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ARPA-E awards $6.6M to two projects for electrolytes for solid-state batteries for EVs 01 December 2015 As part of the$125 million awarded to 41 projects under its 2015 OPEN funding (earlier post), ARPA-E has selected two projects developing manufacturing techniques for ceramic electrolytes for solid-state EV batteries for awards of a combined $6.6 million. Of that,$3.5 million will go to a consortium led by the University of Michigan, and \$3.1 million will go to Corning Incorporated.

Solid-state Li batteries could double the energy density of today’s Li-ion cells and also eliminate the use of conventional flammable electrolytes, increasing abuse tolerance and reducing the need for battery thermal management systems. ARPA-E has already funded a number of solid-state battery projects (e.g., earlier post). Solid-state batteries face conductivity challenges, however.

University of Michigan. In a project labeled “Transitioning Advanced Ceramic Electrolytes into Manufacturable Solid-State EV Batteries,” researchers at the University of Michigan and partners will develop new electrode structures and manufacturing techniques to incorporate Lithium (Li)-conducting ceramic electrolytes into solid-state batteries.

The U-M project is led by Jeff Sakamoto, an associate professor of mechanical engineering, who is also affiliated with with U-M’s Energy Institute.

Like many energy storage researchers, Sakamoto has been exploring solid-state batteries, which don’t use a liquid electrolyte. Liquid electrolyte contributes to many of the lithium-ion battery’s limitations. His APRA-E proposal aims for an inexpensive, highly efficient solid-state battery that is tough and safe enough to power a vehicle.

However, while many researchers feel that new electrolytes must be developed to meet the conductivity threshold required for a true battery breakthrough, Sakamoto posits that existing materials are fine, and what’s really needed is a groundbreaking design and manufacturing process.

Super-ionic conducting oxide (SCO) electrolytes are a unique subset of ceramic electrolytes that offer numerous benefits in the development of solid-state battery technology. First, SCOs are typically synthesize in air and are reasonably stable in ambient air. This stability can simplify cell fabrication, reduce packaging mass, and dramatically improve safety. Second, the thermal stability and increasing conductivity with an increasing temperature can enable an inherently stable cell chemistry that reduces peripheral component such as thermal management and power electronics. In the context of electric vehicles, this can significantly reduce the battery pack mass, cost, and complexity. Third, consisting of stable inorganic materials, solid state oxide batteries can considerably have a longer cycle life compared to liquid electrolyte technology.

Indeed the features offered by solid-state batteries are appealing; however, there are few examples of bulk-scale SCO-enabled technology. Interestingly, it is not due to the lack of available state-of-the-art SCOs, rather it is the numerous processing, scaling and integration challenges that have limited the advancement of bulk-scale solid-state battery technology.

For example, most oxide ceramics require high temperature processing to bond or sinter components together. How the high temperature processing affects reactions with electrodes is a considerable issue. Similarly, maintaining adequate electrolyte/electrode contact area without amplifying chemical reactions is challenging and requires novel cell stack designs and fabrication techniques. In bonding and operating monolithic batteries, mechanical properties are of particular importance since most ceramics are relatively brittle. Lastly, and perhaps most importantly, identifying SCOs with inherent stability against Li or developing technology to enable the use of state-of-the-art SCOs is the key.

—J. Sakamoto in Handbook of Solid State Batteries

Sakamoto will work with other researchers from U-M, Ford Motor Company, Oak Ridge National Laboratory, the Army Research Laboratory, Solid Power, and the University of Rochester.

Corning. In Corning’s project, “Roll-to-Roll Processing Ceramic Battery Electrolyte,” researchers there will develop roll-to-roll manufacturing techniques to produce thin ceramic electrolytes for solid-state batteries. The technology developed in this project is intended to enable solid-state batteries to be produced economically and at high volumes.

Corning holds patents on a number of ceramic electrolyte materials for solid-state batteries, including fluorine-containing lithium-garnet-type oxide ceramics and lithium orthophosphate glasses. The company holds numerous patents in other areas of battery technology as well.

Corning also partnered with PolyPlus in an earlier ARPA-E project on Li-air batteries. (Earlier post.)

Final award amounts may vary.

Resources

• Nakamoto, Jeff. “Chapter 12: Super-ionic Conducting Oxide Electrolytes” Handbook of Solid State Batteries

