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# How to solve $AXB = C$ efficiently (numerically)

This might be a simple question but I can't think of an obvious answer. How do I solve for $X$ in $AXB=C$ efficiently? I don't want to use $(B^T\otimes A) \operatorname{vec}(X) = C$ because $B^T\otimes A$ is very large. Is there a better 1-step solution? Or should I just use gradient descent?

$A$, $B$, $C$, and $X$ are all matrices, with dimensions in the order of 1000 x 1000

Thanks!

• What are the dimensions of each? It looks like $X$ is a vector, but what are $A,B,C$? – Ross Millikan Jul 8 '17 at 0:43
• Good point; I added an edit. So $X$ is also a matrix, everything is a matrix. vec(X) vectorizes X. The dimensions of everything is medium large; basically, it's easy to do A \ B, but not (kron(B.T,A) \ C). – Y. S. Jul 8 '17 at 0:46
• In some cases coarse graining can work well. If you average over, say, 100 by 100 blocks, you get a coarse grained 10 by 10 problem. You can then upscale the solution by merely copying elements to a, say 20 by 20 initial solution. If you then coarse grain the given matrices to 20 by 20 size, then you can use the differences with the upscaled 10 by 10 to 20 by 20 and the real 20 by 20 to get to an efficient way of obtaining the 20 by 20 solution. You can then repeat this doubling (or you can increase by a smaller factor) until you arrive at the original system. – Count Iblis Jul 8 '17 at 1:35
• @whyyes Sorry, I forgot to include additional conditions. We need the existence of solutions to $AY=C$ and $ZB=C$ for $X$ to work. – Sungjin Kim Jul 8 '17 at 3:19
• @whyyes This might be helpful: math.stackexchange.com/questions/585419/… – Sungjin Kim Jul 8 '17 at 7:33

To simplify, you should do a Schur decomposition of $A$ and $B$, i.e. $A=UA'U^*$ and $B=VB'V^*$ with $U,V$ unitary and $A',B'$ upper triangular. Then the system becomes

$$AXB=C \iff U A' U^* X V B' V^* = C \iff A' Y B' =C'$$

with $Y =U^*XV$ and $C' = U^*CV$. The resulting system is efficiently solvable by carefully carrying out the matrix multiplications. Observe that, since $A'$ and $B'$ are upper triangular,

$$A'YB' = \begin{bmatrix}*&*&*\\*&*&* \\ A'_{nn}Y_{n1} & \cdots & A'_{nn}Y_{nn}&\end{bmatrix}\cdot \begin{bmatrix}B'_{11}&\cdots& B'_{1n}\\ &\ddots&\vdots\\&&B'_{nn}\end{bmatrix}$$

Thus $(A'YB')_{n1} = A'_{nn}Y_{n1}B'_{11} = C'_{n1}$ determines $Y_{n1}$. But then we can continue with

$$(A'YB')_{n2} = A'_{nn}Y_{n1}B'_{12} + A'_{nn}Y_{n2}B'_{22} = C'_{n2}$$

which subsequently determines $Y_{n2}$ and so on and so forth.

This idea is basically a variation on the Bartels-Stewart Algorithm which solves Sylvester's equation $AX - XB = C$

• Cool, this is very nice! I'm trying to draw parallels with some previous comments... the "bad" cases (the ones that don't work for the above answers) would correspond to some 0s in the diagonals of $B$ or $A$, which would complicate things? Anyway, I really like the parallel with Sylvester's equation! – Y. S. Jul 8 '17 at 20:11
• Well at some point when following through with this procedure you might run into the issue of division by zero. E.g. if $A'_{nn}=0$ but $C'_{n1}\neq 0$. That being said in this case, since the diagonal entries of the Schur decomposition are the eigenvalues, $A$ is singular and the system is possibly unsolveable to start with. As a further remark, you might be able to find implementations of these algorithms online. A quick google search gave me this dm.unibo.it/~simoncin/matrixeq.pdf which might be worth a read. – Hyperplane Jul 8 '17 at 23:35

To summarize the comments, there are a number of ways of approximating a solution efficiently (using some optimization method, or using a coarse graining / upscaling procedure).

For an exact solution, we can use $X = A^\dagger C B^\dagger$ (or, in matlab, $X = (A \backslash C) / B$) if:

1. if $A$ has full row rank (and therefore a right-inverse), or $C = AU$ for some $U$, and
2. if $B$ has full column rank (and therefore a left-inverse), or $C = VB$ for some $V$.

Then $AXB = C$.

For exact solutions, this seems the best we can do?


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Manifolds of zero dimension and $\mathbb R^0$?

Tu Manifolds Section 5.4

Example 5.13 (Manifolds of dimension zero). In a manifold of dimension zero, every singleton subset is homeomorphic to $$\mathbb R^0$$ and so is open. Thus, a zero-dimensional manifold is a discrete set. By second countability, this discrete set must be countable.

Why exactly is the manifold $$M$$ discrete? I actually proved that the singleton subsets are open in their components but was not able to show they are open in $$M$$ itself.

