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http://math.stackexchange.com/questions/799507/why-is-this-function-neither-injective-nor-surjective | # Why is this function neither injective nor surjective?
Hello guys I am doing some maths revision and I am stuck. I read and I saw videos and I still can't get my head around it.
I have this example:
Determine whenever this function $\mathbb{Z}\to\mathbb{Z}$ is injective, surjective, neither, both:
$$f(x) = n^2 + 1$$
In the answer it is given that is neither without explaining why not.
If I substitute $n$ with $1$, I will get $f(1) = 2$ and, with $-1$, $f(-1) = 2$ again. So that should be surjective but it says is neither?
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The fact that $f(1) = f(-1)$ implies that $f$ is not injective, and says nothing about surjectivity; I would suggest carefully reading the definitions of injective and surjective. – user61527 May 17 '14 at 20:36
Surjective doesn't mean "not injective". A function from $A$ to $B$ is surjective if it produces every value in $B$. – user2357112 May 17 '14 at 20:36
is it because you can't map for example 3 ? Which makes it only a general function ? – mazzer07 May 17 '14 at 20:42
@mazzer07 You can't even map to a negative number. – Eff May 17 '14 at 20:42
Did you mean f(x) = x^2 + 1? – dubiousjim May 18 '14 at 2:28
You're given a function $f$ which takes values in $\mathbb Z$ to values in $\mathbb Z$ by the rule $f(x)=x^2+1$.
Let's recall what injective means. Our function $f$ will be injective if for any $x_1,x_2\in\mathbb Z$ such that $f(x_1)=f(x_2)$, we have $x_1=x_2$. You pointed out that $f(1)=2=f(-1)$, but that $1\neq -1$. We plugged $x_1=1$ and $x_2=-1$ into $f$ and saw that $f(x_1)=f(x_2)$. But $x_1\neq x_2$. Therefore, $f$ cannot be injective. Another intuitive way to see this is to view whether or not $f$ passes the horizontal line test. Does it?
In general, to show that a function $f:X\to Y$ is injective, we can show that $$f(x_1)=f(x_2)\implies x_1=x_2$$ or we can show that $$x_1\neq x_2\implies f(x_1)\neq f(x_2).$$
So let's look at whether or not $f$ is surjective. Remember that $f$ is surjective if for any $y\in\mathbb Z$ (in the target set) there is some $x\in\mathbb Z$ (in the domain) such that $f(x)=y$. But notice that $f(x)=x^2+1>0$ for each $x\in\mathbb Z$. So what happens if we choose $y\leq 0$?
In general, to show that a function $f:X\to Y$ with some rule $f(x)$ is surjective, we pick some arbitrary $y\in Y$. Then we find some value in $x\in X$ in terms of $y$, by looking at the rule of our function, that gives $f(x)=y$.
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I look at injectivity, surjectivity, and bijectivity this way:
A mapping $f : A \to B$ is injective if, for every element in the codomain $B$, there is at most one element in the domain $A$ that maps to it.
A mapping is surjective if, for every element in the codomain, there is at least one element in the domain that maps to it.
A mapping is bijective if, for every element in the codomain, there is exactly one element in the domain that maps to it.
Since your range is $\mathbb Z$, that means that for the function $f(n) = n^2 + 1$ to be injective, no two elements of the domain can map to the same element in the codomain. This is obviously false--$f(-n) = f(n)$ for all integers $n$.
For the function to be surjective, for any integer $m$, there must be another integer $n$ such that $n^2 + 1 = m$. This is also obviously false--if $m = 0$, then there is no integer (or even real) solution to $n^2 + 1 = 0$.
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The fact that $f(1)=f(-1)$ implies that $f$ is not injective.
Then $n^2+1>0, \forall n\in \mathbb{Z}$, so $f(\mathbb{Z}) \subseteq \mathbb{N}$ so f is not surjective.(=there exists a so that for all b $\in \mathbb{Z}$ $f(b) \ne a$ ; ex : $a=-1$)
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Surjective means that every number $n\in \mathbb{Z}$ can be written in the form $n=f(m)=m^2+1$, it is clear that negative numbers $n$ cannot be written like this, so $f$ is not surjective. $f$ injective means that different numbers $n$ and $m$ cannot have the same value by $f$, but $1$ and $-1$ are different and have the same value by $f$: $f(-1)=f(1)=2$, so $f$ is not injective.
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Alright this confused me even more. What about if we had f(x) = x squared for every x element of R. That should be surjective right ? Or again I can't write negative numbers in the form of n squared. ? – mazzer07 May 17 '14 at 20:48
Exactly, you can't write negative numbers in the form of $n^2$, so $f:\mathbb{R}\to \mathbb{R}$ such that $f(x)=x^2$ is also non surjective. – user144542 May 17 '14 at 21:00
@mazze an example of a surjective map is $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x+1$. Because every number can be written in the form $x+1$. This map is also injective. – user144542 May 17 '14 at 21:11
Loosely speaking a function is injective if it cannot map to the same element more than one place. A function is surjective if it maps into all elements (that the function is defined onto). For your example:
First remember that $\mathbb{Z} = \{0,1,-1,2,-2,...\}$. Since the function $f$ maps to $2$ both when $n=1$ and $n=-1$, i.e. $f(1)=f(-1)=2$ it maps to the same element at more than one place. So it is not injective.
Let's check if it is surjective. Then all values of $\mathbb{Z}$ must be mapped to by the function ($f: \mathbb{Z}\rightarrow\mathbb{Z}$). Since there is no $n$ that produces (for example) the value $-2\in\mathbb{Z}$ it cannot be surjective.
Here are some more for you. Is the function $g:[0,\infty)\rightarrow \mathbb{R}$ defined by $g(x)=x^2$ injective, surjective, both or none? Well, $x^2$ is positive or zero for all $x\in[0,\infty)$ so it cannot be surjective since it doesn't map to for example $-3\in\mathbb{R}$. Since $x^2$ maps into a unique element for all $x\in[0,\infty)$ it is, in fact, injective. Now if we look at $h:\mathbb{R}\rightarrow\mathbb{R}$ defined again by $h(x)=x^2$ then it is not surjective for the same reasons. But it is not injective either since we now can take $h(2)=h(-2)=4$. So the function $h$ maps different elements into the same element so it is not injective. As you can see, it is very important which spaces the functions go from and to.
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What about this: The function f(x) = 2x from the set of natural numbers naturals to the set of non-negative even numbers is a surjective function. It says it is a surjective but isn't it also bijective cause f(1) = 2 f(2) = 4 f(3) = 6. So it is one to one mapping and it will be equal 4 inputs 4 outputs maps one one ? – mazzer07 May 17 '14 at 21:13
So $f:\{0,1,2,...\}\rightarrow\{0,2,4,...\}$ by $f(x)=2x$ is surjective, yes. Because all the elements of the set $\{0,2,4,...\}$ are mapped to by $f$. It is also injective, actually, because no matter what 2 elements of $\{0,1,2,...\}$ you choose they will never product the same element using $f$. – Eff May 17 '14 at 21:16
A function is bijective if it is both injective and surjective (per definition). So this particular $f$ is bijective since it is both injective and surjective, as explained above. Of course this depends on your definition of natural numbers. I used $\mathbb{N}=\{0,1,2,...\}$, but it is equally as often used that $\mathbb{N}=\{1,2,3,...\}$. So if we used the function $f:\{1,2,3,...\}\rightarrow\{0,2,4,...\}$ by $f(x)=2x$ then it is not surjective because no element of $\{1,2,3,...\}$ produces the value $0\in\{0,2,4,...\}$. – Eff May 17 '14 at 21:19
I have seen an example in where it is allowed A to point to B and C to point to B and I though that is one way of finding if it is surjective ? Isn't that right ? Also in what exact example it is only injective and not bijective – mazzer07 May 17 '14 at 21:23
I don't completely understand your first question. A function is injective and not bijective if it is not surjective (again from the definition of bijective). The function $g$ in my original answer satisfies this. – Eff May 17 '14 at 21:26 | 2016-06-30 19:59:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9317208528518677, "perplexity": 133.27588324626745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783399117.38/warc/CC-MAIN-20160624154959-00102-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://labs.tib.eu/arxiv/?author=H.%20Yan | • ### A VLT/FORS2 view at $z\sim6$: Lyman-$\alpha$ emitter fraction and galaxy physical properties at the edge of the epoch of cosmic reionization(1710.01784)
Oct. 4, 2017 astro-ph.GA
The fraction of Lyman-$\alpha$ emitters among the galaxy population has been found to increase from $z\sim0$ to $z\sim6$ and drop dramatically at $z>6$. This drop has been interpreted as an effect of an increasingly neutral intergalactic medium with increasing redshift, while a LyC escape fraction evolving with redshift. We report the result of a large VLT/FORS2 program aiming to confirm spectroscopically a large galaxy sample at $z\geq6$ that has been selected in several independent fields through the Lyman Break technique. Combining those data with archival data, we create a large and homogeneous sample of $z\sim6$ galaxies ($N=127$), complete in terms of Ly$\alpha$ detection at $>95\%$ for EW(Ly$\alpha)\geq25\AA$. We use this sample to derive a new measurement of the LAE fraction at $z\sim6$ and derive the physical properties of these galaxies through spectral energy distribution fitting. We find a median LAE fraction at $z\sim6$ lower than in previous studies. The main difference between LAEs and non-LAEs is that the latter are significantly dustier. Using predictions of our SED fitting code accounting for nebular emission, we find an effective Ly$\alpha$ escape fraction $f^{eff}_{esc}(Ly\alpha)=0.23^{+0.36}_{-0.17}$ remarkably consistent with the value derived by comparing UV luminosity function with Ly$\alpha$ luminosity function. We conclude that the drop in the LAE fraction from $z\sim6$ to $z>6$ is less dramatic than previously found and the effect of an increasing IGM neutral fraction is possibly observed at $5<z<6$. Based on our derived $f^{eff}_{esc}(Ly\alpha)$, we find that the IGM has a relatively small impact on Ly$\alpha$ photon visibility at $z\sim6$, with a lower limit for the IGM transmission to \lya\ photons, $T_{IGM}\gtrsim0.20$, likely due to the presence of outflows. [abdridged]
• ### Space Telescope and Optical Reverberation Mapping Project. VII. Understanding the UV anomaly in NGC 5548 with X-Ray Spectroscopy(1704.06345)
Aug. 1, 2017 astro-ph.GA
During the Space Telescope and Optical Reverberation Mapping Project (STORM) observations of NGC 5548, the continuum and emission-line variability became de-correlated during the second half of the 6-month long observing campaign. Here we present Swift and Chandra X-ray spectra of NGC 5548 obtained as a part of the campaign. The Swift spectra show that excess flux (relative to a power-law continuum) in the soft X-ray band appears before the start of the anomalous emission-line behavior, peaks during the period of the anomaly, and then declines. This is a model-independent result suggesting that the soft excess is related to the anomaly. We divide the Swift data into on- and off-anomaly spectra to characterize the soft excess via spectral fitting. The cause of the spectral differences is likely due to a change in the intrinsic spectrum rather than being due to variable obscuration or partial covering. The Chandra spectra have lower signal-to-noise ratios, but are consistent with Swift data. Our preferred model of the soft excess is emission from an optically thick, warm Comptonizing corona, the effective optical depth of which increases during the anomaly. This model simultaneously explains all the three observations: the UV emission line flux decrease, the soft-excess increase, and the emission line anomaly.
• ### Optical Line Emission from z$\sim$6.8 Sources with Deep Constraints on Ly$\alpha$ Visibility(1703.08986)
April 12, 2017 astro-ph.GA
We analyze a sample of $z$-dropout galaxies in the CANDELS GOODS South and UDS fields that have been targeted by a dedicated spectroscopic campaign aimed at detecting their Ly$\alpha$ line. Deep IRAC observations at 3.6 and 4.5 $\mu$m are used to determine the strength of optical emission lines affecting these bands at z$\sim$6.5-6.9 in order to i) investigate possible physical differences between Ly$\alpha$ emitting and non-emitting sources; ii) constrain the escape fraction of ionizing photons; iii) provide an estimate of the specific star-formation rate at high redshifts. We find evidence of strong [OIII]+H$\beta$ emission in the average (stacked) SEDs of galaxies both with and without Ly$\alpha$ emission. The blue IRAC [3.6]-[4.5] color of the stack with detected Ly$\alpha$ line can be converted into a rest-frame equivalent width EW([OIII]+H$\beta$)=1500$^{+530}_{-440}\AA$ assuming a flat intrinsic stellar continuum. This strong optical line emission enables a first estimate of f$_{esc}\lesssim$20% on the escape fraction of ionizing photons from Ly$\alpha$ detected objects. The objects with no Ly$\alpha$ line show less extreme EW([OIII]+H$\beta$)=520$^{+170}_{-150}\AA$ suggesting different physical conditions of the HII regions with respect to Ly$\alpha$-emitting ones, or a larger f$_{esc}$. The latter case is consistent with a combined evolution of f$_{esc}$ and the neutral hydrogen fraction as an explanation of the lack of bright Ly$\alpha$ emission at z$>$6. A lower limit on the specific star formation rate, SSFR$>$9.1$Gyr^{-1}$ for $M_{star}=2 \times 10^9 M_{\odot}$ galaxies at these redshifts can be derived from the spectroscopically confirmed sample.
• ### Resonant Ionization Laser Ion Source (RILIS) Development on Lu and Pr(1704.03101)
April 11, 2017 physics.atom-ph
Resonance ionization laser ion sources are efficient and element selective ion sources, which are particularly well suited for radioactive ion beam facilities. Using TRIUMF's off-line laser ion source test stand with a system of tunable titanium sapphire (Ti:Sa) lasers, laser resonance ionization schemes for lutetium and praseodymium have been investigated with a particular interest to autoionizing states. New ionization schemes via Rydberg states and autoionizing states were found. Their investigation and comparison of ion yields at the off-line test stand will be discussed, and the data of on-line Lu delivery will be presented.
• ### Study of even-parity Rydberg and autoionizing states of lutetium by laser resonance ionization spectroscopy(1702.08589)
March 7, 2017 physics.atom-ph
Multi-step laser resonance ionization spectroscopy of lutetium (Lu) has been performed at TRIUMF's off-line laser ion source test stand. The even-parity Rydberg series $6s^2nd$ $^2D_{3/2}$, $6s^2nd$ $^2D_{5/2}$ and $6s^2ns$ $^2S_{1/2}$ were observed converging to the 6s$^2$ ionization potential. The experimental results has been compared to previous work. 51 levels of Rydberg series $6s^2nd$ $^2D_{5/2}$ and 52 levels of Rydberg series $6s^2ns$ $^2S_{1/2}$ were reported new. Additionally six even-parity autoionization (AI) series converging to Lu ionic states $5d6s$ $^3D_1$ and $5d6s$ $^3D_2$ were observed. The level energies of these AI states were measured. The configurations of the AI states were assigned by relativistic multichannel theory (RMCT) within the framework of multichannel quantum defect theory (MQDT).
• ### The Lyman Continuum escape fraction of faint galaxies at z~3.3 in the CANDELS/GOODS-North, EGS, and COSMOS fields with LBC(1703.00354)
March 1, 2017 astro-ph.GA
The reionization of the Universe is one of the most important topics of present day astrophysical research. The most plausible candidates for the reionization process are star-forming galaxies, which according to the predictions of the majority of the theoretical and semi-analytical models should dominate the HI ionizing background at z~3. We aim at measuring the Lyman continuum escape fraction, which is one of the key parameters to compute the contribution of star-forming galaxies to the UV background. We have used ultra-deep U-band imaging (U=30.2mag at 1sigma) by LBC/LBT in the CANDELS/GOODS-North field, as well as deep imaging in COSMOS and EGS fields, in order to estimate the Lyman continuum escape fraction of 69 star-forming galaxies with secure spectroscopic redshifts at 3.27<z<3.40 to faint magnitude limits (L=0.2L*, or equivalently M1500~-19). We have measured through stacks a stringent upper limit (<1.7% at 1sigma) for the relative escape fraction of HI ionizing photons from bright galaxies (L>L*), while for the faint population (L=0.2L*) the limit to the escape fraction is ~10%. We have computed the contribution of star-forming galaxies to the observed UV background at z~3 and we have found that it is not enough to keep the Universe ionized at these redshifts, unless their escape fraction increases significantly (>10%) at low luminosities (M1500>-19). We compare our results on the Lyman continuum escape fraction of high-z galaxies with recent estimates in the literature and discuss future prospects to shed light on the end of the Dark Ages. In the future, strong gravitational lensing will be fundamental to measure the Lyman continuum escape fraction down to faint magnitudes (M1500~-16) which are inaccessible with the present instrumentation on blank fields.
• ### Space Telescope and Optical Reverberation Mapping Project. V. Optical Spectroscopic Campaign and Emission-Line Analysis for NGC 5548(1702.01177)
Feb. 3, 2017 astro-ph.GA
We present the results of an optical spectroscopic monitoring program targeting NGC 5548 as part of a larger multi-wavelength reverberation mapping campaign. The campaign spanned six months and achieved an almost daily cadence with observations from five ground-based telescopes. The H$\beta$ and He II $\lambda$4686 broad emission-line light curves lag that of the 5100 $\AA$ optical continuum by $4.17^{+0.36}_{-0.36}$ days and $0.79^{+0.35}_{-0.34}$ days, respectively. The H$\beta$ lag relative to the 1158 $\AA$ ultraviolet continuum light curve measured by the Hubble Space Telescope is roughly $\sim$50% longer than that measured against the optical continuum, and the lag difference is consistent with the observed lag between the optical and ultraviolet continua. This suggests that the characteristic radius of the broad-line region is $\sim$50% larger than the value inferred from optical data alone. We also measured velocity-resolved emission-line lags for H$\beta$ and found a complex velocity-lag structure with shorter lags in the line wings, indicative of a broad-line region dominated by Keplerian motion. The responses of both the H$\beta$ and He II $\lambda$4686 emission lines to the driving continuum changed significantly halfway through the campaign, a phenomenon also observed for C IV, Ly $\alpha$, He II(+O III]), and Si IV(+O IV]) during the same monitoring period. Finally, given the optical luminosity of NGC 5548 during our campaign, the measured H$\beta$ lag is a factor of five shorter than the expected value implied by the $R_\mathrm{BLR} - L_\mathrm{AGN}$ relation based on the past behavior of NGC 5548.
• ### CANDELS Multiwavelength Catalogs: Source Identification and Photometry in the CANDELS COSMOS Survey Field(1612.07364)
Dec. 21, 2016 astro-ph.GA
We present a multi-wavelength photometric catalog in the COSMOS field as part of the observations by the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS). The catalog is based on Hubble Space Telescope Wide Field Camera 3 (HST/WFC3) and Advanced Camera for Surveys (ACS) observations of the COSMOS field (centered at RA: $10^h00^m28^s$, Dec:$+02^{\circ}12^{\prime}21^{\prime\prime}$). The final catalog has 38671 sources with photometric data in forty two bands from UV to the infrared ($\rm \sim 0.3-8\,\mu m$). This includes broad-band photometry from the HST, CFHT, Subaru, VISTA and Spitzer Space Telescope in the visible, near infrared and infrared bands along with intermediate and narrow-band photometry from Subaru and medium band data from Mayall NEWFIRM. Source detection was conducted in the WFC3 F160W band (at $\rm 1.6\,\mu m$) and photometry is generated using the Template FITting algorithm. We further present a catalog of the physical properties of sources as identified in the HST F160W band and measured from the multi-band photometry by fitting the observed spectral energy distributions of sources against templates.
• ### Space Telescope and Optical Reverberation Mapping Project VI: reverberating Disk Models for NGC 5548(1611.06051)
Nov. 24, 2016 astro-ph.GA, astro-ph.HE
We conduct a multiwavelength continuum variability study of the Seyfert 1 galaxy NGC 5548 to investigate the temperature structure of its accretion disk. The 19 overlapping continuum light curves (1158 to 9157 angstroms) combine simultaneous HST , Swift , and ground-based observations over a 180 day period from 2014 January to July. Light-curve variability is interpreted as the reverberation response of the accretion disk to irradiation by a central time-varying point source. Our model yields the disk inclination, i, temperature T1 at 1 light day from the black hole, and a temperature-radius slope, alpha. We also infer the driving light curve and find that it correlates poorly with both the hard and soft X-ray light curves, suggesting that the X-rays alone may not drive the ultraviolet and optical variability over the observing period. We also decompose the light curves into bright, faint, and mean accretion-disk spectra. These spectra lie below that expected for a standard blackbody accretion disk accreting at L/LEdd = 0.1
• ### TeV Cosmic-Ray Anisotropy from the Magnetic Field at the Heliospheric Boundary(1610.03097)
Oct. 10, 2016 astro-ph.EP, astro-ph.HE
We performed numerical calculations to test the suggestion by Desiati & Lazarian (2013) that the anisotropies of TeV cosmic rays may arise from their interactions with the heliosphere. For this purpose, we used a magnetic field model of the heliosphere and performed direct numerical calculations of particle trajectories. Unlike earlier papers testing the idea, we did not employ time-reversible techniques that are based on Liouville's theorem. We showed numerically that for scattering by the heliosphere the conditions of Liouville's theorem are not satisfied and the adiabatic approximation and time-reversibility of the particle trajectories are not valid. Our results indicate sensitivity to the magnetic structure of the heliospheric magnetic field, and we expect that this will be useful for probing this structure in future research.
• ### Tracing the reionization epoch with ALMA: [CII] emission in z~7 galaxies(1608.08837)
Aug. 31, 2016 astro-ph.GA
We present new results on [CII]158$\mu$ m emission from four galaxies in the reionization epoch. These galaxies were previously confirmed to be at redshifts between 6.6 and 7.15 from the presence of the Ly$\alpha$ emission line in their spectra. The Ly$\alpha$ emission line is redshifted by 100-200 km/s compared to the systemic redshift given by the [CII] line. These velocity offsets are smaller than what is observed in z~3 Lyman break galaxies with similar UV luminosities and emission line properties. Smaller velocity shifts reduce the visibility of Ly$\alpha$ and hence somewhat alleviate the need for a very neutral IGM at z~7 to explain the drop in the fraction of Ly$\alpha$ emitters observed at this epoch. The galaxies show [CII] emission with L[CII]=0.6-1.6 x10$^8 L_\odot$: these luminosities place them consistently below the SFR-L[CII] relation observed for low redshift star forming and metal poor galaxies and also below z =5.5 Lyman break galaxies with similar star formation rates. We argue that previous undetections of [CII] in z~7 galaxies with similar or smaller star formation rates are due to selection effects: previous targets were mostly strong Ly$\alpha$ emitters and therefore probably metal poor systems, while our galaxies are more representative of the general high redshift star forming population .
• ### Limits on the ions temperature anisotropy in turbulent intracluster medium(1601.03837)
Turbulence in the weakly collisional intracluster medium of galaxies (ICM) is able to generate strong thermal velocity anisotropies in the ions (with respect to the local magnetic field direction), if the magnetic moment of the particles is conserved in the absence of Coulomb collisions. In this scenario, the anisotropic pressure magnetohydrodynamic (AMHD) turbulence shows a very different statistical behaviour from the standard MHD one and is unable to amplify seed magnetic fields, in disagreement with previous cosmological MHD simulations which are successful to explain the observed magnetic fields in the ICM. On the other hand, temperature anisotropies can also drive plasma instabilities which can relax the anisotropy. This work aims to compare the relaxation rate with the growth rate of the anisotropies driven by the turbulence. We employ quasilinear theory to estimate the ions scattering rate due to the parallel firehose, mirror, and ion-cyclotron instabilities, for a set of plasma parameters resulting from AMHD simulations of the turbulent ICM. We show that the ICM turbulence can sustain only anisotropy levels very close to the instabilities thresholds. We argue that the AMHD model which bounds the anisotropies at the marginal stability levels can describe the Alfvenic turbulence cascade in the ICM.
• ### Space Telescope and Optical Reverberation Mapping Project. IV. Anomalous behavior of the broad ultraviolet emission lines in NGC 5548(1603.08741)
March 29, 2016 astro-ph.GA
During an intensive Hubble Space Telescope (HST) Cosmic Origins Spectrograph (COS) UV monitoring campaign of the Seyfert~1 galaxy NGC 5548 performed from 2014 February to July, the normally highly correlated far-UV continuum and broad emission-line variations decorrelated for ~60 to 70 days, starting ~75 days after the first HST/COS observation. Following this anomalous state, the flux and variability of the broad emission lines returned to a more normal state. This transient behavior, characterised by significant deficits in flux and equivalent width of the strong broad UV emission lines, is the first of its kind to be unambiguously identified in an active galactic nucleus reverberation mapping campaign. The largest corresponding emission-line flux deficits occurred for the high-ionization collisionally excited lines, C IV and Si IV(+O IV]), and also He II(+O III]), while the anomaly in Ly-alpha was substantially smaller. This pattern of behavior indicates a depletion in the flux of photons with E_{\rm ph} > 54 eV, relative to those near 13.6 eV. We suggest two plausible mechanisms for the observed behavior: (i) temporary obscuration of the ionizing continuum incident upon BLR clouds by a moving veil of material lying between the inner accretion disk and inner BLR, perhaps resulting from an episodic ejection of material from the disk, or (ii) a temporary change in the intrinsic ionizing continuum spectral energy distribution resulting in a deficit of ionizing photons with energies > 54 eV, possibly due to a transient restructuring of the Comptonizing atmosphere above the disk. Current evidence appears to favor the latter explanation.
• ### Space Telescope and Optical Reverberation Mapping Project. III. Optical Continuum Emission and Broad-Band Time Delays in NGC 5548(1510.05648)
Feb. 29, 2016 astro-ph.GA
We present ground-based optical photometric monitoring data for NGC 5548, part of an extended multi-wavelength reverberation mapping campaign. The light curves have nearly daily cadence from 2014 January to July in nine filters (\emph{BVRI} and \emph{ugriz}). Combined with ultraviolet data from the \emph{Hubble Space Telescope} and \emph{Swift}, we confirm significant time delays between the continuum bands as a function of wavelength, extending the wavelength coverage from 1158\,\AA\ to the $z$ band ($\sim\!9160$\,\AA). We find that the lags at wavelengths longer than the {\it V} band are equal to or greater than the lags of high-ionization-state emission lines (such as He\,{\sc ii}\,$\lambda 1640$ and $\lambda 4686$), suggesting that the continuum-emitting source is of a physical size comparable to the inner broad-line region (BLR). The trend of lag with wavelength is broadly consistent with the prediction for continuum reprocessing by an accretion disk with $\tau \propto \lambda^{4/3}$. However, the lags also imply a disk radius that is 3 times larger than the prediction from standard thin-disk theory, assuming that the bolometric luminosity is 10\% of the Eddington luminosity ($L = 0.1L_{\rm Edd}$). Using optical spectra from the Large Binocular Telescope, we estimate the bias of the interband continuum lags due to BLR emission observed in the filters. We find that the bias for filters with high levels of BLR contamination ($\sim\! 20\%$) can be important for the shortest continuum lags, and likely has a significant impact on the {\it u} and {\it U} bands owing to Balmer continuum emission.
• ### Probe Spin-Velocity Dependent New Interactions by Spin Relaxation Times of Polarized $^{3}He$ Gas(1412.8155)
Sept. 18, 2015 nucl-ex
We have constrained possible new interactions which produce nonrelativistic potentials between polarized neutrons and unpolarized matter proportional to $\alpha\vec{\sigma}\cdot\vec{v}$ where $\vec{\sigma}$ is the neutron spin and $\vec{v}$ is the relative velocity. We use existing data from laboratory measurements on the very long $T_{1}$ and $T_{2}$ spin relaxation times of polarized $^{3}$He gas in glass cells.Using the best available measured $T_{2}$ of polarized $^{3}$He gas atoms as the polarized source and the earth as an unpolarized source, we obtain constraints on two new interactions. We present a new experimental upper bound on possible vector-axial-vector($V_{VA}$) type interactions for ranges between $1\sim10^{8}$m. In combination with previous results, we set the most stringent experiment limits on $g_{V}g_{A}$ ranging from $\sim\mu$m to $\sim10^{8}$m. We also report what is to our knowledge the first experimental upper limit on the possible torsion fields induced by the earth on its surface. Dedicated experiments could further improve these bounds by a factor of $\sim100$. Our method of analysis also makes it possible to probe many velocity dependent interactions which depend on the spins of both neutrons and other particles which have never been searched for before experimentally.
• ### A universal structural and star-forming relation since $z\sim3$: connecting compact star-forming and quiescent galaxies(1509.00469)
Sept. 1, 2015 astro-ph.GA
We study the evolution of the core (r<1 kpc) and effective (r<r_e) stellar-mass surface densities, in star-forming and quiescent galaxies. Since z=3, both populations occupy distinct, linear relations in log(Sigma_e) and log(Sigma_1) vs. log(M). These structural relations exhibit slopes and scatter that remain almost constant with time while their normalizations decline. For SFGs, the normalization declines by less than a factor of 2 from z=3, in both Sigma_e and Sigma_1. Such mild declines suggest that SFGs build dense cores by growing along these relations. We define this evolution as the structural main sequence (Sigma-MS). Quiescent galaxies follow different relations (Sigma^Q_e, Sigma^Q_1) off the Sigma-MS by having higher densities than SFGs of the same mass and redshift. The normalization of Sigma^Q_e declines by a factor of 10 since z=3, but only a factor of 2 in Sigma^Q_1. Thus, the common denominator for quiescent galaxies at all redshifts is the presence of a dense stellar core, and the formation of such cores in SFGs is the main requirement for quenching. Expressed in 2D as deviations off the SFR-MS and off Sigma^Q_1 at each redshift, the distribution of massive galaxies forms a universal, L-shaped sequence that relates two fundamental physical processes: compaction and quenching. Compaction is a process of substantial core-growth in SFGs relative to that in the Sigma-MS. This process increases the core-to-total mass and Sersic index, thereby, making compact SFGs. Quenching occurs once compact SFGs reach a maximum central density above Sigma^Q_1 > 9.5 M_sun/kpc^2. This threshold provides the most effective selection criterion to identify the star-forming progenitors of quiescent galaxies at all redshifts.
• ### A First Experimental Limit on In-matter Torsion from Neutron Spin Rotation in Liquid He-4(1311.0467)
June 25, 2015 hep-th, hep-ph, hep-ex, nucl-ex, gr-qc
We report the first experimental upper bound to our knowledge on possible in-matter torsion interactions of the neutron from a recent search for parity violation in neutron spin rotation in liquid He-4. Our experiment constrains a coefficient $\zeta$ consisting of a linear combination of parameters involving the time components of the torsion fields $T^\mu$ and $A^\mu$ from the nucleons and electrons in helium which violates parity. We report an upper bound of $|\zeta|<9.1x10^{-23}$ GeV at 68% confidence level and indicate other physical processes that could be analyzed to constrain in-matter torsion.
• ### Detecting somatic mutations in genomic sequences by means of Kolmogorov-Arnold analysis(1506.04080)
June 12, 2015 physics.data-an, q-bio.GN
The Kolmogorov-Arnold stochasticity parameter technique is applied for the first time to the study of cancer genome sequencing, to reveal mutations. Using data generated by next generation sequencing technologies, we have analyzed the exome sequences of brain tumor patients with matched tumor and normal blood. We show that mutations contained in sequencing data can be revealed using this technique thus providing a new methodology for determining subsequences of given length containing mutations i.e. its value differs from those of subsequences without mutations. A potential application for this technique involves simplifying the procedure of finding segments with mutations, speeding up genomic research, and accelerating its implementation in clinical diagnostic. Moreover, the prediction of a mutation associated to a family of frequent mutations in numerous types of cancers based purely on the value of the Kolmogorov function, indicates that this applied marker may recognize genomic sequences that are in extremely low abundance and can be used in revealing new types of mutations.
• ### Space Telescope and Optical Reverberation Mapping Project. I. Ultraviolet Observations of the Seyfert 1 Galaxy NGC 5548 with the Cosmic Origins Spectrograph on Hubble Space Telescope(1501.05954)
April 27, 2015 astro-ph.GA
We describe the first results from a six-month long reverberation-mapping experiment in the ultraviolet based on 170 observations of the Seyfert 1 galaxy NGC 5548 with the Cosmic Origins Spectrograph on the Hubble Space Telescope. Significant correlated variability is found in the continuum and broad emission lines, with amplitudes ranging from ~30% to a factor of two in the emission lines and a factor of three in the continuum. The variations of all the strong emission lines lag behind those of the continuum, with He II 1640 lagging behind the continuum by ~2.5 days and Lyman alpha 1215, C IV 1550, and Si IV 1400 lagging by ~5-6 days. The relationship between the continuum and emission lines is complex. In particular, during the second half of the campaign, all emission-line lags increased by a factor of 1.3-2 and differences appear in the detailed structure of the continuum and emission-line light curves. Velocity-resolved cross-correlation analysis shows coherent structure in lag versus line-of-sight velocity for the emission lines; the high-velocity wings of C IV respond to continuum variations more rapidly than the line core, probably indicating higher velocity BLR clouds at smaller distances from the central engine. The velocity-dependent response of Lyman alpha, however, is more complex and will require further analysis.
• ### Space Telescope and Optical Reverberation Mapping Project. II. Swift and HST Reverberation Mapping of the Accretion Disk of NGC 5548(1501.05951)
April 7, 2015 astro-ph.GA, astro-ph.HE
Recent intensive Swift monitoring of the Seyfert 1 galaxy NGC 5548 yielded 282 usable epochs over 125 days across six UV/optical bands and the X-rays. This is the densest extended AGN UV/optical continuum sampling ever obtained, with a mean sampling rate <0.5 day. Approximately daily HST UV sampling was also obtained. The UV/optical light curves show strong correlations (r_max = 0.57 - 0.90) and the clearest measurement to date of interband lags. These lags are well-fit by a \tau propto \lambda^4/3 wavelength dependence, with a normalization that indicates an unexpectedly large disk radius of 0.35 +/- 0.05 lt-day at 1367 A, assuming a simple face-on model. The U-band shows a marginally larger lag than expected from the fit and surrounding bands, which could be due to Balmer continuum emission from the broad-line region as suggested by Korista and Goad. The UV/X-ray correlation is weaker (r_max < 0.45) and less consistent over time. This indicates that while Swift is beginning to measure UV/optical lags in general agreement with accretion disk theory (although the derived size is larger than predicted), the relationship with X-ray variability is less well understood. Combining this accretion disk size estimate with those from quasar microlensing studies suggests that AGN disk sizes scale approximately linearly with central black hole mass over a wide range of masses.
• ### Experimental Observation of Simultaneous Wave and Particle Behaviors in a Narrowband Single Photon's Wave Packe(1412.1549)
Dec. 4, 2014 quant-ph
Light's wave-particle duality is at the heart of quantum mechanics and can be well illustrated by Wheeler's delayed-choice experiment. The choice of inserting or removing the second classical (quantum) beam splitter in a Mach-Zehnder interferometer determines the classical (quantum) wave-particle behaviors of a photon. In this paper, we report our experiment using the classical beam splitter to observe the simultaneous wave-particle behaviors in the wave-packet of a narrowband single photon. This observation suggests that it is necessary to generalize the current quantum wave-particle duality theory. Our experiment demonstrates that the produced wave-particle state can be considered an additional degree of freedom and can be utilized in encoding quantum information.
• ### Absolute measurement of thermal noise in a resonant short-range force experiment(1402.0145)
Planar, double-torsional oscillators are especially suitable for short-range macroscopic force search experiments, since they can be operated at the limit of instrumental thermal noise. As a study of this limit, we report a measurement of the noise kinetic energy of a polycrystalline tungsten oscillator in thermal equilibrium at room temperature. The fluctuations of the oscillator in a high-Q torsional mode with a resonance frequency near 1 kHz are detected with capacitive transducers coupled to a sensitive differential amplifier. The electronic processing is calibrated by means of a known electrostatic force and input from a finite element model. The measured average kinetic energy is in agreement with the expected value of 1/2 kT.
• ### Coulomb Interaction Effects on the Terahertz Photon-Assisted Tunneling through a InAs Quantum Dot(1408.3442)
Aug. 14, 2014 cond-mat.mes-hall
Recently, the terahertz (THz) photon-assisted tunneling (PAT) through a two-level InAs quantum dot (QD) has been successfully realized in experiment [Phys. Rev. Lett. {\bf 109}, 077401 (2012)]. The Coulomb interaction in this device is comparable with the energy difference between the two energy levels. We theoretically explore the effects of Coulomb interaction on the PAT processes and show that the main peaks of the experiment can be well derived by our model analysis. Furthermore, we find additional peaks, which were not addressed in the InAs QD experiment and may be further identified in experiments. In particular, we show that, to observe the interesting photon-induced excited state resonance in InAs QD, the Coulomb interaction should be larger than THz photon frequency.
• ### Hole Spin Coherence in a Ge/Si Heterostructure Nanowire(1403.2093)
June 13, 2014 cond-mat.mes-hall
Relaxation and dephasing of hole spins are measured in a gate-defined Ge/Si nanowire double quantum dot using a fast pulsed-gate method and dispersive readout. An inhomogeneous dephasing time $T_2^* \sim 0.18~\mathrm{\mu s}$ exceeds corresponding measurements in III-V semiconductors by more than an order of magnitude, as expected for predominately nuclear-spin-free materials. Dephasing is observed to be exponential in time, indicating the presence of a broadband noise source, rather than Gaussian, previously seen in systems with nuclear-spin-dominated dephasing.
• ### Antilocalization of Coulomb Blockade in a Ge-Si Nanowire(1401.2948)
Jan. 13, 2014 cond-mat.mes-hall
The distribution of Coulomb blockade peak heights as a function of magnetic field is investigated experimentally in a Ge-Si nanowire quantum dot. Strong spin-orbit coupling in this hole-gas system leads to antilocalization of Coulomb blockade peaks, consistent with theory. In particular, the peak height distribution has its maximum away from zero at zero magnetic field, with an average that decreases with increasing field. Magnetoconductance in the open-wire regime places a bound on the spin-orbit length ($l_{so}$ < 20 nm), consistent with values extracted in the Coulomb blockade regime ($l_{so}$ < 25 nm). | 2021-02-28 14:58:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6588099002838135, "perplexity": 2602.819414463287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361510.12/warc/CC-MAIN-20210228145113-20210228175113-00253.warc.gz"} |
https://www.physicsforums.com/threads/writting-a-circle-in-a-latex-document.346818/ | # Writting a Circle in a Latex Document
1. Oct 18, 2009
### EngWiPy
Hello,
I have the following Latex segment of code:
Code (Text):
\documentclass[a4paper,12pt,openany,oneside]{memoir}
\usepackage{tikz}
\begin{document}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\pgfsetfillopacity{0.5}
\draw [very thick,dashed,fill=gray!60](0,0) circle (3.6cm);
\draw [dashed,fill=gray!20](3,2) circle (3.6cm);
\draw (0,0) circle (6cm);
\node (source) at (0,0) [circle,draw] {$S$};
\node (relay) at (3,2) [circle,draw] {$R$};
\node (destination) at (6,0) [circle,draw] {$D$};
\draw [->] (source.60) to node [auto] {} (relay.west);
\draw [->] (relay.east) to node [auto] {} (destination.120);
\draw [->] (source.east) to node [auto,swap] {} (destination.west);
\end{tikzpicture}
\end{center}
\end{figure}
\end{document}
I want to write above each circle. How can I do that? | 2018-07-20 15:38:22 | {"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": 0.9878872632980347, "perplexity": 11583.050687637979}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591683.78/warc/CC-MAIN-20180720135213-20180720155213-00217.warc.gz"} |
https://community.keysight.com/thread/20311 | AnsweredAssumed Answered
# Matlab Script Engine doesn't start
Question asked by omar on Nov 10, 2010
Hello there,
after installing matlab R2009b I can't start my Matalb Skripts within VEE8, every time i get the matlab command window with the following error meessage:
Warning: Directory access failure: C:\Program Files\Agilent\VEE Pro 8.0\matlab\toolbox\runtime.
Warning: Directory access failure: C:\Program Files\Agilent\VEE Pro 8.0\matlab\work.
To get started, type one of these: helpwin, helpdesk, or demo.
For product information, visit http://www.mathworks.com.
??? Undefined function or variable 'set_param'.
Error in ==> C:\Program Files\Agilent\VEE Pro 8.0\matlab\toolbox\local\matlabrc.m
On line 131 ==>
»
before installing matlab R2009b, i was using Matlab 7.0.1 with the Signal Processing toolbox and i have never had such Problem, Unfortunately now I don't have the license for the Signal Processing toolbox (but if it is the only solution to make my Matlab Skripts work again, i could get it)
I have tried to register the built-in MATLAB Engine by going the directory (CD)<VEE_installation_dir>\MATLAB\bin\win32\ and executing the command "MATLAB.exe /regserver." but without any success ! I still have the same error!!!
If anyone can help with this I would really appreciate
Thanks, | 2021-01-24 19:47:52 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8311702013015747, "perplexity": 11444.106796690274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703550617.50/warc/CC-MAIN-20210124173052-20210124203052-00793.warc.gz"} |
https://k12.libretexts.org/Bookshelves/Mathematics/Analysis/02%3A_Polynomial_and_Rational_Functions/2.06%3A_Finding_Zeros_of_Polynomials/2.6.02%3A_Synthetic_Division_of_Polynomials | # 2.6.2: Synthetic Division of Polynomials
## Synthetic Division of Polynomials
The volume of a rectangular prism is 2x3+5x2−x−6. Determine if 2x+3 is the length of one of the prism's sides.
## Synthetic Division
Synthetic division is an alternative to long division. It can also be used to divide a polynomial by a possible factor, x−k. However, synthetic division cannot be used to divide larger polynomials, like quadratics, into another polynomial.
Let's use synthetic division to divide 2x4−5x3−14x2+47x−30 by x−2.
Using synthetic division, the setup is as follows:
Therefore, 2 is a solution, because the remainder is zero. The factored polynomial is 2x3−x2−16x+15. Notice that when we synthetically divide by k, the “leftover” polynomial is one degree less than the original. We could also write
(x−2)(2x3−x2−16x+15)=2x4−5x3−14x2+47x−30.
Now, let's determine if 4 is a solution to f(x)=5x3+6x2−24x−16.
Using synthetic division, we have:
The remainder is 304, so 4 is not a solution. Notice if we substitute in x=4, also written f(4), we would have f(4)=5(4)3+6(4)2−24(4)−16=304. This leads us to the Remainder Theorem.
Remainder Theorem: If f(k)=r, then r is also the remainder when dividing by (x−k).
This means that if you substitute in x=k or divide by k, what comes out of f(x) is the same. r is the remainder, but it is also the corresponding y−value. Therefore, the point (k,r) would be on the graph of f(x).
Finally, let's determine if (2x−5) is a factor of 4x4−9x2−100.
If you use synthetic division, the factor is not in the form (x−k). We need to solve the possible factor for zero to see what the possible solution would be. Therefore, we need to put $$\ \frac{5}{2}$$ up in the left-hand corner box. Also, not every term is represented in this polynomial. When this happens, you must put in zero placeholders. In this problem, we need zeros for the x3−term and the x−term.
This means that $$\ \frac{5}{2}$$ is a zero and its corresponding binomial, (2x−5), is a factor.
###### Example 1
Earlier, you were asked to determine if 2x+3 is the length of one of the prism's sides.
Solution
If 2x+3 divides evenly into 2x3+5x2−x−6 then it is the length of one of the prism's sides.
If we want to use synthetic division, notice that the factor is not in the form (x−k). Therefore, we need to solve the possible factor for zero to see what the possible solution would be. If 2x+3=0 then x=$$\ -\frac{3}{2}$$. Therefore, we need to put $$\ -\frac{3}{2}$$ up in the left-hand corner box.
When we perform the synthetic division, we get a remainder of 0. This means that (2x+3) is a factor of the volume. Therefore, it is also the length of one of the sides of the rectangular prism.
###### Example 2
Divide x3+9x2+12x−27 by (x+3). Write the resulting polynomial with the remainder (if there is one).
Solution
Using synthetic division, divide by -3.
The answer is $$\ x^{2}+6 x-6-\frac{9}{x+3}$$.
###### Example 3
Divide 2x4−11x3+12x2+9x−2 by (2x+1). Write the resulting polynomial with the remainder (if there is one).
Solution
Using synthetic division, divide by $$\ -\frac{1}{2}$$.
The answer is $$\ 2 x^{3}-12 x^{2}+18 x-\frac{2}{2 x+1}$$
###### Example 4
Is 6 a solution for f(x)=x3−8x2+72? If so, find the real-number zeros (solutions) of the resulting polynomial.
Solution
Put a zero placeholder for the x−term. Divide by 6.
The resulting polynomial is x2−2x−12. While this quadratic does not factor, we can use the Quadratic Formula to find the other roots.
$$\ x=\frac{2 \pm \sqrt{2^{2}-4(1)(-12)}}{2}=\frac{2 \pm \sqrt{4+48}}{2}=\frac{2 \pm 2 \sqrt{13}}{2}=1 \pm \sqrt{13}$$
The solutions to this polynomial are $$\ 6,1+\sqrt{13} \approx 4.61 \text { and } 1-\sqrt{13} \approx-2.61$$.
## Review
Use synthetic division to divide the following polynomials. Write out the remaining polynomial.
1. (x3+6x2+7x+10)÷(x+2)
2. (4x3−15x2−120x−128)÷(x−8)
3. (4x2−5)÷(2x+1)
4. (2x4−15x3−30x2−20x+42)÷(x+9)
5. (x3−3x2−11x+5)÷(x−5)
6. (3x5+4x3−x−2)÷(x−1)
7. Which of the division problems above generate no remainder? What does that mean?
8. What is the difference between a zero and a factor?
9. Find f(−2) if f(x)=2x4−5x3−10x2+21x−4.
10. Now, divide 2x4−5x3−10x2+21x−4 by (x+2) synthetically. What do you notice?
Find all real zeros of the following polynomials, given one zero.
1. 12x3+76x2+107x−20; −4
2. x3−5x2−2x+10; −2
3. 6x3−17x2+11x−2; 2
Find all real zeros of the following polynomials, given two zeros.
1. x4+7x3+6x2−32x−32; −4, −1
2. 6x4+19x3+11x2−6x; 0, −2
## Vocabulary
Term Definition
Oblique Asymptote An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.
Oblique Asymptotes An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.
Remainder Theorem The remainder theorem states that if f(k)=r, then r is the remainder when dividing f(x) by (x−k).
Synthetic Division Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.
This page titled 2.6.2: Synthetic Division of Polynomials is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | 2022-09-29 17:14:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9000734090805054, "perplexity": 942.5028649182107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335362.18/warc/CC-MAIN-20220929163117-20220929193117-00574.warc.gz"} |
http://mathhelpforum.com/calculus/84737-couple-problems-i-have.html | # Math Help - couple problems i have
1. ## couple problems i have
I need help with a few problems I have left.
1. Given : r = csc²Θ
how do I find the range of all possible values of r and also how do I find a Cartesian equation without fractions
2. Given : √(1+3cos²Θ(2-cos²Θ)) dΘ
how do i show that the expression under the radical is positive for all angle
2. ## solved
To solve the first one, it's best to change the problem to Cartesian first, just to see it much easier. In order to do that, you'll want to do this:
$r=1/sin^2\Theta$
$r \times sin^2\Theta=1$
$r^2 \times sin^2\Theta = r$ *multiply both sides by r*
$y^2 = r$ *substitute $y^2$ for $r^2 sin^2\Theta$ *
$y^2 = \pm\sqrt{x^2 + y^2}$ *substitute in $\pm\sqrt{x^2 + y^2}$ for r*
Now you have your Cartesian. So, now let's solve for x, much easier than to solve for y. You should get $\pm y \sqrt{y^2 - 1} = x$
Now, since we want the inverse, just look at the Domain of the above function and you have the Range you wanted originally, which is $\Re/(-1,1)$
to answer part 2, we look at first $cos^2\Theta$ and see that for $cos\Theta$ is negative when $\Theta\pi/2,3\pi/4)" alt="\Theta\pi/2,3\pi/4)" />, but since it's squared, then we don't have to worry about that term. So, since $cos^2\theta$ is positive for all $\Theta$ and since the largest value of $cos^2\Theta$ is 1, then 2-1 is 1, hence positive.
Hope this helps.
3. ## Thanks!!!!!
Thanks soooooo much! You helped me out a ton!! Thanks also for explaining how it works. | 2015-11-27 08:43:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "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": 0.8478987812995911, "perplexity": 628.7489141215977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398448389.58/warc/CC-MAIN-20151124205408-00201-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/31262/deriving-closed-formulas-for-summations | # Deriving closed formulas for summations [duplicate]
Possible Duplicate:
Value of $\sum x^n$
I was wondering how to derive a closed formula for things like $\sum_{i=1}^{n}2^{n}$=$2(2^{n}-1)$ and $\sum_{n=k}^{n}2^{n-k}$=$2^{n-k+1}-1$. I haven't done this in a while, and had wolfram do it for me, and I am not sure what the general tactic in getting these formulas is. Your help is appreciated!
-
## marked as duplicate by Aryabhata, Ross Millikan, Mike Spivey, Akhil MathewApr 6 '11 at 4:56
Well, you can consider three ways, for one, use the inductive method; for two, try to write down what you want to sum, and then multiply it by 2 and subtract them; for three, try to write it as a recurrence relation, and compute it as usual. In each case, just remember that x-1 is a divisor of $x^n -1$ when n is a strictly positive natural number. | 2015-11-27 07:05:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8748767375946045, "perplexity": 367.7033851547228}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398448227.60/warc/CC-MAIN-20151124205408-00147-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://en.m.wiktionary.org/wiki/horosphere | # horosphere
## English
The disk model of the hyperbolic plane, with a horosphere in blue.
### Etymology
A modern scientific coinage, probably from a Latinized spelling of Ancient Greek ὅρος (hóros, boundary) (not to be confused with ὧρος (hôros, year), which gave rise to many time-related terms) + sphere
### Noun
horosphere (plural horospheres)
1. An ${\displaystyle n-1}$ -dimensional hyperplane in hyperbolic ${\displaystyle n}$ -dimensional space: it is (in the Poincaré disk) Euclidean-tangent at infinity to the boundary sphere or (in the upper-half-space model) Euclidean-parallel to the boundary hyperplane. | 2020-09-30 04:09:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8731222152709961, "perplexity": 7906.602562245009}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402101163.62/warc/CC-MAIN-20200930013009-20200930043009-00360.warc.gz"} |
https://www.geogebra.org/material/show/id/17469 | # Example 1
Determine which congruence statement, if any, can be used to show that $\triangle{\it PQR}$ and $\triangle{\it STU}$ are congruent.
Resource Type
Activity
Tags
congruence congruent-triangles corresponding-parts practice
Target Group (Age)
14 – 18
Language
English
• GeoGebra
• Help
• Partners | 2018-12-19 08:17:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2681863605976105, "perplexity": 12124.687460867382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376831715.98/warc/CC-MAIN-20181219065932-20181219091932-00023.warc.gz"} |
https://space.stackexchange.com/tags/nasa/new | # Tag Info
32
The early prototypes of the A4/V2 were painted in the familiar black-and-white roll pattern scheme. This scheme was designed to aid in tracking the rocket after launch. This pattern made it easy to observe any variation or roll of the rocket. The exact pattern was changed many times, and as with the rest of the rocket, the pattern was examined and ...
1
It was important to increase the sample size for medical tests on the astronauts themselves. These still are the only people that have gone outside of low earth orbit, ever. The data, especially on radiation poisoning and Moon EVA human factors, is vital for new missions to the Moon or Mars.
0
The Space Shuttle could have made money as technically it performs very well at delivering large payloads to the ISS (Falcon 9 rockets cost 1.39X more to deliver people and payloads to the ISS according to my table below). The Space Shuttle is probably is not as nimble and adaptable as other smaller systems - both technically and from a pace-of-business ...
5
While this is much granier than the BBC snippet, it does contain the audio overlay and is the same video the BBC clip has, but contains the full event cycle. "Lock the doors" happens around 12:24 Key parts of the full audio (time in the video first) 13:41 - OK all flight controllers on the Flight [Director] loop, we need ...
-2
Because the Handbook told him to was assuredly the proximate cause for him saying the phrase, and evidence preservation is undoubtedly a good reason the manual says to do so, but--so much as a movie and my memory can be trusted--in The Right Stuff, during Mercury, higher ups overruled the Flight Director on an important call during an anomaly. I believe the ...
44
This is indeed part of the procedure that is invoked when a contingency has occurred. It is part of Standard Operation Procedure 2.8 - JSC Contingency Plan, which can be found in the Shuttle Flight Control Operations Handbook (link to 538-page pdf - referenced here) on page 2.8-1. It provides the steps to be taken to secure all data for future investigations ...
22
Does he mean lock the doors in the NASA building, in order to begin some internal investigation, and nobody is allowed to leave, as a matter of policy? Yes, this. It's part of a standard procedure to ensure evidence is preserved for the investigation. It's to prevent people entering as well as leaving.
10
I didn't work on Voyager, but can tell you that deep-space missions tend to retain the original ground hardware, software, language, and build environment, both for continuity/safety as well as budget reasons. There may be little or no funding to continue the mission; it may even fall to outright volunteers. It's amazing and sad to me how much we depend on ...
0
This is too long for a comment. tl;dr: I can't conclude that the lens wasn't used (as @tfb's excellent answer doesn't yet conclude either) but it does seem reasonable that NASA would have gotten a hold of several of the fastest lenses in the world just in case they might have come in handy. Rangers VII, VIII and IX Ranger is mentioned in @Roger's comment ...
32
In 2015, the last original Voyager engineer still on the project, retired. NASA specified that his replacement would have to know FORTRAN. The software was updated regularly after launch: The last true software overhaul was in 1990, after the 1989 Neptune encounter and at the beginning of the interstellar mission. "The flight software was basically ...
13
The Voyager spacecraft are not reprogrammed anymore, so the language in which they are programmed is largely irrelevant. The uplink is only 16 bits/second, just enough to send (simple) commands. How these commands are generated is irrelevant to the spacecraft. Any language that can generate a sequence of bits theoretically suffices. This pdf document ...
3
This isn't a complete answer, but I think there should be at least some doubt cast on this story: it's certainly not as clear as a lot of people think it is. The Wikipedia entry for this lens claims that it was developed in 1966. So we can wonder what it might have been used for. It is generally claimed that it was used (or designed) for pictures of the ...
6
During the Apollo era (1967-1972), 4 NASA astronauts died during training or test flight. Those 4 brave people were the Apollo 1 crew, Gus Grissom, Ed White, and Roger B. Chaffee. The fourth astronaut was Clifton C. Williams who died flying a training jet. However no NASA astronaut died in spaceflight during the Apollo era. Unfortunately, there were ...
7
This is a strange question, but it is worth answering. The ultimate answer is: Space is hard. But let's unpack that. There were many, many factors that led to the various failures and accidents. Some of them were a simple lack of understanding during the design phase - space is hard, that's not a joke but a reality. Simply put, we don't understand how ...
6
It was Voyager 2 (launched before Voyager 1). Technical details on Voyagers are surprisingly hard to find. The most detailed account I found was in Voyage to Jupiter, the official NASA history of the project. During the first minutes of flight, there seemed to be two difficulties with the AACS.The first was a problem with one ...
5
Three more missions were planned but canceled to save some money. The lost and forgotten missions 18, 19 and 20. To get moon dust and rocks samples from more than one spot only. To get some samples of hardware from the Surveyor 3 lander during Apollo 12. To place more than one lunar seismometer. To place more than one lunar ranging retro reflector. To have ...
1
Neil Armstrong was also very cool in the first USA in space mishap, tumbling after docking with the Agena But a better explanation: All these astronauts were ultra high achieving, fighter pilots and test pilots with high ranks. The LMP was actually a co-pilot/flight engineer job (and needed said job very much along with a very competent main pilot). But ...
0
Everything I've seen from Mercury to Apollo has been in imperial units but that doesn't mean they used metric and converted it all over before handing it over. There is a pretty good article here that talks about the Apollo guidance computer. The computer was programed to use SI units, which is basically metric with a new name, but everything entered into it ...
4
Upper left: KSC control room for the Instrument Unit during the launch of Apollo 8. I am not sure if this is the Launch Control Center Firing Room or a back room. I think a back room. (reference) Upper right - the Instrument Unit itself as described in this answer Lower left - Launch Control Center Firing Room at KSC, looking away from the window (the ...
10
When I was trained in Aerospace Engineering at Texas A&M in the 1980s, we were taught to use what has been referred to as the Gravitational FPS system, where distance is measured in feet, force in pounds, and mass in slugs. This was a big disappointment to me at the time, since I had been quite comfortable with metric units in high school physics. ...
80
The Apollo Guidance Computer did use metric/SI units internally for its calculations. But it converted to imperial/USC units when it displayed data on the DSKY. This is probably because the Apollo astronauts (mostly trained as test pilots) had an intuitive "feel" for imperial/USC units. Although data was stored internally in metric units, they were ...
35
NASA used English Engineering Units not Imperial units. (This phrasing is a reply to the original, un-edited question title) They did this because the program was implemented by the US aerospace industry and that industry's industrial base was all in English units. Every manual, tool, data book, milling machine, and fastener used those units. Conversion ...
5
The Apollo 16 Mission Report, section 9.8.2 explains First extravehicular activity.- Following the first sleep period and normal preparation for surface activities, the Commander egressed slightly ahead of schedule. Because of the steerable antenna failure, there was no television coverage of the Commander stepping onto the lunar surface. According to ...
4
It seems like it had to do more with overall politics, funding, goals of human space program than with his German/Nazi background. NASA transferred him to HQ to be responsible for "Planning" in March 1970 and he retired from NASA shortly thereafter i 1972 "after a series of conflicts associated with the truncation of the Apollo program" according to his ...
3
NASA's investments are providing a massive Return on Investment(ROI) This article makes the case that NASA has provided a 1000% ROI A report by the Space Foundation estimated that activities related to space contributed $180 billion to the economy in 2005. More than 60 percent of this came from commercial goods and services created by companies related ... 1 In addition to cooperative programs with tickets, there also have been exchanges of lunar samples, the Soviets offering material obtained from their automated sample-return Luna missions. A brief news article is given here and a photograph is here. This exchange was carried out in Moscow at the Academy of Sciences in 1971. Because the Luna samples were ... 0 1 Unified S Band 2.1 Ghz TV Signal / Voice / Telemetry / Medical Moon to Earth / Earth to Moon <* 2 Two UHF Frequencies 225-380Mhz band used on the moon <* 3 Various radar 8.1 Ghz 4 C-Band 5.6 GHZ 5 2 x Intelsat Satelites for round world & USA TV coverage 6 890mhz tracking 11 Comparison (US expenditures only): \$18 billion is spent per year on simply advertising cosmetics. \$60 billion is spent per year trying to lose weight. \$70 billion is spent per year on pets. \$230 billion is spent per year on advertising. \$0.5 billion is spent per rocket. Just think how many of the world's problems could be solved if we didn't waste so ...
32
It's not mentioned in Apollo documentation. However, these would be the arguments against it: The idea of two spacecraft was already controversial. Now you want three? The original Apollo concept was a "direct" mission. The entire mission would launch as one vehicle, shedding pieces as you go, but never putting pieces back together. All of the ...
2
The RCS thrusters on Apollo come in balanced pairs, one on each side of the vessel's center of gravity. Normally, if you wanted to change your orientation, you'd fire both thrusters in a pair, producing a rotation but not a translation. However, these thrusters have a minimum power level. If you want to create a very small attitude change, you'd fire just ...
6
That made me pause and wonder for myself what it is that's out there that might be worth this money, and reflect upon the many problems on Earth that could be addressed with these resources instead. I would add to the other great answers another very valuable outcome our society gains from space exploration: medical research (and health is arguably one of ...
3
VLBI-experiments were constantly carried out between Soviet and American radio telescopes, despite the political winds. One such experiment is related to astronautics. The VEGA balloons for study Venus atmosphere were tracked by two networks of 20 radio telescopes in total back on Earth: the Soviet network, coordinated by the Soviet Academy of Sciences and ...
14
A communications satellite would have to be in a halo orbit around the Earth-Moon L2 point to have continuous, simultaneous visibility of both the Moon's far side and Earth. The first paper describing this kind of orbit was written in 1968, too late to be used for Apollo. It also described the options for communications satellites in medium lunar orbit. P. ...
Top 50 recent answers are included | 2019-08-20 16:33:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.34975942969322205, "perplexity": 1979.3923646933197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315551.61/warc/CC-MAIN-20190820154633-20190820180633-00559.warc.gz"} |
https://forums.finalgear.com/threads/photography-optics-equations-and-formulae.50628/ | # Photography/Optics Equations and Formulae
#### LP
##### Your Brown Banana for Scale
DONOR
I took a class on campus for photography because I was definitely limiting myself and not fully utilizing all the capabilities of the camera. The class took me from shooting in full program to shooting in Manual and Aperture priority.
However one thing it didn't do was explain why certain things were the way they were. For example, when my teacher wrote down all the f-stop numbers, he explained that "these numbers just are the way the are" and never bothered to explicate because I suspect the class was not math-savvy.
I was inclined to solve this myself, and so I went to wikipedia and looked the equations up and derived the reason myself. I thought I would post that here to get some confirmation on my calculations and to allow others to post their own formulas that they feel are relevant.
$N=\frac{f}{D}=\frac{f}{2R}\Rightarrow R=\frac{f}{2N}$
N = f-stop number, f = focal length in mm, D = diameter of the aperture, R = radius of aperture
Area of light entering the lens is then
$A=\pi R^2=\pi\left (\frac{f}{2N} \right )^2 = \frac{\pi f^2}{4N^2}$
So if you want to determine how much more light you're going to get between 2 different f-stop values:
In another way, if you want to know what aperture will drop you by 1 stop (aka half as much light):
$\frac{A_1}{A_2} = 2 \Rightarrow \left ( \frac{N_2}{N_1} \right )^2 = 2 \Rightarrow N_2 = N_1 \sqrt{2}$
If you want to know what aperture will give you an extra stop of exposure (aka twice as much light):
$\frac{N_1}{\sqrt{2}} = N_2$
Note that none of these require the focal length of the lens, meaning that it doesn't matter what lens you have on, these will apply.
To figure out how many stops a particular f-stop is from another:
$\frac{N_1}{N_2} = \sqrt{2}^x = 2^{x/2} \Rightarrow x = 2 \frac{\log(N_1)-\log(N2)}{\log(2)}$
Obviously it'd be way too fucking hard to be sitting out in the field dividing by square roots and doing logarithms, but if you needed to you could always approximate
$\sqrt{2}\sim 1.4 = \frac{7}{5}$
. I feel it's just nice to know where numbers like f/5.6 come from. The series doesn't make sense otherwise.