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Definition of high frequency in the AudioEnglish.org Dictionary. High frequency (HF) is the ITU designation for the range of radio frequency electromagnetic waves (radio waves) between 3 and 30 megahertz (MHz). (Magyar nyelvű dokumentum - online_elojegyzes.pdf), Online jelentkezés az itt letölthető dokumentumon keresztül. Please update your bookmarks accordingly. Frequency is the number of occurrences of a repeating event per unit of time. If a voltage swings from 1 volt to -1 volt and does that a million times in a second, thats a frequency of 1 megahertz. Proper usage and audio pronunciation (plus IPA phonetic transcription) of the word high frequency. (Magyar nyelvű dokumentum - online_elojegyzes.pdf) high frequency synonyms, high frequency pronunciation, high frequency translation, English dictionary definition of high frequency. The more cycles that occur per second, the higher the frequency. Frequency measures the number of times something occurs in a specific amount of time. The higher the frequency, the shorter the distance between each successive compression (or rarification) in the incoming sound wave. The symbol for wavelength is the Greek letter lambda, λ. Our tips from experts and exam survivors will help you through. It is common for kilohertz (kHz), megahertz (MHz) and gigahertz (GHz) to be used when waves have very high frequencies. I have a data set with 15000 data points. This distance is called the " wavelength ". You could have a high amplitude low frequency signal, which would have a long wavelength but high peaks, but since frequency is defined as the number of wave peaks Research by psychologists has shown that on any given day, 99% of our thoughts are exactly the same as the day before. Can shifting brain waves mean âspiritual awakeningâ? Reflection, refraction, diffraction, interferenceâall of these classic wave behaviors apply to electromagnetic radiation. Define high frequency. The distance between peaks (high points) is called wavelength. Just through the standard formula. BETA WAVEâthe beta wave is known in the spectrum as âhigh frequency low amplitudeâ and are normally present during a waking state. The back-and-forth vibrational motion of the particles of the medium would not be the only observable phenomenon occurring at a given frequency. It is also the number of waves that pass a certain point each second. Köszönjük a türelmét. For example: most people cannot hear a high â¦ The distance between peaks (high points) is called wavelength. For example, if we think of a car that moves at 60 miles per hour, we mean just that. To get different images, the wave height/amplitude can be changed as well as the starting point of the wave, also called the Start studying Physics Chapter 12. The wavelength of a wave is the distance between a point on one wave and the same point on the next wave. Now if I've got a (Remember kilo k = 103 mega M = 106 giga G = 109). A body in periodic motion is said to have undergone one cycle after passing through a series of events or â¦ A frequency of 10 Hz means ten wave cycles per second, where the cycles are much shorter and closer together. It â¦ To be more clear. Since a sound wave is a pressure wave, a detector could be used to detect oscillations in pressure from a high pressure to a low pressure and back to a high â¦ Although frequency is a measurement of rate of movement, it is not identical to velocity. Definition of wave frequency and how it is related to the energy of a wave. Therefore, itâs not enough to look at a frequency response figure like 20Hz-20kHz +/- 3dB, itâs better to be able to see where these swings in emphasis occur and how they are distributed. Higher frequencies have a higher average power for any given section of the wave, than lower frequency waves. high frequency: [noun] a radio frequency between very high frequency and medium frequency â see Radio Frequencies Table. It does depend on the wave length and the period. (Német nyelvű dokumentum - online_anmeldung.pdf). In communications and electronic engineering, an intermediate frequency (IF) is a frequency to which a carrier wave is shifted as an intermediate step in transmission or reception. A smoother frequency response is better Frequency generally refers to a voltage that is changing in the form of a sine wave. This doesn't mean they are 'louder' or have a Adatvédelmi szabályzat But that doesn't mean that it's independent of the wave length and the period. If you were to take As waves travel, they set up patterns of disturbance. Frequency is basically the amount of waves the sound is producing. This MATLAB function estimates the mean normalized frequency, freq, of the power spectrum of a time-domain signal, x. That means that we are trapped in reliving every day, a life that seems not to change. (Német nyelvű dokumentum - online_anmeldung.pdf) [1] It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. If there are many waves, "happen frequently", it is a high frequency. However, if we say that a wave has a frequency of 60 Power spectral density (PSD), specified as a vector or matrix. Frequency, the number of waves that pass a fixed point in unit time; also, the number of cycles undergone during one unit of time by a body in periodic motion. Period: Period of a wave is the time required for a vibration of a wave to make a complete cycle. The frequency of a wave is the number of wave crests passing a point per unit time. The frequency of a wave can be calculated using the equation: $\text{frequency f =}~\frac{\text{number of waves to pass a point}}{\text{time taken in seconds}}$. Learn about how waves are measured according to amplitude, wavelength and frequency. In the case of electrical current, frequency is the number of times a sine wave repeats, or completes, a positive-to-negative cycle. Learn vocabulary, terms, and more with flashcards, games, and other study tools. It is important to note that the amplitude is not the distance between the top and bottom of a wave. For example, if a What does high frequency mean? Whatever frequency we are vibrating at, we attract like frequency. This is a high â¦ The difference in wavelength is the way we tell different kinds of electromagnetic energy apart. When a wave contacts matter, there are several different interactions that can occur. For FFT to be fully fast, it required 2^n data points. We have moved all content for this concept to for better organization. rate of vibration and oscillation measured over a specific period of time (usually one second The intermediate frequency is created by mixing the carrier signal with a local oscillator signal in a process called heterodyning , resulting in a signal at the difference or beat frequency . I want to transform the data to frequency domain using FFT. Example: If an alternating current is said to have a frequency of 3 Hz (see diagram below), that indicates its waveform repeats 3 times in 1 second. It is often easiest to measure this from the trough of one wave to the trough of the next wave, or from thecrest of one wave to the crest of the next wave. Wavelength and frequency are interdependent. Higher frequencies tend to be more 'directional' than lower frequencies. The frequency of a wave is the number of waves produced by a source each second. If the amplitudes of the vibrations are large enough and if natural frequency is within the human frequency range , then the vibrating object will produce sound waves that are audible. Read about our approach to external linking. For example, if someone visits the grocery store twice a week, her shopping frequency is 2 visits per week. most people cannot hear a high-pitched sound above 20 kHz; radio stations broadcast radio waves with frequencies of about 100 MHz; most wireless computer networks operate at 2.4 GHz. Speed is equal to distance divided by time. It is often easiest to measure this from the, of one wave to the trough of the next wave, or from the. It is common for kilohertz (kHz), megahertz (MHz) and gigahertz (GHz) to be used when waves have very high frequencies. Two spectra came from the same powder (in the sense of Chemical pattern) with different shape (one was a nanocubes structures) of particles. Amplitude, wavelength and frequency - CCEA, Reflection and refraction of waves - CCEA, Home Economics: Food and Nutrition (CCEA). High frequency (HF) is an electromagnetic frequency defined by the International Telecommunications Union (ITU) for radar and radio communications. For light and other electromagnetic radiation, it is related to â¦ They allow for logical thought processes, and therefore are instrumental in problem-solving But it doesn't matter where you measure it - as long as it is the same point on each wave. Éppen dolgozunk a weboldalon, de hamarosan visszatérünk. Meaning of high frequency. Second Frequency: Each additional wave or frequency brings along more ripples and as such, more detail. The note A which is above Middle C (more on this later) has a frequency of 440 Hz. Do we have to be a âmonkâ to awake to âlove, compassion and kindness â or over the age of 30 is doable ð? But at this point weâre still dealing with electrical signals, i.e., signals that have not yet been converted by the antenna into electromagnetic radiation, and consequently we only have to concern ourselves with two of these: reflection and interference. of one wave to the crest of the next wave. The frequency or frequencies at which an object tends to vibrate with when hit, struck, plucked, strummed or somehow disturbed is known as the natural frequency of the object. frequency Symbol: f , Î½.The number of oscillations per unit time of a vibrating system. One of those interactions is wave absorption. The amplitude of a wave is its maximum disturbance from its undisturbed position. Frequency, wavelength, amplitude and wave speed You need to know about these quantities used in waves - what they mean, the symbols used for them and the units used to measure them. Online jelentkezés az itt letölthető dokumentumon keresztül. Frequency is measured in units of hertz (Hz) which is equal to one occurrence of a repeating event per second. It is also known as the decameter band or decameter wave as its wavelengths range from one to ten decameters (ten to one hundred meters). Online jelentkezés az itt letölthető dokumentumon keresztül. Sound travels at about 750 miles/hour, so the compression waves between 100 hz and 20,000 hz have wavelengths that range between several feet (for the 100 hz sound) to a fraction of an inch (for the 20,000 hz sound). How i â¦ It operates in a range of three to thirty MHz and as the wavelengths spread over ten to one hundred meters, it is also known as decameter wave and designated as band 7. High-Frequency Trading - HFT: High-frequency trading (HFT) is a program trading platform that uses powerful computers to transact a large number of orders at very fast speeds. n. Abbr. A dokumentumot kitöltve kérjük küldje vissza a következő email címre: info@osztrakmagyarovi.hu, Online jelentkezés az itt letölthető dokumentumon keresztül. If pxx is a matrix, then meanfreq computes the mean frequency of each column of pxx independently. Each wave has a certain shape and length. We generally think of an electrical signal as a one-way phenomenon; it travels from the output of one component to the input of another componâ¦ Frequency is measured in hertz. It is often used as a reference frequency for tuning musical instruments. Are exactly the what does it mean to be a high frequency wave point on one wave and the period per week frequency 440... Wave and the period n't matter where you measure it - as long it! Where the cycles are much shorter and closer together any given day, a life that seems not change., more detail shopping frequency is 2 visits per week crests passing a per! 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### Home > A2C > Chapter 13 > Lesson 13.1.4 > Problem13-73

13-73.

Solve for $x$ in the figure below. Homework Help ✎

You are working with $3$ sides and one angle. Use the Law of Cosines to find the value of $x$.

$x = 9.10$


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# Must two non-isomorphic finite groups have a non-isomorphic Sylow $p$-subgroup?

I'm working through a problem in Rotman which asks for an example of two groups $G\not\cong H$ such that $G$ and $H$ have the same number of elements of order d, for every d.

(Rotman's question has been discussed on MSE, here, for example.)

Intuitively, it seems like I should be able to narrow down my search to groups of order $p^k$ by arguing that if $G$ and $H$ have the above property and $|G| = |H| = p_1^{k_1}\cdots p_n^{k_n}$, then $G$ and $H$ have, in particular, a non-isomorphic Sylow $p_j$-group.

The problem is, I don't know if this is true. Any proof ideas or counter-examples?

• If you narrow your search down to groups of order $p^k$, you will never find a counterexample, as the groups are then their own $p$-Sylow subgroups and therefore isomorphic by assumption. – j.p. Oct 2 '17 at 7:07

• It seems this is the case for any $G$ and $H$ of order $pq$ where $p$ and $q$ are prime because groups of prime order are cyclic. – Peter Kagey Oct 1 '17 at 18:19
• @PeterKagey: There is a difference between $pq=6$ and $pq=15$. In both cases all Sylow subgroups are cyclic, hence $G$ and $H$ have isomorphic Sylow subgroups for all primes, but for $pq=6$ there exist non-isomorphic groups of order $6$, whereas there exists up to isomorphisms only one group of order $15$. So Ted couldn't have written 15 instead of 6. – j.p. Oct 2 '17 at 6:55
• @PeterKagey: If you look at groups of order $2^2\cdot 3$ and $3^2\cdot 5$ you will get the following: for groups of order $12$ there are non-isomorphic groups that have all $p$-Sylow subgroups isomorphic, whereas groups of order $45$ are isomorphic if and only if their $p$-Sylow subgroups are isomorphic for all primes $p$. – j.p. Oct 2 '17 at 6:59
• @PeterKagey: If both groups $G$ and $H$ are abelian (of the same order), then they are isomorphic if and only if their $p$-Sylow subgroups are isomorphic for all primes $p$. So if all groups of the given order are abelian, then you get the implication "all $p$-Sylow subgroup isomorphic" $\implies$ "groups isomorphic", otherwise you can find counterexamples to this statement for the given order. – j.p. Oct 2 '17 at 7:04


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# Weak convergence in $L^2[0,1]$

In $\ell^2$, a sequence converges weakly if it is bounded and each component converges. My question is: How can I think about weak convergence in $L^2[0,1]$?

As $\ell_2(\mathbb N)$ is isometrically isomorphic to any separable infinite-dimensional Hilbert space $\mathcal H$ (such as $L^2[0,1]$), the $\ell_2$-condition for weak convergence of a bounded sequence transfers to every such $\mathcal H$ where

$$\text{"each component"}$$

gets replaced by

$$\text{"each }(\langle e_k,x_n\rangle)_{n\in\mathbb N}\text{ where }(e_k)_{k\in\mathbb N}\text{ is any orthonormal basis of }\mathcal H",$$

e.g. see here (Lemma 8). Then, you can take any orthonormal basis of $L^2[0,1]$ (e.g. $\exp(2\pi inx)$, or the Legendre polynomials on $[0,1]$, or $\ldots$) you like to characterize weak convergence in it.

• Thanks for your answer. If I work in the real version of $L^2[0,1]$, what should I do if I have a concrete sequence of functions? Are there any illuminating examples? The isometric identification won't help me much I fear. Thanks - how would I deal with concrete examples, say $\sin(\alpha_nt)$, where $(\alpha_n)$ is a sequence? – max_zorn Mar 13 '18 at 3:35

A sequence $(u_n)$ in $L^2[0,1]$ converges weakly iff it is bounded and for all intervals $(a,b)\subset [0,1]$ the integrals $\int_a^b u_n$ are converging. Instead of 'each component' converges, we have now each integral on an arbitrary interval converges.