Here is what I have done thus far:

Let $$M$$ be a smooth manifold with dimension zero, which means by definition that all of the connected components of $$M$$'s topological manifold (see here) $$\{C_{\alpha}\}_{\alpha \in J}$$ have dimension zero.

Let $$\alpha \in J$$. $$C_{\alpha}$$ has dimension zero, which means by definition (see here) that $$\forall p \in C_{\alpha}, \exists$$ homeomorphism $$\varphi: U \to V$$ for some $$U$$, a neighborhood of $$p$$ in $$C_{\alpha}$$ and some $$V$$, an open subset of $$\mathbb R^0=\{0\}$$. $$V$$ is either $$\{0\}$$ or $$\emptyset$$. Since $$U$$ contains $$p$$, $$U \ne \emptyset$$. Hence, $$V \ne \emptyset$$ because from nothing comes nothing, so $$V=\mathbb R^0=\{0\}$$. Sets that are homeomorphic to singletons are singletons. Therefore, $$U$$ is a singleton containing p, so $$U=\{p\}$$.

Therefore, we have

• $$\forall p \in M, \exists$$ unique $$\alpha \in J: \{p\}$$ is open in $$C_{\alpha}$$.

I remember the connected components $$C_{\alpha}$$ are:

• closed in $$M$$

• not necessarily open in $$M$$.

• open in $$M$$ if $$J$$ is finite.

I know $$\{p\}$$ is open in one of the connected components of $$M$$. How do we arrive at the conclusion that $$\{p\}$$ is open in $$M$$ itself?

• I don't think you have the right definition of an $n$-dimensional manifold. Being an $n$-dimensional manifold means just that the space is locally homeomorphic to open subsets of ${\mathbf R}^n$. In your other question, you have the hypothesis of "locally Euclidean", which implies in particular that the connected components are open, in which case the two notions of $n$-dimensional coincide. – tomasz Nov 22 '18 at 10:47
• @tomasz I think in Tu, "space is locally homeomorphic to open subsets" is the definition for topological manifold instead of smooth manifold. By $n$-dimensional manifold, do you refer to smooth manifold or topological? – user198044 Nov 22 '18 at 11:15
• Yes. For a smooth manifold you need some further restrictions. But every smooth manifold is in particular a topological manifold. – tomasz Nov 22 '18 at 12:12
• @tomasz I will take it to mean that you refer to a smooth manifold. Tu's definition is different from yours, if I understand correctly. – user198044 Nov 24 '18 at 3:28
• That is my point: it is not (really) different. By the definition you gave 5.2, a manifold is a locally Euclidean space, which implies that connected components are open (because Euclidean spaces are locally connected). Hence, all connected components being $n$-dimensional is the same as being locally Euclidean of dimension $n$. – tomasz Nov 25 '18 at 14:58

You wrote down yourself that you now know that every point in your manifold is clopen (U={p} implies this, since the domain of charts has to be open). But the only clopen subsets of a connected space are the space itself and the empty set, hence $$\{p\}$$ is a maximal connected component, and hence all of $$C_\alpha$$
• Oh, we deduce $\{p\}$ is open in $C_{\alpha}$ and know $\{p\}$ is closed in $C_{\alpha}$ by Hausdorff and so conclude $M$ is totally disconnected, which implies discrete? – user198044 Nov 22 '18 at 11:09
• well, I do not know what totally disconnected means, but we conclude that $C_\alpha$ is just a point, and since $\alpha$ was arbitrary, all connected components are singletons, which are clopen, hence the union of them (i.e. M) is disrcete. – Enkidu Nov 22 '18 at 12:30
I don't know what you know but I would do it like this: Pick a point $$p\in M$$. It has an open neighborhood $$U$$ homeomorphic to $$\mathbf R^0$$. So $$U=\{p\}$$ (it has only one point!). Hence $$\{p\}$$ is open.
• How do you know $p$ has such an open neighborhood in M? I know $p$ has such an open neighborhood in one of the connected components of M, but I don't know how to extend this to $M$ itself. – user198044 Nov 22 '18 at 10:34
• Tom, are you actually tomasz and mistakenly replied here instead of in the comments on the question? If not, I don't understand your reply here. My problem is I know $\{p\}$ is open in a connected component of $M$ but don't know how to conclude $\{p\}$ is open in $M$ itself. – user198044 Nov 25 '18 at 8:44

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Free Version
Easy

# Intervals of Continuity

APCALC-MEZ1JN

The graph of $f$ is given. On which of the following intervals is $f$ continuous?