Last edited:
#### Ramseus
##### Have you been high today?
$N=\frac{f}{D}=\frac{f}{2R}\Rightarrow R=\frac{f}{2N}$
N = f-stop number, f = focal length in mm, D = diameter of the aperture, R = radius of aperture
Theoretically, at least. Magnification and entrance pupil and stuff jumble the math up. How is a constant aperture zoom actually a constant aperture zoom when the aperture remains physically the same size? Magic! When you look at the front of a lens and see the aperture appear to grow/shrink when you zoom in/out it's not a trick to your eye, it's actually relevant to light transmission. I don't get it, optical engineering is weird. | 2019-11-21 15:54:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "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": 0.5002222657203674, "perplexity": 1013.9104398794713}, "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-2019-47/segments/1573496670921.20/warc/CC-MAIN-20191121153204-20191121181204-00499.warc.gz"} |
https://songstudio.info/tech/tech-29/ | Jaewoo Song
### Categories
• Tech
This post is the review of the publication, Holtzman, A., Buys, J., Du, L., Forbes, M., & Choi, Y. (2019). The curious case of neural text degeneration.
This paper compares various decoding algorithms used in NLG(Natural Language Generation) and discusses what should be implemented to make the best outputs with decent quality and diversity.
The authors propose a new suggestion, called Nucleus Sampling.
Currently, it seems to be the best available decoding method from the perspective of both quality and diversity.
What is “degeneration”?
First, let’s specify what the word “degeneration” means in this context.
In NLG, we need to consider two properties of a generated text, “quality” and “diversity”.
Quality means how the output is “well-constructed” considering overall contexts or conditions.
In other words, a good quality text should be relevant and coherent to the context.
On the other hand, diversity means how the generated result has diverse and various expressions or vocabularies.
If the model just imitates learned results from the training set, we cannot say that this generation task is successful especially in open-ended tasks like story generation or text continuation, etc.
So achieving both quality and diversity is the ultimate goal of all generation tasks.
Degeneration is described in this paper as meaning the case in which such quality or diversity is not achieved, which is stated in the paper as “output text that is bland, incoherent, or gets stuck in repetitive loops”.
The failure in quality can be results with incoherent meaning or context and that in diversity can be the case that exact same sentences in the dataset are generated or repetitive phrases come out endlessly at worst.
This is the example of text degeneration situation given by the paper.
As we can see, the left example fell into infinite repetition, which is a drawback of high-probability based decoding and the failure in diversity.
And the right case shows us that too much focus on diversity makes gibberish severely irrelevant to previous contents, which is the failure in quality.
I will discuss the difference among various decoding algorithms in the view of each strength/weakness of quality and diversity later.
Anyway, this paper is proposed to give a new strategy to overcome these degeneration problems.
Decoding algorithms
There are various decoding strategies in NLG, so before going into Nucleus Sampling, let’s see what algorithms can be used in decoding.
There are two representative approaches, maximization-based and sampling-based.
Of course, I cannot say these two categories are completely different since eventually both approaches take partial optimal samples probabilistically, but I will explain them separately for convenience.
• Maximization-based approach
This is the most commonly used decoding method.
The model chooses the next word with the highest probability to make maximum final object probability.
That is, assuming that $y$ is the output sentence and $x$ is the condition, it makes $\prod_{t=1}^{N} P(y_t \mid y_1, …, y_{t-1}, x)$ maximized.
Greedy search and the beam search belong to this category as we saw the last post.
These algorithms might produce a text with good quality but fail to make diverse results.
See the picture below.
Beam search tries to select the next word with the highest probability but actually, humans do not.
The word choices by humans are very varied, diverse and uncertain.
So maximization-based methods have weaknesses in making human-like diverse texts.
On the other hand, maximizing probability has strength in directed generation tasks, which the generated output is a constrained as the transformation of the input such as machine translation, data-to-text generation and summarization.
These tasks make an output with tightly scoped by the input, so repetition and genericness are not that serious problems even if they are vulnerable to the problem of corrupting diversity.
• Sampling-based approach
Sampling-based approach literally samples outputs from the certain partial group.
This can be pure sampling which samples the next word randomly from the whole probability distribution on each time step, or top-$k$ sampling which samples from restricted the top-$k$ most probable words set.
More specifically on top-$k$ sampling, given a distribution $P(x \mid x_{1:i-1})$, top-$k$ vocabulary $V^{(k)} \subset V$ can be defined maximizing $\sum_{x \in V^{(k)}} P(x \mid x_{1:i-1})$.
Random sampling is conducted based on this new distribution.
The problem is how to determine the optimal $k$ value.
Take a look at the below description.
By the above picture, we should notice that having a constant $k$ is not a good idea considering various contexts when conducting top-$k$ decoding.
If the probability distribution is flat, then a small $k$ will cause some problems since it cannot reflect nearly equally distributed word probabilities and fails to improve the diversity of the output.
In contrast, a large $k$ with a peaked distribution is also problematic because it considers extra words with very low probability and this leads to a decrease in quality.
Therefore, the varying $k$ value should be implemented in order to handle these unpredictable distributions.
Additionally, there is a sampling method with temperature that is a hyperparameter for softmax normalization.
Given the logits $u_{1 \colon \lvert V \rvert}$ and temperature $t$, the softmax is re-calculated as $p(x=V_l \mid x_{1:i-1})=\frac{exp(u_l/t)}{\sum_{l^{\prime}}exp(u_{l^{\prime}}/t)}$.
By dividing into $t$, the overall probability distribution is re-normalized to modify relative gaps between words.
Usually $t \in [0,1)$, so overall values become larger and this happens to emphasize the difference between each probability, which makes a little bit more “peaked”.
But controlling the value $t$, we can make this skewing more intense or weaker.
With a small $t$, the emphasis becomes stronger and this makes the model focus on the quality of the output text.
On the other hand, with a large $t$ distribution becomes more uniform and this leads to consideration on the diversity of results.
Nucleus Sampling
The paper proposes a new stochastic method called Nucleus Sampling.
This is also called top-$p$ sampling, which seems similar to top-$k$ sampling but more fluid.
When using Nucleus Sampling, we consider the probability distribution to determine the sample we are willing to get at each time step.
Given a distribution $P(x \mid x_{1:i-1})$, top-$p$ vocabulary $V^{(p)} \subset V$ as the smallest set can be defined as follows.
$\sum_{x \in V^{(p)}} P(x \mid x_{1:i-1}) \geq p$
Now let assume that $p^{\prime} = \sum_{x \in V^{(p)}} P(x \mid x_{1\colon i-1})$.
Then again, we define a new re-scaled probability distribution where the next possible words will be sampled.
$P^{\prime}(x \mid x_{1:i-1}) = \begin{cases} P(x \mid x_{1:i-1})/p^{\prime} & \text{if } x \in V^{(p)} \\ 0 & \text{otherwise.} \end{cases}$
As we can see, this is another similar problem which samples the next probable words with higher probability, but it determines the candidate samples by checking if the cumulative probability mass exceeds the pre-chosen threshold $p$.
So unlike top-$k$ sampling, the size of sampling set changes dynamically.
If $p$ becomes high, then the sample set is a small subset of vocabulary that takes up the vast majority of the entire probability mass.
That’s why this method is called “Nucleus” Sampling.
Experiments & Evaluation metrics
In the paper, the Generatively Pre-trained Transformer(GPT), version 2 was used for experiments.
This model was trained on WebText, a $40$GB collection of text scraped from the web.
The authors conducted several NLG tasks using different decoding algorithms above and evaluated each method with various evaluation metrics.
The metrics used are as follows.
• Perplexity
This indicates the quality of language modeling made by an LM model.
Perplexity is the reciprocal of normalized test data’s probability which is calculated like below.
$PPL(W) = \sqrt[n]{\frac{1}{P(w_1, w_2, ... , w_N)}} = \sqrt[n]{\frac{1}{\prod_{i=1}^{N}P(w_i \mid w_1, w_2, ... ,w_N)}}$
We should notice that the lower the perplexity is, the better the quality of language modeling is.
In other words, the lower perplexity shows us that the LM model was trained to follow the language modeling sequence reflected in the train data.
But that doesn’t mean this model has a good decoding capability.
Lower perplexity also means that the model chooses each next word with relatively small options.
So this does not guarantee the diversity of the decoded outputs.
The paper argues that it is important to produce the perplexity similar to the human-produced sentences and that it is not absolutely good to have low perplexity.
Take a look at the picture below.
We can see that humans do not produce languages with extremely low perplexity as we discussed before.
And algorithms which focus on words with relatively higher probabilities such as beam search and top-$k$ with lower temperature have unnaturally lower perplexities than others.
The rest of the methods like random sampling, top-$k$ with higher temperature and Nucleus sampling reach the perplexity value the human produces.
• Zipf Coefficient
Zipf’s law says that there is an exponential relationship between the rank of a word and its frequency in the text.
Zipf Coeffecient $s$ is used to evaluate this property and $s=1$ means that this is a theoretically perfect exponential curve.
That is, this value indicates how the usage of each word is similar to the human propensity.
The vocabulary distribution and Zipf’s Coefficient of each decoding algorithm is as follows.
The figure shows us that the high-probability based decoding such as beam search is somewhat different from the human tendency to choose words.
And sampling-based methods especially pure sampling and Nucleus sampling are closer to the human distribution as we can see.
But pure sampling overestimates the use of rare words a little bit more, which is quite obvious.
And lower temperature avoids sampling rare words, which is also obvious since it makes the overall probability distribution peakier.
• Self-BLEU
Self-BLEU is another evaluation metric for diversity, which indicates that lower Self-BLEU implies higher diversity.
Self-BLEU is calculated by computing the BLEU score of each generated output using all other generations in the evaluation set as references.
Since the original BLEU score is based on n-gram matching, we can intuitively think that lower matching(score) can have more diversity.
Let’s look at the description below.
Note that higher $k$, $t$ and $p$ makes the score lower, which means an increase of diversity.
And this also means that each algorithm makes output with diversity closer to that of humans.
But this leads to higher perplexity although these high parameters are needed to get close to the reference distribution.
• Repetition
We can also evaluate results by checking repetition to determine the degree of degeneration.
Again, let’s see the figure below.
As we can see, the golden standard obviously has no repetition and sampling methods with higher $p$ or $t$ follow this trend relatively well.
But with lower hyperparameters, these methods are similar to the greedy method and lead to severe repetition.
This indicates that the methods emphasizing the language modeling quality more happen to make repetitive output comparing to those focusing on diversity.
• Human Unified with Statistical Evaluation (HUSE)
The human evaluation can measure the quality of the generated text well but has weakness in detecting the diversity defect.
On the other hand, the statistical evaluations are unable to measure the coherence of generated output properly while they can evaluate the diversity.
HUSE is introduced in Hashimoto, T. B., Zhang, H., & Liang, P. (2019). Unifying human and statistical evaluation for natural language generation. arXiv preprint arXiv:1904.02792, which is a combined evaluation metric of human evaluation and statistical evaluation.
HUSE is computed by training a discriminator to distinguish between text made by human and model distributions, based on two features, the probability assigned by the language model and human judgments of typicality of generations.
To put it simply, the well-produced text which is close to the human distribution should perform well on both evaluations from likelihood and human.
So the overall evaluation results of each algorithm with the above metrics are as follows.
Note that it is ideal that the score is similar to that of humans rather than it is low or high.
And Nucleus sampling made quite good performances although some metrics concluded that other methods are better.
In addition, it is important to know that Nucleus sampling has the highest HUSE score, indicating that it is satisfying to both human evaluation and statistical evaluation.
This is one of the generated examples by experimented decoding algorithms.
Given the initial segment of web text, each method conducted generation from it to complete the text.
You can check other interesting examples in the paper also.
To conclude, this was a very informative and valuable paper.
I think it was of great help to me not only because it presented a new methodology but also because it delivered comprehensive knowledge of various algorithms and evaluation methods for NLG.
I also found that simply extracting high probability words well is not the most important thing in the generation task, but the diversity is also a very important part to consider.
I hope this post is helpful to understand this paper well.
Holtzman, A., Buys, J., Du, L., Forbes, M., & Choi, Y. (2019). The curious case of neural text degeneration. arXiv preprint arXiv:1904.09751. https://arxiv.org/abs/1904.09751. | 2021-10-24 04:21:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "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": 0.7280752062797546, "perplexity": 1279.91930566778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585837.82/warc/CC-MAIN-20211024015104-20211024045104-00716.warc.gz"} |
https://bklein.ece.gatech.edu/laser-photonics/interference/ | # Interference
You’ve probably seen wave interference before in the context of water waves or other waves. In a region with multiple electromagnetic waves, as long as the material is linear (the permittivity and refractive index are independent of the strength of the electromagnetic field), the waves will add according to the principle of linear superposition. If two waves meet that are in phase (in step) with one another, they will interfere constructively, adding to produce a stronger total field. If two waves meet that are $180^o$ out of phase with one another, they will interfere destructively, subtracting to cancel part or all of the field in that region. Of course, we can also have intermediate cases.
Most light sources (the sun, lightbulbs) produce incoherent light, which will be described in the next section. Interference is usually less obvious with incoherent light sources, and more difficult to observe in ordinary life, although not impossible. Laser light, on the other hand, has a high degree of coherence, making interference much easier to observe.
The classic experiment to demonstrate interference is the two-slit experiment. In this experiment, a light wave is passed through two slits, whose dimensions and separation are on the order of the wavelength of the light. In this way, the light wave is split into two light waves, and each of the new waves spreads out due to diffraction. This is diagrammed below.
When the two new waves meet one another, they will interfere constructively in some places, and destructively in others. his can be observed in the following animation.
two_slit_interference_surface
You’ve seen this phenomenon before – for example, when you drop two pebbles into a pond, and observe the water waves spreading out from each pebble and meeting to form a cross-hatch pattern.
Unlike water, we can’t see the light interfering at all the points where the light waves meet – we need something for the light to bounce off of to reach our eyes. So we can put a screen up some distance after the slits, and observe the light bouncing off of the screen to our eyes (effectively ‘sampling’ the intensity in that plane). What we will observe on the screen are bright and dark spots, corresponding to constructive and destructive interference of the two waves in the plane of the screen. This is shown in the next two animations, which show light traveling from the slits to different individual points on the screen. Also shown is the time-average light intensity on the screen.
twoslit_line_constructive
twoslit_line_destructive
As you can see in the animations, the type of interference (constructive or destructive) at a point on the screen is determined by the relative distance the light travels from each slit to that point. If the two waves arrive at a point in phase with each other, we will see constructive interference at that point.
We can treat all of this more mathematically. Below is a diagram of the two slit experiment, defining the coordinate axes and various dimensions.
The waves exiting the slits are not plane waves (although they could be decomposed into a sum of plane waves). A reasonable approximation is to treat them as cylindrical waves. If we take the origin at the location of the bottom slit, we could write the wave leaving that slit as
$\displaystyle E_1 = A \cos{(k \rho - \omega t + \phi_1)}$
where $\rho$ is the cylindrical radius. (Power conservation says that this wave should decrease in amplitude as $1 / \sqrt{\rho}$, but let’s not worry about that for the moment.) If we want to evaluate this wave at location $y = y_s$ on the screen, it will be
$\displaystyle E_1 = A \cos{(k r_1 - \omega t + \phi_1)}$
where $r_1 = \sqrt{s^2 + y_s^2}$ is the distance from the bottom slit to the chosen point on the screen. The wave originating at the top slit evaluated at the same point on the screen is
$\displaystyle E_2 = A \cos{(k r_2 - \omega t + \phi_2)}$
where $r_2 = \sqrt{s^2 + (y_s - d)^2}$ is the distance from the top slit to the chosen point on the screen. We’ve assumed that both waves have the same amplitude $A$. By linear superposition, the total wave is $E_{tot} = E_1 + E_2$. We’re going to use the identity
$\displaystyle \cos{\alpha} + \cos{\beta} = 2 \cos{\frac{\alpha + \beta}{2}} cos{\frac{\alpha - \beta}{2}}$
to obtain
The intensity is proportional to the total field squared. As usual, we are most interested in the time-averaged intensity, which is the intensity integrated over many periods of oscillation and divided by the integration time:
$\displaystyle I_{avg} = \frac{C}{N T} \int_0^{NT} |E_{tot}|^2 dt$
where $C$ is the constant of proportionality, and $N$ is a large integer. The only time dependence in our $E_{tot}$ is the $\omega t$ within one of the cosines. Both cosines will be squared in the intensity equation above. Integrating $\cos^2 (\omega t + blah'')$ over a period $T$ will yield $T/2$ for any time-independent “blah”. Therefore, our time average intensity becomes
$\displaystyle I_{avg} = 2 A^2 C \cos^2 {\left( \frac{k (r_1 - r_2)}{2} + \frac{\phi_1 - \phi_2}{2}\right)}$
$= A^2 C [ \cos{( k (r_1 - r_2) + (\phi_1 - \phi_2))} + 1]$
Let’s assume that the starting phase at each slit is the same, so that $\phi_1 - \phi_2 = 0$. In this case, the intensity will be maximum and we will observe a bright spot (fringe) at any location on the screen where
$k(r_1 - r_2) = 2 \pi m$
where $m$ is any integer. This is equivalent to saying that we obtain a bright spot at locations on the screen where the waves from the two slits are in phase with each other. Since $k = 2*pi / \lambda$, we can also write this as
$r_1 - r_2 = m \lambda$
This says that we obtain a bright spot at locations on the screen where the difference in distance from the two slits equals an integer number of wavelengths. Conversely, the intensity will be zero and we will observe a dark spot at any location on the screen where
$k(r_1 - r_2) = (2 m + 1) \pi$
or
$r_1 - r_2 = (2m + 1) \frac{ \lambda}{2}$
Check out this lovely video of two-slit experiment with incoherent (!) light and water waves: | 2022-09-27 01:47:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 30, "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": 0.7703952789306641, "perplexity": 380.2063588627177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334974.57/warc/CC-MAIN-20220927002241-20220927032241-00480.warc.gz"} |
https://discourse.mc-stan.org/t/is-there-an-easy-way-to-compute-log-p-z-x-using-pystan/14781 | # Is there an easy way to compute log p(z,x) using pystan?
Dear pystan helpers,
We have the model:
model_code =
``````data {
int<lower=0> N;
vector[N] x;
int<lower=0,upper=1> y[N];
}
parameters {
real alpha;
real beta;
}
model {
y ~ bernoulli_logit(alpha + beta * x);
}
``````
We would like to use pystan vb.
sm = StanModel(model_code)
fit_vb = sm.vb(data=data)
Is there an easy way to compute log p(z,y) using pystan?
We can compute this manually, but we would like to know if pystan supports this?
How do we define p(z)? We assume that if we use pystan vb, we know the default is meanfield, so we assume p(z) is a product of Gaussians.
Is there an easy way to compute log p(z,y) using pystan?
Thank you.
I don’t believe the variational distribution is exposed anywhere, if that’s what you’re looking for.
It is possible to compute the log_density and gradients of the likelihood itself (https://pystan.readthedocs.io/en/latest/api.html#pystan.StanFit4model.log_prob). | 2022-05-17 00:51:17 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8875799179077148, "perplexity": 4103.403932623438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662515466.5/warc/CC-MAIN-20220516235937-20220517025937-00035.warc.gz"} |
http://math.stackexchange.com/questions/58025/direct-summand-of-a-free-module | # Direct summand of a free module
Let $M$, $L$, $N$ be $A$-modules and $M=N\oplus L$. If $M$ and $N$ are free, is $L$ necessarily free?
-
Let $M$ be a module over a commutative ring. Recall three properties:
$M$ is projective if it is a direct summand of a free module.
$M$ is weakly stably free if there exists a free module $F$ such that $M \oplus F$ is free.
$M$ is stably free if there exists a finitely generated free module $F$ such that $M \oplus F$ is free.
Then free implies stably free implies weakly stably free implies projective.
But actually projective implies weakly stably free: this is called the Eilenberg swindle. Thus if you don't require $F$ to be finitely generated, then any nonfree projective module gives a negative answer to this question. There are "cheap" examples of this: e.g. let $R_1$ and $R_2$ be any two nonzero rings, let $R = R_1 \times R_2$, and consider $R_1$ as an $R$-module: it is projective but not free.
What about stably free modules which are not free? Here counterexamples lie much deeper. Plop has described what is in fact the most standard counterexample, which reduces the problem to a nontrivial fact in differential topology. A somewhat more comprehensive discussion of this class of examples can be found in $\S 6.4.3$ of my commutative algebra notes. Also, at the end of $\S 3.5$ I give the Eilenberg Swindle as an exercise, with a big hint.
Added: To the best of my recollection, the terminology "weakly stably free" is not standard but something that I made up while writing up my notes. But the point of course is that after this little discussion involving the Eilenberg Swindle, one sees that such terminology is not needed.
-
Is there an easy example of a projective module that is not stably free? – Leon Feb 12 '12 at 17:50
@Leon: a finitely generated projective $R$-module is stably free iff its class in the $K$-group $K_0(R)$ is trivial. So you want examples of rings $R$ with nontrivial $K_0(R)$. Probably the simplest is a Dedekind domain which is not a PID, in which case a nonprincipal ideal gives an example. – Pete L. Clark Feb 12 '12 at 19:28
@Leon: or, if you like topology, by Swan's theorem the Mobius band viewed as a projective module over the ring of continuous functions on the circle is not stably free. – Pete L. Clark Feb 13 '12 at 22:59
No, for example the tangent vector bundle of the sphere $S^2$ is non-trivial, whereas the normal vector bundle is. More concretely, $A=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$, $L=\{(P_1,P_2,P_3)\ |\ xP_1+yP_2+zP_3=0\}$, $N=A \cdot (x,y,z)$, $M=A^3$.
-
May I ask where you learned this material? Could you elaborate on this a bit please. Is $A$, $L$, or $N$ the tangent vector bundle of $S^2$? How did you construct this? Can this be done for any (embedded) smooth manifold? For any variety in $\mathbb{R}^n$? I know there exists a book "Smooth Manifolds and Observables" by Jet Nestruev, which deals with this stuff, but I did not find the things I'm asking you there (I just glanced through). Where should I look? – Leon Feb 11 '12 at 7:31
@Leon: I added some details in the recent duplicate math.stackexchange.com/questions/1000465/… – Hanno Nov 1 '14 at 18:20 | 2015-11-28 11:38:22 | {"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": 0.8338496088981628, "perplexity": 153.67464922324862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398452385.31/warc/CC-MAIN-20151124205412-00123-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/food-for-thought.87877/ | # Food for Thought
1. Sep 7, 2005
### Watts
Assume I have a data set that I am trying to find a distribution that describes how the data is distributed. Assume I have found a function say $f(x) = e^{ - x^2 }$ that describes the distribution of data. Statistics tells me that my first move in doing so is to normalize this function so that $\int\limits_{ - \infty }^\infty {P(x)dx} = 1$. The common approach is to integrate $\int\limits_{ - \infty }^\infty {e^{ - x^2 } dx} = \sqrt \pi$ and multiply the function times the reciprocal of that integral $\frac{1}{{\sqrt \pi }} \cdot \int\limits_{ - \infty }^\infty {e^{ - x^2 } dx} = 1$. But what if I can normalize it a different way say integrate the function $\int\limits_{ - \infty }^\infty {e^{ - x^2 } dx} = \sqrt \pi$ and place the result of that integral in the parentheses beside the variable $P(x) = e^{ - (\sqrt \pi \cdot x)^2 } = e^{ - \pi \cdot x^2 }$ instead of in front of the function. If you now integrate the function $\int\limits_{ - \infty }^\infty {e^{ - \pi \cdot x^2 } dx} = 1$ it still is equal to one. So my question is which PDF do I use? I have normalized the same function two different ways. Any thoughts on this?
2. Sep 7, 2005
### mathman
I think you are missing the point that the normal distribution has 2 parameters which you are not using, the mean and the variance. If you set your problem up with this in mind, you can get a best fit to the data you have, and there won't be any loose ends.
The density is Ke-(x-m)2/2V, where K is defined to make the integral=1.
Last edited: Sep 7, 2005
3. Sep 7, 2005
### Watts
Normal Distribution
The distribution I have chosen is not a normal distribution but simply a gaussian distribution. My point is that statistics has taught every body to normalize a function one single way to make it a true PDF but this doesn’t necessarily have to be the case.
4. Sep 7, 2005
### EnumaElish
Seems to me it's an empirical question. Which of the two CDFs describe your data better? I would use the answer to this question as my selection criterion.
5. Sep 7, 2005
### Watts
Normalization
Not what distribution is appropriate but which PDF has been normalized correctly. That my not even be the case both methods may be correct one or the other or maybe the other. I have successfully done this with other functions several that occur in several aspects of statistics and physics. All of the text shows this being done one way by integrating the function and multiplying it by the integrals reciprocal to normalize. Maybe the conventional method is wrong and you should normalize it the other way I demonstrated or maybe they are both correct. Maybe mathematics has been normalizing distributions incorrectly all along. I don’t know I am just saying maybe there is more than one way.
6. Sep 8, 2005
### EnumaElish
An implicit proportionality assumption could justify the rote method: by dividing the integral by a constant, the proportion of "less than x" to "less than y" is preserved for all functions.
But if there is no such assumption, and the only theoretical criterion is to get the CDF tend to 1, then empirical fit seems to be an obvious tie breaker.
7. Sep 8, 2005
### Watts
Method
Meaning the PDF (1st or 2nd) that contours to the distribution of my data the best.
8. Sep 8, 2005
### mathman
Standard definition has the term "normal distribution" synonomous with "Gaussian distribution".
9. Sep 8, 2005
### Hurkyl
Staff Emeritus
It is usually the case that when one has an unnormalized density function that the true p.d.f. is known to be proportional to it.
For example, suppose I want to pick a random point in the triangle with vertices (0, 0), (0, 1), and (1, 0), and I'm interested in X, the x-coordinate of the point I chose.
I can immediately write down an unnormalized density function for the random variable X: ρ(a) = 1 - a. This is clear because the probability of X = a is obviously proportional to the length of the vertical line segment with x-coordinate a.
The only way to normalize a density function that is known to be correct in this manner is multiplication by a constant. You'll notice that your second method of normalizing a density function does not preserve proprtionality, so it is generally inappropriate to use.
10. Sep 8, 2005
### amcavoy
So this is where the error function came from?
$$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-x^{2}}dx$$
11. Sep 9, 2005
### Watts
Hurkyl
I see your point but either method reduces the unnormalized density function the same proportion of area to equal 1. If this occurs why doesn’t it preserve proportionality? In sense when you multiple by the reciprocal of area the functions height it is reduced by some quantity at the same time if I place the reciprocal area beside the variable the height is preserved but the distribution width is expanded the same proportion of area. Basically instead of squishing it your stretching it the same proportion of area.
12. Sep 9, 2005
### Hurkyl
Staff Emeritus
Let's do an example.
If the random variable X has f(x) = exp(-x²) as an unnormalized density function, then I can state things like the outcome X = 0 happens about 2.7 times more often than the outcome X = 1. We can see this by computing that f(0) / f(1) ~ 2.7.
If we normalize f by multiplying by a constant, then this ratio will remain 2.7.
However, if we use your normalization g(x) = exp(-π x²), then we have g(0) / g(1) ~ 23.1: we see that this normalization greatly distrubs the relative probability!
Now, squishing is a useful concept, but in the following way: you can make the change of variable X = √π Y, and then the random variable Y will have the normalized density function g(x).
Change of variables are used extensively, but the usual goal is to have a random variable whose mean is 0 and whose variance is 1.
Last edited: Sep 9, 2005 | 2018-12-11 14:00:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.8982624411582947, "perplexity": 461.2435097430891}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823621.10/warc/CC-MAIN-20181211125831-20181211151331-00398.warc.gz"} |
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# The probability is 1/2 that a certain coin will turn up head
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26 Dec 2012, 05:53
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The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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26 Dec 2012, 05:59
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The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
P(at last 1 tails) = 1 - P(all heads) = 1 - (1/2)^3 = 7/8.
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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04 Jan 2013, 17:27
8
3
Although Bunuel's approach is, also in my opinion, the best way to go for this sort of question, you could also arrive at the same answer by using the following line of thought:
To get at least 1 tails, you can get one of the 3 configurations (in no particular order):
H H T -> 3 * 1/2*1/2*1/2
H T T -> 3 * 1/2*1/2*1/2
T T T -> 1/2*1/2*1/2
P = 3/8 + 3/8 + 1/8 = 7/8
##### General Discussion
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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16 Sep 2013, 11:09
caioguima wrote:
Although Bunuel's approach is, also in my opinion, the best way to go for this sort of question, you could also arrive at the same answer by using the following line of thought:
To get at least 1 tails, you can get one of the 3 configurations (in no particular order):
H H T -> 3 * 1/2*1/2*1/2
H T T -> 3 * 1/2*1/2*1/2
T T T -> 1/2*1/2*1/2
P = 3/8 + 3/8 + 1/8 = 7/8
Hi,
Kindly explain why it is 3* 1/2 * 1/2 * 1/2.
According to my understanding, the prob of head is 1/2 and tail is 1/2.
So HHT just has to be 1/2 * 1/2 * 1/2.. Isn't it? why multiply by 3?
Kindly clarify.
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Posts: 52431
Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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16 Sep 2013, 11:17
10
abhinaya wrote:
caioguima wrote:
Although Bunuel's approach is, also in my opinion, the best way to go for this sort of question, you could also arrive at the same answer by using the following line of thought:
To get at least 1 tails, you can get one of the 3 configurations (in no particular order):
H H T -> 3 * 1/2*1/2*1/2
H T T -> 3 * 1/2*1/2*1/2
T T T -> 1/2*1/2*1/2
P = 3/8 + 3/8 + 1/8 = 7/8
Hi,
Kindly explain why it is 3* 1/2 * 1/2 * 1/2.
According to my understanding, the prob of head is 1/2 and tail is 1/2.
So HHT just has to be 1/2 * 1/2 * 1/2.. Isn't it? why multiply by 3?
Kindly clarify.
The point is that two heads and a tail can occur in three ways: HHT, HTH, THH. The probability of each case is 1/2*1/2*1/2.
Theory on probability problems: math-probability-87244.html
All DS probability problems to practice: search.php?search_id=tag&tag_id=33
All PS probability problems to practice: search.php?search_id=tag&tag_id=54
Tough probability questions: hardest-area-questions-probability-and-combinations-101361.html
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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20 Nov 2013, 04:41
1
Bunuel wrote:
The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
P(at last 1 tails) = 1 - P(all heads) = 1 - (1/2)^3 = 7/8.
Bunuel, would you mind explaining how you find the power to which you have to raise? What if e.g. there is a bag with three marbles, blue, red and yellow. Now the question is e.g. "What is the probabilty to get a blue marble on at least 1 try if you try 4 times" (putting the marbles back all the time).
Would it be P(at least 1 blue marble) = 1 - P(none blue) = 1 - (2/3)^4 ??
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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20 Nov 2013, 04:51
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2
unceldolan wrote:
Bunuel wrote:
The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
P(at last 1 tails) = 1 - P(all heads) = 1 - (1/2)^3 = 7/8.
Bunuel, would you mind explaining how you find the power to which you have to raise? What if e.g. there is a bag with three marbles, blue, red and yellow. Now the question is e.g. "What is the probabilty to get a blue marble on at least 1 try if you try 4 times" (putting the marbles back all the time).
Would it be P(at least 1 blue marble) = 1 - P(none blue) = 1 - (2/3)^4 ??
Yes, that's correct, the power must be the number of tries.
For the original question: P(at last 1 tails) = 1 - P(all heads) = 1 - (1/2*1/2*1/2)= 1 - (1/2)^3 = 7/8.
For your example: P(at least 1 blue) = 1 - P(no blue) = 1- (2/3*2/3*2/3*2/3) = 1- (2/3)^4.
Hope it's clear.
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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20 Nov 2013, 05:35
ok great thank you!
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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28 Nov 2014, 10:48
1
First thing to do is to come up with the total number of possible outcomes:
The coin is tossed 3 times and there is an equal probability that the coin will turn up heads or tail on each toss (which means that each toss has only two possible outcomes)
_ _ _
2*2*2=2^3=8
Multiply the number of possible outcomes per toss to arrive at the total number of possible outcomes. If the Question would state that the coin is to be tossed four times, the total number of possible outcomes would simply imply another multiplication by 2 or 2^4 which is 16.
Next step is to find the number of scenarios that fulfill the condition the Question stem asks for. (At least one tail)
One can easily recognize that ALL scenarios BUT ONE will include at least one tail. I am talking about the scenario in which all three tosses result in heads.
--> HHH
So 7 out of 8 scenarios will include at least one tail. THH, HTH, HHT etc…
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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07 Jul 2016, 09:16
2
The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
In this problem, there are only two events that could occur for the 3 coin flips. Either the coin will land on tails zero times, or the coin will land on tails at least one time. (Remember that the phrase "at least one time" means "one or more."
Writing this in a probability statement yields:
P(landing on tails at least 1 time) + P(landing on tails zero times) = 1
Thus, we can say:
P(landing on tails at least 1 time) = 1 - P(landing on tails zero times)
Since we are tossing the coin 3 times, the outcome of zero tails in 3 tosses is the same as getting heads on all 3 tosses. We can calculate the probability of zero tails in 3 tosses as the probability of 3 heads in 3 tosses:
½ x ½ x ½ =1/8
Plugging this into our formula we have:
P(landing on tails at least 1 time) = 1 – 1/8
P(landing on tails at least 1 time) = 7/8
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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14 Sep 2016, 00:44
Added info : Formula to solve similar questions
Let 'p' be the probability of getting a head and 'q' be the probability of getting a tail.
If 'n' coins are tossed or one coin is tossed 'n' times
Then $$^nC_r * p ^ r * q^{n-r}$$ gives the probability of having r heads and n-r tails.
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The probability is 1/2 that a certain coin will turn up head [#permalink]
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21 Feb 2018, 11:46
The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
let Tails be T and Heads be H so i need to toss ONE coin three times and the probability is as follows:
first attempt and probability
T
H
H
second attempt and probability
H
T
H
third attempt and probability
H
H
T
so what am i to do next ? after i tossed coin 3 times.... Why do you guys above me all mutiplying ?
we have one coin and it does not effect the next event so we need to add up, no ? hmm like this 1/2+1/2+1/2 =3/2 i thought this was answer...
When do i need to multiply or add in probabality ? Difference?
help niks18
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The probability is 1/2 that a certain coin will turn up head [#permalink]
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21 Feb 2018, 11:57
1
Hey dave13
RULE: The probability of an event occurring can never be greater than 1.
So, whenever during the course of a problem you get a number
greater than 1 for a probability, be assured that you have made a mistake!
You need to multiply the probability when the events are to occur simultaneously, or consecutively.
For example, when we get three heads,
the probability will be $$\frac{1}{2}*\frac{1}{2}*\frac{1}{2} = \frac{1}{8}$$.
We need to add probabilities when the events are alternatives.
For example, when there are three cases that can happen.
You can get HHT, HTT, TTT(for the three coins problem) as possible options for at least one tail.
In order to get the final answer, you need to add the individual probabilities for all the three cases.
Coming back to the problem
An easy way to solve this question is to find the probability that no tail
turns up and reduce that from 1(the maximum probability that an event can occur)
The probability that a tail does not turn up when three coins are tossed is when we get a head
on each of these coin tosses, which is $$\frac{1}{2}*\frac{1}{2}*\frac{1}{2} = \frac{1}{8}$$
Probability (atleast one tail turns up) = 1-P(All heads) = $$1 - \frac{1}{8} = \frac{7}{8}$$(Option D)
Hope this helps you!
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The probability is 1/2 that a certain coin will turn up head [#permalink]
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22 Feb 2018, 08:55
Bunuel wrote:
abhinaya wrote:
caioguima wrote:
Although Bunuel's approach is, also in my opinion, the best way to go for this sort of question, you could also arrive at the same answer by using the following line of thought:
To get at least 1 tails, you can get one of the 3 configurations (in no particular order):
H H T -> 3 * 1/2*1/2*1/2
H T T -> 3 * 1/2*1/2*1/2
T T T -> 1/2*1/2*1/2
P = 3/8 + 3/8 + 1/8 = 7/8
Hi,
Kindly explain why it is 3* 1/2 * 1/2 * 1/2.
According to my understanding, the prob of head is 1/2 and tail is 1/2.
So HHT just has to be 1/2 * 1/2 * 1/2.. Isn't it? why multiply by 3?
Kindly clarify.
The point is that two heads and a tail can occur in three ways: HHT, HTH, THH. The probability of each case is 1/2*1/2*1/2.
Hello Bunuel
if two heads and a tail can occur in three ways: HHT, HTH, THH. The probability of each case is 3* 1/2*1/2*1/2.
then two tails and one heads can occure 3 times as well TTH, THT, HTT right ? 3 * 1/2*1/2*1/2.
and probabilty that all tails and zero heads TTT 1/2*1/2*1/2 (so here we dont need to multiply by 3 because there is only one combination)
am i thinking correctly ? thanks
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Re: The probability is 1/2 that a certain coin will turn up head [#permalink]
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23 Apr 2018, 12:29
Top Contributor
2
The probability is 1/2 that a certain coin will turn up heads on any given toss. If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?
(A) 1/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16
When it comes to probability questions involving "at least," it's best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)
So, here we get: P(getting at least 1 tails) = 1 - P(not getting at least 1 tails)
What does it mean to not get at least 1 tails? It means getting zero tails.
So, we can write: P(getting at least 1 tails) = 1 - P(getting zero tails)
Now let's calculate P(getting zero tails)
What needs to happen in order to get zero tails?
Well, we need heads on the first toss and heads on the second toss and heads on the third toss.
We can write P(getting zero tails) = P(heads on 1st AND heads on 2nd AND heads on 3rd)
This means that P(getting zero tails) = P(heads on 1st) x P(heads on 2nd) x P(heads on 3rd)
Which means P(getting zero tails) = (1/2)x(1/2)x(1/2)= 1/8
P(getting at least 1 tails) = 1 - P(not getting at least 1 tails)
= 1 - 1/8
= 7/8
Cheers,
Brent
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Re: The probability is 1/2 that a certain coin will turn up head &nbs [#permalink] 23 Apr 2018, 12:29
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# The probability is 1/2 that a certain coin will turn up head
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2019-01-24 02:18:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8712338209152222, "perplexity": 1226.825077397321}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584445118.99/warc/CC-MAIN-20190124014810-20190124040810-00044.warc.gz"} |
http://www.verycomputer.com/18_ef623e550819e694_1.htm | ## possible Mathematica/Latex FAQ
### possible Mathematica/Latex FAQ
I would like to put a Mathematica-generated*file in an appendix.
Sure, I can export it with the tex-save option, and actually generate the
dvi with it. However, if I try to use the mathematica latexed file as a
subfile of a master, I have to load the packages for the mathematica file
in the preamble of the master. This tends to mess up the whole rest of
the document. Is there any work-around for this? Anyone have experience
with this? I would like to use txfonts for the rest of the document.
Jeff
--
### possible Mathematica/Latex FAQ
Hi,
I have a small program that acts like a FontEnd. It reads a*
document,
scan for the (new) mathinput environments, send the contents to the
Mathematica
kernel and insert the Mathematica-output into the final TeX document.
You can use any other style file but you must supply environments to
typeset
the Mathematica Input and output. Probably you have to use my virtual
fonts.
You can take a look to a sample document:
http://www.veryComputer.com/~kuska/qkratzer3d1.pdf
Contact me direct if you what the nice commandline TeX FrontEnd.
Regards
Jens
> I would like to put a Mathematica-generated*file in an appendix.
> Sure, I can export it with the tex-save option, and actually generate the
> dvi with it. However, if I try to use the mathematica latexed file as a
> subfile of a master, I have to load the packages for the mathematica file
> in the preamble of the master. This tends to mess up the whole rest of
> the document. Is there any work-around for this? Anyone have experience
> with this? I would like to use txfonts for the rest of the document.
> Jeff
> --
Hello cognoscenti
I am currently learning LaTeX, using \usepackage{amsthm} and would
like to produce a theorem style/type \newtheorem{namedtheorem}
so that
\begin{namedtheorem}{Fermat's Last theorem}[Wiles, late 1900's]
\label{wow}
There are no integers ...
\end{namedtheorem}
%
A version of (\ref{wow}) was first shown...
produces
15 Fermat's Last Theorem (Wiles, late 1900's)
There are no integers ...
A version of (15) was first shown...
================
I'm not particular about whether the "Fermat's Last theorem" is called an
optional or required argument, but I am thinking of it as being
required. The
[Wiles, late 1900's]
though, is definitely to be omittable. Ta, -Jonathan
PS: I have a vague memory of some poster implying that
\usepackage{amsthm}
had been supplanted by another package -if so, where can I get it?
--
Prof. Jonathan King, Mathematics dept, Univ. of Florida | 2020-05-30 00:13:45 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8203109502792358, "perplexity": 5203.815163345835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347406785.66/warc/CC-MAIN-20200529214634-20200530004634-00501.warc.gz"} |
http://crypto.stackexchange.com/questions/19645/encrypted-read-only-message-queue/19646 | I'm creating a prototype for a messaging queue with some custom logic (can't re-use any existing solution).
One requirement is to have all messages encrypted. There are three classes of nodes subscribing to the queue with the following permissions:
The first and obvious thought was to use an RSA private key for group 1. This would allow to encrypt with the private key and decrypt with the public key.
Group 2 would receive the corresponding public key and would be able to only decrypt the content.
Group 3 would have no key and just handles the messages without access to the content.
After doing some further reading, there seem to be a number of problems with this approach (apart from the simple fact that the keys are used in "reverse" in relation to the intended mechanism).
Now the question: Is there some common approach to get this pattern to work? I have searched, but so far have not found anything.
I will have access to someone with more experience in this matter at a later point, but for purposes of the prototype, I just want to get something done that won't make me look like an idiot later. ;-)
-
You should not be encrypting "with the private key". $\:$ See this question and this answer. $\;\;\;\;$ – Ricky Demer Oct 15 '14 at 9:44
Thanks, that did look like the obvious choice, but a little research showed that it's not the way to go. – DeathByPiano Oct 15 '14 at 23:43
Yes. The key for encrypting and decrypting the messages can (and should) be that of a symmetric algorithm, e.g. AES-CTR with suitably random or sequential IV, and known to groups 1 and 2. $\;$ Signing should be on the enciphered data, using an asymmetric signature algorithm (e.g. RSA with PKCS#1v2 signature padding), with the private key known to members of group 1, and the public key known to all (as implied by its name). $\;$ Everyone (except possibly from group 3) should verify signature of enciphered data manipulated. – fgrieu Oct 15 '14 at 9:56
Also: because RSA allows fast signature verification, it is a good candidate for the signature from an efficiency standpoint if reading and forwarding with integrity verification largely predominates writing/signing. If signature size overhead is an issue, RSA signature with message recovery can help reduce the signature overhead (e.g. to 34 bytes with ISO/IEC 9796-2 scheme 3 using SHA-256, with encrypted message of at least 222 bytes and 2048-bit RSA). $\;$ AES-CTR encryption/decryption is fast, and its size overhead is limited to the IV (e.g. 10 bytes for a counter to $2^{80}$). – fgrieu Oct 15 '14 at 10:10 | 2015-04-01 00:12:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3701244592666626, "perplexity": 1122.5387597474596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131302318.88/warc/CC-MAIN-20150323172142-00278-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://mcla.ug/blog/making-an-automatic-chord-recogniser.html | # Making an automatic chord recogniser
I've started working on an automatic chord recogniser for audio. It's something I've wanted to try out for a while but hadn't found the time until recently! It seems like a neat project :) I'm still in the beginnings, but I'm going to talk about what I have so far.
## What is a chromagram?
The feature extraction technique I'm starting with comes from a 1999 paper called "Realtime Chord Recognition of Musical Sound". Nineteen-ninety-nine was a long time ago, but the features are really intuitive and involve some fun simple musicology and DSP. I think it's a really nice place to start – I might try some more exciting (read: time-consuming and complicated) techniques later on, once I've got my teeth into things more.
In Western tonal music, we divide frequencies into semitones which follow a logarithmic scale. Every 12 semitones, we double frequency. A group of such 12 tones, or that distance, is called an octave.
The chromagram feature is a 12 dimensional vector, with each entry giving the relative intensity of a chroma, e.g. A or C#. Chroma simply refers to a note regardless of octave. It's a nice name! Chromagrams are also known as Pitch Class Profiles, but I thought the former was much cooler :D
## How do we calculate chromagrams?
Time for some fun musicology and DSP. To calculate a chromagram, we want to know which chroma are present in a time-sample, and their intensity. This is a perfect job for a Fourier Transform. We take the DFT, then, in short, put the DFT bins into chroma bins. We effectively sum the DFT components that correspond to chroma frequencies (or come close). Here's the maths:
$$M_k = \left[12 \log_2\left(\frac{f_s k}{Nf_{\textrm{ref}}}\right) \mod 12 \right]$$ $$C_c = \sum_{M_k = c} \left|{X_k}\right|^2$$
$M_k$ tells us the closest chroma to the DFT frequency bin:
• $f_{\textrm{ref}}$ is the reference frequency of the 0th chroma (e.g. 27.5 Hz for the lowest A)
• The frequency represented by the bin is easily found by the product of the sampling frequency and the bin index, divided by the DFT length: $f_{\textrm{bin}} = \frac{f_s k}{Nf_{\textrm{ref}}}$
• $\log_2(\frac{f_{\textrm{bin}}}{f_{\textrm{ref}}})$ gives how many octaves the bin frequency is above the chroma reference frequency
• $12 \log_2(\frac{f_{\textrm{bin}}}{f_{\textrm{ref}}})$ gives how many semitones the bin frequency is above the chroma reference frequency
• Rounding this and taking the mod 12 gives the nearest chroma index
$X_k$ is the DFT, and $C_c$ tells us the intensity of the chroma $c$ by summing DFT components where the spectral bins land in the chroma bins.
## My project
I currently have a functioning chromagrammer in Python 3, along with some generic audio processing, and a script for generating tones and chords for testing. There are some issues, and I've had some silly bugs along the way, but I'm enjoying it. Here is one of my favourite pieces of code. All it does is produce overlapping frames of a signal, but it's very neat :D
1 def overlapping_frames(self):
2 """
3 Generates overlapping frames
4
5 Generator that yields a deque containing the current frame of audio
6 data. The deque contents is shifted by the frame size minus the
7 overlap on each iteration, to minimise computation.
8 """
9 frame = collections.deque(maxlen=self.frame_size)
10 frame.extend(self.data[:self.frame_size])
11 yield frame
12
13 for i in range(1, self.num_frames):
14 frame.extend(self.data[i * self.frame_size - self.overlap:
15 (i + 1) * self.frame_size - self.overlap])
16 yield frame
I'm using hypothesis for property based testing, which I'm really getting into. Here's an example test, which actively finds example inputs that will break the test. My tests are so much more useful with this!
1 @hypothesis.given(arrays(float, 100))
2 def test_overlapping_frames_yields_correct_initial_frame(self, data):
3 self.ap.data = np.nan_to_num(data)
4 self.ap.process_data()
5
6 frames = self.ap.overlapping_frames()
7 assert (next(frames) == self.ap.data[:self.ap.frame_size]).all()
It's not a very exciting test, but I can be confident my code works for a variety of values, and it saves me having to make up dummy data.
## Seaborn for pretty graphs!
I've started using seaborn for plotting my graphs, rather than plain old matplotlib. It's amazing! Everything looks so pretty. Here's part of a Mountain Goats song's chromagram plotted for a bunch of time samples. Nice!
I want to try and update my blog with how my project is going. I'm really enjoying it and I'm excited about the different ways it can go. There's a lot of scope for some cool Machine Learning and DSP! I really like the neat intuitive features I'm using at the moment, but I'm excited about what else I could use :D You can check out the code here at the github repo. ::: ::: | 2022-12-07 23:03:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5564352869987488, "perplexity": 2128.783285204795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711221.94/warc/CC-MAIN-20221207221727-20221208011727-00328.warc.gz"} |
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# Clarification about GL_LINE_STRIP and Vertex Arrays
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
10 replies to this topic
### #1Alessandro Members
Posted 09 February 2012 - 12:46 PM
I need to render a bunch of GL_LINE_STRIP primitives and the render function looks like this:
glPushMatrix();
glEnableClientState( GL_VERTEX_ARRAY );
glVertexPointer(3, GL_FLOAT, 0, verts);
glDrawElements( GL_LINE_STRIP, numIndices, GL_UNSIGNED_INT, indices );
glDisableClientState( GL_VERTEX_ARRAY );
glPopMatrix();
I already packed the verts array, and I'm a bit confused about the indices assignment.
The number of indices is actually equal to (number of line strips), or to the (number of line strips * number of line segments) ?
### #2Brother Bob Moderators
Posted 09 February 2012 - 01:05 PM
The number of indices is just what it says; the number of indices to draw. If you have an index array that contains four indices that makes up your line strip, then the parameter is four. The parameter has exactly the same meaning for any type of primitive you can draw.
### #3V-man Members
Posted 09 February 2012 - 02:33 PM
The number of indices is number of strips + 1
Sig: http://glhlib.sourceforge.net
an open source GLU replacement library. Much more modern than GLU.
float matrix[16], inverse_matrix[16];
glhTranslatef2(matrix, 0.0, 0.0, 5.0);
glhScalef2(matrix, 1.0, 1.0, -1.0);
glhQuickInvertMatrixf2(matrix, inverse_matrix);
glUniformMatrix4fv(uniformLocation1, 1, FALSE, matrix);
glUniformMatrix4fv(uniformLocation2, 1, FALSE, inverse_matrix);
### #4Alessandro Members
Posted 09 February 2012 - 03:06 PM
Say you have N line strips, each with V number of vertices.
The number of indices should be N*V (since a line strip draws a connected group of line segments from the first vertex to the last), correct?
But how glDrawElements understand where a line strip ends and a new one begins?
I think I found out that I can't call a unique glDrawElements to draw all the line strips at once. I will have to use glMultiDrawElements.
Here is an example I found:
static GLubyte oneIndices[] = {0, 1, 2, 3, 4, 5, 6};
static GLubyte twoIndices[] = {7, 1, 8, 9, 10, 11};
static GLsizei count[] = {7, 6};
static GLvoid * indices[2] = {oneIndices, twoIndices};
glMultiDrawElements(GL_LINE_STRIP, count, GL_UNSIGNED_BYTE,indices, 2);
I wonder how I'd define the indices[] array in case I have one global array of indices, since the element count it's always the same (in the example they use two separate indices and they have a different number of elements).
### #5Brother Bob Moderators
Posted 09 February 2012 - 04:39 PM
You just put pointers to the parts of the global index array you want to draw from in the indices array instead of pointers to two different arrays.
static GLvoid *indices[] = {globalindices, globalindices+10, globalindices+42};
### #6Alessandro Members
Posted 09 February 2012 - 04:46 PM
Thanks Bob, will try that. I was just playing (that's the correct term) with glDrawRangeElements: obviously this stuff is rocket science for me.
The following call correctly draw 6 points:
glDrawRangeElements( GL_POINTS, 0, 5, 6, GL_UNSIGNED_INT, indices );
Now, (newbie thought) that is supposed to be equivalent:
glDrawRangeElements( GL_POINTS, 0, 2, 3, GL_UNSIGNED_INT, indices );
glDrawRangeElements( GL_POINTS, 3, 5, 3, GL_UNSIGNED_INT, indices );
I mean, I just make two "partial" calls, correctly passing the proper indicies. Instead, to my surprise, glDrawRangeElements( GL_POINTS, 3, 5, 3, GL_UNSIGNED_INT, indices ); it just matches with the glDrawRangeElements( GL_POINTS, 0, 2, 3, GL_UNSIGNED_INT, indices ); Only the first 3 points are drawn...
How is that possible, since I specified different indicies?
### #7Brother Bob Moderators
Posted 09 February 2012 - 05:05 PM
If you want the third line to draw the second half, then you have to point the index array to the second half as well.
glDrawRangeElements( GL_POINTS, 3, 5, 3, GL_UNSIGNED_INT, indices+3 )
Otherwise you're drawing the same thee vertices twice.
The pointer and the number of indices to draw exclusively determine what values to read from the index array. The range-parameters are not what sub-part of the index array to draw, but what values OpenGL can expect when reading the index array. In the third call, 3 and 5 does not mean "draw between indices 3 and 5", it means "when reading index values pointed to by the pointer, the index values are expected to be between 3 and 5".
### #8Alessandro Members
Posted 09 February 2012 - 06:19 PM
Bob thank you for the explanation, I now understood what I was doing wrong. glDrawRangeElements now works fine in my application, and they are much faster than glDrawArrays and also glDrawElements. Tomorrow I'll try to use glDrawRangeElements with VBO's: shouldn't be too difficult now that arrays are setup properly.
static GLvoid *indices[] = {globalindices, globalindices+10, globalindices+42};
one could do something like this:
static GLvoid **indices;
indices[0]=globalIndices;
indices[1]=globalIndices+10;
indices[2]=globalIndices+20;
Thanks again for the precious help and patience. My mind is more oriented to arts and painting, but mathematics and 3D graphics, even if I'm such a rookie at those, they do fascinate me (and scare sometimes ) ...
### #9Brother Bob Moderators
Posted 10 February 2012 - 03:34 AM
Use a dynamic container.
std::vector<void *> indices;
indices.push_back(globalindices);
...
Then pass a pointer to the first element to OpenGL:
glMultiDrawElements(... , &indices[0], indices.size());
### #10Alessandro Members
Posted 10 February 2012 - 05:53 AM
Ok, here is the working code I came up thanks to Bob hints, which renders a bunch of GL_LINE_STRIP primitives using glMultiDrawElements. Maybe someone can find it useful...
// declare stuff
unsigned int *indices; // this array holds all the line strips indices
vector3_t *verts; // this array holds all the line strips vertices
int *globalIndexCount;
std::vector<void *> globalIndices; // this is a pointer to the *indices above
void setup()
{
/* ... fill verts and indices with data ... */
globalIndexCount=new int[NUM_OF_STRIPS];
for (int i=0; i<NUM_OF_STRIPS; i++)
{
globalIndexCount[i]=POINTS_PER_STRIP; // store number of points for a line strip
globalIndices.push_back(indices+i*POINTS_PER_STRIP); // store the reference to a indices data set
}
}
void render()
{
glEnableClientState( GL_VERTEX_ARRAY );
glVertexPointer(3, GL_FLOAT, 0, verts);
glPushMatrix();
glMultiDrawElements(GL_LINE_STRIP, globalIndexCount, GL_UNSIGNED_INT, (const void **) &globalIndices[0], globalIndices.size() );
glPopMatrix();
glDisableClientState( GL_VERTEX_ARRAY );
}
### #11Alessandro Members
Posted 10 February 2012 - 06:00 AM
Hey, not finished yet: how about glMultiDrawElements + VBO's?
I think I've setup correctly the buffers, but I can't figure out how the glMultiDrawElements render call should look like:
void initGL()
{
...
glGenBuffersARB(1, &vboId);
glBindBufferARB(GL_ARRAY_BUFFER_ARB, vboId);
glBufferDataARB(GL_ARRAY_BUFFER_ARB, fullHairCounter*HAIRPOINTS*sizeof(vector3_t), verts, GL_DYNAMIC_DRAW_ARB);
glGenBuffersARB(1, &iboId);
glBindBufferARB(GL_ELEMENT_ARRAY_BUFFER_ARB, iboId);
glBufferDataARB(GL_ELEMENT_ARRAY_BUFFER_ARB, numIndices*sizeof(unsigned int), indices, GL_DYNAMIC_DRAW_ARB);
...
}
void setupVBO()
{
...
glBindBufferARB(GL_ARRAY_BUFFER_ARB, vboId);
glBufferDataARB(GL_ARRAY_BUFFER_ARB, fullHairCounter*HAIRPOINTS*sizeof(vector3_t), verts, GL_DYNAMIC_DRAW_ARB);
glBindBufferARB(GL_ELEMENT_ARRAY_BUFFER_ARB, iboId);
glBufferDataARB(GL_ELEMENT_ARRAY_BUFFER_ARB, globalIndices.size() *sizeof(unsigned int), (const void **) &globalIndices[0], GL_DYNAMIC_DRAW_ARB);
...
}
void renderVBO()
{
glPushMatrix();
glBindBufferARB(GL_ARRAY_BUFFER_ARB, vboId);
glBindBufferARB(GL_ELEMENT_ARRAY_BUFFER_ARB, iboId);
glEnableClientState( GL_VERTEX_ARRAY );
glVertexPointer(3, GL_FLOAT, 0, 0);
glMultiDrawElements(GL_LINE_STRIP, globalIndexCount, GL_UNSIGNED_INT, 0, globalIndices.size() ); // this is clearly wrong: I don't understand how to reference data to be passed to the glMultiDrawElements...
glDisableClientState( GL_VERTEX_ARRAY );
glBindBufferARB(GL_ARRAY_BUFFER_ARB, 0);
glBindBufferARB(GL_ELEMENT_ARRAY_BUFFER_ARB, 0);
glPopMatrix();
}
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic. | 2017-04-24 19:25:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18037207424640656, "perplexity": 4890.395281952632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119782.43/warc/CC-MAIN-20170423031159-00625-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://xcsp.org/specifications/constraints/connection/channel/ | # Constraint channel
The constraint channel ensures that if the ith variable is assigned the value j, then the jth variable must be assigned the value i. The optional attribute startIndex of list gives the number used for indexing the first variable in this list (0, by default).
For the semantics, indexing assumed to start at 1 for simplicity.
channel($X$) with $X=\langle x_1,x_2,\ldots \rangle$, iff $\forall i : 1 \leq i \leq |X|, x_i =j \Rightarrow x_j=i$
#### Syntax
<channel>
<list [ startIndex="integer" ]> (intVar wspace)2+ </list>
</channel>
Note that the opening and closing tags of list are optional when, of course, the attribute startIndex is not necessary, which gives:
#### Syntax
<channel> (intVar wspace)2+ </channel> <!-- Simplified Form -->
Another classical form of channel, sometimes called inverse or assignment in the literature, is defined from two separate lists of variables (that must be of same size). It ensures that the value assigned to the ith variable of the first element list gives the position of the variable of the second element list that is assigned to i, and vice versa. For each list, the optional attribute startIndex gives the number used for indexing the first variable in this list (0, by default).
channel($X$, $Y$) with $X=\langle x_1,x_2,\ldots \rangle$ and $Y=\langle y_1,y_2,\ldots \rangle$, iff $\forall i : 1 \leq i \leq |X|, x_i = j \Leftrightarrow y_j = i$
#### Syntax
<channel>
<list [ startIndex="integer" ]> (intVar wspace)2+ </list>
<list [ startIndex="integer" ]> (intVar wspace)2+ </list>
</channel>
#### Example
<channel>
<list> x1 x2 x3 x4 </list>
<list> y1 y2 y3 y4 </list>
</channel>
Another form is obtained by considering a list of 0/1 variables to be channeled with an integer variable. This third form of constraint channel ensures that the only variable of list that is assigned to 1 is at an index (position) that corresponds to the value assigned to the variable in value.
For the semantics, indexing assumed to start at 1 for simplicity.
channel($X$, $v$) with $X=\langle x_1,x_2,\ldots \rangle$</Latex> iff
• $\forall i : 1 \leq i \leq |X|, x_i = 1 \Leftrightarrow v = i$
• $\exists i : 1 \leq i \leq |X| \land x_i = 1$
#### Syntax
<channel>
<list [ startIndex="integer" ]> (01Var wspace)2+ </list>
<value> intVar </value>
</channel>
#### Example
<channel>
<list> z1 z2 z3 z4 z5 </list>
<value> v </value>
</channel> | 2022-05-27 00:16:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4766947031021118, "perplexity": 3487.6476875369017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662627464.60/warc/CC-MAIN-20220526224902-20220527014902-00421.warc.gz"} |
https://brilliant.org/problems/electrostatics-equilibrium-problem/ | # Electrostatics Equilibrium Problem.
It is required to hold four equal point charges $$+q$$ each in equilibrium at the corners of a square. Find the charge in Coulombs, $$Q$$, that will do this if kept at the centre of the square.
Details and Assumptions
• $$q=1\text{ C}$$
× | 2017-01-19 14:58:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.640867292881012, "perplexity": 757.9058363318684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280718.7/warc/CC-MAIN-20170116095120-00458-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://msp.org/agt/2013/13-4/b20.xhtml | Volume 13, issue 4 (2013)
1 N Bergeron, A Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013) 391 MR3028790 2 D Bertrand, Duality on tori and multiplicative dependence relations, J. Austral. Math. Soc. Ser. A 62 (1997) 198 MR1433209 3 B Clair, K Whyte, Growth of Betti numbers, Topology 42 (2003) 1125 MR1978050 4 G Elek, Abelian coverings, J. Funct. Anal. 141 (1996) 365 MR1418511 5 S Friedl, N Jackson, Approximations to the volume of hyperbolic knots arXiv:1102.3742 6 F González-Acuña, H Short, Cyclic branched coverings of knots and homology spheres, Rev. Mat. Univ. Complut. Madrid 4 (1991) 97 MR1142552 7 J A Hillman, M Sakuma, On the homology of finite abelian coverings of links, Canad. Math. Bull. 40 (1997) 309 MR1464839 8 M Laurent, Équations diophantiennes exponentielles, Invent. Math. 78 (1984) 299 MR767195 9 T T Q Lê, Homology torsion growth and Mahler measure, to appear in Comment. Math. Helv. arXiv:1010.4199 10 T T Q Lê, Hyperbolic volume, Mahler measure, and homology growth, talk at Columbia University (2009) 11 W Lück, Approximating $L^2$-invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994) 455 MR1280122 12 W Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergeb. Math. Grenzgeb. 44, Springer (2002) MR1926649 13 J P Mayberry, K Murasugi, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982) 143 MR648083 14 J W Milnor, Infinite cyclic coverings, from: "Conference on the Topology of Manifolds" (editor J G Hocking), Prindle, Weber & Schmidt (1968) 115 MR0242163 15 J Porti, Mayberry–Murasugi's formula for links in homology 3-spheres, Proc. Amer. Math. Soc. 132 (2004) 3423 MR2073320 16 J Raimbault, Exponential growth of torsion in abelian coverings, Algebr. Geom. Topol. 12 (2012) 1331 MR2966689 17 R Riley, Growth of order of homology of cyclic branched covers of knots, Bull. London Math. Soc. 22 (1990) 287 MR1041145 18 W Schmidt, Heights of points on subvarieties of $\mathbf{G}^n_m$, from: "Number theory" (editors V K Murty, M Waldschmidt), Contemp. Math. 210, Amer. Math. Soc. (1998) 97 MR1478487 19 J P Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer (1977) MR0450380 20 D S Silver, S G Williams, Mahler measure, links and homology growth, Topology 41 (2002) 979 MR1923995 21 V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser (2001) MR1809561 | 2022-05-22 00:34:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.746487021446228, "perplexity": 3174.5155016740514}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662543264.49/warc/CC-MAIN-20220522001016-20220522031016-00094.warc.gz"} |
https://fermatslibrary.com/s/q-2018-01-12-45 | Separability of diagonal symmetric states: a
Jordi Tura
1 2
, Albert Aloy
1
3
, Maciej Lewenstein
1 4
, and Anna Sanpera
3 4
1
ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science an d Technology, 08860 Castelldefels (Barcelona),
Spain
2
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
3
Departament de Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
4
ICREA, Pg. Lluís Companys 23, E-08010 Barcelona, Spain
Decemb er 29, 2017
We study the separability problem in mixtures of Dicke states i.e., the separability
of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the
case of DS in
d
d
(symmetric qudits) can be reformulated as a quadratic conic op-
timization problem. This connection allows us to exchange concepts and ideas between
quantum information and this field of mathematics. For instance, copositive matrices can
be understood as indecomposable entanglement witnesses for DS states. As a consequence,
we show that positivity of the partial transposition (PPT) is sufficient and necessary for
separability of DS states for d 4. Furthermore, for d 5, we provide analytic examples
of PPT-entangled states. Second, we develop new sufficient separability conditions beyond
the PPT criterion for bipartite DS states. Finally, we focus on N-partite DS qubits, where
PPT is known to be necessary and sufficient for separability. In this case, we present a
family of almost DS states that are PPT with respect to each partition but nevertheless
entangled.