• Cool, can you add a reference please? – max_zorn Mar 6 '18 at 8:42
• Thanks - how would I deal with concrete examples, say $\sin(\alpha_nt)$, where $(\alpha_n)$ is a sequence? – max_zorn Mar 13 '18 at 3:35

The space $L^2([0,1])$ equipped with the inner product $$\langle f,g\rangle:=\int^1_0f(x)\overline{g(x)}\,dx$$ is a Hilbert space. By definition of weak convergence we say a sequence $(f_n)\subset L^2([0,1])$ converges weakly to $f\in L^2([0,1])$ whenever $$\lim_n\langle f_n,g\rangle=\langle f,g\rangle\hspace{0.2cm}\text{for all}\hspace{0.2cm}g\in L^2([0,1])$$ On the other hand the spaces $l^2(\mathbb{N})$ and $L^2([0,1])$ are isometric isomorphic to each other. To see this note that if $(e_n)\subset L^2([0,1])$ is a complete orthonormal basis for $L^2([0,1])$ then every $f\in L^2([0,1])$ admits a representation of the form $$f=\sum_{n=1}^{\infty}\langle f,e_n\rangle e_n\Rightarrow ||f||^2_{L^2}=\int_0^1f(x)\overline{f(x)}\,dx=\langle f,f\rangle=\sum^{\infty}_{n=1}|\langle f,e_n\rangle|^2:=||f_n||^2_{l^2}$$ implies $(f_n)\in l^2(\mathbb{N})$ where $f_n:=\langle f,e_n\rangle$. Clearly the operator $A:L^2([0,1])\mapsto l^2(\mathbb{N})$ defined by $Af:=(f_n)$ is a linear bounded operator which preserves the norms (as we can see for the above relation). Now to come back to the initial statement that if $(f^k_n)$ is a bounded sequence for each $k$ and for each $n$ we have $|f^k_n-f_n|\to 0$ as $k\to\infty$ then $(f^k_n)$ converges weakly to $(f_n)\in l^2(\mathbb{N})$. Again by definition $(f^k_n)$ converges weakly to $(f_n)$ whenever $$\lim_k\langle f^k,g\rangle=\langle f,g\rangle$$ where the inner product for $l^2(\mathbb{N})$ is defined by $$\langle f,g\rangle:=\sum_nf_n\overline{g_n}$$ Boundedness of each $(f^k_n)$ implies $(f^k_n)\in l^2(\mathbb{N})$ for each $k$ so their inner product with any other element $g$ of $l^2(\mathbb{N})$ is finite and well defined. On the other hand component wise convergence, $|f^k_n-f_n|\to 0$ as $k\to\infty$ for each $n$, implies $$|\langle f^k,g\rangle-\langle f,g\rangle|=|\langle f^k-f,g\rangle=\Big|\sum_n(f^k_n-f_n)\overline{g_n}\Big|\leqslant \sum_n|f^k_n-f_n||\overline{g_n}|\to 0$$ as $k\to\infty$. Therefore the same definition for both spaces is valid.

• Thanks - how would I deal with concrete examples, say $\sin(\alpha_nt)$, where $(\alpha_n)$ is a sequence? – max_zorn Mar 13 '18 at 3:35
• depends also on your sequence $\alpha_n$. if $\alpha_n=\theta$ for all $n$ then $\sin(\alpha_nt)$ converges trivially to $\sin(\theta t)$. – Arian Mar 15 '18 at 11:40


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# A Desirable Extension of the Nerve Theorem

## Backgroud

The Nerve Theorem (see nLab;) asserts that given a finite collection $\cal K$ of compact sets with the property that all non empty intersections of sets in the family are homotopically trivial, then the union $X$ of the sets in the family is homotopically equivalent to the nerve $N({\cal K})$. Here the nerve of the family is the abstract simplicial complex that records the intersection pattern so if ${\cal K}=\{K_1,K_2\dots,K_n\}$ then $N({\cal K})=\{S \subset \{1,2,\dots,n\}: \cap\{K_i : i \in S\} \ne \emptyset$. The nerve theorem is attributed to Borsuk and Leray (and others). There are various variations. For example if you require that all non empty intersections are homologically trivial then the homology groups of the nerve are the same as those of $X$.

## The desired extension

Suppose that all non empty intersections of sets in $\cal K$ are homotopically equivalent to a single space $Z$. Under what conditions can you conclude that $X$ is homologically the same as the product of Z and the nerve.

This is usually not the case, so the question is to find conditions (as simple as possible) that will imply it.

Perhaps the kind of condition you would need is about sets in K intersecting "nicely". For example, we can define two sets A and B as "nicely intersecting" if the usual long exact Meyer-Vietoris sequence splits into short exact sequences. So you can require that every intersection of some sets in $\cal K$ intersect "nicely" every intersection of some other sets in $\cal K$. I don't have reason to believe that this is the definition required for the conclusion but it just illustrates what type of conditions are possible.

A possible application is indicated in this question.

Edit: Oscar Randal-Williams' answer shows that my suggestion for "nice" intersection is not sufficient for the desired conclusion. Perhaps the most interesting question in view of Oscar's result is this.

Suppose that you have a family $\cal > K$ of compact subsets of $latex R^d$ so that all non empty intersections of members of $\cal K$ are homotopically equivalent to to the same space $Z$ and every intersection of some sets in $\cal K$ intersect "nicely" every intersection of some other sets in $\cal K$, where "nice intersection" means that the Meyer-Vietoris long exact sequence splits. Is it true that the nerve of $\cal K$ has vanishing $i$th homology groups for $i>d$?

I think that a positive answer would suffice for the Helly-type application.

-

The condition you suggest is insufficient. For example, let $F \to E \overset{\pi}\to B$ be a fibre bundle, with compact fibre say, (or more generally a local quasi-fibration), and $\mathcal{K}_B$ be a cover of $B$ satisfying the conditions of the Nerve theorem (i.e. all intersections are empty or contractible). Then $$\mathcal{K} := \{ \pi^{-1}(K) \subset E \vert K \in \mathcal{K}_B \}$$ is a cover of $E$ satisfying your Mayer--Vietoris condition, and it is fairly clear that $$N(\mathcal{K}) = N(\mathcal{K}_B) \simeq B,$$ and $E$ will not typically be equivalent to $N(\mathcal{K}) \times F$ if the original bundle is not trivial.
So, to answer a different question: given a cover $\mathcal{K}$ of a space $X$ where all non-empty intersections are equivalent to $Z$, I think there will typically be a homotopy fibre seqeuence $$Z \to X \to N(\mathcal{K}).$$ Whether or not this is trivial is then another problem. By the example above, every local quasi-fibration arises is this way, so there is unlikely to be an especially simple description of when it is possible.
One condition that will give what you want is the following: if you can find a map $f : X \to Z$ such that for each non-empty intersection $K_\alpha = \cap_{i \in \alpha} K_i$ the map $f\vert_{K_\alpha}: K_\alpha \to Z$ is a homotopy equivalence, then $X \simeq Z \times N(\mathcal{K})$. This is probably much too strong for most applications through.
I should have been more careful. In my second paragraph, I should have said "given a cover $\mathcal{K}$ of a space $X$ such that all intersections are homotopy (or homology) equivalent to a space $Z$, and the inclusions between intersections are all homotopy equivalences, then there is a homotopy (or homology) fibre sequence..." In particular, this essentially forces the M-V condition to hold. – Oscar Randal-Williams Oct 25 '11 at 8:17


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## mamma17 Group Title 29. Is the graph of y=2x-5/x+7 a translation of the graph of an inverse variation? Explain. one year ago one year ago

• This Question is Closed
1. mamma17 Group Title

show how u figured it out plz and ty

2. andriod09 Group Title

is the x negative? that's the question for something like this, this is the equation:$y=2x-\frac{-5}{x}+7$ so then that would me yes, because the equation would be negative on a graph, going from the left to the right.

3. mamma17 Group Title

no it is 2x-5 divded by x+7

4. andriod09 Group Title

oh, then no, it would be positive, not negative. my bad then.

5. mamma17 Group Title

it ok then but how did u figure that out I have to show my work

6. andriod09 Group Title

you really can't, not unless you graph it, and i don't like doing that, sorry. :/

7. mamma17 Group Title

it ok ty though


text:
# Mathematical Diversions

A collection of mathematical miscellany that diverts me from my main research interests. Some of the posts are interesting (to me) mathematical problems with their solutions. Others are mostly recreational things that distracted me at one time or another. The intended audience for posts in this category are my sole self, and so I make no great effort to polish the writing. Do send me an email with questions if you find anything here interesting or confusing.

### Oscillatory Solutions to the Heat Equation

As we all learned in a first course in PDEs, for "reasonable" initial data $u_0$, the solution to the linear heat equation …

### Some Periodic Functions

Some non-intuitive facts about periodic functions

### Traceless general relativity

A quick summary of the traceless version of general relativity.

