rotation of a 2D harmonic oscillator), we have $$L \to (1+\epsilon^2) L = L + \mathcal{O}(\epsilon^2)$$ Thus to first order in ...
1
It sounds like you are interested in symplectic reduction procedures. On of these methods is that of Routh's procedure to eliminate cyclic variables using a Legendre transform to a reduced-variable Hamiltonian called a Routhian. Forming a variational approach may be difficult for some reduction procedures, however we can view conserved quantities as ...
1
While neither the Lagrangian $\mathcal{L}$ nor the action $S$ are invariant under boosts of the form $$\dot{q}(t) \to \dot{q}(t) + v, \quad v \in \mathbb{R},$$ the Euler-Lagrange equations are. The dynamics of the systems are unchanged for any transformation that preserves $\delta S = 0$, i.e. a transformation of the form $$\mathcal{L}(q, \dot{q}, t) \to ... 1 The answer to your question involves the fact that one does not usually know a priori the electric field \textbf{E} (or, for that matter, its direction) of a charge distribution \rho. Gauss's law, in integral form, relates the flux of the electric field through some closed surface S to the charge enclosed within the volume bounded by S. Precisely, ... 1 As your teacher says, it holds for every surface, but a look at the law itself, should clear out why some form of symmetrie is desirable:$$ \iint_S \vec{E} .\mathrm{d}\vec{A}=\iint_S E . \cos\theta . \mathrm{d}A = \frac{Q}{\epsilon_0} $$Here, E and \theta are position-dependent, so to calculate the integral, you need to take care of a position ... 1 As an example, let us suppose that you want to use the Gauss law to evaluate the electric field generated by a body charged in an uniform way. The gauss law tell you that the flux over an arbitrary closed surface around your body is proportional to the total charge:$$\int_{\partial V} \vec{E}\cdot d\vec{S}=\frac{Q}{\epsilon_0} $$but this is an ... 1 I think a classical example is electrical conductivity and resistivity (see Wikipedia), or any physical quantity which is described in the anisotropic case by a tensor (see also elasticity tensor as suggested in the comments by Jon Custer). Consider the Ohm's law in the anisotropic case$$ J_{i}=\sigma_{ij} E_j, \qquad E_i=\rho_{ij} J_j $$The conductivity ... 1 May I ask what text you are reading? My understanding of the stress energy tensor is as follows. The Noether condition is written as,$$\partial _\mu \bigg[\frac{\partial \mathcal L}{\partial (\partial _\mu \phi )}\delta \phi +\mathcal L \delta x^\mu\bigg]=0$$ In the discrete case we can imagine separate infinitesimal time and ... 1 Here we shall only discuss general relativistic diffeomorphism-invariant matter theories in a curved spacetime in the classical limit \hbar\to 0 for simplicity. In particular, we will not discuss the SEM pseudotensor for the gravitational field, but only the stress-energy-momentum (SEM) tensor for matter (m) fields \Phi^A. We emphasize that our ... 1 For an antilinear operator, as the antiunitaries and the complex conjugation, the definition of adjoint is changed:$$\langle U^{a*}\psi,\phi\rangle=\overline{\langle \psi,U\phi\rangle}$$where a* stands for anti-adjoint. It is therefore easy to see that the anti-adjoint of K is K itself (and in general the anti-adjoint of an anti-unitary is ... 1 When you integrate the Lagrangian density over a certain region \Omega, this is in principle allowed to change and this gives you a "boundary" term in the variation. This is well discussed in, e.g., the book of Goldstein (3rd edition), where the correct proof of the Noether theorem is given. 1 The 4 generators of SU(5) are not all "equivalent". In general, the generators of the group/algebra satisfy a defining equation of the form$$[T^i,T^j]=f^{ijk} T^k$$so depending on the structure constants f^{ijk} ,it is possible for example that$$[T^1,T^2]\neq[T^2,T^3]\quad\text{etc,} so it is important which generators are broken. In terms of ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-08-04 03:27:05
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https://zbmath.org/?q=an:0907.05055
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# zbMATH — the first resource for mathematics
Noncommutative symmetric functions. II: Transformations of alphabets. (English) Zbl 0907.05055
In a previous paper these three authors, joined by three more, had developed a theory of noncommutative symmetric functions using the quasi-determinant introduced by Gelfand and Retakh [see I. M. Gelfand, D. Krob, A. Lascoux, B. Leclarc, V. S. Retakh, and J.-Y. Thibon, Adv. Math. 112, No. 2, 218-348 (1995; Zbl 0831.05063)]. They also reinterpreted many elements of the descent algebras of Solomon as noncommutative symmetric functions. In the present work they apply this theory to the combinatorics of free Lie algebras. The idea is that the descent algebra is the noncommutative analog of the character ring, so they extend to the noncommutative case the operations encountered in character computations, such as, for example, the transformation $$A \rightarrow (1-q)A$$ used by Ram in the computation of some character tables. They also study idempotents and nilpotents in descent algebras.
##### MSC:
500000 Symmetric functions and generalizations
##### Keywords:
noncommutative symmetric functions; descent algebras
Full Text:
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2021-01-19 11:45:06
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http://mathhelpforum.com/calculus/176746-integral-evaluation.html
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# Math Help - Integral Evaluation
1. ## Integral Evaluation
I need to evaluate the following integral
$\int{\frac{e^{(-at-bt^2)}}{t}}dt$
It seems there is no closed from expression for this integral. But if we do the power series expansion for either $e^{-at}$ or $e^{-bt^2}$, then the closed form is possible but with an infinite power series. Kindly suggest if any alternative is available?
2. Originally Posted by rpmatlab
But if we do the power series expansion for either $e^{-at}$ or $e^{-bt^2}$, then the closed form is possible but with an infinite power series.
Infinite power series does not constitute as a closed form though, right?
3. Originally Posted by rpmatlab
I need to evaluate the following integral $\int{\frac{e^{(-at-bt^2)}}{t}}dt$
What is the context of that integral?. Perhaps you only need to find the integral from $0$ to $+\infty$ .
4. Hi Resilient, thank you for pointing out that infinite series will not be considered as Closed form. May be after truncating the series appropriately, we will get approximate closed form expression.
5. Originally Posted by Resilient
Infinite power series does not constitute as a closed form though, right?
But it does constitute a symbolic representation that can be further manipulated, which is what I beleive is required in this case.
CB
6. The limits of the integral is from $\sqrt{c}$ to $\inf$, where $c$ is a non negative constant. After solving the integral through series expansion, I got a series of the form
$\sum_{k=0}^{\inf}(\frac{-b}{a^2})^k \frac{1}{k!} \Gamma(2k,b\sqrt{c})$
I want to truncate the series. But the range of $\frac{b}{a^2}$ is $10^{-5}$ to $10^{5}$. Also $\Gamma(2k,b\sqrt{c})$ is increasing with $k$. The series seems to be divergent for $\frac{b}{a^2}\geq 1$. How to truncate this type of series?
Is this correct way to solve this integral or any other alternative is possible?
If there is no other alternative to solve this kind of integral, can you please suggest a method to truncate the divergent series.
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2015-02-28 14:21:52
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https://stefvanbuuren.name/fimd/sec-stepwise.html
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## 5.4 Stepwise model selection
The standard multiple imputation scheme consists of three phases:
1. Imputation of the missing data $$m$$ times;
2. Analysis of the $$m$$ imputed datasets;
3. Pooling of the parameters across $$m$$ analyses.
This scheme is difficult to apply if stepwise model selection is part of the statistical analysis in phase 2. Application of stepwise variable selection methods may result in sets of variables that differ across the $$m$$ datasets. It is not obvious how phase 3 should be done.
### 5.4.1 Variable selection techniques
Brand (1999, chap. 7) was the first to recognize and treat the variable selection problem. He proposed a solution in two steps. The first step involves performing stepwise model selection separately on each imputed dataset, followed by the construction of a new supermodel that contains all variables that were present in at least half of the initial models. The idea is that this criterion excludes variables that were selected accidentally. Moreover, it is a rough correction for multiple testing. Second, a special procedure for backward elimination is applied to all variables present in the supermodel. Each variable is removed in turn, and the pooled likelihood ratio $$p$$-value (Equation (5.14)) is calculated. If the largest $$p$$-value is larger than 0.05, the corresponding variable is removed, and the procedure is repeated on the smaller model. The procedure stops if all $$p \leq 0.05$$. The procedure was found to be a considerable improvement over complete-case analysis.
Yang, Belin, and Boscardin (2005) proposed variable selection techniques using Bayesian model averaging. The authors studied two methods. The first method, called “impute then select,” applies Bayesian variable selection methods on the imputed data. The second method, called “simultaneously impute and select” combines selection and missing data imputation into one Gibbs sampler. Though the latter slightly outperformed the first method, the first method is more broadly applicable. Application of the second method seems to require equivalent imputation and analysis models, thus defeating one of the main advantages of multiple imputation.
Wood, White, and Royston (2008) and Vergouwe et al. (2010) studied several scenarios for variable selection. We distinguish three general approaches:
1. Majority. A method that selects variables in the final that appear in at least half of the models.
2. Stack. Stack the imputed datasets into a single dataset, assign a fixed weight to each record and apply the usual variable selection methods.
3. Wald. Stepwise model selection is based on the Wald statistic calculated from the multiply imputed data.
The majority method is identical to step 1 of Brand (1999), whereas the Wald test method is similar to Brand’s step 2, with the likelihood ratio test replaced by the Wald test. The Wald test method is recommended since it is a well-established approach that follows Rubin’s rules, whereas the majority and stack methods fail to take into account the uncertainty caused by the missing data. Indeed, Wood, White, and Royston (2008) found that the Wald test method is the only procedure that preserved the type I error.
Zhao and Long (2017) review recent work on variable selection on imputed data. These authors favor approaches based on the least absolute shrinkage and selection operator (LASSO) (Tibshirani 1996). The MI-LASSO method by Chen and Wang (2013) tests the coefficients across all the stacked datasets, thus ensuring model consistency across different imputations. Marino, Buxton, and Li (2017) proposed an extension to select covariates in multilevel models.
In practice, it may be useful to combine methods. The Wald test method is computationally intensive, but is now easily available in mice as the D1() function. A strong point of the majority method is that it gives insight into the variability between the imputed datasets. An advantage of the stack method is that only one dataset needs to be analyzed. The discussion of Wood, White, and Royston (2008) contains additional simulations of a two-step method, in which a preselection made by the majority and stack methods is followed by the Wald test. This yielded a faster method with better theoretical properties. In practice, a judicious combination of approaches might turn out best.
### 5.4.2 Computation
The following steps illustrate the main steps involved by implementing a simple majority method to select variables in mice.
data <- boys[boys$age >= 8, -4] imp <- mice(data, seed = 28382, m = 10, print = FALSE) scope <- list(upper = ~ age + hgt + wgt + hc + gen + phb + reg, lower = ~1) expr <- expression(f1 <- lm(tv ~ 1), f2 <- step(f1, scope = scope)) fit <- with(imp, expr) This code imputes the boys data $$m = 10$$ times, fits a stepwise linear model to predict tv (testicular volume) separately to each of the imputed dataset. The following code blocks counts how many times each variable was selected. formulas <- lapply(fit$analyses, formula)
terms <- lapply(formulas, terms)
votes <- unlist(lapply(terms, labels))
table(votes)
votes
age gen hc hgt phb reg wgt
10 9 1 6 9 10 1
The lapply() function is used three times. The first statement extracts the model formulas fitted to the $$m$$ imputed datasets. The second lapply() call decomposes the model formulas into pieces, and the third call extracts the names of the variables included in all $$m$$ models. The table() function counts the number of times that each variable in the 10 replications. Variables age, gen and reg are always included, whereas hc was selected in only one of the models. Since hgt appears in more than 50% of the models, we can use the Wald test to determine whether it should be in the final model.
fit.without <- with(imp, lm(tv ~ age + gen + reg + phb))
fit.with <- with(imp, lm(tv ~ age + gen + reg + phb + hgt))
D1(fit.with, fit.without)
test statistic df1 df2 df.com p.value riv
1 ~~ 2 2.15 1 19.3 409 0.159 0.978
The $$p$$-value is equal to 0.173, so hgt is not needed in the model. If we go one step further, and remove phb, we obtain
fit.without <- with(imp, lm(tv ~ age + gen + reg))
fit.with <- with(imp, lm(tv ~ age + gen + reg + phb))
D1(fit.with, fit.without)
test statistic df1 df2 df.com p.value riv
1 ~~ 2 2.49 5 97.9 410 0.0362 1.29
The significant difference ($$p=0.029$$) between the models implies that phb should be retained. We obtain similar results for the other three variables, so the final model contains age, gen, reg and phb.
### 5.4.3 Model optimism
The main danger of data-driven model building strategies is that the model found may depend highly on the sample at hand. For example, Viallefont, Raftery, and Richardson (2001) showed that of the variables declared to be “significant” with $$p$$-values between 0.01 and 0.05 by stepwise variable selection, only 49% actually were true risk factors. Various solutions have been proposed to counter such model optimism. A popular procedure is bootstrapping the model as developed in Sauerbrei and Schumacher (1992) and Harrell (2001). Although Austin (2008) found it ineffective to identify true predictors, this method has often been found to work well for developing predictive models. The method randomly draws multiple samples with replacement from the observed sample, thus mimicking the sampling variation in the population from which the sample was drawn. Stepwise regression analyses are replayed in each bootstrap sample. The proportion of times that each prognostic variable is retained in the stepwise regression model is known as the inclusion frequency (Sauerbrei and Schumacher 1992). This proportion provides information about the strength of the evidence that an indicator is an independent predictor. In addition, each bootstrap model can be fitted to the original sample. The difference between the apparent performance and the bootstrap performance provides the basis for performance measures that correct for model optimism. Steyerberg (2009, 95) provides an easy-to-follow procedure to calculate such optimism-corrected performance measures.
Clearly, the presence of missing data adds uncertainty to the model building process, so optimism can be expected to be more severe with missing data. It is not yet clear what the best way is to estimate optimism from incomplete data. Heymans et al. (2007) explored the combination of multiple imputation and the bootstrap. There appear to be at least four general procedures:
1. Imputation. Multiple imputation generates 100 imputed datasets. Automatic backward selection is applied to each dataset. Any differences found between the 100 fitted models are due to the missing data.
2. Bootstrap. 200 bootstrap samples are drawn from one singly imputed completed data. Automatic backward selection is applied to each dataset. Any differences found between the 200 fitted models are due to sampling variation.
3. Nested bootstrap. The bootstrap method is applied on each of the multiply imputed datasets. Automatic backward selection is applied to each of the $$100 \times 200$$ datasets. Differences between the fitted model portray both sampling and missing data uncertainty.
4. Nested imputation. The imputation method is applied on each of the bootstrapped datasets.
Heymans et al. (2007) observed that the imputation method produced a wider range of inclusion frequencies than the bootstrap method. This is attractive since a better separation of strong and weak predictors may ease model building. The area under the curve is an overall index of predictive strength. Though the type of method had a substantial effect on the apparent $$c$$-index estimate, the optimism-corrected $$c$$-index estimate was quite similar. The optimism-corrected calibration slope estimates tended to be lower in the methods involving imputation, thus necessitating more shrinkage.
A drawback of the method is the use of classic stepwise variable selection techniques, which do not generalize well to high-dimensional data. Musoro et al. (2014) improved the methods of Heymans et al. (2007) through their use of the LASSO.
Long and Johnson (2015) developed a procedure, called bootstrap imputation and stability selection (BI-SS) , that generates bootstrap samples from the original data, imputes each bootstrap sample by single imputation, obtains the randomized LASSO estimate from each sample, and then selects the active set according to majority. The multiple imputation random LASSO (MIRL) method by Liu et al. (2016) first performs multiple imputation, obtains bootstrap samples from each imputed dataset, estimates regression weights under LASSO, and then selects the active set by majority. It is not yet known how BS-SS and MIRL compare to each other.
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2018-12-14 10:09:42
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http://openstudy.com/updates/4f348e2fe4b0fc0c1a0c1364
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## anonymous 4 years ago 1. A spring has a natural length of 10 in. An 800-lb force stretches the spring to 14 in. (a) Find the force constant. (b) How much work is done in stretching the spring from 10 in. to 12 in.? (c) How far beyond its natural length will a 1600-lb force stretch the spring
1. anonymous
2. anonymous
Wouldn't I need to integrate since it doesn't start from 0?
3. UnkleRhaukus
yeah integrate the force time displacement to get the total work $W_{0 \rightarrow x} =\int_0^xFdx$
4. anonymous
No I mean would I integrate to get the constant K?
5. anonymous
or would it just be 800=4k
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2016-10-28 20:10:03
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https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_12B_Problems/Problem_17&diff=next&oldid=110914
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# Difference between revisions of "2007 AMC 12B Problems/Problem 17"
## Problem 17
If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10}b$, what is the median of the set $\{0,1,a,b,1/b\}$?
$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ a \qquad \mathrm{(D)}\ b \qquad \mathrm{(E)}\ \frac{1}{b}$
## Solution
Note that if $a$ is positive, then, the equation will have no solutions for $b$. This becomes more obvious by noting that at $b=1$, $ab^2 > \log_{10} b$. The LHS quadratic function will increase faster than the RHS logarithmic function, so they will never intersect.
This puts $a$ as the smallest in the set since it must be negative.
Checking the new equation: $-b^2 = \log_{10}b$
Near $b=0$, $-b^2 > \log_{10} b$ but at $b=1$, $-b^2 < \log_{10} b$
This implies that the solution occurs somewhere in between: $0 < b < 1$
This also implies that $\frac{1}{b} > 1$
This makes our set (ordered) $\{a,0,b,1,1/b\}$
The median is $b \Rightarrow \mathrm {(D)}$
## Cheap Solution that most people probably used
Led $b=0.1$. Then $a\cdot0.01 = -1,$ giving $a=-100$. Then the ordered set is $\{-100, 0, 0.1, 1, 10\}$ and the median is $0.1=b,$ so the answer is $\mathrm {(D)}$.