1 Introduction
Entanglement [1] is one of the most striking features of quantum physics, departing entirely from any
classical analogy. Furthermore, entanglement is a key resource for quantum information processing
tasks, such as quantum cryptography [2] or metrology [3]. Importantly, entanglement is a necessary
resource to enable the existence of Bell correlations [4, 5], which are the resource device-independent
quantum information processing is built upon [6]. Despite its both fundamental and applied interest,
the so-called separability problem (i.e., deciding whether a quantum state is entangled or not, given
its description) remains open except for very specific cases. Although this problem has been shown
to be, in the general case, NP-hard [7], it remains unclear whether this is also the case for physical
systems of interest, where symmetries appear in a natural way.
To tackle the separability problem, simple tests have been put forward, which give a partial char-
acterization of entanglement. The most celebrated entanglement detection criterion is the so-called
positivity under partial transposition (PPT) criterion [8]. It states that every state that is not entan-
gled must satisfy the PPT criterion. Therefore, states that break the PPT criterion are entangled.
Unfortunately, the converse is true only in very low-dimensional systems [9], such as two qubit [10] or
qubit-qutrit systems [11]. Examples of entangled states satisfying the PPT criterion have been found
for strictly larger-dimensional systems [12].
Jordi Tura: jordi.tura@mpq.mpg.de
Accepted in Quantum 2017-12-28, click title to verify 1
arXiv:1706.09423v2 [quant-ph] 28 Dec 2017
Symmetries are ubiquitous in Nature and they play a fundamental role in finding an efficient de-
scription of physical systems. The so-called symmetric states constitute an important class of quantum
systems to describe systems of indistinguishable particles [13]. Symmetric states can be mapped to
spin systems that are invariant under the exchange of particles and, moreover, they are spanned solely
by the largest-spin subspace in the Schur-Weyl duality representation [14]. The Dicke states [15] pro-
vide a convenient basis to represent symmetric states. Moreover, Dicke states are also experimentally
available [1618] and they also appear naturally as ground states of physically relevant Hamiltonians,
such as the isotropic Lipkin-Meshkov-Glick model [19]. Much theoretical study has been devoted to
the characterization of entanglement in qubit symmetric states: 3-qubit symmetric states are separable
if, and only if, they satisfy the PPT criterion [13], but this is no longer the case already for N 4
[20, 21]. Despite diverse separability criteria exist for symmetric states (see e.g. [22]), the separability
problem remains still open.
Mixtures of Dicke states are symmetric states that are diagonal in the Dicke basis. These constitute
an important class of quantum states which naturally arise e.g. in dissipative systems such as photonic
or plasmonic one-dimensional waveguides [23]. Mixtures of Dicke states form a small subclass of
the symmetric states. They are the so-called Diagonal Symmetric (DS) states. In this context, the
separability problem has also gained interest. For instance, the best separable approximation (BSA,
[24]) has been found analytically for DS states for N -qubits [25]. In [26], it was conjectured that
N-qubit DS states are separable if, and only if, they satisfy the PPT criterion with respect to every
bipartition. The conjecture was proven by N. Yu in [27] where, moreover, he observed that PPT is a
sufficient and necessary condition for bipartite DS states of qudits with dimension 3 and 4, but becomes
NP-hard for larger dimensions. Within the N -qubit DS set it has been shown in [28] that there is a
family of states that violate the weak Peres conjecture [29]: those states are PPT-bound entangled
with respect to one partition, but they violate a family of permutationally invariant two-body Bell
inequalities [3032].
In experiments, PPT-entangled states have also been recently observed. In the multipartite case,
the Smolin state has been prepared with four photons, using the polarization degree of freedom for
the qubit encoding [33, 34]. Very recently, although bound entanglement is the hardest to detect [35],
the Leiden-Vienna collaboration has reported the observation of bound entanglement in the bipartite
case with two twisted photons, combining ideas of complementarity [36] and Mutually Unbiased Bases
(MUBs) [37].
Here, we independently recover the results of N. Yu [27] by reformulating the problem in terms
of optimization in the cone of completely-positive
1
matrices. First, we revisit the problem of deter-
mining separability of two DS qudits in arbitrary dimensions. We show that it can be reformulated
in terms of a quadratic conic optimization problem [38]. In particular, we show that separability in
DS states is equivalent to the membership problem in the set of completely-positive matrices. The
equivalence between these two problems allows us to import/export ideas between entanglement theory
2
. Second, we provide examples of entangled PPTDS states and
entanglement witnesses detecting them. Third, we give further characterization criteria for separability
in DS states in terms of the best diagonal dominant decomposition. Finally, we present a family of
N-qubit almost-DS states that are PPT with respect to each bipartition, but nevertheless entangled.
The word almost here means that by adding an arbitrarily small off-diagonal term (GHZ coherence)
to a family of separable DS N-qubits, the state becomes PPT-entangled.
1
Throughout this paper, the term completely-positive corresponds to the definition given in Def. 3.4 and it is not to
be confounded with the concept of a completely positive map that arises typically in a quantum information context.
2
Quadratic conic optimization problems appear naturally in many situations (see [38] and references therein). These
include economic modelling [39], block designs [40], maximin efficiency-robust tests [41], even Markovian models of DNA
evolution [42]. Recently, they have found their application in data mining and clustering [43], as well as in dynamical
systems and control [44, 45].
Accepted in Quantum 2017-12-28, click title to verify 2
The paper is organized as follows. In Section 2 we establish the notation and the basic definitions
that we are going to use in the next sections. In Section 3 we discuss the separability problem
for bipartite DS states of arbitrary dimension, with particular emphasis in their connection to non-
convex quadratic optimization problems. In Section 4 we provide sufficient criteria to certify either
separability or entanglement. In Section 5 we present a class of PPT-entangled multipartite qubit
almost-diagonal symmetric states. In Section 6 we conclude and discuss further research directions.
Finally, in the Appendix we present some proofs, examples and counterexamples that complement the
results discussed in the text.
2 Preliminaries
In this section we set the notation and define the basic concepts that we are going to use throughout
the paper.
2.1 The separability problem
Definition 2.1. Consider a bipartite quantum state ρ acting on
d
d
0
. The state ρ is positive
semi-definite (ρ 0) and normalized (Trρ = 1). A state ρ is separable if it can be written as
ρ =
X
i
p
i
ρ
A
i
ρ
B
i
, (1)
where p
i
form a convex combination (p
i
0 and
P
i
p
i
= 1) and ρ
A
i
(ρ
B
i
) are quantum states acting on
Alice’s (Bob’s) subsystem; i.e., they are positive semidefinite operators of trace one. If a decomposition
of ρ of the form of Eq. (1) does not exist, then ρ is entangled.
The separability problem; i.e., deciding whether a quantum state ρ admits a decomposition of
the form of Eq. (1) is, in general, an NP-hard problem [7]. However, there exist simple tests that
provide sufficient conditions to certify that ρ is entangled [1]. One of the most renowned separability
criteria is the positivity under partial transposition (PPT) criterion [8]. It states that, if ρ can be
decomposed into the form of Eq. (1), then the state ( T )[ρ] must be positive semi-definite, where T
is the transposition with respect to the canonical basis of
d
0
. Such state is denoted ρ
T
B
, the partial
transposition of ρ on Bob’s side. Because (ρ
T
B
)
T
= ρ
T
A
, the PPT criterion does not depend on which
side of the bipartite system the transposition operation is applied on. Breaking PPT criterion is a
necessary and sufficient condition for entanglement only in the two qubit [10] and qubit-qutrit [11]
cases, and there exist counterexamples for states of strictly higher physical dimension [12].
In the multipartite case, the definition of separability given in Eq. (1) naturally generalizes to N
subsystems.
Definition 2.2. A quantum state ρ acting on
d
1
···
d
N
is fully separable if it can be written as
ρ =
X
i
p
i
ρ
(A
1
)
i
··· ρ
(A
N
)
i
, (2)
where ρ
(A
k
)
i
are quantum states acting on the k-th subsystem and p
i
form a convex combination.
Therefore, the PPT criterion also generalizes to 2
bN/2c
criteria, where b·c is the floor function,
depending on which subsystems one chooses to transpose.
2.1.1 Entanglement witnesses
Let us denote by D
sep
the set of separable sates (cf. Eqs. (1), (2)). This set is closed and convex.
Therefore it admits a dual description in terms of its dual cone, which we denote
P = {W = W
s. t. hW, ρi 0 ρ D
sep
},
Accepted in Quantum 2017-12-28, click title to verify 3
where the usual Hilbert-Schmidt scalar product hW, ρi = Tr(W
ρ) is taken. The elements of P can
be thus viewed as half-spaces containing D
sep
. Of course, not every operator in P is useful to detect
entangled states. In order to be non-trivial, one requires that W has at least one negative eigenvalue.
Such operators are called entanglement witnesses (EW) [46] and they form a non-convex set, denoted
W = {W P s. t. W 6 0}. A state ρ is then separable if, and only if, Tr(W ρ) 0 for all W W.
Among EWs, it is worth to make a distinction that relates them to the PPT criterion: decomposable
and indecomposable EWs.
Definition 2.3. Decomposable EWs (DEWs) in a bipartite quantum system are those W W of the
form
W = P + Q
T
B
, (3)
with P 0 and Q 0. Indecomposable EWs (IEWs) are those EWs that are not of the form of Eq.
(3).
Although DEWs are easier to characterize [47], they do not detect PPT-entangled states, because
Tr(W ρ) = Tr(P ρ) + Tr(Q
T
B
ρ) = Tr(P ρ) + Tr(
T
B
) 0. (4)
In Section 4.1 we construct EWs which detect entangled PPTDS states, therefore they correspond to
indecomposable witnesses.
3 Separability in diagonal symmetric states acting on
d
d
.
In this section, we characterize the bipartite diagonal symmetric two-qudit states in terms of the
separability and the PPT properties. We establish an equivalence between: (i) separability and the
PPT property in DS states and (ii) quadratic conic optimization problems and their relaxations,
respectively.
We first introduce the Dicke basis in its full generality and then we move to the two particular cases
of interest to this paper: the case of N -qubits and the case of 2-qudits. One can think of the space
spanned by the Dicke states as the linear subspace of (
d
)
N
containing all permutationally invariant
states.
Definition 3.1. Consider a multipartite Hilbert space (
d
)
N
of N qudits. The Dicke basis in that
space consists of all vectors which are equal superpositions of k
0
qudits in the state |0i, k
1
qudits in
the state |1i, etc., where the multiindex variable k = (k
0
, . . . , k
d1
) forms a partition of N; i.e., k
i
0
and
P
d1
i=0
k
i
= N. They can be written as
|D
k
i
X
σS
N
σ(|0i
k
0
|1i
k
1
···|d 1i
k
d1
), (5)
where σ runs over all permutations of N elements.
The Dicke state has
N
k
different elements, where the quantity follows from the multinomial com-
binatorial quantity
N
k
!
=
N!
k
0
!k
1
! ···k
d1
!
. (6)
Finally, recall that there are as many Dicke states as partitions of N into d (possibly empty) subsets;
therefore, the dimension of the subspace of (
d
)
N
is given by
dim[{|D
k
i : k N}] =
N + d 1
d 1
!
, (7)
where denotes partition of.
In this paper, we are particularly interested in the case of N-qubits and 2-qudits:
Accepted in Quantum 2017-12-28, click title to verify 4
d = 2. For N-qubit states we shall use the notation |D
k
i |D
N
k
i, where k = k
1
denotes the
number of qubits in the excited (|1i) state. Mixtures of Dicke states correspond to
ρ =
N
X
k=0
p
k
|D
N
k
ihD
N
k
|. (8)
N = 2. For bipartite d-level systems, we are going to denote the Dicke states by |D
k
i |D
ij
i,
where i and j are the indices (possibly repeated) of the non-zero k
i
and k
j
. Since the terminology
Dicke states is often reserved for the multipartite case, we call |D
ij
i simply symmetric states.
In the bipartite case (Sections 3 and 4), we focus on diagonal symmetric states, given by Def. 3.2:
Definition 3.2. Let ρ be a state acting on a bipartite Hilbert space H
A
H
B
=
d
d
. The state
ρ is said to be diagonal symmetric (DS) if, and only if, ρ can be written in the form
ρ =
X
0ij<d
p
ij
|D
ij
ihD
ij
|, (9)
where p
ij
0,
P
ij
p
ij
= 1, |D
ii
i := |iii and |D
ij
i = (|iji + |jii)/
2.
In the computational basis, a DS ρ is a d
2
× d
2
matrix that is highly sparse. Therefore, it will
be useful to associate a d × d matrix to ρ that captures all its relevant information. We define the
M-matrix of ρ to be
Definition 3.3. To every DS ρ acting on
d
d
, there is an associated d × d matrix M(ρ), with
non-negative entries
M(ρ) :=
p
00
p
01
/2 ··· p
0,d1
/2
p
01
/2 p
11
··· p
1,d1
/2
.
.
.
.
.
.
.
.
.
.
.
.
p
0,d1
/2 p
1,d1
/2 ··· p
d1,d1
, (10)
which arises from the partially transposed matrix ρ
T
B
.
Notice that, while a DS state ρ is always diagonal in the Dicke basis, its partial transposition (which
is defined with respect to the computational basis) scrambles its elements. Then ρ
T
B
is block-diagonal
in the Dicke basis and its blocks are 1 × 1 elements corresponding to p
ij
with i < j, and M(ρ). One
can see the effect of the partial transposition operation on a DS state by inspecting the action of T
B
onto the elements |D
ij
ihD
ij
| that compose Eq. (9):
If i = j, then (|D
ii
ihD
ii
|)
T
B
= |D
ii
ihD
ii
|, because |D
ii
i = |iii.
If i 6= j, the action of the partial transposition is best seen by expanding |D
ij
i onto the computa-
tional basis: (|D
ij
ihD
ij
|) =
1
2
(|ijihij|+|ijihji|+|jiihij|+|jiihji|). Therefore, two of the terms are
left invariant and the remaining two are to be mapped as (|ijihji|+|jiihij|)
T
B
= |iiihjj|+|jjihii|.
Thus, M(ρ) is the submatrix corresponding to the elements indexed by |iiihjj| for 0 i, j < d of ρ
T
B
.
Because there is no mixing between other rows or columns, we have that ρ
T
B
decomposes as the direct
sum
ρ
T
B
= M(ρ)
M
0i6=j<d
p
ij
2
. (11)
Since p
ij
= p
ji
, we find that the 1 × 1 blocks appear all with multiplicity 2.
Therefore, each M(ρ) with non-negative entries summing 1 is in one-to-one correspondence to a DS
state ρ. In this section we characterize the separability properties of ρ in terms of equivalent properties
of M(ρ), which are naturally related to quadratic conic optimization.
In quadratic conic optimization, one is interested in characterizing the so-called completely positive
(CP) matrices, which are defined as follows
Accepted in Quantum 2017-12-28, click title to verify 5
Definition 3.4. Let A be a d × d matrix. A is completely positive (CP) if, and only if, it admits a
decomposition A = B · B
T
, where B is a d ×k matrix, for some k 1, such that B
ij
0.
Matrices which are CP form a proper
3
cone, which is denoted by CP
d
. Note that the sum of two
CP matrices is a CP matrix and the multiplication of a CP matrix by a non-negative scalar is a CP
matrix.
Given a non-convex optimization problem over the simplex, which is NP-hard in general, CP
matrices translate the complexity of the problem by reformulating it as a linear problem in matrix
variables over CP
d
. Therefore, they allow to shift all the difficulty of the original problem into the cone
constraint. Precisely, every non-convex quadratic optimization problem over the simplex (LHS of Eq.
(12)) has an equivalent CP formulation (RHS of Eq. (12)):
max
x
i
0, hu|xi=1
hx|Q|xi = max
X∈CP
d
, hu|X|ui=1
Tr(XQ), (12)
where |ui is the unnormalized vector of ones and Q is, without loss of generality
4
, symmetric and
positive semi-definite. Therefore, deciding membership in CP
d
is NP-hard [38].
One can obtain, however, an upper bound on the optimization in Eq. (12) by observing that every
CP matrix A is positive semi-definite, because it allows for a factorization A = B · B
T
. Moreover, it
is also entry-wise non-negative: A
ij
0. This motivates Definition 3.5:
Definition 3.5. Let A be a d ×d matrix. A is doubly non-negative (DNN) if, and only if, A 0 and
A
ij
0.
We are now ready to introduce the equivalences between the separability problem in DS states and
quadratic conic optimization. After producing our results, we learned that these equivalences were
independently observed by Nengkun Yu in [27]. We nevertheless prove them in a different way.
Theorem 3.1. Let ρ be a DS state acting on
d
d
.
ρ is separable M(ρ) is CP. (13)
We prove Theorem 3.1 in Appendix A.
By virtue of Theorem 3.1, we recover the result of [27]: Because it is NP-Hard to decide whether
a matrix admits a CP decomposition [38], the separability problem in
d
d
DS states is NP-Hard.
We remark that the NP-hardness result that we obtain holds under polynomial-time Turing reduc-
tions
5
, as opposed to poly-time many-one
6
reductions [48]. For instance, this is the case for Gurvits’
initial reduction of the weak membership problem
7
in the set of separable states from the NP-complete
problem PARTITION
8
[7]. In the case we present here, the reduction holds because the NP-hardness
of deciding membership in the CP
d
set follows via a Turing reduction, which is the result we use as
our starting point. The part of the reduction that we provide here, however, is many-one.
3
Closed, convex, pointed and full-dimensional.
4
Q can be assumed to be symmetric because hx|Q|xi = (hx|Q|xi)
T
= hx|Q
T
|xi. It can be assumed to be positive
semi-definite because adding to (Q + Q
T
)/2 does not change the optimal |xi; it only adds a bias to the maximum.
5
Intuitively speaking, a Turing reduction describes how to solve problem A by running an algorithm for a second
problem B, possibly multiple times.
6
A many-one reduction is a special case of a Turing reduction, with the particularity that the algorithm for problem
B can be called only one time, and its output is immediately returned as the output of problem A.
7
Weak in the sense that it allows for error in points at a given Euclidean distance from the border of the set.
8
The PARTITION problem is a decision problem corresponding to whether a given set of integer numbers can be
partitioned into two sets of equal sum. This problem is efficiently solvable with a dynamic programming procedure [49],
but becomes NP-hard when the magnitudes of the input integers become exponentially large with the input size.
Accepted in Quantum 2017-12-28, click title to verify 6
We here briefly discuss the steps that would be required to make this result completely rigurous from
a computer science point of view. One would need to embed the NP-hardness into the formalism of the
weak membership problem [7, 50]. This requires, for instance, showing that the convex set of separable
DS states has some desirable properties such as being well-bounded or p-centered. We refer the reader
to [48] for the technical aspects of these definitions. On the other hand, the completely positive cone is
known to be well-bounded and p-centered: in [51] it was proved that the weak membership problem in
the completely positive cone is NP-hard. By using the one-to-one correspondence between DS states
and CP matrices given by M(ρ) in Def. 3.3, then the result is mapped onto the DS set
9
.
Geometrically, the set of separable DS states is convex. Hence, it is fully characterized by its ex-
tremal elements (those that cannot be written as a non-trivial convex combination of other separable
DS states). Identifying such elements is of great importance towards the characterization of the sepa-
rability properties of DS states. For instance, in the set of all separable density matrices, the extremal
ones are the rank-1 projectors onto product vectors. However, this property may be lost when re-
stricting our search in a subspace: observe that the set of separable DS states states is obtained as the
intersection of the subspace of DS states with the convex set of separable density matrices. Therefore,
the set of extremal separable DS states states may contain states that are separable, but not extremal
in the set of separable density matrices (see Fig. 1). Theorem 3.1 allows us to fully characterize
extremality in the set of separable DS states in terms of extremal CP matrices, thus obtaining the
following corollary:
Corollary 3.1. The extremal separable DS states ρ fulfill
p
ij
= 2
p
ii
p
jj
, i < j. (14)
Proof. Since the extremal rays of the CP
d
cone are the rank-1 matrices
~
b
~
b
T
where b
i
0 [42], by
normalizing and comparing to Eq. (10) we obtain Eq. (14).
Separable
Figure 1: Cartoon picture of the set of separable states SEP (cylinder) and its intersection with the subspace of
diagonal symmetric states DS (ellipse). The intersection of the subspace of DS states with the set of separable states
gives rise to the set of separable DS states, which is represented by the green ellipse, including its interior. Only the
states of the form |iiihii| are extremal in both sets (represented by the black dot in the figure). However, states that
were in the boundary of SEP, could now be extremal when viewed in DS (represented by the border of the green
ellipse in the figure).
9
The technical requirement of full dimensionality [48, 51] depends on the set in which one embeds the problem. Recall
that we are interested in solving the separability problem within DS states. The set of DS separable states is of course
not full-dimensional when embedded in the whole two-qudit Hilbert space. However, it is full-dimensional when viewed
in the DS subspace (cf. Figure 1).
Accepted in Quantum 2017-12-28, click title to verify 7
Theorem 3.2. Let ρ be a DS state acting on
d
d
.
ρ is PPT M(ρ) is DNN. (15)
Proof. Let us assume that ρ is PPT. Note that the partial transposition of ρ can be written as
ρ
Γ
=
M
0a<b<d
p
ab
2
p
ab
2
M(ρ). (16)
Since ρ is PPT, Eq. (16) implies that M(ρ) 0. Since ρ is a valid quantum state, then p
ab
0 for all
0 a b < d. Hence, all the entries of M(ρ) are also non-negative. Thus, M(ρ) is DNN.
Conversely, if M(ρ) is DNN then we have that all its entries are non-negative; i.e., p
ab
0 for
0 a b < d. These conditions guarantee that ρ 0. Additionally, as M(ρ) 0, these conditions
imply that ρ
Γ
0. Hence, ρ is PPT.
Figure 2: For a two qudit PPTDS state ρ, if its corresponding M(ρ) is in CP
d
then ρ is separable, if M (ρ) is in
DNN
d
then ρ is P P T and if M(ρ) is in DNN
d
but not CP
d
then ρ is P P T but entangled.
Recall (cf. Definitions 3.4 and 3.5, also Fig. 2) that CP
d
DNN
d
. However, the inclusion is strict
for d 5: It is known that CP
d
= DNN
d
for d 4 and CP
d
( DNN
d
for d 5 [38]. This yields a
full characterization of the bipartite separable DS states in terms of the PPT criterion:
Theorem 3.3. Let ρ be a DS state acting on
d
d
, with d 4.
ρ is separable ρ is PPT. (17)
Proof. The result follows from the identity CP
d
= DNN
d
, which holds for d 4 [38]. In Example
C.1 we provide a constructive proof for d = 3.
Finally, we end this section by giving a sufficient separability criteria for any d in terms of the ranks
of M(ρ).
Theorem 3.4. Let ρ be a PPTDS state with M(ρ) of rank at most 2. Then, ρ is separable.
Proof. Since ρ is PPT, M (ρ) 0. Therefore, it admits a factorization M(ρ) = V V
T
, where V is
a d × 2 or a d × 1 matrix. Geometrically, every row of V can be seen as a vector in
2
(or a scalar
if the rank of M(ρ) is one). Therefore, M(ρ) can be seen as the Gram matrix of those vectors; each
element being their scalar product. Since M(ρ) is doubly non-negative, it implies that all these scalar
products must be positive; therefore, the angle between each pair of vectors is smaller or equal than
π/2. Thus, the geometrical interpretation is that M(ρ) is CP if, and only if, they can be isometrically
embedded into the positive orthant of
k
for some k. This is always possible to do for k = 2 (see Fig.
3), which corresponds to M(ρ) having rank at most 2.
Accepted in Quantum 2017-12-28, click title to verify 8
Figure 3: Visual representation of the proof for Theorem 3.4. When the angle between each pair of vectors is smaller
or equal than π/2, meaning that M (ρ) is CP
d
, all vectors can be isometrically embedded in the nonnegative orthant.
4 Sufficient criteria for entanglement and separability
In this section we further characterize the bipartite DS states by providing sufficient criteria to cer-
tify entanglement by means of Entanglement Witnesses for DS states, and by providing sufficient
separability conditions in terms of M(ρ).
4.1 Entanglement Witnesses for DS states
We begin by introducing the concept of copositive matrix:
Definition 4.1. A matrix A is called copositive if, and only if, ~x
T
A~x 0 for all ~x with non-negative
entries.
The set of d × d copositive matrices also forms a proper cone, denoted COP
d
. The cones CP
d
and COP
d
are dual to each other with respect to the trace inner product. It is also easy to see that
PSD
d
+ N
d
COP
d
, where PSD
d
is the set of positive-semidefinite d × d matrices and N
d
is the
set of symmetric entry-wise non-negative matrix. Actually, we have DNN
d
= PSD
d
N
d
and the
observation follows from the inclusion CP
d
DNN
d
.
Therefore, one can view copositive matrices as EWs for DS states. Furthermore, one could think of
PSD
d
+ N
d
as the set of DEWs for DS states, in the sense that they do not detect entangled PPTDS
states.
In Examples 4.1 and D.1 we provide some M(ρ) DNN
d
\CP
d
for d 5, therefore corresponding to
entangled PPTDS states. The paradigmatic example of a copositive matrix detecting matrices DNN,
but not CP, (i.e., PPT, but entangled DS states) is the Horn matrix [40], which is defined as
H :=
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
. (18)
It is proven that H COP
5
\ (PSD
5
+ N
5
) in [40]. As Tr(HM(ρ)) = 1 < 0, H corresponds to an
(indecomposable) entanglement witness for the state corresponding to M(ρ).
Although the boundary of the set of copositive matrices remains uncharacterized for arbitrary
dimensions, COP
5
was fully characterized in [51]:
COP
5
= {DAD : D is positive diagonal, A s. t. p(A, ~x) is a sum of squares}, (19)
Accepted in Quantum 2017-12-28, click title to verify 9
where
p(A, ~x) :=
X
i,j
A
ij
x
2
i
x
2
j
X
k
x
2
k
!
.
Furthermore, the extremal rays of COP
d
have been fully characterized for d 5, divided into classes,
but this also remains an open problem for higher d [38].
In Appendix B we discuss exposedness properties of the sets of completely positive and co-positive
matrices and their relation to the separability problem and its geometry.
Examples of entangled PPTDS states for d = 5. Let us provide an example of a bipartite
PPTDS entangled state for d = 5.
˜
M(ρ) =
1 1 0 0 1
1 2 1 0 0
0 1 2 1 0
0 0 1 1 1
1 0 0 1 3
. (20)
It can be easily seen using the Range criterion, as in Section C.2, that ρ is entangled. By Theorem
3.1, it is equivalent to show [38] that M (ρ) DNN
5
\ CP
5
.
Finally, it can be appreciated how the Horn matrix can be used as an EW and certify entanglement
Tr(H
˜
M(ρ)) = 1 < 0.
4.2 Sufficient separability conditions for diagonal symmetric states
In the spirit of best separable approximations (BSA) [24], in this section we provide sufficient sepa-
rability conditions for bipartite PPTDS states. In the same way that the BSA allows one to express
any PPTDS state as a sum of a separable part and an entangled one with maximal weight on the
separable one
10
. In this section we introduce Best Diagonal Dominant (BDD) approximations, which
give a sufficient criterion to certify that a PPTDS state is separable. The idea is that although checking
membership in CP
d
is NP-hard, it is actually easy to (i) characterize the extremal elements in CP
d
(cf. Corollary 3.1) and (ii) check for membership in a subset of DD
d
CP
d
that is formed of those
matrices A N
d
that are diagonal dominant. In [52] the inclusion DD
d
CP
d
was proven. Therefore,
to show that CP
d
\DD
d
is nonempty we study when the decomposition of a potential element in CP
d
as a convex combination of an extremal element of CP
d
and an element of DD
d
is possible (see Figure
4).
Let us start by stating a lemma that gives an explicit separable decomposition of a quantum state.
Lemma 4.1. Let I be the unnormalized quantum state defined as
I =
d1
X
i=0
|iiihii| +
X
0i<j<d
2|D
ij
ihD
ji
|, (21)
where |D
ij
i = (|iji + |jii)/
2. For instance, for d = 3,
I =
1 · · · · · · · ·
· 1 · 1 · · · · ·
· · 1 · · · 1 · ·
· 1 · 1 · · · · ·
· · · · 1 · · · ·
· · · · · 1 · 1 ·
· · 1 · · · 1 · ·
· · · · · 1 · 1 ·
· · · · · · · · 1
. (22)
10
In [25], the BSA was found analytically for N-qubit DS states
Accepted in Quantum 2017-12-28, click title to verify 10
Figure 4: Given a two-qudit PPTDS state ρ, if we can decompose M(ρ) in terms of M (I) (an extremal element of
CP
d
) and M(˜ρ) (an element of DD
d
) we can certify that M(ρ) is in CP
d
and therefore certify that ρ is separable.
Then, I is separable.
Proof. Let |e(~ϕ)i = |0i + e
ϕ
1
|1i+ ···+ e
ϕ
d1
|d 1i. A separable decomposition of I is given by
I =
Z
[0,2π]
d
d~ϕ
(2π)
d
(|e(~ϕ)ihe(~ϕ)|)
2
. (23)
Indeed,
I =
X
ijlk
|ijihkl|
Z
[0,2π]
d
d~ϕ
(2π)
d
e
(ϕ
i
+ϕ
j
ϕ
k
ϕ
l
)
=
X
ijlk
|ijihkl|(δ
i,k
δ
j,l
+ δ
i,l
δ
j,k
δ
i,j,k,l
), (24)
where δ is the Kronecker delta function.
Lemma 4.1 allows us to give a sufficient condition for a state ρ to be separable. The idea is to
subtract εI from ρ in such a way that it remains a valid diagonal symmetric state, PPT, and close
enough to the interior of the separable set such that it is easy to certify that the state is separable (see
Fig. 4).
Theorem 4.1. Let ρ be a two-qudit PPTDS state with associated M(ρ). If there exists ε 0 such
that
1. ε ρ
ij
for all i, j such that 0 i, j < d.
2. εd (hu|
1
M(ρ)
|ui)
1
and |ui R(M(ρ)), R(M(ρ)) is the range of M(ρ) and
1
M(ρ)
is the pseudo-
inverse of M(ρ). Here |ui is a normalized vector of ones.
3. for all i such that 0 i < d, ρ
ii
+ ε(d 2)
P
j6=i
ρ
ji
.
Then, ρ is separable.
See the proof in Appendix D.2.
A few comments are in order: The first condition on Theorem 4.1 ensures that I can be subtracted
from ρ and ˜ρ will remain in the DS subspace. The second condition requires that I can be subtracted
from ρ while maintaining the PPT property of ˜ρ. If |ui / R(M(ρ)), then ˜ρ would not be PPT for any
ε 6= 0. Therefore, the second condition gives the maximal value of ε that can be subtracted such that ˜ρ
remains PPT. Finally, the third condition relies on guaranteeing that ˜ρ is separable, which is ensured
by M(˜ρ) to be diagonal dominant. This means that one might need to subtract a minimal amount of
I to accomplish such a property (unless ρ is already diagonal dominant).
In Example D.3 we show that CP
d
\ DD
d
6= using the approach of Theorem 4.1.
The above result can be now normalized and generalized to any other extremal element I in CP
d
:
Accepted in Quantum 2017-12-28, click title to verify 11
Lemma 4.2. Let x
d
with x
i
0 and ||x|| > 0. Let I
x
be the quantum state defined as
I
x
=
1
||x||
2
1
d1
X
i=0
x
2
i
|iiihii| +
X
0i<j<d
2x
i
x
j
|D
ij
ihD
ij
|
.
Then, the quantum state I
x
is separable.
See the proof in Appendix D.4.
Note that the corresponding M(I
x
) is given by |u
x
ihu
x
|, where |u
x
i = x/||x||
1
. The sum of the
elements of M(I
x
) is then one; i.e., ||M(I
x
)||
1
= 1.
Lemma 4.2 allows us to give a sufficient condition for a state ρ is separable. This time the idea is
to decompose ρ as a convex combination between I
x
, which is a state that is extremal in the set of
separable DS states, and a state ˜ρ which is deep enough in the interior of the set of separable states,
such that we can certify its separability by other means (by showing that M(˜ρ) is diagonal dominant
and doubly non-negative; therefore completely positive [52]).
Theorem 4.2. Let ρ be a two-qudit PPTDS state with associated M(ρ). Let x
d
with x
i
> 0. If
there exists a λ [0, 1) such that
1. λ (M(ρ))
ij
||x||
2
1
/x
i
x
j
for all i and j,
2. λ 1/hu
x
|
1
M(ρ)
|u
x
i and |u
x
i R(M(ρ)), where R(M(ρ)) is the range of M(ρ) and
1
M(ρ)
denotes
the pseudo-inverse of M(ρ),
3. λx
i
(||x||
1
2x
i
) ||x||
2
1
h
P
j6=i
(M(ρ))
ij
(M(ρ))
ii
i
for all i,
then ρ is separable. Equivalently, then M(ρ) is completely positive.
See the proof in Appendix D.5 (i.e., write ρ = (1 λ)˜ρ + λI
x
and ensure that the associated M (˜ρ)
is completely positive).
Notice that Theorem 4.2 provides an advantage over Theorem 4.1 since the parameters of I
x
are not
fixed which allows to consider a bigger family of decompositions M(ρ) CP
d
parametrized by x. In
Example 4.2 we attempt to apply both Theorems in order to guarantee separability and illustrate such
Example. In this example we provide a PPTDS state with associated M(ρ) CP
d
\DD
d
and we
show how to apply Theorem 4.2 to guarantee separability. Furthermore, we also apply Theorem 4.1
to illustrate the advantage of Theorem 4.2.
Take the following DS state ρ
3
with associated
M(ρ) =
α β γ
β δ β
γ β
=
1
100
19 8 11.5
8 6.4 8
11.5 8 19.6
, (25)
where it can be checked that the state ρ is normalized. A priori we do not know if this state is
separable, the goal is to apply Theorems 4.1 and 4.2 in order to see if separability can be guaranteed.
For both Theorems the more restrictive between conditions 1 and 2 provides an upper bound for the
corresponding decomposition and condition 3 a lower bound but, as mentioned, Theorem 4.2 offers
more flexibility since such bound can be varied by fitting I
x
. This example illustrates this fact since
we will see that Theorem 4.2 guarantees separability but Theorem 4.1 does not.
Lets start with Theorem 4.2. For the given case (25) we want to find if a convex decomposition
M(ρ) = (1 λ)M(˜ρ) + λM (I
x
) exists while fulfilling the conditions of the theorem. For instance,
for illustrative purposes we fix λ = 0.8 and by numerical means we obtain that a possible convex
combination would be with an M(I
x
) = |u
x
ihu
x
| given by |u
x
i = 1/100(37.46|0i+ 25.16|1i+ 37.38|2i).
Accepted in Quantum 2017-12-28, click title to verify 12
Lets proceed to show that the given M(ρ) and M(I
x
) meet the conditions of Theorem 4.2. Condition
1 provides the more restrictive upper bound given by
λ
γ||x||
2
1
x
1
x
3
= 0.8213, (26)
while the lower bound will be given by the following case of Condition 3
λ
||x||
2
1
[2β δ]
x
2
(||x||
1
δ)
= 0.7681. (27)
Therefore, there exists a range of values λ [0.7681, 0.8213] that satisfy the conditions of Theorem
4.2 and certifies that the state ρ is separable. Notice that, for illustrative purposes, once we found an
M(I
x
) fulfilling the conditions we fixed it to find a range of values for λ but we could have allowed for
more freedom and find a bigger range of possible decompositions.
Now lets see what happens with Theorem 4.1. In this case the most restrictive upper bound is
given by Condition 2
ε (hu|
1
M(ρ)
|ui)
1
= 0.06, (28)
while the most restrictive lower bound will be given by the following case of Condition 3
ε 2β δ = 0.096. (29)
Thus, for this case there does not exist an ε satisfying the conditions for Theorem 4.1, while there
exists a range of λ satisfying conditions for Theorem 4.2 and therefore illustrating its advantage by
being able to certify separability.
5 A class of PPT-entangled quasi-DS states
In this Section, we introduce a uni-parametric class of N -qubit PPTESS, for an odd number of qubits.
As it has been shown in [27, 28], N-qubit PPTDS states are fully separable. The class we introduce
can be seen as a N-qubit PPTDS state with slight GHZ coherences. Surprisingly, in the family of
states we provide, an arbitrarily small weight on the non-diagonal elements (in the Dicke basis) allows
the state to be genuinely multipartite entangled while maintaining the PPT property.
The procedure we have chosen to derive this class of states is based on the iterative algorithm for
finding extremal PPT symmetric states [20, 21] (see also [53]), which we briefly recall here in the interest
of completeness. One starts with an initial symmetric state ρ
0
that is fully separable; for instance,
the symmetric completely mixed state. Then, one picks a random direction σ
0
in the set of quantum
states and subtracts it from the initial state while preserving the PPT property, therefore obtaining
ρ
0
x
0
σ
0
, x
0
> 0. One necessarily finds a critical x
0
such that one arrives at the boundary of the PPT
set, where the rank of ρ
0
x
0
σ
0
or one of its partial transpositions must have decreased. Hence, at least
one new vector appears in the kernel of the state or in the kernel of some of its partial transpositions.
This state with lower ranks is set as the initial state for the next iteration ρ
1
= ρ
0
x
0
σ
0
. The new
direction σ
1
is chosen such that it preserves all the vectors present in the kernels of both the state
and its partial transpositions. This process is repeated until no new improving direction can be found,
yielding an extremal state ρ
k
in the PPT set. As the PPT set contains all separable states, we note
that if the rank of such extremal PPT state is greater than one, then it cannot be extremal in the set
of separable states (because these are pure product vectors, which have rank one), therefore it must
be entangled. The study carried out in [20, 21] looked for typical extremal PPT states by exploring
random directions every time. However, by carefully picking these directions, one can look for classes
of states of different forms, such as the ones presented in Theorem 5.1.
In Example E.1 we present a 4-qubit PPT-entangled symmetric state whose density matrix is sparse
with real entries when represented in the computational basis and has a closed analytical form.
Accepted in | 2020-08-03 20:39:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8668910264968872, "perplexity": 2115.960845946515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735833.83/warc/CC-MAIN-20200803195435-20200803225435-00282.warc.gz"} |
https://repository.cihe.edu.hk/jspui/handle/cihe/1315 | Please use this identifier to cite or link to this item: https://repository.cihe.edu.hk/jspui/handle/cihe/1315
Title: Preliminary tests of a newly developed superconducting gravity gradiometer Author(s): Chan, Anthony Hing Hung Author(s): Moody, M. V.Paik, H. J. Issue Date: 1983 Publisher: IEEE Journal: IEEE Transactions on Magnetics Volume: 19 Issue: 3 Start page: 461 End page: 464 Abstract: We have recently completed a series of tests on a newly developed superconducting gravity gradiometer consisting of two 400 g niobium proof masses separated by a distance of 15 cm. The resonance frequency of the proof mass suspension systems is 25 Hz. The sensitivity of this gradiometer obtained during these tests was 1 to 2 Eötvös Hz -1/2 (1 Eötvös = 10 -9 s -2 \equiv1 E) below 1Hz, and 0.2 E Hz -1/2 between 15 and 20 Hz. Using this gradiometer, a prototype experiment has been performed to demonstrate a new source-independent null test of the inverse square law of gravitation. The result, assuming a potential of the form \phi(r) = -(GM/r) [1 + \alpha \exp(-\mur)] , is α = + 0.024 ± 0.036 at μ -1 = 1m. URI: https://repository.cihe.edu.hk/jspui/handle/cihe/1315 DOI: 10.1109/TMAG.1983.1062415 CIHE Affiliated Publication: No Appears in Collections: CIS Publication | 2022-06-25 13:48:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4326506555080414, "perplexity": 5380.564414497215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103035636.10/warc/CC-MAIN-20220625125944-20220625155944-00668.warc.gz"} |
https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Organic_Chemistry_Lab_Techniques_(Nichols)/01%3A_General_Techniques/1.05%3A_Filtering_Methods/1.5B%3A_Decanting | # 1.5B: Decanting
When there is a need to separate a solid-liquid mixture, on occasion it is possible to pour off the liquid while leaving the solid behind. This process is called decanting, and is the simplest separation method. Decanting is often used to remove hydrated sodium sulfate ($$\ce{Na2SO4}$$) from an organic solution. The sodium sulfate often clings to the glassware (Figure 1.67a), enabling the liquid to be poured off (Figure 1.67b). If liquid is to be poured into a small vessel, a funnel could be used or liquid poured down a glass stirring rod to direct the flow (Figure 1.67c). Unfortunately, there are many mixtures that do not decant well.
This page titled 1.5B: Decanting is shared under a CC BY-NC-ND 4.0 license and was authored, remixed, and/or curated by Lisa Nichols via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | 2022-12-05 11:46:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8902427554130554, "perplexity": 2209.8786158639477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711016.32/warc/CC-MAIN-20221205100449-20221205130449-00629.warc.gz"} |
http://vase.essex.ac.uk/quizzes/index.cgi?quiz=human-visual-system | # A quiz on the human visual system
1. Why are the decks of ships usually illuminated using red light at night?
• To reduce the amount of light pollution from the ship
• So that only one type of cone is used, leaving the others free for looking out
• So that the dark-adaption of the rods is affected as little as possible
• Because ships may not show any other colour lights
2. Blue cones are much less responsive than red- and green-sensitive ones. How does the brain handle this?
• The rods contribute to the blue response too
• The brain amplifies' the blue signal much more than the red and green ones
• There are more blue-sensitive cones in the fovea
• Sunlight is much more intense in the blue
3. Signals from the eye travel via the optic nerve to which of the following?
• laterate geniculate nucleus
• visual striate cortex
• V4
• V1
4. What type of cell is responsible for vision in low-light conditions?
• fovea
• retina
• rod
• cone
5. Where does the "where" or "how" pathway run?
• V2, V3, V4, V5
• V2, V4
• V2, V3
• V2, V5
6. Roughly, what is the range of wavelengths of visible light?
• 100--1000 nm
• 400--700 nm
• 0--100 nm
• 1000--5000 nm
7. What is "continuity" in the context of visual processing?
• filling in missing boundaries
• a continuous stream of pixels
• an analogue signal
• a derivative exists
8. What type of cell is responsible for colour vision?
• rod
• fovea
• cone
• retina
9. To which of the following do the cells in V1 respond to?
• motion
• orientation, edge
• orientation, edge, motion
• edge, motion
10. What is the fovea?'
• A depression in the retina which has many more cones than elsewhere on the retina
• The part of the eye that focuses light
• Where the optic nerve leaves the eye
• The part of the eye that controls how much light is admitted into the eye
11. | 2018-03-25 05:11:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4252663850784302, "perplexity": 5825.182699029415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651820.82/warc/CC-MAIN-20180325044627-20180325064627-00489.warc.gz"} |
http://galileo.phys.virginia.edu/classes/252/spectra.html | # Spectra
Michael Fowler, University of Virginia
## Early Work
The first person to realize that white light was made up of the colors of the rainbow was Isaac Newton, who in 1666 passed sunlight through a narrow slit, then a prism, to project the colored spectrum on to a wall. This effect had been noticed previously, of course, not least in the sky, but previous attempts to explain it, by Descartes and others, had suggested that the white light became colored when it was refracted, the color depending on the angle of refraction. Newton clarified the situation by using a second prism to reconstitute the white light, making much more plausible the idea that the white light was composed of the separate colors. He then took a monochromatic component from the spectrum generated by one prism and passed it through a second prism, establishing that no further colors were generated. That is, light of a single color did not change color on refraction. He concluded that white light was made up of all the colors of the rainbow, and that on passing through a prism, these different colors were refracted through slightly different angles, thus separating them into the observed spectrum.
In 1752, the Scottish physicist Thomas Melvill discovered that putting different substances in flames, and passing the light through a prism, gave differently patterned spectra. Ordinary table salt, for example, generated a "bright yellow". Furthermore, not all the colors of the rainbow appeared— there were dark gaps in the spectrum, in fact for some materials there were just a few patches of light. By the 1820's, Herschel had recognized that spectra provided an excellent way to detect and identify small quantities of an element in a powder put into a flame.
Meanwhile, the white light of the sun was coming in for more detailed scrutiny. In 1802, William Wollaston in England had discovered (perhaps by using a thinner slit or a better prism) that in fact the solar spectrum itself had tiny gaps— there were many thin dark lines in the rainbow of colors. These were investigated much more systematically by Joseph von Fraunhofer, beginning in 1814. He increased the dispersion by using more than one prism. He found an "almost countless number" of lines. He labeled the strongest dark lines A, B, C, D, etc.
## Foucault Connects Melvill's Bright Lines and Fraunhofer's Dark Lines
In 1849, Foucault (of speed of light and pendulum fame) examined the spectrum of light from a voltaic arc between carbon poles. He saw a bright double yellow line at exactly the same wavelength as Fraunhofer's dark D line in the solar spectrum. Investigating further, Foucault passed the sun's light through the arc, then through a prism. He observed that the D lines in the spectrum were even darker than usual. After testing with other sources, he concluded that the arc, which emitted light at the D line frequency, would also absorb light from another source at that frequency.
This discovery did not surprise Sir George Stokes in Cambridge. He pointed out that any mechanical system with a natural frequency of oscillation will emit at that frequency if disturbed, but will also absorb most readily at that frequency from incoming disturbances, the phenomenon of resonance (Dampier page 241).
Question: In a total eclipse of the sun, the only sunlight reaching earth comes from the hot gases of the sun's atmosphere, light from the sun's main disc being blocked by our moon. The light from these hot gases was analyzed during an eclipse in 1870. How do you think the spectrum observed related to that of full sunlight?
The spectrum of hydrogen, which turned out to be crucial in providing the first insight into atomic structure over half a century later, was first observed by Anders Angström in Uppsala, Sweden, in 1853. His communication was translated into English in 1855. Angström, the son of a country minister, was a reserved person, not interested in the social life that centered around the court. Consequently, it was many years before his achievements were recognized, at home or abroad (most of his results were published in Swedish). Meanwhile, in Freeport, Pennsylvania, in 1855, David Alter described the spectrum of hydrogen and other gases. In the 1840's, Alter had started the first commercial production of bromine from brines. He also found a way to extract oil from coal, but that proved uneconomic after the discovery of oil in Pennsylvania. His work was not widely recognized, either. (Dampier, BDS)
## Bunsen and Kirchhoff
The first really systematic investigation of spectra was that of Bunsen and Kirchhoff, in Heidelberg, between 1855 and 1863. They used several techniques. For one thing, they introduced various salts into -- what else? -- the flame of a Bunsen burner. This was a very effective way of viewing spectra, because the Bunsen burner flame itself gave out practically no light. They also used the cooler flame of alcohol burning mixed with water to generate a vapor to study absorption spectra. Finally, they studied the spectra of electric arcs between electrodes of different materials. Using iron electrodes gave a spectrum that coincided with dark lines in the sun's spectrum. Copper electrodes did not. They concluded that the sun's atmosphere contained iron, but not much copper, and that, they said, seemed very plausible since there is so much iron in the earth, and in meteors.
(Cautionary note to philosophers: In 1835, the French philosopher Auguste Comte (the founder of positivism) wrote: "…Our knowledge concerning the gaseous envelopes [of stars] is necessarily limited to their existence, size … and refractive power, we shall not at all be able to determine their chemical composition or even their density … I regard any notion concerning the true mean temperature of the various stars as forever denied to us." (Pais, IB, page 155))
The collaboration of Kirchhoff and Bunsen was a major research effort, even by modern standards. They determined thousands of spectral lines, each to an accuracy of one part in ten thousand. They spectroscopically discovered new elements: rubidium and cesium. Their method was used to find fifteen more new elements before the end of the century. In 1869, Joseph Lockyer studied the spectra of solar prominences (in eclipses). He found the spectra to be slightly Doppler shifted, so was able to deduce the speeds of the gases whirling around the sunspots. He also found a spectrum never seen before, and conjectured that it came from a new element he named Helium.
In fact, helium was later discovered on earth in 1895, by Ramsay. At that time, it had just become evident that there was an inert component, argon, in the earth's atmosphere. Earlier, an inert gas had been observed to emanate from uranium salts when they were heated. Ramsay assumed this would be the same gas, but decided to check. On heating uranium salts and performing a spectral analysis of the emitted gas, much to his surprise he found it to be helium. (Yet another example of the Scientific Method at work: viz., almost all important discoveries are made accidentally while looking for something else.)
## The Balmer Series
It is clear from the above that a tremendous amount of scientific progress was made using spectral lines, yet no-one had the slightest idea why atoms emitted at the frequencies they did. It was appreciated that spectra implied that atoms had structure. In 1852, Stokes had stated that probably the vibrations that produced light were vibrations among the constituent parts of molecules (a term which also included atoms at that time) and in 1875 Maxwell, in enumerating properties atoms must have, included the capability of internal motion or vibration. This worried Maxwell, though. As he said, the spectroscopic evidence forces the conclusion that the atom is quite complex, with many internal degrees of freedom. Yet apparently all these modes of vibration, or almost all of them, are not excited by heat, since if they were this extra capacity of the atom to absorb energy would be reflected in its specific heat (Pais, IB page 175).
Obviously, if any pattern could be discerned in the spectral lines for an atom, that might be a clue as to the internal structure of the atom. One might be able to build a model. A great deal of effort went into analyzing the spectral data from the 1860's on. The big breakthrough was made by Johann Balmer, a math and Latin teacher at a girls' school in Basel, Switzerland. Balmer had done no physics before, and made his great discovery when he was almost sixty. He decided that the most likely atom to show simple spectral patterns was the lightest atom, hydrogen. Angström had measured the four visible spectral lines to have wavelengths 6562.10, 4860.74, 4340.1 and 4101.2 in Angstrom units (10-10 meters). Balmer concentrated on just these four numbers, and found they were given by the formula:
$\lambda =b\left(\frac{{n}^{2}}{{n}^{2}-4}\right)$
where b = 3645.6 Angströms, and n = 3, 4, 5, 6. Balmer suggested that there would be other lines— in the infrared— corresponding to n = 7, 8, etc., and in fact some of them had already been observed, unbeknownst to Balmer. He further conjectured that the 4 could be replaced by 9, 16, 25, … and this also turned out to be true— but these lines, further into the infrared, were not detected until the early twentieth century, along with the ultraviolet lines generated by replacing the 4 by 1.
It is instructive to write Balmer's general formula in terms of the inverse wavelength. This is called the wave number— the number of waves that fit in one unit of length.
$\frac{1}{\lambda }=R\left(\frac{1}{{n}^{2}}-\frac{1}{{m}^{2}}\right).$
where $n,m$ are integers, and $R$ is the Rydberg constant, 109,737 cm-1.
This constant is named after the Swedish physicist Rydberg who (in 1888) presented a generalization of Balmer's formula, in which the integer $n$ was replaced by $n+$ constant, the constant being less than unity. Rydberg suggested that all atomic spectra formed families with this pattern. (He also said he was unaware of Balmer's work.) It turns out that there are families of spectra following Rydberg's pattern, notably in the alkali metals, sodium, potassium, etc., but not with the precision the hydrogen atom lines fit the Balmer formula, and low values of $n$ give lines that deviate considerably.
(Modern footnote: atoms having spectral lines following Rydberg's formula are called Rydberg atoms. As we shall see later, these Rydberg atoms have one electron orbiting at a much greater distance from the nucleus than the others. Consequently, Rydberg atoms can only survive in a gas at very low pressure, otherwise that outermost electron gets knocked off. Prof. Tom Gallagher in our Department is a world expert on these atoms, which have proven a rich source of information on the quantum mechanics of atomic structure.)
One pattern that was noticed in the spectra ofmany atoms isRitz' Combination Principle:
If for a given atom there are spectral lines at two wave numbers, there is sometimes another spectral line at the precise sum of those two wave numbers.
It is easy to see from Balmer's formula that this is true for some pairs of lines in the hydrogen spectrum. It also turns out to be true for atoms where the spectral lines have no other discernible pattern.
Books I used in preparing this lecture:
Inward Bound, Abraham Pais, Oxford, 1986
Collins' Biographical Dictionary of Scientists, HarperCollins, Glasgow, 1994
A History of Science, W. C. Dampier, Cambridge, 1929. | 2018-04-23 15:00:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6723787784576416, "perplexity": 1135.9852648419153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946077.4/warc/CC-MAIN-20180423144933-20180423164933-00256.warc.gz"} |
https://www.physicsforums.com/threads/toppling-cylinder-angular-velocity.568480/ | # Toppling cylinder angular velocity
1. Jan 17, 2012
### h4nsm0l3m4n
1. The problem statement, all variables and given/known data
I am approximating an object as a cylinder (h=6, r=1, mass=200lb). The cylinder is in the process of toppling over from a vertical position. I will assume that the cylinder can pivot freely on the bottom surface. At some angle theta, the cylinder will strike an object, we'll assume with the top edge of the cylinder. I would like the estimate the impact load exerted on that object.
2. Relevant equations
F=m*a
$\tau$=r*F*sin($\theta$)
$\tau$=I*$\alpha$
Erot=$\frac{1}{2}$*I*$\omega$2
3. The attempt at a solution
I would like to use the impact loading case (basic summary here http://www.pdhcenter.com/courses/s164/s164content.pdf) to help find an equivalent load on impact.
Basically, in this process I will approximate the object as a spring do an energy balance between the rotation energy of the cylinder and the energy stored in the spring.
However I needed to find the angular velocity to calculate the rotational potential energy.
I figured the gravitational force exerts a torque on the cylinder from the center of mass
F=m*g
$\tau$=r*F*sin($\theta$)
r in this case is the distance from the pivot point to the center of mass of the cylinder
$\tau$=$\frac{h}{2}$*m*g*sin($\theta$)
Since, $\tau$=I*$\alpha$
$\alpha$=$\frac{h*m*g*sin(\theta)}{2*I}$
This is the angular acceleration. I thought perhaps I could integrate it with respect to t to find the angular velocity. However, it would appear that with $\theta$ in the equation I cannot integrate since theta is not a constant with respect to time.
Am I missing something important here, or just going about solving this problem totally the wrong way? Any help would be great, and I apologize in advance in case I put this in the wrong forum.
Last edited: Jan 17, 2012
2. Jan 17, 2012
### Simon Bridge
You have to put the terms with theta in them on the LHS before integrating.$$\frac{1}{\sin\theta}\frac{d^2\theta}{dt^2} = \frac{hmg}{2I}$$... if you want. Second-order non-homogenious DE,
Last edited: Jan 17, 2012 | 2017-11-25 04:36:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8978960514068604, "perplexity": 404.68328388468916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934809392.94/warc/CC-MAIN-20171125032456-20171125052456-00127.warc.gz"} |
https://no.overleaf.com/latex/templates/lecture-note-template/dwyrjrnthdcz | # Lecture Note Template
Author
Naijia Fan
AbstractThis template can be useful when creating lecture notes. Clear theorem style and useful equations and tikz example. Easy to add chapters. | 2020-04-10 03:38:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4904923141002655, "perplexity": 8296.016467111373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371883359.91/warc/CC-MAIN-20200410012405-20200410042905-00182.warc.gz"} |
https://brilliant.org/problems/getting-closer-and-closer/ | # Getting Closer And Closer
Algebra Level 1
True or False?
$$\quad$$ A graph of a function can cross an asymptote.
× | 2017-05-29 06:02:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8913628458976746, "perplexity": 7595.9439214102795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612018.97/warc/CC-MAIN-20170529053338-20170529073338-00478.warc.gz"} |
https://www.studyadda.com/question-bank/mensuration_q4/4591/362432 | • # question_answer The bricks of dimension $\mathbf{15cm}\times \mathbf{25cm}\times \mathbf{7cm}$ required to build a wall $\left( \mathbf{3}.\mathbf{5}\times \mathbf{10}\times \mathbf{12} \right){{\mathbf{m}}^{\mathbf{3}}}$are A) 160000B) 170000 C) 180000 D) 200000
(a): no. of bricks = $\frac{\left( 3.5\times 100 \right)\times \left( 10\times 100 \right)\times \left( 12\times 1400 \right)}{15\times 25\times 7}$ (All dimensions converted into cm) $=\frac{3.5\times 10\times 12}{15\times 25\times 7}=1000000$ $=\frac{60}{15\times 25}\times 1000000=160000.$ | 2020-07-07 00:22:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33953601121902466, "perplexity": 9538.25805820049}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890566.2/warc/CC-MAIN-20200706222442-20200707012442-00016.warc.gz"} |
http://mathoverflow.net/feeds/question/36974 | Reference for model structure on CosimplicialAbelianGroups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:29:02Z http://mathoverflow.net/feeds/question/36974 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36974/reference-for-model-structure-on-cosimplicialabeliangroups Reference for model structure on CosimplicialAbelianGroups Urs Schreiber 2010-08-28T15:44:54Z 2010-08-28T15:46:26Z <p>There is a standard (simplicial) model category structure on the category \$Ab^\Delta \simeq Ch^\bullet_+(Ab)\$ of <em>co</em>-simplicial abelian groups, whose fibrations are the <em>degreewise surjections</em> (and weak equivalences are the usual quasi-isos).</p> <p>My question is about proper attribution: where does this first appear in citable form?. What is the canonical reference you would cite?</p> | 2013-05-21 07:29:05 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9101923108100891, "perplexity": 8768.021606119477}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368699776315/warc/CC-MAIN-20130516102256-00033-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://search.r-project.org/CRAN/refmans/brms/html/Wiener.html | Wiener {brms} R Documentation
## The Wiener Diffusion Model Distribution
### Description
Density function and random generation for the Wiener diffusion model distribution with boundary separation alpha, non-decision time tau, bias beta and drift rate delta.
### Usage
dwiener(
x,
alpha,
tau,
beta,
delta,
resp = 1,
log = FALSE,
backend = getOption("wiener_backend", "Rwiener")
)
rwiener(
n,
alpha,
tau,
beta,
delta,
types = c("q", "resp"),
backend = getOption("wiener_backend", "Rwiener")
)
### Arguments
x Vector of quantiles. alpha Boundary separation parameter. tau Non-decision time parameter. beta Bias parameter. delta Drift rate parameter. resp Response: "upper" or "lower". If no character vector, it is coerced to logical where TRUE indicates "upper" and FALSE indicates "lower". log Logical; If TRUE, values are returned on the log scale. backend Name of the package to use as backend for the computations. Either "Rwiener" (the default) or "rtdists". Can be set globally for the current R session via the "wiener_backend" option (see options). n Number of draws to sample from the distribution. types Which types of responses to return? By default, return both the response times "q" and the dichotomous responses "resp". If either "q" or "resp", return only one of the two types.
### Details
These are wrappers around functions of the RWiener or rtdists package (depending on the chosen backend). See vignette("brms_families") for details on the parameterization.
wienerdist, Diffusion | 2022-07-02 15:08:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7325528860092163, "perplexity": 12403.28392251681}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104141372.60/warc/CC-MAIN-20220702131941-20220702161941-00161.warc.gz"} |
https://raysect.github.io/documentation/api_reference/primitives/meshes.html | # 2.2. Meshes¶
class raysect.primitive.mesh.mesh.Mesh
This primitive defines a polyhedral surface with triangular faces.
To define a new mesh, a list of vertices and triangles must be supplied. A set of vertex normals, used for smoothing calculations may also be provided.
The mesh vertices are supplied as an Nx3 list/array of floating point values. For each Vertex, x, y and z coordinates must be supplied. e.g.
vertices = [[0.0, 0.0, 1.0], [1.0, 0.0, 0.0], ...]
Vertex normals are similarly defined. Note that vertex normals must be correctly normalised.
The triangle array is either Mx3 or Mx6 - Mx3 if only vertices are defined or Mx6 if both vertices and vertex normals are defined. Triangles are defined by indexing into the vertex and vertex normal arrays. i.e:
triangles = [[v1, v2, v3, n1, n2, n3], ...]
where v1, v2, v3 are the vertex array indices specifying the triangle’s vertices and n1, n2, n3 are the normal array indices specifying the triangle’s surface normals at each vertex location. Where normals are not defined, n1, n2 and n3 are omitted.
The mesh may be an open surface (which does not enclose a volume) or a closed surface (which defines a volume). The nature of the mesh must be specified using the closed argument. If closed is True (default) then the mesh must be watertight and the face normals must be facing so they point out of the volume. If the mesh is open then closed must be set to False. Incorrectly setting the closed argument may result in undefined behaviour, depending on the application of the ray-tracer.
If vertex normals are defined for some or all of the triangles of the mesh then normal interpolation may be enabled for the mesh. For optical models this will result in a (suitably defined) mesh appearing smooth rather than faceted. If the triangles do not have vertex normals defined, the smoothing argument is ignored.
An alternate option for creating a new mesh is to create an instance of an existing mesh. An instance is a “clone” of the original mesh. Mesh instances hold references to the internal data of the target mesh, they are therefore very memory efficient (particularly for detailed meshes) compared to creating a new mesh from scratch. A new instance of a mesh can be created using the instance() method.
If a mesh contains degenerate triangles (common for meshes generated from CAD models), enable tolerant mode to automatically remove them during mesh initialisation. A degenerate triangle is one where two or more vertices are coincident or all the vertices lie on the same line. Degenerate triangles will produce rendering error if encountered even though they are “infinitesimally” thin. A ray can still intersect them if they perfectly align as the triangle edges are treated as part of the triangle surface).
The kdtree_* arguments are tuning parameters for the kd-tree construction. For more information see the documentation of KDTree3D. The default values should result in efficient construction of the mesh’s internal kd-tree. Generally there is no need to modify these parameters unless the memory used by the kd-tree must be controlled. This may occur if very large meshes are used.