### The Nirenberg Trick and Wave Maps

The "Nirenberg trick" or "Nirenberg transformation" is the observation1 that the semilinear wave equation …

### Convergence to Planewaves for Solutions to Schrödinger's equation III

In this installment, we look at Schrödinger's equation with barriers, and study the decay of the solutions.

### Convergence to Planewaves for Solutions to Schrödinger's equation II

We use Laplace transform arguments to look at the convergence of infinite energy solutions of the free 1D Schrödinger's equation …

### Convergence to Planewaves for Solutions to Schrödinger's equation

We look at the convergence of infinite energy solutions of the free 1D Schrödinger's equation to plane waves.

### Poohsticks and differential geometry

Random thought: Poohsticks illustrates well Lie differentiation

### Joris' Theorem

A proof is given of Joris' theorem, which states that if two coprime powers of a function are both smooth, then so is the original …

### When are curves tangent to hypersurfaces?

Given a curve and a hypersurface, both in Euclidean space, how can we tell if the curve is tangent to the hypersurface?


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Equivalent Units and Related Costs; Cost of Production Report ; Entries Wilmington Chemical... 1 answer below »

Equivalent Units and Related Costs; Cost of Production Report; Entries

Wilmington Chemical Company manufactures specialty chemicals by a series of three processes, all materials being introduced in the Distilling Department. From the Distilling Department, the materials pass through the Reaction and Filling departments, emerging as finished chemicals.

The balance in the account Work in Process—Filling was as follows on December 1, 2014:

 Work in Process—Filling Department (4,600 units, 80% completed): Direct materials (4,600 x $18.50)$85,100 Conversion (4,600 x 80% x $12.00) 44,160$129,260

The following costs were charged to Work in Process—Filling during December:

 Direct materials transferred from Reaction Department: 59,300 units at $18.20 a unit$1,079,260 Direct labor 358,560 Factory overhead 344,508

During December, 58,800 units of specialty chemicals were completed. Work in Process—Filling Department on December 31 was 5,100 units, 40% completed.

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# band diagonal Matrix

How can I represent this matrix in latex ?

• Just as any other matrix with \begin{bmatrix} ... \end{bmatrix}. For the braces this might be helpful: tex.stackexchange.com/questions/130798/braces-over-matrix – Henri Menke Mar 14 '17 at 10:03
• You'll get better help if you explain what you have got so far, and what specifically you are trying to achieve. – Seamus Mar 14 '17 at 10:29
• Do you need the matrix for fixed n and k or do you need exactly what is shown on your drawing? – Fabian Pijcke Mar 15 '17 at 16:43

You might find this useful. As for adding a big 0, you can overlay with \smash and \rlap (or use a tikzpicture).

\documentclass{article}
\usepackage{mathtools}

%from page 117 of symbols.pdf
\makeatletter
\def\revddots{\mathinner{\mkern1mu\raise\p@
\vbox{\kern7\p@\hbox{.}}\mkern2mu
\raise4\p@\hbox{.}\mkern2mu\raise7\p@\hbox{.}\mkern1mu}}
\makeatother

\begin{document}
$$\begin{bmatrix} 0 & \cdots & 1\\ \vdots & \revddots & \vdots\\ 1 & \cdots & 0 \end{bmatrix}$$
\end{document}


text:
# Mathematical Sciences Research Institute

Home » Seminar » Schedules » Quantitative Floer theory working group: "Cyclic symmetry, enumerative invariants and mirror symmetry"


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Recommendation

# Estimating the entropy of neural data by saving them as a .png file

, and based on reviews by Federico Stella and 2 anonymous reviewers
A recommendation of:

### A quick and easy way to estimate entropy and mutual information for neuroscience

Codes used in this study
Scripts used to obtain or analyze results
Abstract
Submitted: 06 August 2020, Recommended: 20 April 2021

#### Recommendation

Entropy and mutual information are useful metrics for quantitative analyses of various signals across the sciences including neuroscience (Verdú, 2019). The information that a neuron transfers about a sensory stimulus is just one of many examples of this. However, estimating the entropy of neural data is often difficult due to limited sampling (Tovée et al., 1993; Treves and Panzeri, 1995). This manuscript overcomes this problem with a 'quick and dirty' trick: just save the corresponding plots as PNG files and measure the file sizes! The idea is that the size of the PNG file obtained by saving a particular set of data will reflect the amount of variability present in the data and will therefore provide an indirect estimation of the entropy content of the data.

The method the study employs is based on Shannon’s Source Coding Theorem - an approach used in the field of compressed sensing - which is still not widely used in neuroscience. The resulting algorithm is very straightforward, essentially consisting of just saving a figure of your data as a PNG file. Therefore it provides a useful tool for a fast and computationally efficient evaluation of the information content of a signal, without having to resort to more math-heavy methods (as the computation is done “for free” by the PNG compression software). It also opens up the possibility to pursue a similar strategy with other (than PNG) image compression software. The main limitation is that the PNG conversion method presented here allows only a relative entropy estimation: the size of the file is not the absolute value of entropy, due to the fact that the PNG algorithm also involves filtering for 2D images.

The study comprehensively reviews the use of entropy estimation in circuit neuroscience, and then tests the PNG method against other math-heavy methods, which have also been made accessible elsewhere (Ince et al., 2010). The study demonstrates use of the method in several applications.  First, the mutual information between stimulus and neural response in whole-cell and unit recordings is estimated. Second, the study applies the method to experimental situations with less experimental control - such as recordings of hippocampal place cells (O’Keefe & Dostrovsky,1971) as animals freely explore an environment. The study shows the method can replicate previously established metrics in the field (e.g. Skaggs information, Skaggs et al. 1993). Importantly, it does this while making fewer assumptions on the data than traditional methods. Third, he study extends the use of the method to imaging data of neuronal morphology, such as charting the growth stage of neuronal cultures. However, the radial entropy of a dendritic tree seems at first more difficult to interpret than the common Sholl analysis of radial crossings of dendrite segments (Figure 6Ac of Zbili and Rama, 2021). As the authors note, a similar technique is used in paleobiology to discriminate pictures of biogenic rocks from abiogenic ones (Wagstaff and Corsetti, 2010). Perhaps neuronal subtypes could also be easily distinguished through PNG file size (Yuste et al., 2020). These examples are generally promising and creative applications.The authors used open source software and openly shared their code so anyone can give it a spin (https://github.com/Sylvain-Deposit/PNG-Entropy).

We were inspired by the wide applicability of the presented back-of-the-envelope technique, so we used it in a situation that the study had not tested: namely,  the dissection of microcircuits via optogenetic tagging of target neurons. In this process, one is often confronted with the problem that not only the opsin-carrying cells will spike in response to light, but also other nearby neurons which are activated synaptically (via the opto-tagged cell). Separating these two types of responses is typically done using a latency or jitter analysis, which requires the experimenter subjectively searching for detection parameters. Therefore a rapid and objective technique is preferable. The PNG rate difference method on slice whole cell recordings of opsin tagged neurons revealed higher mutual information metrics for direct optogenetic activation than for postsynaptic responses, showing the method can be easily used to objectively segregate different spike triggers.

Figure caption: Using a PNG entropy metric to distinguish between direct optogenetic responses and postsynaptic excitatory responses. Left, PNG rate difference calculated for whole cell recordings of optogenetic activation in brain slices. About 20 consecutive 60ms sweeps were analysed from each of 7 postsynaptic cells and 8 directly activated cells. Analysis was performed as in Fig4B of the preprint (https://doi.org/10.1101/2020.08.04.236174) using code from https://github.com/Sylvain-Deposit/PNG-Entropy/blob/master/BatchSaveAsPNG.py. Right, six example traces from a cell carrying channelrhodopsin (black, top) and a cell that was excited synaptically (gray, bottom).

References

Ince, R.A.A., Mazzoni, A., Petersen, R.S., and Panzeri, S. (2010). Open source tools for the information theoretic analysis of neural data. Front Neurosci 4. https://doi.org/10.3389/neuro.01.011.2010

O'Keefe, J., & Dostrovsky, J. (1971). The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Research, 34(1), 171-175. https://doi.org/10.1016/0006-8993(71)90358-1

Skaggs, M. E., McNaughton, B. L., Gothard, K. M., and Markus, E. J. (1993). An information-theoretic approach to deciphering the hippocampal code. Adv. Neural Inform. Process Syst. 5, 1030-1037.