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2021-12-09 07:47:30
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https://msp.org/agt/2009/9-4/p06.xhtml
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#### Volume 9, issue 4 (2009)
Recent Issues
The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Amalgamations of Heegaard splittings in $3$–manifolds without some essential surfaces
### Guoqiu Yang and Fengchun Lei
Algebraic & Geometric Topology 9 (2009) 2041–2054
##### Abstract
Let $M$ be a compact, orientable, $\partial$–irreducible $3$–manifold and $F$ be a connected closed essential surface in $M$ with $g\left(F\right)\ge 1$ which cuts $M$ into ${M}_{1}$ and ${M}_{2}$. In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary $\left({Q}_{i},\partial {Q}_{i}\right)$ in $\left({M}_{i},F\right)$ satisfying $\chi \left({Q}_{i}\right)\ge 2+g\left(F\right)-2g\left({M}_{i}\right)+1$, $i=1,2$. Then $g\left(M\right)=g\left({M}_{1}\right)+g\left({M}_{2}\right)-g\left(F\right)$. As a consequence, we further show that if ${M}_{i}$ has a Heegaard splitting ${V}_{i}{\cup }_{{S}_{i}}{W}_{i}$ with distance $D\left({S}_{i}\right)\ge 2g\left({M}_{i}\right)-g\left(F\right)$, $i=1,2$, then $g\left(M\right)=g\left({M}_{1}\right)+g\left({M}_{2}\right)-g\left(F\right)$.
The main results follow from a new technique which is a stronger version of Schultens’ Lemma.
##### Keywords
essential surface, Heegaard genus
##### Mathematical Subject Classification 2000
Primary: 57M99, 57N10
Secondary: 57M27
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2020-09-18 11:57:47
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https://gre.kmf.com/question/all/181?keyword=&page=9
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#### 题目列表
A deck can be built by 5 workers in 6 hours. Working at the same rate, how many hours would it take 8 workers to build the same deck?
The average of 1, 1, 3, 2, 5, 4, x,y is 3. What is the value of x and y?
Indicate all possible choices.
$x\neq-y$
x+y
#### Quantity B
||$\frac{1}{x+y}$
In △ABC, ∠A=62°, ∠B=58°
AB
AC
90°
#### Quantity B
Measure of angle Q
$(x^{2})(y^{3})>0 (x^{3})(y^{2})<0$
x
#### Quantity B
y
The radius of circle M is less than the diameter of circle N
#### Quantity A
The circumference of circle N
#### Quantity B
The circumference of circle M
xy>0
$xy^{2}$<0
x
#### Quantity B
y
|x|≥1
$|y|<(\frac{1}{2})$
#### Quantity A
$\frac{x}{y}$
#### Quantity B
$\frac{y}{x}$
A rectangle has a side of 10 and a diagonal of 26.
What is the area of each right triangle formed by the diagonal of the rectangle?
_____
The average (arithmetic mean) of 8, j, 21, and 24 is k. The average of k, 25, and 30 is 24. What is the value of j?
The equation of l1 is y=2x+ 6. If l2 is parallel to A. which of the following could be the equation of l2?
If x, y, and z are consecutive odd integers, which of the following must be odd?
If 2 is the remainder when m is divided by 5, what is the remainder when 3m is divided by 5?
Jasmine drives the first 150 miles of her trip at an average speed of 50 miles per hour. If she drives the remaining 80 miles of her trip at an average of 40 miles per hour, what is her average speed, in miles per hour, for the entire trip?
If a and b are prime numbers, such that a > b, which of the following cannot be true?
If 5x + y = 2x + 4y, what is x in terms of y?
Amanda goes to the toy store to buy 1 ball and 3 different board games.
If the toy store is stocked with 3 types of balls and 6 types of board games, how many different selections of the 4 items can Amanda make?
The diagonal of a square with side 4 is a radius of a circle. What is the area of the circle?
#### Quantity A
The original price of a gaming system
#### Quantity B
The final price of the gaming system after a 20% price decrease then a 25% price increase
1 2 ... 6 7 8 9 10 11 12 ... 22 23
25000 +道题目
137本备考书籍
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2021-10-25 11:07:23
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https://freakonometrics.hypotheses.org/tag/carter
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# Reinterpreting Lee-Carter Mortality Model
Last week, while I was giving my crash course on R for insurance, we’ve been discussing possible extensions of Lee & Carter (1992) model. If we look at the seminal paper, the model is defined as follows
# Smoothing mortality rates
This morning, I was working with Julie, a student of mine, coming from Rennes, on mortality tables. Actually, we work on genealogical datasets from a small region in Québec, and we can observe a lot of volatiliy. If I borrow one of her graph, we get something like
Since we have some missing data, we wanted to use some Generalized Nonlinear Models. So let us see how to get a smooth estimator of the mortality surface. We will write some code that we can use on our data later on (the dataset we have has been obtained after signing a lot of official documents, and I guess I cannot upload it here, even partially).
"http://freakonometrics.free.fr/Deces-France.txt",
"http://freakonometrics.free.fr/Exposures-France.txt",
library(gnm)
D=DEATH$Male E=EXPO$Male
A=as.numeric(as.character(DEATH$Age)) Y=DEATH$Year
I=(A<100)
base=data.frame(D=D,E=E,Y=Y,A=A)
subbase=base[I,]
subbase=subbase[!is.na(subbase$A),] The first idea can be to use a Poisson model, where the mortality rate is a smooth function of the age and the year, something like $D_{x,t}\sim\mathcal{P}(E_{x,t}\cdot \exp[{\color{blue}s(x,t)}])$that can be estimated using library(mgcv) regbsp=gam(D~s(A,Y,bs="cr")+offset(log(E)),data=subbase,family=quasipoisson) predmodel=function(a,y) predict(regbsp,newdata=data.frame(A=a,Y=y,E=1)) vX=trunc(seq(0,99,length=41)) vY=trunc(seq(1900,2005,length=41)) vZ=outer(vX,vY,predmodel) persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)", ylab="Years (1900-2005)",zlab="Mortality rate (log)") The mortality surface is here It is also possible to extract the average value of the years, which is the interpretation of the $a_x$ coefficient in the Lee-Carter model, predAx=function(a) mean(predict(regbsp,newdata=data.frame(A=a, Y=seq(min(subbase$Y),max(subbase$Y)),E=1))) plot(seq(0,99),Vectorize(predAx)(seq(0,99)),col="red",lwd=3,type="l") We have the following smoothed mortality rate Recall that the Lee-Carter model is $D_{x,t}\sim\mathcal{P}(E_{x,t}\cdot \exp[{\color{blue}a_x+b_x\cdot k_t}])$ where parameter estimates can be obtained using regnp=gnm(D~factor(A)+Mult(factor(A),factor(Y))+offset(log(E)), data=subbase,family=quasipoisson) predmodel=function(a,y) predict(regnp,newdata=data.frame(A=a,Y=y,E=1)) vZ=outer(vX,vY,predmodel) persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)", ylab="Years (1900-2005)",zlab="Mortality rate (log)") The (crude) mortality surface is with the following $a_x$ coefficients. plot(seq(1,99),coefficients(regnp)[2:100],col="red",lwd=3,type="l") Here we have a lot of coefficients, and unfortunately, on a smaller dataset, we have much more variability. Can we smooth our Lee-Carter model ? To get something which looks like $D_{x,t}\sim\mathcal{P}(E_{x,t}\cdot \exp[{\color{blue}s_a(x)+s_b(x)\cdot s_k(t)}])$ Actually, we can, and the code is rather simple library(splines) knotsA=c(20,40,60,80) knotsY=c(1920,1945,1980,2000) regsp=gnm(D~bs(subbase$A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3)+ Mult(bs(subbase$A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3), bs(subbase$Y,knots=knotsY,Boundary.knots=range(subbase$Y),degre=3))+ offset(log(E)),data=subbase, family=quasipoisson) BpA=bs(seq(0,99),knots=knotsA,Boundary.knots=range(subbase$A),degre=3)
BpY=bs(seq(min(subbase$Y),max(subbase$Y)),knots=knotsY,Boundary.knots= range(subbase$Y),degre=3) predmodel=function(a,y) predict(regsp,newdata=data.frame(A=a,Y=y,E=1)) v Z=outer(vX,vY,predmodel) persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)", ylab="Years (1900-2005)",zlab="Mortality rate (log)") The mortality surface is now and again, it is possible to extract the average mortality rate, as a function of the age, over the years, BpA=bs(seq(0,99),knots=knotsA,Boundary.knots=range(subbase$A),degre=3)
Ax=BpA%*%coefficients(regsp)[2:8]
plot(seq(0,99),Ax,col="red",lwd=3,type="l")
We can then play with the smoothing parameters of the spline functions, and see the impact on the mortality surface
knotsA=seq(5,95,by=5)
knotsY=seq(1910,2000,by=10)
regsp=gnm(D~bs(A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3)+ Mult(bs(A,knots=knotsA,Boundary.knots=range(subbase$A),degre=3),
bs(Y,knots=knotsY,Boundary.knots=range(subbase$Y),degre=3)) +offset(log(E)),data=subbase,family=quasipoisson) predmodel=function(a,y) predict(regsp,newdata=data.frame(A=a,Y=y,E=1)) vZ=outer(vX,vY,predmodel) persp(vZ,theta=-30,col="green",shade=TRUE,xlab="Ages (0-100)", ylab="Years (1900-2005)",zlab="Mortality rate (log)") We now have to use those functions our our small data sample ! That should be fun…. # Course on risk, insurance, and uncertainty The course on risk and insurance in Luminy, starts at 10.30 on Friday (here) instead of Thursday (I switched with Patrice Bertail). The slides can be found here, Then, it will be time to leave Marseille, # Modélisation des tables de mortalité En cours d’actuariat, nous commencerons à parler des tables de mortalité prospectives. Côté théorie, je peux renvoyer vers quelques slides de Pierre Devolder, ici. Je renvoie ici pour un court papier de vulgarisation sur le sujet. Sur l’ajustement de lois de mortalité prospectives, je renvoie au très beau mémoire en ligne ici (ou pour des choses très proches en anglais). Pour l’approche statique, je renvoie également ici pour une méthode simple d’ajustement de loi de Makeham. Pour les aspects dynamique, il y a des documents techniques ici ou . Sinon pour les données que nous utiliserons en TD, elles sont ici (pour l’exposition, i.e. le nombre de vivants) et (pour les décès). Il faut retravailler un peu les données, à cause d’une valeur “110+”, mais rien de plus simple, > DEATH=read.table("https://perso.univ-rennes1.fr/arthur.charpentier/R/FR-Deaths_1x1.txt",header=TRUE) > EXPOSURE=read.table("https://perso.univ-rennes1.fr/arthur.charpentier/R/FR-Exposures_1x1.txt",header=TRUE) > DEATH$Age=as.numeric(as.character(DEATH$Age)) Message d'avis : NAs introduits lors de la conversion automatique > DEATH$Age[is.na(DEATH$Age)]=110 > EXPOSURE$Age=as.numeric(as.character(EXPOSURE$Age)) Message d'avis : NAs introduits lors de la conversion automatique > EXPOSURE$Age[is.na(EXPOSURE$Age)]=110 Coté code, je renvoie à un précédant billet sur le sujet, ici, et pour le package (non officiel) de démographie développé par Rob Hyndman. Enfin, à la fin du cours, on parlera un peu de modèles de durée. J’essayerai de taper quelques choses dans les jours à venir, mais côté TD, on utilisera ce que fait Frédéric Planchet (qui connaît fort bien le sujet), et qui se trouve en ligne ici. Les données sont en ligne ici, avec un peu de code R pour retraiter les donnéesici et , et un peu de code pour des modèles classiques, comme du modèle de Cox (ici) ou du Kaplan Meier (). Les codes R utilisés en TD sont les suivants (avec en plus les graphiques obtenus), > D=DEATH[DEATH$Year==an,]
> E=EXPOSURE[EXPOSURE$Year==an,] > MU = D[,3:5]/E[,3:5] > plot(0:110,log(MU[,1]),type="l",col="red") > lines(0:110,log(MU[,2]),col="blue") Avec ci-dessus les log des taux de décès instantannés pour l’année 1986. > PH=matrix(NA,111,111) # ligne X, colonne H > PF=PH > for(x in 0:110){ + PH[x+1,1:(111-x)]=exp(-cumsum(MU[(x+1):111,2])) + PF[x+1,1:(111-x)]=exp(-cumsum(MU[(x+1):111,1])) + } > x=0 > plot(1:111,PH[x+1,],ylim=c(0,1),type="l",col="blue") > lines(1:111,PF[x+1,],col="red") pour la fonction de survie à la naissance (ci-dessus) ou à 50 ans (ci-dessous), > somme=function(x){sum(x,na.rm=TRUE)} > EH=EF=rep(NA,111) > EH=apply(PH,1,somme) > EF=apply(PF,1,somme) > plot(0:110,EH,type="l",col="blue") > lines(0:110,EF,col="red") Ce qui correspond aux espérances de vie résiduelles. > for(k in 1:3){ + Z=MU[,k] + Z[is.nan(Z)==TRUE]=NA + MU[,k]=Z + } > AGE=unique(DEATH$Age)
> ANNEE=unique(DEATH$Year) > MUF=matrix(MU[,1],length(AGE),length(ANNEE)) > MUH=matrix(MU[,2],length(AGE),length(ANNEE)) > persp(AGE[1:100],ANNEE,log(MUH[1:100,]), + theta=-30,col="light green",shade=TRUE) > persp(AGE[1:100],ANNEE,log(MUH[1:100,107:1]), + theta=-30,col="light green",shade=TRUE) > image(AGE,ANNEE,log(MUH)) On a retourné le graph en mettant les années à l’envers, mais on peut aussi regarder la surface pour les femmes, ainsi que l’allure des courbes de niveau > persp(AGE[1:100],ANNEE,log(MUF[1:100,107:1]), + theta=-30,col="light green",shade=TRUE) > image(AGE,ANNEE,log(MUF)) Mais nous n’avons ici qu’une lecture statique des tables, et nous ne suivons pas vraiment un individu: en même temps que la personne vieilli, le temps passe…. Considérons une personne née en 1900, et regardons (ex post) l’évolution de sa probabilité de décès (en regardant non pas la probabilité de décéder à 18 d’une personne de 18 ans en 1900, mais en tenant compte du fait que la personne a fêté ses 18 ans en 1918). > MUFP=MUHP=rep(NA,111) # MU PROSPECTIF > for(i in 1:111){ + MUHP[i]=MUH[i,colonne+i] + MUFP[i]=MUF[i,colonne+i] + } > plot(0:110,log(MUHP),col="purple",type="l") > D=DEATH[DEATH$Year==annee.naissance,]
> E=EXPOSURE[EXPOSURE$Year==annee.naissance,] > MU = D[,3:5]/E[,3:5] > lines(0:110,log(MU[,2]),col="blue") On peut aussi l’ajustement par un modèle à la Lee-Carter (via la fonction fournie par LifeMetrics) > source("https://perso.univ-rennes1.fr/arthur.charpentier/fitModels.r") > DEATH=DEATH[DEATH$Age<90,] # virer age>90
> EXPOSURE=EXPOSURE[EXPOSURE$Age<90,] # virer age>90 > XV=unique(DEATH$Age)
> YV=unique(DEATH$Year) > ETF=t(matrix(EXPOSURE[,3],length(XV),length(YV))) > DTF=t(matrix(DEATH[,3],length(XV),length(YV))) > ETH=t(matrix(EXPOSURE[,4],length(XV),length(YV))) > DTH=t(matrix(DEATH[,4],length(XV),length(YV))) > WA=matrix(1,length(YV),length(XV)) > LCF = fit701(xv=XV,yv=YV,etx=ETF,dtx=DTF,wa=WA) -1166100 -> -263940.2 ->-204175.2 -188375.9 -> -185822.4 ->-184403.5 -183223.3 -> -183069.8 ->-182350.3 -181701.1 -> -181600.6 ->-181145.5 -180729.6 -> -180671.5 ->-180371.8 -180101.9 -> -180068.6 ->-179871.2 > LCH = fit701(xv=XV,yv=YV,etx=ETH,dtx=DTH,wa=WA) -2564403 -> -1028776 ->-953629.9 -881334.2 -> -868279.3 ->-834413.6 -804946.8 -> -802817.2 ->-786343 -772775 -> -772007.5 ->-764501.7 -758653.8 -> -758399 ->-755212 -752931 -> -752851 ->-751620.2 -750811.1 -> -750786.7 ->-750351.4 > plot(LCF$x,LCF$beta1,type="l",col="red") > lines(LCH$x,LCH$beta1,col="blue") > plot(LCF$x,LCF$beta2,type="l",col="red") > lines(LCH$x,LCH$beta2,col="blue") > plot(LCF$y,LCF$kappa2,type="l",col="red") > lines(LCH$y,LCH$kappa2,col="blue") > MUF = DTF/ETF > MUH = DTH/ETH > MUFMODEL = MUF; MUHMODEL = MUH > for(i in 1:nrow(MUF)){ + for(j in 1:ncol(MUF)){ + MUFMODEL[i,j] = exp(LCF$beta1[j]
+ +LCF$beta2[j]*LCF$kappa2[i])
+ MUHMODEL[i,j] = exp(LCH$beta1[j] + +LCH$beta2[j]*LCH$kappa2[i]) + }} soit, pour avoir un dessin, > persp(LCH$y,LCH$x,log(MUH), + theta=45,col="light green",shade=TRUE) > persp(LCH$y,LCH\$x,log(MUHMODEL),
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2020-11-27 00:35:07
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https://learn.microsoft.com/en-us/dotnet/framework/deployment/how-the-runtime-locates-assemblies
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# How the Runtime Locates Assemblies
To successfully deploy your .NET Framework application, you must understand how the common language runtime locates and binds to the assemblies that make up your application. By default, the runtime attempts to bind with the exact version of an assembly that the application was built with. This default behavior can be overridden by configuration file settings.