A

$-2\leq x < 0$ and $0 \leq x < 3$

B

$-2\leq x < 3$

C

$-2 \leq x < 0$ and $0 < x \leq 3$

D

$-2\leq x < 0$ and $0 < x < 3$


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deiteresfp

2022-01-17

Solve the following equation?
$\left(x-2xy-{y}^{2}\right)\frac{dy}{dx}+{y}^{2}=0$

Piosellisf

Expert

For this one we look for an integrating factor which will make the equation exact.
Let . Then .
Thus $\frac{{M}_{x}-{N}_{y}}{N}=\frac{1-2y}{{y}^{2}}=\frac{1}{{y}^{2}}-2\frac{1}{y}$
which is purely a function of y. This tells us that there is an integrating factor $\mu \left(y\right)$ which makes the equation exact which satisfies
$\frac{d\mu }{dy}=\left(\frac{1}{{y}^{2}}-2\frac{1}{y}\right)\mu$
Clearly $\mu \left(y\right)={e}^{\int \left(\frac{1}{{y}^{2}}-2\frac{1}{y}\right)dy}={e}^{\frac{-1}{y}-2\mathrm{log}y}={y}^{-2}{e}^{y-1}$.
Try (first checking that my algebra is correct) and then multiplying the differential equation by this integrating factor, it should become an exact differential equation which I'm guessing you know how to solve if you are asking this question.

Cleveland Walters

Expert

Hint: Your equation can be turned into a first order linear differential equation in $\frac{dx}{dy}$ (i.e., just multiply the equation by $\frac{dx}{dy}$ and it is a standard form....)

alenahelenash

Expert

As it looks some people misunderstood (I apologize, the fault may be my side) my first answer, here is the complete solution after reducing the problem into standard form: As I wrote in the first answer, multiplying both sides by $\frac{dx}{dy}$ (the hint was that it will make the equation linear in x), $\frac{dx}{dy}+\left(\frac{1-2y}{{y}^{2}}\right)\cdot x=1$. This is a linear equation (in x) of the standard form $\frac{dx}{dy}+P\left(y\right)\cdot x=Q\left(x\right)$. So, ${e}^{\int P\left(y\right)dy}={e}^{\int \left(\frac{1-2y}{{y}^{2}}\right)dy}=\frac{{e}^{\frac{-1}{y}}}{{y}^{2}}$ is an integrating factor (IF). So, multiplying by the IF and integrating the general solution is given by $x\cdot \frac{{e}^{\frac{-1}{y}}}{{y}^{2}}=\int \frac{{e}^{\frac{-1}{y}}}{{y}^{2}}dy+c={e}^{\frac{-1}{y}}+c$ Or, $x={y}^{2}\left(1+c{e}^{\frac{1}{y}}\right)$.


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# If its possible to derive the public key from a private key, why can't we go in reverse?

I'm looking at source code for BitcoinJ that derives a public key from the private key.

  /**
* Create the public key from the private key
*
* @param       privKey             Private key
* @param       compressed          TRUE to generate a compressed public key
* @return                          Public key
*/
private byte[] pubKeyFromPrivKey(BigInteger privKey, boolean compressed) {
ECPoint point = ecParams.getG().multiply(privKey);
return point.getEncoded(compressed);
}

Since "G" is known, and published for each given curve, what prevents someone from dividing (or multiple consecutive subtractions) to determine the private key?

If possible, would you explain it in layman's terms, and also provide additional concepts I can research the relevant background.

-
The short answer is that there is no (known) efficient algorithm that can solve $A = a G$ for $a$, even when $A$ and $G$ are known. –  CodesInChaos Feb 11 '14 at 16:26
@CodesInChaos Thanks. From my algebra background I see A = aG as simple variables. Is the difficulty inherent in the fact we're looking at a multiplicative cyclic group? My intent with this question is to study the simplest approaches to key recovery with crypto the following categories: 1: integer factorization, 2: discrete logarithm, 3: ECC discrete logarithms. –  makerofthings7 Feb 11 '14 at 16:34
I'm not sure if this kind of key recovery applies to the 3 scenarios. Again this is all self study so if you have a better path to learning, please do share. –  makerofthings7 Feb 11 '14 at 16:35
I used additive notation. If you prefer multiplicative notation you'd say it's hard to solve $A=G^a$ for $a$. This problem (with either notation) is the discrete logarithm problem, which is believed to be hard on (the commonly used) finite fields and elliptic curves). DLP is easy in some groups and hard in others. We use those where it's hard, because else this kind of crypto would be trivially broken. –  CodesInChaos Feb 11 '14 at 16:37
@makerofthings7 DLOG and ECC DLOG are conceptually the same Problem. It is 'logarithms in groups'. You can see an elliptic curve as a fancy way to specify a group. As this group is abelian it is common to denote it additively. –  gmoktop Feb 11 '14 at 18:41

In the basic finite fields, however, people use mostly multiplicative notation to imply, that the DLOG problem is hard, because the equivalent problem based on the addition of numbers is quite easy (given numbers $a$ and $b$, everyone can calculate $a\cdot b$).


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by

# S520, up through 2/1

Here’s some more notes for you, again in PDF format.  $LaTeX$ is actually getting kind of fun, now that I’m figuring it out.  I’ve even figured out how to draw simple diagrams and the like from within $LaTeX$, so I don’t have to keep switching back and forth between programs, which is kind of nice.  Anyway, here you go.