Tovée, M.J., Rolls, E.T., Treves, A., and Bellis, R.P. (1993). Information encoding and the responses of single neurons in the primate temporal visual cortex. J Neurophysiol 70, 640-654. https://doi.org/10.1152/jn.1993.70.2.640

Treves, A., and Panzeri, S. (1995). The Upward Bias in Measures of Information Derived from Limited Data Samples. Neural Computation 7, 399-407. https://doi.org/10.1162/neco.1995.7.2.399

Verdú, S. (2019). Empirical Estimation of Information Measures: A Literature Guide. Entropy (Basel) 21. https://doi.org/10.3390/e21080720

Wagstaff, K.L., and Corsetti, F.A. (2010). An evaluation of information-theoretic methods for detecting structural microbial biosignatures. Astrobiology 10, 363-379. https://doi.org/10.1089/ast.2008.0301

Yuste, R., Hawrylycz, M., Aalling, N., Aguilar-Valles, A., Arendt, D., Armañanzas, R., Ascoli, G.A., Bielza, C., Bokharaie, V., Bergmann, T.B., et al. (2020). A community-based transcriptomics classification and nomenclature of neocortical cell types. Nat Neurosci 23, 1456-1468. https://doi.org/10.1038/s41593-020-0685-8

Zbili, M., and Rama, S. (2021). A quick and easy way to estimate entropy and mutual information for neuroscience. BioRxiv 2020.08.04.236174. https://doi.org/10.1101/2020.08.04.236174

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Cite this recommendation as:
Haudur Freyja Olafsdottir, Mahesh Karnani and Fleur Zeldenrust (2021) Estimating the entropy of neural data by saving them as a .png file. Peer Community in Neuroscience, 100001. 10.24072/pci.cneuro.100001

#### Reviewed by Federico Stella, 16 Mar 2021

The authors answered my concerns and I think the paper does not need further additions.

#### Reviewed by anonymous reviewer, 28 Feb 2021

The second submission version of the manuscript addresses most of the remarks that I made in the previous round. I appreaciate also the additions included in the new version, namely the introduction of the smart PNG-rate concept, and the new possible application examples. However, some of the issues I pointed out about the original version have not been dealt with completely, and would still need minor revision.

The main remark is still about the definitions and terminology. The authors have clarified the presentation and use of the terms well, but I still find that there are a few more clarifications needed.

For one, the concept of entropy rate still pops up out of the blue in the presentation. It appears first on line 86, then on lines 141-142, and then in Section 2.2. It should preferably be defined before all of these in a formal way. A somewhat formal definition is given on lines 247-248 with "The entropy rate R is defined as $R = \frac{1}{T}H$ with T being the sampling of the signal." This should indeed appear earlier, with also preferably a bit of more elaboration.

Moreover, the term 'sampling of the signal' for $T$ in the above quote is inconsistent with the earlier terminology. Also, I still hope for more clarification of the term 'size'. This also applies to its use on lines 84-86; i.e. what does "a signal of size S" exactly mean?

In addition, I still find a few smaller remarks about the manuscript. The authors have dealt only partly with my remark about italicization of math symbols. For instance, around Equation 1, the $xi$-terms are only in italic in the equation, but not in the body text. The symbols should be in italic everywhere, not only inside equations. This should also apply also to the variables $T$, $v$, $RS$, $R_N$, the probabilities $p(x)$, etc.

There still appear some language issues and typos, e.g. two in "In this study, the authors showed that a larger synapse drived the postsynaptic spiking in a greater manner, which increase the SIE." At least some of these mistakes should be possible to detect even with proof-reading software. Correcting them would make the presentation more convincing and readable.

## Evaluation round #1

DOI or URL of the preprint:

Version of the preprint: 1

#### Decision by Haudur Freyja Olafsdottir, Mahesh Karnani and Fleur Zeldenrust, 05 Nov 2020

Dear Sylvain and Mickael, Thanks for submitting your preprint to PCI C Neuro, and apologies again for the delays - it took us some time to reach outside our usual neuroscientist pool in order to get feedback on the mathematics as well. The reviews of your manuscript "A quick and easy way to estimate entropy and mutual information for neuroscience" are below. The manuscript was evaluated by 2 reviewers and 2 of us (reviewers 3 and 4). Everyone was enthusiastic about the work, but felt there were some crucial edits to be made. Furthermore, the reviewers would like you to make explicit under what experimental conditions the method can be used, which could enhance the use of the presented method. If you could write a response to the reviews and upload a revised preprint, we would be happy to evaluate a revision for recommendation, this time with a much faster turnaround time. We hope this feedback helps in the preparation of a revised manuscript. Thank you for participating in the PCI Circuit Neuroscience initiative. Best regards, Freyja, Mahesh and Fleur Reviewer 1: The manuscript presents a novel method for entropy estimation based on leveraging on optimal noiseless compression algorithms, already developed for image processing, and in particular on the PNG format. The idea is that the size of the PNG file obtained by saving a particular set of data will reflect the amount of variability present in the data and will therefore provide with an indirect estimation of the entropy content of the data. The method is based on Shannon’s Source Coding Theorem, therefore approaching entropy estimation problem from the field of compressed sensing, something that as the authors rightly state, has not yet found wide use at least in the field of neuroscience. The resulting algorithm is extremely straightforward, essentially consisting in just the PNG saving step. Therefore it provides a useful tool for a fast and computationally efficient evaluation of the complexity of a signal, without having to resort to more math-heavy methods (as the math is done “for free” by the PNG compression software). The main issue with this method is that in its present form is essentially limited to a private use by the experimenter, when needing to have a on-the-go assessment of the variation of entropy content in some preparation. Especially, it seems to me that it is particularly suitable for well controlled recordings with multiple trials of fixed length or for continuous imaging of cell cultures (and indeed, these are the examples presented in the manuscript). It would problematic to apply the method in its present form either to experiments with a larger behavioral component or, on the other hand, to compare different recordings This comes from the fact that the PNG conversion method presented here allows only for relative entropy estimation. The authors rightly note how the size of the file has no direct relationship to the value of entropy. And indeed all the examples presented in the manuscript deal with the estimation of entropy or information content in data of fixed size obtained from the same system. While this is no doubt a very handy approach to obtain quick measures of entropy for large dataset, it is hard for me to see how it could be generalized to allow for comparison across different experimental setup, or the same experimental setup in different conditions. Taking the case of mutual information, consider as an example the mutual information between a place cell activity and the position of the animal in space. The conditional entropy component of the mutual information requires to consider different bins in space, for which variable amount of data would be available, given the heterogeneities of the animal behavior. Moreover comparing two neurons recorded in two different sessions will also present the problem of different exploration time and amount of data. I am not sure wether the authors have already considered cases of this type and wether it would be possible to include a normalization step in their method, so that, for example PNG size is compared to an image of the same size obtained from a random process. As this would greatly extend the applicability of the PNG conversion, the paper would benefit from a discussion of possible solutions to this sort of situation. Reviewer 2: The article proposes quantifying the entropy and mutual information of neuroscientific data based on file size after compressing the data as a PNG-file. The idea is intriguing and is presented and studied in the article fairly thoroughly. The paper has some merits, but there are multiple issues as well, which are listed below. The authors correctly note the issue that file size does not correspond to absolute entropy values. Nevertheless, they state, already in the Abstract, that "we can reliably estimate entropy" with the method, which seems a bit like false advertising. Perhaps using "the level of entropy" in place of just "entropy" would be more accurate. The paper contains some repetition, especially in Sections 2.2 and 3.1. However, even the parts that are explained twice are not explained clearly enough; in particular, the quadratic extrapolation method requires some clarification. The terminology should be defined and stated more clearly and rigorously in the beginning; for instance, the authors use the terms "sampling", "bin" and "word" related to the variable T, but the meaning of these terms and the variable remain nevertheless unclear. The authors state the formula of Shannon entropy in Equation 1, but they do not discuss what is the probability space on which the entropy is considered in the context of the paper. It would be nice to have a discussion about this at least on a descriptive and intuitive level. The authors use the terms entropy and entropy rate in an interchangeable manner, although they are two quite different concepts. Which one are the authors actually interested in computing? This especially causes confusion when interpreting Figure 2A. Figure 2B has a couple of issues: First, the authors state that the plot of the PNG file size "follows the same curve than the entropy". However, although they appear similar, the values of the curves differ quite substantially at ~25% white pixels, where the entropy is around 50% of the maximum entropy but the file size is above 60% of the maximum. Secondly, the file size curve seems slightly asymmetric, which is surprising, as the file size should be the same regardless of whether compressing an image containing x percent white pixels or x percent black pixels. Perhaps the file size curve should be averaged over more runs of the compression? The authors should discuss the fact that, in addition to compression, the PNG algorithm also involves filtering for 2D images, (see e.g. http://www.libpng.org/pub/png/book/chapter09.html ), which affects the compression size of 2D images. The authors should note that this also affects the results gained with the 2D-images. Currently the 2D images are simply transformed into 1D signals. There are quite a few grammatical errors and typos throughout the text, so I would urge the authors to read through and edit the manuscript with care in this respect. The authors talk about "white noise with amplitude x". Is this a common convention for the use of the term amplitude? Perhaps some other terminology would be more standard? The mathematical notation has some issues. Most crucially, Equation 5 for the conditional entropy is simply incorrect. The conditional probability should be denoted with a vertical line "|", not a forward slash "/". The mathematical variables should always be italicized (e.g. on lines 41-42). The two (unnumbered) Equations between lines 161 and 166 are repeated in Equations 2 and 8 unnecessarily. Also, the Equation below line 161 has the term T in all three limits. Reviewer 3: This preprint presents a png-file compression based method for estimating relative entropy of time series signals and 2-d images. The authors start with a very useful presentation of another entropy estimation method and careful demonstration of better performance of the png method. They also explore the limitations in Fig2. After this, they present the use of the png method for analysing neural data by obtaining a metric similar to mutual information for repetitive trials of electrophysiological data, and analysing dendritic complexity in micrographs of neurons. The method appears useful and simple to apply, though parts of the application should be clarified and it would be more useful to present the questions that can be addressed with the method rather than metrics that can be approximated. Only the last use case seems to be addressing a clear question, ‘what is the growth state of a neural culture?’. I have major and minor suggestions for improvement below. Major: 1) Line 296 states that conditional entropy of signal X given signal Y can be interpreted as the noise entropy of X across trials. This seems incorrect and needs justification. Conditional entropy should express the entropy that remains in X given the knowledge of Y. It makes some intuitive sense that the noise entropy of X across trials will be affected by the driving signal Y, but to equate it with conditional entropy as defined in eq.5 seems a step too far. The main usefulness of the png method presented in the paper, estimating mutual information, relies on this interpretation. Perhaps an overly ambitious assumption like this explains the deviations between Fig3C middle and right panels. If the authors could come up with a more convincing method of estimating mutual information with the png approach, that would enhance the usefulness of the paper. However, given that the png method only returns relative entropy of one signal at a time it appears not suited for estimating conditional entropies. Other electrophysiological signal entropy metrics may also be equally useful for the community, such as transfer entropy, and just the relative entropy of signals, but it may be helpful to demonstrate an example of what questions can be answered with these. 2) Entropy of images is presented as another key use of the method in figure 4. The estimation of culture growth stage (Fig4C) is useful. However, the usefulness of the other two presented instances is dubious. Fig4A shows a use in estimating layers in a Cajal drawing. It is unclear why one would need to do this, as the boundaries of layers 2-5 appear to be obscured in the curve, compared to just looking at the drawing. Fig4B shows a use in estimating a curve similar to Sholl analysis (but different enough to not serve the same purpose). As the Sholl analysis curve describes the number of neural branches as a function of radial distance from soma, it gives us an immediately useful metric which needs no interpretation. The png metric however seems more difficult to interpret. It means something about the uncertainty/complexity of dendritic patterns at a given radius. Can the authors explain more why they believe this is useful? Demonstrating an improved identification of cell types based on the png metric over the Sholl curve would make a very strong case for usefulness. As the authors pointed out, astrobiologists can use png metrics to help distinguish between biogenic and nonbiogenic rocks – could the technique offer an equally useful categorization assay for neuroscience? Minor: 1) Authors explain on lines 64-70 that the quadratic extrapolation method is not the only alternative for estimating signal entropy. Why was the quadratic extrapolation method singled out as the comparison for the png method? Would the other methods perform as well or better than the png method? 2) Could the authors mention what computer was used for the study? This is important because, e.g., line 424, it is mentioned that speed is an advantage of the png method because getting the same metric with quadratic extrapolation took 2h. This depends on the hardware. 3) Equation between lines 161 and 162 should be the same as eq.2 on line 222. Reviewer 4: Estimating entropy/mutual information is often a difficult and computationally expensive process, and the simple method the authors propose is attractive, because it offers a quick and easy way to compare the entropy between conditions. However, I think there is an issue, namely that the "file size does not correspond to absolute entropy values", possibly because "the PNG algorithm also involves filtering for 2D images, (see e.g. http://www.libpng.org/pub/png/book/chapter09.html ), which affects the compression size of 2D images". Since entropy and mutual information are almost always heavily dependent on the estimation method (and direct measurement is almost never possible), this is not a problem per se, but I think the authors should be extremely clear on when their method does or does not apply. The authors make strong claims in the abstract ("By simply saving the signal in PNG picture format and measuring the size of the file on the hard drive, we can reliably estimate entropy through different conditions") and although they mention the limitations themselves, both in the abstract and in the text, I think they have to be careful here. So I believe this context should be made clearer, and it should be made explicit when to use (and NOT use) the method. The authors claim in the discussion that "PNG files must be all of the same dimensions, of the same dynamic range and saved with the same software" -- maybe they could think of a few typical experimental conditions where this does or does not apply? Secondly, I also wonder whether the size of a PNG file correlates linearly with the entropy of non-white noise files. The authors do not show this explicitly, only for white nose files, but aren't any signals that we are interested in non-white noise? Finally, personally I would be quite curious where the differences in png file size come from, if it is not the entropy. Do the authors have an opinion on that (because that could help others to judge when or when not one can use the method)?