Parameters: vertices (object) – An N x 3 list of vertices. triangles (object) – An M x 3 or N x 6 list of vertex/normal indices defining the mesh triangles. normals (object) – An K x 3 list of vertex normals or None (default=None). smoothing (bool) – True to enable normal interpolation (default=True). closed (bool) – True is the mesh defines a closed volume (default=True). tolerant (bool) – Mesh will automatically correct meshes with degenerate triangles if set to True (default=True). kdtree_max_depth (int) – The maximum tree depth (automatic if set to 0, default=0). kdtree_min_items (int) – The item count threshold for forcing creation of a new leaf node (default=1). kdtree_hit_cost (double) – The relative computational cost of item hit evaluations vs kd-tree traversal (default=20.0). kdtree_empty_bonus (double) – The bonus applied to node splits that generate empty leaves (default=0.2). parent (Node) – Attaches the mesh to the specified scene-graph node (default=None). transform (AffineMatrix3D) – The co-ordinate transform between the mesh and its parent (default=unity matrix). material (Material) – The surface/volume material (default=Material() instance). name (str) – A human friendly name to identity the mesh in the scene-graph (default=””).
bounding_box()
Returns a world space bounding box that encloses the mesh.
The box is padded by a small margin to reduce the risk of numerical accuracy problems between the mesh and box representations following coordinate transforms.
Returns: A BoundingBox3D object.
contains()
Identifies if the point lies in the volume defined by the mesh.
If a mesh is open, this method will always return False.
This method will fail if the face normals of the mesh triangles are not oriented to be pointing out of the volume surface.
Parameters: p – The point to test. True if the point lies in the volume, False otherwise.
from_file()
Instances a new Mesh using data from a file object or filename.
The mesh must be stored in a RaySect Mesh (RSM) format file. RSM files are created with the Mesh save() method.
Parameters: file (object) – File object or string path. parent (Node) – Attaches the mesh to the specified scene-graph node. transform (AffineMatrix3D) – The co-ordinate transform between the mesh and its parent. material (Material) – The surface/volume material. name (str) – A human friendly name to identity the mesh in the scene-graph.
hit()
Returns the first intersection with the mesh surface.
If an intersection occurs this method will return an Intersection object. The Intersection object will contain the details of the ray-surface intersection, such as the surface normal and intersection point.
If no intersection occurs None is returned.
Parameters: ray – A world-space ray. An Intersection or None.
load()
Loads the mesh specified by a file object or filename.
The mesh must be stored in a RaySect Mesh (RSM) format file. RSM files are created with the Mesh save() method.
Parameters: file – File object or string path.
next_intersection()
Returns the next intersection of the ray with the mesh along the ray path.
This method may only be called following a call to hit(). If the ray has further intersections with the mesh, these may be obtained by repeatedly calling the next_intersection() method. Each call to next_intersection() will return the next ray-mesh intersection along the ray’s path. If no further intersections are found or intersections lie outside the ray parameters then next_intersection() will return None.
Returns: An Intersection or None.
save()
Saves the mesh to the specified file object or filename.
The mesh in written in RaySect Mesh (RSM) format. The RSM format contains the mesh geometry and the mesh acceleration structures.
Parameters: file – File object or string path.
raysect.primitive.mesh.obj.import_obj(cls, filename, scaling=1.0, **kwargs)
Create a mesh instance from a Wavefront OBJ mesh file (.obj).
Some engineering meshes are exported in different units (mm for example) whereas Raysect units are in m. Applying a scale factor of 0.001 would convert the mesh into m for use in Raysect.
Parameters: filename (str) – Mesh file path. scaling (double) – Scale the mesh by this factor (default=1.0). **kwargs – Accepts optional keyword arguments from the Mesh class. Mesh
raysect.primitive.mesh.stl.import_stl(cls, filename, scaling=1.0, mode=0, **kwargs)
Create a mesh instance from a STereoLithography (STL) mesh file (.stl).
Some engineering meshes are exported in different units (mm for example) whereas Raysect units are in m. Applying a scale factor of 0.001 would convert the mesh into m for use in Raysect.
Parameters: filename (str) – Mesh file path. scaling (double) – Scale the mesh by this factor (default=1.0). **kwargs – Accepts optional keyword arguments from the Mesh class. Mesh | 2018-02-24 12:03:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1830647736787796, "perplexity": 2630.842755178952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815560.92/warc/CC-MAIN-20180224112708-20180224132708-00488.warc.gz"} |
https://en.wikipedia.org/wiki/Liquid_fuel | # Liquid fuel
A flaming cocktail works by burning ethanol (grain alcohol), a type of liquid fuel also found in all alcoholic drinks
Liquid fuels are combustible or energy-generating molecules that can be harnessed to create mechanical energy, usually producing kinetic energy; they also must take the shape of their container. It is the fumes of liquid fuels that are flammable instead of the fluid. Most liquid fuels in widespread use are derived from fossil fuels; however, there are several types, such as hydrogen fuel (for automotive uses), ethanol, and biodiesel, which are also categorized as a liquid fuel. Many liquid fuels play a primary role in transportation and the economy.
Liquid fuels are contrasted with solid fuels and gaseous fuels.
## General properties
Some common properties of liquid fuels are that they are easy to transport, and can be handled with relative ease. Physical properties of liquid fuels vary by temperature, though not as greatly as for gaseous fuels. Some of these properties are: flash point, the lowest temperature at which a flammable concentration of vapor is produced; fire point, the temperature at which sustained burning of vapor will occur; cloud point for diesel fuels, the temperature at which dissolved waxy compounds begin to coalesce, and pour point, the temperature below which the fuel is too thick to pour freely. These properties affect the safety and handling of the fuel.
## Petroleum fuels
A worker collecting fuel samples for testing aboard a ship
Most liquid fuels used currently are produced from petroleum. The most notable of these is gasoline. Scientists generally accept that petroleum formed from the fossilized remains of dead plants and animals by exposure to heat and pressure in the Earth's crust.
### Gasoline
Gasoline is the most widely used liquid fuel. Gasoline, as it is known in United States and Canada, or petrol virtually everywhere else, is made of hydrocarbon molecules (compounds that contain hydrogen and carbon only) forming aliphatic compounds, or chains of carbons with hydrogen atoms attached. However, many aromatic compounds (carbon chains forming rings) such as benzene are found naturally in gasoline and cause the health risks associated with prolonged exposure to the fuel.
Production of gasoline is achieved by distillation of crude oil. The desirable liquid is separated from the crude oil in refineries. Crude oil is extracted from the ground in several processes, the most commonly seen may be beam pumps. To create gasoline, petroleum must first be removed from crude oil.
Liquid gasoline itself is not actually burned, but its fumes ignite, causing the remaining liquid to evaporate and then burn. Gasoline is extremely volatile and easily combusts, making any leakage potentially extremely dangerous. Gasoline sold in most countries carries a published octane rating. The octane number is an empirical measure of the resistance of gasoline to combusting prematurely, known as knocking. The higher the octane rating, the more resistant the fuel is to autoignition under high pressures, which allows for a higher compression ratio. Engines with a higher compression ratio, commonly used in race cars and high-performance regular-production automobiles, can produce more power; however, such engines require a higher octane fuel. Increasing the octane rating has, in the past, been achieved by adding 'anti-knock' additives such as lead-tetra-ethyl. Because of the environmental impact of lead additives, the octane rating is increased today by refining out the impurities that cause knocking.
### Diesel
Conventional diesel is similar to gasoline in that it is a mixture of aliphatic hydrocarbons extracted from petroleum. Diesel may cost more or less than gasoline, but generally costs less to produce because the extraction processes used are simpler. Some countries (particularly Canada, India and Italy) also have lower tax rates on diesel fuels.
After distillation, the diesel fraction is normally processed to reduce the amount of sulfur in the fuel. Sulfur causes corrosion in vehicles, acid rain and higher emissions of soot from the tail pipe (exhaust pipe). Historically, in Europe lower sulfur levels than in the United States were legally required. However, recent US legislation reduced the maximum sulfur content of diesel from 3,000 ppm to 500 ppm in 2007, and 15 ppm by 2010. Similar changes are also underway in Canada, Australia, New Zealand and several Asian countries. See also Ultra-low-sulfur diesel.
A diesel engine is a type of internal combustion engine which ignites fuel by injecting it into a combustion chamber previously compressed with air (which in turn raises the temperature) as opposed to using an outside ignition source, such as a spark plug.
### Kerosene
Kerosene is used in kerosene lamps and as a fuel for cooking, heating, and small engines. It displaced whale oil for lighting use. Jet fuel for jet engines is made in several grades (Avtur, Jet A, Jet A-1, Jet B, JP-4, JP-5, JP-7 or JP-8) that are kerosene-type mixtures. One form of the fuel known as RP-1 is burned with liquid oxygen as rocket fuel. These fuel grade kerosenes meet specifications for smoke points and freeze points.
In the mid-20th century, kerosene or "TVO" (Tractor Vaporising Oil) was used as a cheap fuel for tractors. The engine would start on gasoline, then switch over to kerosene once the engine warmed up. A "heat valve" on the manifold would route the exhaust gases around the intake pipe, heating the kerosene to the point where it can be ignited by an electric spark.
Kerosene is sometimes used as an additive in diesel fuel to prevent gelling or waxing in cold temperatures. However, this is not advisable in some recent vehicle diesel engines, as doing so may interfere with the engine's emissions regulation equipment.
### Liquefied petroleum gas (LPG)
LP gas is a mixture of propane and butane, both of which are easily compressible gases under standard atmospheric conditions. It offers many of the advantages of compressed natural gas (CNG), but does not burn as cleanly, is denser than air and is much more easily compressed. Commonly used for cooking and space heating, LP gas and compressed propane are seeing increased use in motorized vehicles; propane is the third most commonly used motor fuel globally.
### Carbon dioxide formation from petroleum fuels.
Petroleum fuels, when burnt, release carbon dioxide that is harmful to world climate. The amount of carbon dioxide released when one liter of fuel is combusted can be estimated:[1] As a good approximation the chemical formula of e.g. diesel is C
n
H
2n
. Note that diesel is a mixture of different molecules. As carbon has a molar mass of 12 g/mol and hydrogen (atomic!) has a molar mass of about 1 g/mol, so the fraction by weight of carbon in diesel is roughly 12/14. The reaction of diesel combustion is given by:
2C
n
H
2n
+ 3nO
2
⇌ 2nCO
2
+ 2nH
2
O
Carbon dioxide has a molar mass of 44g/mol as it consists of 2 atoms of oxygen (16 g/mol) and 1 atom of carbon (12 g/mol). So 12 g of carbon yield 44 g of Carbon dioxide. Diesel has a density of 0.838 kg per liter. Putting everything together the mass of carbon dioxide that is produced by burning 1 liter of diesel can be calculated as:
${\displaystyle 0.838kg/L\cdot {\frac {12}{14}}\cdot {\frac {44}{12}}=2.63kg/L}$
The number of 2.63 kg of carbon dioxide from 1 liter of Diesel is close to the values found in the literature.
For gasoline, with a density of 0.75 kg/L and a ratio of carbon to hydrogen atoms of about 6 to 14, the estimated value of carbon emission if 1 liter of gasoline is burnt gives:
${\displaystyle 0.75kg/L\cdot {{\frac {6\cdot 12}{6\cdot 12+14}}\cdot 1}\cdot {\frac {44}{12}}=2.3kg/L}$
## Non-petroleum fossil fuels
When petroleum is not easily available, chemical processes such as the Fischer–Tropsch process can be used to produce liquid fuels from coal or natural gas. Synthetic fuels from coal were strategically important during World War II for the German military. Today synthetic fuels produced from natural gas are manufactured, to take advantage of the higher value of liquid fuels in transportation.
## Natural gas
Natural gas, composed chiefly of methane, can be compressed to a liquid and used as a substitute for other traditional liquid fuels. Its combustion is very clean compared to other hydrocarbon fuels, but the fuel's low boiling point requires the fuel to be kept at high pressures to keep it in the liquid state. Though it has a much lower flash point than fuels such as gasoline, it is in many ways safer due to its higher autoignition temperature and its low density, which causes it to dissipate when released in air.
## Hydrogen
Liquefied hydrogen is the liquid state of the element hydrogen. It is a common liquid rocket fuel for rocket applications and can be used as a fuel in an internal combustion engine or fuel cell. Various concept hydrogen vehicles have been lower volumetric energy, the hydrogen volumes needed for combustion are large. Hydrogen was liquefied for the first time by James Dewar in 1898.
## Ammonia
Ammonia (NH3) has been used as a fuel before at times when gasoline is unavailable (e.g. for buses in Belgium during WWII). It has a volumetric energy density of 17 Megajoules per liter (compared to 10 for hydrogen, 18 for methanol, 21 for dimethyl ether and 34 for gasoline). It must be compressed or cooled to be a liquid fuel, although it does not require cryogenic cooling as hydrogen does to be liquefied.[4]
## References
1. ^ Hilgers, Michael (2020). The Diesel Engine, in series: commercial vehicle technology. Berlin/Heidelberg/New York: Springer. ISBN 978-3-662-60856-2.
2. ^ "AccessScience | Encyclopedia Article | Alcohol fuel". Accessscience.com. Retrieved 2008-11-06.
3. ^ E85
4. ^ "Ammonia FAQs". Retrieved 9 August 2012. | 2023-01-31 20:36:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7665424942970276, "perplexity": 3152.866023160628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499890.39/warc/CC-MAIN-20230131190543-20230131220543-00257.warc.gz"} |
https://techwhiff.com/learn/give-a-description-of-what-you-believe-to-be-the/81741 | # Give a description of what you believe to be the consequences of a healthcare organization not...
###### Question:
Give a description of what you believe to be the consequences of a healthcare organization not involving nurses in each stage of the SDLC when purchasing and implementing a new health information technology system. Provide specific examples of potential issues at each stage of the SDLC and explain how the inclusion of nurses may help address these issues.
explain whether you had any input in the selection and planning of new health information technology systems in your nursing practice or healthcare organization and explain potential impacts of being included or not in the decision-making process. Provide examples
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##### How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola y^2=36+4x^2?
How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola y^2=36+4x^2?... | 2023-03-28 22:27:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19395631551742554, "perplexity": 2922.595090006662}, "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-2023-14/segments/1679296948871.42/warc/CC-MAIN-20230328201715-20230328231715-00359.warc.gz"} |
http://www.popflock.com/learn?s=Standardized_coefficient | Standardized Coefficient
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Standardized Coefficient
In statistics, standardized (regression) coefficients, also called beta coefficients or beta weights, are the estimates resulting from a regression analysis where the underlying data have been standardized so that the variances of dependent and independent variables are equal to 1.[1] Therefore, standardized coefficients are unitless and refer to how many standard deviations a dependent variable will change, per standard deviation increase in the predictor variable.
## Usage
Standardization of the coefficient is usually done to answer the question of which of the independent variables have a greater effect on the dependent variable in a multiple regression analysis, when the variables are measured in different units of measurement (for example, income measured in dollars and family size measured in number of individuals). It may also be considered a general measure of effect size, quantifying the "magnitude" of the effect of one variable on another. For simple linear regression with orthogonal predictors, the standardized regression coefficient equals the correlation between the independent and dependent variables.
## Implementation
A regression carried out on original (unstandardized) variables produces unstandardized coefficients. A regression carried out on standardized variables produces standardized coefficients. Values for standardized and unstandardized coefficients can also be re-scaled to one another subsequent to either type of analysis. Suppose that ${\displaystyle \beta }$ is the regression coefficient resulting from a linear regression (predicting ${\displaystyle y}$ by ${\displaystyle x}$). The standardized coefficient the simply results as ${\displaystyle \beta ^{\ast }={\frac {s_{x}}{s_{y}}}\beta }$, where ${\displaystyle s_{x}}$ and ${\displaystyle s_{y}}$ are the (estimated) standard deviations of ${\displaystyle x}$ and ${\displaystyle y}$, respectively.[1]
Sometimes, standardization is done only with respect to the standard deviation of the regressor (the independent variable ${\displaystyle x}$).[2][3]
Standardized coefficients' advocates note that the coefficients are independent of the involved variables' units of measurement (i.e., standardized coefficients are unitless), which makes comparisons easy.[3]
Critics voice concerns that such a standardization can be very misleading.[2][4] Due to the re-scaling based on sample standard deviations, any effect apparent in the standardized coefficient may be due to confounding with the particularities (especially: variability) of the involved data sample(s). Also, the interpretation or meaning of a "one standard deviation change" in the regressor ${\displaystyle x}$ may vary markedly between non-normal distributions (e.g., when skewed, asymmetric or multimodal).
## Terminology
Some statistical software packages like PSPP, SPSS and SYSTAT label the standardized regression coefficients as "Beta" while the unstandardized coefficients are labeled "B". Others, like DAP/SAS label them "Standardized Coefficient". Sometimes the unstandardized variables are also labeled as "b".
## References
1. ^ a b Menard, S. (2004), "Standardized regression coefficients", in Lewis-Beck, M.S.; Bryman, A.; Liao, T.F. (eds.), The Sage Encyclopedia of Sociel Science Research Methods, Thousand Oaks, CA, USA: Sage Publications, pp. 1069-1070, doi:10.4135/9781412950589.n959
2. ^ a b Greenland, S.; Schlesselman, J. J.; Criqui, M. H. (1986). "The fallacy of employing standardized regression coefficients and correlations as measures of effect". American Journal of Epidemiology. 123 (2): 203-208. doi:10.1093/oxfordjournals.aje.a114229.
3. ^ a b Newman, T. B.; Browner, W. S. (1991). "In defense of standardized regression coefficients". Epidemiology. 2 (5): 383-386. doi:10.1097/00001648-199109000-00014.
4. ^ Greenland, S.; Maclure, M.; Schlesselman, J. J.; Poole, C.; Morgenstern, H. (1991). "Standardized regression coefficients: A further critique and review of some alternatives". Epidemiology. 2 (5): 387-392. doi:10.1097/00001648-199109000-00016. | 2020-11-29 23:50:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7383459806442261, "perplexity": 3397.5923248773183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141203418.47/warc/CC-MAIN-20201129214615-20201130004615-00380.warc.gz"} |
http://mathoverflow.net/questions/29302/reasons-to-believe-vopenkas-principle-huge-cardinals-are-consistent?sort=newest | # Reasons to believe Vopenka's principle/huge cardinals are consistent
There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large cardinal axioms, like the existence of an infinite number of Grothendieck universes.
Are there any such heuristic arguments for the existence of Vopenka cardinals or huge cardinals? I'd very much like to believe them, mainly because they simplify a great deal of trouble one has to go through when working with accessible categories and localization (every localizer is accessible on a presheaf category, for instance).
For Vopenka's principle, the category-theoretic definition is that every full complete (cocomplete) subcategory of a locally presentable category is reflective (coreflective). This seems rather unintuitive to me (and I don't even understand the model-theoretic definition of Vopenka's principle).
What reason is there to believe that ZFC+VP (or ZFC+HC, which implies the consistency of VP) is consistent? Obviously, I am willing to accept heuristic or informal arguments (since a formal proof is impossible).
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Harry, what are the informal heuristic arguments for the consistency of ZFC? (A reference would be great.) – danseetea Jun 24 '10 at 22:55
You'll notice that I never said I've actually read any of them. =D! – Harry Gindi Jun 25 '10 at 0:44
Most of the arguments previously presented take a set-theoretic/logical point of view and apply to large cardinal axioms in general. There's a lot of good stuff there, but I think there are additional things to be said about Vopěnka's principle specifically from a category-theoretic point of view.
One formulation of Vopěnka's principle (which is the one that I'm used to calling "the" category-theoretic definition, and the one used as the definition in Adamek&Rosicky's book, although there are many category-theoretic statements equivalent to VP) is that there does not exist a large (= proper-class-sized) full discrete (= having no nonidentity morphims between its objects) subcategory of any locally presentable category. I think there is a good argument to be made for the naturalness of this from a category-theoretic perspective.
To explain why, let me back up a bit. To a category theorist of a certain philosophical bent, one thing that category theory teaches us is to avoid talking about equalities between objects of a category, rather than isomorphism. For instance, in doing group theory, we never talk about when two groups are equal, only when they are isomorphic. Likewise in doing topology, we never talk about when two spaces are equal, only when they are homeomorphic. Once you get used to this, it starts to feel like an accident that it even makes sense to ask whether two groups are equal, rather than merely isomorphic. And in fact, it is an accident, or at least dependent on the particular choice of axioms for a set-theoretic foundation; one can give other axiomatizations of set theory, provably equivalent to ZFC, in which it doesn't make sense to ask whether two sets are equal, only whether two elements of a given ambient set are equal. These are sometimes called "categorial" set theories, since the first example was Lawvere's ETCS which axiomatizes the category of sets, but I prefer to call them structural set theories, since there are other versions, like SEAR, which don't require any category theory.
Now there do exist categories in which it does make sense to talk about "equality" of objects. For instance, any set X can be regarded as a discrete category $X_d$, whose objects are the elements of X and in which the only morphisms are identities. Moreover, a category is equivalent to one of the form $X_d$, for some set X, iff it is both a groupoid and a preorder, i.e. every morphism is invertible and any parallel pair of morphisms are equal. I call such a category a "discrete category," although some people use that only for the stricter notion of a category isomorphic to some $X_d$. So it becomes tempting to think that one might instead consider "category" to be a fundamental notion, and define "set" to mean a discrete category.
Unfortunately, however, what I wrote in the previous paragraph is false: a category is equivalent to one of the form $X_d$, for some set X, iff it is a groupoid and a preorder and small. We can just as well construct a category $X_d$ when X is a proper class, and it will of course still be discrete. In fact, just as a set is the same thing as a small discrete category, a proper class is the same thing as a large discrete category. However, this feels kind of bizarre, because the large categories that arise in practice are almost never of the sort that admit a meaningful notion of "equality" between their objects, and in particular they are almost never discrete. Consider the categories of groups, or rings, or topological spaces, or sets for that matter. Outside of set theory, proper classes usually only arise as the class of objects of some large category, which is almost never discrete. The world would make much more sense, from a category-theoretic point of view, if there were no such things as proper classes, a.k.a. discrete large categories --- then we could define "set" to mean "discrete category" and life would be beautiful.
Unfortunately, we can't have large categories without having large discrete categories, at least not without restricting the rest of mathematics fairly severly. This is obviously true if we found mathematics on ZFC or NBG or some other traditional "membership-based" or "material" set theory, since there we need a proper class of objects before we can even define a large category. But it's also true if we use a structural set theory, since there are a few naturally and structurally defined large categories that are discrete, such as the category of well-orderings and all isomorphisms between them (the core of the full subcategory of Poset on the well-orderings).
Thus Vopěnka's principle, as I stated it above, is a weakened version of the thesis that large discrete categories don't exist: it says that at least they can't exist as full subcategories of locally presentable categories. Since locally presentable categories are otherwise very well-behaved, this is at least reasonable to hope for. In fact, from this perspective, if Vopěnka's principle turns out to be inconsistent with ZFC, then maybe it is ZFC that is at fault! (-:
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It should be noted that Petr Vopěnka himself did not believe in the principle! Here is the story, taken from Adámek and Rosický Locally Presentable and Accessible Categories (p. 278-279).
The story of Vopěnka's principle (as related to the authors by Petr Vopěnka) is that of a practical joke which misfired: In the 1960's P. Vopěnka was repelled by the multitude of large cardinals which emerged in set theory. When he constructed, in collaboration with Z. Hedrlín and A. Pultr, a rigid graph on every set (see Lemma 2.64), he came to the conclusion that, with some more effort, a large rigid class of graphs must surely be also constructible. He then decided to tease set-theorists: he introduced a new principle (known today as Vopěnka's principle), and proved some consequences concerning large cardinals. He hoped that some set-theorists would continue this line of research (which they did) until somebody showed that the principle was nonsense. However the latter never materialized — after a number of unsuccessful attempts at constructing a large rigid class of graphs, Vopěnka's principle received its name from Vopěnka's disciples. One of them, T. J. Jech, made Vopěnka's principle widely known.
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William Reinhardt gave heuristic reasons for some very large cardinal axioms in a paper in the proceedings of the 1967 UCLA set theory meeting. I don't know whether he considered cardinals as large as the ones you want, but some of the ideas there might be useful for you. (Disclaimer: I'm not at all convinced that such large cardinals exist. My belief in their consistency is based on the fact that very smart people, like Jack Silver, have looked seriously for inconsistencies and haven't found any.)
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Do you happen to know where to find this paper (or perhaps the title)? – Harry Gindi Jun 24 '10 at 2:09
Here's the header from the MathSciNet review: MR0401475 (53 #5302) Reinhardt, W. N. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), pp. 189--205. Amer. Math. Soc., Providence, R. I., 1974. – Andreas Blass Jun 24 '10 at 2:17
Andreas, I worry about the very-smart-people argument. After all, before Wiles, very smart people had looked seriously and been unable to refute FLT, but $\neg$FLT turned out to be inconsistent anyway... – Joel David Hamkins Jun 24 '10 at 4:00
Joel, I agree that the smart-people argument is not very strong, and I should have phrased my answer to make that clear. I didn't mean "I believe that very large cardinals are consistent, and the reason is ..."; rather I meant "To the extent that I believe that very large cardinals are consistent, the only reason I have is ...." In the case of small large cardinals (inaccessible, or even indescribable of various levels), I think the reflection idea gives some additional plausibility to the axioms (though I don't think it makes them "clearly true"); I don't see that for measurables and up. – Andreas Blass Jun 24 '10 at 6:43
Without implying anything about my personal beliefs, I'm surprised no one has yet mentioned Penelope Maddy's pair of papers called "Believing the Axioms" (I and II), which I enjoyed as a discussion of reasons to believe (or not believe) in various axioms beyond ZFC. – Mike Shulman Jun 25 '10 at 5:22
Because of Goedel's Incompleteness Theorems, we know that we cannot describe a complete axiomatization of mathematics. Any proposed axiomatization $T$, if consistent, will be unable to prove the principle Con(T) asserting that $T$ itself is consistent, although we have reason to desire this principle once we have committed ourselves to $T$. Adding the consistency principle Con(T) simply puts off the question to Con(T+Con(T)), and so on, in a process that proceeds into the transfinite.
Thus, we come to know that there should be a transfinite tower of theories above our favorite theories, transcending them in consistency strength. The incompleteness theorems imply that there is a tower of theories above PA, above ZFC, each level transcending the consistency strength of the prior levels.
How fortunate and wonderful that we have also independently come upon such a tower of theories: the large cardinal hierarchy. Numerous large cardinal concepts arose very early in set theory, from the time of Cantor, before Goedel's theorems and before the notion of consistency strength was formulated. These large cardinal concepts arose from natural set-theoretic questions in infinite combinatorics: Can there be a regular limit cardinal? Can there be a countably-complete measure measuring all subsets of a set? Does every $\kappa$-complete filter on a set extend to a $\kappa$-complete ultrafilter? And so on.
Eventually, it was realized that these large cardinal notions separate into a very tall hierarchy, with the property that from the larger cardinals, one can prove the consistency of the smaller cardinals. For example, if $\kappa$ is the least Mahlo cardinal, then the universe $H_\kappa$ is a model of ZFC + there is a stationary proper class of inaccessible cardinals + there are no Mahlo cardinals. If $\delta$ is the least measurable cardinal, then $H_\delta$ satisfies ZFC + there are a proper class of Ramsey cardinals, but no measurable cardinal.
Thus, the large cardinal hierarchy provides exactly the tower of theories, whose levels transcend consistency strength, that we knew should exist. And it does so in a way that is mathematically robust and interesting, with its foundations arising, not in some syntactic diagonalization, but in mathematically fulfilling and meaningful questions in infinite combinatorics.
The case of Vopenka's principle is just like this. VP is a large cardinal axiom at the higher end of the large cardinal hierarchy, implying the consistency of the existence of supercompact cardinals, say, which are far stronger than strong cardinals, which imply entire towers of measurable cardinals, which imply numerous Ramsey cardinals and so on down the line.
Illustrating the essential large cardinal nature, the VP axiom is elegantly stated: for every proper class sequence $\langle M_\alpha | \alpha\in\text{ORD}\rangle$ of first order structures, there is a pair of ordinals $\alpha\lt\beta$ for which $M_\alpha$ embeds elementarily into $M_\beta$. (It is equivalently stated in terms just of graphs, if you like.) It's simple and clear---beautiful! And the consequences are far-reaching and often profound, as you have observed in category theory, in the way that VP implies that the set-theoretic universe is regular and organized.
These are the reasons you should be attracted to Vopenka's principle. It is an elegant combinatorial principle, with far-reaching consequences that interest you, which has not yet been refuted.
In contrast, I find the philosophical heuristics that seek to justify the large cardinal axioms, on the grounds of reflection or some other means, to be so much hot air ultimately unsatisfying. These arguments are not mathematically sound, and cannot be made to be, by the Incompleteness Theorems. Philosophically, they seem much more like rationalizations after the fact. For example, even at the much lower (and therefore seemingly easier-to-justify) level of inaccessible cardinals, one sometimes hears an appeal to reflection type views, that since we have no definable unbounded map from a set into the ordinals, that there should be a level $V_\kappa$ of the universe also with this feature, and that such a level would be inaccessible cardinal. Of course, the conclusion outstrips the argument, with the conclusion seeming to justify at most $V_\kappa\models$ZFC, which is a weaker notion, and the meta-reflection principle appealed to amounts anyway to a large cardinal principle of its own.
Ultimately, we must recognize the uncertain nature of all our mathematical enterprise. As our hypotheses rise higher in the large cardinal hierarchy, we must become less sure of consistency---perhaps they will be shown to be inconsistent. This issue arises even at the lowest levels of our mathematical axiomatizations, for we may find at any time (as mentioned in a recent MO question) that even PA is inconsistent. As Woodin says, we all have in our minds the image of a railway line, lined by a sequence of telegraph poles, proceeding into infinity; but when the physicists tell us that the universe is finite, we realize that this picture is pure imagination. Perhaps it is simply inconsistent? So skepticism about consistency has nothing especially to do with the infinite.
Meanwhile, the large cardinal axioms are fascinating and have fascinating consequences. Let's seek out the boundary of consistency, with an attitude tempered by the realization that we may find inconsistency.
In summary, we cannot ever be sure that our axioms are consistent, and we know that above the mathematical theory about which we may be sure, there is a tall tower of theories whose levels transcend consistency. Among them are fascinating theories that are elegantly stated with far-reaching consequences, and which we have not yet refuted. So let's study them! Let's find the boundary between consistency and inconsistency!
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As a historical footnote, I'd like to mention that arguments for incompleteness, in the form "if an inaccessible $\kappa$ exists then its existence is not provable" (because $V_\kappa$ satisfies "there is no inaccessible") actually preceded Gödel's theorems. Kuratowski gave such an argument in 1925 and Zermelo in 1928, though neither was rigorous by today's standards. – John Stillwell Jun 24 '10 at 3:26
This is bound to be stupid, but when $M_\alpha$ is a graph, what does $M_\alpha \models \varphi(\vec{x})$ mean? Also, what is $\phi$, and what are its source and target (if that even makes any sense)? – Harry Gindi Jun 24 '10 at 5:19
Joel, your remark about graphs instead of arbitrary structures is nice for non-logicians. An alternative way to be nice to them is to replace "elementary embedding" with mere "embedding". (It makes no real difference, since you can Skolemize the structures.) But I don't see that both sorts of niceness can be done simultaneously. Does anyone know an equivalent formulation of Vopenka's principle that uses mere embeddings but for a "simple sort of structure (simpler than a graph plus all its Skolem functions)? – Andreas Blass Jun 24 '10 at 6:49
Harry, when $G$ is a graph, then $G\models\varphi$ simply refers to first-order satisfaction for assertions $\varphi$ in the language of graph theory, where you can quantify over vertices and use the edge-relation. You can express things like "G is triangle-free" and "G has total diameter 5" this way, but other assertions, such as "G is connected," are not first order expressible. Andreas, that is a very interesting point. – Joel David Hamkins Jun 24 '10 at 12:08
I think I shall delete those words, since I surely don't mean to offend anyone. I just meant to express my frustration with the justifications that are sometimes provided for large cardinals on the basis of reflection ideas, in those instances when to the contrary they are attempts to justify our confidence in the axioms. It is, of course, the reflection principles themselves, suitably formalized, that carry a large cardinal interest. – Joel David Hamkins Jun 25 '10 at 1:31
Here is a practical argument. Set theorists like to solve various problems and large cardinals help as either (rarely) they imply a positive answer or (more usually) their consistency implies the consistency of a positive answer. Under "positive" I mean an answer that is not a counterexample, an answer that does not give an object with strange properties. For example, determinacy at the n-th level of the projective hierarchy solves most problems on sets of that level positively (if blah are sets of that level then blah blah vs there is a sequence of blah sets for which this and this hold but this and this do not) and said determinacy is equiconsistent to the existence of n Woodin cardinals.
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The standard heuristic argument for large cardinal axioms (including huge cardinals) is the reflection principle. The intuitive idea is that $V$ is "absolutely infinite" and so cannot be defined as the collection which satisfies $\varphi$; there will always be some smaller $V_\alpha$ that already satisfies $\varphi$. See the paper Higher Order Reflection Principles by M. Victoria Marshall R. for more details.
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Does this heuristic fail for Reinhardt cardinals, which are known to be inconsistent with ZFC? – Harry Gindi Jun 24 '10 at 1:48
I think you're asking if every plausible-looking reflection principle can be trusted. The answer is no; there is a reflection principle corresponding to Reinhardt cardinals (this may even have motivated the definition of Reinhardt cardinals in the first place, but I'm unsure of my history), and in fact there are other reflection principles which are even more easily shown to be inconsistent. So why trust a heuristic that doesn't always work? There have been attempts to delimit the acceptable reflection principles; see people.fas.harvard.edu/~koellner/papers/ORP_final.pdf – Timothy Chow Jun 24 '10 at 2:18
The link people.fas.harvard.edu/~koellner/papers/ORP_final.pdf is no longer working (at least I get "access forbidden 403" there). Is there another place to download this pdf from? I'd be curious about the contents. Thanks – Desiato Oct 10 '14 at 1:09
– Timothy Chow Oct 10 '14 at 2:04 | 2016-02-07 13:14:57 | {"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": 0.8461100459098816, "perplexity": 458.04766337709503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701149377.17/warc/CC-MAIN-20160205193909-00079-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://socratic.org/questions/56337ce2581e2a79ddea6256 | # Question a6256
Oct 31, 2015
${\text{6.8 dm}}^{3}$
#### Explanation:
When dealing with gaseous products, mroe specifically with the volume occupied by a sample of gas, you need to have some information about the pressure and temperature at which the reaction takes place.
Usually, when no mention of pressure and temperature is made, you can assume that the reactiontakes place at STP - Standard Temperature and Pressure.
At STP conditions, which imply a pressure of $\text{100 kPa}$ and a temperature of ${0}^{\circ} \text{C}$, one mole of any ideal gas occupies exactly $\text{22.7 L}$ - this is known as the molar volume of a gas at STP.
Basically, if you know how many moles of gas a reaction produces at STP, you can use the molar volume of a gas to determine what volume would that sample occupy.
So, the balanced chemical equation for this reaction looks like this
${\text{C"_3"H"_text(8(g]) + 5"O"_text(2(g]) -> color(red)(3)"CO"_text(2(g]) + 4"H"_2"O}}_{\textrm{\left(l\right]}}$
The gaseous product in this case will be carbon dioxide, ${\text{CO}}_{2}$.
Notice that you have a $1 : \textcolor{red}{2}$ mole ratio between propane and carbon dioxide, which means that the reaction will produce $\textcolor{red}{3}$ times as many moles of ${\text{CO}}_{2}$ as you have moles of ${\text{C"_3"H}}_{8}$ that take part in the reaction.
So, if $0.1$ moles of propane react completely, you will get
0.1color(red)(cancel(color(black)("moles C"_3"H"_8))) * (color(red)(3)" moles CO"_2)/(1color(red)(cancel(color(black)("mole C"_3"H"_8)))) = "0.3 moles CO"_2
This means that the volume occupied by the carbon dioxide will be
0.3color(red)(cancel(color(black)("moles"))) * "22.7 L"/(1color(red)(cancel(color(black)("mole")))) = "6.81 L"
Now, since you have ${\text{1 L" = "1 dm}}^{3}$, the answer will be
$V = \textcolor{g r e e n}{{\text{6.8 dm}}^{3}}$
SIDE NOTE Most textbooks and onmline resources still use the old values for STP, which correspond to a pressure of 1 atm and a temperature of 0 degrees Celsius.
At these conditions, the molar volume of a gas is equal to 22.4 L. If you use that value, you will get
0.3color(red)(cancel(color(black)("moles"))) * "22.4 L"/(1color(red)(cancel(color(black)("mole")))) = "6.72 L"#
which rounds to
$V = {\text{6.7 dm}}^{3} \to$ option C | 2020-08-13 02:55:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 17, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7831358313560486, "perplexity": 953.1563017479772}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738950.61/warc/CC-MAIN-20200813014639-20200813044639-00534.warc.gz"} |
https://z.maynards.site/raw/n/18459013869377224982r.html | [:open map:] • A function $f : X \to Y$ between two topological spaces is called an open map exactly when: • $\forall U \in \mathcal T_X : f(X) \in \mathcal T_Y$ Referenced by: | 2022-09-28 16:47:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7271443009376526, "perplexity": 1412.2341311651282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335257.60/warc/CC-MAIN-20220928145118-20220928175118-00718.warc.gz"} |
https://physics.stackexchange.com/questions/250752/how-can-you-tell-if-the-work-done-by-a-force-is-negative/376906 | # How can you tell if the work done by a force is negative?
This is kind of confusing to me. I'm guessing that it's specific to the problem. Is the work done by friction always negative? Is the work done by gravity always negative? Spring as well?
It seems like work done by friction should be negative, but then there was a problem that said "Determine the increase in internal energy of the crate–incline system owing to friction." Basically, the answer was a positive work in friction. It's confusing to me.
• Work is positive when the object is moved in the same direction as the force, and negative when it's moved against the force. – Jold Apr 20 '16 at 1:54
• tutor4physics.com/positivenegativework.htm – Robert Harvey Apr 20 '16 at 2:14
• Say you are standing on a bus which is accelerating. Static friction of the floor on your feet does positive work on you giving you kinetic energy. – M. Enns Apr 20 '16 at 2:37
• Possible duplicates: physics.stackexchange.com/q/65813/2451 and links therein. – Qmechanic Dec 29 '17 at 19:36
Whenever a given force and displacement produced by that force has an angle >90 degrees,the work done is said to be negative. Since by the definition of work done,W= F.s = Fs cos(angle between f and s),thus whenever angle is >90, cos(angle) is negative and hence work done is negative.
Coming on your question Is the work done by friction always negative ,my answer is no.Consider a rolling tyre,the direction of friction on the tyre is same as direction of motion so here the work done by friction is positive.
Is the work done by gravity always negative The answer again is no,when a body is falling down, the direction of gravitational force and motion is same so work done is positive.
Is the work done by spring force is always negative The answer is again no,when you stretch/compress a spring the work done by spring force is negetive but when you release the stretched/compressed spring work done by spring force is positive.
All you have to do is to identify direction of force acting and the direction in which body is displaced.
Force is a vector, meaning magnitude and direction. Work done by a force is relative to the direction of a force is the scalar value obtained by performing the vector dot product of the force and the displacement (which is also a vector). If something isn't coming out to what you expect when you compute work, make sure you have the right magnitude and direction for everything. The direction of force is not always intuitive.
In the case of a displacement that does not have a uniform force along its length, you would have to use the calculus integral to compute the scalar work done by said force.
The reason work can be negative is because it's possible for something to move counter to one of the forces that is exerted on it (because another force overpowered it or it was already moving in such a way the force didn't get a chance to overpower it). In fact work can be negative, positive or $0$. If you have (2) people playing tug of war the losing team performed negative work while the winning team performed positive work along the axis of the rope. If those people didn't change their elevation relative to the center of earth's mass, the gravity exerted by the center of mass for performed $0$ work while they were playing, even if they temporarily were dug into the ground (having a net displacement of $0$ results in a net work of $0$ - even if you temporarily had some displacement).
You could probably look at work as "when push came to shove how effective was the force at displacing"?
Assuming a stationary medium relative to an external observer, if you're sliding (relative to the stationary medium/observer) along some +x axis then any friction you experience would be accelerating you in the opposite direction, then the force friction delivers would be along the -x axis. Friction would only act on something moving to accelerate it in the opposite direction its current velocity (which is a vector, because velocity is speed and direction), so friction, from its perspective, always has a negative displacement it works over, so the work friction does is always negative.
If something is already moving really fast it's also possible that all forces perform negative work (at in a simplified model where you have a direction such that no force acts positively in it). Force has a say in the change in momentum once it starts acting on something, but doesn't have any say over what the momentum was prior to it acting on it, and for that reason the work done by that force can be $0$, negative or positive (as oppose to the net change in momentum which is in the same direction as the net force). Of course if you started with $0$ momentum, then the work done by a positive net force cannot be negative, and if there was any change in momentum the work by that net force will be positive.
Since work done by a force $\vec F$ undergoing a displacement $d\vec r$ is defined as $\vec F \cdot d\vec r$ when this dot product is positive the force and displacement are in the same direction and is negative when they are in opposite directions.
The work done by a frictional force does not always have to be negative.
Imagine a block $A$ on top of block $B$ and a force is applied to block $B$ to make both blocks increase their speed in a horizontal direction.
The frictional force on block $B$ due to block $A$ certainly does negative work because force is in the opposite direction to the displacement of block $B$.
However the frictional force on block $A$ due to block $B$ does positive work on block $A$ (increasing its kinetic energy) because the frictional force and displacement are in the same direction.
So decide on the direction of the force and the direction of its displacement and the definition of work done will do the rest.
You pull a spring to extend it. The force you exert on the spring is in the same direction as your displacement. You have done positive work on the spring.
The force that the spring exerts on you is in the opposite direction to its displacement so the spring has done negative work.
In physics when a force is applied on a body, work is done by the force on the body. It is a dot product of force and displacement it causes (say $d$). Then $$W = F\cdot d = F d \cos (\theta)$$ where $\theta$ is the angle between the force and displacement vectors.
If the force is acting in one direction but displacement is in the opposite direction, the angle between these 2 vectors would be 180 degrees and $\cos(180^\circ)=-1$. Hence $$W = F d \cos (180^\circ) = -Fd$$
You see that the work done is negative. However if the force and displacement were both in the same direction, the angle would have been zero and $\cos (0^\circ) = +1$. Work done would have been $W= Fd$ or a positive value. Hence positive work would have been done by the force.
A more tangible way of comprehending this topic (that can be a little confusing) is to imagine force acting in one direction but displacement happening in opposite direction. Such a situation can arise when a body is moving in a certain direction or the displacement is happening in a certain direction but a force comes and acts in the opposite direction. The force, you can imagine will reduce the velocity of the body. If this happens, the kinetic energy (KE) of the body will reduce. Since there is a reduction in KE, we say force has done negative work.
Alternately, if force and displacement are in the same direction, the force must be increasing the velocity of the body. Therefore the KE must be increasing. This increase in KE is associated with positive work done by the Force
Watch this video made by me for more clarity
What is Negative Work
When work done is in direction of force $\implies \theta<90$ is positive and work done is negative when $θ>90$ . In case of friction , you states that internal energy is increased. That internal energy is not necessary to be kinetic energy. Body gains internal energy by heat!
$\theta<90 \implies \cos\theta$ is positive. $90<\theta<180\implies \cos\theta$ is negative.
It is interesting to know while ( applying brakes ) to stop a moving vehicle , work done by friction is positive. | 2019-10-19 12:38:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6247018575668335, "perplexity": 225.55553247224483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986693979.65/warc/CC-MAIN-20191019114429-20191019141929-00419.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/find-derivative-following-functions-it-be-understood-that-a-b-c-d-p-q-r-s-are-fixed-non-zero-constants-m-n-are-integers-ax-b-px-2-qx-r-derivative-algebra-derivative-functions_13987 | Share
# Find the Derivative of the Following Functions (It is to Be Understood That A, B, C, D, P, Q, R And S Are Fixed Non-zero Constants And M And N Are Integers) : (Ax + B)/(Px^2 + Qx + R) - Mathematics
#### Question
Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers) : (ax + b)/(px^2 + qx + r)
#### Solution
Is there an error in this question or solution?
#### APPEARS IN
NCERT Solution for Mathematics Textbook for Class 11 (2018 (Latest))
Chapter 13: Limits and Derivatives
Q: 8 | Page no. 317
#### Video TutorialsVIEW ALL [1]
Solution Find the Derivative of the Following Functions (It is to Be Understood That A, B, C, D, P, Q, R And S Are Fixed Non-zero Constants And M And N Are Integers) : (Ax + B)/(Px^2 + Qx + R) Concept: Derivative - Algebra of Derivative of Functions.
S | 2020-06-01 22:56:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6939800977706909, "perplexity": 698.7791577225406}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347419639.53/warc/CC-MAIN-20200601211310-20200602001310-00075.warc.gz"} |
http://www.perimeterinstitute.ca/video-library/collection/mathematical-physics?qt-seminar_series=0 | # Mathematical Physics
This series consists of talks in the area of Mathematical Physics.
## Seminar Series Events/Videos
Feb 25 2019 - 2:00pm
Room #: 400
Speaker(s):
Scientific Areas:
## Categorical Bernstein Operators and the Boson-Fermion correspondence.
Thursday Jan 17, 2019
Bernstein operators are vertex operators that create and annihilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category.
Collection/Series:
Scientific Areas:
## REPRESENTATIONS OF THE ELLIPTIC QUANTUM GROUP AND RELATED GEOMETRY
Monday Jan 14, 2019
Speaker(s):
Collection/Series:
Scientific Areas:
## Categories of line operators in 3d N=4 gauge theories
Monday Oct 29, 2018
Speaker(s):
A 3d N=4 gauge theory admits two topological twists, which we'll simply call A and B. The two twists are exchanged by 3d mirror symmetry. It is known that local operators in the A (resp. B) twist include the Coulomb-branch (resp. Higgs-branch) chiral rings. In this talk I will discuss the *line* operators preserved by the two twists, which in each case should have the structure of a braided tensor category.
Collection/Series:
Scientific Areas:
## Unification of integrability in supersymmetric gauge theories
Monday Oct 22, 2018
Speaker(s):
The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory. I will explain how string theory unifies these phenomena. This is based on my work with Kevin Costello [arXiv:1810.01970].
Collection/Series:
Scientific Areas:
## Positive geometries and the amplituhedron
Monday Oct 22, 2018
Speaker(s):
Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic canonical form" whose residues reflect
the boundary structure of the space. Familiar examples include
polytopes and the positive parts of toric varieties. A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.
Collection/Series:
Scientific Areas:
## Yangians from Nakajima quiver varieties
Monday Oct 15, 2018
Speaker(s):
One can associate to any finite graph Q the skew-symmetic Kac-Moody Lie algebra g_Q. While this algebra is always infinite, unless Q is a Dynkin diagram of type ADE, g_Q shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to Q gives a geometric representations of g_Q. Encouraged by this story, one could hope to define the Yangian of g_Q, for general Q, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties.
Collection/Series:
Scientific Areas:
## Towards a categorification of a projection from the affine to the finite Hecke algebra in type A
Monday Sep 24, 2018
Work of Bezrukavnikov on local geometric Langlands correspondence and works of Gorsky, Neguţ, Rasmussen and Oblomkov, Rozansky on knot homology and matrix factorizations suggest that there should be a categorical version of a certain natural homomorphism from the affine Hecke algebra to the finite Hecke algebra in type A, sending basis lattice elements on the affine side to Jucys-Murphy elements on the finite side.
Collection/Series:
Scientific Areas:
## Langlands duality and self-duality for Hitchin systems
Monday Sep 24, 2018
Speaker(s):
I will sketch why self-dual versions of the moduli of G-Higgs bundles are expected to arise physically from the study of 4d theories of class S. I will then describe an extension of the Langlands duality results of Hausel-Thaddeus (G=SL(n)) and Donagi-Pantev (arbitrary reductive G) that yields self-dual moduli spaces as a corollary.
Collection/Series:
Scientific Areas:
## Toward AGT for general algebraic surfaces
Thursday Sep 20, 2018
Speaker(s):
The Alday-Gaiotto-Tachikawa correspondence connects gauge theory on a fourfold with conformal field theory. We are interested in a certain algebro-geometric incarnation of this framework, where the fourfold is an algebraic surface and instantons/differential geometry are replaced with sheaves/algebraic geometry. In this talk, we will present a certain approach to AGT that yields partial results for quite general surfaces, and ask questions about what still needs to be done to state and prove the full correspondence in the language of algebraic geometry.
Collection/Series:
Scientific Areas:
## Localization theory for W-algebras
Monday Sep 10, 2018
Speaker(s):
The localization theorem, which has played a central role in representation theory since its discovery in the 1980s, identifies a regular block of Category O for a semisimple Lie algebra with certain D-modules on its flag variety. In this talk we will explain work in progress which produces a similar picture for the Virasoro algebra and more generally for affine W-algebras.
Collection/Series:
Scientific Areas:
## LECTURES ON-DEMAND
### Jocelyn Bell Burnell: University of Oxford
Speaker: Jocelyn Bell Burnell | 2019-01-19 20:21:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6565184593200684, "perplexity": 1441.7970708045361}, "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-2019-04/segments/1547583681597.51/warc/CC-MAIN-20190119201117-20190119223117-00557.warc.gz"} |
http://mathhelpforum.com/calculus/115918-simple-geometric-series-print.html | # Simple Geometric Series
• Nov 21st 2009, 09:41 AM
WhoCares357
Simple Geometric Series
$\sum_{k=1}^\infty (\frac{2}{3})^{k+2}$
This is what I did.
$\sum_{k=1}^\infty \frac{4}{9}*(\frac{2}{3})^{k}$
$a=\frac{4}{9}, r=\frac{2}{3}<1$
Threfore series converges to
$\frac{4}{3}$
This seems correct to me. However when I try to solve it thorugh maple (software) it outputs the result of $\frac89$. Did I do something wrong?
• Nov 21st 2009, 09:58 AM
e^(i*pi)
Quote:
Originally Posted by WhoCares357
$\sum_{k=1}^\infty (\frac{2}{3})^{k+2}$
This is what I did.
$\sum_{k=1}^\infty \frac{4}{9}*(\frac{2}{3})^{k}$
$a=\frac{4}{9}, r=\frac{2}{3}<1$
Threfore series converges to
$\frac{4}{3}$
This seems correct to me. However when I try to solve it thorugh maple (software) it outputs the result of $\frac89$. Did I do something wrong?
Your value of $a$ is incorrect. As the sum starts at $k=1$ you'd need to do
$\frac{4}{9} \times \left(\frac{2}{3}\right)^1 = \frac{8}{27}$
Putting in this value of a should give $\frac{8}{9}$
• Nov 21st 2009, 10:00 AM
WhoCares357
Quote:
Originally Posted by e^(i*pi)
Your value of $a$ is incorrect. As the sum starts at $k=1$ you'd need to do
$\frac{4}{9} \times \left(\frac{2}{3}\right)^1 = \frac{8}{27}$
Putting in this value of a should give $\frac{8}{9}$
Can you explain why? Wouldn't that force the series to start from 2?
• Nov 21st 2009, 10:06 AM
e^(i*pi)
Quote:
Originally Posted by WhoCares357
Can you explain why? Wouldn't that force the series to start from 2?
As I understood it as it's the first term a would be the value of f(k) at k=1 as that's where the sum starts from.
I don't think the series starts at 2 as a is the initial value before anything is added
• Nov 21st 2009, 01:28 PM
RockHard
Just a question I am new to series, because this geometric series starts at n=1 would you not try to get the power in terms of n - 1? | 2016-09-28 03:00:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 18, "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": 0.9577929973602295, "perplexity": 1059.2324062990408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661289.57/warc/CC-MAIN-20160924173741-00273-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.2014.51.1.129 | - Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
Origin-symmetric convex bodies with minimal Mahler volume in $\mathbb{R}^2$ Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 129-137 https://doi.org/10.4134/BKMS.2014.51.1.129Printed January 1, 2014 Youjiang Lin and Gangsong Leng Shanghai University, Shanghai University Abstract : In this paper, a new proof of the following result is given: The product of the volumes of an origin-symmetric convex bodies $K$ in $\mathbb{R}^2$ and of its polar body is minimal if and only if $K$ is a parallelogram. Keywords : convex body, polar body, Mahler conjecture, polytopes MSC numbers : 52A10, 52A40 Downloads: Full-text PDF | 2021-07-27 12:30:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3046053647994995, "perplexity": 2555.324325962832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153391.5/warc/CC-MAIN-20210727103626-20210727133626-00628.warc.gz"} |
http://dtaaffe9.yolasite.com/dtaaffe9.php | ## Ball Of Foot Aches
Knee pain comes in as many types of conditions as the foot pain that we have thus far discussed. The causes of knee pain are also vastly different. Arthritis is one of those causes, but there are several others, such as bursitis, gout, or ligament injuries. In some cases, knee pain can be the result of improperly aligned feet. When this is the case, inadequate arch support provided by footwear can be a contributor. High-heeled footwear can also lead to knee pain because this type of footwear does not provide adequate balance of your body weight, thus contributing to an improper posture.
Corns. A corn is actually a form of a callus - a protective layer of dead skin cells composed of a tough protein called keratin. A corn itself is cone-shaped and usually develops if a shoe rubs against the toes for a prolonged period. As the skin thickens, the corn forms a knobby core that points inward. Hard corns develop on toe joints, usually on the little toe. A shoe that squeezes the front of the foot may cause one toe to rub against another forming a corn between the toes, which is usually soft. These corns can be painful, however, if they harden and rub against each other.
The complete structure of the foot is composed of 24 bones forming a double crossing arch in the foot. There are multiple joints, ligaments and tissues intertwined around each other, contributing to the anatomy of the foot. One of the most important muscles of the foot is the plantar fascia that provides secondary support to the foot. Weight absorbing pads are also essential structures of the foot that protects it from getting hurt while walking or running. Foot pain is an indication of some problems affecting the interaction of internal muscles and its contact with the external surrounding.
Side of your foot pain may be accompanied by abnormal growths of tissue as well. This can occur on any part of the foot that is susceptible to abnormal pressure and agitation. When corns and calluses plague your foot due to over activity, this swells the area and produces pain at the site of damage. Flat Feet Shoes that are tight fitting will irritate these foot abnormalities leading to loss of function and possible even infections. Abnormal growths and foot deformations place pressure on the side of the foot, and cause it to rub against the side of shoes. This can cause quite intense side of foot pain
This is among the most common of foot injuries and is signaled by pain on the front of the heel or all along the arch. You probably have a lot of pain when you first get out of bed - it's probably murder just to walk out of your bedroom. Ditto on standing up after a long time of being seated. The pain may range anywhere from a vague pull along the arch to the impression that your arch is ripping in half (don't worry, it's not actually doing that). If you feel pain, it's plantar fasciitis.
Avoiding foot pain before it begins is the most effective way to prevent expensive surgeries later on. Despite the fact that many people disregard foot pain for weeks, pretending that the pain isn't there or hoping it would just go away are inadequate reactions to the foot pain situation. There are plenty of ways to help your feet stay strong, healthy, and pain-free with only a few changes to your daily routine. Here are a few reasons why metatarsalgia occurs, how to cure foot conditions such as Achilles tendinitis, and how minor changes, for instance using high heel insoles, can help.
A hot and cold foot treatment can help in easing sudden foot pain To do this, fill a bucket with hot water and another bucket with cold water. Soak your feet in the cold water for a few minutes and then soak them in hot water. Alternate between hot and cold water soaks for a few times and finish with a cold water soak. The hot water dilates the blood vessels in your feet and the cold water constricts it. This compress will help in boosting the blood circulation, giving you relief from foot pain.
## How To Use Increase Height Inserts
If you have tried other kinds of insoles or lifts before but have found that they are noticeable and not comfortable to wear, the Levitator shoe lifts are something that you should try if you want to increase your height. They are designed to easily fit into your shoes and provide you with comfort while making you look taller. You can increase your height easily when you find the right kind of insoles to use. You can buy the insoles and shoe lifts online and have them ready to use.
Patients with heel pain caused by plantar fasciitis are often advised by doctors to wear supportive footwear whenever they bear weight so as not to strain and further injure the plantar fascia. The problem is that most footwear lacks good arch support. Podiatrists sometimes advise expensive custom made orthotic inserts for such shoes; other times they recommend patients try over-the-counter insoles. For those patients, therapeutic shoe inserts can replace the manufacturer's footbeds included in shoes with removable insoles. Unless otherwise directed by a health care provider, use orthopedic insoles in both shoes, even if the heel pain occurs only in one foot. Brands of Shoe Inserts for Heel Pain
If you are feeling plantar pain, a simple approach to assist in alleviating the stress on the fascia is to make use of arch support inserts and do plantar fasciitis exercises, that will strengthen the plantar fascia as well. One of the prescribed plantar fasciitis stretching exercises is stretching the foot before rising from bed. The Plantar Fascia will tighten while you sleep, which causes the intense pain that people feel when they get out of bed. By stretching the plantar fascia before you put pressure on it, you can help reduce the pain that you experience with your first morning steps.
While buying shoes for yourself, make sure that you don't make the decision on the visual appeal of the shoes. Buy shoes that not only fit you well, but also have other added features enabling improved arch support. If you are looking for a cheaper alternative, you could even buy good quality orthotic shoe inserts. It is also essential that any structural foot abnormality one may be suffering from is considered while one goes shopping for footwear. One can check out custom-made orthopedic shoes that are specifically designed as per one's size. There is a great need to provide support to all parts of the foot.
Most people experience pain on the heel when they wake up in the morning and begin to walk. There is less pain and stiffness after a while; however, the pain may increase during the day. The pain can occur when you stand or sit for a long time too. The illness is caused when there is strain on the ligament that provides support to the arch. Tiny ligament tears are caused when there are repeated strains resulting in swelling and pain. Continued stretching of the plantar fascia can result in heel spur which is a bone-like development on the heel. Flat feet or high arches can be a cause.
Flat feet (pes planus) are very common across a broad range of people. The strain placed on the foot and body due to the flatness of the feet can cause numerous foot problems. Unfortunately, the necessary use of shoes at an early age to protect our feet does not allow those with flat feet grow taller to develop the muscular and structural adjustment needed to function better. The result are feet that need added structural support to prevent tendon and ligament strain, as well as delay the progression of foot deformities that are related to flat feet, such as bunions and hammertoes.
The repetitive pulling and tightening of the arch and the resultant biomechanical damage leads to a variety of foot problems. Orthotics help in reducing the over-stretching or overpronation of the arch. Shoe inserts or foot supports decrease the tension on the ligament by distributing the weight evenly. So, the bones and tissues are no longer under an immense strain. These provide support to the arches, stability to the ankle and provide the much-needed padding or cushioning for an improved shock absorption. Unlike the rubber footbeds that just act as a cushion, orthotic devices are designed to correct any type of misalignment in the foot.
Stretching the calf muscles is an important part to reducing plantar fascia tension. The calf muscles, including the gastroc and soleus, attach to the back of the heel. Tension from these muscles pulls back on the heel, causing tension on the plantar fascia. A great way to stretch the calf the first thing in the morning, before you take your first steps, is to hook a towel on the ball of your foot and while keeping the knee straight pull back on the towel. Hold the stretch for a minimum of 30 seconds and repeat a few times.
By using this type of material, the weight of the body is resisted and the insert will retain its stiffness, and therefore it's ability to alter the foot structure. Comfort is maintained despite its hardness because the insert is an exact match to the individual foot. Plastizote coverings are added to the inserts to reduce pressure and friction even further. They are commonly used in many individuals with and without diabetes to support excessive foot flattening, reduce arch ligament injury (plantar fasciitis), and treat serious foot tendonitis and spraining. For diabetics with significant foot structural abnormalities, prescription orthotics are the preferred way of protecting the foot.
Apple cider vinegar is an effective home remedy. Dip a cotton towel or cotton bandage in apple cider vinegar, fold it up and place it on your heel. Secure the towel or bandage in place with tape. Wear it regularly for a few days. Make a mixture of apple cider vinegar, molasses, and baking soda and rub it on your heel spur. Applying ice to the heel and massaging the area with coconut oil are also good home remedies. A best way to reduce the pain and discomfort associated with this bony growth is to immerse your feet in a warm bath of Epsom salts.
## Strategies for Height Inserts
Do you often want to be taller and are not happy with the height that you are? You may have tried a lot of different products and techniques to try to grow taller and if you are ready to try something that really works, you can find shoe insoles that will make you look inches taller, all without having to take any kind of supplement or other procedure that is designed to make you look taller. The insoles that you can try are called the Levitator shoe lifts, and they are made to easily fit into your shoes without being able to notice that they are there.
If you have tried other kinds of insoles or lifts before but have found that they are noticeable and not comfortable to wear, the Levitator shoe lifts are something that you should try if you want to increase your height. They are designed to easily fit into your shoes and provide you with comfort while making you look taller. You can increase your height easily when you find the right kind of insoles to use. You can buy the insoles and shoe lifts online and have them ready to use.
Patients with heel pain caused by plantar fasciitis are often advised by doctors to wear supportive footwear whenever they bear weight so as not to strain and further injure the plantar fascia. The problem is that most footwear lacks good arch support. Podiatrists sometimes advise expensive custom made orthotic inserts for such shoes ; other times they recommend patients try over-the-counter insoles. For those patients, therapeutic shoe inserts can replace the manufacturer's footbeds included in shoes with removable insoles. Unless otherwise directed by a health care provider, use orthopedic insoles in both shoes , even if the heel pain occurs only in one foot.
If your foot pronates too much, the ankle rolls too far downward and inward with each step. This can cause the arch to collapse and muscles and supporting structures to overstretch and lengthen. The ligaments running from the heel to the toes become strained, often leading to inflammation along the bottom of the foot. This inflammation and heel pain is known as Plantar Fasciitis. In response, the bone may also develop a bony growth (heel spur), right in the center of the heel. This too causes heel pain and can prevent an individual from walking with ease. The best shoes for heel pain (plantar fasciitis)
If you are in search of inserts for kids, then you can consult the doctor and buy Spenco Total Support Gel - Arch Support Insoles. These supports are made of special thermoplastic gel and they also have special forefoot impact zone. These are metatarsal arch supports that provide stability and support to arches and heel of the kids. The best thing about these arch support insoles is that they have unique 3 pod cushioning system that are shock absorbent. However, all this is only possible when you will select shoes, slipper or sandals with arch supports that fits the exact foot size of the kids.
No one can imagine his/her life without feet because it is the important part of our body. Foot care is very necessary in our life. The heel lifts have been popular in the market can be attributed to the fact that the majority of the consumers has come to know using this kind of shoe embellishments as an enormous way to treat the most common foot problems. Unbelievably, using shoe insoles has proved to be the best way for people to evade foot pain and other structural defects. You can easily find shoe inserts for various kinds of occasion like for official purpose, sports day, weddings and casual wear.