The common language runtime performs a number of steps when attempting to locate an assembly and resolve an assembly reference. Each step is explained in the following sections. The term probing is often used when describing how the runtime locates assemblies; it refers to the set of heuristics used to locate the assembly based on its name and culture.
Note
You can view binding information in the log file using the Assembly Binding Log Viewer (Fuslogvw.exe), which is included in the Windows SDK.
## Initiating the Bind
The process of locating and binding to an assembly begins when the runtime attempts to resolve a reference to another assembly. This reference can be either static or dynamic. The compiler records static references in the assembly manifest's metadata at build time. Dynamic references are constructed on the fly as a result of calling various methods, such as Assembly.Load.
The preferred way to reference an assembly is to use a full reference, including the assembly name, version, culture, and public key token (if one exists). The runtime uses this information to locate the assembly, following the steps described later in this section. The runtime uses the same resolution process regardless of whether the reference is for a static or dynamic assembly.
You can also make a dynamic reference to an assembly by providing the calling method with only partial information about the assembly, such as specifying only the assembly name. In this case, only the application directory is searched for the assembly, and no other checking occurs. You make a partial reference using any of the various methods for loading assemblies such as Assembly.Load or AppDomain.Load.
Finally, you can make a dynamic reference using a method such as Assembly.Load and provide only partial information; you then qualify the reference using the <qualifyAssembly> element in the application configuration file. This element allows you to provide the full reference information (name, version, culture and, if applicable, the public key token) in your application configuration file instead of in your code. You would use this technique if you wanted to fully qualify a reference to an assembly outside the application directory, or if you wanted to reference an assembly in the global assembly cache but you wanted the convenience of specifying the full reference in the configuration file instead of in your code.
Note
This type of partial reference should not be used with assemblies that are shared among several applications. Because configuration settings are applied per application and not per assembly, a shared assembly using this type of partial reference would require each application using the shared assembly to have the qualifying information in its configuration file.
The runtime uses the following steps to resolve an assembly reference:
1. Determines the correct assembly version by examining applicable configuration files, including the application configuration file, publisher policy file, and machine configuration file. If the configuration file is located on a remote machine, the runtime must locate and download the application configuration file first.
2. Checks whether the assembly name has been bound to before and, if so, uses the previously loaded assembly. If a previous request to load the assembly failed, the request is failed immediately without attempting to load the assembly.
Note
The caching of assembly binding failures is new in .NET Framework version 2.0.
3. Checks the global assembly cache. If the assembly is found there, the runtime uses this assembly.
4. Probes for the assembly using the following steps:
1. If configuration and publisher policy do not affect the original reference and if the bind request was created using the Assembly.LoadFrom method, the runtime checks for location hints.
2. If a codebase is found in the configuration files, the runtime checks only this location. If this probe fails, the runtime determines that the binding request failed and no other probing occurs.
3. Probes for the assembly using the heuristics described in the probing section. If the assembly is not found after probing, the runtime requests the Windows Installer to provide the assembly. This acts as an install-on-demand feature.
Note
There is no version checking for assemblies without strong names, nor does the runtime check in the global assembly cache for assemblies without strong names.
## Step 1: Examining the Configuration Files
Assembly binding behavior can be configured at different levels based on three XML files:
• Application configuration file.
• Publisher policy file.
• Machine configuration file.
These files follow the same syntax and provide information such as binding redirects, the location of code, and binding modes for particular assemblies. Each configuration file can contain an <assemblyBinding> element that redirects the binding process. The child elements of the <assemblyBinding> element include the <dependentAssembly> element. The children of <dependentAssembly> element include the <assemblyIdentity> element, the <bindingRedirect> element, and the <codeBase> element.
Note
Configuration information can be found in the three configuration files; not all elements are valid in all configuration files. For example, binding mode and private path information can only be in the application configuration file. For a complete list of the information that is contained in each file, see Configuring Apps by Using Configuration Files.
### Application Configuration File
First, the common language runtime checks the application configuration file for information that overrides the version information stored in the calling assembly's manifest. The application configuration file can be deployed with an application, but is not required for application execution. Usually the retrieval of this file is almost instantaneous, but in situations where the application base is on a remote computer, such as in an Internet Explorer Web-based scenario, the configuration file must be downloaded.
For client executables, the application configuration file resides in the same directory as the application's executable and has the same base name as the executable with a .config extension. For example, the configuration file for C:\Program Files\Myapp\Myapp.exe is C:\Program Files\Myapp\Myapp.exe.config. In a browser-based scenario, the HTML file must use the <link> element to explicitly point to the configuration file.
The following code provides a simple example of an application configuration file. This example adds a TextWriterTraceListener to the Listeners collection to enable recording debug information to a file.
<configuration>
<system.diagnostics>
<trace useGlobalLock="false" autoflush="true" indentsize="0">
<listeners>
<add name="myListener" type="System.Diagnostics.TextWriterTraceListener, system version=1.0.3300.0, Culture=neutral, PublicKeyToken=b77a5c561934e089" initializeData="c:\myListener.log" />
</listeners>
</trace>
</system.diagnostics>
</configuration>
### Publisher Policy File
Second, the runtime examines the publisher policy file, if one exists. Publisher policy files are distributed by a component publisher as a fix or update to a shared component. These files contain compatibility information issued by the publisher of the shared component that directs an assembly reference to a new version. Unlike application and machine configuration files, publisher policy files are contained in their own assembly that must be installed in the global assembly cache.
The following is an example of a Publisher Policy configuration file:
<configuration>
<runtime>
<assemblyBinding xmlns="urn:schemas-microsoft-com:asm.v1">
<dependentAssembly>
<assemblyIdentity name="asm6" publicKeyToken="c0305c36380ba429" />
<bindingRedirect oldVersion="3.0.0.0" newVersion="2.0.0.0"/>
</dependentAssembly>
</assemblyBinding>
</runtime>
</configuration>
To create an assembly, you can use the Al.exe (Assembly Linker) tool with a command such as the following:
Al.exe /link:asm6.exe.config /out:policy.3.0.asm6.dll /keyfile: compatkey.dat /v:3.0.0.0
compatkey.dat is a strong-name key file. This command creates a strong-named assembly you can place in the global assembly cache.
Note
Publisher policy affects all applications that use a shared component.
The publisher policy configuration file overrides version information that comes from the application (that is, from the assembly manifest or from the application configuration file). If there is no statement in the application configuration file to redirect the version specified in the assembly manifest, the publisher policy file overrides the version specified in the assembly manifest. However, if there is a redirecting statement in the application configuration file, publisher policy overrides that version rather than the one specified in the manifest.
A publisher policy file is used when a shared component is updated and the new version of the shared component should be picked up by all applications using that component. The settings in the publisher policy file override settings in the application configuration file, unless the application configuration file enforces safe mode.
#### Safe Mode
Publisher policy files are usually explicitly installed as part of a service pack or program update. If there is any problem with the upgraded shared component, you can ignore the overrides in the publisher policy file using safe mode. Safe mode is determined by the <publisherPolicy apply="yes|no"/> element, located only in the application configuration file. It specifies whether the publisher policy configuration information should be removed from the binding process.
Safe mode can be set for the entire application or for selected assemblies. That is, you can turn off the policy for all assemblies that make up the application, or turn it on for some assemblies but not others. To selectively apply publisher policy to assemblies that make up an application, set <publisherPolicy apply=no/> and specify which assemblies you want to be affected using the <dependentAssembly> element. To apply publisher policy to all assemblies that make up the application, set <publisherPolicy apply=no/> with no dependent assembly elements. For more about configuration, see Configuring Apps by using Configuration Files.
### Machine Configuration File
Third, the runtime examines the machine configuration file. This file, called Machine.config, resides on the local computer in the Config subdirectory of the root directory where the runtime is installed. This file can be used by administrators to specify assembly binding restrictions that are local to that computer. The settings in the machine configuration file take precedence over all other configuration settings; however, this does not mean that all configuration settings should be put in this file. The version determined by the administrator policy file is final, and cannot be overridden. Overrides specified in the Machine.config file affect all applications. For more information about configuration files, see Configuring Apps by using Configuration Files.
## Step 2: Checking for Previously Referenced Assemblies
If the requested assembly has also been requested in previous calls, the common language runtime uses the assembly that is already loaded. This can have ramifications when naming assemblies that make up an application. For more information about naming assemblies, see Assembly Names.
If a previous request for the assembly failed, subsequent requests for the assembly are failed immediately without attempting to load the assembly. Starting with .NET Framework version 2.0, assembly binding failures are cached, and the cached information is used to determine whether to attempt to load the assembly.
Note
To revert to the behavior of the .NET Framework versions 1.0 and 1.1, which did not cache binding failures, include the <disableCachingBindingFailures> Element in your configuration file.
## Step 3: Checking the Global Assembly Cache
For strong-named assemblies, the binding process continues by looking in the global assembly cache. The global assembly cache stores assemblies that can be used by several applications on a computer. All assemblies in the global assembly cache must have strong names.
## Step 4: Locating the Assembly through Codebases or Probing
After the correct assembly version has been determined by using the information in the calling assembly's reference and in the configuration files, and after it has checked in the global assembly cache (only for strong-named assemblies), the common language runtime attempts to find the assembly. The process of locating an assembly involves the following steps:
1. If a <codeBase> element is found in the application configuration file, the runtime checks the specified location. If a match is found, that assembly is used and no probing occurs. If the assembly is not found there, the binding request fails.
2. The runtime then probes for the referenced assembly using the rules specified later in this section.
Note
If you have multiple versions of an assembly in a directory and you want to reference a particular version of that assembly, you must use the <codeBase> element instead of the privatePath attribute of the <probing> element. If you use the <probing> element, the runtime stops probing the first time it finds an assembly that matches the simple assembly name referenced, whether it is a correct match or not. If it is a correct match, that assembly is used. If it is not a correct match, probing stops and binding fails.
### Locating the Assembly through Codebases
Codebase information can be provided by using a <codeBase> element in a configuration file. This codebase is always checked before the runtime attempts to probe for the referenced assembly. If a publisher policy file containing the final version redirect also contains a <codeBase> element, that <codeBase> element is the one that is used. For example, if your application configuration file specifies a <codeBase> element, and a publisher policy file that is overriding the application information also specifies a <codeBase> element, the <codeBase> element in the publisher policy file is used.
If no match is found at the location specified by the <codeBase> element, the bind request fails and no further steps are taken. If the runtime determines that an assembly matches the calling assembly's criteria, it uses that assembly. When the file specified by the given <codeBase> element is loaded, the runtime checks to make sure that the name, version, culture, and public key match the calling assembly's reference.
Note
Referenced assemblies outside the application's root directory must have strong names and must either be installed in the global assembly cache or specified using the <codeBase> element.
### Locating the Assembly through Probing
If there is no <codeBase> element in the application configuration file, the runtime probes for the assembly using four criteria:
• Application base, which is the root location where the application is being executed.
• Culture, which is the culture attribute of the assembly being referenced.
• Name, which is the name of the referenced assembly.
• The privatePath attribute of the <probing> element, which is the user-defined list of subdirectories under the root location. This location can be specified in the application configuration file and in managed code using the AppDomainSetup.PrivateBinPath property for an application domain. When specified in managed code, the managed code privatePath is probed first, followed by the path specified in the application configuration file.
#### Probing the Application Base and Culture Directories
The runtime always begins probing in the application's base, which can be either a URL or the application's root directory on a computer. If the referenced assembly is not found in the application base and no culture information is provided, the runtime searches any subdirectories with the assembly name. The directories probed include:
• [application base] / [assembly name].dll
• [application base] / [assembly name] / [assembly name].dll
If culture information is specified for the referenced assembly, only the following directories are probed:
• [application base] / [culture] / [assembly name].dll
• [application base] / [culture] / [assembly name] / [assembly name].dll
#### Probing with the privatePath Attribute
In addition to the culture subdirectories and the subdirectories named for the referenced assembly, the runtime also probes directories specified using the privatePath attribute of the <probing> element. The directories specified using the privatePath attribute must be subdirectories of the application's root directory. The directories probed vary depending on whether culture information is included in the referenced assembly request.
The runtime stops probing the first time it finds an assembly that matches the simple assembly name referenced, whether it is a correct match or not. If it is a correct match, that assembly is used. If it is not a correct match, probing stops and binding fails.
If culture is included, the following directories are probed:
• [application base] / [binpath] / [culture] / [assembly name].dll
• [application base] / [binpath] / [culture] / [assembly name] / [assembly name].dll
If culture information is not included, the following directories are probed:
• [application base] / [binpath] / [assembly name].dll
• [application base] / [binpath] / [assembly name] / [assembly name].dll
#### Probing Examples
Given the following information:
• Referenced assembly name: myAssembly
• Application root directory: http://www.code.microsoft.com
• <probing> element in configuration file specifies: bin
• Culture: de
The runtime probes the following URLs:
• http://www.code.microsoft.com/de/myAssembly.dll
• http://www.code.microsoft.com/de/myAssembly/myAssembly.dll
• http://www.code.microsoft.com/bin/de/myAssembly.dll
• http://www.code.microsoft.com/bin/de/myAssembly/myAssembly.dll
##### Multiple Assemblies with the Same Name
The following example shows how to configure multiple assemblies with the same name.
<dependentAssembly>
<assemblyIdentity name="Server" publicKeyToken="c0305c36380ba429" />
<codeBase version="1.0.0.0" href="v1/Server.dll" />
<codeBase version="2.0.0.0" href="v2/Server.dll" />
</dependentAssembly>
#### Other Locations Probed
Assembly location can also be determined using the current binding context. This most often occurs when the Assembly.LoadFrom method is used and in COM interop scenarios. If an assembly uses the LoadFrom method to reference another assembly, the calling assembly's location is considered to be a hint about where to find the referenced assembly. If a match is found, that assembly is loaded. If no match is found, the runtime continues with its search semantics and then queries the Windows Installer to provide the assembly. If no assembly is provided that matches the binding request, an exception is thrown. This exception is a TypeLoadException in managed code if a type was referenced, or a FileNotFoundException if an assembly being loaded was not found.
For example, if Assembly1 references Assembly2 and Assembly1 was downloaded from http://www.code.microsoft.com/utils, that location is considered to be a hint about where to find Assembly2.dll. The runtime then probes for the assembly in http://www.code.microsoft.com/utils/Assembly2.dll and http://www.code.microsoft.com/utils/Assembly2/Assembly2.dll. If Assembly2 is not found at either of those locations, the runtime queries the Windows Installer.
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2023-03-27 05:33:46
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http://noenthuda.com/blog/category/computer-science/
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Hill Climbing in real life
Fifteen years back, I enrolled for a course on Artificial Intelligence as part of my B.Tech. programme at IIT Madras. It was well before stuff like “machine learning” and “data science” became big, and the course was mostly devoted to heuristics. Incidentally, that term, we had to pick between this course and one on Artificial Neural Networks (I guess nowadays that one is more popular given the hype about Deep Learning?), which meant that I didn’t learn about neural networks until last year or so.
A little googling tells me that Deepak Khemani, who taught us AI in 2002, has put up his lectures online, as part of the NPTEL programme. The first one is here:
In fact, the whole course is available here.
Anyways, one of the classes of problems we dealt with in the course was “search”. Basically, how does a computer “search” for the solution to a problem within a large “search space”?
One of the simplest heuristic is what has come to be known as “hill climbing” (too lazy to look through all of Khemani’s lectures and find where he’s spoken about this). I love computer science because a lot of computer scientists like to describe ideas in terms of intuitive metaphors. Hill climbing is definitely one such!
Let me explain it from the point of view of my weekend vacation in Edinburgh. One of my friends who had lived there a long time back recommended that I hike up this volcanic hill in the city called “Arthur’s Peak“.
On Saturday evening, I left my wife and daughter and wife’s parents (who I had travelled with) in our AirBnB and walked across town (some 3-4 km) to reach Holyrood Palace, from where Arthur’s Seat became visible. This is what I saw:
Basically, what you see is the side of a hill, and if you see closely, there are people walking up the sides. So what you guess is that you need to make your way to the bottom of the hill and then just climb.
But then you make your way to the base of the hill and see several paths leading up. Which one do you take? You take the path that seems steepest, believing that’s the one that will take you to the top quickest. And so you take a step along that path. And then see which direction to go to climb up steepest. Take another step. Rinse. Repeat. Until you reach a point where you can no longer find a way up. Hopefully that’s the peak.
Most of the time, you are likely to end up on the top of a smaller rock. In any case, this is the hill climbing algorithm.
So back to my story. I reached the base of the hill and set off on the steepest marked path.
I puffed and panted, but I kept going. It was rather windy that day, and it was threatening to rain. I held my folded umbrella and camera tight, and went on. I got beautiful views of Edinburgh city, and captured some of them on camera. And after a while, I got tired, and decided to call my wife using Facetime.
In any case, it appeared that I had a long way to go, given the rocks that went upwards just to my left (I was using a modified version of hill climbing in that I used only marked paths. As I was to rediscover the following day, I have a fear of heights). And I told that to my wife. And then suddenly the climb got easier. And before I knew it I was descending. And soon enough I was at the bottom all over again!
And then I saw the peak. Basically what I had been climbing all along was not the main hill at all! It was a “side hill”, which I later learnt is called the “Salisbury Crags”. I got down to the middle of the two hills, and stared at the valley there. I realised that was a “saddle point”, and hungry and tired and not wanting to get soaked in rain, I made my way out, hailed a cab and went home.
I wasn’t done yet. Determined to climb the “real peak”, I returned the next morning. Again I walked all the way to the base of the hill, and started my climb at the saddle point. It was a tough climb – while there were rough steps in some places, in others there was none. I kept climbing a few steps at a time, taking short breaks.
One such break happened to be too long, though, and gave me enough time to look down and feel scared. For a long time now I’ve had a massive fear of heights. Panic hit. I was afraid of going too close to the edge and falling off the hill. I decided to play it safe and turn back.
I came down and walked across the valley you see in the last picture above. Energised, I had another go. From what was possibly a relatively easier direction. But I was too tired. And I had to get back to the apartment and check out that morning. So I gave up once again.