#### Reviewed by anonymous reviewer, 26 Oct 2020

The manuscript presents a novel method for entropy estimation based on leveraging on optimal noiseless compression algorithms, already developed for image processing, and in particular on the PNG format. The idea is that the size of the PNG file obtained by saving a particular set of data will reflect the amount of variability present in the data and will therefore provide with an indirect estimation of the entropy content of the data. The method is based on Shannon’s Source Coding Theorem, therefore approaching entropy estimation problem from the field of compressed sensing, something that as the authors rightly state, has not yet found wide use at least in the field of neuroscience. The resulting algorithm is extremely straightforward, essentially consisting in just the PNG saving step. Therefore it provides a useful tool for a fast and computationally efficient evaluation of the complexity of a signal, without having to resort to more math-heavy methods (as the math is done “for free” by the PNG compression software).

The main issue with this method is that in its present form is essentially limited to a private use by the experimenter, when needing to have a on-the-go assessment of the variation of entropy content in some preparation. Especially, it seems to me that it is particularly suitable for well controlled recordings with multiple trials of fixed length or for continuous imaging of cell cultures (and indeed, these are the examples presented in the manuscript). It would problematic to apply the method in its present form either to experiments with a larger behavioral component or, on the other hand, to compare different recordings

This comes from the fact that the PNG conversion method presented here allows only for relative entropy estimation. The authors rightly note how the size of the file has no direct relationship to the value of entropy. And indeed all the examples presented in the manuscript deal with the estimation of entropy or information content in data of fixed size obtained from the same system. While this is no doubt a very handy approach to obtain quick measures of entropy for large dataset, it is hard for me to see how it could be generalized to allow for comparison across different experimental setup, or the same experimental setup in different conditions.

Taking the case of mutual information, consider as an example the mutual information between a place cell activity and the position of the animal in space. The conditional entropy component of the mutual information requires to consider different bins in space, for which variable amount of data would be available, given the heterogeneities of the animal behavior. Moreover comparing two neurons recorded in two different sessions will also present the problem of different exploration time and amount of data. I am not sure wether the authors have already considered cases of this type and wether it would be possible to include a normalization step in their method, so that, for example PNG size is compared to an image of the same size obtained from a random process. As this would greatly extend the applicability of the PNG conversion, the paper would benefit from a discussion of possible solutions to this sort of situation.

#### Reviewed by anonymous reviewer, 29 Sep 2020

The article proposes quantifying the entropy and mutual information of neuroscientific data based on file size after compressing the data as a PNG-file. The idea is intriguing and is presented and studied in the article fairly thoroughly. The paper has some merits, but there are multiple issues as well, which are listed below.

• The authors correctly note the issue that file size does not correspond to absolute entropy values. Nevertheless, they state, already in the Abstract, that "we can reliably estimate entropy" with the method, which seems a bit like false advertising. Perhaps using "the level of entropy" in place of just "entropy" would be more accurate.

• The paper contains some repetition, especially in Sections 2.2 and 3.1. However, even the parts that are explained twice are not explained clearly enough; in particular, the quadratic extrapolation method requires some clarification. The terminology should be defined and stated more clearly and rigorously in the beginning; for instance, the authors use the terms "sampling", "bin" and "word" related to the variable T, but the meaning of these terms and the variable remain nevertheless unclear.

• The authors state the formula of Shannon entropy in Equation 1, but they do not discuss what is the probability space on which the entropy is considered in the context of the paper. It would be nice to have a discussion about this at least on a descriptive and intuitive level.

• The authors use the terms entropy and entropy rate in an interchangeable manner, although they are two quite different concepts. Which one are the authors actually interested in computing? This especially causes confusion when interpreting Figure 2A.

• Figure 2B has a couple of issues: First, the authors state that the plot of the PNG file size "follows the same curve than the entropy". However, although they appear similar, the values of the curves differ quite substantially at ~25% white pixels, where the entropy is around 50% of the maximum entropy but the file size is above 60% of the maximum. Secondly, the file size curve seems slightly asymmetric, which is surprising, as the file size should be the same regardless of whether compressing an image containing x percent white pixels or x percent black pixels. Perhaps the file size curve should be averaged over more runs of the compression?

• The authors should discuss the fact that, in addition to compression, the PNG algorithm also involves filtering for 2D images, (see e.g. http://www.libpng.org/pub/png/book/chapter09.html ), which affects the compression size of 2D images. The authors should note that this also affects the results gained with the 2D-images. Currently the 2D images are simply transformed into 1D signals, which

• There are quite a few grammatical errors and typos throughout the text, so I would urge the authors to read through and edit the manuscript with care in this respect.

• The authors talk about "white noise with amplitude x". Is this a common convention for the use of the term amplitude? Perhaps some other terminology would be more standard?