I’m a construction worker and have to be on my feet most of the time. My feet never gave me any trouble until a few weeks ago when I had to carry lumber up and down stairs several times a day. After about a week on the job I noticed pain in my heel to the extent that I could barely walk. I had a vacation coming up so I thought I’d give it a rest to see if it got better. But it’s just as bad and even worse first thing in the morning. What can possibly be causing it?”
It is advisable to use proper shoes because running or walking involves strenuous movements. You can consult a physiotherapist to know the kind of motion control needed. Running shoes for plantar fasciitis provide special structural support around the heel which is not found in normal running shoes. They also have an inbuilt scientific technique to deal with pain. Some shoes do not have any supplementary devices, but instead, have an inbuilt mechanism in the sole which treats the problem. But, these shoes are very costly with a limited number of designs and styles.
These arch supports are generally available over the shelf and can be bought either online or offline depending on the choice of the person. Ideally, these come in form of shoe insoles, which can replace the existing show insoles of your sports shoes The other problem that needs to be taken a look at is how the shape of the leg changes when the person walks, the height of the arch of the leg also goes up and down with the motion. The arch support that is bought should be able to provide this required dynamic support that the feet needs.
Plantar fascia is a band of connective tissue that runs along the bottom of the foot. Since this band of fibrous tissue provides support to the arch of the foot, repeated trauma to it can cause inflammation of the plantar fascia. This condition is medically referred to as plantar fasciitis. People suffering from this condition often suffer from heel pain. The pain is felt the most when one takes the first few steps in the morning. Since our feet carry our entire body weight, stress to the heel bone is most likely in case of obese people. Plantar fasciitis could also be a self-inflicted problem.
## Lex Fenwick Out As Dow Jones CEO
Lex Fenwick Out As Dow Jones CEO
NEW YORK (AP) -- In an unexpected shakeup, News Corp said Tuesday that it is replacing Lex Fenwick, CEO of its Dow Jones subsidiary. The media company controlled by Rupert Murdoch said it plans to review how Dow Jones serves institutional clients.
The company named William Lewis Dow Jones' interim CEO.
News Corp. CEO Robert Thomson said the review will include improvements to DJX, a product for institutions that includes a variety of services such as Factiva and the Dow Jones newswire.
Thomson said in a statement that "greater flexibility in its product offerings is likely in the short term." DJX, which has been in beta testing since its launch last year, has had a single price per user for its omnibus offering.
The product was meant to compete with rival Bloomberg's financial data and news services and was largely developed by Fenwick, who joined Dow Jones in February 2012 after 25 years at Bloomberg LP, where he held positions including that of CEO at Bloomberg Ventures.
The company had high hopes for DJX. Dow Jones' institutional clients account for about 30 percent of Dow Jones' revenue. The rest comes from consumer publications like The Wall Street Journal and Barron's magazine.
Thomson said in November that the institutional part of the business "hasn't performed to our expectations or to its potential for really almost two decades."
## Sale Insider: Shoes, fashion, and books sales - The Denver Post
Sale Insider: Shoes, fashion, and books sales - The Denver Post
Murrays shoe store (provided by murray shoe store, The Denver Post)
Feet first
Murray's Shoes in Littleton continues its Semi-annual Clearance Sale through Feb. 15. Just look for the bargain tables laden with brands such as Dansko, Earth, Orthaheel, Klogs, Merrell, Clarks and others --discounted 25 to 75 percent. The family-owned store also carries accessories, handbags and socks. Hours are 10 a.m. to 6 p.m. Monday through Saturday. 8116 W. Bowles Ave., Unit F, Littleton, 303-972-8436,murraysshoes.com
Dot to dot
Books with red ribbon (The Denver Post | photos.com)
Eccentricity, the boutique that carries women's fashions, footwear, accessories and jewelry, has discounted its entire store at least 20 percent in January.Savings are indicated by dots --items with a red dot equal a20 percent discount, green is 30, yellow is 40 and blue is 50 percent off. Hours of the Cherry Creek North store are 10 a.m. to 5:30 p.m. Monday through Friday, 10 a.m. to 5 p.m. Saturday. 290 Fillmore St., 303-388-8877, eccentricity.com
Hot reads
"Sizzlin' Smoky Sale" is the theme of the three-day book sale at the Smoky Hill Library Jan. 23-25. Find books for all ages including hardback, paperback and collectibles. Hours are 9 a.m. to 8:30 p.m. Jan. 23, 9 a.m. to 4:30 p.m. Jan. 24 and 9 a.m. to 4 p.m. Jan. 25. Proceeds support projects and programs in the Arapahoe Library District along with equipment, materials and events.5430 S. Biscay Circle, Centennial, 303-542-7279
Free days this week
Jan. 20: Denver Botanic Gardens, 1007 York St.,botanicgardens.org
Jan. 22: Denver Zoo, denverzoo.org
Each Monday, "Sale Insider" gives shoppers the inside scoop on store discounts and bargain events. Preference is given to retailers and events that have not been listed recently. To include a sale, e-mail living@denverpost.com 10-14 days in advance.
## Six Established Solutions To Strengthen Human Growth Hormones Production Naturally
Nighttime also seems to be the best time to take additional supplements to increase the flow of growth hormone. UCLA researchers have found that taking the amino acid arginine and orthinine together on an empty stomach right before bedtime can boost growth hormone levels significantly. However, the amounts required to see a difference were between 40 and 60 grams, dosages too large to take in any version besides injectable form. Other factors for the height increase could be an increase in per capita income, more sanitary living conditions, better education, better health and nutrition for all family members and better social services and health systems being available.
There are principally two methods, that are used to measure the ratio of height to weight, namely, the common formula and the second one is BMI or Body Mass Index. For example, if a person is 150 cm tall then his ideal weight should be somewhere near 50 kg plus or minus 5 kg. This general notion is the best example for all formulas. There are some other formulas that have been developed on the basis of different logic. The body mass index is a value that indicates the excessive amount of mass in the human body. There are three different ways to calculate this value.
Just two days after releasing revised recommendations for changes to the Height Act that would have opened the window to future changes to D.C.'s building height limits, the National Capital Planning Commission voted to strip the proposal of its only meaningful alteration to the 1910 law and send a watered-down recommendation to Congress. The recommendations the NCPC will send to Congress would leave the Height Act almost entirely intact. The only proposed changes are a small alteration to the allowed occupancy of penthouses and the removal of some obsolete language relating to fire safety.
The size of a subtree at height $k$ is $2^k-1$.The probability of #1 is therefore $\left(\frac2^k-12^h - 1\right)^2$. The probability of #2 is $2 \left(\frac2^k-1-12^h - 1\right)^2$. The probability of the event that #1 happens and #2 does not (since #2 is a subset of #1) is thus\beginalign\Pr\text$w$ at height $k$ is lca &= \frac\left(2^k-1\right)^2 - 2\left(2^k-1-1\right)^2\left(2^h - 1\right)^2 \\&=\frac2^2k-1 - 1\left(2^h -1 \right)^2\endalignonce you work it out. (You can check that these probabilities sum to $1$.) There are $2^h-k$ nodes at height $k$. So the probability that the lca is at height $k$ is $\frac2^h+k-1 - 2^h-k\left(2^h - 1\right)^2$.
It is safe to assume that the creator implied the first interpretation, but you can never be too sure on anything. The user would have had to describe it into human years because they probably simply did not understand how old dwarves are. But because dwarves live longer than humans, it is an error to describe a character as such, because a dwarf at the physical and mental equivilent of a human at twenty will still experience and learn more than a human could within those spans. Metabolic Syndrome or Syndrome X – Apple shaped abdomen or abdominal obesity, high triglyceride level, lower HDL levels, Higher Fasting Blood Sugar.
Meanwhile, blacks still experience more fat related disease and than whites because of the sheer number of overweight black people. By giving blacks correct BMI information at a young age, perhaps this would prevent starvation diets by healthy young people who's numbers don't fit the standard. By preventing the wieght loss and gain spiral, perhaps less blacks would end up seriously overwieght to begin with. It would also bring down the cost of health insurance for black people as weight is considered a factor in the price of health insurance policies.
We have seen the differences in perception result in different people seeing different things and attaching different meanings to the same stimuli. Every person sees things in his or her own way and as perception becomes a person’s reality this can lead to misunderstandings. The ways which we organize and make judgements about what we have perceived is to a large extent based on our previous experiences and learning. There are cultural differences in the way the body language is perceived and interpreted. Although these explanations are rich in detail they leave unexplored the origin of women’s roles as looked at by psychologists and sociologists.
Professor Byrne told BBC News that studying elephants has helped construct a map of part of the evolutionary tree that is very distant from humans. The results form the study might also help explain the historically close bond that elephants share with humans and the ease with which they can be tamed despite the potential danger they pose to humans. The next step in the research is to determine the role an elephant's trunk plays in the animal's ability to gesture. HealthDay)—The adult obesity rate in the United States remains as high as ever, with one in three Americans carrying unhealthy amounts of weight, according to a new federal report.
## Stand Up for Better Health? Maybe Not - Yahoo
Stand Up for Better Health? Maybe Not - Yahoo
Bill Seaver stands at his UpDesk most workday mornings, tapping away at his keyboard and manning the phones. The 37-year-old marketing consultant from Nashville, Tenn., bought the adjustable height desk for his home office about a year ago so he could move between sitting and standing whenever he felt like it.
"I bought it to help ease my back pain," he said. "I haven't messed up my back since I started using it."
Seaver said he loves his standing desk, which he believes has also helped him feel more fit. And if the thousands of glowing reviews on Amazon and UpDesk's website are any indication, many other Americans feel the same way.
Standing desks and their slow-moving cousins, treadmill desks, allow people to take a stand against prolonged sitting, which is so bad for your health, said experts. Some scientists have compared it to smoking. With so much movement engineered out of modern life, the idea is that rising up from your office chair and prying yourself from the couch for a few extra hours a day should lead to better health and lower the risk of heart disease, stroke and cancer.
Marc Hamilton, a professor of inactivity physiology at Pennington Biomedical Research Center in Baton Rouge, La., has led numerous studies that demonstrate the health dangers of sloth. He said there is little doubt that long periods of sitting carries health risks, but he has never seen evidence directly linking the use of standing or moving desks to improved health.
"It's a creative idea, but it's not been scientifically proven," he said. "As of now, there's really no research to show they do any good."
Hamilton doesn't believe there's any harm in using a standing desk but suspects it would supply borderline health benefits at best. He doubts whether it would cause the small muscles throughout the body to contract often enough to change the bad biochemistry that arises from too little movement.
Get Outta That Chair!
"When you sit for long periods of time, there is very little contractile activity going on in the skeletal muscle, and after more than a thousand studies, we believe that small, low level contractions are very important for good health," he said.
Activities that initiate low level muscle contractions are the ones you do as you go about your everyday life without raising your heart rate or breaking a sweat, Hamilton explained. On their own, activities like standing in line, strolling to work and doing housework burn a minuscule number of calories, and just barely move the muscles, but over the course of the day, they add up.
After several hours without any such activity, studies find that the genes and enzymes regulating the amount of glucose and fat in the body diminish, resulting in fat from the bloodstream being captured and stored by fat cells all throughout the body.
The fat that wraps around the organs appears to be particularly damaging to your health. It's been linked to a wide range of major diseases, including heart disease, stroke, diabetes, cancer and osteoporosis.
Sitting for too long is not the same as exercising too little, Hamilton pointed out. A regular exercise program doesn't seem to cancel out the negative effects of minimal lifestyle movement. For example, a series of investigations done at the University of South Carolina found that men who were sedentary more than 23 hours a week had a 64 percent greater risk of dying from heart disease compared with those who reported less than 11 hours a week of sedentary activity, even if they exercised regularly.
9 Habits Worth Rethinking
Standing is probably somewhat better than sitting, Hamilton conceded. He's just not sure that it's enough to nudge someone out of the sedentary category. And if it does offer some benefit, he said that much like taking a medication, studies need to determine the correct dosage for maximum health benefits.
Kamron Kunce, a spokesman for UpDesk, stressed that the desk was meant to increase movement, not just standing.
"The height is adjustable, so you are constantly moving it up and down. That's what the consumer should be aiming for," he said.
As for moving desks like the TreadDesk in which the user strolls along at one to two miles per hour while working at a computer or doing paperwork, Hamilton said, again, there's no evidence that it provides a counterbalance to a largely couch potato existence. But he said a moving desk might lead to health improvements, provided a person uses it often enough.
"An individual might succeed with a walking desk, but is it a viable solution at the population level?" There have been a lot of home treadmills out there for a long time," he said, "and many of them are gathering dust in a basement."
Walk Don't Run For Better Health
Jerry Carr, president of TreadDesk, said he understood Hamilton's concerns over the lack of scientific evidence for standing and moving desks, but even without a scientific paper to quote, he said his clients seemed convinced.
"From the feedback we get from our customers, there is no question there are many benefits, both physical and mental, when you are up and moving instead of sitting still," he said.
Prices for standing desks start at around $500. Moving desks start at around$800 for the treadmill component, plus the cost of the desk.
Despite the cost, Seaver said buying his UpDesk was a great investment. Not only does his back feel better but he said he has more energy and has lost weight without a formal workout program.
"I'm a little surprised. I'm holding steady on my weight without any exercise," he said. "I think standing up has contributed to that."
Also Read
## Eminem Acting Career Over? Marshall Mathers Drops ‘Southpaw’ Movie
Eminem Acting Career Over? Marshall Mathers Drops ‘Southpaw’ Movie
Is Eminem's acting career already over?
As previously reported by The Inquisitr, Eminem's hit single Berzerk is being featured on Saturday Night Football.
Eminem, or Marshall Mathers, has not had much luck at all with the Southpaw movie project. Almost a year ago filming was put on hold so Eminem could get back to recording music. He even turned down Neill Blomkamp's Elysium, which was a pretty good SciFi flick, and also bailed on a movie project called Shady Talez, which apparently would have been good for a Halloween release.
Southpaw tells "the story of left-handed boxer who wins a title but soon after suffers a personal tragedy, forcing him to reexamine and rebuild his life and win the respect of his young daughter." With Eminem gone for good Jake Gyllenhaal may take up the slack by taking the lead role.
So where does that leave Eminem's acting career? He doesn't have any other film projects or offers set up for the moment. The last time we actually saw Marshall Mathers in a movie was in 2008 for the release of Funny People.
Still, Hypable seems to think Eminem's acting career could still be a good idea:
"No one would be blamed for thinking Eminem's acting career would go the way of Ice Cube. Rappers get notoriously terrible movies (or maybe they're just notoriously terrible actors), and why would anyone think any different about 8 Mile, a movie loosely based off of Eminem's own life? Except that 8 Mile is actually good. And not only was it good, but Marshall Mathers led the movie like a veteran actor, easily switching between the comedy and the drama."
Of course, it's hard to judge how good an actor Eminem could be when he was essentially playing as himself instead of a role that's dramatically out of character for Marshall Mathers. What do you think? | 2021-03-08 11:47:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21176296472549438, "perplexity": 2817.7507922214645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178375439.77/warc/CC-MAIN-20210308112849-20210308142849-00511.warc.gz"} |
https://robotics.stackexchange.com/questions/15336/state-space-model | # State space model
I want to design a LQR controller for my two wheeled mobile robot. While designing the controller, I need to find gain, K for the LQR which this required the state space model of wheeled mobile robot. My question is how to find the matrix A, B, C and D for this nonlinear system by using the equations below.
The outputs are x, y and theta while inputs are v and w.
• A bit pushed for time right now, so can't really write a decent answer. Does this paper help? – sempaiscuba Mar 13 '18 at 4:59
• Thanks sempaiscuba! But my wheeled robot is not a bicycle. – CK1994 Mar 13 '18 at 5:09
• You can't. This system is nonlinear. You have to options. The first one is to linearize the system. The second is to design nonlinear controller. Also, it seems $v,w$ are the inputs and $x,y,\theta$ are the outputs. – CroCo Mar 13 '18 at 14:25
• Yes sorry my bad. The input is v and w while output is x,y and theta. Croco I know there is a way to linearise the equation but how to linearise it? – CK1994 Mar 13 '18 at 15:19
As an approximation, you could linearize the system. If the nonlinearities are small it's common to linearize about a nominal point such as $\theta=0$, but in your case this might not work well since it would eliminate any lateral dynamics. \begin{align} \dot{x} = v \\ \dot{y} = 0 \\ \dot{\theta} = \omega \end{align}
Another option would be to linearize every timestep. This could get expensive though as your feedback gain matrix, $K$, also needs to be recalculated each time.
Finally, a middle ground could be to linearize about several predefined points, e.g. $\theta=[-20,-15,-10,0,10,15,20]$. At each timestep you would separately compute the control action using the $K_j$ computed for each of these linearization points and then interpolate between those values based on the current $\theta$. | 2019-08-20 05:29:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9382607340812683, "perplexity": 612.5807609135223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315222.56/warc/CC-MAIN-20190820045314-20190820071314-00423.warc.gz"} |
http://mathhelpforum.com/geometry/230414-mathematical-induction.html | # Math Help - Mathematical induction
1. ## Mathematical induction
Use mathematical induction to prove that
4 x 6^(2n) + 3 x 2^(3n) is divisible by 7 for n=1,2,3...
I roughly know the method but I can't seem to get there with this. Please could someone take the time to post a detailed answer? Thank you
2. ## Re: Mathematical induction
Just out of interest, why did you post this in the Geometry forum?
Anyway, can you at least prove the base case?
3. ## Re: Mathematical induction
I wasn't sure which forum it should be in...where would you suggest I post it?
And yes, I have proved the base case, and I know I have to assume that p(k) is true and use this to prove p(k+1) but I am unsure of how to proceed from there on.
4. ## Re: Mathematical induction
Well what is your P(k) statement?
What are you trying to prove in your P(k+1) statement? Can you use P(k) to help you?
Actually show us what you have done and we can give you some guidance.
5. ## Re: Mathematical induction
I have p(k) is 4 x 6^(2k) + 3 x 2^(3k) = 7m where m is an integer.
And I have to use that to show that p(k+1) is divisible by 7.
So I have 4 x 6^2(k+1) + 3 x 2^3(k+1)
So far I have written it as 4 x 6^(2k+2) + 3 x 2^(3k+3)
And from there I have split the powers to get
4 x 6 x 6^(2k+1) +3 x 4 x 2^(3k+1) ... this is the line that I am unsure of and I don't know if it's right or not. If it is, I may need some guidance as to where to go from here.
6. ## Re: Mathematical induction
The final step is wrong, you should have
\displaystyle \begin{align*} 4 \times 6^{2k + 2} + 3\times 2^{3k + 3} &= 4 \times 6^2 \times 6^{2k} + 3 \times 2^3 \times 2^{3k} \end{align*}
7. ## Re: Mathematical induction
Ok thanks, I understand that.
So do I know multiply out the 4x6^2 and the 3x2^3 ?
8. ## Re: Mathematical induction
No, you need to somehow get \displaystyle \begin{align*} 4 \times 6^{2k} + 3\times 2^{3k} \end{align*} so that you can replace it with 7m.
9. ## Re: Mathematical induction
I know I somehow have to get it into the form of p(k) to show it is divisible by 7, but I'm playing around with it and I can't see how it works?
10. ## Re: Mathematical induction
Yes I know that but I can't see how?!
11. ## Re: Mathematical induction
Originally Posted by alexlbrown59
Yes I know that but I can't see how?!
We are trying to reduce the expression for the case of k + 1 to either the case for k or a combination of the cases for 1 and k.
I have found that it helps sometimes in finding a proof to actually work out the cases for n = 1, n = 2, n = 3. You have to do that for n = 1 to do a proof by induction anyway.
$4 * 6^{(2 * 1)} + 3 * 2^{(3 * 1)} = 4 * 36 + 3 * 8 = 144 + 24 = 168 = 7 * 24.$
$4 * 6^{(2 * 2)} + 3 * 2^{(3 * 2)} = 4 * 6^4 + 3 * 2^6 = 4 * 1296 + 3 * 64 = 5284 + 192 = 5376 = 7 * 768.$
$4 * 6^{(2 * 3)} + 3 * 2^{(3 * 3)} = 4 * 6^6 + 3 * 2^9 = 4 * 46656 + 3 * 512 = 186624 + 1536 = 188160 = 7 * 26680.$
Now that is initially disappointing because I at least do not see an obvious relationship between the various multiples of 7. But I now have some confidence that what I am trying to prove may actually be true.
Now I play around a bit 5376 - 168 = 5208 = 7 * 744. And 188160 - 5376 = 182784 = 7 * 26112. Well of course, the difference between two multiples of 7 is itself a multiple of 7. Maybe that will give me a proof.
$k \in \mathbb Z\ and\ k > 0\ and\ 4 * 6^{2k} + 3 * 2^{3k} = 7m_k,\ where\ m_k \in \mathbb Z\ and\ m_k > 0.$
$4 * 6^{2(k + 1)} + 3 * 2^{3(k + 1)} = 4 * 6^{2k} * 6^2 + 3 * 2^{3k} * 2^3 = 36(4 * 6^{2k}) + 8(3 * 2^{3k}) .$
$So\ 4 * 6^{2(k + 1)} + 3 * 2^{3(k + 1)} - (4 * 6^{2k} + 3 * 2^{3k}) =$
$36(4 * 6^{2k}) + 8(3 * 2^{3k}) - (4 * 6^{2k} + 3 * 2^{3k}) =$
$35(4 * 6^{2k}) + 7(3 * 2^{3k}) = 7(20 * 6^{2k} + 3 * 2^{3k}).$
$But\ obviously\ 20 * 6^{2k} + 3 * 2^{3k}\ is\ an\ integer > 0\ because\ k\ is\ a\ positive\ integer.$
$So\ \exists\ j_{k + 1} \in \mathbb Z\ such\ that\ j_{k + 1} = 20 * 6^{2k} + 3 * 2^{3k} > 0.$
$\therefore 4 * 6^{2(k + 1)} + 3 * 2^{3(k + 1)} - (4 * 6^{2k} + 3 * 2^{3k}) = 7j_{k + 1} \implies$
$4 * 6^{2(k + 1)} + 3 * 2^{3(k + 1)} = 7j_{k + 1} + 4 * 6^{2k} + 3 * 2^{3k} = 7j_{k + 1} + 7m_k = 7(j_{k + 1} + m_k).$
There is no guaranteed route to a proof. You just have to try various things.
12. ## Re: Mathematical induction
Wow Jeff! Many thanks for such a clear and detailed answer. I would never have thought about doing it that way but it does make sense, and I reckon with a bit more practice I'll get the hang of it. You're a life saver
13. ## Re: Mathematical induction
I'm reviewing this topic and worked out your problem, hope this helps
14. ## Re: Mathematical induction
4×6^2×6^(2k)+3×2^3×2^(3k)
Man it's so close if you do it using modulus 7.
Then 4×6^2×6^(2k)+3×2^3×2^(3k) mod 7 = 4*1*6^(2k) + 3*1*2^(3k) mod 7 = 4*6^(2k)+3*2^(3k) mod 7 = (according to inductive hypothesis) 0 | 2015-10-09 01:54:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9762658476829529, "perplexity": 570.9132721142414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737913039.71/warc/CC-MAIN-20151001221833-00178-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://mathhelpforum.com/number-theory/65636-divisibility-gcd-9-a-print.html | # Divisibility (gcd) 9
• December 19th 2008, 12:47 PM
Sea
Divisibility (gcd) 9
$n \geq 2$ and $k \in \mathbf{Z^{+}}$
Show that:
$(n-1)^{2}|n^{k}-1 \Leftrightarrow n-1|k$
• December 19th 2008, 01:12 PM
JaneBennet
This is the same as saying that $n^2\mid (n+1)^k-1\ \Leftrightarrow\ n\mid k$ for all $n,k\in\mathbb{Z}^+.$
Note that $(n+1)^k-1\equiv kn\pmod{n^2}.$
Hence $n^2\mid(n+1)^k-1\ \Leftrightarrow\ n^2\mid kn\ \Leftrightarrow\ n\mid k.$ | 2015-04-21 13:00:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "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": 0.9103916883468628, "perplexity": 3311.6073906511615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246641393.36/warc/CC-MAIN-20150417045721-00057-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://brilliant.org/problems/1234525286/ | # 1234525286
Number Theory Level 3
How many ordered pairs of positive integers $$(x,y)$$ satisfy $$x^2-y^2= 1234525286$$?
× | 2016-10-24 14:22:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.26398760080337524, "perplexity": 2035.4349970665555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719638.55/warc/CC-MAIN-20161020183839-00058-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.coursehero.com/file/5630170/ss-10/ | ss_10
# ss_10 - 1 A typical 64 Mbyte dynamic RAM chip can be...
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Unformatted text preview: 1?. A typical 64 Mbyte dynamic RAM chip can be carried on a piece of silicon approximately 101112 in size. Estimate the lateral dimension associated with an individual transistor - storing a single data bit - on this chip. Ma page W\$‘z£ \ \ ‘1 5 -‘° ‘2. ‘1‘ =_ on — 0 mm x \o m Kim “I _ 4A \e— ébknoé 7% :3) 0_0q142 Mo {U ‘: 0."\‘47?JAM ¢________... 21. Consider a specimen of pure silicon with a gap energy of 1.12 eV. a) Sketch a well-labeled energy-band diagram for pure silicon. b) At a temperature of O K, would you expect the conductivity of this silicon to be more like that of: (i) SiOz at 300 K; (ii) Si at 300 K; or (iii) Cu at 300 K? Briefly explain your answer. 0) Suppose the silicon at 0 K is illuminated by a beam of light whose wavelength is 2 pm. Will the silicon absorb or transmit this light? Quantitativer justify your answer. Will the adsorptionftransmission property change if the silicon is allowed to warm to room temperature? Briefly explain your answer. d) Suppose the silicon is doped with arsenic whose donor level lies 0.054 eV below the conduction band. Will this dopant affect the adsorptionftransmission behavior of the silicon for 2 Min wavelength light? Briefly explain your answer. ® ———-—————-—— EEO a,va \‘n m mow WW :9: WM, 8 H: mm “To N Rafi: "Riv CE“; “3 “RD/ms“: gx © E'st him ELECTQOKX‘ Mix WE CWXBJ/Evfiy wak$35. R .. "'" @6796?“ 3-w\%zm°“%\l‘“t‘$\ he”: not NT 6;. _ W L '2 FR; M. '7' 0-6?— c\l ' m %x mm WMMT "was 2» WW5 UN 3%— N s; @013. x": 09W N “Jim‘w S- \Wwfi‘lt W: Mam 6* _ wwmwza‘ m» u- WWE‘UW a} 2/“ ha“ _' 5-H: e'Q/x WNW: Mkk PD\ ‘5‘ To Volta-i2, e“: W955: WE. gnaw. Wh'm M... MM 22mm w\\\ my”; Lit-VIEW MV’W To 2 a. Lt \. ® fiulcfiz, RT‘$09M merc‘fii MOST Etttm \H Wt. Wow. mm. use, MMA-Lk‘f Emmi}; nary WE 6.29:,s m has? MM Eh, swarms ts» EYE.an Matt. limusm. Us :2. s. 24. Using well-labeled energy-band diagramgfidescfibe the differences between intrinsic silic0n, p-type silicon, and n-type silicon. “M‘WmnqmmvwmflWflW imam-WW E , WWW '“” W OE: Ea: mud mam mm MM 15' Ucufig MW /v///i/ W5 \DWgssL in m“ WWW \$~ \s we» e:me WWW“ VDEMWW www- .W 25. Contrast the effect on the electrical conductivity of doping pure Si with 1 wt% Al to that of alloying pure A] with 1 wt% Si. What is the approximate increase or decrease in electrical conductivity in each case? M \S is: W TEL Eta—«c»:— mmai \Q\\L {\0' RS 9‘ ‘5-me haema- mm mick To 3:, (fibbml: \ (304, \mu. macaw: 6;; VbY Msbxdtr 3“ TD Momva LmU. bar/mat crfi V9; 9(th '3 \o games: at Wm memo smTchca \ 28. Outline 3 sequence of processing steps that could be used to create a pad of pure copper metal with lateral dimensions of 100 um x100 inn and a thickness of 100 nm to be deposited on a 1 inch diameter silicon wafer so that the pad is electrically isolated from the silicon except for a single connecting lead which has lateral dimensions of 10 um x 10 um. '- fl,» yer/#2325137 ‘ {L9 ' y 0 1t “3 \J/u-s I/Ah '0 C). u at, was ‘Fdwsm 8' Z ' .22” L ._. / ” M s: C? most-r C» “Y0 “W afloat» WCQ. W04 Wham: \ELEAST . (3 many Yél-tSVST’ + spams: LMLD‘; 5M8: ...
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Jill Tulane University ‘16, Course Hero Intern | 2018-06-21 23:04:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4198560416698456, "perplexity": 14743.922099646265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864300.98/warc/CC-MAIN-20180621211603-20180621231603-00185.warc.gz"} |
http://tex.stackexchange.com/questions/103723/vbox-overflow-when-using-titlegm-causing-a-blank-page-to-preceed-the-title-page | # vbox overflow when using TitleGM causing a blank page to preceed the title page
I'm getting a blank page that precedes my title page, and I suspect that it is due to the vbox overflow warning that I'm getting. The problem is that I don't know how to properly adjust the vbox setting! Can anyone help? Here an example document that shows the problem:
\documentclass[10pt,ebook,oneside,openright]{memoir}
\usepackage{color}
\usepackage{graphicx}
\usepackage{rotfloat}
\usepackage[scaled]{berasans}
\usepackage[T1]{fontenc}
\usepackage{type1cm}
\usepackage{eso-pic}
\usepackage{setspace}
\usepackage{listings}
\usepackage{lmodern}
\usepackage{tikz}
\usepackage{framed}
\ifpdf
\usepackage{pdfcolmk}
\fi
\pagestyle{ruled}
\setsecnumdepth{subsection}
\renewcommand*\familydefault{\sfdefault}
\floatstyle{boxed}
\restylefloat{figure}
\newlength{\drop}
\newcommand*{\titleGM}{
\thispagestyle{empty}
\begingroup
\drop = 0.1\textheight
\vspace*{\baselineskip}
\vfill
\hbox{
\hspace*{0.2\textwidth}
\rule{1pt}{\dimexpr\textheight-28pt\relax} % Vertical line
\hspace*{0.05\textwidth}
\parbox[b]{0.75\textwidth}{
\vbox{
\vspace{\drop}
{\noindent\HUGE\bfseries Growing The\\[0.5\baselineskip]Money Tree}\\[2 \baselineskip]
{\Large\itshape Cultivating Financial Freedom One Leaf At A Time}\\[.37 \baselineskip]
{\Large \textsc{John Svazic}}\par
\vspace{0.5\textheight}
{\noindent ARM Trading Press}\\[\baselineskip]
}% end of vbox
}
}
\endgroup}
\begin{document}
\titleGM
\frontmatter
\mainmatter
It was a dark and stormy night...
\backmatter
\end{document}
-
The culprits were \vspace{0.5\textheight} and \linespread{1.2}. I reduced the length used for the \vspace and moved the \linespread line somewhere after building the title. I also removed some spurious blank spaces (by inserting % at appropriate locations (I hope I have remove all of them)), and changed to roman font for "John Svazic" (latin modern doesn't provide small caps in the sans serif family):
\documentclass[10pt,ebook,oneside,openright]{memoir}
\usepackage{color}
\usepackage{graphicx}
\usepackage{rotfloat}
\usepackage[scaled]{berasans}
\usepackage[T1]{fontenc}
\usepackage{type1cm}
\usepackage{eso-pic}
\usepackage{setspace}
\usepackage{listings}
\usepackage{lmodern}
\usepackage{tikz}
\usepackage{framed}
\ifpdf
\usepackage{pdfcolmk}
\fi
\pagestyle{ruled}
\setsecnumdepth{subsection}
\renewcommand*\familydefault{\sfdefault}
\floatstyle{boxed}
\restylefloat{figure}
\newlength{\drop}
\newcommand*{\titleGM}{%
\thispagestyle{empty}
\begingroup
\drop=0.1\textheight
\vspace*{\baselineskip}
\vfill
\hbox{%
\hspace*{0.2\textwidth}%
\rule{1pt}{\dimexpr\textheight-28pt\relax} % Vertical line
\hspace*{0.05\textwidth}%
\parbox[b]{0.75\textwidth}{%
\vbox{%
\vspace{\drop}
{\noindent\HUGE\bfseries Growing The\\[0.5\baselineskip]Money Tree}\\[2 \baselineskip]
{\Large\itshape Cultivating Financial Freedom One Leaf At A Time}\\[.37 \baselineskip]
{\Large\rmfamily\textsc{John Svazic}}\par
\vspace{0.47\textheight}
\noindent ARM Trading Press
\vspace*{\baselineskip}
}% end of vbox
}%
}
\endgroup}
\begin{document}
\titleGM | 2014-10-25 00:02:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6587514877319336, "perplexity": 10440.60631489437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119646554.36/warc/CC-MAIN-20141024030046-00295-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://openstudy.com/updates/5088b00ae4b0524190915770 | ## anonymous 3 years ago find the range of function (i)cos x/3
1. anonymous
$(ii) \frac{ x+2 }{ |x+2| }$
2. anonymous
(1) it is a cosine function so range always lies [-1,1] for all values of x (2) {-1,1} for all values of x except x=-2 for which function is not defined:) | 2016-09-27 15:47:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5423189401626587, "perplexity": 2240.666233843531}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661123.53/warc/CC-MAIN-20160924173741-00266-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://itectec.com/ubuntu/ubuntu-no-sound-after-pulseaudio-reinstall/ | # Ubuntu – No sound after PulseAudio reinstall
pulseaudiore-installationsoundvirtualbox
I encountered an audio bug and reported it on Launchpad. There I was asked to reinstall PulseAudio.
That sounded dangerous, so I tried it on a VirtualBox virtual machine I had lying around. (this VM had no sound problems at all). And indeed, after I did a naive sudo apt-get remove pulseaudio -> reboot -> sudo apt-get install pulseaudio, the VM has no sound any more.
Strangely, everything looks OK. There is nothing muted (I unmuted every single playback chanel in alsamixer just in case). The driver for the sound card is recognized and loaded, aplay plays sound files without producing an error, and when I installed and opened PulseAudio Volume Meter, playing a sound resulted in the volume indicator changing with the sound. Only there is nothing coming out of the loudspeakers. This isn't a hardware problem either, because the sound in the host system is functioning.
I tried to configure PulseAudio as described on this page of the Ubuntu wiki, but had no success.
I am afraid to try the reinstall on my real system until I know how to do it without causing a problem. Did I miss a configuration step somewhere? Should I have reinstalled in some other way?
Both guest and host systems are Ubuntu 10.10 64 bit.
• Please amend your question to include the most current information from pactl stat in a Terminal and from the instructions at https://wiki.ubuntu.com/Audio/AlsaInfo. As an aside, I've contacted Fabio (the person who asked you to reinstall pulse) with clarifications to his triaging process. Thanks for your patience. | 2021-06-14 09:01:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.250511109828949, "perplexity": 2816.24575732065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487611641.26/warc/CC-MAIN-20210614074543-20210614104543-00221.warc.gz"} |
https://mathzsolution.com/algebra-what-allows-us-to-do-the-same-thing-to-both-sides-of-an-equation/ | # Algebra: What allows us to do the same thing to both sides of an equation?
I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. What I do not understand is why making a new equation (adding or taking away from an expression) allows one to know what an unknown represents. What about equations lets us do this?
## Equations
Here’s one way of looking at it: if $$a=b$$, then $$f(a)=f(b)$$, no matter the function $$f(x)$$.
From this point of view, solving an equation amounts to applying a sequence of functions in order to generate an equation whose solutions are easy to read off. What this sort of argument shows is that the new equations are logical consequences of the original equation.
So, for example, suppose you’re given that $$x+1=2$$
and you want to subtract one from both sides. Then just apply the function $$f(s)=s-1$$, yielding $$x=f(x+1)=f(2)=1$$
Aren’t we done? We’ve isolated $$x$$, so what more is there to do? The problem is that high school trains us to stop thinking once we get to this point. (Actually it trains us not to think about what we’re doing at all and instead to rely on the process to do the thinking for us. But manipulating symbols, by itself, doesn’t constitute a mathematical argument. Experts often omit the details, but that’s because they know how to fill them in if they needed to; beginners should be taught how to fill in the logical details above and beyond the symbol manipulation.)
Indeed there are some subtleties here. First of all, just because this last equation is a consequence of the original equation doesn’t mean the last implies the original. (That would amount to the very common mistake of thinking a conditional and its converse are logically equivalent.) In other words, for an arbitrary function $$f(a)=f(b)$$ need not imply that $$a=b$$: the operation you perform on both sides might not be reversible. (It was in the example I just gave because the function I applied was linear, and all (non-constant) linear functions have inverses that don’t require domain restrictions, which makes the transformation “reversible.” Unfortunately in school almost all the examples we start out with are linear, so we have our intuition about equation-solving trained on a very special set of examples, which don’t illustrate what can happen in general.)
The failure of $$f(a)=f(b)$$ to imply $$a=b$$ explains why certain operations—for example, squaring both sides—might generate “extraneous solutions.” I put quotation marks around that phrase, because it’s something of a misnomer: they aren’t actually solutions to the (original) equation, precisely because they’re extraneous. So, for example, if you apply the function $$f(s)=s^2$$ to the equation $$x=1$$ you deduce that $$x^2=1$$. You could then apply the function $$g(s)=\sqrt{s}$$ to deduce that $$|x|=1$$. At this stage you could analyze the problem into cases (depending on whether $$x$$ is positive or negative) using the definition of absolute value and deduce that either $$x=1$$ or $$x=-1$$. But this doesn’t mean that either answer is a solution to the original equation. (Obviously $$x=-1$$ doesn’t satisfy the original equation!) That’s because the second step, of squaring both sides, isn’t reversible. The chain of implication doesn’t flow all the way backwards.
Another subtlety is that applying a certain transformation to both sides may require you to make an assumption without even realizing it. In other words, some operations, such as dividing by $$x$$, tacitly carry certain restrictions. The function $$f(s)=\frac{s}{x}$$, for example, requires that $$x\neq0$$; otherwise the value of the function doesn’t make sense. So if you have $$x^2=x$$ and you apply the function $$f(s)=\frac{s}{x}$$ to both sides, you’re tacitly assuming that $$x\neq0$$. That’s why in other cases you might lose solutions rather than generate extraneous ones.
Of course, not all equations have solutions. For example, applying $$f(s)=s-x$$ to the equation $$x=x+1$$
yields $$0=1$$. What this argument shows is that $$(\exists x)(x=x+1)\implies0=1$$
By contraposition we conclude that $$\lnot(\exists x)(x=x+1)$$
or in other words that there is no value of $$x$$ that satisfies the equation $$x=x+1$$, because assuming there is such a value leads us into a contradiction. (As this example illustrates, if we’re being totally rigorous we should really pay attention to quantifiers. But that’s more than you’re asking.)
And some equations are, in fact, true for all values of the variables. Such equations are called identities. A silly example is
$$x=x$$
but a slightly more interesting example is
$$x^2-1=(x+1)(x-1)$$
If you try to solve identities like this one, you’ll produce a tautology like $$0=0$$.
## Inequalities
For what it’s worth, you can also extend this idea to the logic of inequalities. If you apply a function $$f(x)$$ to the statement $$a, you’d typically like to conclude something like $$f(a) or $$f(a)>f(b)$$. In other words, you want to know whether the process preserves the direction of the inequality or reverses it.
But to draw such a conclusion, you generally need to know whether $$f(x)$$ is
• increasing, i.e., $$a or
• decreasing, i.e., $$af(b)$$
on the interval from $$a$$ to $$b$$. So, for example, $$f(x)=x+2$$ is always increasing, and $$g(x)=-x$$ is always decreasing, so applying $$f$$ to $$a yields
$$a+2=f(a)
but applying $$g$$ yields
$$-a=g(a)>g(b)=-b$$
This second fact is just what we mean when we say “multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality.” It’s simply a consequence of the fact that the function $$f(x)=-x$$ is decreasing.
How about squaring both sides of an inequality? In that case we’re dealing with the function $$h(x)=x^2$$. Well, $$h(x)$$ is decreasing on $$(-\infty,0)$$ and increasing on $$(0,\infty)$$, so you have to be careful about “squaring both sides” of an inequality. If $$a and $$b<0$$, then $$h$$ is decreasing on $$(a,b)$$, so
$$a^2=h(a)>h(b)=b^2$$
but if $$a>0$$ then the inequality is reversed.
## Summary
The moral of this story is:
when you do something to both sides of an equation or inequality, think carefully about what function you’re applying. In particular, for equations, think about whether
• It’s invertible on the same domain you start out with.
• Its application requires you to make any assumptions.
For inequalities, you also want to think about whether
• the function is increasing or decreasing on the interval defined by the inequality you’re starting from. | 2023-01-30 18:02:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 63, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7980484366416931, "perplexity": 205.68677529131688}, "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-2023-06/segments/1674764499826.71/warc/CC-MAIN-20230130165437-20230130195437-00723.warc.gz"} |
https://sourceforge.net/p/jmol/mailman/jmol-developers/?viewmonth=201308 | ## jmol-developers — Detailed discussions about the source code by and for Jmol developers
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[Jmol-developers] code revision of user interface for the JSmol pop-up menu From: Angel Herráez - 2013-08-31 23:54:46 Attachments: text/plain coremenu_AH.zip Dear Bob, I've worked on some retouching of the user interface of the JSmol pop-up menu. I couldn't find what is the source file that produces this menu, so I downloaded the production file, http://chemapps.stolaf.edu/jmol/jsmol/j2s/core/coremenu.z.js dated 19-Aug-2013_17:25 and worked from that I am attaching - the original file (unpacked in some online beautifier´) - my modified file (unpacked) - a Unix diff file between them, for guidance I have tested in localhost and everything seems to behave well. My motivations: - make the CSS rules isolated, i.e. applied only to jmolPopupMenu and not to other jQuery.menu instances that might be in the page - make it more compact (less tall, much less wide) - avoid the separators being visually active elements (on hover) - apply greyed-out style to disabled elements (rather than translucency), and make them visually inactive to mouse hover - avoid the # change in the url (by removing the href; this also seems to make the checkboxes more responsive) (Note) - added a ">" sign to expandable menu items - removed some css code that is not used by jmolPopupMenu - a more standard layout of html label tags, ie not surrounding the input tag The CSS is rather simplified, which should make easier for page developers to override some rules if they need (e.g. for finger-based devices). Still, there's quite a deal of mystery in some rules inherited from jQueryUI. I cleared up all that looked safe. Hope you can incorporate the changes. I think it feels quite better. --Note--: I tried to replace the a tags by span with a specific class (after all, they are not hyperlinks nor anchors), but it does not work. Seems that other code in the js file (coming from JQuery UI menu I believe) relies on them being 'a' tags, and I'd rather not mess up with that code. A flat seems to be fine, even if not semantically pure.
Re: [Jmol-developers] JSmol and java.lang.Error From: Gareth Oakes - 2013-08-21 22:52:51 > From: Robert Hanson > Subject: Re: [Jmol-developers] JSmol and java.lang.Error > > It's a pretty simple solution. Really what the problem is is that both >Jmol > and MathJax are competing for the AJAX processing. For compatibility with > MathJax, make sure: [snip] Thanks Bob we'll try your suggestion! -Gareth
Re: [Jmol-developers] JSmol and java.lang.Error From: Gareth Oakes - 2013-08-21 22:19:13 > Date: Tue, 20 Aug 2013 09:28:37 -0500 > From: Robert Hanson > > Yes, that's an important issue. I'm not satisfied, either, for just this > reason. It's not an easy one, because it relates to the Java2Script > compiler we use. There are many extensions we do need to use - especially > in the number area. Not sure about JavaScript error processing, although > obviously we need to catch those for our own purposes. I think the problem is to do with the fact that for some reason the java.lang.Error object is over-writing the core Javascript Error object. I would expect the Java2Script should place java.lang.Error into its own object (eg. a javascript object called "Java"). I don't know anything about Java2Script though so there may be good reasons why that isn't the case. > If you point me to a problem page and point me to the bit of your MathJax > code that is problematic, I can see what I can do. Thanks for the offer Bob. I will send some samples to you off-list. Cheers, Gareth
Re: [Jmol-developers] JSmol and java.lang.Error From: Robert Hanson - 2013-08-20 16:27:23 Attachments: Message as HTML It's a pretty simple solution. Really what the problem is is that both Jmol and MathJax are competing for the AJAX processing. For compatibility with MathJax, make sure: 1) you are using dynamic loading of the applet -- no
Re: [Jmol-developers] Jmol integration into Java application (scripting question) From: Robert Hanson - 2013-08-20 14:36:33 Attachments: Message as HTML Oh, yes, sure. If you want to use the file dialog, just change the setting of defaultDropScript to your liking. Something like: set defaultDropScript 'load "%FILE"; wireframe -0.2; vibration on see http://chemapps.stolaf.edu/jmol/docs/?ver=13.2#set_defaultdropscriptfor a more complex example. If you are using the JmolViewer interface, just add whatever you want to the load script and use JmolViewer.script() Bob On Tue, Aug 20, 2013 at 5:35 AM, Andrei Ionut Apopei < andrei_ionut1987@...> wrote: > > Dear Jmol developers, > > Currently I'm trying to integrate a Jmol functionality into a Java > application. So far I founded a good starting point at this link: > http://old.nabble.com/Integrating-Jmol-into-an-application-td31583044.html > . > > My OS: Windows 7 > Jmol version: 13.2.3 > > It works great, but I wanna know if it is possible to put some scripts > (specific zoom, atom selection, vibration on, and so on) to load at the > beginning (when the file load e.g. Caffeine.xyz). How can I achieve this? > > Thanks, > Andrei > > > ------------------------------------------------------------------------------ > Introducing Performance Central, a new site from SourceForge and > AppDynamics. Performance Central is your source for news, insights, > analysis and resources for efficient Application Performance Management. > Visit us today! > http://pubads.g.doubleclick.net/gampad/clk?id=48897511&iu=/4140/ostg.clktrk > _______________________________________________ > Jmol-developers mailing list > Jmol-developers@... > https://lists.sourceforge.net/lists/listinfo/jmol-developers > > -- Robert M. Hanson Larson-Anderson Professor of Chemistry St. Olaf College Northfield, MN http://www.stolaf.edu/people/hansonr If nature does not answer first what we want, it is better to take what answer we get. -- Josiah Willard Gibbs, Lecture XXX, Monday, February 5, 1900
Re: [Jmol-developers] JSmol and java.lang.Error From: Robert Hanson - 2013-08-20 14:31:54 Attachments: Message as HTML [you must be working with Gareth, right?] So what is MathJax, and what does it have to do with Jmol? Sounds interesting.... There are some general browser page-reloading issues with JavaScript in general. I think it can depend upon whether you press ENTER in the URL line or press whatever button does a reload. Looks to me like it handles memory differently. Not sure. Bob On Tue, Aug 20, 2013 at 4:52 AM, Geoffrey Rowland wrote: > Have not investigated this in any detail, but have also noticed a clash > between JSmol and MathJax in some circumstances - with MathJax displaying a > [Math Processing Error]. > > In case it helps with the diagnosis, here are the symptoms I have > observed. Behaviour is somewhat variable. Firefox appears least > susceptible, Chrome and IE more so. Sometimes a page reload fixes the > issue, sometimes it breaks a working display. The problem appears worse for > pages with more, or more complex, JSmol instances. So, perhaps related to > the relative load/processing times for JSmol and MathJax? Get the problem > whether MathJax is loaded locally or from CDN. > > Cheers > > Geoff > > > ------------------------------------------------------------------------------ > Introducing Performance Central, a new site from SourceForge and > AppDynamics. Performance Central is your source for news, insights, > analysis and resources for efficient Application Performance Management. > Visit us today! > http://pubads.g.doubleclick.net/gampad/clk?id=48897511&iu=/4140/ostg.clktrk > _______________________________________________ > Jmol-developers mailing list > Jmol-developers@... > https://lists.sourceforge.net/lists/listinfo/jmol-developers > > -- Robert M. Hanson Larson-Anderson Professor of Chemistry St. Olaf College Northfield, MN http://www.stolaf.edu/people/hansonr If nature does not answer first what we want, it is better to take what answer we get. -- Josiah Willard Gibbs, Lecture XXX, Monday, February 5, 1900
Re: [Jmol-developers] JSmol and java.lang.Error From: Robert Hanson - 2013-08-20 14:28:45 Attachments: Message as HTML Gareth, Right list for sure. Yes, that's an important issue. I'm not satisfied, either, for just this reason. It's not an easy one, because it relates to the Java2Script compiler we use. There are many extensions we do need to use - especially in the number area. Not sure about JavaScript error processing, although obviously we need to catch those for our own purposes. If you point me to a problem page and point me to the bit of your MathJax code that is problematic, I can see what I can do. Bob On Tue, Aug 20, 2013 at 12:12 AM, Gareth Oakes wrote: > Hi there, > > I'm not sure if this is the correction mailing list or not, but wanted to > report an issue with JSmol. We are working on integrating JSmol into a web > site which already uses MathJax and we ran into an incompatibility issue. > We have a crude workaround but I wanted to explore the root cause. > > I have discovered that upon execution of the JSmol getApplet() method, the > java2script Clazz system seems to replace the standard Javascript Error > object with a copy of java.lang.Error. This causes MathJax to fail because > it expects to have the standard Javascript Error object available. > > Why are objects in the global Javascript namespace being modified and is > there a way to prevent it? > > BTW I discovered this in test file "jsmol.htm" which ships with jsmol-13.3.3.zip. > My test environment was Chrome 28.0.1500.95 on Windows XP. > > Cheers, > Gareth Oakes > > > ------------------------------------------------------------------------------ > Introducing Performance Central, a new site from SourceForge and > AppDynamics. Performance Central is your source for news, insights, > analysis and resources for efficient Application Performance Management. > Visit us today! > http://pubads.g.doubleclick.net/gampad/clk?id=48897511&iu=/4140/ostg.clktrk > _______________________________________________ > Jmol-developers mailing list > Jmol-developers@... > https://lists.sourceforge.net/lists/listinfo/jmol-developers > > -- Robert M. Hanson Larson-Anderson Professor of Chemistry St. Olaf College Northfield, MN http://www.stolaf.edu/people/hansonr If nature does not answer first what we want, it is better to take what answer we get. -- Josiah Willard Gibbs, Lecture XXX, Monday, February 5, 1900
[Jmol-developers] Jmol integration into Java application (scripting question) From: Andrei Ionut Apopei - 2013-08-20 10:35:15 Attachments: Message as HTML Dear Jmol developers, Currently I'm trying to integrate a Jmol functionality into a Java application. So far I founded a good starting point at this link: http://old.nabble.com/Integrating-Jmol-into-an-application-td31583044.html. My OS: Windows 7 Jmol version: 13.2.3 It works great, but I wanna know if it is possible to put some scripts (specific zoom, atom selection, vibration on, and so on) to load at the beginning (when the file load e.g. Caffeine.xyz). How can I achieve this? Thanks, Andrei
[Jmol-developers] JSmol and java.lang.Error From: Geoffrey Rowland - 2013-08-20 09:52:20 Attachments: Message as HTML Have not investigated this in any detail, but have also noticed a clash between JSmol and MathJax in some circumstances - with MathJax displaying a [Math Processing Error]. In case it helps with the diagnosis, here are the symptoms I have observed. Behaviour is somewhat variable. Firefox appears least susceptible, Chrome and IE more so. Sometimes a page reload fixes the issue, sometimes it breaks a working display. The problem appears worse for pages with more, or more complex, JSmol instances. So, perhaps related to the relative load/processing times for JSmol and MathJax? Get the problem whether MathJax is loaded locally or from CDN. Cheers Geoff
[Jmol-developers] JSmol and java.lang.Error From: Gareth Oakes - 2013-08-20 05:12:21 Attachments: Message as HTML Hi there, I'm not sure if this is the correction mailing list or not, but wanted to report an issue with JSmol. We are working on integrating JSmol into a web site which already uses MathJax and we ran into an incompatibility issue. We have a crude workaround but I wanted to explore the root cause. I have discovered that upon execution of the JSmol getApplet() method, the java2script Clazz system seems to replace the standard Javascript Error object with a copy of java.lang.Error. This causes MathJax to fail because it expects to have the standard Javascript Error object available. Why are objects in the global Javascript namespace being modified and is there a way to prevent it? BTW I discovered this in test file "jsmol.htm" which ships with jsmol-13.3.3.zip. My test environment was Chrome 28.0.1500.95 on Windows XP. Cheers, Gareth Oakes
Showing 10 results of 10 | 2018-02-22 09:31:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2405421882867813, "perplexity": 8252.19573091978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814079.59/warc/CC-MAIN-20180222081525-20180222101525-00216.warc.gz"} |
https://socratic.org/questions/what-is-the-domain-of-y-tan-3-x-3 | What is the domain of y=tan^3(x) +3?
Jul 29, 2018
Domain: $x \ne \left(2 k + 1\right) \frac{\pi}{2} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$
Explanation:
The period of $y = a {\tan}^{n} \left(b x + c\right) + d , n = 1 , 2 , 3 , \ldots$ is $\frac{\pi}{\left\mid b \right\mid}$ #.
The asymptotes are given by
$b x + c = \left(2 k + 1\right) \frac{\pi}{2} \Rightarrow x = \frac{1}{b} \left(\left(2 k + 1\right) \frac{\pi}{2} - c\right) ,$
$k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$
So,
the period of $y = {\tan}^{3} x + 3 : \pi$
The asymptotes: $x = \left(2 k + 1\right) \frac{\pi}{2} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$
$\Rightarrow$ the domain is given by
$x \ne \left(2 k + 1\right) \frac{\pi}{2} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$
See graph, with asymptotes.
graph{(y - (tan(x ))^3 - 3 )(x-1/2pi+0.001y)=0} | 2019-03-19 00:25:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8785733580589294, "perplexity": 11351.194941353928}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201812.2/warc/CC-MAIN-20190318232014-20190319013523-00078.warc.gz"} |
http://www.lmfdb.org/LocalNumberField/?p=11&n=13 | Polynomial $p$ $e$ $f$ $c$ Galois group Slope content
x13 - x + 3 11 1 13 0 $C_{13}$ (as 13T1) $[\ ]^{13}$
x13 - 11 11 13 1 12 $F_{13}$ (as 13T6) $[\ ]_{13}^{12}$ | 2019-08-25 20:07:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8105462193489075, "perplexity": 991.4547954882001}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330800.17/warc/CC-MAIN-20190825194252-20190825220252-00326.warc.gz"} |
https://www.borealisai.com/research-blogs/machine-learning-certification-approaches-and-challenges/ | Over the last decade, machine learning (ML) has revolutionized many industries, with ML models finding success in a variety of settings such as recommendation systems, speech and image recognition. However, research has shown that these models may be vulnerable to a plethora of failures such as fairness, bias, robustness issues and vulnerability to privacy leaks. Figure 1 outlines several of these failures and where they may affect an ML model during its pipeline. For more information on some of these failures and ways to mitigate them, you can find our posts on fairness and bias in AI, privacy and adversarial robustness. As a consequence, countries and international organizations are looking at ways to ensure ML and AI, in general, are developed in a responsible manner.
Figure 1. ML model failures, sourced from finlayson et al., 2019, with modifications.
Machine learning certification is the process by which we ensure that an ML model is robust to one or more of these failures. While ML certification is currently at its infancy with many open questions, it has many potential benefits such as mitigating the harmful effects of the aforementioned failures, establishing trust in the model, improving explainability, and helping comply with any regulatory and legal requirements. This can be of paramount importance in numerous settings, especially in ones where the ML models have life-changing impact – such as autonomous vehicles, medical and financial domains. ML Certification is a holistic process that should be ingrained into the entire pipeline of an ML project, from the requirements gathering phase to monitoring the deployed model during production. At its core, ML Certification can be regarded as filling out the statement below:
## Certification Statement
“The submitted machine learning model is known to abide by the property of …(a)… up to a threshold of …(b)… over the following input range: …(c)… .”
Here, property (a) in the template above refers to the absence of a failure. For instance adversarial attack is a well-studied failure in ML models, and the corresponding property of adversarial robustness refers to robustness of these attacks. While in the ideal world, we would wish to certify the model for each property against all possible thresholds and input ranges, certifying an ML model is computationally expensive. Therefore, we identify one or more thresholds (b) and one or more input ranges (c) in the template above, making this process more tractable. As a result, we end up with several filled templates, otherwise known as specifications, each with a unique combination of property, threshold and input range to certify an ML model against. Figure 2 outlines the steps in the ML certification process. Broadly, this process consists of two steps:
Figure 2. Flowchart outlining the ML certification process.
1. Determine the specification, which consists of a unique combination of property, threshold and input range for which to certify the ML model against.
2. Verify whether the ML model satisfies the given specification.
In step 1 of the ML certification process, we fill in the blanks (a, b, and c) in the template above, resulting in one or more specifications (filled statements) that we wish to certify an ML model against. Subsequently, in step 2 we verify whether the ML model satisfies the specifications.
## Determine specification
The first step of certification consists of determining the property, threshold and input range (collectively known as the specification) for which to certify the ML model against. It is a holistic process integrated into the project life cycle, and should begin very early in the project life cycle. Ideally, this process starts during the requirements gathering phase of the project in order to ensure clarity of the security requirements of the desired ML model like all other requirements. In addition, more specifications may need to be added or current ones may need to be revised based on any changes in project development.
Determining specification requires significant subject matter expertise on the use case, the limitations of the ML model architecture, including the parameters and hyper-parameters used, and the limitations of the deployment environment, including the type of users this ML model is exposed to. As a consequence, identifying these specifications, including the careful selection of thresholds, should be done in collaboration with all stakeholders of the project. Currently, there is limited literature on how to identify specifications, including ways on explaining the significance of the chosen threshold and input range.
#### A simple specification example
As an example to motivate the next part of our post, consider a simple example where a ‘model’ is represented by the equation $y = x^2$, and a specification that defines the property of adversarial robustness, which states that a small change in the input should not drastically change the output. More specifically, given a model M, input value x and threshold ϵ, this simplistic definition of adversarial robustness states that $\forall x’ s.t. |x – x’| < \epsilon, |M(x) – M(x’)| < \epsilon$. In other words, all possible inputs $x′$ within the range of x should result in the model’s output to be within ϵ range of it’s output on $x$. We use a threshold of 0.3, and an input value of 1, which results in the input range of [0.7,1.3]. Figure 3 provides a visual of the ‘model’, as well as outlines the bounding box that represents the input and output values for which we wish to certify the property of adversarial robustness.
Figure 3. Target ‘model’ $y = x^2$ with bounding box that outlines the input and output range determined based on the threshold (0.3) and input value (1) provided. This bounding box is the region where we wish to certify the property (adversarial robustness).
## Verify specification
The second step of certification consists of verifying whether the ML model satisfies the specifications that have been determined in the previous step. In other words, we check if the ML model is robust to the failures identified in the property for the given thresholds and input ranges in this step. This should usually be done after model training but before the final evaluation of the model on the test set to ensure that we are able to modify the model without introducing bias since typically the model should not be modified after the final evaluation. There are three main approaches identified in the literature that can verify whether a model satisfies a given specification:
1. Constraint-based verification, where the model, as well as the specifications, are converted to a set of constraints and a solver is used to exhaustively search for violations of the specification.
2. Abstraction-based verification, where abstraction of the input region that encapsulates the specification in propagated through the ML model and the output region is checked for violations of the specification.
3. Duality-based verification, where verification is viewed as an optimization problem and a corresponding dual is computed to determine the worst case violation, which is compared against the specification.
We next provide an overview of each of these approaches.
#### Constraint-based verification
Main idea: Figure 4 shows the main idea of the constraint-based verification approach. In this approach, the ML model, as well as the specification, is converted to a set of constraints – often using boolean or some other formal logic such as mixed integer linear programming. The resulting set of constraints is designed such that finding a solution results in finding an input to the ML model that is within the identified input range but violates the specification (or in other words, a counter-example proving that the model does not satisfy the specification).
Figure 4. Flowchart outlining the constraint-based verification process.
These constraints are provided to a formal logic solver, which exhaustively searches over the input range provided in the specification in order to find whether violations exist. If a violation is found, then that is proof that the model does not satisfy the specification. However, if the solver is unable to find a violation, then that is proof that one does not exist since the solver exhaustively searches over the input region. While exhaustively searching over the input range requires exponential time w.r.t the size of the input region in the worst case, solvers use many optimization techniques in order to more efficiently search over the input region in practice. In order to get a better understanding of this approach, you can find our excellent tutorials that cover SAT solvers.
Verifying the example specification: In order to verify the example specification using the constraints-based approach, the model $y = x^2$, as well as the specification of adversarial robustness, are first converted to a set of constraints. Let us denote the constraints encoding the model as F and constraints encoding the specification as G. We provide as input to the formal solver the formula F ∧¬G where the variables are defined over the input range identified in the specification. The solver exhaustively searches over this range for a satisfiable assignment. If it finds one, that assignment refers to an input that is a successful adversarial attack on the model – proving that the model does not satisfy the property. In case the solver does not find a satisfying input to this formula, this is a guarantee that one does not exist.
Advantages: There are several advantages to this approach. Firstly, formal language solvers are used in many settings, and there is a lot of literature and tools available which can be leveraged to verify ML models. Secondly, this approach can provide guarantees since it exhaustively searches over the entire input space. Finally, since formal logic is a rich language, we can encode a broad range of specifications in order to verify an ML model.
Challenges: There are, however, some key limitations to this approach currently. The resulting formula of the ML model and specification can be very large and pose significant scalability and storage issues. Further, in the worst case, this approach requires exponential time with respect to the input range and can therefore take an exceptionally long time to solve (in the order of many years). Finally, while theoretically, many types of ML models can be converted to constraints, there is a considerable gap in the available tools to convert many types of ML models (e.g. no tools exist to convert various neural network architectures such as sigmoid activation functions).
Further reading: An introduction to constraints-based verification of machine learning (specifically neural networks) can be found here.
## Abstraction-based Verification
Figure 5. Flowchart outlining the abstraction-based verification process.
Main idea: Figure 5 shows the main idea of the abstraction-based verification approach. First, an abstraction of the input region identified in the specification is defined using a geometric object. Then, functions (known as abstract transformers) are used to propagate this geometric region through the ML model, which results in a corresponding output region. This (potentially infinite) output region is searched to find any violations of the specification. Because this approach uses an abstraction of the input region and the model, it usually over-approximates the search area. What this means is that if a violation is found in the output region, it does not guarantee that this violation is caused by a value in the true input region (identified by the specification), since it may have been from an input in the over-approximated region. In such a case, a refinement of the abstraction is necessary, which results in an increase in computational overhead. However, if no violation is found in the output region, then we can guarantee that the ML model satisfies the specification. There are several types of geometric regions that are used in this approach, which balance degree of over-approximation with computational resources required.