I still have unfinished business in Edinburgh!
Maths, machine learning, brute force and elegance
Back when I was at the International Maths Olympiad Training Camp in Mumbai in 1999, the biggest insult one could hurl at a peer was to describe the latter’s solution to a problem as being a “brute force solution”. Brute force solutions, which were often ungainly, laboured and unintuitive were supposed to be the last resort, to be used only if one were thoroughly unable to implement an “elegant solution” to the problem.
Mathematicians love and value elegance. While they might be comfortable with esoteric formulae and the Greek alphabet, they are always on the lookout for solutions that are, at least to the trained eye, intuitive to perceive and understand. Among other things, it is the belief that it is much easier to get an intuitive understanding for an elegant solution.
When all the parts of the solution seem to fit so well into each other, with no loose ends, it is far easier to accept the solution as being correct (even if you don’t understand it fully). Brute force solutions, on the other hand, inevitably leave loose ends and appreciating them can be a fairly massive task, even to trained mathematicians.
In the conventional view, though, non-mathematicians don’t have much fondness for elegance. A solution is a solution, and a problem solved is a problem solved.
With the coming of big data and increased computational power, however, the tables are getting turned. In this case, the more mathematical people, who are more likely to appreciate “machine learning” algorithms recommend “leaving it to the system” – to unleash the brute force of computational power at the problem so that the “best model” can be found, and later implemented.
And in this case, it is the “half-blood mathematicians” like me, who are aware of complex algorithms but are unsure of letting the system take over stuff end-to-end, who bat for elegance – to look at data, massage it, analyse it and then find that one simple method or transformation that can throw immense light on the problem, effectively solving it!
The world moves in strange ways.
Schoolkid fights, blockchain and smart contracts
So I’ve been trying to understand the whole blockchain thing better, since people nowadays seem to be wanting to use it for all kinds of contracts (even the investment bankers are taking interest, which suggests there’s some potential out there 😛 ).
One of the things I’ve been doing is to read this book (PDF) on Blockchain by Arvind Narayanan and co at Princeton. It’s an easy to read, yet comprehensive, take on bitcoin and cryptocurrency technologies, the maths behind it and so on.
And as I’ve been reading it, I’ve been developing my own oversimplified model of what blockchain and smart contracts are, and this is my take at explaining it.
Imagine that Alice and Bob are two schoolkids and they’ve entered into a contract which states that if Alice manages to climb a particular tree, Bob will give her a bar of chocolate. Alice duly climbs the tree and claims the chocolate, at which point Bob flatly denies that she climbed it and refuses to give her the chocolate. What is Alice to do?
In the conventional “contract world”, all that Alice can do is to take the contract that she and Bob had signed (assume they had formalised it) and take it to a court of law (a schoolteacher, perhaps, in this case), which will do its best possible in order to determine whether she actually climbed the tree, and then deliver the judgment.
As you may imagine, in the normal schoolkid world, going to a teacher for adjudicating on whether someone climbed a tree (most likely an “illegal” activity by school rules) is not the greatest way to resolve the fight. Instead, either Alice and Bob will try to resolve it by themselves, or call upon their classmates to do the same. This is where the blockchain comes in.
Simply put, in terms of the blockchain “register”, as long as more than half of Alice and Bob’s classmates agree that she climbed the tree, she is considered to have climbed the tree, and Bob will be liable to give her chocolate. In other words, the central “trusted third party” gets replaced by a decentralised crowd of third parties where the majority decision is taken to be the “truth”.
Smart contracts take this one step further. Bob will give the bar of chocolates to the collective trust of his classmates (the adjudicators). And if a majority of them agree that Alice did climb the tree, the chocolate will be automatically given to her. If not, it will come back to Bob. What blockchain technologies allow for is to write code in a clever manner so that this can get executed automatically.
This might be a gross oversimplification, but this is exactly how the blockchain works. Each transaction is considered “valid” and put into the blockchain if a majority of nodes agrees it’s valid. And in order to ensure that this voting doesn’t get rigged, the nodes (or judges) need to perform a difficult computational puzzle in order to be able to vote – this imposes an artificial cost of voting which makes sure that it’s not possible to rig the polls unless you can take over more than half the nodes – and in a global blockchain where you have a really large number of nodes, this is not feasible.
So when you see that someone is building a blockchain based solution for this or that, you might wonder whether it actually makes sense. All you need to do is to come back to this schoolkid problem – for the kind of dispute that is likely to arise from this problem, would the parties prefer to go to a mutually trusted third party, or leave it to the larger peer group to adjudicate? Using the blockchain is a solution if and only if the latter case is true.
Programming back to the 1970s
I learnt to write computer code circa 1998, at a time when resources were plenty. I had a computer of my own – an assembled desktop with a 386 processor and RAM that was measured in MBs. It wasn’t particularly powerful, but it was more than adequate to handle the programs I was trying to write.
I wasn’t trying to process large amounts of data. Even when the algorithms were complex, they weren’t that complex. Most code ran in a matter of minutes, which meant that I didn’t need to bother about getting the code right the first time round – apart from for examination purposes. I could iterate and slowly get things right.
This was markedly different from how people programmed back in the 1970s, when computing resource was scarce and people had to mostly write code on paper. Time had to be booked at computer terminals, when the code would be copied onto the computers, and then run. The amount of time it took for the code to run meant that you had to get it right the first time round. Any mistake meant standing in line at the terminal again, and further time to run the code.
The problem was particularly dire in the USSR, where the planned economy meant that the shortages of computer resources were shorter. This has been cited as a reason as to why Russian programmers who migrated to the US were prized – they had practice in writing code that worked for the first time.
Anyway, the point of this post is that coding became progressively easier through the second half of the 20th century, when Moore’s Law was in operation, and computers became faster, smaller and significantly more abundant.
This process continues – computers continue to become better and more abundant – smartphones are nothing but computers. On the other side, however, as storage has gotten cheap and data capture has gotten easier, data sources are significantly larger now than they were a decade or two back.
So if you are trying to write code that uses a large amount of data, it means that each run can take a significant amount of time. When the data size reaches big data proportions (when it all can’t be processed on a single computer), the problem is more complex.
And in that sense, every time you want to run a piece of code, however simple it is, execution takes a long time. This has made bugs much more expensive again – the amount of time programs take to run means that you lose a lot of time in debugging and rewriting your code.
It’s like being in the 1970s all over again!
I don’t know if I’ve written about this before (that might explain how I crossed 2000 blogposts last year – multiple posts about the same thing), but anyway – I’m writing this listening to Aerosmith’s Dream On.
I don’t recall when the first time was that I heard the song, but I somehow decided that it sounded like Led Zeppelin. It was before 2006, so I had no access to services such as Shazam to search effectively. So for a long time I continued to believe it was by Led Zep, and kept going through their archives to locate the song.
And then in 2006, Pandora happened. It became my full time work time listening (bless those offshored offices with fast internet and US proxies). I would seed stations with songs I liked (back then there was no option to directly play songs you liked – you could only seed stations). I discovered plenty of awesome music that way.
And then one day I had put on a Led Zeppelin station and started work. The first song was by Led Zeppelin itself. And then came Dream On. And I figured it was a song by Aerosmith. While I chided myself for not having identified the band correctly, I was happy that I hadn’t been that wrong – given that Pandora uses machine learning on song patterns to identify similar songs, that Dream On had appeared in a LedZep playlist meant that I hadn’t been too far off identifying it with that band.
Ten years on, I’m not sure why I thought Dream On was by Led Zeppelin – I don’t see any similarities any more. But maybe the algorithms know better!
Coin change problem with change – Dijkstra’s Algorithm
The coin change problem is a well studied problem in Computer Science, and is a popular example given for teaching students Dynamic Programming. The problem is simple – given an amount and a set of coins, what is the minimum number of coins that can be used to pay that amount?
So, for example, if we have coins for 1,2,5,10,20,50,100 (like we do now in India), the easiest way to pay Rs. 11 is by using two coins – 10 and 1. If you have to pay Rs. 16, you can break it up as 10+5+1 and pay it using three coins.
The problem with the traditional formulation of the coin change problem is that it doesn’t involve “change” – the payer is not allowed to take back coins from the payee. So, for example, if you’ve to pay Rs. 99, you need to use 6 coins (50+20+20+5+2+2). On the other hand, if change is allowed, Rs. 99 can be paid using just 2 coins – pay Rs. 100 and get back Re. 1.
So how do you determine the way to pay using fewest coins when change is allowed? In other words, what happens to the coin change problems when negative coins can be used? (Paying 100 and getting back 1 is the same as paying 100 and (-1) ) .
Unfortunately, dynamic programming doesn’t work in this case, since we cannot process in a linear order. For example, the optimal way to pay 9 rupees when negatives are allowed is to break it up as (+10,-1), and calculating from 0 onwards (as we do in the DP) is not efficient.
For this reason, I’ve used an implementation of Dijkstra’s algorithm to determine the minimum number of coins to be used to pay any amount when cash back is allowed. Each amount is a node in the graph, with an edge between two amounts if the difference in amounts can be paid using a single coin. So there is an edge between 1 and 11 because the difference (10) can be paid using a single coin. Since cash back is allowed, the graph need not be directed.
So all we need to do to determine the way to pay each amount most optimally is to run Dijkstra’s algorithm starting from 0. The breadth first search has complexity \$latex O(M^2 n)\$ where $M$ is the maximum amount we want to pay, while $n$ is the number of coins.
I’ve implemented this algorithm using R, and the code can be found here. I’ve also used the algorithm to compute the number of coins to be used to pay all numbers between 1 and 10000 under different scenarios, and the results of that can be found here.
You can feel free to use this algorithm or code or results in any of your work, but make sure you provide appropriate credit!
PS: I’ve used “coin” here in a generic sense, in that it can mean “note” as well.
The Birthday Party Problem
Next Tuesday is my happy birthday. As of now, I’m not planning to have a party. And based on some deep graph theoretic analysis that the wife and I just did over the last hour, it’s unlikely I will – for forming a coherent set of people to invite is an NP-hard problem, it seems like.
So five birthdays back we had a party, organised by the wife and meant as a surprise to me. On all counts it seemed like a great party. Except that the guests decided to divide themselves into one large clique and one smaller clique (of 2 people), leaving me as the cut vertex trying to bridge these cliques. That meant the onus was on me to make sure the tiny clique felt included in the party, and it wasn’t a lot of fun.
The problem is this – how do you invite a subset of friends for a party so that intervention by the host to keep guests entertained is minimised?
Let’s try and model this. Assume your friends network can be represented by an unweighted undirected graph, with a pair of friends being connected by an edge if they know (and get along with) each other already. Also assume you have full information about this graph (not always necessary).
The problem lies in selecting a subgraph of this graph such that you can be confident that it won’t break into smaller pieces (since that will mean you bonding with each such sub-group), and no guest feels left out (since the onus of making them comfortable will fall on you).
Firstly, the subgraph needs to be connected. Then, we can safely eliminate all guests who have degree of either zero or one (former is obvious, latter since they’ll be too needy on their only friend). In fact, we can impose a condition that each guest should have a minimum degree of two even in the subgraph.
Then we need to impose conditions on a group in the party breaking away. We can assume that for a group of people to break away, they need to be a clique (it is not a robust requirement, since you and someone you find at a party can suddenly decide to find a room, but reasonable enough).
We can also assume that for a group to break away, the strength of their mutual connections should outweigh the strength of their connections to the rest of the group. Since we’re using unweighted graphs here, we can simply assume that a group can break away if the number of edges between this group and the rest of the network is less than the size of the group.
So if there is a group of three who, put together, have two connections to the rest of the group, the group can break away. Similarly, a clique of four will break away from the main group if they have three or less edges going over. And let’s assume that the host is not a part of this subgroup of guests.
Given these constraints, and constraints on party size (minimum and maximum number of guests to invite), how can we identify an appropriate subset of friends to invite for the party? And I’m assuming this problem is NP-Hard (without thinking too much about it) – so can we think of a good heuristic to solve this problem
Do let me know the answer before next Tuesday, else I may not be able to have a party this time as well!
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2017-07-26 20:26:46
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https://www.techwhiff.com/learn/ed-in-a-water-tank-in-two-configraions-b-block-2/142430
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# Ed in a water tank in two configraions() (B) block 2 sits on top of block...
###### Question:
ed in a water tank in two configraions() (B) block 2 sits on top of block 1; The water level will rise proportional to 1.Two blocks (V-2 m2, SG, 04 V2 Im, SG2-1.1) are to be plac block 2 sits separately on the tank bottom; the displaced volumes in each configuration. Compute the displaced water volumes AVA and A, configurations, clearly indicating which configuration has the greater displacement. Approx Ans : AVA 1.7 m3 and AVB 1.6 m3 for the two
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2023-02-06 22:44:23
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https://www.rosettacommons.org/docs/latest/scripting_documentation/RosettaScripts/Movers/movers_pages/IteratedConvergenceMover
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Back to Mover page.
## IteratedConvergence
Repeatedly applies a sub-mover until the given filter returns a value within the given delta for the given number of cycles
<IteratedConvergence name="(&string)" mover="(&string)" filter="(&string)" delta="(0.1 &real)" cycles="(1 &integer)" maxcycles="(1000 &integer)" />
• mover - the mover to repeatedly apply
• filter - the filter to use when assaying for convergence (should return a reasonable value from report_sm())
• delta - how close do the filter values have to be to count as converged
• cycles - for how many mover applications does the filter value have to fall within delta of the reference value before counting as converged. If the filter is outside of the range, the reference value is reset to the new filter value.
• maxcycles - exit regardless if filter doesn't converge within this many applications of the mover - intended mainly as a safety check to prevent infinite recursion.
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2019-01-21 07:09:26
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https://www.freemathhelp.com/forum/threads/113695-Simple-Integral-quot-Find-what-s-wrong-quot-question?s=c6baf0838fb31169ceee84a60ce07983&p=440787
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Thread: Simple Integral "Find what's wrong" question
1. Simple Integral "Find what's wrong" question
dsfsf.jpg
Question in attached picture.
Thanks for the help, kinda forgot the simple rules of splitting integrals.
All that came to mind was no dx attached to each new integral.
Also side question: if the prompt says "Evaluate the integral using a substitution prior to integration by parts." how do I approach that, especially since the tabular method is easily applicable to the derivative.
Appreciate it <3333333
2. Originally Posted by tesstess
dsfsf.jpg
Question in attached picture.
Thanks for the help, kinda forgot the simple rules of splitting integrals.
All that came to mind was no dx attached to each new integral.
Also side question: if the prompt says "Evaluate the integral using a substitution prior to integration by parts." how do I approach that, especially since the tabular method is easily applicable to the derivative.
Appreciate it <3333333
"dx" is superfluous, anyway, so long as the context is clear.
Why do you suppose there is anything "wrong" with it? Really, though, just check the parentheses.
3. Originally Posted by tesstess
dsfsf.jpg
Question in attached picture.
Thanks for the help, kinda forgot the simple rules of splitting integrals.
All that came to mind was no dx attached to each new integral.
Also side question: if the prompt says "Evaluate the integral using a substitution prior to integration by parts." how do I approach that, especially since the tabular method is easily applicable to the derivative.
Appreciate it <3333333
$\int$(f(x)+/-g(x))dx = $\int$f(x)dx +/- $\int$g(x)dx and $\int$kf(x)dx = k$\int$f(x)dx, for some constant k.
You need these two to do problems like your. One problem is that there are no dx's but there is another problem.
4. Originally Posted by tesstess
dsfsf.jpg
Question in attached picture.
Thanks for the help, kinda forgot the simple rules of splitting integrals.
All that came to mind was no dx attached to each new integral.
I agree with tkh that $dx$ is totally irrelevant.
This is a poor question, it is a bad 'grouping' question.
$\int {\left[ {fg + 4h - g} \right]} = \int {fg} + 4\int h - \int g$
5. Originally Posted by tkhunny
"dx" is superfluous, anyway, so long as the context is clear.
Why do you suppose there is anything "wrong" with it? Really, though, just check the parentheses.
The homework question prompt asked for whats wrong with it.
So I'm assuming through your response technically there should be "dx"'s and I need to comment on the solution's use of parentheses?
Thanks for the response! <3
6. Originally Posted by tesstess
dsfsf.jpg
Question in attached picture.
Thanks for the help, kinda forgot the simple rules of splitting integrals.
All that came to mind was no dx attached to each new integral.
Also side question: if the prompt says "Evaluate the integral using a substitution prior to integration by parts." how do I approach that, especially since the tabular method is easily applicable to the derivative.
Appreciate it <3333333
The statement will be correct if the square brackets on the RHS are removed.
7. I think it should be mentioned that the dx is not entirely irrelevant; many teachers will insist on it, though it is not nearly as important as the parenthesization in this example. So it may well be expected in the answer to this question.
It has several purposes, varying according to context:
• merely identifying the variable with respect to which you are integrating;
• (often) marking the end of the integrand;
• making the connection to definite integrals as sums (in fact, sums (∫ = S) of small differences (d));
• clarifying the fact that, as an antiderivative, the integral operates on (and undoes) a differential, so that ∫du = u;
• making the dimensional analysis work out.
It becomes much more important in multiple integrals, and is not needed when you just write ∫f as opposed to ∫f(x); but teachers who insist on it do have good reason to do so.
8. Originally Posted by Harry_the_cat
The statement will be correct if the square brackets on the RHS are removed.
Fair enough, the dx is not TOTALLY irrelevant, but
There is NO difference: $\int\;f(x)\;dx = \int\;dx\;f(x) = \int\;f(x)$
The context must be CLEAR.
Textbook convention and/or teacher requirements my disagree.
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2018-12-19 09:25:23
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https://tex.stackexchange.com/questions/580747/how-can-i-type-text-above-a-matrix
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# How can I type text above a matrix?