• The mathematical notation has some issues. Most crucially, Equation 5 for the conditional entropy is simply incorrect.

• The conditional probability should be denoted with a vertical line "|", not a forward slash "/".

• The mathematical variables should always be italicized (e.g. on lines 41-42).

• The two (unnumbered) Equations between lines 161 and 166 are repeated in Equations 2 and 8 unnecessarily. Also, the Equation below line 161 has the term T in all three limits.

No user comments yet


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# Changing system shell

What I'd like to do is to have a Mathematica script use some user-defined bash functions before scheduling some calculations on the results generated by by these functions. Since I have them written in bash, I use

$SystemShell = "/bin/bash"  and get the error Set::wrsym: Symbol$SystemShell is Protected. >>

and all subsequent lines naturally crash if they need bash to work. Funny thing is, if I go back and Shift+Enter the same line again,

$SystemShell = "/bin/bash"  this and all subsequent lines work fine. I am doing this on the login node of a small cluster, using PUTTY+XMing to forward X11 to my Windows system so I can use the GUI. So, to summarize, changing the SystemShell fails if it is part of the first set of commands being Shift+Enter-ed together, but not on subsequent attempts. Could someone help me understand what is happening and how I could get this to work in one go so that I can make a .m script out of it and submit it as a job to the compute nodes? Thanks a bunch! Sample script: $SystemShell = "/bin/bash"
bash = StartProcess[$SystemShell] WriteLine[bash, "! submitjob () { some code }"]; ReadString[bash, EndOfBuffer]  • When I get a Protected warning the first thing I try, if I really do want to change it, is Unprotect – Bill Jul 10 '16 at 17:47 • Thanks! I didn't know about this feature. It's puzzling why it's protected on the first go, but not the second time, though. Is this unique to my situation, or is someone able to reproduce this? – Tamaghna Jul 10 '16 at 22:35 ## 1 Answer There is no obligation to use the $SystemShell symbol. It does not have any special meaning and one could just as well do

myShell = "/bin/bash";
bash = StartProcess[myShell]

$SystemShell is defined only for convenience, and on Windows it is always "cmd" while on Unix it is always "/bin/sh". Of course a different shell can be launched like any other executable. As for the Protected attribute behavior observed, that is a consequence of how the kernel autoloading mechanism works. Initially $SystemShell has only a special ActivateLoad definition (which can be seen by evaluating OwnValues[$SystemShell] in a fresh session) that allows the implementation to be loaded at first use. It is very common for a symbol to get protected again after all its real definitions are read in, though this is not the case for $SystemShell.


text:
# Discontinuous current operation (Boost Converter)

Discussion in 'Homework Help' started by jegues, Oct 19, 2012.

1. ### jegues Thread Starter Well-Known Member

Sep 13, 2010
735
43
Hello all,

Attached below are the pages from my textbook for which I am concerned.

On page 245 we can calculate the average inductor current as follows,

$I_{L} = \frac{1}{2}I_{max}(D+D_{1})$

Now what I thought an equivalent expression for the diode current would be,

$I_{D} = \frac{I_{L}D_{1}T}{T} = I_{L}D_{1}$

Is this incorrect? If so, why?

It is clear from the graph of the diode current on page 245 that indeed,

$I_{D} = \frac{\frac{1}{2}I_{max}D_{1}T}{T} = \frac{1}{2}I_{max}D_{1}$

but instead of calculating the area of a triangle, I've always "stretched" that triangle into a rectangle since I know that the current flowing through the diode for that time frame (i.e. D1T) is the average inductor current. With this in mind, the area of my rectangle would be,

$I_{D} = \frac{I_{L}D_{1}T}{T}$

Is this intuition incorrect?

This is thought of stretching the area of said triangle into a equivalent rectangle has previously worked for me when doing the analysis for the Buck converter under discontinuous conduction mode. In particular, I wrote

$I_{s} = I_{L}D$

where Is is the source current.

Does the source of my confusion make sense? Can you see where I'm going wrong?

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• ###### Q-_0132_NEW.jpg
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## torsdag 27 mars 2014

### How to Make Schrödinger's Equation Physically Meaningful + Computable

The derivation of Schrödinger's equation as the basic mathematical model of quantum mechanics is hidden in mystery: The idea is somehow to start considering a classical Hamiltonian $H(q,p)$ as the total energy equal to the sum of kinetic and potential energy:
• $H(q,p)=\frac{p^2}{2m} + V(q)$,
where $q(t)$ is position and $p=m\dot q= m\frac{dq}{dt}$ momentum of a moving particle of mass $m$, and make the formal ad hoc substitution with $\bar h =\frac{h}{2\pi}$ and $h$ Planck's constant:
• $p = -i\bar h\nabla$ with formally $\frac{p^2}{2m} = - \frac{\bar h^2}{2m}\nabla^2 = - \frac{\bar h^2} {2m}\Delta$,
to get Schrödinger's equation in time dependent form
• $i\bar h\frac{\partial\psi}{\partial t}=H\psi$,
with now $H$ a differential operator acting on a wave function $\psi (x,t)$ with $x$ a space coordinate and $t$ time, given by
• $H\psi \equiv -\frac{\bar h^2}{2m}\Delta \psi + V\psi$,
where now $V(x)$ acts as a given potential function. As a time independent eigenvalue problem Schrödinger's equation then takes the form:
•  $-\frac{\bar h^2}{2m}\Delta \psi + V\psi = E\psi$,
with $E$ an eigenvalue, as a stationary value for the total energy
• $K(\psi ) + W(\psi )\equiv\frac{\bar h^2}{2m}\int\vert\nabla\psi\vert^2\, dx +\int V\psi^2\, dx$,
as the sum of kinetic energy $K(\psi )$ and potential energy $W(\psi )$, under the normalization $\int\psi^2\, dx = 1$.  The ground state then corresponds to minimal total energy,

We see that the total energy $K(\psi ) + W(\psi)$ can be seen as smoothed version of $H(q,p)$ with
• $V(q)$ replaced by $\int V\psi^2\, dx$,
• $\frac{p^2}{2m}=\frac{m\dot q^2}{2}$ replaced by  $\frac{\bar h^2}{2m}\int\vert\nabla\psi\vert^2\, dx$,
and Schrödinger's equation as expressing stationarity of the total energy as an analog the classical equations of motion expressing stationarity of the Hamiltonian $H(p,q)$ under variations of the path $q(t)$.

We conclude that Schrödinger's equation for a one electron system can be seen as a smoothed version of the equation of motion for a classical particle acted upon by a potential force, with Planck's constant serving as a smoothing parameter.

Similarly it is natural to consider smoothed versions of classical many-particle systems as quantum mechanical models resembling Hartree variants of Schrödinger's equation for many-electrons systems, that is quantum mechanics as smoothed particle mechanics, thereby (maybe) reducing some of the mystery of Schrödinger's equation and opening to computable quantum mechanical models.

We see Schrödinger's equation arising from a Hamiltonian as total energy kinetic energy + potential energy, rather that from a Lagrangian as kinetic energy - potential energy. The reason is a confusing terminology with $K(\psi )$ named kinetic energy even though it does not involve time differentiation, while it more naturally should occur in a Lagrangian as a form of the potential energy like elastic energy in classical mechanics.

#### 17 kommentarer:

1. and make the formal ad hoc substitution

QM has a lot deeper theoretical justification than you give it credit for.

It is possible that the earlier derivations used an ad hoc substitution like this (I don't really know since Schroedingers original derivation differs from a more modern one).

Introductory text books certainly use to include the operator ad hoc.

If you read Sakurai's Modern Quantum Mechanics, ch 1.7, you will see how the operator p in a plane wave basis is derived including powers of p.

Further, I can't really understand your point in the last paragraphs. Have you read the different formulations of QM and the Ehrenfest theorem? If so you should be aware of the connections between a Hamiltonian or Lagrangian approach and the connection to a classical picture. What is new in that you write here??

2. The idea is to seek a formulation of QM which is not an ad hoc uncomputable linear multidim Schrödinger equation requiring a statistical interpretation.

3. And you don't think anybody has tried that...? ;-)

4. Of course, but if you insist to use statistics it means that you have given up physical interpretation.

5. Would you say that electrons are distinguishable particles?

If so, how come that all experiments including some numbers of electrons, without exceptions, show that electrons follows a Fermi-Dirac distribution and not a Maxwell-Boltzmann distribution?

6. Of course electrons in different locations in space are distinguishable particles, as in different atoms. Electrons within atoms partly loose identity and mix into some form of electron clouds.

7. You need to motivate that position a lot more.

If you you have two electrons in a system, say el1 and el2, at location x1 and x2. Call this (x1,el1) and (x2,el2).

If we change the positions of the electrons, that is (x1,el2) and (x2,el1), what difference, in a physical sense, would this exchange make?

8. This is the same as rotating new identical tires on your car. What difference in a physical sense would that exchange make according to your view?

9. You're missing the point. Your tires are not, of course, identical, other then as an idealized construction! You can also decide to label one of them with a magic marker or whatever.