Notes from January 25

Notes from January 27

Notes from January 29

Notes from February 1


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Research article| Volume 9, ISSUE 2, e13358, February 2023
• PDF [1 MB]PDF [1 MB]
• Top

# High sensitivity temperature sensor based on polymer-liquid modified anti-resonant reflecting guidance in silica capillary

Open AccessPublished:January 31, 2023

## Abstract

A high sensitivity temperature sensor based on polymer and liquid modified anti-resonant reflection principle is proposed. The sensor is made of a section of liquid-filled and UV-curable adhesive coated silica capillary, sandwiched between single mode fibers. The guiding mechanism of the waveguide is affected by both the liquid inside the hollow core, and the adhesive outside the silica cladding. The location of the resonant dips are influenced by the refractive index (RI) of the liquid, while the fringe visibility (resonant strength) is tuned by the both the liquid and the adhesive. By carefully selecting the refractive index value of the infiltrated liquid, and controlling the curing time of the adhesive, sharp resonant dips can be obtained. The sensor exhibits very high temperature sensitivity against temperature. A high sensitivity of 3.369 nm/°C is achieved with an infiltration index of 1.39, in the temperature range from 25 °C to 60 °C. The detection limit of the proposed sensors are 0.04 °C. Moreover, the temperature sensitivity of the sensor is found to be decreasing with the order of resonance. The sensor is easy to fabricate and cost effective, and is applicable in both dry and wet environments, which can find wide applications in biochemical and pharmaceutical industry.