Verifying the example specification: To verify the example specification using an abstraction-based approach, we construct an approximation of the identified input region using a geometric object. This region is propagated through the model in order to obtain the corresponding abstract output region, which is then checked for violation of the specification. Figure 5 shows the result of three types of geometric regions that can be used to create this abstraction: interval, zonotope, and polyhedron domains. These regions are ordered in terms of increasing computational resources needed, which allow for an increase in the precision of the abstraction region, as can be seen in the figure. Given the definition of adversarial robustness used in the specification, we can check for violations in the output region by identifying whether there are any values in the output region that differ from the expected output region by ϵ. In other words, we check if the corresponding output region obtained via these geometric regions is a strict super-set of the expected output region (which is a square).
Figure 6. Result of abstraction-based verification on the example specification using interval (left), zonotope (center) and polyhedron (right) domains.
Advantages: Since the propagation is done in one forward pass of the model, abstraction-based verification is much faster than the constrained-based approach. In addition, this approach allows to effectively balance the trade-off between computation and precision by first using simpler geometric objects in order to check whether violations exist and refining that region using more complex objects as needed.
Challenges: This approach does pose certain challenges. Due to the possible over-approximation, the identified output region can extensively deviate from the true output region, resulting in situations where this approach may not find a solution. Further, the types of specifications it allows is restricted to ones where the input regions are contiguous, and so not all specifications may be verified using this approach.
Further reading: An introduction to abstraction-based verification of machine learning models (particularly neural networks) can be found here.
## Duality-based Verification
Main idea: Figure 7 provides an outline of duality-based verification. In this approach, verification is viewed as an optimization problem and a dual function is first constructed. The dual – or more specifically – the Lagrangian dual function relates to the original ‘primal’ function in such a way that the decision variables in the primal problem correspond to the constraints in the dual problem, and every constraint in the primal problem corresponds to a decision variable in the dual problem. An (upper-bound) optimal solution to the dual serves as a lower bound for the primal function. Because this dual function is constructed to be simpler to solve, we find its optimal solution and compare that with the threshold provided in the specification.
Figure 7. Flowchart outlining the duality-based verification process.
Verifying the example specification: In order to verify the example specification using duality-based approach,
Figure 8. Visualizing the primal and corresponding dual functions. Note that the axis in the primal (x) and dual ($\lambda$) are different, but the graphs of the two functions are given together for sake of clarity. Left: weak duality where there is a gap between the optimal solution of the primal and dual functions. Right: strong duality where the optimal solutions of primal and dual functions are equal, providing tighter bounds.
we first construct the corresponding dual to the primal problem of verifying the specification. Consider Figure 8, which showcases two possible duality functions. Note that the axis for the primal and dual function differ (x versus λ); however, we provide both functions on the same dimensions for better clarity. While the model in the specification is the equation y = x2, the primal and dual functions may not necessarily be the same equation. Nevertheless, for the sake of simplicity, we assume that both the primal and dual problems here are quadratic equations. The optimal solution to the dual function computed on the left figure is lower than the optimal solution to the primal problem. This type of dual function is known as weak duality, and the gap between the optimal functions is known as the dual gap. On the other hand, the optimal solution for the primal and dual functions are equal in the figure to the right, and this is known as strong duality. Ideally, we wish to construct a strong duality in order to get a tighter bound, but depending on the use case, we may wish to allow for a duality gap for the sake of finding a simpler dual function.
Once we compute the optimal solution for the dual function, we compare that against the specification. If the optimal solution is below the threshold, that means the model satisfies the given specification since the specification is better than even the worst-case bounds. On the other hand, if the worst case is greater than the specification, then we are not sure if the model violates the specification since there may be a significant duality gap.
Advantages: There are several advantages to the Duality-based verification approach. Firstly, this approach is computationally efficient, especially in the case of strong convexity in the dual function where gradient-based optimization techniques can be used. In addition, this approach allows to trade computation with precision by controlling the complexity of the dual function as well as the duality gap. Finally, since this approach uses optimization to find a solution, we can leverage algorithms that can speed up optimization, such as branch-and-bound techniques.
Challenges: As with other approaches, duality-based verification is not without its challenges. Similar to the abstraction-based verification, this approach can over approximate the solution due to the duality gap, and as a result, it can falsely state that there is a violation of the specification when that is not the case. In addition, depending on the choice of the solver, this approach may not provide complete guarantees. Finally, we may not be able to construct dual functions for all types of constraints and may only be able to apply it to a limited number of specifications.
Further reading: An introduction to duality-based verification of machine learning (emphasizing neural networks) can be found here.
## Conclusion
This concludes our post on machine learning certification, a process that ensures that an ML model is robust to one or more failures. This process consists of first determining the specification consisting of the property, threshold and input range for which to certify the model against and then verifying the specification to check whether the model satisfies the identified specifications using one of the identified verification approaches.
ML certification is an active area of research with a rich array of new works that aim to improve the number of properties, as well as the size and types of models that can be certified. A notable initiative here is the verification of neural networks competition (VNNCOMP) that aims to bring together the ML verification community, where researchers can submit tools to compete against others on verifying against a benchmark of ML models and specifications. In addition, a recent line of research (such as Adel et al., 2019) aims to either correct or replace the ML model with another of similar performance but known to abide by the desired specifications.
However, there are several open problems that need to be addressed in ML certification. Firstly, there are limited systematic ways of determining the specification to certify the ML model against. Conversely, there is a gap between mathematical definitions of specifications that are required to certify a model and explaining their real-world significance. For instance, effectively explaining to stakeholders the significance of the threshold and definition used for a given property to certify an ML model is an area that needs significant improvement. Further, due to the dangers of domain shift and drift, where the distribution of input data or relationship between data and labels alter over time – the ML model may need to be recertified for some or all specifications periodically, and we currently do not know how to determine the frequency of re-certification, as well as what that process should entail. Finally, if the verification process fails (i.e. the model is found not to abide by the specification), it is not clear what is the best remediation process. Perhaps the model architecture does not allow it to satisfy both the business and security objectives, in which case a new architecture should be selected instead of modifying the current model. Or perhaps the specification is too stringent and may need to be adjusted. Identifying the root cause in case of a failure is non-trivial and requires more investigation.
With governments and regulatory boards providing new guidelines on developing ML models, the landscape of ML certification is constantly shifting. ML certification is important because it helps improve the explainability of the model’s properties, establishes trust in the model, and helps comply with any internal or external regulations. More importantly, as firm believers in responsible AI, ML certification is the right thing to do.
In this blog post, we outline various approaches that provide some form of guarantee that a model satisfies the identified specifications. There are other approaches that do not offer any sort of guarantees – known as testing approaches. These approaches search for counter-examples, which are violations that show that the model does not satisfy a specification. The search can range from simple (random-based fuzzing) to complicated (natural evolutionary strategies). A survey of machine learning testing approaches can be found here. | 2022-12-07 20:16:06 | {"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": 0.5968025922775269, "perplexity": 483.1514718323822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00326.warc.gz"} |
http://www.wikihow.com/Calculate-Real-Estate-Commissions | Expert Reviewed
# wikiHow to Calculate Real Estate Commissions
If you buy or sell a real estate property, a commission will be owed to the brokers involved in buying or selling the property. Commissions are often paid by the seller, and the overall commission is split by the agent who worked on behalf of the seller and the agent who represented the buyer. Learning how commissions work and how they are calculated can help you to determine the cost of your property, or how much you will receive from a sale.
### Part 1 Calculating Common Real Estate Commissions
1. 1
Multiply the commission percentage by the value of the sale to find out your total commission. Because the commission is a percent amount, you will not know the dollar amount until the house sale is finalized. To estimate commission, however, simply multiply the percentage by the value of the property. Remember, to convert percentage to decimal first by dividing it by 100.
• Rate: 5.5%; House: $200,000 → .055 x 200,000 =$11,000
• Rate: 4.75%; Property: $325,000 → .0475 x 325,000 =$15,437.50
• Rate: 6.3%; House $132,000 → .063 x 132,000 =$8,316
• Formula = ${\displaystyle {\frac {Comm\%}{100}}*PropertyValue}$
2. 2
Familiarize yourself with common commission amounts. When you buy or sell a home, the broker receives a percentage of the sale value as commission. This is their payment for helping your buy or sell the house. This percentage typically ranges between 5% and 7%, with the average currently around 5.5%.[1]
3. 3
Discuss your specific commissions before signing any paperwork. Some brokers have arrangements where there will be a certain percentage charged on the first $100,000 of the home value, and a smaller percentage charged on the remainder of the house. On rare occasions, the commission is a flat fee.[2] If you buy a house for$225,000, and your realtor has a mixed commission (7% for the first $100,000, 3% for the rest), you would simply break the price up and calculate separately: •$225,000 - $100,000 =$125,000
• ($100,000 x 7% ) + (125,000 x 3%) • ($7,000) + ($3,750) • Total Commission =$10,750
4. 4
Remember that commission is already tacked onto final sale price. A commission is always deducted from the total value of the home sale price, not added to it. The seller pays it in one sense (since it comes out of the proceeds they receive for their sale), but the funds come from the buyer. For example, if you are selling a home for $200,000, and if the dollar value of the commission is$10,000, you will receive $190,000 for your purchase. • If you sold a house for$150,000 at 5% commission, you receive $142,500 on the sale, or$150,000 - $7,500 in commission. • If you buy a house for$225,000, and your realtor's commission is 4.6%, then you'll be paying your realtor $10,350, already included in the sale. • Note that these are the final sale prices. The house would be on the market for less before fees are added. 5. 5 Understand how commissions are split between brokers. The standard arrangement is that the broker representing the buyer and the broker representing the seller will each split the commission 50/50. At this point the broker would then split the commission 60/40 with the agent (this is the most common split). However, this changes from office to office, and you don't need to do the math as a buyer or seller. Note that if you choose not to use a broker, the seller's broker would receive the entire commission. It is possible, however, to negotiate with the broker to reduce the fee to the half they would usually receive in order to save money for yourself. • If you had$10,000 commission, $5,000 would go to the buyer's broker, and$5,000 would go to the seller's broker.
### Part 2 Calculating Total Cost of A Sale
1. 1
Settle on the commission amount ahead of the sale. Before selling a property, make sure to determine exactly what the commission will be in percentage form. Commissions are often negotiable, and do no not be afraid to ask for a reduction in commissions, especially if you are selling a high-value property.
• For this section, assume you settled on a 5% commission with your realtor for a 2-story ranch house in Georgia.
2. 2
Determine the property's gross sale price. Once the commission is determined, you need to determine the sale price of your property. Ask your agent for help in understanding the sale price of your home. The commission will be based on the total price of the home, not the amount the seller gets to keep after a mortgage or other lien is paid off. The sale price will only be finalized once you have agreed to an offer from a buyer and the appropriate legal documents have been signed and confirmed.
• Continuing the example, pretend this GA ranch is worth $200,000. • Note that gross sales price refers to the price of your home before any deductions are taken off. This means before any taxes, commissions, fees, etc. 3. 3 Calculate the commission by multiplying the gross sales price of the property by the commission percentage that was agreed upon. For example, our ranch that sold for$200,000 with a 5% commission rate would result in a $10,000 agent commission. Remember to convert the percentage to a decimal (by dividing by 100) before multiplying if your calculator does not have a "%" button.[3] • ${\displaystyle 200,000*5\%=10,000}$ 4. 4 Add taxes to the commission amount. Since commission is being paid in exchange for a service, the commission amount is often taxed just like any other purchase with a sales tax. Sales tax rates vary between states and countries. To calculate this, simply find out what the sales tax amount is (say, 4%), and multiply 4% by the commission amount. This will tell you the amount of tax that is owed, and you can simply add this amount to the total commission owing to obtain the total cost of the commission. • For example, multiply 4% (or 0.04) by our$10,000 commission and you get $400 in sales tax. This means your total commission would be$10,400.
5. 5
Subtract the commission from the total sale to determine your cut. To determine the net value you will receive for your home after commission, simply subtract the gross value from the commission paid. If the ranch's gross value is $200,000, and you have a total commission of$10,400, you would be left with $189,600 afterwards. ## Community Q&A Search Add New Question • How do I calculate the commission given the total price and the percentage of commission? wikiHow Contributor Multiply the percent of the commission by the total price. Remember, percent is parts per hundred. For example, if the commission is 5 percent, multiply by .05 (move the decimal place over two spots). Many calculators or phone apps have a percent key, which makes that easy. Example: Price is$40,000, commission 6 percent. Commission: .06 X $40,000=$2,400
• How can I figure out the cost of a home if the commission is 8% of the price of the home, and the commission is $6,240? wikiHow Contributor$6,240 Amount of Commission ÷ 8% Rate of Commission = $78,000 Sale Price of Home Math Check:$78,000 x 8% = $6,240 Unanswered Questions • How do commissions work if two people are trading/buying each others homes and both have agents? • How do I figure out$3 per every 1000?
• If the commission total was 13,350.00 and the listing office was holding deposits in the amount of 6,000. And the co broke was 2.5%, how would I write that up on a commission statement?
• If the buyer assumes the buyer's closing, will this reduce the selling price?
• The buyer's agent is getting 2.5% of the net sale of a home selling for 200,000. How much can the agent expect to get after the house gets their portion?
200 characters left
## Tips
• Ask agents if they would be willing to reduce their commission. Many real estate agents are willing to do this in tight markets, or if the house does not sell in a reasonable amount of time.
• Consider the amount of commission you will be paying when you sign a contract with a real estate agent. The commission comes out of the seller's profits, so you will want to compare what kind of service you get for a higher commission amount with those agents who offer to take a smaller commission.
## Things You'll Need
• Listing agreement
• Calculator
## Article Info
Categories: Real Estate
Thanks to all authors for creating a page that has been read 122,778 times. | 2017-03-26 17:06:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.21719814836978912, "perplexity": 2332.9797287998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189244.95/warc/CC-MAIN-20170322212949-00194-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://blog.marmakoide.org/?cat=6 | # Julia, Pyglet, and a GPU
A Julia fractal. Note the smooth, anti-aliased rendering. The color is picked as function of the estimated function to the Julia boundary.
Pyglet is a very good Python module, wrapping OpenGL and providing access to mouse and keyboards events. The API is really easy to learn, well-documented, with that "clean & lean" feel to it. I started to use Pyglet for work, and I positively love it.
So, Pyglet, allows you to summon the power of your graphic card using Python. And those days, even a puny low-end graphic card packs a lot of power : GPU, massively parallel computing. Although it's been around for years now, I never bothered tinkering with GPU, lack of time, other interests. Now that I feel comfortable with OpenGL, Python and Pyglet, however, harnessing this GPU goodness require very little effort. With years of delay, let's jump in that band-wagon already !
For a first try, I went for the done, done again, and overdone : fractals. More precisely, Julia fractals. You can see this fractal as function to iterate for each pixel of a picture, that takes as input the coordinate of the pixel.
$Z_n$ is a complex variable, with $Z_0$ being the coordinate of the pixel in the complex plane, and $C$ a complex constant. This function is iterated until $|Z_N|<2$ or a maximum count of iterations is reached. You then color the pixel depending on the number of iterations spent. Consider visiting the Wikipedia page about Julia fractals, there is a lot to learn about it there.
And you get a nice picture, a fractal. What happen if you play with the parameter $C$ ? The Julia fractal is fairly intensive to compute, rendering it in real-time requires some significant engineering... or a GPU, as the computations for each pixel are completely parallel. So what I did is to render the Julia fractal as a fragment shader, and control the $C$ parameter with the mouse. The code for this turns out to be dead simple, assuming you are familiar with the OpenGL API
import pyglet
from pyglet.gl import *
class JuliaWindow(pyglet.window.Window):
def __init__(self):
super(JuliaWindow, self).__init__(caption = 'julia', width = 512, height = 512)
self.C = (-0.70176, -0.3842)
)
def on_mouse_motion(self, x, y, dx, dy):
self.C = (6. * ((float(x) / window.width) - .5), 6 * ((float(y) / window.height) - .5))
def on_draw(self):
glMatrixMode(GL_PROJECTION)
glOrtho(-1., 1., 1., -1., 0., 1.)
glMatrixMode(GL_MODELVIEW)
glVertex2i(-1, -1)
glTexCoord2i(-2, -2)
glVertex2f(1, -1)
glTexCoord2i(2, -2)
glVertex2i(1, 1)
glTexCoord2i(2, 2)
glVertex2i(-1, 1)
glTexCoord2i(-2, 2)
glEnd()
window = JuliaWindow()
pyglet.app.run()
This code create a window, and display a quad polygon that fills the window. The texture for the quad is a procedural one, computed by a shader. Here, I use a neat utility class that encapsulate OpenGL API for the shaders : shader.py, by Tristam MacDonald. The $C$ parameter of the fractal is controlled by the mouse position in the window.
Of course, most of the magic happens inside the shaders. The vertex shader, julia.v.glsl, is not exactly rocket science.
#version 110
varying vec2 uv;
void
main() {
gl_Position = gl_ModelViewProjectionMatrix * gl_Vertex;
uv = vec2(gl_MultiTexCoord0);
}
All the actual job is done in the fragment shader, julia.f.glsl. Basically, I count how much iterations are required to have $|Z_N|<2$, and use this to pick a color. $C$ can be controlled from outside the shader, as uniforn variable. For sake of simplicity, I hard-coded the color scheme. One nice improvement would be to pick the color from a 1D texture, itself procedurally generated.
#version 110
varying vec2 uv;
int max_iter_count;
uniform vec2 C;
int iter_count;
vec2 Z, dZ;
vec2 sqr_Z;
float sqr_norm_Z, sqr_norm_dZ;
void
main() {
max_iter_count = 1024;
Z = uv;
dZ = vec2(1., 0.);
for(iter_count = 0; iter_count < max_iter_count; ++iter_count) {
sqr_Z = vec2(Z.x * Z.x, Z.y * Z.y);
if (sqr_Z.x + sqr_Z.y > 1e7)
break;
dZ = vec2(
2. * (Z.x * dZ.x - Z.y * dZ.y) + 1.,
2. * (Z.x * dZ.y + Z.y * dZ.x));
Z = vec2(
Z.x * Z.x - Z.y * Z.y,
2. * Z.x * Z.y) + C;
}
sqr_norm_Z = Z.x * Z.x + Z.y * Z.y;
sqr_norm_dZ = dZ.x * dZ.x + dZ.y * dZ.y;
vec4 color0 = vec4(1.0, 0.0, 0.0, 1.0);
vec4 color1 = vec4(1.0, 1.0, 0.0, 1.0);
gl_FragColor = mix(
color0,
color1,
sqrt(sqrt(sqr_norm_Z / sqr_norm_dZ) * .5 * log(sqr_norm_Z))
);
}
Note that here I use a variant on the classical Julia : I render a distance to the boundary of the fractal, the trick being from the amazing Inigo Quilez. It gives me anti-aliasing at a much cheaper cost than super-sampling. A few minutes of coding, and I got that instant enlightenment, with a feeling of achievement on the top of it. Those GPU are indeed great little pieces of silicon !
# Multiple pages PDF documents with Matplotlib
Matplotlib, what is that ? It is a software package to make plots, yet another one... but a really good one. Since Matplotlib is a Python module, plots are described in Python, rather than a (usually clumsy) custom language. So a script using Matplotlib can harness the full power of Python and its nice modules like Numpy. Say, reading compressed data file, doing some fancy statistics, etc. Moreover, Matplotlib is rather complete, providing a wide diversity of plots and render targets such as PNG, PDF, EPS. Finally, the quality of the plots is up to the standard of scientific publication. Originally a Gnuplot user, it's been a couple of years with Matplotlib and me, and I am very happy with it. Perfect ? Well, documentation for Matplotlib is a little bit scattered. I'm working on a tutorial, a Sisyphus-grade task... Rather than waiting for completing that looooong tutorial, a short recipe today !
# The problem
The problem : you want to have several plots on a single figure, you want to fit them on a A4 format, and you want to have this as a PDF document. The latter constraint, PDF document, is welcome, as the recipe is for PDF output only. It works for Matplotlib 1.1.x, and it mostly works for Matplotlib 1.0.x. It's a two step trick.
# First step: multiple pages & PDF
Only one Matplotlib back-end seems to support multiple pages, the PDF back-end. We have to explicitly instantiate that back-end and tells it that we want to go multiple pages. Here's how it's done
from matplotlib.backends.backend_pdf import PdfPages
pdf_pages = PdfPages('my-fancy-document.pdf')
Now, for each new page we want to create, we have to create a new Figure instance. It's simple, one page, one figure. We want a A4 format, so we can eventually print our nice document, for instance. That's not too hard, we have just to specify the proper dimensions and resolutions when creating a Figure. Once we are done with a page, we have to explicitly tell it to the back-end. In this example, the script creates 3 pages.
from matplotlib import pyplot as plot
from matplotlib.backends.backend_pdf import PdfPages
# The PDF document
pdf_pages = PdfPages('my-fancy-document.pdf')
for i in xrange(3):
# Create a figure instance (ie. a new page)
fig = plot.figure(figsize=(8.27, 11.69), dpi=100)
# Plot whatever you wish to plot
# Done with the page
pdf_pages.savefig(fig)
# Write the PDF document to the disk
pdf_pages.close()
# Second step: multiple plots on a figure
One A4 page is a lot of space, we can squeeze a lot of plots in one single figure. Recent Matplotlib versions (1.x.x and later) have now a nice feature to easily layout plots on one figure : grid layout.
plot.subplot2grid((5, 2), (2, 0), rowspan=1, colspan=1)
The first parameter of subplot2grid is the size of the grid. The second parameter is the coordinate of one element of the grid. That element will be filled with a plot. Finally, we can specify the span of the plot, so that its spans over several elements of the grid. This gives a great deal of control over the layout, without too much headaches. There is a small gotcha however: the coordinates are in row, col order. Here's an example of subplot2grid in action.
import numpy
from matplotlib import pyplot as plot
# Prepare the data
t = numpy.linspace(-numpy.pi, numpy.pi, 1024)
s = numpy.random.randn(2, 256)
#
# Do the plot
#
grid_size = (5, 2)
# Plot 1
plot.subplot2grid(grid_size, (0, 0), rowspan=2, colspan=2)
plot.plot(t, numpy.sinc(t), c= '#000000')
# Plot 2
plot.subplot2grid(grid_size, (2, 0), rowspan=3, colspan=1)
plot.scatter(s[0], s[1], c= '#000000')
# Plot 2
plot.subplot2grid(grid_size, (2, 1), rowspan=3, colspan=1)
plot.plot(numpy.sin(2 * t), numpy.cos(0.5 * t), c= '#000000')
# Automagically fits things together
plot.tight_layout()
# Done !
plot.show()
Grid layout in action
Note the tight_layout call, which pack all the plots within a figure automatically, and does a nice job at it. But it is in Matplotlib 1.1.x only... With Matplotlib 1.0.x I don't know yet how to do it. Why bothering ? Because say, at work, you might have to deal with Linux distributions are providing rather outdated versions of software in their official packages repositories, and even the unofficial ones are lagging. Yes, CentOS, I'm looking at you !
# Putting it together
Now, we can combine subplot2grid with the multiple page trick, and get multiple pages PDF documents which look good.
import numpy
from matplotlib import pyplot as plot
from matplotlib.backends.backend_pdf import PdfPages
# Generate the data
data = numpy.random.randn(7, 1024)
# The PDF document
pdf_pages = PdfPages('histograms.pdf')
# Generate the pages
nb_plots = data.shape[0]
nb_plots_per_page = 5
nb_pages = int(numpy.ceil(nb_plots / float(nb_plots_per_page)))
grid_size = (nb_plots_per_page, 1)
for i, samples in enumerate(data):
# Create a figure instance (ie. a new page) if needed
if i % nb_plots_per_page == 0:
fig = plot.figure(figsize=(8.27, 11.69), dpi=100)
# Plot stuffs !
plot.subplot2grid(grid_size, (i % nb_plots_per_page, 0))
plot.hist(samples, 32, normed=1, facecolor='#808080', alpha=0.75)
# Close the page if needed
if (i + 1) % nb_plots_per_page == 0 or (i + 1) == nb_plots:
plot.tight_layout()
pdf_pages.savefig(fig)
# Write the PDF document to the disk
pdf_pages.close()
A preview of the PDF document you should get from the example. Do not mind the ugly antialiasing, it's me who fumbled with the PDF to PNG conversion.
As you can see, a little bit of arithmetic is used to have the proper number of pages and create them at the right moment. It's a minimal example, but by now, you got the recipe. | 2018-10-19 14:04:26 | {"extraction_info": {"found_math": true, "script_math_tex": 9, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 10, "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": 0.47870776057243347, "perplexity": 4196.426792175642}, "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-2018-43/segments/1539583512400.59/warc/CC-MAIN-20181019124748-20181019150248-00186.warc.gz"} |
http://physics.stackexchange.com/questions/62256/confused-on-calculating-resistance-distance-matrix | # Confused on Calculating Resistance Distance Matrix
I am trying to create a computer program to compute the equivalent resistance over any points on any rectangular set of resistors (all with a resistance of 1 ohm). It seems that the resistance distance matrix is exactly what I need. I understand how to compute the inverse and what a Laplacian matrix is, but what is the difference between $\Gamma_{ii}$, $\Gamma_{jj}$, and $\Gamma_{ij}$ in $\Omega_{ij} = \Gamma_{ii}$ + $\Gamma_{jj} - 2\Gamma_{ij}$?
-
$\Gamma_{ii}$ is the $i$-th entry on the diagonal of $\Gamma = L^+ = (D-A)^+$, $\Gamma_{jj}$ is the $j$-th entry, and $\Gamma_{ij}$ is the entry located at row $i$, column $j$. Thus $\Omega_{ij}$ is a scalar, but you could assemble all such values into a matrix $\Omega$ that gives the resistances between all pairs of vertices. | 2016-02-09 23:01:54 | {"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": 0.9585679769515991, "perplexity": 70.14569790788244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701158130.6/warc/CC-MAIN-20160205193918-00088-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://zbmath.org/authors/?q=ai%3Awen.zaiwen | # zbMATH — the first resource for mathematics
## Wen, Zaiwen
Compute Distance To:
Author ID: wen.zaiwen Published as: Wen, Zaiwen; Wen, Zai-Wen; Wen, ZaiWen; Wen, Z.
Documents Indexed: 60 Publications since 2005
all top 5
#### Co-Authors
0 single-authored 12 Liu, Xin 12 Yuan, Ya-xiang 11 Yin, Wotao 10 Zhang, Yin 6 Goldfarb, Donald 4 Hu, Jiang 4 Ulbrich, Michael 3 Jiang, Bo 3 Li, Yongfeng 3 Liu, Haoyang 3 Milzarek, Andre 3 Xiao, Xiantao 2 Huang, JianChao 2 Jia, Zeyu 2 Lai, Rongjie 2 Wang, Xiao 2 Wang, Yanfei 2 Wu, Xinming 2 Xu, Fangfang 2 Zhang, Hongchao 2 Zhang, Junyu 1 Bao, Weizhu 1 Baraniuk, Richard G. 1 Cai, Yongyong 1 Cen, Shicong 1 Chen, Cheng 1 Dai, Yu-Hong 1 Dang, Cong D. 1 Dong, Qian 1 Duan, Yaqi 1 Gu, Xianfeng 1 Gu, Xiaoyi 1 Klöckner, Dennis 1 Lan, Guanghui 1 Laska, Jason N. 1 Li, Qingna 1 Lin, Chen 1 Lin, Lin 1 Ling, Qing 1 Liu, Yafeng 1 Lu, Zhaosong 1 Lui, Lok Ming 1 Ma, Chao 1 Marchesini, Stefano 1 Nashed, Zuhair 1 Nie, Jiawang 1 Pan, Shaohua 1 Pang, Tongyao 1 Qian, Peng-Yu 1 Scheinberg, Katya 1 Shen, Zuowei 1 Singer, Amit 1 Sun, Qiyu 1 Tian, Tonghua 1 Wang, Lanhui 1 Wang, Mengdi 1 Wang, Yifei 1 Wang, Zizhuo 1 Wen, Shouwen 1 Wu, Leqin 1 Xu, Yangyang 1 Yang, Yu 1 Ye, Yinyu 1 Yuan, Honglin 1 Yuan, Xiaoming 1 Zhang, Liwei 1 Zhang, Shuzhong 1 Zhou, Aihui 1 Zhu, Jinwei
all top 5
#### Serials
11 SIAM Journal on Scientific Computing 6 Journal of Scientific Computing 6 Journal of the Operations Research Society of China 5 Optimization Methods & Software 4 Journal of Computational Mathematics 4 SIAM Journal on Matrix Analysis and Applications 4 SIAM Journal on Optimization 3 Science China. Mathematics 2 Inverse Problems 2 IEEE Transactions on Signal Processing 2 SIAM Journal on Imaging Sciences 2 Mathematical Programming Computation 1 IEEE Transactions on Information Theory 1 SIAM Journal on Numerical Analysis 1 Science in China. Series A 1 Mathematical Programming. Series A. Series B 1 Applied and Computational Harmonic Analysis 1 International Journal of Wavelets, Multiresolution and Information Processing 1 Frontiers of Mathematics in China 1 Operations Research Transactions
all top 5
#### Fields
45 Operations research, mathematical programming (90-XX) 41 Numerical analysis (65-XX) 11 Linear and multilinear algebra; matrix theory (15-XX) 9 Calculus of variations and optimal control; optimization (49-XX) 7 Information and communication theory, circuits (94-XX) 4 Statistical mechanics, structure of matter (82-XX) 3 General and overarching topics; collections (00-XX) 3 Statistics (62-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Partial differential equations (35-XX) 2 Operator theory (47-XX) 2 Optics, electromagnetic theory (78-XX) 2 Quantum theory (81-XX) 2 Biology and other natural sciences (92-XX) 2 Systems theory; control (93-XX) 1 History and biography (01-XX) 1 Functions of a complex variable (30-XX) 1 Computer science (68-XX)
#### Citations contained in zbMATH Open
40 Publications have been cited 702 times in 518 Documents Cited by Year
Alternating direction augmented Lagrangian methods for semidefinite programming. Zbl 1206.90088
Wen, Zaiwen; Goldfarb, Donald; Yin, Wotao
2010
A feasible method for optimization with orthogonality constraints. Zbl 1281.49030
Wen, Zaiwen; Yin, Wotao
2013
Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Zbl 1271.65083
Wen, Zaiwen; Yin, Wotao; Zhang, Yin
2012
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization, and continuation. Zbl 1215.49039
Wen, Zaiwen; Yin, Wotao; Goldfarb, Donald; Zhang, Yin
2010
An alternating direction algorithm for matrix completion with nonnegative factors. Zbl 1323.65044
Xu, Yangyang; Yin, Wotao; Wen, Zaiwen; Zhang, Yin
2012
Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization. Zbl 1285.90068
Shen, Y.; Wen, Z.; Zhang, Yin
2014
Alternating direction methods for classical and ptychographic phase retrieval. Zbl 1254.78037
Wen, Zaiwen; Yang, Chao; Liu, Xin; Marchesini, Stefano
2012
On the convergence of an active-set method for $$\ell_1$$ minimization. Zbl 1244.49055
Wen, Zaiwen; Yin, Wotao; Zhang, Hongchao; Goldfarb, Donald
2012
Trust, buy verify: fast and accurate signal recovery from 1-bit compressive measurements. Zbl 1393.94314
Laska, Jason N.; Wen, Zaiwen; Yin, Wotao; Baraniuk, Richard G.
2011
Block coordinate descent methods for semidefinite programming. Zbl 1334.90118
Wen, Zaiwen; Goldfarb, Donald; Scheinberg, Katya
2012
Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization. Zbl 1298.30007
Lai, Rongjie; Wen, Zaiwen; Yin, Wotao; Gu, Xianfeng; Lui, Lok Ming
2014
A curvilinear search method for $$p$$-harmonic flows on spheres. Zbl 1193.49030
Goldfarb, Donald; Wen, Zaiwen; Yin, Wotao
2009
On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory. Zbl 1319.65041
Liu, Xin; Wang, Xiao; Wen, Zaiwen; Yuan, Yaxiang
2014
Limited memory block Krylov subspace optimization for computing dominant singular value decompositions. Zbl 1278.65045
Liu, Xin; Wen, Zaiwen; Zhang, Yin
2013
On the analysis of the discretized Kohn-Sham density functional theory. Zbl 1317.15008
Liu, Xin; Wen, Zaiwen; Wang, Xiao; Ulbrich, Michael; Yuan, Yaxiang
2015
A regularized Newton method for computing ground states of Bose-Einstein condensates. Zbl 1433.82025
Wu, Xinming; Wen, Zaiwen; Bao, Weizhu
2017
A line search multigrid method for large-scale nonlinear optimization. Zbl 1203.65095
Wen, Zaiwen; Goldfarb, Donald
2009
Trace-penalty minimization for large-scale eigenspace computation. Zbl 1373.65026
Wen, Zaiwen; Yang, Chao; Liu, Xin; Zhang, Yin
2016
Adaptive regularized self-consistent field iteration with exact Hessian for electronic structure calculation. Zbl 1273.82004
Wen, Zaiwen; Milzarek, Andre; Ulbrich, Michael; Zhang, Hongchao
2013
Orientation determination of cryo-EM images using least unsquared deviations. Zbl 1402.92449
Wang, Lanhui; Singer, Amit; Wen, Zaiwen
2013
An efficient Gauss-Newton algorithm for symmetric low-rank product matrix approximations. Zbl 1321.65060
Liu, Xin; Wen, Zaiwen; Zhang, Yin
2015
Gradient type optimization methods for electronic structure calculations. Zbl 1300.82006
Zhang, Xin; Zhu, Jinwei; Wen, Zaiwen; Zhou, Aihui
2014
A proximal gradient method for ensemble density functional theory. Zbl 1325.65095
Ulbrich, Michael; Wen, Zaiwen; Yang, Chao; Klöckner, Dennis; Lu, Zhaosong
2015
A regularized semi-smooth Newton method with projection steps for composite convex programs. Zbl 1394.90534
Xiao, Xiantao; Li, Yongfeng; Wen, Zaiwen; Zhang, Liwei
2018
A brief introduction to manifold optimization. Zbl 07292555
Hu, Jiang; Liu, Xin; Wen, Zai-Wen; Yuan, Ya-Xiang
2020
Adaptive quadratically regularized Newton method for Riemannian optimization. Zbl 1415.65139
Hu, Jiang; Milzarek, Andre; Wen, Zaiwen; Yuan, Yaxiang
2018
A new trust region algorithm for image restoration. Zbl 1078.94506
Wen, Zaiwen; Wang, Yanfei
2005
$$L_p$$-norm regularization algorithms for optimization over permutation matrices. Zbl 1353.65055
Jiang, Bo; Liu, Ya-Feng; Wen, Zaiwen
2016
A note on semidefinite programming relaxations for polynomial optimization over a single sphere. Zbl 1354.65123
Hu, Jiang; Jiang, Bo; Liu, Xin; Wen, ZaiWen
2016
A stochastic semismooth Newton method for nonsmooth nonconvex optimization. Zbl 1434.90108
Milzarek, Andre; Xiao, Xiantao; Cen, Shicong; Wen, Zaiwen; Ulbrich, Michael
2019
A sparse completely positive relaxation of the modularity maximization for community detection. Zbl 1405.90082
Zhang, Junyu; Liu, Haoyang; Wen, Zaiwen; Zhang, Shuzhong
2018
Global optimization with orthogonality constraints via stochastic diffusion on manifold. Zbl 1418.90206
Yuan, Honglin; Gu, Xiaoyi; Lai, Rongjie; Wen, Zaiwen
2019
Decentralized jointly sparse optimization by reweighted $$\ell_q$$ minimization. Zbl 1393.94338
Ling, Qing; Wen, Zaiwen; Yin, Wotao
2013
Robust linear optimization under matrix completion. Zbl 1308.65099
Wen, Shouwen; Xu, Fangfang; Wen, Zaiwen; Lin, Chen
2014
Subspace methods with local refinements for eigenvalue computation using low-rank tensor-train format. Zbl 1361.65021
Zhang, Junyu; Wen, Zaiwen; Zhang, Yin
2017
Accelerating convergence by augmented Rayleigh-Ritz projections for large-scale eigenpair computation. Zbl 1365.65096
Wen, Zaiwen; Zhang, Yin
2017
Globally convergent Levenberg-Marquardt method for phase retrieval. Zbl 1431.94018
Ma, Chao; Liu, Xin; Wen, Zaiwen
2019
A parallel line search subspace correction method for composite convex optimization. Zbl 1317.90234
Dong, Qian; Liu, Xin; Wen, Zai-Wen; Yuan, Ya-Xiang
2015
Structured quasi-Newton methods for optimization with orthogonality constraints. Zbl 1431.65085
Hu, Jiang; Jiang, Bo; Lin, Lin; Wen, Zaiwen; Yuan, Ya-Xiang
2019
A composite risk measure framework for decision making under uncertainty. Zbl 1424.90141
Qian, Peng-Yu; Wang, Zi-Zhuo; Wen, Zai-Wen
2019
A brief introduction to manifold optimization. Zbl 07292555
Hu, Jiang; Liu, Xin; Wen, Zai-Wen; Yuan, Ya-Xiang
2020
A stochastic semismooth Newton method for nonsmooth nonconvex optimization. Zbl 1434.90108
Milzarek, Andre; Xiao, Xiantao; Cen, Shicong; Wen, Zaiwen; Ulbrich, Michael
2019
Global optimization with orthogonality constraints via stochastic diffusion on manifold. Zbl 1418.90206
Yuan, Honglin; Gu, Xiaoyi; Lai, Rongjie; Wen, Zaiwen
2019
Globally convergent Levenberg-Marquardt method for phase retrieval. Zbl 1431.94018
Ma, Chao; Liu, Xin; Wen, Zaiwen
2019
Structured quasi-Newton methods for optimization with orthogonality constraints. Zbl 1431.65085
Hu, Jiang; Jiang, Bo; Lin, Lin; Wen, Zaiwen; Yuan, Ya-Xiang
2019
A composite risk measure framework for decision making under uncertainty. Zbl 1424.90141
Qian, Peng-Yu; Wang, Zi-Zhuo; Wen, Zai-Wen
2019
A regularized semi-smooth Newton method with projection steps for composite convex programs. Zbl 1394.90534
Xiao, Xiantao; Li, Yongfeng; Wen, Zaiwen; Zhang, Liwei
2018
Adaptive quadratically regularized Newton method for Riemannian optimization. Zbl 1415.65139
Hu, Jiang; Milzarek, Andre; Wen, Zaiwen; Yuan, Yaxiang
2018
A sparse completely positive relaxation of the modularity maximization for community detection. Zbl 1405.90082
Zhang, Junyu; Liu, Haoyang; Wen, Zaiwen; Zhang, Shuzhong
2018
A regularized Newton method for computing ground states of Bose-Einstein condensates. Zbl 1433.82025
Wu, Xinming; Wen, Zaiwen; Bao, Weizhu
2017
Subspace methods with local refinements for eigenvalue computation using low-rank tensor-train format. Zbl 1361.65021
Zhang, Junyu; Wen, Zaiwen; Zhang, Yin
2017
Accelerating convergence by augmented Rayleigh-Ritz projections for large-scale eigenpair computation. Zbl 1365.65096
Wen, Zaiwen; Zhang, Yin
2017
Trace-penalty minimization for large-scale eigenspace computation. Zbl 1373.65026
Wen, Zaiwen; Yang, Chao; Liu, Xin; Zhang, Yin
2016
$$L_p$$-norm regularization algorithms for optimization over permutation matrices. Zbl 1353.65055
Jiang, Bo; Liu, Ya-Feng; Wen, Zaiwen
2016
A note on semidefinite programming relaxations for polynomial optimization over a single sphere. Zbl 1354.65123
Hu, Jiang; Jiang, Bo; Liu, Xin; Wen, ZaiWen
2016
On the analysis of the discretized Kohn-Sham density functional theory. Zbl 1317.15008
Liu, Xin; Wen, Zaiwen; Wang, Xiao; Ulbrich, Michael; Yuan, Yaxiang
2015
An efficient Gauss-Newton algorithm for symmetric low-rank product matrix approximations. Zbl 1321.65060
Liu, Xin; Wen, Zaiwen; Zhang, Yin
2015
A proximal gradient method for ensemble density functional theory. Zbl 1325.65095
Ulbrich, Michael; Wen, Zaiwen; Yang, Chao; Klöckner, Dennis; Lu, Zhaosong
2015
A parallel line search subspace correction method for composite convex optimization. Zbl 1317.90234
Dong, Qian; Liu, Xin; Wen, Zai-Wen; Yuan, Ya-Xiang
2015
Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization. Zbl 1285.90068
Shen, Y.; Wen, Z.; Zhang, Yin
2014
Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization. Zbl 1298.30007
Lai, Rongjie; Wen, Zaiwen; Yin, Wotao; Gu, Xianfeng; Lui, Lok Ming
2014
On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory. Zbl 1319.65041
Liu, Xin; Wang, Xiao; Wen, Zaiwen; Yuan, Yaxiang
2014
Gradient type optimization methods for electronic structure calculations. Zbl 1300.82006
Zhang, Xin; Zhu, Jinwei; Wen, Zaiwen; Zhou, Aihui
2014
Robust linear optimization under matrix completion. Zbl 1308.65099
Wen, Shouwen; Xu, Fangfang; Wen, Zaiwen; Lin, Chen
2014
A feasible method for optimization with orthogonality constraints. Zbl 1281.49030
Wen, Zaiwen; Yin, Wotao
2013
Limited memory block Krylov subspace optimization for computing dominant singular value decompositions. Zbl 1278.65045
Liu, Xin; Wen, Zaiwen; Zhang, Yin
2013
Adaptive regularized self-consistent field iteration with exact Hessian for electronic structure calculation. Zbl 1273.82004
Wen, Zaiwen; Milzarek, Andre; Ulbrich, Michael; Zhang, Hongchao
2013
Orientation determination of cryo-EM images using least unsquared deviations. Zbl 1402.92449
Wang, Lanhui; Singer, Amit; Wen, Zaiwen
2013
Decentralized jointly sparse optimization by reweighted $$\ell_q$$ minimization. Zbl 1393.94338
Ling, Qing; Wen, Zaiwen; Yin, Wotao
2013
Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Zbl 1271.65083
Wen, Zaiwen; Yin, Wotao; Zhang, Yin
2012
An alternating direction algorithm for matrix completion with nonnegative factors. Zbl 1323.65044
Xu, Yangyang; Yin, Wotao; Wen, Zaiwen; Zhang, Yin
2012
Alternating direction methods for classical and ptychographic phase retrieval. Zbl 1254.78037
Wen, Zaiwen; Yang, Chao; Liu, Xin; Marchesini, Stefano
2012
On the convergence of an active-set method for $$\ell_1$$ minimization. Zbl 1244.49055
Wen, Zaiwen; Yin, Wotao; Zhang, Hongchao; Goldfarb, Donald
2012
Block coordinate descent methods for semidefinite programming. Zbl 1334.90118
Wen, Zaiwen; Goldfarb, Donald; Scheinberg, Katya
2012
Trust, buy verify: fast and accurate signal recovery from 1-bit compressive measurements. Zbl 1393.94314
Laska, Jason N.; Wen, Zaiwen; Yin, Wotao; Baraniuk, Richard G.
2011
Alternating direction augmented Lagrangian methods for semidefinite programming. Zbl 1206.90088
Wen, Zaiwen; Goldfarb, Donald; Yin, Wotao
2010
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization, and continuation. Zbl 1215.49039
Wen, Zaiwen; Yin, Wotao; Goldfarb, Donald; Zhang, Yin
2010
A curvilinear search method for $$p$$-harmonic flows on spheres. Zbl 1193.49030
Goldfarb, Donald; Wen, Zaiwen; Yin, Wotao
2009
A line search multigrid method for large-scale nonlinear optimization. Zbl 1203.65095
Wen, Zaiwen; Goldfarb, Donald
2009
A new trust region algorithm for image restoration. Zbl 1078.94506
Wen, Zaiwen; Wang, Yanfei
2005
all top 5
#### Cited by 961 Authors
28 Wen, Zaiwen 18 Ma, Shiqian 11 Yin, Wotao 11 Yuan, Ya-xiang 9 Goldfarb, Donald 9 Lai, Rongjie 9 Lu, Zhaosong 8 Huang, Ting-Zhu 8 Liu, Xin 7 Han, Deren 7 Jiang, Bo 7 Lui, Lok Ming 7 Sun, Defeng 7 Toh, Kimchuan 7 Yuan, Xiaoming 7 Zhang, Shuzhong 7 Zhao, Xile 6 Cheng, Wanyou 6 Steidl, Gabriele 6 Xiao, Yunhai 6 Xu, Yangyang 6 Zhu, Xiaojing 5 Boyd, Stephen Poythress 5 Chang, Huibin 5 Deng, Liangjian 5 Ji, Teng-Yu 5 Ke, Yifen 5 Li, Donghui 5 Lou, Yifei 5 Ma, Changfeng 5 Marchesini, Stefano 5 Osher, Stanley Joel 5 Qi, Liqun 5 Shen, Yuan 5 Xiu, Naihua 5 Zhang, Lei-Hong 5 Zhang, Yin 4 Absil, Pierre-Antoine 4 Chen, Yannan 4 Dai, Yu-Hong 4 He, Bingsheng 4 He, Hongjin 4 Hu, Jiang 4 Jiang, Tai-Xiang 4 Li, Rencang 4 Lin, Tianyi 4 Lu, Jianfeng 4 Ma, Feng 4 Ma, Tian-Hui 4 Parpas, Panos 4 So, Anthony Man-Cho 4 Tao, Min 4 Xiao, Lin 4 Zeng, Tie Yong 4 Zhang, Hongchao 4 Zhu, Hong 3 Bai, Yanqin 3 Bai, Zhaojun 3 Becker, Stephen R. 3 Cai, Xingju 3 Cheng, Lizhi 3 Choi, Gary Pui-Tung 3 Dalmau, Oscar 3 Francisco, Juliano B. 3 Gao, Bin 3 Gu, Xianfeng 3 Hager, William W. 3 Jiao, Yuling 3 Koutsourelakis, Phaedon-Stelios 3 Lara, Hugo J. 3 Li, Jiaofen 3 Li, Yongfeng 3 Lin, Lin 3 Ling, Qing 3 Mohan, Manil Thankamani 3 Nocedal, Jorge 3 Oviedo, Harry 3 Saab, Rayan 3 Setzer, Simon 3 Singer, Amit 3 Sun, Tao 3 Sun, Wenyu 3 Ulbrich, Michael 3 Viloche Bazán, Fermín S. 3 Wang, Kai 3 Wang, Yanfei 3 Wei, Ke 3 Wu, Soon-Yi 3 Xu, Fangfang 3 Yang, Junfeng 3 Ye, Yinyu 3 Yu, Bo 3 Zhou, Aihui 2 Andén, Joakim 2 Arasu, Krishnasamy Thiru 2 Bai, Zhengjian 2 Bao, Weizhu 2 Barnett, Alex H. 2 Bi, Shujun 2 Birtea, Petre ...and 861 more Authors
all top 5
#### Cited in 119 Serials
33 Computational Optimization and Applications 32 Journal of Scientific Computing 32 SIAM Journal on Optimization 32 SIAM Journal on Scientific Computing 24 Mathematical Programming. Series A. Series B 22 Optimization Methods & Software 19 SIAM Journal on Imaging Sciences 15 Science China. Mathematics 12 Journal of Computational and Applied Mathematics 12 SIAM Journal on Matrix Analysis and Applications 10 Journal of Computational Physics 10 Journal of Optimization Theory and Applications 10 Journal of Global Optimization 10 Mathematical Programming Computation 9 Numerical Algorithms 8 Inverse Problems 8 Applied Mathematics and Computation 8 Information Sciences 8 Journal of the Operations Research Society of China 7 Advances in Computational Mathematics 7 Journal of Machine Learning Research (JMLR) 7 Inverse Problems and Imaging 6 Neural Networks 5 Applied Mathematical Modelling 5 Journal of Mathematical Imaging and Vision 5 Applied and Computational Harmonic Analysis 5 International Journal of Computer Vision 4 Numerische Mathematik 4 Optimization 4 Machine Learning 4 Computational and Applied Mathematics 4 Abstract and Applied Analysis 4 Optimization Letters 3 Computer Methods in Applied Mechanics and Engineering 3 International Journal of Control 3 Journal of Multivariate Analysis 3 Computational Statistics 3 European Journal of Operational Research 3 INFORMS Journal on Computing 3 Mathematical Problems in Engineering 3 Journal of Applied Mathematics 3 Journal of Industrial and Management Optimization 3 Frontiers of Mathematics in China 3 SIAM Journal on Mathematics of Data Science 2 Computers & Mathematics with Applications 2 Journal of the Franklin Institute 2 Mathematics of Computation 2 BIT 2 SIAM Journal on Numerical Analysis 2 Operations Research Letters 2 Applied Numerical Mathematics 2 Annals of Operations Research 2 Journal of Statistical Computation and Simulation 2 Linear Algebra and its Applications 2 Numerical Linear Algebra with Applications 2 Nonlinear Dynamics 2 Soft Computing 2 Mathematical Methods of Operations Research 2 Acta Mathematica Sinica. English Series 2 Multiscale Modeling & Simulation 2 Inverse Problems in Science and Engineering 2 Numerical Algebra, Control and Optimization 1 Communications on Pure and Applied Mathematics 1 Journal of Mathematical Analysis and Applications 1 Linear and Multilinear Algebra 1 Psychometrika 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Journal of Geometry and Physics 1 ACM Transactions on Mathematical Software 1 The Annals of Statistics 1 Automatica 1 Calcolo 1 Journal of the American Statistical Association 1 Metron 1 Numerical Functional Analysis and Optimization 1 SIAM Journal on Control and Optimization 1 Bulletin of the Korean Mathematical Society 1 Circuits, Systems, and Signal Processing 1 Acta Mathematicae Applicatae Sinica. English Series 1 Computers & Operations Research 1 Asia-Pacific Journal of Operational Research 1 European Journal of Applied Mathematics 1 Multidimensional Systems and Signal Processing 1 Neural Computation 1 Japan Journal of Industrial and Applied Mathematics 1 International Journal of Computer Mathematics 1 Pattern Recognition 1 Computational Statistics and Data Analysis 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Mathematical Methods of Statistics 1 ETNA. Electronic Transactions on Numerical Analysis 1 Bulletin des Sciences Mathématiques 1 The Journal of Fourier Analysis and Applications 1 Journal of Nonparametric Statistics 1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 1 Journal of Scheduling 1 Data Mining and Knowledge Discovery 1 Discrete Dynamics in Nature and Society 1 Optimization and Engineering 1 RAIRO. Operations Research ...and 19 more Serials
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#### Cited in 35 Fields
318 Operations research, mathematical programming (90-XX) 297 Numerical analysis (65-XX) 78 Linear and multilinear algebra; matrix theory (15-XX) 72 Computer science (68-XX) 67 Calculus of variations and optimal control; optimization (49-XX) 64 Statistics (62-XX) 57 Information and communication theory, circuits (94-XX) 16 Partial differential equations (35-XX) 13 Biology and other natural sciences (92-XX) 10 Global analysis, analysis on manifolds (58-XX) 9 Differential geometry (53-XX) 8 Statistical mechanics, structure of matter (82-XX) 8 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 8 Systems theory; control (93-XX) 7 Functional analysis (46-XX) 7 Quantum theory (81-XX) 6 Combinatorics (05-XX) 6 Operator theory (47-XX) 5 Probability theory and stochastic processes (60-XX) 4 Dynamical systems and ergodic theory (37-XX) 4 Mechanics of particles and systems (70-XX) 4 Optics, electromagnetic theory (78-XX) 3 Convex and discrete geometry (52-XX) 3 Fluid mechanics (76-XX) 2 Ordinary differential equations (34-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Integral transforms, operational calculus (44-XX) 2 Mechanics of deformable solids (74-XX) 2 Geophysics (86-XX) 1 General and overarching topics; collections (00-XX) 1 Functions of a complex variable (30-XX) 1 Difference and functional equations (39-XX) 1 Approximations and expansions (41-XX) 1 Algebraic topology (55-XX) 1 Mathematics education (97-XX) | 2021-10-22 20:33:54 | {"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": 0.5848032236099243, "perplexity": 14857.787265570722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585518.54/warc/CC-MAIN-20211022181017-20211022211017-00316.warc.gz"} |
https://www.zbmath.org/authors/?q=ai%3Ahsu.shu-yu | # zbMATH — the first resource for mathematics
## Hsu, Shu-Yu
Compute Distance To:
Author ID: hsu.shu-yu Published as: Hsu, S.; Hsu, S. Y.; Hsu, S.-Y.; Hsu, Shu-Yu External Links: MGP · Wikidata
Documents Indexed: 43 Publications since 1990 Reviewing Activity: 86 Reviews
#### Co-Authors
34 single-authored
all top 5
#### Serials
6 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 4 Differential and Integral Equations 4 Discrete and Continuous Dynamical Systems 3 Mathematische Annalen 3 Advances in Differential Equations 2 Geometriae Dedicata 2 Manuscripta Mathematica 1 Journal of Mathematical Analysis and Applications 1 Journal of Mathematics of Kyoto University 1 Pacific Journal of Mathematics 1 Proceedings of the American Mathematical Society 1 Tokyo Journal of Mathematics 1 Advances in Mathematical Sciences and Applications 1 Calculus of Variations and Partial Differential Equations 1 Communications in Applied Analysis 1 International Mathematical Forum 1 Bulletin of Mathematical Sciences
#### Fields
28 Partial differential equations (35-XX) 14 Differential geometry (53-XX) 10 Global analysis, analysis on manifolds (58-XX) 2 Ordinary differential equations (34-XX) 1 Functional analysis (46-XX)
#### Citations contained in zbMATH
36 Publications have been cited 208 times in 121 Documents Cited by Year
A note on compact gradient Yamabe solitons. Zbl 1242.53044
Hsu, Shu-Yu
2012
Large time behaviour of solutions of the Ricci flow equation on $${\mathbb{R}}^ 2$$. Zbl 1053.53045
Hsu, Shu-Yu
2001
Singular limit and exact decay rate of a nonlinear elliptic equation. Zbl 1244.35051
Hsu, Shu-Yu
2012
Asymptotic behaviour of solutions of the equation $$u_t=\Delta\log u$$ near the extinction time. Zbl 1028.35079
Hsu, Shu-Yu
2003
Gradient estimates for a nonlinear parabolic equation under Ricci flow. Zbl 1249.53083
Hsu, Shu-Yu
2011
Asymptotic profile of solutions of a singular diffusion equation as $$t\to\infty$$. Zbl 1019.35055
Hsu, Shu-Yu
2002
Due date assignment using artificial neural networks under different shop floor control strategies. Zbl 1094.90528
Hsu, S. Y.; Sha, D. Y.
2004
Dynamics near extinction time of a singular diffusion equation. Zbl 1007.35009
Hsu, Shu-Yu
2002
Some properties of the Yamabe soliton and the related nonlinear elliptic equation. Zbl 1285.35046
Hsu, S. Y.
2014
Existence and asymptotic behaviour of solutions of the very fast diffusion equation. Zbl 1270.35105
Hsu, Shu-Yu
2013
Global existence and uniqueness of solutions of the Ricci flow equation. Zbl 1011.35085
Hsu, Shu-Yu
2001
Behaviour of solutions of a singular diffusion equation near the extinction time. Zbl 1232.35085
Hsu, Shu-Yu
2004
Uniqueness of solutions of a singular diffusion equation. Zbl 1036.35103
Hsu, Shu-Yu
2003
Removable singularities and non-uniqueness of solutions of a singular diffusion equation. Zbl 1024.35053
Hsu, Shu-Yu
2003
Dynamics of solutions of a singular diffusion equation. Zbl 1027.35055
Hsu, Shu-Yu
2002
On the initiation and growth of kink bands in fiber composites. II: Analysis. Zbl 0967.74582
Vogler, T. J.; Hsu, S.-Y.; Kyriakides, S.
2001
Composite failure under combined compression and shear. Zbl 0973.74627
Vogler, T. J.; Hsu, S.-Y.; Kyriakides, S.
2000
Removable singularities of semilinear parabolic equations. Zbl 1196.35021
Hsu, Shu-Yu
2010
Classification of radially symmetric self-similar solutions of $$u_t=\Delta \log u$$ in higher dimensions. Zbl 1212.35150
Hsu, Shu-Yu
2005
Removable singularity of the polyharmonic equation. Zbl 1184.35008
Hsu, Shu-Yu
2010
Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow. Zbl 1303.35036
Hsu, Shu-Yu
2014
Sobolev inequalities for manifolds evolving by Ricci flow. Zbl 1187.53070
Hsu, Shu-Yu
2008
Extinction profile of solutions of a singular diffusion equation. Zbl 1188.35103
Hsu, Shu-Yu
2005
Lypunov stable penalty methods for imposing holonomic constraints in multibody system dynamics. Zbl 0708.70013
Kurdila, A. J.; Junkins, J. L.; Hsu, S.
1990
Maximum principle and convergence of fundamental solutions for the Ricci flow. Zbl 1210.58022
Hsu, Shu-Yu
2009
A simple proof on the non-existence of shrinking breathers for the Ricci flow. Zbl 1096.58008
Hsu, Shu-Yu
2006
Large time behaviour of solutions of a singular diffusion equation in $$\mathbb R^n$$. Zbl 1082.35030
Hsu, Shu-Yu
2005
Inelastic behavior of an AS4/PEEK composite under combined transverse compression and shear. II: Modeling,. Zbl 0938.74505
Hsu, S.-Y.; Vogler, T. J.; Kyriakides, S.
1999
On the axial propagation of kink bands in fiber composites. II: Analysis. Zbl 0967.74585
Hsu, S.-Y.; Vogler, T. J.; Kyriakides, S.
1999
Global behaviour of solutions of the fast diffusion equation. Zbl 1406.35055
Hsu, Shu-Yu
2019
Minimizer of an isoperimetric ratio on a metric on $$\mathbb{R}^2$$ with finite total area. Zbl 1404.53082
Hsu, Shu-Yu
2018
Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature. Zbl 1256.53042
Hsu, Shu-Yu
2013
Generalized $$\mathcal L$$-geodesic and monotonicity of the generalized reduced volume in the Ricci flow. Zbl 1195.53092
Hsu, Shu-Yu
2009
Existence of singular solutions of a degenerate equation in $$\mathbb R^2$$. Zbl 1207.35184
Hsu, Shu-Yu
2006
Non-existence and behaviour at infinity of solutions of some elliptic equations. Zbl 1058.35014
Hsu, Shu-Yu
2004
Optimal motion estimation from multiple images by normalized epipolar constraint. Zbl 1111.68712
Ma, Y.; Vidal, R.; Hsu, S.; Sastry, S.
2001
Global behaviour of solutions of the fast diffusion equation. Zbl 1406.35055
Hsu, Shu-Yu
2019
Minimizer of an isoperimetric ratio on a metric on $$\mathbb{R}^2$$ with finite total area. Zbl 1404.53082
Hsu, Shu-Yu
2018
Some properties of the Yamabe soliton and the related nonlinear elliptic equation. Zbl 1285.35046
Hsu, S. Y.
2014
Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow. Zbl 1303.35036
Hsu, Shu-Yu
2014
Existence and asymptotic behaviour of solutions of the very fast diffusion equation. Zbl 1270.35105
Hsu, Shu-Yu
2013
Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature. Zbl 1256.53042
Hsu, Shu-Yu
2013
A note on compact gradient Yamabe solitons. Zbl 1242.53044
Hsu, Shu-Yu
2012
Singular limit and exact decay rate of a nonlinear elliptic equation. Zbl 1244.35051
Hsu, Shu-Yu
2012
Gradient estimates for a nonlinear parabolic equation under Ricci flow. Zbl 1249.53083
Hsu, Shu-Yu
2011
Removable singularities of semilinear parabolic equations. Zbl 1196.35021
Hsu, Shu-Yu
2010
Removable singularity of the polyharmonic equation. Zbl 1184.35008
Hsu, Shu-Yu
2010
Maximum principle and convergence of fundamental solutions for the Ricci flow. Zbl 1210.58022
Hsu, Shu-Yu
2009
Generalized $$\mathcal L$$-geodesic and monotonicity of the generalized reduced volume in the Ricci flow. Zbl 1195.53092
Hsu, Shu-Yu
2009
Sobolev inequalities for manifolds evolving by Ricci flow. Zbl 1187.53070
Hsu, Shu-Yu
2008
A simple proof on the non-existence of shrinking breathers for the Ricci flow. Zbl 1096.58008
Hsu, Shu-Yu
2006
Existence of singular solutions of a degenerate equation in $$\mathbb R^2$$. Zbl 1207.35184
Hsu, Shu-Yu
2006
Classification of radially symmetric self-similar solutions of $$u_t=\Delta \log u$$ in higher dimensions. Zbl 1212.35150
Hsu, Shu-Yu
2005
Extinction profile of solutions of a singular diffusion equation. Zbl 1188.35103
Hsu, Shu-Yu
2005
Large time behaviour of solutions of a singular diffusion equation in $$\mathbb R^n$$. Zbl 1082.35030
Hsu, Shu-Yu
2005
Due date assignment using artificial neural networks under different shop floor control strategies. Zbl 1094.90528
Hsu, S. Y.; Sha, D. Y.
2004
Behaviour of solutions of a singular diffusion equation near the extinction time. Zbl 1232.35085
Hsu, Shu-Yu
2004
Non-existence and behaviour at infinity of solutions of some elliptic equations. Zbl 1058.35014
Hsu, Shu-Yu
2004
Asymptotic behaviour of solutions of the equation $$u_t=\Delta\log u$$ near the extinction time. Zbl 1028.35079
Hsu, Shu-Yu
2003
Uniqueness of solutions of a singular diffusion equation. Zbl 1036.35103
Hsu, Shu-Yu
2003
Removable singularities and non-uniqueness of solutions of a singular diffusion equation. Zbl 1024.35053
Hsu, Shu-Yu
2003
Asymptotic profile of solutions of a singular diffusion equation as $$t\to\infty$$. Zbl 1019.35055
Hsu, Shu-Yu
2002
Dynamics near extinction time of a singular diffusion equation. Zbl 1007.35009
Hsu, Shu-Yu
2002
Dynamics of solutions of a singular diffusion equation. Zbl 1027.35055
Hsu, Shu-Yu
2002
Large time behaviour of solutions of the Ricci flow equation on $${\mathbb{R}}^ 2$$. Zbl 1053.53045
Hsu, Shu-Yu
2001
Global existence and uniqueness of solutions of the Ricci flow equation. Zbl 1011.35085
Hsu, Shu-Yu
2001
On the initiation and growth of kink bands in fiber composites. II: Analysis. Zbl 0967.74582
Vogler, T. J.; Hsu, S.-Y.; Kyriakides, S.