But say that you have two electrons in your system, you decide to name them Frank and Kurt.

You look away for an instance and then examine the system again. Which electron is Frank and which is Kurt. Can you tell a difference? Are they distinguishable? Is there a physical difference if Frank does Kurts job and vice verse?

10. And if you don't mark the tires, can you then tell which is which?

11. That's a silly question.

You can mark tires, but you can't mark electrons. So how do you distinguish them if they are distinguishable?

12. Why should I mark electrons?
Why should I mark new identical tires?

13. Have you used your theory on molecular Hydrogen and showed that the theory manages to bind in the most simplest case?

If so, what bond length did you get?

14. I guess that you haven't done this then.

Are you planning on doing it?

15. Yes, I have many plans.

16. What have you done so far, when it comes to real calculations?

How does it compare to real data?

17. Any progress?


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# Appending URL parameters to links

I've got some pages that may or may not load with a ReturnUrl parameter, and some links that will be generated without that parameter (though I've applied a class to them).

The site uses jQuery, and I think I've found the most elegant and least breaky solution as follows:

$(document).ready(function() {$('a.my_returnlink').each(function(){
$(this).attr('href',$(this).attr('href')+window.location.search);
});
});

That is, given a page url like http://example.com/page.aspx?ReturnUrl=blahblahblah and the following link on the page: Some link, that link will end up pointing to http://example.com/someotherpage.aspx?ReturnUrl=blahblahblah.

Is there any reason not to do it this way?

Generally speaking, doing something with jQuery will perform worse than doing that same thing directly through the DOM. In this case, using the following code inside the each will yield better performance:
this.href += window.location.search;

Note, however, that code will create an invalid URL if there is already a query on the original href.


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# 15th Varenna Conference on Nuclear Reaction Mechanisms

Jun 11 – 15, 2018
Villa Monastero
Europe/Zurich timezone

## Aggregate Decay Behavior of Fission Products in Nuclear Reactors -Decay Heat, Reactor Antineutrino and the Pandemonium Problem-

Jun 12, 2018, 12:20 PM
20m
Villa Monastero

Varenna (Italy)
Fission

### Speaker

Dr Tadashi Yoshida (Tokyo Institute of Technology)

### Description

The decisive importance of fission-product (FP) decay heat (DH) was far and widely recognized after the core-melt accident at the TMI nuclear power station in 1979. Even before this, in the late 1970s, a lot of experimental and theoretical studies on FPDH were motivated and initiated in Japan, the US and Europe. In these countries, extensive FP decay data libraries to be used in DH summation calculations were elaborated from the up-to-date FP decay data which had been taken from the high-resolution $\gamma$-ray measurements. Optimistic expectation for good agreement between the experimental and the theoretical results, however, were not materialized with one exception. That is the JNDC (Japanese Nuclear Data Committee) Library where the gross theory of $\beta$-decay was employed for supplementing the insufficiency of the experimental decay data which was not yet perceived clearly then. In due course beyond the TMI accident, the origin of the disagreement was identified as the Pandemonium problem which had been warned of by Hardy et al. based on computer simulation. The best experimental way to overcome to this difficulty, which is intrinsic to high $Q_\beta$-value decay, is now recognized as the total absorption $\gamma$-ray spectroscopy (TAGS). After the extensive effort at Idaho in the 1990s, TAGS was further motivated by series of the IAEA meetings on this subject since 2006 up to the present from the view points both of nuclear science and technology. Another important aspect of the aggregate FP in reactor cores is as the emitter of antineutrinos ($\bar{\nu}_e$).　It is stressed that the gross theory of $\beta$-decay still have potential to play an important role in investigating the reactor $\bar{\nu}_e$ flux and its energy spectrum along with the steadily accumulating TAGS experimental data at present.

### Primary author

Dr Tadashi Yoshida (Tokyo Institute of Technology)


text:
# Multiprocessing best practices¶

torch.multiprocessing is a drop in replacement for Python’s multiprocessing module. It supports the exact same operations, but extends it, so that all tensors sent through a multiprocessing.Queue, will have their data moved into shared memory and will only send a handle to another process.

Note

When a Tensor is sent to another process, the Tensor data is shared. If torch.Tensor.grad is not None, it is also shared. After a Tensor without a torch.Tensor.grad field is sent to the other process, it creates a standard process-specific .grad Tensor that is not automatically shared across all processes, unlike how the Tensor’s data has been shared.

This allows to implement various training methods, like Hogwild, A3C, or any others that require asynchronous operation.

## Sharing CUDA tensors¶

Sharing CUDA tensors between processes is supported only in Python 3, using a spawn or forkserver start methods. multiprocessing in Python 2 can only create subprocesses using fork, and it’s not supported by the CUDA runtime.

Warning

CUDA API requires that the allocation exported to other processes remains valid as long as it’s used by them. You should be careful and ensure that CUDA tensors you shared don’t go out of scope as long as it’s necessary. This shouldn’t be a problem for sharing model parameters, but passing other kinds of data should be done with care. Note that this restriction doesn’t apply to shared CPU memory.

## Best practices and tips¶

There are a lot of things that can go wrong when a new process is spawned, with the most common cause of deadlocks being background threads. If there’s any thread that holds a lock or imports a module, and fork is called, it’s very likely that the subprocess will be in a corrupted state and will deadlock or fail in a different way. Note that even if you don’t, Python built in libraries do - no need to look further than multiprocessing. multiprocessing.Queue is actually a very complex class, that spawns multiple threads used to serialize, send and receive objects, and they can cause aforementioned problems too. If you find yourself in such situation try using a multiprocessing.queues.SimpleQueue, that doesn’t use any additional threads.

We’re trying our best to make it easy for you and ensure these deadlocks don’t happen but some things are out of our control. If you have any issues you can’t cope with for a while, try reaching out on forums, and we’ll see if it’s an issue we can fix.

### Reuse buffers passed through a Queue¶

Remember that each time you put a Tensor into a multiprocessing.Queue, it has to be moved into shared memory. If it’s already shared, it is a no-op, otherwise it will incur an additional memory copy that can slow down the whole process. Even if you have a pool of processes sending data to a single one, make it send the buffers back - this is nearly free and will let you avoid a copy when sending next batch.

### Asynchronous multiprocess training (e.g. Hogwild)¶

Using torch.multiprocessing, it is possible to train a model asynchronously, with parameters either shared all the time, or being periodically synchronized. In the first case, we recommend sending over the whole model object, while in the latter, we advise to only send the state_dict().

We recommend using multiprocessing.Queue for passing all kinds of PyTorch objects between processes. It is possible to e.g. inherit the tensors and storages already in shared memory, when using the fork start method, however it is very bug prone and should be used with care, and only by advanced users. Queues, even though they’re sometimes a less elegant solution, will work properly in all cases.

Warning

You should be careful about having global statements, that are not guarded with an if __name__ == '__main__'. If a different start method than fork is used, they will be executed in all subprocesses.

#### Hogwild¶

A concrete Hogwild implementation can be found in the examples repository, but to showcase the overall structure of the code, there’s also a minimal example below as well:

import torch.multiprocessing as mp
from model import MyModel

def train(model):
loss_fn(model(data), labels).backward()
optimizer.step()  # This will update the shared parameters

if __name__ == '__main__':
num_processes = 4
model = MyModel()
# NOTE: this is required for the fork method to work
model.share_memory()
processes = []
for rank in range(num_processes):
p = mp.Process(target=train, args=(model,))
p.start()
processes.append(p)
for p in processes:
p.join()


text:
Okay, this is the final part in the above series of posts on some identities related to definite integrals (before I get too lazy and forget to post the same).

So, what is the magic identity? Here it is.

$\displaystyle (3) \int_{-a}^{a} f(x) \, dx = \int_0^a \left( f(x) + f(-x)\right) \, dx$

Proof: Let $t = -x$ in the second integral on the right hand side. Then, we have $\displaystyle \int_0^a f(-x) \, dx = - \int_0^{-a} f(t) \, dt = \int_{-a}^0 f(x) \, dx$, and combining this with the first integral on the right hand side yields the desired result.

Now, apply the above identity to the “difficult” integral in problem $(6)$ from the Integration Bee, Challenging Integrals post to evaluate the integral. The solution turns out to be an easy one. The answer is $\pi /4$, just in case you need to verify.


text:
# 'S' in s domain

1. ### pforpashya

12
Can somebody tell me exactly what is 'S' in frequency domain?

I know that s=σ+jw

where σ is neper frequency and jw is complex frequency but i want the precise information on s

you can suggest me a book in which it is clearly explained.

Thank you.

71
3. ### meBigGuy

This is the way I think of it, but I may be completely wrong. Think of s as having a magnitude and phase (when expressed exponentially). If you express that rectangularly it can be seen as the sum of two orthogonal components. Call them what you will. One is the cosine component and the other is the sine component. I view complex numbers this way since the rectangular components are often difficult to visualize. Writing a polynomial with s and t allows you to express sine and cosine components that vary in time. Or, said another way, frequency and phase varying in time. If I said this incorrectly, we will soon find out :)