I'm trying to write a matrix like the one in the image, but I don't know how to do it (I tried using the align environment, but I had no luck). Any help will be really appreciated. :)
• Welcome to TeX SX! You could do that with the blkarray package. – Bernard Jan 27 at 22:57
• Also nicematrix is promising candidate. – Zarko Jan 27 at 22:58
• Welcome to the TeX.SE community. You can try with kbordermatrix, spalign, nicematrix package for example. But I'm scared of the photo is not that you come from Dante Alighieri's hell? Italy celebrates 700 years since his birth.:-)))))) I'm joking. – Sebastiano Jan 27 at 22:59
• @Sebastiano: A Cat in hell ??? ;o) – Bernard Jan 27 at 23:14
• Thank you guys sooo much, I was struggling a lot with this one :( I'll try with those packages, thank youuu <3 – Doja Cat Jan 27 at 23:28
Here's a solution that uses only some very basic LaTeX packages, along with center, tabular, and bmatrix environments. I've tried as much as possible to mimic the layout in the screenshot you posted.
\documentclass{article}
\usepackage{array,xcolor,amsmath,multirow}
\begin{document}
\begin{center}
\begin{tabular}{@{} r @{} l c >{\hspace{7mm}}l @{}}
& \multicolumn{2}{c}{\textcolor{cyan}{From:}} & \\
& \textcolor{cyan}{City}
& \textcolor{cyan}{Subu\rlap{rbs}} % ok, the use of '\rlap' is kludgy...
& \textcolor{cyan}{To:} \\[0.75ex]
\multirow{2}{*}{$M{=}$}
& \multicolumn{2}{@{}l}{\multirow{2}{*}{%
$\begin{bmatrix} 0.95 & 0.03 \\ 0.05 & 0.97 \end{bmatrix}$}}
& \textcolor{cyan}{City} \\
& & & \textcolor{cyan}{Suburbs}
\end{tabular}
\end{center}
\end{document}
• It looks great, thank you very much! – Doja Cat Jan 27 at 23:30
Here is a way to do that with {NiceTabular} of nicematrix.
You put all the elements (text and numbers) in a great array and you put the brackets where you want with the command \SubMatrix in the \CodeAfter.
With the key baseline, you put the baseline where you want (for the alignment with M=).
\documentclass{article}
\usepackage{nicematrix}
\begin{document}
\newcommand{\cyan}{\color{cyan}}
$M = \begin{NiceTabular}{cc>{\cyan}l}[baseline=line-4] \Block{1-2}{\cyan From:} \\ \cyan City & \cyan Suburbs & To: \\$.95$&$.03$& City \\$.05$&$.97$& Suburbs \\ \CodeAfter \SubMatrix[{3-1}{4-2}][slim] \end{NiceTabular}$
\end{document}
You need several compilations (because nicematrix uses PGF/Tikz under the hood).
• Excellent your package :-) – Sebastiano Mar 13 at 13:19
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2021-05-06 12:48:00
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https://socratic.org/questions/what-is-the-z-score-of-x-if-n-135-mu-74-sd-3-and-x-73
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# What is the z-score of X, if n = 135, mu= 74, SD =3, and X =73?
Jan 23, 2016
$Z = \frac{73 - 74}{\frac{3}{\sqrt{135}}} = - \frac{\sqrt{135}}{3}$
We can use: $z = \frac{x - \mu}{\sigma}$ assuming we have $\sigma$
$z = \frac{x - \mu}{\frac{s}{\sqrt{n}}}$; where n is sample size...
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2019-04-21 20:32:12
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https://conwaylife.com/forums/viewtopic.php?f=12&t=2992&start=175
|
A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.
fluffykitty
Posts: 653
Joined: June 14th, 2014, 5:03 pm
Contact:
That changes nothing since |e| is already illegal so |e b||c d| is also illegal. I'm not sure if there's any good way to fix MATRIXPARTY without making it boring, but reducing the number of dimensions to 1 probably makes the function finite. With that restriction, VECTORPARTY(1)=1, VECTORPARTY(2)=3, and VECTORPARTY(3) is at least 15 (2, 10, 001, 0111, 011, 01, 0, 11111111...1). Also, you really should join the Googology Discord server at https://discord.gg/V6R4hRJ
Edit: VECTORPARTY is equivalent to TREE but restricted so every node has at most one child, so its maximum possible FGH level is the SVO (however, it is most likely sub-e0)
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fluffykitty wrote:All versions of MATRIXPARTY(3) and CUBEPARTY(3) are infinite. The sequence is
Code: Select all
|2|
|1 1|
|1 1|
followed by extensions of
Code: Select all
|1 1 0 0 0|
|1 0 1 0 0|
|0 1 0 1 0|
|0 0 1 0 1|
|0 0 0 1 1|
(Element (x,y) is 1 iff x=y=0, x=y=max, or abs(x-y)=1)
No, cubeparty is finite.
I said you can take the cube, CUT it, TRANSLATE it, and ROTATE it, and if after some cutting/rotating/translating one piece (one cut piece) matches a previous cube it's disallowed.
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testitemqlstudop wrote:
fluffykitty wrote:All versions of MATRIXPARTY(3) and CUBEPARTY(3) are infinite. The sequence is
Code: Select all
|2|
|1 1|
|1 1|
followed by extensions of
Code: Select all
|1 1 0 0 0|
|1 0 1 0 0|
|0 1 0 1 0|
|0 0 1 0 1|
|0 0 0 1 1|
(Element (x,y) is 1 iff x=y=0, x=y=max, or abs(x-y)=1)
No, cubeparty is finite.
I said you can take the cube, CUT it, TRANSLATE it, and ROTATE it, and if after some cutting/rotating/translating one piece (one cut piece) matches a previous cube it's disallowed.
I’m afraid that doesn’t change anything. It still is infinite, unless you can cut out of the middle, which you don’t allow.
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Yes you can cut out the middle through this process: cut front side, cut behind side, cut top, cut bottom, cut left, cut right, and then you have the middle.
Moosey
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testitemqlstudop wrote:Yes you can cut out the middle through this process: cut front side, cut behind side, cut top, cut bottom, cut left, cut right, and then you have the middle.
I mean remove the middle to have the outside, somehow turning
Code: Select all
| a b c |
| d e f |
| g h i |
To
Code: Select all
| a c |
| g i |
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That doesn't help either:
Code: Select all
x = 20, y = 5, rule = //3
2B3A2.2BA.A4.2B2A$BAB2A2.BAB.A4.BABA$ABABA2.ABA.A4.AB2A$2ABAB11.3AB$
3A2B2.3A.B!
If you use symmetric matrices with entries of 0 and 1 only, it reduces to the subcubic graph function without the subcubic restriction and with a much more relaxed definition of embedding. With this interpretation, the sequence of matrices after the first becomes a path of length n-2 with loops at the ends.
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Consider an extension of TREE(N), which we will call TREE*(M,N), in which instead of requiring each tree in the sequence to have at most i nodes, where i is the tree's 1-index in the sequence, we require each tree in the sequence to have at most i+m nodes. (This is in effect a combination of both tree(N), equivalent to TREE*(N,1) and TREE(N), equivalent to TREE*(0,N).) It's fairly trivial to show that TREE*(1,N) = TREE*(0,N+1)-1 = TREE(N+1)-1, and only slightly less trivial to furthermore show that TREE*(M,N) < TREE*(0,M+N) = TREE(M+N) for all nonzero M and all N. It's also simple to derive a stronger bound ensuring that TREE*(0,M+N) ≥ TREE(M) + TREE*(TREE(M),N) for all nonzero M and all N. What other bounds can be derived?
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
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Can somebody explain BEAF to me?
I get that {x 1 anything} = x
And {x y} = {x y [any amount of 1s] } = x^y
But that does the other rule mean?
I’m also guessing that {anything 1} = {[that same anything]}, like on BAN (which I need an explanation on too)
EDIT:
After reading the introduction to BEAF article on the Googology Wiki, I have a less shaky idea of it (though it would be more intuitive once I read it without skimming or going off to find that ζ_0 is (it’s the first c such that c = ε_c))
Why do I use nested parentheses?!?
Also, if you want to have a more intuitive grasp on some of those ordinal things (Γ_0 in a nutshell! It’s the first c such that c=Φ_c(0), or, according to the googology wiki, {ω,ω,1,2} in ordinal BEAF), I highly recommend reading this series of blog posts that I found on the web by John Carlos Baez
Last edited by Moosey on June 8th, 2019, 10:17 am, edited 1 time in total.
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Similarly to the difference between tree(N) and TREE(N), can we define a function LSSCG(N) (standing for labeled SSCG) which is SSCG(0) calculated with N available labels for nodes? If this function is finite, I can derive that LSSCG(1) = SSCG(0) = 2, LSSCG(2) = SSCG(1)+1 = 6, and LSSCG(3) ≥ SSCG(SSCG(3)+2)+SSCG(3)+2. I don't know enough about the topic to know whether the results that conclude that TREE(N) and SSCG(N) are finite still hold for labeled graphs, so I don't know for sure whether LSSCG is finite (although I am fairly sure LSSCG(3) is).
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
http://conwaylife.com/wiki/A_for_all
Aidan F. Pierce
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I don't think anyone has thought of that before and I'm not sure if it's finite. If it is, it's probably much stronger than SSCG(n). One method to demonstrate FGH lower bounds for this type of function is to create a transfinite list of graphs/trees/vectors/whatever such that no element is contained in an earlier element. If you can construct a list of length x, then the function is stronger than f_y for any y<x (this does not imply that it is stronger than f_x itself however). This method has been applied to SSCG and SCG to prove growth rate lower bounds on the Googology Wiki, and the SSCG results can also be applied to LSSCG fairly easily.
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Is ggç (See bottom of this post) ω^ω in the FGH?
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It's probably at most w^3.
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It occurred to me that LSSCG(3) (if it is finite) is actually much larger than the lower bound I calculated previously — I had been operating under the assumption that the longest sequences would be of the form
Code: Select all
3
Code: Select all
1--2
followed by the optimal SSCG(3) sequence using label 2 and then the optimal SSCG(SSCG(3)+2) sequence using label 1. However, there are two things I missed:
1. That the second term in the sequence is not a minor of graphs containing isolated islands each consisting of a single label, but for which different islands can have different labels (as in the following example):
Code: Select all
2--2--2--2--2
1--1--1
2 2 2
1 1
Hence the sequence that would have generated SSCG(3) now generates a far larger number, although probably not approaching the order of SSCG(4) or even TREE(3).
And,
2. That taking the second term to be
Code: Select all
2-2
can actually lead to very long sequences consisting of vanilla-SSCG-type graphs with a single substituted 2-labeled node, similarly to the following:
Code: Select all
3
Code: Select all
2--2
Code: Select all
2 2 2
Code: Select all
2 2
, followed by the optimal SSCG(5) sequence with one arbitrary node in each graph labeled as 2 instead of 1 (for instance). This gives a phenomenally weak lower bound for LSSCG(3) as SSCG(SSCG(5)+4)+SSCG(5)+4. (Obviously the singly-substituted version of SSCG(5) will almost certainly be much larger than the standard SSCG(5).)
Eliminating the
Code: Select all
2 2
term and continuing with a doubly-substituted sequence yields a much stronger lower bound (possibly an actual approximate value) for LSSCG(3) on the order of SSCG(SSCG_s(SSCG_s(4, 2),1)), with SSCG_s(M, N) referring to the exactly-N-times-substituted version of SSCG(M) with at most one substitution per island, but even determining any kind of bound on SSCG_s(4, 2) or whether or not it is greater than SSCG(4) is too much for my ability to visualize.
All the above of course assumes LSSCG(3) is finite, which it may or may not be. If it is, though, I am fairly confident that it is much larger than any value that can reasonably be obtained using the vanilla SSCG function. If it turns out that LSSCG in general is finite, then LSSCG(3) itself will obviously pale in comparison to LSSCG(4), LSSCG(5) and so on in much the same way that SSCG(3) pales in comparison to SSCG(4). Of course, I have no real idea whether LSSCG is finite — my best guess would be that LSSCG(3) at least is (since the number of inhomogeneous nodes is bounded at two unless I missed a possible starting sequence, and it seems like that wouldn't be enough to allow infinite sequences), but since my guess is based on nothing but intuition with very little experience to back it up, I could easily be wrong. Even LSSCG(4) has such a rich variety of possibilities that I find it difficult to even begin to contemplate whether or not it is finite.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
http://conwaylife.com/wiki/A_for_all
Aidan F. Pierce
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SSCG(3) with 2 node colors and 1--2 forbidden is at least SSCG(SSCG(3)) since you can construct SSCG(3) graphs using only color 1 and SSCG(SSCG(3)) graphs using only color 2, which is far larger than SSCG(4). (Also, SSCG(3)>TREE(3) as demonstrated by APG in a cp4space post) My best sequence for LSSCG(3) is 3, 1--2, (triangle of 2s), (T tetromino of 2s), 2--2--2--2--2, (SSCG(2) graphs containing only 1s except for a 2--2--2--2 in each graph), (SSCG(3) graphs containing only 1s except for as many 2--2--2s as possible in each graph)... with the multisets of 2 graphs being computed using a similar procedure to the one APG used in https://cp4space.wordpress.com/2013/01/13/graph-minors/ to compute SSCG(2), except with a few nodes reserved for an SSCG run using a few initial nodes (x+2 on the xth iteration, although other options may work better). This obviously gets at least 3*2^(2*2^95-1)-9 iterations of SSCG (which is what would happen if you used SSCG(2)'s sequence for the 2 graphs), but goes much farther since in each step the maximum nodes gains SSCG(n) instead of just 1. However, this probably doesn't even beat f_((whatever SSCG's ordinal is)+w) for small arguments. (The maximum possible FGH level for this technique would be f_(SSCG^n) for LSSCG(n+1)) To go farther, we could use a mapping from ordinals to LSSCG(3) graphs such that if x<y then f(y) does not contain f(x) for ordinals x,y.
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fluffykitty wrote:It's probably at most w^3.
Oh. How do I get to ω^ω level?
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You either need infinitely many arguments (which will lead you down the array notation path) or more complex methods.
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Moosey
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fluffykitty wrote:You either need infinitely many arguments (which will lead you down the array notation path) or more complex methods.
I choose the latter.
What would I do?
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Well, you can do something like my f function, where you encode FGH levels as numbers, or do stuff like VECTORPARTY or TREE. I don't think there's any other way to get high FGH levels without doing one of those or using array notation methods like in BEAF.
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Moosey
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fluffykitty wrote:Well, you can do something like my f function, where you encode FGH levels as numbers, or do stuff like VECTORPARTY or TREE. I don't think there's any other way to get high FGH levels without doing one of those or using array notation methods like in BEAF.
How would I encode a level as a number?
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One thing that may be useful is that p(a,b)=2^a*(2b+1) is different for all (nonnegative integer) values of a and b, so you can encode two numbers into one. Also, it never outputs 0, so you can do something different in that case. Using that, you can encode arbitrary binary trees into nonnegative integers.
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Moosey
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fluffykitty wrote:One thing that may be useful is that p(a,b)=2^a*(2b+1) is different for all (nonnegative integer) values of a and b, so you can encode two numbers into one. Also, it never outputs 0, so you can do something different in that case. Using that, you can encode arbitrary binary trees into nonnegative integers.
Okay, sow how would I turn that into a vgç_n(x,y,z) (very generalized ç)?
Obviously I’d want something along the lines of a function which is f_n (x,y,z) = f_inverse p of its inputs (x,y,z) to expand back the inputs, but here I’m a little clueless.
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One possible method of encoding ordinals as binary trees is to use a leaf node to represent 0 and a branch node to represent b+w^a, where a and b are its children. For example, 1=0+w^0 would be (()()) using a linear representation of trees. 2=1+w^0=(()(()())), w=0+w^1=((()())()), w+1=w+w^0=(((()())())()), etc. Using this representation, you can get a function that grows faster than any FGH level below e0 (=w^w^w^...). For levels of the form 0 or a+1, you can do what you normally do, and for other levels you can use fundamental sequences. The rules for fundamental sequences are (b+w^(a+1))[n+1]=((b+w^a)+w^(a+1))[n], (b+w^a)[n]=b+w^(a[n]) if the previous rule doesn't apply, and (b+w^a)[0]=b if the previous rules don't apply. The useful property of fundamental sequences is that for any ordinal a, an ordinal b is less than a if and only if it is less than a[n] for some n. If you set n=x, then the resulting function will eventually outgrow any instance with lower ordinal input for large enough x, since a[x] will eventually be greater than any ordinal less than a. A method like this one is used in my f, but with a much stronger ordinal notation based on ordinal collapsing functions.
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fluffykitty wrote:One possible method of encoding ordinals as binary trees is to use a leaf node to represent 0 and a branch node to represent b+w^a, where a and b are its children. For example, 1=0+w^0 would be (()()) using a linear representation of trees. 2=1+w^0=(()(()())), w=0+w^1=((()())()), w+1=w+w^0=(((()())())()), etc. Using this representation, you can get a function that grows faster than any FGH level below e0 (=w^w^w^...). For levels of the form 0 or a+1, you can do what you normally do, and for other levels you can use fundamental sequences. The rules for fundamental sequences are (b+w^(a+1))[n+1]=((b+w^a)+w^(a+1))[n], (b+w^a)[n]=b+w^(a[n]) if the previous rule doesn't apply, and (b+w^a)[0]=b if the previous rules don't apply. The useful property of fundamental sequences is that for any ordinal a, an ordinal b is less than a if and only if it is less than a[n] for some n. If you set n=x, then the resulting function will eventually outgrow any instance with lower ordinal input for large enough x, since a[x] will eventually be greater than any ordinal less than a. A method like this one is used in my f, but with a much stronger ordinal notation based on ordinal collapsing functions.
So how would I use that to make a vgç function? Or would it be better just to define a new, simpler function?
Just to verify that I understand this, w^w^w=
((((()())())())())
If I typed it correctly.
I think I still need help in how to do all this.
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Moosey wrote:
fluffykitty wrote:trees
So how would I use that to make a vgç function? Or would it be better just to define a new, simpler function?
A good start would be to make a function for computing fundamental sequences in this tree representation. Also, if possible, simplifying functions is usually a good idea.
[quote="Moosey]
Just to verify that I understand this, w^w^w=
((((()())())())())
If I typed it correctly.