2001
Optimal motion estimation from multiple images by normalized epipolar constraint. Zbl 1111.68712
Ma, Y.; Vidal, R.; Hsu, S.; Sastry, S.
2001
Composite failure under combined compression and shear. Zbl 0973.74627
Vogler, T. J.; Hsu, S.-Y.; Kyriakides, S.
2000
Inelastic behavior of an AS4/PEEK composite under combined transverse compression and shear. II: Modeling,. Zbl 0938.74505
Hsu, S.-Y.; Vogler, T. J.; Kyriakides, S.
1999
On the axial propagation of kink bands in fiber composites. II: Analysis. Zbl 0967.74585
Hsu, S.-Y.; Vogler, T. J.; Kyriakides, S.
1999
Lypunov stable penalty methods for imposing holonomic constraints in multibody system dynamics. Zbl 0708.70013
Kurdila, A. J.; Junkins, J. L.; Hsu, S.
1990
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#### Cited by 170 Authors
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The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata. | 2021-02-25 08:30:54 | {"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": 0.46391722559928894, "perplexity": 4108.491225867411}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178350846.9/warc/CC-MAIN-20210225065836-20210225095836-00214.warc.gz"} |
https://www.bartleby.com/essay/Thank-You-for-Arguing-Notes-FKCC5FRCTC | # Thank You for Arguing Notes Essay
1221 WordsAug 21, 20135 Pages
Thank You for arguing chapter notes Ch.2-Set your goal 1) The goal is to change your audience’s mood, mind, or willingness 2) An argument should be focused on winning over an audience rather than beating them 3) Decide what you want at the end of the argument Ch.3-Control the Tense 1) Future tense is the best in an argument 2) A good strategy is to switch tenses in an argument depending on the situation 3) Changing the tense can sometimes result in a smaller conflict with a simpler solution Ch.4-Soften them up 1) Logos is argument by logic, ethos is argument by character, and pathos is argument by emotion 2) Logos involves winning your audience over with facts and using your opponent’s argument to your own advantage 3) Using…show more content… | 2019-10-23 23:03:03 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8275617957115173, "perplexity": 3283.953732980747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987836368.96/warc/CC-MAIN-20191023225038-20191024012538-00439.warc.gz"} |
http://faculty.gvsu.edu/boelkinm/Home/AC/sec-1-3-derivative-pt.html | Skip to main content
# Section1.3The derivative of a function at a point¶ permalink
##### Motivating Questions
• How is the average rate of change of a function on a given interval defined, and what does this quantity measure?
• How is the instantaneous rate of change of a function at a particular point defined? How is the instantaneous rate of change linked to average rate of change?
• What is the derivative of a function at a given point? What does this derivative value measure? How do we interpret the derivative value graphically?
• How are limits used formally in the computation of derivatives?
An idea that sits at the foundations of calculus is the instantaneous rate of change of a function. This rate of change is always considered with respect to change in the input variable, often at a particular fixed input value. This is a generalization of the notion of instantaneous velocity and essentially allows us to consider the question “how do we measure how fast a particular function is changing at a given point?” When the original function represents the position of a moving object, this instantaneous rate of change is precisely velocity, and might be measured in units such as feet per second. But in other contexts, instantaneous rate of change could measure the number of cells added to a bacteria culture per day, the number of additional gallons of gasoline consumed by going one mile per additional mile per hour in a car's velocity, or the number of dollars added to a mortgage payment for each percentage increase in interest rate. Regardless of the presence of a physical or practical interpretation of a function, the instantaneous rate of change may also be interpreted geometrically in connection to the function's graph, and this connection is also foundational to many of the main ideas in calculus.
In what follows, we will introduce terminology and notation that makes it easier to talk about the instantaneous rate of change of a function at a point. In addition, just as instantaneous velocity is defined in terms of average velocity, the more general instantaneous rate of change will be connected to the more general average rate of change. Recall that for a moving object with position function $s\text{,}$ its average velocity on the time interval $t = a$ to $t = a+h$ is given by the quotient
\begin{equation*} AV_{[a,a+h]} = \frac{s(a+h)-s(a)}{h}. \end{equation*}
In a similar way, we make the following definition for an arbitrary function $y = f(x).$
##### Definition1.3.1
For a function $f\text{,}$ the average rate of change of $f$ on the interval $[a,a+h]$ is given by the value
\begin{equation*} AV_{[a,a+h]} = \frac{f(a+h)-f(a)}{h}. \end{equation*}
Equivalently, if we want to consider the average rate of change of $f$ on $[a,b]\text{,}$ we compute
\begin{equation*} AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}. \end{equation*}
It is essential to understand how the average rate of change of $f$ on an interval is connected to its graph.
##### Preview Activity1.3.1
Suppose that $f$ is the function given by the graph below and that $a$ and $a+h$ are the input values as labeled on the $x$-axis. Use the graph in Figure 1.3.2 to answer the following questions.
1. Locate and label the points $(a,f(a))$ and $(a+h, f(a+h))$ on the graph.
2. Construct a right triangle whose hypotenuse is the line segment from $(a,f(a))$ to $(a+h,f(a+h))\text{.}$ What are the lengths of the respective legs of this triangle?
3. What is the slope of the line that connects the points $(a,f(a))$ and $(a+h, f(a+h))\text{?}$
4. Write a meaningful sentence that explains how the average rate of change of the function on a given interval and the slope of a related line are connected.
# Subsection1.3.1The Derivative of a Function at a Point
Just as we defined instantaneous velocity in terms of average velocity, we now define the instantaneous rate of change of a function at a point in terms of the average rate of change of the function $f$ over related intervals. In addition, we give a special name to “the instantaneous rate of change of $f$ at $a\text{,}$” calling this quantity “the derivative of $f$ at $a\text{,}$” with this value being represented by the shorthand notation $f'(a)\text{.}$ Specifically, we make the following definition.
##### Definition1.3.3
Let $f$ be a function and $x = a$ a value in the function's domain. We define the derivative of $f$ with respect to $x$ evaluated at $x = a$, denoted $f'(a)\text{,}$ by the formula
\begin{equation*} f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}, \end{equation*}
provided this limit exists.
Aloud, we read the symbol $f'(a)$ as either “$f$-prime at $a$” or “the derivative of $f$ evaluated at $x = a\text{.}$” Much of the next several chapters will be devoted to understanding, computing, applying, and interpreting derivatives. For now, we observe the following important things.
##### Note1.3.4
• The derivative of $f$ at the value $x = a$ is defined as the limit of the average rate of change of $f$ on the interval $[a,a+h]$ as $h \to 0\text{.}$ It is possible for this limit not to exist, so not every function has a derivative at every point.
• We say that a function that has a derivative at $x = a$ is differentiable at $x = a\text{.}$
• The derivative is a generalization of the instantaneous velocity of a position function: when $y = s(t)$ is a position function of a moving body, $s'(a)$ tells us the instantaneous velocity of the body at time $t=a\text{.}$
• Because the units on $\frac{f(a+h)-f(a)}{h}$ are “units of $f$ per unit of $x\text{,}$” the derivative has these very same units. For instance, if $s$ measures position in feet and $t$ measures time in seconds, the units on $s'(a)$ are feet per second.
• Because the quantity $\frac{f(a+h)-f(a)}{h}$ represents the slope of the line through $(a,f(a))$ and $(a+h, f(a+h))\text{,}$ when we compute the derivative we are taking the limit of a collection of slopes of lines, and thus the derivative itself represents the slope of a particularly important line.
While all of the above ideas are important and we will add depth and perspective to them through additional time and study, for now it is most essential to recognize how the derivative of a function at a given value represents the slope of a certain line. Thus, we expand upon the last bullet item above.
As we move from an average rate of change to an instantaneous one, we can think of one point as “sliding towards” another. In particular, provided the function has a derivative at $(a,f(a))\text{,}$ the point $(a+h,f(a+h))$ will approach $(a,f(a))$ as $h \to 0\text{.}$ Because this process of taking a limit is a dynamic one, it can be helpful to use computing technology to visualize what the limit is accomplishing. While there are many different options, one of the best is a java applet in which the user is able to control the point that is moving. For a helpful collection of examples, consider the work of David Austin of Grand Valley State University, and this particularly relevant example. For applets that have been built in Geogebra 1 , see Marc Renault's library via Shippensburg University, with this example being especially fitting for our work in this section.
In Figure 1.3.5, we provide a sequence of figures with several different lines through the points $(a, f(a))$ and $(a+h,f(a+h))$ that are generated by different values of $h\text{.}$ These lines (shown in the first three figures in magenta), are often called secant lines to the curve $y = f(x)\text{.}$ A secant line to a curve is simply a line that passes through two points that lie on the curve. For each such line, the slope of the secant line is $m = \frac{f(a+h) - f(a)}{h}\text{,}$ where the value of $h$ depends on the location of the point we choose. We can see in the diagram how, as $h \to 0\text{,}$ the secant lines start to approach a single line that passes through the point $(a,f(a))\text{.}$ In the situation where the limit of the slopes of the secant lines exists, we say that the resulting value is the slope of the tangent line to the curve. This tangent line (shown in the right-most figure in green) to the graph of $y = f(x)$ at the point $(a,f(a))$ is the line through $(a,f(a))$ whose slope is $m = f'(a)\text{.}$
As we will see in subsequent study, the existence of the tangent line at $x = a$ is connected to whether or not the function $f$ looks like a straight line when viewed up close at $(a,f(a))\text{,}$ which can also be seen in Figure 1.3.6, where we combine the four graphs in Figure 1.3.5 into the single one on the left, and then we zoom in on the box centered at $(a,f(a))\text{,}$ with that view expanded on the right (with two of the secant lines omitted). Note how the tangent line sits relative to the curve $y = f(x)$ at $(a,f(a))$ and how closely it resembles the curve near $x = a\text{.}$
##### Note1.3.7
The instantaneous rate of change of $f$ with respect to $x$ at $x = a\text{,}$ $f'(a)\text{,}$ also measures the slope of the tangent line to the curve $y = f(x)$ at $(a,f(a))\text{.}$
The following example demonstrates several key ideas involving the derivative of a function.
##### Example1.3.8Using the limit definition of the derivative
For the function given by $f(x) = x - x^2\text{,}$ use the limit definition of the derivative to compute $f'(2)\text{.}$ In addition, discuss the meaning of this value and draw a labeled graph that supports your explanation.
Solution
The following activities will help you explore a variety of key ideas related to derivatives.
##### Activity1.3.2
Consider the function $f$ whose formula is $\displaystyle f(x) = 3 - 2x\text{.}$
1. What familiar type of function is $f\text{?}$ What can you say about the slope of $f$ at every value of $x\text{?}$
2. Compute the average rate of change of $f$ on the intervals $[1,4]\text{,}$ $[3,7]\text{,}$ and $[5,5+h]\text{;}$ simplify each result as much as possible. What do you notice about these quantities?
3. Use the limit definition of the derivative to compute the exact instantaneous rate of change of $f$ with respect to $x$ at the value $a = 1\text{.}$ That is, compute $f'(1)$ using the limit definition. Show your work. Is your result surprising?
4. Without doing any additional computations, what are the values of $f'(2)\text{,}$ $f'(\pi)\text{,}$ and $f'(-\sqrt{2})\text{?}$ Why?
##### Activity1.3.3
A water balloon is tossed vertically in the air from a window. The balloon's height in feet at time $t$ in seconds after being launched is given by $s(t) = -16t^2 + 16t + 32\text{.}$ Use this function to respond to each of the following questions.
1. Sketch an accurate, labeled graph of $s$ on the axes provided in Figure 1.3.10. You should be able to do this without using computing technology.
2. Compute the average rate of change of $s$ on the time interval $[1,2]\text{.}$ Include units on your answer and write one sentence to explain the meaning of the value you found.
3. Use the limit definition to compute the instantaneous rate of change of $s$ with respect to time, $t\text{,}$ at the instant $a = 1\text{.}$ Show your work using proper notation, include units on your answer, and write one sentence to explain the meaning of the value you found.
4. On your graph in (a), sketch two lines: one whose slope represents the average rate of change of $s$ on $[1,2]\text{,}$ the other whose slope represents the instantaneous rate of change of $s$ at the instant $a=1\text{.}$ Label each line clearly.
5. For what values of $a$ do you expect $s'(a)$ to be positive? Why? Answer the same questions when “positive” is replaced by “negative” and “zero.”
##### Activity1.3.4
A rapidly growing city in Arizona has its population $P$ at time $t\text{,}$ where $t$ is the number of decades after the year 2010, modeled by the formula $P(t) = 25000 e^{t/5}\text{.}$ Use this function to respond to the following questions.
1. Sketch an accurate graph of $P$ for $t = 0$ to $t = 5$ on the axes provided in Figure 1.3.12. Label the scale on the axes carefully.
2. Compute the average rate of change of $P$ between 2030 and 2050. Include units on your answer and write one sentence to explain the meaning (in everyday language) of the value you found.
3. Use the limit definition to write an expression for the instantaneous rate of change of $P$ with respect to time, $t\text{,}$ at the instant $a = 2\text{.}$ Explain why this limit is difficult to evaluate exactly.
4. Estimate the limit in (c) for the instantaneous rate of change of $P$ at the instant $a = 2$ by using several small $h$ values. Once you have determined an accurate estimate of $P'(2)\text{,}$ include units on your answer, and write one sentence (using everyday language) to explain the meaning of the value you found.
5. On your graph above, sketch two lines: one whose slope represents the average rate of change of $P$ on $[2,4]\text{,}$ the other whose slope represents the instantaneous rate of change of $P$ at the instant $a=2\text{.}$
6. In a carefully-worded sentence, describe the behavior of $P'(a)$ as $a$ increases in value. What does this reflect about the behavior of the given function $P\text{?}$
# Subsection1.3.2Summary
• The average rate of change of a function $f$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b-a}\text{.}$ The units on the average rate of change are units of $f$ per unit of $x\text{,}$ and the numerical value of the average rate of change represents the slope of the secant line between the points $(a,f(a))$ and $(b,f(b))$ on the graph of $y = f(x)\text{.}$ If we view the interval as being $[a,a+h]$ instead of $[a,b]\text{,}$ the meaning is still the same, but the average rate of change is now computed by $\frac{f(a+h)-f(a)}{h}\text{.}$
• The instantaneous rate of change with respect to $x$ of a function $f$ at a value $x = a$ is denoted $f'(a)$ (read “the derivative of $f$ evaluated at $a$” or “$f$-prime at $a$”) and is defined by the formula \begin{equation*} f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}, \end{equation*} provided the limit exists. Note particularly that the instantaneous rate of change at $x = a$ is the limit of the average rate of change on $[a,a+h]$ as $h \to 0\text{.}$
• Provided the derivative $f'(a)$ exists, its value tells us the instantaneous rate of change of $f$ with respect to $x$ at $x = a\text{,}$ which geometrically is the slope of the tangent line to the curve $y = f(x)$ at the point $(a,f(a))\text{.}$ We even say that $f'(a)$ is the “slope of the curve” at the point $(a,f(a))\text{.}$
• Limits are the link between average rate of change and instantaneous rate of change: they allow us to move from the rate of change over an interval to the rate of change at a single point.
# Subsection1.3.3Exercises¶ permalink
##### 6
Consider the graph of $y = f(x)$ provided in Figure 1.3.13.
1. On the graph of $y = f(x)\text{,}$ sketch and label the following quantities:
• the secant line to $y = f(x)$ on the interval $[-3,-1]$ and the secant line to $y = f(x)$ on the interval $[0,2]\text{.}$
• the tangent line to $y = f(x)$ at $x = -3$ and the tangent line to $y = f(x)$ at $x = 0\text{.}$
2. What is the approximate value of the average rate of change of $f$ on $[-3,-1]\text{?}$ On $[0,2]\text{?}$ How are these values related to your work in (a)?
3. What is the approximate value of the instantaneous rate of change of $f$ at $x = -3\text{?}$ At $x = 0\text{?}$ How are these values related to your work in (a)?
##### 7
For each of the following prompts, sketch a graph on the provided axes in Figure 1.3.14 of a function that has the stated properties.
1. $y = f(x)$ such that
• the average rate of change of $f$ on $[-3,0]$ is $-2$ and the average rate of change of $f$ on $[1,3]$ is 0.5, and
• the instantaneous rate of change of $f$ at $x = -1$ is $-1$ and the instantaneous rate of change of $f$ at $x = 2$ is 1.
2. $y = g(x)$ such that
• $\frac{g(3)-g(-2)}{5} = 0$ and $\frac{g(1)-g(-1)}{2} = -1\text{,}$ and
• $g'(2) = 1$ and $g'(-1) = 0$
##### 8
Suppose that the population, $P\text{,}$ of China (in billions) can be approximated by the function $P(t) = 1.15(1.014)^t$ where $t$ is the number of years since the start of 1993.
1. According to the model, what was the total change in the population of China between January 1, 1993 and January 1, 2000? What will be the average rate of change of the population over this time period? Is this average rate of change greater or less than the instantaneous rate of change of the population on January 1, 2000? Explain and justify, being sure to include proper units on all your answers.
2. According to the model, what is the average rate of change of the population of China in the ten-year period starting on January 1, 2012?
3. Write an expression involving limits that, if evaluated, would give the exact instantaneous rate of change of the population on today's date. Then estimate the value of this limit (discuss how you chose to do so) and explain the meaning (including units) of the value you have found.
4. Find an equation for the tangent line to the function $y = P(t)$ at the point where the $t$-value is given by today's date.
##### 9
The goal of this problem is to compute the value of the derivative at a point for several different functions, where for each one we do so in three different ways, and then to compare the results to see that each produces the same value.
For each of the following functions, use the limit definition of the derivative to compute the value of $f'(a)$ using three different approaches: strive to use the algebraic approach first (to compute the limit exactly), then test your result using numerical evidence (with small values of $h$), and finally plot the graph of $y = f(x)$ near $(a,f(a))$ along with the appropriate tangent line to estimate the value of $f'(a)$ visually. Compare your findings among all three approaches; if you are unable to complete the algebraic approach, still work numerically and graphically.
1. $f(x) = x^2 - 3x\text{,}$ $a = 2$
2. $f(x) = \frac{1}{x}\text{,}$ $a = 1$
3. $f(x) = \sqrt{x}\text{,}$ $a = 1$
4. $f(x) = 2 - |x-1|\text{,}$ $a = 1$
5. $f(x) = \sin(x)\text{,}$ $a = \frac{\pi}{2}$ | 2017-09-26 17:55:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8474099636077881, "perplexity": 136.17490519841928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818696677.93/warc/CC-MAIN-20170926175208-20170926195208-00641.warc.gz"} |
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https://projecteuclid.org/euclid.tjm/1358951330 | ## Tokyo Journal of Mathematics
### Isomorphism among Families of Weighted $K3$ Hypersurfaces
#### Abstract
It is known that there are exactly 95 weighted projective spaces having Gorenstein $K3$ surfaces as anticanonical divisors, some of which have isometric Picard lattices for generic members. For each set of such families, an explicit birational correspondence coming from a torus action is constructed in this paper. As a result the number of `essentially different' families of weighted Gorenstein $K3$ surfaces is 75.
#### Article information
Source
Tokyo J. Math., Volume 35, Number 2 (2012), 461-467.
Dates
First available in Project Euclid: 23 January 2013
Permanent link to this document
https://projecteuclid.org/euclid.tjm/1358951330
Digital Object Identifier
doi:10.3836/tjm/1358951330
Mathematical Reviews number (MathSciNet)
MR3058718
Zentralblatt MATH identifier
1262.14046
#### Citation
KOBAYASHI, Masanori; MASE, Makiko. Isomorphism among Families of Weighted $K3$ Hypersurfaces. Tokyo J. Math. 35 (2012), no. 2, 461--467. doi:10.3836/tjm/1358951330. https://projecteuclid.org/euclid.tjm/1358951330
#### References
• Batyrev, V. V., Dual polyhedra and the mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994), 493–535.
• Belcastro, S.-M., Picard lattices of families of $K3$ surfaces, Commun. in Algebra 30 (2002), 61–82.
• Ebeling, W., Mirror symmetry, Kobayashi's duality, and Saito's duality, Kodai Math. J. 29 (2006), 319–336.
• Iano-Fletcher, A. R., Working with weighted complete intersections, in Explicit Birational Geometry of $3$-folds, Alessio Corti and Miles Reid (eds.), London Mathematical Society Lecture Note Series No. 281 (2000), 101–173.
• Kobayashi, M., Duality of weights, mirror symmetry and Arnold's strange duality, Tokyo J. Math. 31 (2008), 225–251.
• Yonemura, T., Hypersurface simple $K3$ singularities, Tôhoku Math. J. 42 (1990), 351–380. | 2019-07-19 14:33:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7241804599761963, "perplexity": 3391.0340063772396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526254.26/warc/CC-MAIN-20190719140355-20190719162355-00502.warc.gz"} |
http://blancosilva.github.io/forum/2017/09/06/MA524ch3.html | ## Basic, Intermediate and CAS problems
The following is the list of non-advanced problems for Chapter 3 of the class notes. There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.
1. The following sequences all converge to zero. %
Indicate the type of convergence of each sequence.
2. Find an example of a function $$f\colon \mathbb{R} \to \mathbb{R}$$ with a unique root at $$x=0$$ for which the Newton-Raphson sequence is a loop no matter the initial guess $$x_0\neq 0$$: $$x_{2n}=x_0$$, $$x_{2n+1}=-x_0$$ for all $$n \in \mathbb{N}.$$ Bonus points is your function is trigonometric.
3. Consider the equation $$x = \cos x.$$
• Show graphically that there exists a unique positive root $$x^\star.$$ Indicate approximately where it is located.
• Show that Newton’s method applied to $$f(x) = x-\cos x$$ converges for any initial guess $$x_0 \in \big[0,\frac{\pi}{2}\big].$$
4. Consider the equation $$\tan x + \lambda x = 0, \quad (0 < \lambda < 1).$$
• Show graphically, as simply as possible, that there is exactly one root $$x^\star$$ in the interval $$\big[ \frac{1}{2}\pi, \pi\big].$$
• Does Newton’s method converge to the root $$x^\star \in \big[ \frac{1}{2}\pi, \pi\big]$$ if the initial approximation is taken to be $$x_0 = \pi$$? Justify your answer.
5. Consider the equation $$\log^2 x - x -1 = 0, (x > 0).$$
• Graphical considerations suggest that there is exactly one positive solution $$x^\star$$, and that $$0 <x^\star < 1.$$ Prove this.
• What is the largest positive $$0<x_0\leq 1$$ such that Newton’s method with $$f(x) = \log^2 x - x -1$$ started at $$x_0$$ converges to $$x^\star$$?
6. Consider the two equivalent equations
• Show that there is exactly one positive root and find a rough interval containing it.
• For both equations above, determine the largest interval on which Newton’s method converges.
Hint: Investigate the convexity of the functions involved.
7. Design a process in desmos.com to test the search for critical points given by the recursion formulas produced by Newton’s method.
8. In a computer language or CAS of your choice, design a routine that gathers the following as input:
• The definition of a generic real-valued function $$f\colon \mathbb{R} \to \mathbb{R},$$
• The derivative $$f’$$ of that function,
• An initial guess $$x_0 \in \mathbb{R},$$
• A number $$N$$ of steps,
and produces the first $$N+1$$ terms of the Newton-Raphson sequence to approximate a root of $$f.$$
Modify the previous routine to receive as input, instead of a number of steps, a tolerance tol indicating the accuracy of the solution. For example, if we require a root of the equation $$f(x)=0$$ accurate to the first 16 correct decimal places, we use tol = 1e-16.
9. The purpose of this problem is the design of Horner’s method to evaluate polynomials effectively. Given a polynomial
where $$a_0, a_1, \dotsc, a_n$$ are real numbers, and given $$x_0 \in \mathbb{R}$$, we define the Horner’s scheme $$\{ b_0, b_1, \dotsc, b_n \}$$ to evaluate $$p(x_0)$$ as follows:
• Prove that $$b_0 = p(x_0)$$
• Use Horner’s method to evaluate $$p(x) = 2x^3 - 6x^2 +2x -1$$ at $$x=3.$$ Illustrate all steps, and count the number of basic operations (addition, subtraction, multiplication, division) used.
• Employ the usual method of evaluation of polynomials to evaluate $$p(x) = 2x^3 - 6x^2 +2x -1$$ at $$x=3.$$ Count the number of basic operations (note that a raising to the cube counts as two multiplications, e.g.)
10. In a computer language or CAS of your choice, write a routine to apply Horner’s scheme to evaluate polynomials. Your routine should gather the following inputs:
• A list of coefficients [a0, a1, ..., an] representing the polynomial $$p(x) = a_0 + a_1 x + \dotsb + a_n x^n.$$
• A value x0
The output of your routine should be $$p(x_0).$$
11. Use any of the routines that you wrote in Problem 8 to produce a table and a visual representation for the numerical solution of the following equations, with the given initial guesses and steps.
• $$f(x) = \sin x$$, with $$x_0=0.5$$, 5 steps.
• $$f(x) = \sin x$$, with $$x=3$$, enough steps to obtain accurately the first 16 correct decimal places of $$\pi.$$
• $$f(x) = -1+\log x$$, with $$x=2$$, enough steps to obtain accurately the first 16 correct decimal places of $$e.$$
12. The objective of this problem is to use Newton’s method to find an approximation to the golden ratio $$\phi=\frac{1}{2}(1+\sqrt{5})$$ accurate to the first 16 decimal places. Find first an appropriate polynomial $$p(x)$$ with integer coefficients for which $$\phi$$ is a root. Employ any of the routines that you wrote in Problem 9 with a good initial guess to guarantee the required result.
13. Consider the function $$f(x) = 9x -4\log(x-7).$$ We wish to study the behavior of Newton-Raphson to find approximations to the critical points of this function.
• Find the domain $$D$$ of $$f.$$
• Find the global minimum of $$f$$ analytically.
• Compute an exact formula for the Newton-Raphson iterate $$x_{n+1}$$ for an initial guess $$x_0 \in D.$$
• Compute five iterations of the Newton-Raphson method starting at each of the following initial guesses:
• $$x_0 = 7.4.$$
• $$x_0 = 7.2.$$
• $$x_0 = 7.01.$$
• $$x_0 = 7.8.$$
• $$x_0 = 7.88.$$
• Prove that the Newton-Raphson method converges to the optimal solution for any initial guess $$x_0 \in (7,7.8888).$$
• What is the behavior of the Newton-Raphson method if the initial guess is not in the interval $$(7,7.8888)?$$
14. Consider the function $$f(x) = 6x -4\log(x-2) -3\log(25-x).$$ We wish to study the behavior of Newton-Raphson to find approximations to the critical points of this function.
• Find the domain $$D$$ of $$f.$$
• Find the global minimum of $$f$$ analytically.
• Compute an exact formula for the Newton-Raphson iterate $$x_{n+1}$$ for an initial guess $$x_0 \in D.$$
• Compute five iterations of the Newton-Raphson method starting at each of the following initial guesses:
• $$x_0 = 2.6.$$
• $$x_0 = 2.7.$$
• $$x_0 = 2.4.$$
• $$x_0 = 2.8.$$
• $$x_0 = 3.$$
• Prove that the Newton-Raphson method converges to the optimal solution for any initial guess $$x_0 \in (2,3.05).$$
• What is the behavior of the Newton-Raphson method if the initial guess is not in the interval $$(2,3.05)?$$
15. Approximate the solution of the following system by computing two steps of Newton-Raphson’s method for an appropriate function $$\boldsymbol{g} \colon \mathbb{R}^3 \to \mathbb{R}^3:$$ and initial guess $$\boldsymbol{x}_0 = (1,0,1)$$.
16. The purpose of this exercise is to show that Newton’s method is unaffected by linear scaling of the variables. Consider a linear invertible transformation of variables $$\boldsymbol{x}^\intercal = \boldsymbol{A} \cdot \boldsymbol{y}^\intercal$$. Write Newton’s method in the space of the variables $$\boldsymbol{y}$$ and show that it generates the sequence $$\boldsymbol{y}_n^\intercal = \boldsymbol{A}^{-1} \cdot \boldsymbol{x}_n^\intercal$$, where $$\{ \boldsymbol{x}n \}{n \in \mathbb{N}}$$ is the sequence generated by Newton’s method in the space of variables $$\boldsymbol{x}$$.
17. Let $$\boldsymbol{A}$$ be a square matrix. A LU-decomposition is a factorization of $$\boldsymbol{A}=\boldsymbol{L} \cdot \boldsymbol{U}$$ into a lower triangular matrix $$\boldsymbol{L}$$ and an upper triangular matrix $$\boldsymbol{U}$$, both of which have non-zero entries in their diagonals. For example, the general case for $$3 \times 3$$ square matrices:
• Find an LU-decomposition of the following matrix
that satisfies that all diagonal entries of $$\boldsymbol{L}$$ are ones.
• Find an example of a $$2\times 2$$ square matrix for which there is not any possible LU-decomposition.
18. In a computer language or CAS of your choice, design a routine that solves a linear system
by performing first a LU-decomposition $$\boldsymbol{A} = \boldsymbol{L} \cdot \boldsymbol{U}$$ (provided this is possible!) and manipulating the resulting equation to solve instead the two (faster) systems:
• Find $$\boldsymbol{y}$$ that solves the system $$\boldsymbol{L}\cdot \boldsymbol{y}^\intercal = \boldsymbol{c}^\intercal$$ by Gaussian elimination.
• Find $$\boldsymbol{x}$$ that solves the system $$\boldsymbol{U}\cdot \boldsymbol{x}^\intercal = \boldsymbol{y}^\intercal$$ by Gaussian elimination.
You may design a routine that computes LU-decompositions, or you may use a built-in routine for that purpose.
19. In a computer language or CAS of your choice, design a routine that gathers the following as input:
• The definition of a generic real-valued function $$f\colon \mathbb{R}^d \to \mathbb{R},$$
• The gradient $$\nabla f$$ of that function,
• An initial guess $$\boldsymbol{x}_0 \in \mathbb{R},$$
• A number $$N$$ of steps,
and produces the first $$N+1$$ terms of the Newton-Raphson sequence to approximate a root of $$f.$$
20. Approximate the solution for the system in Problem 15 by computing the first two iterations of a Broyden method with $$\boldsymbol{A}_0 = \nabla \boldsymbol{g} (\boldsymbol{x}_0)$$ for an appropriate function $$\boldsymbol{g} \colon \mathbb{R}^d \to \mathbb{R}$$ and initial guess $$\boldsymbol{x}_0 = (1,0,1).$$
21. Compute the first two iterations of Broyden method with initial guess $$(1,4)$$ to search for the critical points of the function $$f(x,y) = 2x^2+y^2-xy$$
• Using $$\boldsymbol{A}_0 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}.$$
• using $$\boldsymbol{A}_0 = \mathrm{Hess} f(1,4).$$
22. Approximate the solution for the system in Problem 15 by computing the first two iterations of a Broyden method for an appropriate function $$\boldsymbol{g} \colon \mathbb{R}^d \to \mathbb{R}$$ and initial guess $$\boldsymbol{x}_0 = (1,0,1).$$
23. Compute the first two iterations of the method of Steepest descent with initial guess $$(1,4)$$ to search for the critical points of the function $$f(x,y) = 2x^2+y^2-xy.$$
24. Consider the quadratic polynomial $$p(x,y) = \tfrac{1}{2} \mathcal{Q}_{\boldsymbol{Q}}(x,y) + \langle D, [x,y] \rangle + 13,$$ with
• Find the global minimum value of $$p$$, and its location.
• Compute the eigenvalues of $$Q$$. Is $$Q$$ positive definite?
• What is the worst-case scenario rate of convergence of sequences of steepest descent for this function?
• Compute sequences of steepest descent for this function with the initial guesses below.
• $$(0,0)$$
• $$(-0.4, 0)$$
• $$(10,0)$$
• $$(11, 0)$$
25. Consider the quadratic polynomial $$p(x,y,z) = \tfrac{1}{2} \mathcal{Q}_{\boldsymbol{Q}}(x,y,z) + \langle D, [x,y,z] \rangle,$$ with
• Find the global minimum value of $$p$$, and its location.
• Compute the eigenvalues of $$Q$$. Is $$Q$$ positive definite?
• What is the worst-case scenario rate of convergence of sequences of steepest descent for this function?
• Compute sequences of steepest descent for this function with the initial guesses below.
• $$(0,0,0)$$
• $$(15.09, 7.66, -6.56)$$
• $$(11.77, 6.42, -4.28)$$
• $$(4.46, 2.25, 1.85)$$ | 2021-07-25 05:05:05 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8709360361099243, "perplexity": 281.61368648683714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151638.93/warc/CC-MAIN-20210725045638-20210725075638-00125.warc.gz"} |
https://www.gradesaver.com/textbooks/math/geometry/CLONE-68e52840-b25a-488c-a775-8f1d0bdf0669/chapter-10-section-10-3-preparing-to-do-analytic-proofs-exercises-page-458/2a | ## Elementary Geometry for College Students (6th Edition)
Published by Brooks Cole
# Chapter 10 - Section 10.3 - Preparing to Do Analytic Proofs - Exercises - Page 458: 2a
#### Answer
($\frac{a}{2}$,$\frac{b}{2}$)
#### Work Step by Step
Given Points(a,0) and (0,b) if x1=a,y1=0,x2=0,y2=b Mid point of these two points is($\frac{x1+x2}{2}$,$\frac{y1+y2}{2}$) ($\frac{a+0}{2}$,$\frac{0+b}{2}$) = ($\frac{a}{2}$,$\frac{b}{2}$)
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2022-08-09 08:43:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.653333306312561, "perplexity": 5714.014325631724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570913.16/warc/CC-MAIN-20220809064307-20220809094307-00283.warc.gz"} |
https://www.physicsforums.com/threads/schrodingers-equation-measurement.543469/ | # Schrodinger's Equation & Measurement
Hi there,
Ok, so if you know the interactions, etc., you can calculate from time0 the probabilities for certain results at say time8.
But say, however, we perform a measurement before time8. Will the original predictions for time8 still hold if a measurement is performed prior to that time? Or will it be different - in that case, what information do you need from the measurement, to plug into the unitary operator (which will include interactions, etc.) to find the new probabilities at time8? | 2020-09-20 22:54:37 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9141359925270081, "perplexity": 823.5114479247808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400198868.29/warc/CC-MAIN-20200920223634-20200921013634-00003.warc.gz"} |
https://askdev.io/questions/335978/how-do-i-acquire-a-domain-owned-by-someone-else | # How do I acquire a domain owned by someone else?
We've been fancying a dashless domain for a long period of time. It is signed up to somebody else that does not utilize it as a public internet site, the whois information claims it runs out in a couple of years.
What should I do to acquire this domain name, what should I stay clear of doing so as not to fire myself in the foot or jack the rate up and also what does a domain name backordering solution do?
13
2022-06-07 14:33:59
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when it comes to your "what needs to I stay clear of doing so as not to fire myself in the foot or jack the rate up and also what does a domain name backordering solution do?" component of the inquiry: I simply did the "backordering" point for the very first time with GoDaddy a month approximately earlier.
BRIEF ANSWER: do not make use of GoDaddy make use of something like SnapNames or Pool.
LENGTHY ANSWER: I saw the name I desired on among those "soon to expire" internet sites, so I mosted likely to backorder at GoDaddy. Ends up the domain name was signed up with GoDaddy, so every little thing I read claimed it needs to be more probable I will certainly get it due to the fact that we are both at the very same registrar. I paid $18 to "backorder it". What did that suggest? I COULDN'T TELL due to the fact that GoDaddy does not actually clarify the information from what I can establish, as quickly as the domain name ran out after that they would certainly place in a proposal for$10 on my part.
so after that I head over to snapnames.com where I understand that individuals can place in quiet proposals for running out domain names therefore below I am with this foolish backorder from GoDaddy that is mosting likely to be a pitiful $10, and also individuals on SnapNames can include proposals for$100 npls. After that involved figure out there are great deals of these websites (like pool.com).
So I call GoDaddy and also I chat for life with some schmuck that maintained entering circles. Either he really did not recognize what he was speaking about, or the whole procedure actually is an enigma. I asked him if there was a person there that recognized just how points functioned, and also he claimed "I'm an elderly representative here" ugh, all right so you do not recognize
and also hence, from what I've found out, you require to select among the huge "drop auction" websites like both I stated over and also you'll require to be all set to bet large cash money if it is a reputation!
yet like the others claimed, if you can get ahold of the proprietor and also conspire in this way it would possibly be best.
1
2022-06-07 15:05:50
Source
Most registrars will certainly supply the solutions of trying to get the domain name for you. When you do a WhoIS lookup on Network Solutions, and also the domain name returns as being signed up, they will certainly supply you alternatives on just how to get it.
2
2022-06-07 15:01:10
Source
Try emailing the e-mail in the whois address.
I periodically do this for my company and also concerning 75% of the moment, we receive a reply back with a person going to market. If you are bothered with utilizing your firm is domain name in the e-mail, make a gmail account. That is what we do, yet I can not claim whether it influences the rate.
6
2022-06-07 15:00:43
Source
If you do submit a domain enrollment disagreement with ICANN, be definitely particular your case is uncompromising. Additionally remember that this will certainly (last I looked) price you around \$1500, which is usually way greater than the individual that possesses the domain name would certainly ask to market it to you.
If you do plan to submit a disagreement, see if you can get the individual that presently has the domain name to make a deal to market it to you first. This usually aids confirm breach of contract if they are additionally not utilizing it (it is parked or something).
If it runs out in a couple of years, do not also think of tinkering the domain name backordering solutions. Their function is to register the domain name for you faster than the various other domain name backordering solutions, and/or the type - in public. They are usually a wild-goose chase to start with, given that they can just aid you after both the registrant and also the existing registrar have actually permitted the name to go down. If it is a reputation, several registrars nowadays do not permit the name to go down to the general public degree at all, rather maintaining it and also placing it on public auction.
Additionally, if you have not currently, stop going and also seeing the domain in your internet browser. Any kind of sort of typin website traffic raises the value of the domain dramatically. If your consumers are additionally keying in the domain name, do your ideal to get them to type your existing one effectively. If you can lower the quantity of website traffic to the domain name prior to you make a deal, you'll likely get a far better bargain.
If you do intend to make a deal, attempt examining to see what various other domain names the proprietor has, and also start by asking about a few other one. When they estimate you an actually high rate, act withdrawn, and also ask about numerous others, consisting of the real one you desire. As others have actually claimed, stay clear of any kind of organization with your actual domain name.
If you do connect with the proprietor, and also they agree to market, generally they'll price estimate some expensive beginning number (usually 5 - 10k for middling - negative domain names). This is great, supply them a whole lot much less. An excellent general rule is to go for a last asking price of 10% of the first demand.
8
2022-06-07 14:53:56
Source
There are just a number of means you are mosting likely to have the ability to acquire that domain name. To start with, by obtaining the proprietor to market it to you. Conversely, if you can confirm that there is a reputable reason that you need to possess that domain name as opposed to the here and now proprietor, such as a hallmark, you can submit a Domain Name Registration Dispute with ICANN and also allow them determine that need to possess it.
Nonetheless, if the only factor you desire this domain name is due to the fact that you like it, and also the proprietor does not intend to market it, after that you are rather unfortunate. The proprietor might not be utilizing it, yet that is his selection.
13
2022-06-07 14:53:09
Source | 2022-12-02 22:47:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3417341709136963, "perplexity": 1716.707754080622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710916.70/warc/CC-MAIN-20221202215443-20221203005443-00708.warc.gz"} |
https://www.azdictionary.com/definition/paraphrase | • Definition for "paraphrase"
• A restatement of a text or passage in…
• Sentence for "paraphrase"
• The Ben Franklin paraphrase is particularly…
• Synonym for "paraphrase"
• repetition
• Antonym for "paraphrase"
• metaphrase
• Verb Forms for "paraphrase"
• paraphrased
• Phrases for "paraphrase"
• To paraphrase
• Cross Reference for "paraphrase"
• chaldee paraphrases
• Form for "paraphrase"
• paraphrased
• Same Context for "paraphrase"
• render
# paraphrase definition
• noun:
• A restatement of a text or passage in another form or other words, often to simplify meaning.
• The restatement of texts to put it differently as a studying or training product.
• A restatement of a text in different words, often to clarify definition.
• A restatement of a text, passage, or work, expressing the meaning regarding the initial an additional type, typically with regard to its clearer and fuller exposition; a setting forth the signification of a text various other and ampler terms; a free of charge translation or rendering; -- opposed to metaphrase.
• A restatement of a text or passage, offering the sense of the initial this means that, typically in fuller terms in accordance with increased detail, with regard to clearer and more complete exposition: against metaphrase. Whenever initial is in a foreign language, interpretation and paraphrase might combined.
• particularly, in Scotland, among sixty-seven versified renderings of as numerous selected passages of Scripture, frequently bound up with the metrical psalms, and like all of them sung in church, etc.
• In instrumental songs, a transcription; a variation. Also paraphrasis.
• rewording for the purpose of clarification
• verb-transitive:
• To restate in a paraphrase.
• expressing, understand, or convert with latitude; to give the meaning of a passage in other language.
• verb-intransitive:
• To compose a paraphrase.
• to produce a paraphrase.
• verb:
• To restate some thing as, or even to create a paraphrase.
• express equivalent message in numerous terms
• others:
• To restate or convert with latitude; interpret; construe; unfold and show the sense of (an author) with better clearness and particularity by replacing other terms for his very own.
• To understand or amplify by change of terms; make a paraphrase. | 2017-05-26 20:56:46 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.803248941898346, "perplexity": 8719.667300193194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608684.93/warc/CC-MAIN-20170526203511-20170526223511-00303.warc.gz"} |
https://www.statistics-lab.com/%E6%95%B0%E5%AD%A6%E4%BB%A3%E5%86%99%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95%E4%BB%A3%E5%86%99optimization-algorithms%E4%BB%A3%E8%80%83/ | ### 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory
statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization algorithms相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory
Since every year the complexity of scientific and applied problems is increasing, there is an uprise of the need of a large amount of process of input data with the use of computing machinery for the numerical computation, and optimization of computations by all rational methods and means is a topical and important question. As a rule, the solution of the applied problems reduces to solving the typical problems of computational mathematics. Computational mathematics is a science about the methods of numerical solving problems of applied mathematics using modern computing machinery. The subject of computational mathematics is numerical methods (NM) or, that is the same thing, the multitude of computational algorithms (CA) and the question of their substantiation: the convergence and speed of NM convergence, their stability and error, the optimality on different criterion, the implementation time on the computing machinery, the necessary memory of computing machinery, etc.
The purpose of this chapter is to explain the main concepts and some results of computational mathematics, which are repeatedly used in the monograph, and at the same time, they have an independent value. The presentation of this information, in our opinion, is necessary for a confident possession of the given material.
In the first chapter:
• Certain supporting data in the theory of computing is given for determining the accuracy of the approximate solving problem, the quality of the used algorithm, and the comparative study of various algorithms according to some criteria.
• The principal features of the CA are determined (accuracy, execution time, necessary memory of the computing machinery), and the methods of their estimate are provided.
• There is a definition of optimal (inaccuracy and high speed), asymptotically optimal, and optimal in the order algorithms.
• The questions of identifying and clarifying a priori information on the problem are considered in cases when a priori information is not incorrectly set in the order to improve the quality of the obtained approximate solutions of a problem.
## 数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations
One of the main directions of development of modern computational mathematics is the improvement of the theory of error computations, the research of comparative study of CA, and the optimal algorithm development in the solving problems of computational and applied mathematics.
The solution of the majority of the problems with modern computing tools is based on the computational experiment (CE), which seamlessly unites the mathematical model (MM), the computational algorithm (CA), the computations on the computing machinery, and the experiment. CE is necessary for obtaining an adequate quantitative description of the studied phenomenon or process itself with the necessary accuracy for the least possible machine time. The process of observations and comparing them with the matched model of MM is conducted on the computing machinery to check the validity of MM. The implementation of MM is performed with the use of numerical methods (NM), which approximate the input model and make it suitable for practical use. The errors of this approximation, as well as the rounding errors in the implementation of NM on the computing machinery and the errors of measurement or observation of implementation of the studied process, must be considered in determining errors of MM or the adequacy of MM and in the process itself. The errors of input MM must be considered in the process of definition of the requirements for the accuracy of the solution of various problems that are solved within this model.
In this monograph, the most important computational aspects of the determination of the error of MM and constructing its optimal implementation are analyzed. The multifaceted approach is based on the analysis of the three main characteristics of the corresponding NM-accuracy, implementation time, and necessary memory on computing machinery. According to these characteristics, the comparative study and the optimization of the corresponding NM are performed.
Main Characteristics of CA Let the problem $P(I)$ is solved by the algorithm $A(X)$ on the computing machinery $C(Y)$, where $I, X, Y$ are finite multitudes (vectors) of parameters from which essentially $P, A, C$ are, respectively, dependent on. The components of the vector $I$ may include data on a priori qualities of solving a problem, for example, constant that constrains the absolute values of the order of the derivatives from the given functions, data on the accuracy of the input values, etc. A number of iterations of the algorithm, the degree of approximation, the step of the grid, etc., can be the vector $X$ components. Vector $Y$ can contain the number of cell digits of the computing machinery memory, the total volume of its RAM, the run time of the computing machinery, the quantity of the computing machinery
processors, the characteristics of the used operational systems and compiler programs, etc.
In the practice of computational solving problems on the computing machinery, the following are the characteristics of problems, algorithms, and computing machinery: $E(I, X, Y)$ is a global error of solution $E$ of the problem $P$ on the computing machinery $C$ with algorithm $A, T(I, X, Y)$ is the time that is required to obtain a solution of the problem, $M(I, X, Y)$ is a required computing machinery memory, and fef is a coefficient of technical and economic efficiency. Explain the characteristics of $T, M$, and fef..
Total time $T$ – The period of time from setting the problem $P(I)$ to its solution of CA $A(X)$ on computing machinery $C(Y)$ can be estimated as follows:
$$T=T_{1}+T_{2}+T_{3}+T_{4}$$
where $T_{1}$ is the time to set the problem and develop or select CA $A$ and computing machinery $C ; T_{2}$ is the programming time, transmission motion, and debugging $\mathrm{CA}$ $A ; T_{3}$ is the implementation time of $\mathrm{CA}, A$ on the computing machinery $C$; and $T_{4}$ is the time of interpretation of the obtained solution and its comprehension. Practically when estimating $T_{3}$, only the essential operators are often considered to the number and time of the operation performing of the computing machinery. In further detail, the given characteristic Tis described below and in the works $[2,85,97,114]$.
## 数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation
Optimization of computations lies in the optimization of one of the introduced characteristics (in a general way on $I, X, Y$ ) in accordance with certain constraints on others $[118,111]$.
The first primary problem is to minimize time $T(I, X, Y)$ while upholding the real (Re) constraints on $M, E$, and $f e f$ :
$$\begin{gathered} T(I, X, Y)=\min {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, E(I, X, Y) \leq E_{\mathrm{Re}}, f e f(I, X, Y) \leq f e f_{\mathrm{Re}} . \end{gathered}$$
The second primal problem is to minimize the global error $E(I, X, Y)$ with constraints on $M, T$, and $f e f$.
$$\begin{gathered} E(I, X, Y)=\min {l, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re},} \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad f e f(I, X, Y) \leq f e f_{\mathrm{Re}} \end{gathered}$$
The third primal problem is to maximize a coefficient of the technical and economic effectiveness $f e f(I, X, Y)$ :
$$\begin{gathered} f e f(I, X, Y)=\max {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad E(I, X, Y) \leq E_{\mathrm{Re}} \end{gathered}$$
We make some remarks on the mentioned statements of the primal problems. Let the computing machinery $C(Y)$ be fixed. Then $T, M, E$, and fef depend only on $I, X$. It is convenient to consider $I$ to be a random value and consider the probabilistic characteristics of the values $T, M, E$, and $f e f$, which will also be the characteristics of CA $A$ and will depend only on $X$. We designate each of the characteristics $T, M, E$, and fef through $H(I, X)$ and designate the frequency distribution accordingly to $H, I$ through $P(H), P(I)$. The essential characteristics of $\mathrm{CA} A(X)$ is the mathematical expectation $M_{H}(X)$ and the dispersion $D_{H}(X)$ :
\begin{aligned} &M_{H}(X)=\int_{G} H P(I) d I=\int_{-\infty}^{\infty} H P(H) d H \ &D_{H}(X)=\int_{G}\left(H-M_{H}\right)^{2} P(I) d I=\int_{-\infty}^{\infty}\left(H-M_{H}\right)^{2} P(H) d H \end{aligned}
where $G$ is a domain of possible values of $I$.
## 数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory
• 给出了计算理论中的一定支持数据，用于确定近似求解问题的准确性、所用算法的质量，以及按一定标准对各种算法进行比较研究。
• 确定了 CA 的主要特征（准确性、执行时间、计算机的必要内存），并提供了它们的估计方法。
• 在顺序算法中有最优（不准确和高速）、渐近最优和最优的定义。
• 在没有错误设置先验信息的情况下，考虑识别和澄清有关问题的先验信息的问题，以提高获得的问题近似解的质量。
## 数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations
CA的主要特点让问题磷(一世)由算法解决一种(X)在计算机器上C(是)， 在哪里一世,X,是是有限数量的参数（向量），它们本质上是磷,一种,C分别依赖于。向量的组成部分一世可能包括有关解决问题的先验质量的数据，例如，约束给定函数的导数阶的绝对值的常数、有关输入值准确性的数据等。 算法的多次迭代，逼近的程度，网格的步长等，可以是向量X组件。向量是可以包含计算机内存的单元位数，其RAM的总容量，计算机的运行时间，计算机的数量
## 数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation
F和F(一世,X,是)=最大限度一世,X,是 米(一世,X,是)≤米R和,吨(一世,X,是)≤吨R和,和(一世,X,是)≤和R和
## 有限元方法代写
tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。
## MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 | 2023-03-24 12:16:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.7087296843528748, "perplexity": 771.7276520400883}, "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-2023-14/segments/1679296945282.33/warc/CC-MAIN-20230324113500-20230324143500-00302.warc.gz"} |
https://buboflash.eu/bubo5/show-dao2?d=1431900917004 | Tags
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A discrete quantity can only take on [...] For example, the number of students in a classroom can only be a positive integer and is thus a discrete quantity since the range of possible values is not continuous.
certain values.
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A discrete quantity can only take on [...] For example, the number of students in a classroom can only be a positive integer and is thus a discrete quantity since the range of possible values is not continuous.
?
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A discrete quantity can only take on [...] For example, the number of students in a classroom can only be a positive integer and is thus a discrete quantity since the range of possible values is not continuous.
certain values.
#### Summary
status measured difficulty not learned 37% [default] 0
No repetitions | 2022-08-09 14:10:41 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9116397500038147, "perplexity": 286.51892979425264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570977.50/warc/CC-MAIN-20220809124724-20220809154724-00538.warc.gz"} |
https://www.deepdyve.com/lp/aps_physical/improved-quark-coalescence-for-a-multi-phase-transport-model-9mi9KfP0I8 | # Improved quark coalescence for a multi-phase transport model
Improved quark coalescence for a multi-phase transport model The string melting version of a multi-phase transport model is often applied to high-energy heavy-ion collisions since the dense matter thus formed is expected to be in parton degrees of freedom. In this work we improve its quark coalescence component, which describes the hadronization of the partonic matter to a hadronic matter. We removed the previous constraint that forced the numbers of mesons, baryons, and antibaryons in an event to be separately conserved through the quark coalescence process. A quark now could form either a meson or a baryon depending on the distance to its coalescence partner(s). We then compare results from the improved model with the experimental data on hadron dN/dy,pT spectra, and v2 in heavy-ion collisions from sNN=62.4 GeV to 5.02 TeV. We show that, besides being able to describe these observables for low-pT pions and kaons, the improved model also better describes the low-pT baryon observables in general, especially the baryon pT spectra and antibaryon-to-baryon ratios for multistrange baryons. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review C American Physical Society (APS)
# Improved quark coalescence for a multi-phase transport model
, Volume 96 (1) – Jul 17, 2017
## Improved quark coalescence for a multi-phase transport model
Abstract
The string melting version of a multi-phase transport model is often applied to high-energy heavy-ion collisions since the dense matter thus formed is expected to be in parton degrees of freedom. In this work we improve its quark coalescence component, which describes the hadronization of the partonic matter to a hadronic matter. We removed the previous constraint that forced the numbers of mesons, baryons, and antibaryons in an event to be separately conserved through the quark coalescence process. A quark now could form either a meson or a baryon depending on the distance to its coalescence partner(s). We then compare results from the improved model with the experimental data on hadron dN/dy,pT spectra, and v2 in heavy-ion collisions from sNN=62.4 GeV to 5.02 TeV. We show that, besides being able to describe these observables for low-pT pions and kaons, the improved model also better describes the low-pT baryon observables in general, especially the baryon pT spectra and antibaryon-to-baryon ratios for multistrange baryons.
/lp/aps_physical/improved-quark-coalescence-for-a-multi-phase-transport-model-9mi9KfP0I8
Publisher
The American Physical Society
ISSN
0556-2813
eISSN
1089-490X
D.O.I.
10.1103/PhysRevC.96.014910
Publisher site
See Article on Publisher Site
### Abstract
The string melting version of a multi-phase transport model is often applied to high-energy heavy-ion collisions since the dense matter thus formed is expected to be in parton degrees of freedom. In this work we improve its quark coalescence component, which describes the hadronization of the partonic matter to a hadronic matter. We removed the previous constraint that forced the numbers of mesons, baryons, and antibaryons in an event to be separately conserved through the quark coalescence process. A quark now could form either a meson or a baryon depending on the distance to its coalescence partner(s). We then compare results from the improved model with the experimental data on hadron dN/dy,pT spectra, and v2 in heavy-ion collisions from sNN=62.4 GeV to 5.02 TeV. We show that, besides being able to describe these observables for low-pT pions and kaons, the improved model also better describes the low-pT baryon observables in general, especially the baryon pT spectra and antibaryon-to-baryon ratios for multistrange baryons.
### Journal
Physical Review CAmerican Physical Society (APS)
Published: Jul 17, 2017
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Export lists, citations | 2018-07-19 13:56:59 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8802520632743835, "perplexity": 3400.7715475063806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590901.10/warc/CC-MAIN-20180719125339-20180719145339-00433.warc.gz"} |
https://codereview.stackexchange.com/questions/216634/custom-hashset-in-java | # Custom Hashset in Java
I have written this custom hashset and though it isn't completed yet, I would like to know if there is anything I am overlooking in terms of clean code conventions. My aim was also to apply generics to it. So, would also want any input regarding that.
public class CustomHashset<T> {
private static final int SIZE = 100;
private Entry<T>[] buckets;
private int size;
public CustomHashset() {
this.buckets = new Entry[SIZE];
this.size= 0;
}
private int hash(T element) {
return element.hashCode() % buckets.length;
}
int index = hash(element);
Entry<T> current = buckets[index];
while (current!=null) {
if (current.key.equals(element)) return false;
current = current.next;
}
Entry<T> entry = new Entry<T>();
entry.setKey(element);
entry.setNext(buckets[index]);
buckets[index] = entry;
size++;
return true;
}
public int size() {
return size;
}
private static class Entry<T> {
private T key;
private Entry next;
public T getKey() {
return key;
}
public void setKey(T element) {
this.key = element;
}
public Entry getNext () {
return next;
}
public void setNext(Entry next) {
this.next = next;
}
}
}
• Request for Clarification: do you want your CustomHashSet to be a HashSet? i.e., do you want this to be valid: HashSet<T> hashSet = new CustomHashSet<>(); – DapperDan Apr 1 '19 at 15:05
• @Wood Glass, no. – user200188 Apr 1 '19 at 16:01
• What are your design goals? Is there a reason that java.util.HashSet will not suffice? Do you want to integrate nicely with the Java Collections Framework, or avoid it entirely? It is hard to review this code without more context. – Benjamin Kuykendall Apr 1 '19 at 22:10
• @BenjaminKuykendall, simply learning how to implement these data structures using generics. – user200188 Apr 2 '19 at 6:38
## Problems with generics
• Entry<T>.next should have type Entry<T>.
• CustomHashset<T>.buckets has type Entry<T>[] but is initialized to an Entry[]. Initializing a generic array properly is annoying, but this StackOverflow question explores a few solutions. At the end of the day, there's only so much you can do about it; you can at least suppress the compiler warning though.
## Interface of Entry
You never use the getters and setters. Remove them.
As you implement more methods, it should become apparent that you are using the chain of Entry<T> objects as a linked list. Thus the interface it presents could be more abstract: in fact, I would create a LinkedList<T> class with add(T t) and contains(T t) methods. Then make buckets an array of LinkedList<T>. This way, the hash set never has to deal with individual entries.
## Small problems
• Use or omit this more consistently. I would omit it unless needed.
• Return boolean not Boolean.
• Currently, a null key causes a NullPointerException when you call hash(). You should detect null inputs to add and explicitly throw an IllegalArgumentException instead.
• Use default visibility instead of public to expose methods of private inner classes.
• Thanks for your review. Very interesting take. 1. I really didn't have any problem initializing the generic array. Maybe there is something you can point to in the way I have initialized it for a better understanding. I took a look at the link you added and it appears I was doing something close to the checked except I wasn't wrapping it in the way it was explained over there. 2. I did use the setters but not the getters. So, I will remove the getters. Thanks for that. Continuation ...below – user200188 Apr 4 '19 at 7:27
• 3. Could you elaborate further on the 'Abstract out linked list'? I get you want me to make the Entry<T> a LinkedList<T> because, under the hood, I was using node chaining. I don't get making buckets an array of LinkedList<T>. Even though I could do that, would I not be relying on an existing implementation? I could very well have just added a Java HashMap internally and not bother about implementing the hashing and the add contains e.t.c as I would just be calling the hashMap operators under the hood. So, just want to know if this is Ok to do. – user200188 Apr 4 '19 at 7:28
• I would also ditch the setters in favor of a constructor: Entry<T>(T element, Entry<T> next). If you do this, you can even make both fields final. If you don't want to use java.util.LinkedList, writing your own CustomLinkedList that supports add and contains is really easy; in fact you have most of the code there, it's just a mater of reorganizing it. – Benjamin Kuykendall Apr 4 '19 at 12:44
Improving names:
• Your class has 2 size variables, which looks kinda strange. Better to rename this private static one to something like INITIAL_CAPACITY (or DEFAULT_INITIAL_CAPACITY, if you have plans adding constructor with initialCapacity parameter), because size = how many elements are stored, but this variable denotes initial array length.
• method hash in reality returns bucket index for given element, thus, should be renamed. For example, to indexFor or bucketIndex.
Other:
• in Entry - next is not parameterized (as well as its getter and setter).
• Thanks I will look into those. – user200188 Apr 2 '19 at 19:04 | 2020-09-24 11:46:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2969067394733429, "perplexity": 3349.1546578417265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400217623.41/warc/CC-MAIN-20200924100829-20200924130829-00344.warc.gz"} |
https://www.physicsoverflow.org/43508/construct-an-rotation-inside-the-two-fundamental-rotations | # Construct an SO(3) rotation inside the two SU(2) fundamental rotations
+ 4 like - 0 dislike
45 views
We know that two SU(2) fundamentals have multiplication decompositions, such that $$2 \otimes 2= 1 \oplus 3.$$ In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial representation of SU(2).
I hope to see the precise SO(3) rotation from the two SU(2) fundamental rotations.
-
So let us first write two SU(2) fundamental objects in terms of an SO(3) object. In particular, we can consider the following three:
$$|1,1\rangle= \begin{pmatrix} 1\\ 0 \end{pmatrix}\begin{pmatrix} 1\\ 0 \end{pmatrix}= | \uparrow \uparrow \rangle,$$ $$|1,0\rangle ={1 \over \sqrt{2} } (\begin{pmatrix} 1\\ 0 \end{pmatrix} \begin{pmatrix} 0\\ 1 \end{pmatrix} + \begin{pmatrix} 0\\ 1 \end{pmatrix} \begin{pmatrix} 1\\ 0 \end{pmatrix})={1 \over \sqrt{2} }(| \uparrow \downarrow \rangle+ | \downarrow \uparrow \rangle) ,$$ $$|1,-1\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix}\begin{pmatrix} 0\\ 1 \end{pmatrix}= | \downarrow \downarrow \rangle.$$
where the $| \uparrow \rangle$ and $\downarrow \rangle$ are in SU(2) fundamentals. And we shothand $| \uparrow \uparrow \rangle \equiv | \uparrow \rangle |\uparrow \rangle$ and so on.
question: How do we rotate between $|1,1\rangle$, $|1,0\rangle$, $|1,-1\rangle$, via two SU(2) rotations acting on two SU(2) fundamentals? Namely, that is, construct an SO(3) rotation inside the two SU(2) fundamental rotations? The SU(2) has three generators, parametrized by $m_x,m_y,m_z$; how do we write down the generic SO(3) rotations from two SU(2) rotations?
Let us consider an example, an SU(2) rotation $U$ acting on the SU(2) fundamental $\begin{pmatrix} 1\\ 0 \end{pmatrix}$ give rise to $$U \begin{pmatrix} 1\\ 0 \end{pmatrix}= \begin{pmatrix} \cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2}) & (i m_x -m_y) \sin(\frac{\theta}{2}) \\ (i m_x +m_y) \sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2})-{i m_z} \sin(\frac{\theta}{2}) \\ \end{pmatrix} \begin{pmatrix} 1\\ 0 \end{pmatrix}= \begin{pmatrix} \cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\ (i m_x +m_y) \sin(\frac{\theta}{2}) \end{pmatrix} \equiv\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2}) \begin{pmatrix} 1\\ 0 \end{pmatrix} + (i m_x +m_y) \sin(\frac{\theta}{2}) \begin{pmatrix} 0\\ 1 \end{pmatrix}$$
In other words, the SU(2) rotation $U$ (with the $|\vec{m}|^2=1$) rotates SU(2) fundamentals. Two SU(2) rotations act as $$UU|1,1\rangle = U \begin{pmatrix} 1\\ 0 \end{pmatrix}U \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} \cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\ (i m_x +m_y) \sin(\frac{\theta}{2}) \end{pmatrix}\begin{pmatrix} \cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\ (i m_x +m_y) \sin(\frac{\theta}{2}) \end{pmatrix}$$
Hint: Naively, we like to construct $$L_{\pm} =L_{x} \pm i L_y,$$ such that $L_{\pm}$ is an operator out of two SU(2) rotations acting on two SU(2) fundamentals, such that it raises/lowers between $|1,1\rangle$, $|1,0\rangle$, $|1,-1\rangle$.
question 2: Is it plausible that two SU(2) are impossible to perform such SO(3) rotations, but we need two U(2) rotations?
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
+ 2 like - 0 dislike
The following solution originates from the theory of geometric quantization. I'll not explain the full theory behind it, but I'll give here the solution, then briefly discuss how to check that this is the required solution.