## 1. Introduction

Fiber-optic temperature sensors offer unique advantages, such as immunity to electromagnetic interferences, repeatability, high stability, high resolution, and fast response. Measurement of temperature using fiber optics has been extensively investigated using various techniques, such as fiber gratings and fiber interferometers. Fiber Bragg gratings (FBG) [
• Lai Y.
• Zhou K.
• Sugden K.
• Bennion I.
Point-by-point inscription of first-order fiber Bragg grating for C-band applications.
,
• Li Y.
• Chen W.
• Wang H.
• Liu N.
• Lu P.
Bragg gratings in all-solid bragg photonic crystal fiber written with femtosecond pulses.
,
• Koo K.P.
• Kersey A.D.
Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing.
] have attracted much attention due to wavelength multiplexing features, but they suffer from relatively low temperature sensitivities (∼0.01 nm/°C). Long-period fiber gratings (LPFG), which exhibit higher sensitivity (∼0.1 nm/°C) as a temperature sensor, however have undesirable changes in the spectral response due to the high bending sensitivity [
• Bhatia V.
• Vengsarkar A.M.
Optical fiber long-period grating sensors.
]. Other temperature sensors based on fiber interferometers, such as thin polarization maintaining fiber (PMF) based Sagnac loop [
• Zhang G.
• Wu X.
• Zhang W.
• Li S.
• Shi J.
• Zuo C.
• Fang S.
• Yu B.
High temperature vernier probe utilizing photonic crystal fiber–based Fabry–Perot interferometers.
,
• Zhang G.
• Wu X.
• Li S.
• Liu W.
• Fang S.
• Zuo C.
• Zhang W.
• Yu B.
Miniaturized Fabry–Perot probe utilizing PMPCF for high temperature measurement.
,
• Ge Q.
• Zhu J.
• Cui Y.
• Zhang G.
• Wu X.
• Li S.
• Wang H.
• Yu B.
Fiber optic temperature sensor utilizing thin PMF based Sagnac loop.
], have relatively lower sensitivities. Photonic Crystal fibers generally have periodic air holes running through the entire fiber length, in which various materials can be filled to achieve different sensing functions [
• Amiri I.S.
• Paul B.K.
• Ahmed K.
• Aly A.H.
• Zakaria R.
• Yupapin P.
• Vigneswaran D.
Tri-core photonic crystal fiber based refractive index dual sensor for salinity and temperature detection.
,
• Alam M.S.
• Akter S.
• Paul B.K.
• Ahmed K.
• Vigneswaran D.
• Aktar M.N.
FEM based highly sensitive dual core temperature sensor: design and analysis.
,
• Jabin M.A.
• et al.
Design and fabrication of amoeba faced photonic crystal fiber for biosensing application.
,
• Ahmed K.
• et al.
Tetra-core surface plasmon resonance based biosensor for alcohol sensing.
,
• Jabin M.
• et al.
Titanium-coated dual-core D-shaped SPR-based PCF for hemoglobin sensing.
,
• Rifat A.A.
• Ahmed K.
• Paul B.K.
• Ahmed R.
Development of photonic crystal fiber-based gas/chemical sensors.
,
• Ahmed K.
• et al.
Refractive index-based blood components sensing in terahertz spectrum.
]. Introducing material into fiber structures is an effective method to improve the temperature sensitivity of optical fiber sensors [
• Wang H.
• Liao M.
• Xiao H.
• Han X.
• Jiang Y.
• Tan J.
• Zhang P.
• Shao J.
• Tian Y.
• Yang J.
High sensitivity temperature sensor based on a PDMS-assisted bow-shaped fiber structure.
,
• Hasan M.M.
• Taher H.J.
• Mohammed S.A.
Highly sensitive fiber-optic temperature sensor based on tapered no-core fiber for biomedical and biomechanical applications.
,
• Wang S.
• Yang Y.
• Niu P.
• Wu S.
• Liu S.
• Jin R.-B.
• Lu P.
• Hu X.
• Dai N.
Fiber tip Michelson interferometer for temperature sensing based on polymer-filled suspended core fiber.
,
• Zhao J.
• Zhao Y.
• Lv R.-Q.
• Chen M.-Q.
• Hu X.-G.
• Zhao Q.
A fiber ring cavity laser temperature sensor based on polymer-coated No-core fiber as tunable filter.
,
• Han T.
• Liu Y.
• Wang Z.
• Guo J.
• Wu Z.
• Wang S.
• Li Z.
• Zhou W.
Unique characteristics of a selective-filling photonic crystal fiber Sagnac interferometer and its application as high sensitivity sensor.
,
• Liang H.
• Zhang W.
• Geng P.
• Liu Y.
• Wang Z.
• Guo J.
• Gao S.
• Yan S.
Simultaneous measurement of temperature and force with high sensitivities based on filling different index liquids into photonic crystal fiber.
,
• Han T.
• Liu Y.
• Wang Z.
• Wu Z.
• Wang S.
• Li S.
Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber.
,
• Zhao C.
• Wang Z.
• Zhang S.
• Qi L.
• Zhong C.
• Zhang Z.
• Jin S.
• Guo J.
• Wei H.
Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer.
,
• Yang M.
• Wang D.N.
• Wang Y.
• R Liao C.
Fiber in-line Mach–Zehnder interferometer constructed by selective infiltration of two air holes in photonic crystal fiber.
]. By utilizing the high thermo-optic coefficient of the materials, the temperature sensitivity of these sensors usually increases over an order of magnitude compared with those sensors without the materials using the same sensing principle [
• Qian W.
• Zhao C.
• Wang Y.
• Chan C.
• Liu S.
• Jin W.
Partially liquid-filled hollow-core photonic crystal fiber polarizer.
]. For example, by infiltrating a solid core photonic crystal fiber (PCF) with liquids of the high index value, bandgap-like effect is introduced to the fiber, so that very high temperature sensitivity is achieved [
• Peng Y.
• Hou J.
• Zhang Y.
• Huang Z.
• Xiao R.
• Lu Q.
Temperature sensing using the bandgap-like effect in a selectively liquid-filled photonic crystal fiber.
]. By selectively filled the cladding hole of a simplified hollow core PCF with alcohol to modify the anti-resonant reflecting guidance of the fiber, and a temperature sensitivity of −0.46 nm/°C is obtained [
• Liu S.
• Hou M.
• Guo J.
• Li Z.
• Lu P.
Anti-resonant reflecting guidance in alcohol-filled hollow core photonic crystal fiber for sensing applications.
]. However, infiltrating liquids into a specific air hole of a PCF usually requires complicated process like femtosecond laser assisted drilling to selectively open the target air hole, which increases the cost of the sensors.
In our previous work, the transmission mechanism of a silica capillary is altered by introducing liquid inside and outside the silica capillary, and high sensing performance is achieved [
• Niu P.
• Liu S.
• Cheng W.
Liquid modified anti-resonant reflecting guidance in silica capillary for sensing applications.
]. However, liquid environment is required for the operation of the sensor, which restricts its application in dry fields. In this work, we presents a highly sensitive temperature sensor based on a modified fiber structure. The sensor consists of a section of silica capillary spliced between single mode fibers. The capillary is filled with refractive index matching liquids (RIML), and is also coated with UV-curable adhesive. The RIML together with the adhesive modifies the anti-resonant reflecting guiding property of the silica capillary by changing the resonant wavelength and resonant strength. Experimental results show that the transmission spectrum of the sensor exhibits redshift response when the ambient temperature rises, and a sensitivity of 3.369 nm/°C is achieved with a device filled with the liquid index value of 1.39, in the temperature range of 25 °C–60 °C. The influence of the index value of the infiltrate liquids on the sensitivity is also investigated. The proposed temperature sensor features a simple fabrication process, low cost and high sensitivity, which can find wide applications in the biochemical and pharmaceutical industry, where high precision temperature measurement is needed.