[/quote]
Yes.
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(a,b)=
(ω^a)+b, ω^a >= b
b+(ω^a), ω^a < b
()= 0
Would be one way to do it, in a function notation.
fluffykitty wrote: A good start would be to make a function for computing fundamental sequences in this tree representation. Also, if possible, simplifying functions is usually a good idea.
I’m not sure I understand.
If we do make a thing like this, how would one use the p(a,b) = (2^a)(2b+1) to encode an fgh level?
I get that presumably we’d need the inverse operation, which would make binary trees. Presumably, we would do this to all leaves with a value > 0, then iterate (a,b) to get the ordinal encoded by the integer.
That, I suppose, would let you encode an fgh level, but then what?
How does one turn the fgh level into a function without using the fgh itself?
Regardless, I guess we have a function which returns an ordinal given a finite number, sort of like a reverse ordinal collapsing function (but ordinal collapsing functions take uncountable values and make them countable rather than taking countable values and making them finite.)
Call it ord(n). It seems a little difficult to represent using function notation.
ord(0) =
Code: Select all
()
= 0
ord(1)=
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(()())
=1
ord(2)=
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(ord(1), ord(0))
= ω
ord(3) =
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(ord(0), ord(1))
= 2
ord(4)=
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(ord(2), ord(0))
= ω^ω
ord(5) =
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(ord(0),ord(2))
= ω+1
ord(6)=
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(ord(1), ord(1))
= ω+1 again, I think
ord(7)=
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(ord(0),ord(3))
= 3
I think every ordinal, finite or infinite, but < ε_0, can be expressed this way.
ord(8)=
Code: Select all
(ord(3),ord(0))
= ω^2
Every finite number is, I think, mapped to a different tree (since every finite number has a unique prime factorization.)
I think every finite number c is represented exactly once (or perhaps 0 times) as ord(k), that is, c = ord(k) for only one k if c is finite.
This is because n = (0, n-1) — thus no n = (0, n-c) for any c un= 1 since n-c+1 = (0, n-c) and n-c+1 = n iff c = 1
And no finite number n is (c, k) for any c > 0 and any k >= 0, otherwise n >= ω
Also only certain odd primes n make ord(n) finite (as well as 0 and 1 which are not primes)— namely, at least a subset such that, if n = 2c+1 c is prime (or 0 or 1) and satisfies the same property.
3 and 7 are some of these, I’m not sure there are any others. However, other non-prime numbers work, such as 15.
In general, are these just powers of 2, -1?
It seems that for k= (2^n)-1, ord(k) = n
EDIT:
ord(9) =
Code: Select all
(ord(0),ord(4))
= ω^ω +1
ord(10)=
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(ord(1),ord(2))
=ω2
I think there’s a set, call it Š, of finite numbers and countable ordinals such that any value k in the set has exactly one corresponding finite n such that ord(n)=k. This is true for all finite numbers. How many countably infinite values are there in it though? 0? Infinitely many?
Every finite number maps to a unique tree, if that helps. Ergo:
If there is only a single way to represent a value as a binary tree in fluffykitty’s way, then that value is in š if you can find an integer that maps to that tree. That is do not need to worry about extra integers mapping to that tree.
I’m not sure whether every tree is mapped to by some integer, but I feel that there is a proof for that. Something along the lines of:
There is a countably infinite amount of binary trees, and there is a countably infinite amount of integers, thus every tree is mapped to.
This is not a good proof, since you can define a function which maps all integers to the same tree, but for our purposes I think some proof like that would work.
Anyways, now:
ord(11)=
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(ord(0),ord(5))
= ω+2
ord(12) =
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(ord(2)ord(1))
= ω^ω + 1
ord(13)=
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(ord(0),ord(6))
= ω+2
I feel the omega+3s will have three or four things in between.
ord(14)=
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(ord(1),ord(3))
= ω+2
Or not; maybe there’ll be 4 of them though
ord(15)=
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we all know what’s going on for a Mersenne not-necessarily-a-prime
=4
ord(16)=
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(ord(4),ord(0))
ω^(ω^ω)
In general, ord(2^^n) = ω^^n
ord(17)=
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(ord(0),ord(8))
=(w^2)+1
This is getting to be a whole lotta fun
ord(18)=
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(ord(1),ord(4))
=(w^w) + w
ord(19) =
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(ord(0),ord(9))
=w^w +2
Just one more
ord(20) =
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(ord(2),ord(2))
= (w^w) + w
I think Š= Positive integers U limit ordinals, if I understand limit ordinals correctly.
Edit:
I think (w^w)+w, for instance, might be a limit ordinal, in which case Š ⊂ (Positive integers U limit ordinals)
Ord(currently largest known prime)=
82589933
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My CA rules can be found here
Also, the tree game
Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"
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2020-05-25 12:01:55
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http://codeforces.com/blog/entry/85030
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### Lain's blog
By Lain, history, 2 months ago,
The above image is a problem from a recent hiring test by CodeNation. Any clues on how to solve it?
P.S. This test is over, it's been a couple days now
• +28
» 2 months ago, # | -28 Don't ask questions about running contests/hiring tests.
• » » 2 months ago, # ^ | ← Rev. 2 → +8 It was a hiring test held a few days ago on 21st November, it's over.
• » » 2 months ago, # ^ | +12 I'll clarify this in the post. I thought it was clear form the word recent
» 2 months ago, # | +19 I tried sqrt root decomposition but got MLE. Anyway here is my idea: Consider vertices with degree $> \sqrt{n}$ heavy. We create array $sum[v]$ for light nodes that stores value on node $v$, and for heavy nodes, $neigh[h]$ to denote lazy share to neighbors, $ans[h]$ — contribution for second query from light nodes (as second neighbor), and $cnt[h][v]-$ number of common neighbors between $h$ and $v$, note that it can be precomputed for all $sqrt(n)$ heavy nodes using dfs. Type1 Queries: If the vertex $v$ is light, then simply add $x$ to all it's neighbor's $sum[]$, and for each heavy node $h$ add $cnt[h][v] * x$ to $ans[h]$. Otherwise if the vertex is heavy just add $x$ to $neigh[v]$.Type2 Queries: If the vertex $v$ is light, then iterate through all its neighbors and add their $sum[]$. Also for each heavy node $h$ add $cnt[h][v] * neigh[h]$. On the other hand, then just add to $ans[h]$ $cnt[h][v] * neigh[h]$ for all heavy nodes.
• » » 2 months ago, # ^ | 0 That's an interesting solution. I guess the MLE is because you can use $O(N^2)$ memory?
• » » » 2 months ago, # ^ | 0 No, I tried allocating just an array of size $10^6$ and it got MLE too.
• » » » » 2 months ago, # ^ | 0 Actually, I think it may not blow up to $O(N^2)$ memory. I think the worst case is a complete graph, which would be $O(N)$ for the $cnt$ array. Not sure if there is any real proof of this though. Perhaps your solution was just at the edge, or needed some optimizations.
• » » » » » 2 months ago, # ^ | 0 I don't understand why you think it'll blow up. It takes exactly $O(n\sqrt{n})$ memory for storing $cnt[h][v]$ and takes $O(\sqrt{n})$ time per query.
• » » » » » » 2 months ago, # ^ | 0 You're right, it does not blow up. A misunderstanding on my part.
• » » » » » » 2 months ago, # ^ | ← Rev. 2 → 0 I believe you are wrong about $\mathcal{O}(\sqrt n)$ part. If the bound for a vertex being heavy is $\sqrt n$, then you can have $\Omega(\frac{E}{\sqrt n})$ such vertices, which can be big. The bound should be $\sqrt E$, and then you would have $\mathcal{O}(\sqrt E)$ complexity for a query, and $\mathcal{O}(E \sqrt E)$ memory usage.
• » » 2 months ago, # ^ | 0 Did you try using a constant threshold for degree instead of $\sqrt {n}$ ? May be that could have helped.
• » » » 2 months ago, # ^ | 0 Yes. I tried optimizing it in all the ways I could but as I said it even MLE'd on declaring an array of 10^6 ints.
• » » 2 months ago, # ^ | ← Rev. 2 → 0 a problem that uses a similar idea is problem E from AMPPZ 2019 (a Polish ICPC contest [i'm not really sure if this is a regional contest or what])
» 2 months ago, # | +39 I've never seen any company with such difficult "hiring test" questions.Anyways, a solution similar to fugazi's should work, as long as the constant factor isn't too high. I'm not if it can be improved to be better than square root, though.
• » » 2 months ago, # ^ | +10 They're well known in India for having tests that are challenging for even Div 1 participants. Not sure what the goal is with that, but the problems are interesting and fun.
• » » 2 months ago, # ^ | +19 In case you're interested, here are the other two questions they asked (I haven't tested my solution ideas, they may be wrong). Problem 1Given two numbers A and B, 1<=A<=B<=1e9, find the number of special numbers in the range A to B inclusive. A number is called special if the sum of pairwise product of digits is prime. For example, if the number is "abc", then the sum of pairwise product of digits is a*b+b*c+c*a. My IdeaDigit DP, the dp state is on the current number of digits, current sum of digits (which is at most 90), and the current pairwise product (which is at most 9C2*81). Problem 2Find the expected number of segments in a string of length A in a language having alphabet size B. A segment is defined as the maximum contiguous substring containing the same character. For example, "10011" has 3 segments: "1", "00", and "11". My IdeaIt reduces to 1+the number of positions such that A[i]!=A[i+1], which can be worked out with math, the final answer will be something like 1+(A-1)*(B-1)/B.
• » » 2 months ago, # ^ | +15 Yeah codenation tests are much tougher than usual.Once when I wrote a test from the same company, there was a question... You have 2 sequences of length 10^5 each and the second sequence has all distinct elements. You have to find the LCS (longest common subsequence).I solved it but it's much tougher than usual hiring tests.
• » » » 2 months ago, # ^ | 0 What's the approach of the problem?
• » » » » 2 months ago, # ^ | +39 Consider first sequence as A and second sequence as B. map i_th element of second sequence(B[i]) with i [for 1 <= i <= n(means consider B[i] as i)]now replace every element in A with it's mapped value. if some A[i] doesn't have any value to be mapped then replace it with some small constant value.Now LCS = length of longest increasing subsequence in A. [ strictly increasing ] Time Complexity — nlogn
• » » » » » 2 months ago, # ^ | 0 Yes that is exactly what I did.
• » » » 2 months ago, # ^ | ← Rev. 2 → 0 That was the problem I got in my Google hiring test. I don't do a lot of competitive coding but I somehow cleared CodeAgon. Any tips for the rounds that are going to come?
• » » 2 months ago, # ^ | +1 This was most likely the last question of the round which is usually completely irrelevant to qualifying. For reference when I gave their internship test the coding round had 3 problems, the 3rd was a somewhat interesting DFS order LCA problem, I couldn't handle an overcounting case and just submitted the obvious idea using all pairs LCA for partial points and still easily qualified. If I'm not mistaken in that test those who didn't even attempt that specific question still qualified if they got the other two problems (easy greedy problem and a slightly tricky (1700ish) prime factorization or dp + combinatorics problem).So while they usually have at least 1 2000+ rated question just solving the 1800 or below rated questions is usually enough to qualify.
• » » 2 months ago, # ^ | 0 Actually the main point is just to filter because so many people are there in test, rest after qualifying they will conduct one 30 min telephonic round which is totally based on your projects and then rest interview rounds are normal like 1600-1800 guys will qualify easily, and final round is in depth project discussion, most people who got rejected are those because of their projects but not due to cp skills.
» 2 months ago, # | 0 Was this an offcampus hiring drive? Or it was oncampus?
• » » 2 months ago, # ^ | +12 "On campus" placements at BITS Pilani
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2021-01-24 15:00:15
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https://www.catalyzex.com/paper/arxiv:1101.5672
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Get our free extension to see links to code for papers anywhere online!
##### On the Local Correctness of L^1 Minimization for Dictionary Learning
Jan 29, 2011
Quan Geng, Huan Wang, John Wright
The idea that many important classes of signals can be well-represented by linear combinations of a small set of atoms selected from a given dictionary has had dramatic impact on the theory and practice of signal processing. For practical problems in which an appropriate sparsifying dictionary is not known ahead of time, a very popular and successful heuristic is to search for a dictionary that minimizes an appropriate sparsity surrogate over a given set of sample data. While this idea is appealing, the behavior of these algorithms is largely a mystery; although there is a body of empirical evidence suggesting they do learn very effective representations, there is little theory to guarantee when they will behave correctly, or when the learned dictionary can be expected to generalize. In this paper, we take a step towards such a theory. We show that under mild hypotheses, the dictionary learning problem is locally well-posed: the desired solution is indeed a local minimum of the $\ell^1$ norm. Namely, if $\mb A \in \Re^{m \times n}$ is an incoherent (and possibly overcomplete) dictionary, and the coefficients $\mb X \in \Re^{n \times p}$ follow a random sparse model, then with high probability $(\mb A,\mb X)$ is a local minimum of the $\ell^1$ norm over the manifold of factorizations $(\mb A',\mb X')$ satisfying $\mb A' \mb X' = \mb Y$, provided the number of samples $p = \Omega(n^3 k)$. For overcomplete $\mb A$, this is the first result showing that the dictionary learning problem is locally solvable. Our analysis draws on tools developed for the problem of completing a low-rank matrix from a small subset of its entries, which allow us to overcome a number of technical obstacles; in particular, the absence of the restricted isometry property.
* 37 pages, 1 figure
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2020-12-01 03:05:15
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https://tex.stackexchange.com/questions/485446/biblatex-with-multiple-bibliographies-on-documentclass-book
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Biblatex with multiple bibliographies on documentclass book
I'm writing a book-like document, for which I need to use the LaTeX documentclass book (I'm using chapters, etc.). At the end of the document I want a unique bibliography, broken in several themes, just like this:
I've produced the following minimal working example:
\documentclass{article}
\usepackage[backend=biber,style=alphabetic]{biblatex}
\begin{document}
\section{Introduction}
Notes for logic and the theory of computation.
\nocite{Ben12}\nocite{GJ79}\nocite{Gor16}\nocite{HMU06}
\nocite{HR04}\nocite{Sip12}
\section*{Bibliography}
\vspace{5mm}
\printbibliography[keyword=Logic, title={\normalsize\normalfont Logic:}]
\printbibliography[keyword=Computation,
title={\normalsize\normalfont Theory of Computation:}]
\end{document}
which does exactly what I want when the documentclass is article. The entries of the .bib file are the following:
@BOOK{Ben12,
AUTHOR="M. Ben-Ari",
TITLE="Mathematical Logic for Computer Science",
PUBLISHER="Springer",
EDITION="3rd edition",
keywords="Logic",
YEAR={2012} }
@BOOK{GJ79,
AUTHOR="M. R. Garey and D. S. Johnson",
TITLE="Computers and Intractability: a Guide to the Theory of {NP}-Completeness",
PUBLISHER="W. H. Freeman \& Co.",
keywords="Computation",
YEAR={1979} }
@BOOK{Gor16,
AUTHOR="V. Goranko",
TITLE="Logic as a Tool: A Guide to Formal Logical Reasoning",
PUBLISHER="Wiley",
keywords="Logic",
YEAR={2016} }
@BOOK{HMU06,
AUTHOR="J. E. Hopcroft and R. Motwani and J. D. Ullman",
TITLE="Introduction to Automata Theory, Languages, and Computation",
EDITION="3rd edition",
keywords="Computation",
YEAR={2006} }
@BOOK{HR04,
AUTHOR="M. Huth and M. Ryan",
TITLE="Logic in Computer Science: Modelling and Reasoning about Systems",
PUBLISHER="Cambridge University Press",
EDITION="2nd edition",
keywords="Logic",
YEAR={2004} }
@BOOK{Sip12,
AUTHOR="M. Sipser",
TITLE="Introduction to the Theory of Computation",
PUBLISHER="Cengage Learning",
keywords="Computation",
EDITION="3rd edition",
YEAR={2012} }
However, as soon as I change the documentclass to book and the \section commands to \chapter, I run into trouble: the header "Bibliography" appears alone in a page, followed by an empty page, followed by a page only with the "Logic" entries, then followed by another empty page, followed by a page only with the "Theory of Computation" entries.
How can I manage to get in the documentclass book a similar formatting for the bibliography as the one I get when the documentclass is article?
• It would be slightly easier to reproduce what you are seeing if you could include all necessary .bib entries from the example (maybe reduce the number to two or four, to keep things shorter)... Right now I have to make up additional entries myself to get things working. – moewe Apr 18 '19 at 11:32
• I've just included all the .bib entries I used. – Daniel Apr 18 '19 at 13:03
Don't put formatting commands into the title key. It should only contain the title text. Any additional formatting should be performed via bibliography headings. You can use a particluar heading style with the heading option and define a new one with \defbibheading.
\documentclass{book}
\usepackage[backend=biber,style=alphabetic]{biblatex}
\begin{document}
\nocite{aristotle:anima,aristotle:physics,nussbaum}
\end{document}
seems natural, where subbibliography is a predefined heading that uses a sectioning command one level below the normal bibliography heading (for the book class bibliography uses \chapter* and subbibliography uses \section*).
\documentclass{book}
\usepackage[backend=biber,style=alphabetic]{biblatex}
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2020-02-17 12:33:28
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https://db.barbanon.org/source/00016257.html
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Type Game 2020-10-01 Glulx IFComp 2020 interactive fiction
# Last House on the Block
Mr. Harrison was a quiet old man who lived in the house at the end of the block. No one really knew him, but everyone said he was rich; anyone who lived like as much like a hermit as he did had to be a millionaire. After lunch one day when you overheard your parents talking about how the old man had died with no family and how the city was coming to take the house and everything inside you decided on the spot that they wouldn't take everything...if there was any cash in there it was coming home with you.
In this game, you wander around the departed Mr. Harrison's house, searching for any hidden treasures. For the bulk of the game, you don't find any--no gold and jewels, at any rate. Instead, you find the sort of things you'd expect: old furniture and similar clutter of a home long lived-in.