A general $SU(2)$ group element in the fundamental representation can be written as: $$g = \begin{bmatrix} \alpha & \beta\\ -\bar{\beta}& \bar{\alpha} \end{bmatrix}$$ with $$|\alpha|^2+\beta|^2=1$$ The three dimensional Hilbert space of the three dimensional representation can be parametrized by: $$\psi (z) = a + b z + cz^2 \quad (1)$$
where $x$ is an indeterminate
The action of $SU(2)$ on this vector space is given by:
$$(g\cdot \psi)(z) = (-\bar{\beta} z + \bar{\alpha})^2 \psi( \frac{\alpha z + \beta}{-\bar{\beta} z + \bar{\alpha}}) \quad (2)$$
1. To see that this is a representation, one can check that the composition of the action of two group elements coincides with the action of their product.
2. To see that this is a faithful $SO(3)$ representation but not a faithful $SU(2)$, we can easily see that for the nontrivial element of the center: $$g_c = \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}$$ We have for every $\psi$ $$(g_c\cdot \psi)(z) = \psi(z)$$
3. Although, the "spherical" components $a, b, c$ are complex. To see that the representation is real, one can see that the "Cartesian" components $(a+c), b, i^{-1}(a-c)$ transform by means of only real combinations of $\alpha$ and $\beta$.
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user David Bar Moshe
answered Apr 22, 2018 by (4,345 points)
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
Then I guess maybe you can also solve this or at least share your opinions: math.stackexchange.com/q/2745276 - thanks.
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
@ David, it is easy to see from $$UU|1,1\rangle = U \begin{pmatrix} 1\\ 0 \end{pmatrix}U \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} \cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\ (i m_x +m_y) \sin(\frac{\theta}{2}) \end{pmatrix}\begin{pmatrix} \cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\ (i m_x +m_y) \sin(\frac{\theta}{2}) \end{pmatrix}$$ when $m_y=\pm 1, m_x=m_z=1$ and $\theta=\pi$, we have $$UU|1,1\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix} \begin{pmatrix} 0\\ 1 \end{pmatrix}=|1,-1\rangle$$
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
Do you have any precise answer how to find $U$, such that $$UU|1,1\rangle = \frac{1}{\sqrt 2} \begin{pmatrix} 1\\ 0 \end{pmatrix}\begin{pmatrix} 0\\ 1 \end{pmatrix} + \frac{1}{\sqrt 2} \begin{pmatrix} 1\\ 0 \end{pmatrix}\begin{pmatrix} 0\\ 1 \end{pmatrix}=|1,0\rangle ?$$
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
@annie heart there is no such a $U$. Please observe that if you act by some $U \otimes U$ on the highest weight vector: $\begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix}$, then the result will always be a separable vector of the form $\begin{pmatrix} a\\ b\end{pmatrix} \otimes \begin{pmatrix} a\\ b \end{pmatrix}$. But the vector that you want to reach is entangled, and there is no way to change the entanglement state by a local transformation.
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user David Bar Moshe
Indeed, the entanglement structure is different, then how about construct the rotations out of two U(2) rotations, instead of two SU(2)?
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
No, $U(2)$ will not change the entanglement structure either, An $SU(3)$ element acting on the three dimensional vector space will be needed.
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user David Bar Moshe
+ 1 like - 0 dislike
1. OP's eq. (1) is to be understood as a relation between complex representations of $SU(2)$, i.e complex vector spaces. Recalling that the fundamental $SU(2)$ representation ${\bf 2}\cong \overline{\bf 2}$ is isomorphic to the complex conjugate representation, let us instead consider the isomorphism $${\bf 2}\otimes \overline{\bf 2}~\cong~{\bf 1}\oplus {\bf 3}. \tag{A}$$
2. The left-hand side of eq. (A) can be realized as the real vector space $u(2)$ of $2\times 2$ Hermitian matrices. The group $SU(2)$ acts on $u(2)$ via conjugation. Given a spinor $| \psi\rangle\in {\bf 2}$, then $${\bf 1}\oplus su(2)~\cong~ u(2)~\ni~| \psi\rangle \langle\psi | ~=~\frac{1}{2}\sum_{\mu=0}^3x^{\mu} \sigma_{\mu}, \qquad (x^0,x^1,x^2,x^3) ~\in~\mathbb{R}^4. \tag{B}$$ The triplet ${\bf 3}$ corresponds to the traceless part, that is: $su(2)$. Hence the spinor $| \psi\rangle$ represents the 3-vector $\vec{r}=(x^1,x^2,x^3)$. See also this related Phys.SE post.
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user Qmechanic
answered Apr 22, 2018 by (2,900 points)
Thanks! +1, the same question as to David, do you have a precise rotation from how to find U, such that UU|1,1⟩=(1/√2) (10)(01)+(1/√2)√(10)(01)=|1,0⟩ by some U?
This post imported from StackExchange Physics at 2020-11-06 18:48 (UTC), posted by SE-user annie marie heart
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https://www.physicsforums.com/threads/derivation-of-sackur-tetrode-equation.540183/ | # Derivation of sackur-tetrode equation
1. Oct 14, 2011
### ralden
how sackur-tetrode equation derive?, can it derive without the use of macrocanonical ensemble? only by classical thermodynamics? thank you.
2. Oct 14, 2011
### vanhees71
The Sackur-Tetrode formula can only be derived properly as the classical limit of quantum statistics, which is Boltzmann statistics for both fermions and bosons, taking into account the indistinguishability of particles, which is a specific quantum-mechanical phenomenon. Classical statistics leads to the Gibbs paradoxon and a wrong (non-extensive) expression for the entropy which is solved by the Sackur-Tetrode formula.
3. Oct 14, 2011
### wbandersonjr
You can derive the Sackur-Tetrode equation by solving for the entropy of an ideal gas using Stirling's approximation applied to the multiplicity formula. My thermodynamics text does not go through the whole derivation, but that is how it says to derive it.
4. Oct 15, 2011
### vanhees71
Yes, that you can do, but you have to assume the indistinguishability of particles, which leads to an additional factor $1/N!$ compared to classical mechanics. This factor cannot justified without the indistinguishability argument that is generically quantum theoretical. | 2017-11-21 16:43:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6113661527633667, "perplexity": 1057.4297738729742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806419.21/warc/CC-MAIN-20171121151133-20171121171133-00680.warc.gz"} |
https://plover.readthedocs.io/en/latest/system-dev/system.html | Implementing the System#
For more details on creating a system plugin on Plover, check out this guide.
Orthography#
Some systems, including English stenotype, includes orthography rules. These are spelling rules that are encoded in the system outside of the dictionary to ensure that suffixes are attached correctly. Here are some of the orthography rules in English:
• sh + s = shes (publish, publishes)
• ie + ing = ying (lie, lying)
There are a few things to take note when making these orthography rules:
• Don’t make an orthography rule just for a single word. If your rule only applies to a single word, it’s better to include it in the dictionary instead. For instance, the suffix {^nance}, when attached to the word “maintain”, gives “maintenance”. We don’t need to include the orthography rule ain + nance = enance; simply including it in the dictionary is good enough.
• Don’t make your orthography rule too general. It is much better to write orthography rules that are more specific. For instance in English, instead of writing the rule y + i = i, it is far better to write rules for individual cases, such as y + ial = ial, y + ical = ical and so on. This prevents rules from changing words that they shouldn’t change.
• Orthography rules can have exceptions. If there are only a few exceptions, you can include them in the dictionary. If there are too many exceptions, consider making the rule more specific if possible to reduce the number of exceptions.
• Orthography rules only apply to {^suffix}, not {^}suffix. It might help to know that you can explicitly define suffixes in your dictionary that are not affected by orthography rules.
Generating the Dictionary#
If you chose to make a phonetic system, then you’ll have to generate a basic dictionary before the system is usable. The process of dictionary-building can be slow, especially if you choose to refine your dictionary manually over the span of several months. You may also choose to generate your dictionary automatically, and there are many tools to help you with that. Here are some tools to help you with making a dictionary:
• Plover Dictionary Builder: A dictionary builder plugin that allows you to add dictionary entries for multiple words in one go.
• Plovary: A dictionary generation tool written in Python.
If you are making an orthographic system, then you’ll find Python Dictionaries useful. Python dictionaries allow you to programmatically translate input strokes into words without explicitly defining the translation of every possible stroke like in a traditional dictionary.
Here are some tips for generating a dictionary:
• It is not always a good idea to blindly include as many words as possible. Often, when system designers try to include as many words as they possibly can in the dictionary, they might end up including rare words in the dictionary that interfere with common words. For instance, including the word “Tob” might prevent you from including an easy way to write “to be”. In such cases, you might choose to add a disambiguation key to the rare word entry (such as the asterisk key on the English layout), or make it such that the word cannot be written normally unless the user is intentionally trying to write it.
• You might want to include word stems in your dictionary, even if they’re not actual words by themselves.
• Try to categorize your dictionary. Separate the words from the symbols and phrases, and misstrokes from the correct strokes. This will help you and potential learners tremendously and will ensure that you have a clean and maintainable dictionary. | 2023-03-28 02:56:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2389942854642868, "perplexity": 1250.0820464882704}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00503.warc.gz"} |
https://chemistry.stackexchange.com/questions/134893/how-can-one-reclaim-isotopic-cadmium-from-a-solution-of-acid-i-e-h2so4-or-hcl | # How can one reclaim isotopic cadmium from a solution of acid, i.e. H2SO4 or HCl etc.?
I have dissolved isotopic cadmium-114 with $$\ce{H2SO4}$$ and or $$\ce{HCl}$$. Now that I have the cadmium-114 in solution, precipitating out $$\ce{CdSO4}$$ and or $$\ce{CdCl2}$$, how can I reduce these simply to leave the pure cadmium metal isotope? I have dissolved $$\ce{^{114}Cd}$$ with acid in very complex electronic tubes and I wish to reduce the precipitated compound(s) to pure metal for reuse.
$$\ce{^{114}Cd}$$ or is running about \$ 1,500 per gram. So extremely useful to reclaim it. Can anyone tell me a simple way to do this?
• What is "isotopic" cadmium? – Karl Jun 6 '20 at 7:06
• What "isotopic cadmium"? There are several isotopes of cadmium. Which one? – Nilay Ghosh Jun 6 '20 at 7:37
• The usual way is electrolytic from a solution of the sulfate, Al as anode. – Karl Jun 6 '20 at 10:26
• Again, Thank you everyone for your responses. I am a Laser manufacture and also a plasma tube re-furbisher where possible. I make new tubes, however, they are very complex glass work. So, saving a tube when possible is economical. The Helium Cadmium Laser gains 30% more power by using a single isotope of Cadmium. Cd114 is the best option and produces the most usable power, especially on 325nm UV lines. 441.6nm lines are not as sensitive but do benefit. I have no need of Cd salts. Just pure metal. Cleaning the tubes for reuse is a standard process. So, dissolving the Cd out of the tube is on st – Dale Harder Jun 6 '20 at 20:01
• I´m not sure that you will find a procedure in the literature that is well suited to your application. But you surely have a lot of contacts into academia. Talk to one who has a wire into inorganic chemistry, convince them to write a small grant application for a one-year project to devise a safe and efficient small scale electrolytic Cd recycling process. There are a lot of good buzzwords in the subject. ;-) – Karl Jun 6 '20 at 22:41
Thank you everyone for the discussion. Sorry, I have been away for a few days, so was unable to follow up. Reading the ideas, it is a toss up on a procedure. Scraping the Cd off of an Aluminum electrode might work, but Al contamination would be very bad for the laser oscillation. It might work for awhile, but would die from contaminants in short order.
The Al oxide, though only a molecular layer, is extremely tough and can withstand a great deal of heat and abuse. I make Aluminum edge-wound voice coils for my speaker company and we depend on the oxide layer to act as an insulator from winding to winding. Our coils can withstand 600 degrees and will melt before they fail.
Stripping the oxide gives you only a few seconds before it reforms in the normal atmosphere. Any other thoughts? Perhaps reduction of the salt by heating?
Thanks Dale
• Perhaps the answer lies here...I understand a lot of it, but not everything. Please look. onlinelibrary.wiley.com/doi/pdf/10.1002/pip.402 Thanks, Dale. – user94922 Jun 15 '20 at 4:19
• The reference is paywalled. In the old days, electrolysis into a mercury cathode, followed by distilling the mercury off, might have been done. Of course, nowadays that would be something only for a specialist laboratory. But maybe use a glassy carbon cathode? Then no aluminum to contaminate the cadmium. – Ed V Jul 15 '20 at 12:26
From a solution of an aqueous cadmium salt ($$\ce{CdSO4}$$ or $$\ce{CdCl2}$$), try inserting a metal plate of say Aluminum.
With time and warming of the solution, a coating of cadmium should appear on the metal sheet.
Try scrapping off the deposited cadmium metal, per the replacement reaction:
[EDIT]$$\ce{2 Al(s) + 3 CdSO4 (aq) -> 3 Cd (s) + Al2(SO4)3 (aq)}$$
Note: depending on your tools for scrapping, you may induce some metallic impurities.
[EDIT] Importantly, my standard chemistry approach (based on a single-displacement reaction) is different from an electrolysis or electrochemical cell approach, employing an Aluminum electrode, as I suspect, that the electrode may quickly become coated and effectively inoperative. Also, an electrochemical cell with a large surface area anode (here Al) would require an even larger surface area cathode to operate efficiently. Further, the ability to facilely recover the Cd metal is key to the cadmium recycling process. My proposed process may fulfill these requirements.
• @ Ajkoer 1) Your equation is wrong ! You would need 3 CdSO4 to balance the equation. And it would produce Al2(SO4)3. 2) Are you sure that this reaction works ? Usually aluminum is covered by a protective layer of aluminum oxide, which prevents the metal from touching the solution. If it does it does not produce aluminum sulfite... – Maurice Jun 7 '20 at 10:25
• Thanks, I worked on it late and was originally thinking of using CdCl2. Will edit. However, more fundamental does my path work in practice? – AJKOER Jun 7 '20 at 14:03
• Why would cadmium ions not get reduced on a Cd cathode in electrochemical cell? And why should Al be used as a cathode in the first place? – Karl Jun 7 '20 at 17:12
• Cadmium can be reduced by metals like zinc and magnesium. With aluminum, it will be theoretically possible, if the protective layer of aluminum oxide would be removed. This protective layer is transparent, continuous and nearly insoluble in diluted acids. That is why for example Aluminum is used as container for handling concentrated nitric acid. The alumina layer protects the metals from corrosion. I don't know how aluminum behaves with cadmium salts. But I think there will be no reaction, because of the protective layer of alumina. Anyway cadmium may be reduced by electrolysis. – Maurice Jun 7 '20 at 19:25 | 2021-04-21 13:49:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45651596784591675, "perplexity": 2465.300479601543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039544239.84/warc/CC-MAIN-20210421130234-20210421160234-00064.warc.gz"} |
https://money.stackexchange.com/questions/54223/why-is-the-breakdown-of-a-loan-repayment-into-principal-and-interest-of-any-impo/54228 | # Why is the breakdown of a loan repayment into principal and interest of any importance?
Repaying a loan is a type of cash outflow. Why should I care which part of each payment goes towards the interest and which part towards the principal? "A loan is a loan", no?
I know that this breakdown is considered when calculating income tax, but is there any other significance?
(source)
• Why would this be down voted? It's a reasonable financial question that many people have and is important to have answered! – Dave Sep 24 '15 at 23:11
The breakdown between how much of your payment is going toward principal and interest is very important.
The principal balance remaining on your loan is the payoff amount. Once the principal is paid off, your loan is finished. Each month, some of your payment goes to pay off the principal, and some goes to pay interest (profit for the bank).
Using your example image, let's say that you've just taken out a $300k mortgage at 5% interest for 30 years. You can click here to see the amortization schedule on that loan. The monthly payment is$1610.46.
On your first payment, only $360 went to pay off your principal. The rest ($1250) went to interest. That money is lost. If you were to pay off your $300k mortgage after making one payment, it would cost you$299,640, even though you had just made a payment of $1250. Interest accrues on the principal balance, so as time goes on and more of the principal has been paid, the interest payment is less, meaning that more of your monthly payment can go toward the principal. 15 years into your 30-year mortgage, your monthly payment is paying$762 of your principal, and only $849 is going toward interest. Your principal balance at that time would be about$203k. Even though you are halfway done with your mortgage in terms of time, you've only paid off about a third of your house.
Toward the end of your mortgage, when your principal balance is very low, almost all of your payment goes toward principal. In the last year, only \$513 of your payments goes toward interest for the whole year.
You can think of your monthly loan payment as a minimum payment. If you continue to make the regular monthly payments, your mortgage will be paid off in 30 years. However, if you pay more than that, your mortgage will be paid off much sooner. The extra that you pay above your regular monthly payment all goes toward principal.
Even if you have no plans to pay your mortgage ahead of schedule, there are other situations where the principal balance matters. The principal balance of your mortgage affects the amount of equity that you have in your home, which is important if you sell the house. If you decide to refinance your mortgage, the principal balance is the amount that will need to be paid off by the new loan to close the old loan.
• So the only scenario in which I care about the breakdown is if I decide to repay the loan before schedule? – Sparkler Sep 24 '15 at 18:37
• Even if you have no plans to pay ahead of schedule, you might sell the house or refinance. If you refinance after 10 years, the amount of principal left in the loan will determine how big of a new mortgage you need. – Ben Miller - Reinstate Monica Sep 24 '15 at 18:45
• yeah, I'm seeing selling the house as repaying ahead of schedule. Do you mind adding this bit to your answer? – Sparkler Sep 24 '15 at 19:12
• @Sparkler Sure. I've expanded my answer. Let me know if it is still unclear. – Ben Miller - Reinstate Monica Sep 24 '15 at 19:19
• @Sparkler, as per your previous question as well, by paying down the principal you will reduce the interest payable on the subsequent months thus reduce your expenses on the property and increase your net rental return if it is an investment property. – user9722 Sep 24 '15 at 21:44
It's important because it shows that the amount you owe does not decrease linearly with each payment, and you gain equity as a correspondingly slower rate at the beginning of the loan and faster at the end. This has to be figured in when considering refinancing, or when you sell the place and pay off the mortgage.
It also shows why making extra payments toward principal (if your loan permits doing so) is so advantageous -- unlike a normal payment that lowers the whole curve by a notch, reducing the length of time over which interest is due and thus saving you money in the long run. (Modulo possible lost-opportnity costs, of course.)
Yes, the distinction between how your funds are applied to principal vs interest is very important. The interest amount charged each period (probably monthly) is not just one fixed sum calculated at the origination, but rather is a dynamically calculated amount that changes each period relative to how much principal is remaining (amount you owe).
The picture you posted showing principal and interest assumes the payer always paid their minimum payment and never made any extra payments of principal.
Take a look at the following graph and play around with the extra payment fields. You will see some pretty drastic differences in the Total Interest Paid (green lines) when extra payments are made.
http://mortgagevista.com/#m=2&a=240000&b=4.5&c=30y&e=200&f=1/2020&g=10000&h=1/2025&G&H&J&M&N&P&n&o&p&q&x
The other answers have touched on amortization, early payment, computation of interest, etc, which are all very important, but I think there's another way to understand the importance of knowing the P/I breakdown.
The question mentions the loan payment as "cash outflow". That is true, but from an accounting perspective (disclaimer: I am not an accountant, but I know enough of the basics to be dangerous), the outflow needs to be directed to different accounts.
The loan principal appears as a liability on your personal balance sheet, which you could use, for example, in determining net worth. The principal amount in your payment should be applied to reduce the liability account. The interest payment goes into the expense account.
Another way to look at it is that the principal, while it does reduce your cash account, can be thought of as an internal transfer to the liability account, thus reducing the size of the liability. The interest payment cannot.
Aside: From this perspective, the value of the home is an asset, and the difference between the asset account and the loan liability account is the equity in the house (as pointed out in different language by the accepted answer). Of course, precisely determining the value of an illiquid asset like a house at any given moment pretty much requires you to actually sell it, so those accounts are hard to maintain in real-time (the liability of the loan is much easier to track).
It's important because you may be able to reduce the total amount of interest paid (by paying the loan faster); but you can do nothing to reduce the total of your principal repayments.
The distinction can also affect the amount of tax you have to pay. Some kinds of interest payments can be counted as business expenses, which means that they reduce the amount of income you have to pay tax on. But this is not generally the case for money used to repay the loan principal.
The reason it's broken out is very specific: this is showing you how much interest accrued during the month. It is the only place that's shown, typically.
Each month's (minimum) payment is the sum of [the interest accrued during that month] and [some principal], say M=I+P, and B is your total loan balance. That I is fixed at the amount of interest that accrued that month - you always must pay off the accrued interest. It changes each month as some of the principal is reduced; if you have a 3% daily interest rate, you owe (0.03*B*31) approximately (plus a bit as the interest on the interest accrues) each month (or *30 or *28). Since B is going down constantly as principal is paid off, I is also going down.
The P is most commonly calculated based on an amortization table, such that you have a fixed payment amount each month and pay the loan off after a certain period of time. That's why P changes each month - because it's easier for people to have a constant monthly payment M, than to have a fixed P and variable I for a variable M.
As such, it's important to show you the I amount, both so you can verify that the loan is being correctly charged/paid, and for your tax purposes. | 2019-11-22 00:06:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4108671545982361, "perplexity": 994.4818095102187}, "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-2019-47/segments/1573496671053.31/warc/CC-MAIN-20191121231600-20191122015600-00180.warc.gz"} |
https://kwant-project.org/doc/dev/reference/kwant.operator | # 3.6. kwant.operator – Operators and Observables¶
## Observables¶
Density(syst[, onsite, where, …]) An operator for calculating general densities. Current(syst[, onsite, where, …]) An operator for calculating general currents. Source(syst[, onsite, where, …]) An operator for calculating general sources.
kwant.solvers.mumps – High performance sparse solver based on MUMPS | 2021-12-01 06:10:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7638204097747803, "perplexity": 2869.4270270138877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964359093.97/warc/CC-MAIN-20211201052655-20211201082655-00142.warc.gz"} |
https://socratic.org/questions/what-temperature-scale-is-used-in-science | # What temperature scale is used in science?
Jun 18, 2018
Well, what temperature scale is appropriate....?
#### Explanation:
Several experimental gas laws established the dependence between a gaseous volume and temperature. And from the development of ideas of atoms and molecules, the experimental gas laws enabled chemists and physicists to form a pretty shrewd idea about the nature of gaseous particles...and of course from these ideas the molecular nature of matter, especially with regard to gases, were developed.
Old $\text{Charles' Law}$ established the proportionality of volume and temperature....i.e. $V \propto T$...and this law proposed that there was a limit with regard to temperature...what today we would call $\text{absolute zero...}$. And thus at absolute zero, $- 273.15$ ""^@C or $0 \cdot K$ all molecular motion ceases (a simplification!), and this temperature represents the LIMIT of temperature...
Today $- 273.15$ ""^@C-=0*K...the $K$ stands for $\text{Kelvin Temperature}$, after William Thomson, later the Lord Kelvin (1824-1907)…
Most of the Gas Laws...$\text{Charles' Law}$, the $\text{Ideal Gas Law}$ specify the use of the Kelvin scale...
As a curiosity much of the gas laws owe to railway development....i.e. steam driven locomotives directly exploit gaseous pressure to provide motion... | 2019-12-06 05:25:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8076520562171936, "perplexity": 2678.0356043276997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540484815.34/warc/CC-MAIN-20191206050236-20191206074236-00219.warc.gz"} |
https://economics.stackexchange.com/questions/34004/locally-nonsatiated-preferences | # locally nonsatiated preferences
what does this symbol mean in the discuss of locally nonsatiated preferences: $$\varepsilon > 0$$ and $$||y-x||<\varepsilon$$.
If you take the vector $$\mathbf{x}$$ as the center and an $$\epsilon > 0$$ sufficiently small as the radius, then the open ball (local neighborhood of the vector $$\mathbf{x}$$) is the set of all vectors $$\mathbf{y} \in \mathbb{R}^{n}$$ whose distance to the center $$\mathbf{x}$$ is less than $$\epsilon$$ measured w.r.t. the metric $$\Arrowvert \cdot \Arrowvert$$, i.e., more formally as
$$\mathbf{B}(\mathbf{x},\epsilon) := \big\{\mathbf{y} \in \mathbb{R}^{n}\,\Big\arrowvert\, \Arrowvert \mathbf{y} - \mathbf{x} \Arrowvert < \epsilon \big\}.$$
Notice that measuring a distance by a metric is only possible within a metric space or normed space like Hausdorff or Banach. Topological spaces do not have in general a metric, in this case the above concept must be generalized. Fortunately, all finite dimensional spaces have a canonical norm, i.e., the Euclidean norm.
$$||v||$$ is the norm symbol, it means the length of the vector $$v$$.
• Surely you mean "norm". The norm is a special case of distance-functions or "metrics". – heh Feb 14 at 19:16
• Indeed I do. Thank you! – Giskard Feb 14 at 19:52
Depends on what exactly you mean with "this symbol":
1. $$\varepsilon$$ denotes a real number.
2. $$>$$ stands for the strictly-greater-than relation in the real numbers.
3. $$0$$ is the number zero, i.e., the neutral element with respect to addition.
4. $$x$$ and $$y$$ are consumption bundles, i.e., vectors of the same finite dimension with non-negative real entries.
5. $$-$$ is the minus sign, the inverse of the $$+$$ operation. The operation is performed element-wise for vectors.
6. $$||v||$$ is the norm (i.e., length) of the vector $$v$$. If $$v\in\mathbb{R}^n$$, then $$||v||=(v_1^2+\cdots +v_n^2)^\frac{1}{2}$$.
7. $$<$$ stands for the strictly-less-than relation in the real numbers. | 2020-04-08 03:19:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9797102808952332, "perplexity": 402.52591141105154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371807538.83/warc/CC-MAIN-20200408010207-20200408040707-00121.warc.gz"} |
https://brilliant.org/problems/tricky-2/ | # Tricky!
Algebra Level 3
$\large 2^x = 2^{46} - 2^{45} - 2^{44} - \ldots - 2^{-44} - 2^{-45} - 2^{-46}$
What is the value of $$x$$ that satisfy the equation above?
× | 2017-10-21 10:42:57 | {"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": 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": 0.8325300216674805, "perplexity": 1652.71081446854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824733.32/warc/CC-MAIN-20171021095939-20171021115939-00827.warc.gz"} |
http://encyclopedia.kids.net.au/page/co/Conn_Smythe_Trophy | ## Encyclopedia > Conn Smythe Trophy
Article Content
# Conn Smythe Trophy
The Conn Smythe Trophy is awarded annually to the most valuable player during the National Hockey League's Stanley Cup playoffs. The winner is chosen by the vote of members of the Professional Hockey Writers' Association.
The trophy was named for the late Conn Smythe[?] (1895-1980), a Hockey Hall of Fame builder and owner of the Toronto Maple Leafs. The first ever trophy was won by Jean Beliveau[?] of the Montreal Canadiens for his performance in the 1965 Stanley Cup playoffs. Since then, Patrick Roy[?] has won it 3 times, more than any other player in history and Wayne Gretzky, Mario Lemieux, Bobby Orr, and Bernie Parent[?] have each won it twice.
(#) indicates multiple winner
++ indicates player from losing team
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# How many different-sized circles with positive integer radii have area
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How many different-sized circles with positive integer radii have areas less than 100?
(A) Four
(B) Five
(C) Six
(D) Ten
(E) Fifteen
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Re: How many different-sized circles with positive integer radii have area [#permalink]
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28 Nov 2017, 21:29
B
For radius=6 the area is 113.09>100. Hence only upto 5
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29 Nov 2017, 16:13
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Bunuel wrote:
How many different-sized circles with positive integer radii have areas less than 100?
(A) Four
(B) Five
(C) Six
(D) Ten
(E) Fifteen
I can see potential trouble with area = 100.
Area of circle is $$πr^2$$.
So it must be $$r^2 * 3.14 < 100$$, where r is an integer.
Radii 1, 2, 3, 4, and 5 (25 * 3.14 = approx 78) work.
Radius 6 does not: 36 * 3 = 108, so with 3.14 it would be higher.
Five values satisfy the conditions.
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Re: How many different-sized circles with positive integer radii have area [#permalink]
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01 Dec 2017, 07:54
Bunuel wrote:
How many different-sized circles with positive integer radii have areas less than 100?
(A) Four
(B) Five
(C) Six
(D) Ten
(E) Fifteen
If r = 1, then the area of the circle is π(1)^2 = π.
If r = 2, then the area is π(2)^2 = 4π.
If r = 3, then the area is π(3)^2 = 9π.
If r = 4, then the area is π(4)^2 = 16π.
If r = 5, then the area is π(5)^2 = 25π.
Since π < 4, the area of any of the circles above will be less than 100. However, if r = 6, then the area of the circle will be π(6)^2 = 36π. However, since π > 3, 36π will be greater than 100. Thus, there are five different-sized circles with positive integer radii with area less than 100.
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Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2018-07-21 15:48:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6387928128242493, "perplexity": 10317.192385836592}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592636.68/warc/CC-MAIN-20180721145209-20180721165209-00247.warc.gz"} |
https://homework.cpm.org/category/CCI_CT/textbook/calc/chapter/4/lesson/4.3.1/problem/4-102 | ### Home > CALC > Chapter 4 > Lesson 4.3.1 > Problem4-102
4-102.
If a function is increasing, then its slopes are positive.
FInd where the derivative is greater than 0. | 2020-10-29 07:28:02 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8712432980537415, "perplexity": 2265.862756975508}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107903419.77/warc/CC-MAIN-20201029065424-20201029095424-00604.warc.gz"} |
https://math.stackexchange.com/questions/3847832/functional-analysis-finding-finite-functions-over-different-norms | # Functional Analysis: Finding finite functions over different norms
The hint in the exercise is that you can define your function piece-wise.
I know that the three norms look like this: $$\norm f_1 = \sum_{t \in T}|f(t)|$$ $$\norm f_2 = \left(\sum_{t \in T}|f(t)|^2\right)^{1/2}$$ $$\norm f_{\infty} = \sup_{t \in T}|f(t)|$$ Where $$T$$ is the time domain (I believe in this exercise it would be the continuous real-valued signals, but I'm not 100% sure).
I'm not too sure what the functions should look like.
• What's $T$? ${}{}{}$ Oct 1 '20 at 17:43
• Should the last norm in the first paragraph be the sup norm? Oct 1 '20 at 17:58
• @jejove2096 which functions have you tried? Were the norms finite or infinite? Oct 1 '20 at 18:38
• Sorry what is the time domain? Indeed I am confused since you have real-valued signal defined on $(0,\infty)$, but then your norms seem to say that your signal is defined on a discrete set $T$. Oct 1 '20 at 19:02
• Please don't self-delete your question with a high-quality answer again. The loss of good content, especially the well-written answer, from the network has a negative impact on this site's SEO. Jan 15 at 8:33
$$\norm{f}_p=\left(\int_0^\infty |f|^pdt\right)^{1/p}$$
and
$$\norm{f}_\infty=\sup_{(0,\infty)}|f|$$
(if they exist)
$$p=1):$$ Consider the function
$$f(t)=\max\left(\frac{1}{\sqrt{t}}-1,0\right)$$
Then
$$\norm{f}_1=\int_0^\infty |f|dt=\int_0^1\left(\frac{1}{\sqrt{t}}-1\right)dt=1$$
$$\norm{f}_2^2=\int_0^\infty |f|^2dt=\int_0^1\left(\frac{1}{\sqrt{t}}-1\right)^2dt=\int_0^1\left(-\frac{2}{\sqrt{t}}+\frac{1}{t}+1\right)dt=\infty$$
$$\norm{f}_\infty=\infty$$
$$p=2):$$ Consider the function
$$f(t)=\begin{cases} \frac{1}{t^{1/4}} & t\leq 1 \\ \frac{1}{t} & t\geq 1 \end{cases}$$
Then
$$\norm{f}_1=\int_0^\infty |f|dt\geq \int_1^\infty \frac{1}{t}dt=\infty$$
$$\norm{f}_2^2=\int_0^\infty |f|^2dt=\int_0^1\frac{1}{\sqrt{t}}dt+\int_1^\infty\frac{1}{t^2}dt=\frac{5}{3}$$
$$\norm{f}_\infty=\infty$$
$$p=\infty):$$ Consider the function
$$f(t)=\frac{1}{\sqrt{t+1}}$$
Then
$$\norm{f}_1=\int_0^\infty |f|dt\geq \int_0^\infty \frac{1}{\sqrt{t+1}}dt=\infty$$
$$\norm{f}_2^2=\int_0^\infty |f|^2dt\geq \int_0^\infty \frac{1}{t}dt=\infty$$
$$\norm{f}_\infty=1$$
For the second portion, assume that both $$\norm{f}_1$$ and $$\norm{f}_\infty$$ exist. From this, we know that $$f(t)$$ is bounded on $$(0,\infty)$$ (this bound is $$M=\norm{f}_\infty$$) and that
$$\norm{f}_1=\int_0^\infty |f|dt<\infty$$
This implies that
$$\frac{|f|^2}{M^2}=\left(\frac{|f|}{M}\right)^2<\frac{|f|}{M}$$
(since $$\frac{|f|}{M}\leq 1$$). Therefore
$$\norm{f}_2^2=\int_0^\infty |f|^2dt=M^2\int_0^\infty \frac{|f|^2}{M^2}dt
We conclude $$\norm{f}_2$$ is also finite. | 2021-10-22 04:31:20 | {"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": 42, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.93498295545578, "perplexity": 425.6748631407962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585450.39/warc/CC-MAIN-20211022021705-20211022051705-00038.warc.gz"} |
https://magento.stackexchange.com/questions/61935/qty-at-0-doesnt-change-product-to-out-of-stock | # QTY at 0 doesn't change product to Out of Stock
UPDATE: I figured out that the issue is that while this script updates the item in the cataloginventory_stock_item table, it fails to update it in the cataloginventory_stock_status table. Added an answer for how to fix this.
The following script is being used to update the inventory levels of items in my store:
<?
define('MAGENTO', realpath(dirname(__FILE__)));
require_once MAGENTO . '/app/Mage.php';
$count = 0;$file = fopen(MAGENTO . '/var/import/updateStockLevels.csv', 'r');
while (($line = fgetcsv($file)) !== FALSE) {
if ($count == 0) { foreach ($line as $key=>$value) {
$cols[$value] = $key; } }$count++;
if ($count == 1) continue; #Convert the lines to cols if ($count > 0) {
foreach($cols as$col=>$value) { unset(${$col});${$col} =$line[$value]; } } // Check if SKU exists$product = Mage::getModel('catalog/product')->loadByAttribute('sku',$sku); if ($product ) {
$productId =$product->getId();
$stockItem = Mage::getModel('cataloginventory/stock_item')->loadByProduct($productId);
$stockItemId =$stockItem->getId();
$stock = array(); if (!$stockItemId) {
$stockItem->setData('product_id',$product->getId());
$stockItem->setData('stock_id', 1); } else {$stock = $stockItem->getData(); } foreach($cols as $col=>$value) {
$stock[$col] = $line[$value];
}
foreach($stock as$field => $value) {$stockItem->setData($field,$value?$value:0); }$stockItem->save();
unset($stockItem); unset($product);
}
echo "<br />Stock updated $sku"; } fclose($file);
?>
This script works great and all qty levels are updated accurately. The issue is with the in stock check.
If I go to an item page on the frontend that has qty zero after running the script, it shows it as In Stock. However, when I go to check that item in the backend, the Inventory->Stock Availability field correctly shows as Out of Stock, and the database itself has the correct value.
If I go to the product page in the admin, and click to save the product (since it already shows it correctly being out of stock in the availability field), its fixes the problem. The frontend will now show it properly as out of stock.
Since qty set to 0 and is_in_stock set to false (0) is not enough to mark it properly as Out of Stock on the frontend, I assume that there must be some field that I am not properly updating in my script, that is only updated properly when I save the item from the admin backend. However I don't know what field(s) I could be missing.
I confirmed that the issue was in fact that some fields were not being updated that needed to be. These fields ended up being qty and stock_status in the table cataloginventory_stock_status. I found a couple ways to fix this. You can update each item in the script using the assign function that the Mage_CatalogInventory_Model_Stock_Status has. Simply place it in the if ( $product ) loop. The other way is to use the rebuild function that the Mage_CatalogInventory_Model_Stock_Status has. You call this outside the if loop and it will rebuild all the stock statuses on the site based on their newly updated qty. This is the method I used, and my edit was simply adding this function call after the if loop, right after the fclose line. fclose($handle);
$stockStatus = Mage::getModel('cataloginventory/stock_status')->rebuild($website_Id); | 2019-10-16 19:14:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4364619255065918, "perplexity": 2217.547261912885}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986669546.24/warc/CC-MAIN-20191016190431-20191016213931-00269.warc.gz"} |
https://www.physicsforums.com/threads/mechanics-of-fluids.749376/ | Mechanics of fluids
1. Apr 18, 2014
LagrangeEuler
1. The problem statement, all variables and given/known data
Small part of fluid which in the moment $t=0$ was at the point $(X_1,X_2,X_3)$ in some other moment is in the point $(x_1,x_2,x_3)$. Where
$$x_1(t)=X_1$$
$$x_2(t)=X_2+\sin \pi t\sin \pi X_1$$
$$x_3(t)=X_3$$
We know that the piece of the fluid is in the t=0 in line which connects points $(0,0,0)$ i $(1,0,0)$. Plot piece of the fluid at the moment $t=\frac{1}{2},t=1$ and $t=\frac{3}{2}$.
2. Relevant equations
As far as I undestand this is Lagrange formalism.
3. The attempt at a solution
First of all I want to try to understand the problem. So that small part is one of the point in the line? Right? And every point in the line which connects points $(0,0,0)$ i $(1,0,0)$ changes with equations
$$x_1(t)=X_1$$
$$x_2(t)=X_2+\sin \pi t\sin \pi X_1$$
$$x_3(t)=X_3$$?
Right?
2. Apr 18, 2014
Staff: Mentor
Imagine a vector ΔS joining the point (0,0,0) to the point (1,0,0) at time t = 0. As time progresses the material points at the two ends of the vector move to new locations, so ΔS changes. You want to find the vector ΔS joining these same two material points at later times. Note that the material point at the origin stays at the origin. So you only need to determine where the other material point is located as a function of time. You need to plot this, and show the vector joining the two material points at the different times. This will illustrate visually a little bit about how the fluid is deforming.
Chet | 2017-10-21 20:52:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7197038531303406, "perplexity": 338.5133358083618}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824894.98/warc/CC-MAIN-20171021190701-20171021210701-00404.warc.gz"} |
https://www.allanswered.com/post/wwgw/supervised-modeling-q3-d-cant-create-decision-boundary/ | ### [Supervised Modeling] Q3.d: Can't create decision boundary
401
views
2
19 months ago by
Having a difficult time plotting the decision boundary on this question.
d) Now use all features. How does this affect the confusion matrix? Draw the decision boundary (Since we cannot really visualize a 4-dimensional plot, use only 'Sepal.Length' and 'Sepal.Width' as X for the boundary function but use the predictor trained on all features)
I think this is the problematic part. I consistently receive this error:
ValueError: operands could not be broadcast together with shapes (10000,2) (4,)
I believe this is because I am using a model that is trained on more than 2 dimensions (which is what the question specifies). This is what my code looks like:
iris_no_setosa = iris[iris.Species != 'setosa']
iris_train, iris_test = train_test_split(iris_no_setosa, test_size=0.3)
all_feat_nb = GaussianNB()
predictions = all_feat_nb.fit(iris_train.drop('Species',1) , iris_train.Species).predict(iris_test.drop('Species',1))
#Confusion Matrix
cm= confusion_matrix(y_pred=predictions, y_true=iris_test.Species)
helper_functions.plot_confusion_matrix(cm, iris_test.Species.unique())
#Drop extra features so decision boundary can work
iris_train = iris_train[['Sepal.Width','Sepal.Length','Species']]
iris_test = iris_test[['Sepal.Width','Sepal.Length','Species']]
#Decision Boundary -- This is where the problem arises
helper_functions.decision_boundary(X=iris_test.drop('Species',1) , Y=iris_test.Species, model=all_feat_nb)
The problem specifically occurs when I plot the decision boundary, the confusion matrix works just fine.
Anyone else understand this?
Community: ITC Fellows 16-17
5
18 months ago by
Hi,
I got the same error and you're right it is because we are using a model that is trained on more than 2 dimensions. To solve it I think we should train the model with 2 dimensions but it is not what we were asked ...
# Training data
X_train = train.drop('Species', 1)
# Target value
y_train = train.Species
# Fit the training data
gnb = GaussianNB()
gnb.fit(X_train, y_train)
# Make a prediction on the test set
X_test = test.drop('Species', 1)
y_test = test.Species
p_test = gnb.predict(X_test)
# Confusion matrix
cm = confusion_matrix(y_test,p_test)
# Plot the confusion matrix
classes = test.Species.unique()
plot_confusion_matrix(cm,classes)
# Decision boundary
y_test = y_test.reshape(y_test.size, 1)
#decision_boundary(X_test[['Sepal.Length','Sepal.Width']],y_test,gnb)
I spoke with some others about this, and they seem to agree. There must be a mistake with the question.
written 18 months ago by Alex Reibman | 2018-07-19 15:33:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5119171738624573, "perplexity": 4465.690721058715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591140.45/warc/CC-MAIN-20180719144851-20180719164851-00624.warc.gz"} |
http://www.lmfdb.org/EllipticCurve/5.5.65657.1/43.1/ | ## Results (displaying both matches)
curve label base field conductor norm conductor label isogeny class label Weierstrass coefficients
43.1-a1 5.5.65657.1 43 43.1 43.1-a $$\bigl[1$$ , $$-2 a^{4} + 3 a^{3} + 9 a^{2} - 10 a - 8$$ , $$a^{4} - a^{3} - 4 a^{2} + 3 a + 2$$ , $$-a^{4} - 2 a^{3} + 4 a^{2} + 11 a + 10$$ , $$2 a^{4} + 3 a^{3} - 9 a^{2} - 20 a - 9\bigr]$$
43.1-a2 5.5.65657.1 43 43.1 43.1-a $$\bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 6$$ , $$a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 2$$ , $$a + 1$$ , $$5 a^{4} - 3 a^{3} - 21 a^{2} - 3 a + 6$$ , $$9 a^{4} + 2 a^{3} - 40 a^{2} - 33 a - 4\bigr]$$ | 2019-08-23 17:31:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9931701421737671, "perplexity": 4178.635496022433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318952.90/warc/CC-MAIN-20190823172507-20190823194507-00376.warc.gz"} |
https://stats.stackexchange.com/questions/215696/do-all-machine-learning-algorithms-separate-data-linearly | # Do all machine learning algorithms separate data linearly?
I am an enthusiast of programming and machine learning. Only a few months back I started learning about machine learning programming. Like many who don't have a quantitative science background I also started learning about ML by tinkering with the algorithms and datasets in the widely used ML package(caret R).
A while back I read a blog in which the author talks about usage of linear regression in ML. If I am remembering correct he talked about how all machine learning in the end uses some kind of "linear regression"(not sure whether he used this exact term) even for linear or non-linear problems. That time I didn't understood what he meant by that.
My understanding of using machine learning for non-linear data is to use a non linear algorithm to separate the data.
This was my thinking
Let's say to classify linear data we used linear equation $y=mx+c$ and for non linear data we use non-linear equation say $y=sin(x)$
This image is taken from sikit learn website of support vector machine. In SVM we used different kernels for ML purpose. So my initial thinking was linear kernel separates the data using a linear function and RBF kernel uses a non-linear function to separate the data.
But then I saw this blog where the author talks about Neural networks.
To classify the non linear problem in left subplot, the neural network transforms the data in such a way that in the end we can use simple linear separation to the transformed data in the right sub-plot
My question is whether all machine learning algorithms in the end uses a linear separation to classifiction(linear /non-linear dataset)?
• – Sycorax
Jun 1, 2016 at 15:24
• Your non-linear model $\sin(x)$ is also a linear. introduce a new variable $s=\sin(x)$, then your problem becomes $y=\theta_0+\theta_1 s$ - a linear one. In this sense a lot of ML algos are linear indeed. Jun 22, 2016 at 14:03
• I like mbq's answer on this thread as well, Help me understand support vector machines. Jun 24, 2016 at 14:48
The answer is No. user20160 has a perfect answer, I will add 3 examples with visualization to illustrate the idea. Note, these plots may not be helpful for you to see if the "final decision" is in linear form but give you some sense about tree, boosting and KNN.
We will start with decision trees. With many splits, it is a non-linear decision boundary. And we cannot think all the previous splits are "feature transformations" and there are a final decision line at the end.
Another example is the boosting model, which aggregates many "weak classifiers" and the final decision boundary is not linear. You can think about it is a complicated code/algorithm to make the final prediction.
Finally, think about K Nearest Neighbors (KNN). It is also not a linear decision function at the end layer. in addition, there are no "feature transformations" in KNN.
Here are three visualizations in 2D space (Tree, Boosting and KNN from top to bottom). The ground truth is 2 spirals represent two classes, and the left subplot is the predictions from the model and the right subplot is the decision boundaries from the model.
EDIT: @ssdecontrol's answer in this post gives another perspective.
It depends on how we define the "transformation".
Any function that partitions the data into two pieces can be transformed into a linear model of this form, with an intercept and a single input (an indicator of which "side" of the partition the data point is on). It is important to take note of the difference between a decision function and a decision boundary.
• I don't want to critic, but the boosting seems a bit rough, no? Is it not possible to get a smoother result with different parameters? Sorry to be pernickety, because I find the all explanation very good.
– YCR
Jun 1, 2016 at 15:56
• @YCR I think that is the point of boosting where you have a rough decision boundary. The roughness is caused by aggregating many weak classifiers (in this example,they are trees). But I agree with you that the second example is not a good model, and it is overfitting :) Jun 1, 2016 at 16:53
• (+1) Great visualization (I also use spirals a lot in my experimentations). A suggestion: plot the decision boundaries as image, and perhaps add probabiliity levels (if you are using probabilistic outputs) with contour. Jun 24, 2016 at 14:22
• @Firebug great suggestion! these plot are generated in a grid and only can tell you the final label. Contour is much better. Jun 24, 2016 at 14:27
• Look at my answer here: stats.stackexchange.com/a/218578/60613 Jun 24, 2016 at 14:45
Some algorithms use a hyperplane (i.e. linear function) to separate the data. A prominent example is logistic regression. Others use a hyperplane to separate the data after a nonlinear transformation (e.g. neural networks and support vector machines with nonlinear kernels). In this case, the decision boundary is nonlinear in the original data space, but linear in the feature space into which the data are mapped. In the case of SVMs, the kernel formulation defines this mapping implicitly. Other algorithms use multiple splitting hyperplanes in local regions of data space (e.g. decision trees). In this case, the decision boundary is piecewise linear (but nonlinear overall).
However, other algorithms have nonlinear decision boundaries, and are not formulated in terms of hyperplanes. A prominent example is k nearest neighbors classification. Ensemble classifiers (e.g. produced by boosting or bagging other classifiers) are generally nonlinear.
• @CagdasOzgenc Let's consider the case of binary classification and a network w/ sigmoidal output, as you're suggesting. This is equivalent to logistic regression on the activations of the previous layer (using softmax outputs would be equivalent to multinomial logistic regression). So, the decision boundary is a hyperplane in feature space. The picture in the original question shows a nice example of this. Jun 22, 2016 at 21:06
• Is f the output neuron's activation function and x the output of the previous layer? Not sure I understand what you're asking. Jun 23, 2016 at 11:25
• Say we determine the class label by comparing the output to some threshold. If $f$ is non-invertible, it seems like that could induce multiple, parallel hyperplanes (all orthogonal to $Ax$). That would certainly be a unique output neuron. Jun 23, 2016 at 11:44
• The decision boundaries should still be linear no matter what f is, as long as it operates on Ax. But it's true, we could imagine functions that operate on x more generally, which could give nonlinear boundaries. Fun to think about. Jun 23, 2016 at 12:18
• How about decision trees? Jun 27, 2016 at 18:35 | 2022-05-26 05:53:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5754880309104919, "perplexity": 781.4574819680128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662601401.72/warc/CC-MAIN-20220526035036-20220526065036-00121.warc.gz"} |
http://physics.stackexchange.com/tags/stars/new | # Tag Info
6
How opaque is that -- would we be able to see a couple of meters, some kilometers, or nothing at all? The photosphere of our sun is somewhere on the order of 500 km thick. For a quick ballpark, you can imagine an exponential decrease in the transmission of light which about this characteristic thickness. It might be a little less, but it's still ...
2
Well, I can give you a definitive answer to Q1, but my answer to Q2 would only be educated speculation. Perhaps some of the astrophysicists on here can be more help with that one. However, before I tackle Q1, a very important disclaimer: Temperature is a measure of the average kinetic energy of the particles of an object, and cannot be used all by itself ...
4
A brief history of what science thought about the sun can be found here . It is reasonable that once thermodynamics advanced to the point of measuring and calculating energies the discrepancy between heat output of the sun and the age of the earth had to be explained. They tried with gravitation, but until the discovery of nuclear energy and E=m*c^2 it ...
6
I like to explain this using a figure from a talk by Marco Limongi some years ago. Based on a given set of models, the $x$-axis shows the initial mass of the models and the $y$-axis the final mass. The different coloured layers show the composition of the star at the moment of collapse. The mass ejected in the supernova is the difference between the curve ...
65
It has more to do with physiology of the eye rather than the spectrum of light produced by stars. Stars emit light over the full range of visible wavelengths. Hot stars emit more blue/violet light, cool stars emit more red light. The Sun is relatively neutral in that regard, so does not have a strong colouration, but many other stars in the sky have ...
10
Short answer: Many of them are not. Stars can be modeled as black-bodies. That means that the spectrum of light that they emit depends on their temperature, and the color of a star can be described as its Color Temperature. The color temperatures of stars can vary quite greatly, because stars have very different temperatures and emission spectras. One of ...
5
The sun is white. I've seen people say that it just looks yellow because the blue light is being scattered by the atmosphere. Since starlight is also scattered your question is still valid. But based on these pictures, the sun still seems to look pretty white through our atmosphere. My guess is that, since you can safely look directly at a star, you learned ...
2
Our sun is actually white. Sun seen from space. It's just that when the sun rays enter the earth, our atmosphere scatters the white light resulting in different colors. The reason why the Sun appears to be white sometimes is usually because it's directly overhead. Then the rays coming from the sun have to travel the least distance and hence encounter less ...
4
By estimating the distance is the most obvious method, but you are correct, the parallax will be too small to measure. If we can tell what type of star it is (by measuring its spectrum, or using its colour(s)), then we know roughly how intrinsically luminous the star is. The actual brightness then tells us how far away it is. Fortunately, the diameter of ...
3
Lyman Alpha absorption systems The Lyman-$\alpha$ Forest and Gunn-Peterson troughs are two extremes on the scale of absorption features that are left by neutral hydrogen in intergalactic space. When ultraviolet light from a background source, typically a Quasar or a young, strongly star forming galaxy, travels through intergalactic space, it is ...
4
Black body radiation is given by Planck's formula (see link for variables) Here is the measured irradiance of the sun and the attempt to fit it with the black body formula: The effective temperature, or black body temperature, of the Sun (5,777 K) is the temperature a black body of the same size must have to yield the same total emissive power. ...
8
Maybe the simplest way to think about this is that the Sun is in approximate thermal equilibrium and would absorb any photon, of any frequency, that is incident upon it. This is essentially the definition of a BB. There are many radiative processes that can absorb (and hence emit) radiation at all frequencies, not just those corresponding to atomic ...
2
You would be very interested in one of the recent Kepler discoveries - Kepler 444. The star is estimated to be 11.2 billion years old (using asteroseismology) and is surrounded by a number of rocky exoplanets. These planets are all too close to their parent K-dwarf star to be in the habitable zone, but there is no reason there couldn't be planets further ...
4
I think you already know the answer... Pop III stars, by definition, are born from primordial gas that is basically Hydrogen, Helium with trace amounts of deuterium, tritium, lithium and beryllium; they initially contain almost no C, N, or O. Therefore the primary fusion in massive Pop III stars has to be (well, initially the deuterium is burned but this is ...
1
After (re)combination (I never understand why the "re" is used) and the formation of the CMB, the universe was transparent and the only light in it was from the rapidly cooling CMB. The baryonic universe was composed almost entirely of neutral hydrogen and helium. After perhaps 100 million years, the first galaxies and stars (assisted by dark matter) were ...
0
the answer is actually very simple. Electrons, protons and neutrons and other subatomic particles don't exist in the degree of proximity that a collapsing start forced them into. The massive gravitational force overcomes the equilibrium forces that exist in matter in its "normal" state ("normal" here refers to the state of the gas/dust cloud before ...
4
I'm not going to attempt to usurp Chris White's perfectly good answer - but just fill in some detail and answer the edit. For a star like the Sun, the collapse proceeds in 4 basic stages, each takes about 10 times as long as the previous one. Pseudo-spherical collapse of the cloud - not far from a free fall timescale, often quoted as a few $10^4$ years. ...
5
Short answer: gravitational potential energy is converted into heat. Let's look at the Sun as an example. Its mass is $M_\odot = 2.0\times10^{30}\ \mathrm{kg}$ and its radius is $R_\odot = 7.0\times10^8\ \mathrm{m}$. If its density were uniform, its gravitational binding energy would be U_{\odot,\,\text{uniform}} = -\frac{3GM_\odot^2}{5R_\odot} = ...
1
A star is neither "flaming" nor "fire" in the sense that we use those words about things on Earth. It's just a big, hot ball of ionized gas. The only thing that happens "to" it is that it gets hotter and denser. At some point the temperature rises high enough to ionize the gas. Later still fusion becomes possible at non-vanishing rates. The energy for the ...
Top 50 recent answers are included | 2015-03-29 07:41:12 | {"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": 0.634288489818573, "perplexity": 491.12984577916967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131298387.35/warc/CC-MAIN-20150323172138-00213-ip-10-168-14-71.ec2.internal.warc.gz"} |
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Mat. Zametki: Year: Volume: Issue: Page: Find
Mat. Zametki, 2017, Volume 102, Issue 6, Pages 828–835 (Mi mz11716)
On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory
S. Yu. Dobrokhotovab, V. E. Nazaikinskiiab
a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
Abstract: Rapidly oscillating integrals of the form
\begin{equation*} I(r,h)=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{\tfrac ih F(r\cos\phi)} G(r\cos\phi) d\phi, \end{equation*}
where $F(r)$ is a real-valued function with nonvanishing derivative, arise when constructing asymptotic solutions of problems with nonstandard characteristics such as the Cauchy problem with spatially localized initial data for the wave equation with velocity degenerating on the boundary of the domain; this problem describes the run-up of tsunami waves on a shallow beach in the linear approximation. The computation of the asymptotics of this integral as $h\to0$ encounters difficulties owing to the fact that the stationary points of the phase function $F(r\cos\phi)$ become degenerate for $r=0$. For this integral, we construct an asymptotics uniform with respect to $r$ in terms of the Bessel functions $\mathbf{J}_0(z)$ and $\mathbf{J}_1(z)$ of the first kind.
Keywords: rapidly oscillating integral, degeneration of stationary points, uniform asymptotics, Bessel function, wave equation.
Funding Agency Grant Number Russian Science Foundation 16-11-10282 This work was supported by the Russian Science Foundation under grant 16-11-10282.
DOI: https://doi.org/10.4213/mzm11716
Full text: PDF file (488 kB)
First page: PDF file
References: PDF file HTML file
English version:
Mathematical Notes, 2017, 102:6, 756–762
Bibliographic databases:
Document Type: Article
UDC: 517.9
Citation: S. Yu. Dobrokhotov, V. E. Nazaikinskii, “On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory”, Mat. Zametki, 102:6 (2017), 828–835; Math. Notes, 102:6 (2017), 756–762
Citation in format AMSBIB
\Bibitem{DobNaz17} \by S.~Yu.~Dobrokhotov, V.~E.~Nazaikinskii \paper On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory \jour Mat. Zametki \yr 2017 \vol 102 \issue 6 \pages 828--835 \mathnet{http://mi.mathnet.ru/mz11716} \crossref{https://doi.org/10.4213/mzm11716} \elib{http://elibrary.ru/item.asp?id=30737867} \transl \jour Math. Notes \yr 2017 \vol 102 \issue 6 \pages 756--762 \crossref{https://doi.org/10.1134/S0001434617110141} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000418838500014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85039455205}
• http://mi.mathnet.ru/eng/mz11716
• https://doi.org/10.4213/mzm11716
• http://mi.mathnet.ru/eng/mz/v102/i6/p828
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
This publication is cited in the following articles:
1. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem”, Math. Notes, 104:4 (2018), 471–488
2. Dobrokhotov S.Yu. Tolstova O.L. Sekerzh-Zenkovich S.Ya. Vargas C.A., “Influence of the Elastic Base of a Basin on the Propagation of Waves on the Water Surface”, Russ. J. Math. Phys., 25:4 (2018), 459–469
3. S. I. Kabanikhin, O. I. Krivorotko, “An algorithm for source reconstruction in nonlinear shallow-water equations”, Comput. Math. Math. Phys., 58:8 (2018), 1334–1343
• Number of views: This page: 148 References: 18 First page: 21 | 2019-04-19 07:53:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3344113528728485, "perplexity": 5472.384224604681}, "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/1555578527148.46/warc/CC-MAIN-20190419061412-20190419083412-00082.warc.gz"} |
https://physicscatalyst.com/Class9/heron_formula.php | # Class 9 Maths notes for Heron Formula
Table of Content
## Mensuration
• It is branch of mathematics which is concerned about the measurement of length ,area and Volume of plane and Solid figure
## Perimeter
• The perimeter of plane figure is defined as the length of the boundary
• It units is same as that of length i.e. m ,cm,km
Note:
You have read from Left to right to get the conversion formula
1 m= 10 dm = 100 cm
1 dm= 10 cm = 100 mm
1 km= 10 hm = 100 decameters
1 decameter= 10 m = 1000 cm
## Area
• The area of the plane figure is the surface enclosed by its boundary
• It unit is square of length unit. i.e. m2 , km2
Note:
You have read from Left to right to get the conversion formula
$1 m^2 = 100 dm^2 = 10^4 cm^2$
$1 dm^2 = 100 cm^2 = 10^4 mm^2$
$1 \ hectare = 100 \ square \ Decameter = 10^2 m^2$
$1 \ square \ myraimeter=100 km^2 = 10^8 m^2$
## Perimeter and Area of Different Figure
N Shape Perimeter/height Area 1 Right angle triangle Base =b, Height =h Hypotenuse=d $P=b+h+d$ Height =h $A=\frac {1}{2}BH$ 2 Isosceles right angled triangle Equal side =a $P= a(2+ \sqrt 2)$ $A=\frac {1}{2} a^2$ 3 Isosceles triangle Equal side =a and base as b $P= 2a +b$ $A=\frac {1}{2} a \sqrt {a^2 - \frac {b^2}{4}}$ 4 Any Triangle Any triangle of sides a,b ,c $P=a+b+c$ $A=\sqrt {s(s-a)(s-b)(s-c)}$ $s=\frac {a+b+c}{2}$ This is called Heron's formula (sometimes called Hero's formula) is named after Hero of Alexandria 5 Square Side =a P=4a $A=a^2$ 6 Rectangle of Length and breath L and B respectively $P=2L +2B$ $A=L \times B$ 7 Parallelograms Two sides are given as a and b $P=2a+2b$ $A= Base \times height$ When the diagonal is also given ,say d Then $A= \sqrt {s(s-a)(s-b)(s-d)}$ $s=\frac {a+b+d}{2}$ 8 Rhombus Diagonal d1 and d2 are given $p=2 \sqrt {d_1^2+d_2^2}$ Each side=$\frac {1}{2} \sqrt {d_1^2+d_2^2}$ $A=\frac {1}{2} d_1 d_2$ 9 Quadrilateral a. All the sides are given a,b,c ,d b. Both the diagonal are perpendicular to each other c. When a diagonal and perpendicular to diagonal are given a. $P=a+b+c+d$ a. $A=\sqrt {s(s-a)(s-b)(s-c)(s-d)}$ $s=\frac {(a+b+c+d)}{2}$ b. $A=\frac {1}{2} d_1 d_2$ where d1 and d2 are the diagonal c. $A=\frac {1}{2} d(h_1+h_2)$ where d is diagonal and h1 and h2 are perpendicular to that
## How to solve the Area and Perimeter problems
1. We must remember the formula for all the common figures as given above the table
2. Find out what all is given in the problem
3. Convert all the given quantities in the same unit
4. Sometimes Perimeter is given and some side is unknown,So you can calculate the sides using the Perimeter
5. If it is a complex figure ,break down into common know figures like square,rectangle,triangle
6. Sometimes we can find another side using Pythagoras theorem in the complex figure
7. If common figure, apply the formula given above and calculate the area.
8. If complex figure, calculate the area for each common figure in it and sum all the area at the end to calculate the total area of the figure
## Solved Examples
Question 1.
A right angle triangle has base 20 cm and height as 10 cm, What is the area of the triangle?
Solution
Given values B=20 cm
H=10 cm
Both are in same units
$A=\frac {1}{2}BH=100 \ cm^2$
Question 2.
Sides of triangles are in the ratio 12:17:25. The perimeter of the triangle is 540 cm. Find out the area of the triangle?
Solution
Let the common ration between the sides be y,then sides are 12y,17y,25 y
Now we know the perimeter of the triangle is given by
P=a+b+c
540=12y+17y+25y
or y=10 cm
Now Area of triangle is given as $A=\sqrt {s(s-a)(s-b)(s-c)}$
Where s=(a+b+c)/2
Here s=270 cm
a=120 cm
b=170cm
c=250cm
Substituting all these values in the area equation,we get
A=9000 cm2
Question 3.
A equilateral triangle is having side 2 cm. What is the area of the triangle?
Solution
We know that Are of equilateral triangle is given by
$A= \frac {a^2 \sqrt 3}{4}$
Substituting the values given above
$A=\sqrt 3 \ cm^2$
## We can summarize various method to calculate the Area of the Triangle
If you know the altitude and Base $A =\frac {1}{2}BH$ If you all the three sides $A=\sqrt {s(s-a)(s-b)(s-c)}$ $s=\frac {a+b+c}{2}$ If it is isosceles right angle triangle with equal side a $A=\frac {1}{2}a^2$ If it is isosceles triangle with equal side a and other side as $A=\frac {1}{2} a \sqrt {a^2 - \frac {b^2}{4}}$ If it is equilateral triangle with equal side a $A=\frac {a^2 \sqrt 3}{4}$ If it is right angle triangle with Base B and Height H $A =\frac {1}{2}BH$ | 2022-10-04 20:33:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.7279186844825745, "perplexity": 867.9681206767118}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337524.47/warc/CC-MAIN-20221004184523-20221004214523-00515.warc.gz"} |