## 2. Fabrication and principle of the fiber sensor

The silica capillary (Polymicron Ltd) we used to construct the sensor has an outer diameter of 125 μm, and a hollow core diameter of 75 μm. To infiltrate the silica tube with refractive index matching liquid, one end of the silica tube is immersed in the liquid. Due to the capillary effect, the liquid quickly fills the silica capillary. A liquid with a refractive index of 1.39 was chosen from a series of RIML developed by Cargille Lab for the fabrication of the sensor. The index value of the chosen liquid is close to that of the cured UV adhesive, which could help to improve the spectrum quality of the device. After the infiltration, the liquid-filled capillary is spliced to single mode fibers at both ends with a fusion splicer (Fujikura 80s), the splicing parameter of which is optimized to minimize the deformation of the hollow core at the splicing joints. It can also be seen from Fig. 1(a) that there is a small amount of liquid evaporation at the arc discharge splice. Then the capillary is dipped in a UV-curable adhesive (NOA13685, Norland Products, Inc) and then exposed to UV light for 20 min to make sure that the sharp resonant dips can be obtained. The refractive index (RI) of the adhesive is 1.3685 before cured, and the energy density required for full curing is 6 J/cm2. Fig. 1(b) illustrates the schematic structure of the proposed sensor. The transmission spectrum of the sensor is collected by a broadband source and an optical spectrum analyzer (Yokogawa 6370C).
For an empty silica capillary (not infiltrated or coated), the guiding principle of the waveguide is based on the anti-resonant reflecting mechanism. The ring cladding of the silica capillary is considered as an F–P etalon, and waves that satisfy the resonant condition of the etalon will leak out of the cladding, while those that do not satisfy the resonant condition can be guided through the hollow core [
• Liu S.
• Tian J.
• Wang S.
• Wang Z.
• Lu P.
Anti-resonant reflecting guidance in silica tube for high temperature sensing.
]. However, when RIML (RI = 1.39) is filled into the hollow core of the silica capillary, the resonant condition of the etalon is altered, leading to the change of the resonant wavelength, as well as the resonant strength. The anti-resonant reflecting fringe is not much obvious compared with that of the empty capillary. This is because the reflection coefficient at the liquid-silica interface is much smaller compared to the air-silica interface, which weakens the strength of the resonance. The light intensity at resonant wavelength can be expressed as [
• Liu S.
• Tian J.
• Liu N.
• Xia J.
• Lu P.
Temperature insensitive liquid level sensor based on anti-resonant reflecting guidance in silica tube.
]:
$Equation 1.$
(1)

where r and r’ respectively represent the reflection coefficients of the inner and outer surfaces of capillary cladding, and Iresonant is the input light intensity. Eq. (1) shows when r has a close absolute value as r’, the value of Tresonant is relatively small, and those lossy dips in the transmission spectrum are sharp.
As shown in Fig. 2(a), when we only filled the capillary with RIML, we could not distinguish the position of the resonant dip due to the large difference between the absolute values of the reflection coefficients r and r’ on the inner and outer walls of the capillary. To improve the contrast of the fringe, UV adhesive is introduced to the outer surface of the capillary, so as to bring balance to the absolute values of r and r’. Before curing, the index value of the adhesive is 1.3685. In this case, the absolute value of r’ is still smaller than that of r, and there is not much improvement in the spectrum, as seen in Fig. 2(b). However, with the increase of curing time, the refractive index of UV adhesive is increased to a higher value. The increase of the RI can be explained by the volume shrinkage of the UV-curing adhesive, which originates from the shortening of inter-atomic spacing [
• Liu S.
• Cao S.
• Zhang Z.
• Wang Y.
• Liao C.
• Wang Y.
Temperature sensor based on side-polished fiber SPR device coated with polymer.
]. As shown in Fig. 3, the main component of adhesive is acrylate. During polymerization, the chemical bond between acrylate monomer molecules changes from van der Waals force to covalent bond, which leads to the closer distance between molecules [
• Liu S.
• Cao S.
• Zhang Z.
• Wang Y.
• Liao C.
• Wang Y.
Temperature sensor based on side-polished fiber SPR device coated with polymer.
]. The density of the adhesive increases as the molecules get closer, and leads to higher index value.
In order to investigate the effect of UV curing time on the transmission spectrum of the samples, the samples were cured for 60 min and the data were recorded every 10min interval. Fig. 4 shows the change of the transmission spectrum of the sample near 1400 nm band with the increase of UV curing time. It can be found from Fig. 4 that the resonant dip becomes more and more sharp within 20min of UV curing, but becomes gentle as the UV curing time continues to increase. This is because the refractive index of UV adhesive increases with the increase of curing time, and the refractive index is closest to 1.39 when curing for 20min. It is worth noting that from Fig. 4, we can find that the position of the resonant peak has shifted, which is because the ambient temperature of the sensor has slightly increased under the ultraviolet lamp irradiation for a long time. For the above reasons, we controlled the UV curing time at 20min in order to obtain sharp resonant dip in the transmission spectrum. In this way, the resonant strength is strong in the silica cladding, and the fringe contrast is much better, as can be seen from Fig. 2(c).
Fig. 5(a) gives the cross section illustration of the modified waveguide, and Fig. 5(b) gives the optical path of the modified anti-resonant reflecting guidance. The location of the resonant wavelength can be given by,
$Equation 2.$
(2)

where n1 and n2 are the refractive indices of the RIML and silica, and m is the resonance order, respectively. The thickness of the ring cladding is 25 μm. By taking these values into Eq. (2), and also consider the influence of dispersion on the value od n2, the resonant wavelength in the measurement range can be calculated to be at 1254 nm, 1324 nm, and 1402 nm. There is difference between the theoretical value and experimental value. This is because the values of n1 and the thickness d might still not be so accurate comparied with their real values, which causes the difference.