For some of these items, you are given (brief) descriptions that give you a little insight into the old man's life, but for most items there is no description given. There's something particularly disappointing about reading "You see nothing special about the unusual rock." And there is a profusion of such useless items throughout the game. For example:
I'm reminded somewhat of Hill 160, which similarly included rather too many individual objects:
Implementation aside, the puzzle design is disappointing. Most of the objects really are useless, and there's only one real puzzle in the game: how to open the trapdoor in the closet. And its solution, unfortunately, involves guessing the correct objects to stack on each other in order to climb high enough to reach it. And to do this while subject to a timer in the form of a dying cell phone battery. Not a pleasant puzzle.
The game has a nice little gimmick: at the beginning, you select a companion who accompanies you throughout the game by picking up an object associated with them. This would be great, except that the characters have very little personality and there's no way to interact with them. They're as underdescribed as the clutter in the house.
The concept of an IF game that uses the generic conventions in a realistic setting is nice. Here we have exploration, puzzle solving, a search for a treasure. This has been done better in other games, but the idea is a good one. And the final payoff at the end is suitable: fitting in with the cold war era environment, you eventually find a bunker in which a 'treasure' is stashed.
With deeper implementation and better puzzle design, the basic outline of Last House on the Block could make a very good game, but the game as it stands feels aimless and unsatisfying.
Play time: 58 minutes.
Last House on the Block
An Interactive Fiction by Jason Olson
Release 1 / Serial number 201013 / Inform 7 build 6M62 (I6/v6.33 lib 6/12N)
Despite the "Release 1" above, I played the updated version of the game released 2020-10-14.
Name Role
Jason Olson Developer
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2021-05-15 11:07:29
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https://brilliant.org/problems/9-16-25-is-one-of-them/
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# 9 + 16 = 25 is one of them
$n^2+(n+1)^2+\cdots +(n+k)^2=(n+k+1)^2+\cdots +(n+2k)^2$
How many integers $$n$$ with $$1\leq n \leq 2016$$ are there such that the equation above is fulfilled for some positive integer $$k$$?
For example, with $$n=3,k=1$$ we have the familiar Pythagorean triple $$3^2+4^2=5^2$$.
Precursor
×
Problem Loading...
Note Loading...
Set Loading...
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2017-10-22 06:41:06
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http://proteinsandwavefunctions.blogspot.com/2013/04/presenting-data-in-histograms.html
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Monday, April 22, 2013
Presenting data in histograms
I've recently been working on a relatively large amount of data sets of the error between different measurements of the chemical shifts from a specific atom in a protein. A problem I encountered was how to determine the bin width of the histograms without manually choosing a unique value for each set. And if you have to fit a function to the distribution, then the chosen bin width might greatly affect the resulting fit.
The python module astroML contains an improved version of pylab's hist, where the form of the histogram can be automatically chosen based on different statistical models. Two noteworthy models are the Freedman-Diaconis Rule and the Bayesian block method discussed here, with examples of usage shown here.
The following python code calculates the optimal bin number based on the Freedman-Diaconis Rule without the use of the astroML module:
def bins(t):
t.sort()
n=len(t)
width = 2*(t[3*n/4]-t[n/4])*n**(-1./3)
return int((t[-1]-t[0])/width)
An interesting alternative to histograms for estimating a distribution is Kernel Density Estimation, where kernels (a non-skewed function that integrates to one e.g. a normal distribution) are placed at each datapoint and the sum of the kernels give the kernel density estimate.
As an example, the following code generates a data set x, and plots the data by the three mentioned methods:
import numpy as np
from astroML.plotting import hist
from scipy.stats.kde import gaussian_kde
import pylab
#generate data from two gaussians
x1 = np.random.normal(0,0.5,size=1000)
x2 = np.random.normal(1.5,0.3, size=1000)
x = np.concatenate((x1,x2))
#plot histogram from the Freedman-Diaconis Rule (filled opaque blue bins)
hist(x, normed=1, alpha = 0.2, bins='freedman')
#plot histogram using bayesian blocks (green step line)
hist(x, normed=1, bins='blocks', histtype = 'step')
#plot KDE using gaussian kernels (red):
my_pdf = gaussian_kde(x)
data_range = np.linspace(min(x),max(x),100)
pylab.plot(data_range,my_pdf(data_range))
pylab.savefig('test')
Which produces the following plot, where the filled bins are from the Freedman-Diaconis Rule, the green step line is the bayesian block method and the red line is the KDE.
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2018-02-19 23:24:21
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https://mathoverflow.net/questions/337709/representing-a-number-as-a-sum-of-four-squares-and-factorization
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# Representing a number as a sum of four squares and factorization
Rabin and Shallit have a randomized polynomial-time algorithm to express an integer $$n$$ as a sum of four squares $$n=a^2+b^2+c^2+d^2$$ (in time $$\log(n)^2$$ assuming the Extended Riemann Hypothesis).
I'm wondering why this does not give an efficient factorization algorithm? Here's what one could try: run their algorithm $$m$$ times, with different random steps. This should give expressions $$n=a_l^2+b_l^2+c_l^2+d_l^2, l\leq m$$, presumably with many distinct representations as a sum of four squares (cf. Jacobi's theorem). We can think of these as factorizations of $$n$$ over the Lipschitz integers, so $$n=|a+bi+cj+dk|^2=(a+bi+cj+dk)(a-bi-cj-dk)$$.
The Lipschitz integers do not admit a Euclidean algorithm, but the Hurwitz quaternions do. Hence one should be able to take the $$\gcd$$ of Hurwitz quaternions efficiently. I.e., for $$N,D$$ Hurwitz quaternions, there should be an efficient algorithm to find $$N=QR, D=PR$$, with $$|R|< |N|,|D|$$.
Now, take $$\gcd(a_l+b_li+c_lj+d_lk,a_p+b_pi+c_pj+d_pk)$$, $$1\leq l , where the $$\gcd$$ is taken in the Hurwitz quaternions. It should be efficient to find the $$\gcd$$ since the Hurwitz quaternions admit a Euclidean algorithm. In turn, this should give further factorizations of $$a_l+b_li+c_lj+d_lk$$ into Hurwitz quaternions, and hence the norms of these factors will give factors of $$n$$.
This approach will fail if it turns out that all of these Hurwitz $$\gcd$$ factors differ by Hurwitz units, for example if $$n$$ is prime. Of course, we could initially run a polynomial-time primality test to make sure $$n$$ is not prime.
Question: Are there certain composite $$n$$ for which one will not obtain a factorization this way with high enough probability to give a fast algorithm (i.e. $$m$$ has to be too large to get a pair with non-trivial $$\gcd$$ with high probability)? Maybe I just need to think a bit more about the proof of Jacobi's theorem...
I think the reason is that there are $$p+1$$ distinct ways of writing an odd prime $$p$$ as the sum of four squares up to sign changes; these correspond to the same number of elements of the Lipschitz order up to units. If you take two different Lipschitz elements of reduced norm $$p$$ up to units, their greatest common divisor is $$1$$.
So if we take two random Lipschitz elements of reduced norm $$n=pq$$, then their greatest common divisor will be $$1$$ with probability $$(1-1/(p+1))(1-1/(q+1))$$, and I don't see how you win with this. (These aren't significantly different odds than trying a random element modulo $$n$$ and hoping for a factor in common!)
• Sorry for blindness, but what are $6$ ways of writing $p=5$? – Ilya Bogdanov Sep 12 '19 at 15:33
• Starting with $5 = 2^2 + 1^2 + 0^2 + 0^2 = \mathrm{nrd}(2+i)$, we obtain $48 = 4 \cdot (24/2)$ elements of the Lipschitz order with reduced norm $5$ obtained by permuting the order of summands and allowing signs: e.g. $-2i + ij$, $j+2ij$, etc. The action of Lipschitz units $\langle \pm 1, \pm i, \pm j, \pm ij\rangle$ divides this by $8$, leaving $6$ representatives: $2 \pm i$, $2 \pm j$, $2 \pm ij$. – John Voight Sep 13 '19 at 1:27
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2020-02-20 12:29:28
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http://www.campusgate.co.in/2011/12/cubes.html
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### Cubes
A cube is a 3-dimensional diagram with all sides equal. If we divide it into the size $\left( {\displaystyle\frac{{\rm{1}}}{{\rm{n}}}} \right)^{{\rm{th}}}$ part of its side, we get ${\rm{n}}^3$ smaller cubes.
Shown below is a cube which is painted on all the sides and the cut into $\left( {\displaystyle\frac{{\rm{1}}}{{\rm{4}}}} \right)^{{\rm{th}}}$ of its original side.
Some observations: A cube has 6 faces, 12 edges and 8 corners. We can see that the cubes which got all the three sides painting lies at the corners. So the number of cubes which got painted all the three sides is equal to 8. Cubes with 2 sides painting lie on the edges (see the diagram). But the cubes which are on the left and right side of the edge matches with the corners. So we have to substract these two cubes from the number of cubes lying on the edge to get the number of cubes with 2 sides painting. Cubes with 1 side painting lies on the surfaces. Since, the top row, bottom row, left column, and right column matches with the edges, We must exclude these cubes while calculating the single side painted cubes.
The following rules may be helpful: If a cube is divided into the size $\left( {\displaystyle\frac{{\rm{1}}}{{\rm{n}}}} \right)^{{\rm{th}}}$ of its original side after get painted all the sides, Then
Total number of cubes = ${\rm{n}}^3$
Cubes with 3 sides painting = 8
Cubes with 2 sides painting = $12 \times (n - 2)$
Cubes with 1 sied painting = ${\rm{6 \times (n - 2)}}^{\rm{2}}$
Cubes with no painting = ${\rm{(n - 2)}}^{\rm{3}}$
Solved Examples (Level - 1)
1. A cube whose two adjacent faces are coloured is cut into 64 identical small cubes. How many of these small cubes are not coloured at all?
Assume the top face of the cube and its right side are colored green and orange respectively.
Now If we remove the colored faces, we left with a cuboid, whose front face is indicated with dots.
So on the front face there are 9 cubes, and behind it lies 4 stacks. So total 9 x 4 = 36
2. A cube, painted yellow on all-faces is cut into 27 small cubes of equal size. How many small cubes got no painting?
Assume we have taken out the front 9 cubes. Then the cube looks like the one below.
Now the cube which is in the middle has not got any painting. The cubes on the Top row, bottom row, left column and right column all got painting on atleast one face.
Alternative method:
Use formula: ${\rm{(n - 2)}}^{\rm{3}}$ Here n = 3 So ${\rm{(3 - 2)}}^3$ = 1
3. All surfaces of a cube are coloured. If a number of smaller cubes are taken out from it, each side 1/4 the size of the original cube's side, Find the number of cubes with only one side painted.
The original (coloured) cube is divided into 64 smaller cubes as shown in the figure. The four central cubes on each face of the larger cube, have only one side painted. Since, there are six faces, therefore total number of such cubes = 4 x 6 = 24.
Alternative Method:
Use formula : ${\rm{6 \times (n - 2)}}^{\rm{2}}$ = ${\rm{6}} \times {\rm{(4 - 2)}}^2$ = 24
Level - 2
4. Directions: One hundred and twenty-five cubes of the same size are arranged in the form of a cube on a table. Then a column of five cubes is removed from each of the four corners. All the exposed faces of the rest of the solid (except the face touching the table) are coloured red. Now, answer these questions based on the above statement:
(1) How many small cubes are there in the solid after the removal of the columns?
(2) How many cubes do not have any coloured face?
(3) How many cubes have only one red face each?
(4) How many cubes have two coloured faces each?
(5) How many cubes have more than 3 coloured faces each?
The following figure shows the arrangement of 125 cubes to form a single cube followed by the removal of 4 columns of five cubes each.
When the corner columns of the original cube are removed , and the resulting block is coloured on all the exposed faces (except the base) then we get the right hand side diagram. We labelled the various columns from a to u as shown in the figure
(1): Since out of 125 total number of cubes, we removed 4 columns of 5 cubes each, the remaining number of cubes = 125 - (4 x 5) = 125 - 20 = 105.
(2): Cubes with no painting lie in the middle. So cubes which are blow the cubes named as s, t, u, p, q, r, m, n, o got no painting. Since there are 4 rown below the top layer, total cubes with no painting are (9 x 4) = 36.
(3): There are 9 cubes namaed as m, n, o, p, q, r, s, t and u in layer 1, and 4 cubes (in columns b, e, h and k) in each of the layers 2, 3, 4 and 5 got one red face. Thus, there are 9 + (4 x 4) = 25 cuebs.
(4) the columns (a, c, d, f, g, i, j, l) each got 4 cubes in the layers 2, 3, 4, 5. Also in the layer 1, h, k, b, e cubes got 2 faces coloured. so total cubes are 32 + 4 = 36
(5): There is no cube in the block having more than three coloured faces. There are 8 cubes (in the columns a, c, d, f, g, i, j and l) in layer 1 which have 3 coloured faces. Thus, there are 8 such cubes.
Thus, there are 8 such cubes.
5. Directions: A cube of side 10 cm is coloured red with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into 125 smaller cubes of equal size. Answer the following questions based on this statement:
(1) How many cubes have three green faces each?
(2) How many cubes have one face red and an adjacent face green?
(3) How many cubes have at least one face coloured?
(4) How many cubes have at least two green faces each?
Clearly, upon colouring the cube as stated and then cutting it into 125 smaller cubes of equal size we get a stack of cubes as shown in the following figure.
The figure can be analysed by assuming the stack to be composed of 5 horizontal layers.
(1): All the corner cubes are painted green. So there are 8 cubes with 3 sides painted green.
(2): There is no cube having one face red and an adjacent face green as all the green painted cubes got paint on atleast 2 faces.
(3): Let us calculate the number of cubes with no painting. By formula, $\left( {{\rm{n - 2}}} \right)^{\rm{3}}$ = $\left( {{\rm{5 - 2}}} \right)^{\rm{3}}$ = 27
Therefore, there are 125 - 27 = 98 cubes having at least one face coloured.
(4): From the total cubes, Let us substract the cubes with red painting, cubes with no painting.
125 - (9 x 6) - 27 = 44
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2014-10-31 17:40:24
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https://computergraphics.stackexchange.com/questions/5562/do-i-need-to-use-glmemorybarrier-with-atomic-counters
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# Do I need to use glMemoryBarrier with atomic counters?
The OpenGL SuperBible 7th Edition points out that
the glMemoryBarrier() function supports a bit specifically for synchronizing access to atomic counters with other parts of the OpenGL pipeline
and describes its usage.
The OpenGL Wiki says:
Atomic counter memory access is not incoherent. So it follows the same rules as for texture reads, framebuffer writes, and so forth, not the Image Load/Store rules. You do not need to use a glMemoryBarrier to synchronize counter accesses.
So now I am confused:
Do I (and if so when do I) need to use glMemoryBarrier when using an atomic counter?
From the OpenGL 4.6 specification, section 7.12: Shader Memory Access:
As described in the OpenGL Shading Language Specification, shaders may perform random-access reads and writes to buffer object memory by reading from, assigning to, or performing atomic memory operation on shader buffer variables, or to texture or buffer object memory by using built-in image load, store, and atomic functions operating on shader image variables. The ability to perform such random-access reads and writes in systems that may be highly pipelined results in ordering and synchronization issues discussed in the sections below.
Notice something missing from that list? It mentions "shader buffer variables", and "shader image variables". But atomic counter variables are not mentioned. And atomic counter variables are a different kind of thing from either of those.
Therefore, that entire section (which is what explains the behavior of glMemoryBarrier) does not apply to atomic counters. Well, except where it specifically says otherwise, but that's only where it defines that helper FS invocations don't have side-effects. Note that even this explicitly calls out atomic counters as being something different from regular buffer or image operations: "stores, atomics, and atomic counter updates".
To lend greater credence to this view, the ARB_atomic_counter_buffer_object extension does not mention barriers at all. It doesn't require ARB_shader_image_load_store as a companion extension either, which is what defines glMemoryBarrier. Indeed, it has no interactions whatsoever with image load/store, while image load/store does have specified interactions with atomic counters.
Given the weight of the evidence in the specifications, I would have to say that the Wiki is right. In the interest of full disclosure, I did write that Wiki article. Though I also did all this research before I wrote it.
Basically, what the API seems to be saying is that buffer updates from atomic counters are treated like buffer updates from feedback operations. The implementation is required to track when you attempt to use that buffer for a read operation, then issue any synchronization needed to make sure that you can read the value. Or if you write to the value in the buffer (like clearing it to a value), then the implementation must automatically synchronize this.
The likely reason the specification permits this for atomic counters but not for image load/store is that the range of data for atomic counters is fixed. SSBOs and image load/store through buffer textures can write to arbitrary parts of the bound range of buffer data. By contrast, atomic counters write to 4 bytes times the number of atomic counter variables used by the shader.
Note however that atomic counters are affected by the in-shader memoryBarrier function. That is, atomic counters are subject to the requirements of, as the Wiki calls it, "Internal Visibility". Granted, that's not very useful most of the time, as the primary use of atomic counters is for when multiple invocations are all updating the same one.
On the other hand:
the glMemoryBarrier() function supports a bit specifically for synchronizing access to atomic counters with other parts of the OpenGL pipeline
The specification describes this particular barrier as:
Memory accesses using shader atomic counter built-in functions issued after the barrier will reflect data written by shaders prior to the barrier. Additionally, atomic counter function invocations after the barrier will not execute until all memory accesses (e.g., loads, stores, texture fetches, vertex fetches) initiated prior to the barrier complete.
What's curious about this is the "all memory accesses" part. This specific wording is only used in two other bits: shader storage and shader image. Other barriers for coherent operations like query buffers and transform feedbacks tend to use the term "all shader writes".
So... I would say that there is confusion at the specification level.