## 3. Sensing performance of the fiber sensor

Fig. 6 shows the temperature sensing set-up for our proposed sensor. Broadband source (BBS) and spectral analyzer (OSA) were used to monitor the output spectrum of the sensor. The sensor's thermal response is investigated by employing a heating and cooling treatment with a column oven (LCO 102). The manufactured sensor is put into the oven to monitor the transmission spectrum under different temperatures. The temperature is increased from room temperature to 60 °C, with a step of 5°. Each temperature is kept for 10 min for stabilization before recording the spectrum. Then the sensor is cooled down to room temperature in the same way as the heating process. Fig. 7(a) shows the transmission spectra of the sensor at different temperatures during the heating process.
The transmission dips are found to shift toward longer wavelength as temperature rises. The wavelength responses of the heating and cooling process are plotted in Fig. 7(b). The sensor exhibits a very linear response against temperature, and achieves a temperature sensitivity of 3.369 nm/°C. The temperature sensitivity of the sensor is given as,
$Equation 3.$
(3)

where C1 and C2 are the thermo-optic coefficients of RIML and silica. The value of C1 and C2 are −3.44 × 10−4 and 1 × 10−5, respectively. Equation (3) indicates that the temperature sensitivity is mainly affected by the thickness of the capillary(d), the index value of the infiltrated liquid (n1) and the thermal-optic coefficient of the liquid, as well as the order of the resonance (m). The thermal response of different resonance orders is plotted in Fig. 8. The temperature sensitivities for m = 16, 17 and 18 are 3.369 nm/°C, 3.165 nm/°C and 2.941 nm/°C, respectively. The sensitivity decreases with resonance orders, which is in agreement with Eq. (3). The melting point of the adhesive restricts the measurement range of the sensor, however, comparied with other sensor devices based on similar scheme (introduction of liquids), the measurement range of which is also small, our sensitivity relatively higher, and the structure is simple and cost effective.
The sensitivity of a sample does not fully quantify the performance of a sensor. The detection limit [
• Liu S.
• Tian J.
• Wang S.
• Wang Z.
• Lu P.
Anti-resonant reflecting guidance in silica tube for high temperature sensing.
] represents the smallest amount of change that a sensor can detect, which is be defined as [
• Wu D.K.C.
• Kuhlmey B.T.
• Eggleton B.J.
Ultrasensitive photonic crystal fiber refractive index sensor.
],
$Equation 4.$
(4)

For the monitored dip, the full width at half maximum (FWHM) is 3.59 nm. The resolution of the OSA is 0.02 nm. Assuming a signal-to-noise ratio of 50 dB, the detection limit is estimated to be 0.04 °C by Eq. (4).
In order to test whether the designed sensor is affected by the external environment, we put the prepared sensor in the refractive index matching liquids with the different index values (1.3, 1.4, 1.5 and 1.6), to observe the change of the transmission spectrum. As shown in Fig. 9(a), the change of external refractive index has little influence on the location of the resonant dip in the transmission spectrum. This is because the UV adhesive after curing isolate the waveguide from the surrounding wet environment. The stability of the sensor is also tested by recording the sensor spectrum in time scale of 24 h. Fig. 9(b) gives the spectrum at different time. The location of the transmission dips is quite stable during the test. The repeatability of the sensor is performed with three temperature tests on the same sample under the same measurement conditions, as shown in Fig. 10(a)∼(c). The results show good repeatability of the temperature sensitivity with an average error of less than 3.6％.

## 4. Conclusions

In conclusion, we reported a highly sensitive temperature sensor based on modified anti-resonant reflecting guidance mechanism in silica capillary. The infiltration of liquids into the hollow core of the silica capillary alters the resonant condition of the waveguide, and causes the resonant wavelength to change. On the other hand, by coating the infiltrated capillary with UV cured adhesive, the reflectivities at both interfaces of the silica cladding is closer to each other, which enhances the resonant strength, and leads to improvement of the fringe visibility. The device is found to be exhibiting very high sensitivity against temperature. The transmission dips shift toward longer wavelength as temperature rises. Linear wavelength response is observed during the heating and cooling process, and a temperature sensitivity of 3.369 nm/°C is obtained from room temperature to 60 °C. The detection limit of the proposed sensor is 0.04 °C. The sensitivity is also found to be related to the resonant order. The proposed sensor is cost effective, easy to fabricate, with good reproducibility and high sensitivity, and is applicable in both dry and wet environments, which can be applicated in fields like biochemistry industry and pharmacy industry.

## Author contribution statement

Wei Cheng: performed the experiments; wrote the paper.
Shuhui Liu: conceived and designed the experiments; analysis and interpreted the data.
Panting Niu: contributed reagents, materials, analysis tools or data; analyzed and interpreted the data.
Ting Chen: analyzed and interpreted the data.

## Funding statement

Shuhui Liu was supported by National Natural Science Foundation of China [12004290; 51909195], Hubei Provincial Natural Science Foundation of China [2020CFB251], Scientific Research Project of Education Department of Hubei Province [Q20181501; Q20191512], State Key Laboratory of Advanced Optical Communication Systems and Networks [2020GZKF007], Headmaster Foundation of Wuhan Institute of Technology [2020133].

## Data availability statement

Data will be made available on request.

## Declaration of interest’s statement

The authors declare no competing interests.