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2019-08-26 00:36:19
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https://bookdown.org/andreabellavia/mixtures/ordinary-least-squares-ols-regression.html
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## 3.1 Ordinary Least Squares (OLS) regression
### 3.1.1 Chemical-specific regression (EWAS)
A simple way to assess the association between a set of $$p$$ environmental exposures ($$X_1 - X_p$$) and a given outcome $$Y$$ is to build $$p$$ different regression models, one for each exposure (the approach that we previously described as “one-at-the-time”). Each model can be further adjusted for potential confounders of each exposure-outcome association. For example, is $$Y$$ was a continuous exposure, we could fit a set of linear regression models such as: $$E[Y|X_1,C]=\beta_0+\beta_1 \cdot X_1 + \beta\cdot C$$. The implicit assumption of this modeling procedure is that, for each element of the mixture, the other components do not act as confounders of the exposure-outcome association, as depicted in Figure 3.1.
When evaluating a set of environmental exposures, this procedure of fitting a set of independent regression models is usually referred to as environment-wide association study (EWAS, Patel et al. 2010). This approach usually requires correcting for multiple comparisons using either the Bonferroni approach or the false discovery rate (FDR).
Table 3.1 reports results from independent linear regression models (here without any adjustment for multiple comparisons) in selected exposures from our illustrative example.
Table 3.1: Single regressions for selected exposures in the simulated dataset
Estimate p.value
x3 0.078 0.007
x4 0.089 0.003
x5 0.068 0.005
x12 0.294 0.000
x13 0.238 0.000
These results see m to indicate that all exposures are independently associated with the outcome.
### 3.1.2 Multiple regression
Results from independent linear regression are hampered by the strong assumption that mixture components do not act as confounders of the association between each other component and the outcome of interest. This assumption is very seldom met in practice. A common situation, for example, is that two or more constituents of the mixture share one or more source, which usually results in moderate to high levels of correlation between exposures. Using DAGs, we can depict this situation as in Figure 3.2.
In this situation, a statistical model evaluating the association between $$X_1$$ and $$Y$$ will need to adjust for $$X_2$$ to reduce the impact of bias due to residual confounding. In general, when any level of correlation exists between two mixture components, we do expect them to act as confounders of the association between the other exposure and the outcome. This implies that results from independent linear regressions are likely biased due to uncontrolled confounding. In our illustrative example, for instance, we know that $$X_{12}$$ and $$X_{13}$$ are highly correlated; results from independent linear regressions indicated that both exposures are positively associated with the outcome (Table 3.1), but these coefficients are probably biased. Mutually adjusting for the two exposures in the same statistical model is therefore required to account for such confounding and possibly identify whether both exposures are really associated with the outcome, or if the real driver of the association is just one of the two. Note that both situations are realistic: we might have settings where a specific exposure is biologically harmful (say $$X_{12}$$), and the association between the correlated one ($$X_{13}$$) and the outcome was a spurious result due to this high correlation, as well as settings where both exposures are really associated with the outcome (maybe because it is the source of exposure to have a direct effect). We need statistical methodologies that are able to detect and distinguish these possible scenarios.
The most intuitive way to account for co-confounding between mixture components is to mutually adjust for all exposures in the same regression model:
$E[Y|X,C]=\beta_0+\sum_{i=1}^p\beta_i \cdot X_i + \beta \cdot C$
Table 3.2 presents results from a multiple regression that includes the 14 exposures in our example, as well as results from the chemical-specific models.
Table 3.2: Multiple and single regression results from the simulated dataset
Estimate - multiple p.value - multiple Estimate - single p.value - single
x1 0.058 0.080 0.106 0.001
x2 0.018 0.554 0.073 0.012
x3 -0.030 0.774 0.078 0.007
x4 0.053 0.644 0.089 0.003
x5 0.004 0.923 0.068 0.005
x6 0.060 0.047 0.120 0.000
x7 -0.031 0.620 0.153 0.000
x8 0.017 0.679 0.137 0.000
x9 0.025 0.673 0.160 0.000
x10 0.052 0.260 0.125 0.000
x11 0.049 0.341 0.149 0.000
x12 0.222 0.138 0.294 0.000
x13 -0.083 0.586 0.238 0.000
x14 0.054 0.293 0.185 0.000
### 3.1.3 The problem of multicollinearity
Results from the multiple regression are not consistent with those obtained from independent regression models, especially (and unsurprisingly) for those exposures that showed high levels of correlations. For example, within the exposure cluster $$X_{12}-X_{13}$$, the multiple regression model suggests that only $$X_{12}$$ is associated with the outcome, while the coefficient of $$X_{13}$$ is strongly reduced. Something similar happens for the $$X_3-X_4-X_5$$ cluster, where only $$X_4$$ remains associated with $$Y$$. Can we safely conclude that $$X_{12}$$ and $$X_4$$ are associated with $$Y$$ and that the other results were biased due to uncontrolled confounders? Before addressing this question, let’s take a look at a published paper where we evaluated the performance of several statistical models to evaluate the association between a mixture of 8 phthalate metabolites and birth weight in a pregnancy cohort (). Figure 3.3 presents results from the 8 independent regressions and a multiple regression model. Figure 3.4 presents instead the correlation plot of the 8 metabolites.
While we were expecting results from the two approaches to be different in the presence of high correlations, the coefficients obtained from the multiple regression leave room to a lot of skepticism. For example, the coefficients for MEOHP and MEHHP, when evaluated together, change respectively from -24 to 247, and from -28 to -127. Are these results reliable? Are we getting any improvement from to the biased results that we obtained from independent linear regressions?
The most common problem that arises when using multiple regression to investigate mixture-outcome association is multicollinearity (or simply collinearity). This occurs when independent variables in a regression model are correlated, with stronger consequences the higher the correlation. More specifically, a high correlation between two predictors simultaneously included in a regression model will decrease the precision of their estimates and increase their standard errors. If the correlation between two covariates (say $$X_1$$ and $$X_2$$) is very high, then one is a pretty accurate linear predictor of the other. Collinearity does not influence the overall performance of the model, but has an important impact on individual predictors. In general (as a rule of thumb), given two predictors $$X_1$$ and $$X_2$$ that are associated with the outcome ($$\beta=0.2$$ for both) when their correlation is equal to 0, the estimates in a linear model will be impacted by $$\rho(X_1, X_2)$$ as presented in Figure 3.5.
This issue, usually referred to as reverse paradox (the coefficients of 2 correlated covariates will inflate in opposite directions), is clearly affecting results from the paper presented above (the coefficients of highly correlated phthalate metabolites are either extremely large or extremely small), and possibly also results from the illustrative example (coefficients from correlated variables have opposite signs). Nevertheless, it should be noted that high correlation does not automatically imply that coefficients will be inflated. In another example (), for instance, we evaluated a mixture of three highly correlated parabens compounds, yet results from multiple regression were in line to those obtained from other mixture modeling techniques.
To quantify the severity of multicollinearity in a regression analysis one should calculate the Variance Inflation Factor (VIF). The VIF provides a measure of how much the variance of an estimated regression coefficient is increased because of collinearity. For example, if the VIF for a given predictors were 4, than the standard error of that predictors would be 2 times larger than if that predictor had 0 correlation with other variables. As a rule of thumb, VIFs above 4 should set the alarm off, as they indicate that those coefficients are likely affected by the high correlations between the corresponding predictor and other covariates in the model. Table 3.3 shows VIFs in our illustrative example, indicating that our results are deeply affected by multicollinearity. In this situation, alternative modeling options should be pursued.
Table 3.3: VIFs from multiple regression results presented in Table 3.2
x
x1 1.235658
x2 1.317951
x3 49.479946
x4 58.241935
x5 11.256382
x6 2.271043
x7 2.722583
x8 3.892965
x9 2.553431
x10 2.810535
x11 3.694404
x12 6.085748
x13 6.557098
x14 3.152092
z1 1.139690
z2 4.784064
z3 1.135437
### References
Bellavia, Andrea, Yu-Han Chiu, Florence M Brown, Lidia Mı́nguez-Alarcón, Jennifer B Ford, Myra Keller, John Petrozza, et al. 2019. “Urinary Concentrations of Parabens Mixture and Pregnancy Glucose Levels Among Women from a Fertility Clinic.” Environmental Research 168: 389–96.
Chiu, Yu-Han, Andrea Bellavia, Tamarra James-Todd, Katharine F Correia, Linda Valeri, Carmen Messerlian, Jennifer B Ford, et al. 2018. “Evaluating Effects of Prenatal Exposure to Phthalate Mixtures on Birth Weight: A Comparison of Three Statistical Approaches.” Environment International 113: 231–39.
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2022-06-26 15:02:57
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https://www.physicsforums.com/threads/springs-and-hookes-law.215834/
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Springs and Hooke's law
1. Feb 16, 2008
rolodexx
[SOLVED] Springs and Hooke's law
1. The problem statement, all variables and given/known data
An unstretched spring has a force constant of 1200 N/m. How large a force and how much work are required to stretch the spring by 1.00 m from its unstretched length?
2. Relevant equations
F= -k*s
W= F * s
3. The attempt at a solution
I used Hooke's law and obtained a force of 1200 N (which was correct). But the displacement is only 1 meter, so work should have also been 1200 (J). But it's wrong anyway.
The second part of the problem asks "How large a force and how much work are required to stretch the spring by 1.00 m beyond the length reached in part (a)?" so I multiplied the force constant by 2 to get 2400 N, and it was right. However, multiplying 2400 by 2 meters to give W of 4800 J was also incorrect. I don't know what I'm misunderstanding.
Last edited: Feb 16, 2008
2. Feb 16, 2008
Staff: Mentor
As you do work on the spring to stretch it, the force is not constant. So you can't just multiply the final force times the displacement. (You can integrate, if you know how.)
Hint: Have you studied elastic potential energy? How much energy is stored in a stretched spring?
3. Feb 16, 2008
rolodexx
The course I'm in is *supposed* to be algebra and trig-based, but I am quickly learning that using calculus would make my life a lot simpler, if I only remembered how to do that stuff. We also haven't studied elastic potential energy yet... I looked on Wikipedia and it said this was found by integrating Hooke's law (which I would love to do if I knew how) and gave a formula. U = 1/2 (kx$$^{2}$$) So I plugged in my values for the force constant and displacement, and got elastic potential energy of 600. I used this as my value for the work done in part a), and 600 J was correct.
However... I tried the same strategy on part b), by using the same force constant and a displacement of 2 meters, but the resulting 2400 J was also wrong. What did you mean by not being able to multiply final force by displacement?
4. Feb 16, 2008
Staff: Mentor
That's because the question asked for the additional work needed to stretch the spring from 1 to 2 meters, not from 0 to 2. (Subtract.)
I was referring to your earlier idea for calculating work. For example, you multiplied 1200 N x 1 m, which is incorrect because the force actually varies from 0 to 1200 N as you stretch the spring.
5. Feb 16, 2008
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2018-01-22 14:58:44
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https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Book%3A_Delftse_Foundations_of_Computation/02%3A_Proof/2.07%3A_Application_-_Recursion_and_Induction/2.7.02%3A_Towers_of_Hanoi
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# 2.7.2: Towers of Hanoi
Another standard example of recursion is the Towers of Hanoi problem. Let n be a pos- itive integer. Imagine a set of n discs of decreasing size, piled up in order of size, with the largest disc on the bottom and the smallest disc on top. The problem is to move this tower of discs to a second pile, following certain rules: Only one disc can be moved at a time, and a disc can only be placed on top of another disc if the disc on top is smaller. While the discs are being moved from the first pile to the second pile, discs can be kept in a third, spare pile. All the discs must at all times be in one of the three piles.
The Towers of Hanoi puzzle was first published by Édouard Lucas in 1883. The puzzle is based on a legend of temple wherein there initially was one pile of discs neatly sorted from largest to smallest. In Lucas’s story, monks have since been continuously moving discs from this pile of 64 discs accord- ing to the rules of the puzzle to again created a sorted stack at the other end of the temple. It is said that when the last disc is placed, the world will end. But on the positive side, even if the monks move one disc every second, it will take approximately 42 times the age of the universe until they are done. And that is assuming they are using the optimal strategy...
Source: en.Wikipedia.org/wiki/Tower_of_Hanoi
For example, if there are two discs, the problem can be solved by the following sequence of moves:
Move disc 1 from pile 1 to pile 3
Move disc 2 from pile 1 to pile 2
Move disc 1 from pile 3 to pile 2
A simple recursive subroutine can be used to write out the list of moves to solve the problem for any value of n. The recursion is based on the observation that for n > 1, the problem can be solved as follows: Move n − 1 discs from pile number 1 to pile number 3 (using pile number 2 as a spare). Then move the largest disc, disc number n, from pile number 1 to pile number 2. Finally, move the n − 1 discs from pile number 3 to pile number 2, putting them on top of the nth disc (using pile number 1 as a spare). In both cases, the problem of moving n − 1 discs is a smaller version of the original problem and so can be done by recursion. Here is the subroutine, written in Java:
void Hanoi(int n, int A, int B, int C) {
// List the moves for moving n discs from
// pile number A to pile number B, using
// pile number C as a spare. Assume n > 0.
if (n == 1) {
System.out.println("Move disc 1 from pile " + A + " to pile " + B);
}
else{
Hanoi(n-1, A, C, B);
System.out.println("Move disc " + n + " from pile " + A + " to pile " + B);
Hanoi(n-1, C, B, A);
}
}
This problem and its fame have led to implementations in a variety of languages, including a language called Brain f*ck. In the Computer Organisation course, you can implement an interpreter for this language and test it on the implementation of the Hanoi algorithm.
We can use induction to prove that this subroutine does in fact solve the Towers of Hanoi problem.
Theorem 3.12.
The sequence of moves printed by the Hanoi subroutine as given above correctly solves the Towers of Hanoi problem for any integer n ≥ 1.
Proof. We prove by induction that whenever n is a positive integer and A,B, and C are the numbers 1, 2, and 3 in some order, the subroutine call Hanoi(n, A, B, C) prints a sequence of moves that will move n discs from pile A to pile B, following all the rules of the Towers of Hanoi problem.
In the base case, n = 1, the subroutine call Hanoi(1, A, B, C) prints out the single step “Move disc 1 from pile A to pile B”, and this move does solve the problem for 1 disc.
Let k be an arbitrary positive integer, and suppose that Hanoi(k, A, B, C) correctly solves the problem of moving the k discs from pile A to pile B using pile C as the spare, whenever A, B, and C are the numbers 1, 2, and 3 in some order. We need to show that Hanoi(k + 1, A, B, C) correctly solves the problem for k + 1 discs. Since k + 1 > 1,Hanoi(k + 1, A, B, C) begins by calling Hanoi(k, A, C, B). By the induction hypothesis, this correctly moves k discs from pile A to pile C. disc number k + 1 is not moved during this process. At that point, pile C contains the k smallest discs and pile A still contains the(k + 1)st disc, which has not yet been moved. So the next move printed by the subroutine,
“Move disc (k + 1) from pile A to pile B”, is legal because pile B is empty. Finally, the subroutine calls Hanoi(k, C, B, A), which, by the induction hypothesis, correctly moves the $$k$$ smallest discs from pile $$C$$ to pile $$B,$$ putting them on top of the $$(k+1)^{\text { st }}$$ disc, which does not move during this process. At that point, all (k + 1) discs are on pile B, so the problem for k + 1 discs has been correctly solved.
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2021-07-25 22:34:20
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https://classnotes.ng/lesson/measure-of-dispersion-of-variation-of-grouped-data-economics-ss2/
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# Measure of Central Tendency of Group Data
Welcome to class!
In today’s class, we will be talking about the measure of the central tendency of group data. Enjoy the class!
### MEASURES OF CENTRAL TENDENCY
CONTENT
• MEAN
• MODE
• MEDIAN
They are the values which show the degree to which a given data or any given set of values will converge toward the central point of the data.
Measures of central tendency, also called measures of location, is the statistical information that gives the middle or centre or average of a set of data. Measures of central tendency include arithmetic mean, median and mode.
#### MEAN
This is the average of variables obtained in a study. It is the most common kind of average.
For group data the formula for calculating the mean is
$\frac{\sum _{}^{}fx}{\sum _{}^{}f}$
Where
$\sum _{}^{}$
=Summation
F=frequency
X=observation
#### MEDIAN
It is the middle number in any given distribution. The formula is
Median =
Where; L = Lower class limit.
N = Summation 0f the frequency.
Fb = Cumulative frequency before the median class.
f = frequency of the median class.
c= Class size.
#### MODE
It is the number that appears most in any given distribution, i.e the number with the greatest frequency. When a series has more than one mode, say two, it is said to be bi-modal or tri-modal for three.
Mode =
Where M=mode
L = the lower class boundary of the modal class.
D1 = the frequency of the modal class minus the frequency of the class before the modal class.
D2 = the frequency of the modal class minus the frequency of the class after it.
C = the width of the modal class.
###### Example:
The table below shows the marks of students of JSS 3 mathematics.
Marks 1-5 6-10 11-15 16-20 21-25 26-30 Frequency 2 3 4 5 6 7
Use the information above to calculate the following:
1. the mean
2. the median
3. the mode
Solution
mark frequency mid-point fx
1-5 2 3 6 6-10 3 8 24 11-15 4 13 52 16-20 5 18 90 21-25 6 23 138 26-30 7 28 196
27 506
Mean=
B. Median
Mark F Cf 1-5 2 2 6-10 3 5 11-15 4 9 16-20 5 14 21-25 6 20 26-30 7 27
Median =
###### General evaluation
The table below shows the weekly profit in naira from a mini-market.
You are required to calculate:
1. The mean.
2. The median.
3. The mode.
Weekly profit(#) 1-10 11-20 21-30 31-40 41-50 51-60 Frequency 6 6 12 11 10 5
1. Amplified and Simplified Economics for SSS by Femi Alonge page 29-30.
2. Further Mathematics Scholastics Series page 265-265.
In our next class, we will be talking about the Measure of Dispersion of Variation of Grouped Data. We hope you enjoyed the class.
Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.
LEARN TO CODE IN 8 WEEKS. Pay Only ₦25000 To Join Class💃
Access Fun Video Lessons to Pass WAEC, NECO, JAMB, POST-UTME in One Sitting💃
Don`t copy text!
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2022-11-28 04:05:29
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