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text:
Friday 4 March, 12:00 – 13:00, M113

In this talk we discuss minimisation problems in $L^\infty$ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Via the method of $L^p$ approximations as $p\to \infty$, we illustrate the existence of constrained minimisers in $L^p$ for all large $p< \infty$, which in turn converge to a desired $L^\infty$ minimiser as $p\to \infty$. We further show that the special $L^\infty$ minimiser we construct solves a divergence PDE system which involves certain auxiliary measures as coefficients. This can be seen as a divergence form counterpart of the Aronsson-Euler PDE system which is associated with the $L^\infty$ constrained minimisation problem. This talk is based on joint work with Nikos Katzourakis.

Ed Clark (PhD student)


text:
# What would a black hole look like?

1. May 25, 2007

### MonstersFromTheId

If you were close enough to see one with the naked eye, what would you see?

Would you see a "black hole" in the center of the accretion disk?

Or would any view of the black hole itself be completely obscured by light emitted by matter near the black hole as it's torn to shreds by tidal forces?

Would you be able to see the jets at the poles with the naked eye?

Would intense gravitational lensing distort or color stars in the background?

Does anyone know of any artistic renderings of what a black hole would look like to the naked eye, that lean more toward scientific accuracy and less toward dramatic Hollywood glitz?

2. May 25, 2007

### Chris Hillman

The view from near a black hole

Ah good, you want astrophysical accuracy--- be careful, since many websites which purport to show visualizations are highly misleading in various ways. Sites from respectable physicists are more likely to be accurate--- or rather, to announce their limiting assumptions up front. I tried to collect some examples here:

http://math.ucr.edu/home/baez/RelWWW/HTML/visual.html

This rather depends upon whether you have in mind a specific astrophysical object such as the supermassive black hole near the center of our galaxy, or an idealized isolated black hole, with only "distant stars" providing background light. Even the simplest model, the Schwarzschild vacuum (nonrotating isolated hole) provides spectacular effects close up (strong field gravitational lensing), which are sufficiently complicated to explain that you should probably begin with that.

You'd see a dark disk against whatever background (I'd suggest not worrying about accretion disks at first). This does not represent the horizon but the locus $r=3m$ sometimes called the "photon sphere", because this is the location of unstable circular photon orbits. Just around this dark disk you'd see amplified and smeared images of background galaxies and stars and around that some concentric rings of n-ary, (n-1)-ary, .. 2-ary images of stars whose primary images are seen somewhere else on your celestial sphere.

As I said, the dark disk is the locus $r=3m$, not $r=2m$. But you can see glowing matter falling into the hole--- even the "back face" since light rays this close to a hole wind around it several times before escaping to infinity (if indeed they do escape). But you'd see this falling matter rapidly redshift out of visibility at it nears the hole.

Light bending is independent of frequency, but light from distant stars would tend to be blueshifted as it falls toward you "from infinity".

At this point, you should ask: how does the picture I sketched change for observers orbiting the hole? For observers who fall into it? The details depend sensitively upon the motion of the observer, but some simple special cases are studied in detail in Andrew Hamilton's website (see the link at the above cited web page).

AFAIK, few serious attempts to render scientifically accurate pictures of realistic black holes have been done; the closest I am aware of is an attempt to visualize how any accretion disk would appear warped and how different parts would appear frequency shifted. But there are so many things to consider (not least: human color vision is quite tricky to model!) that nothing I have seen seems sufficiently careful. But I am suprised that (apparently) noone has commissioned a state-of-the-art IMAX simulation with all the bells and whistles.

3. May 29, 2007

### MonstersFromTheId

Thanks so much Chris!
Some of those applets within the links on the first site you mensioned are really cool.

"But I am suprised that (apparently) no one has commissioned a state-of-the-art IMAX simulation with all the bells and whistles."

Ya know?
And the thing is I have a very hard time figuring out why that's the case.

I can see it in regard to things like asteroid fields in entertainment, at least to an extent. To entertain you need a bit of drama, thus "Star Wars" style asteroid fields so thick with debris it's a hair raising ride racing through one.

But with black holes, (not to mention a whole host of other astronomical features) it seems to me that, reality is likely to provide a whole heck of a lot more in the way of dramatic spectacle than the wildest imaginings of the very best concept artists.

I've got high hopes for THE UNIVERSE which premiers tonight. Guess we'll see if they get most of the details right (in fact I should post a thread to remind people that it's on tonight). But as far as I've been able to find on the show's official website, the only rendering they've got is of a supermassive black hole in the center of a galaxy. Cool, but a bit of a far off shot considering it's done from the point of view of someone well outside the galaxy rendered.

4. May 29, 2007

### chroot

Staff Emeritus
Excellent post, Chris.

I'd actually suspect that a ray-tracing program like Povray could be shoehorned into doing all the complex math for you. After all, the point of a ray-tracer is to literally trace the path of photons leaving a source and arriving at your eye.

A little Googling indicates this has already been done, to various degrees of accuracy. You'd be a much better judge of accuracy than me, Chris.

http://astro.ago.uni-lj.si/bh/pmwiki.php?n=Main.Lensing

http://www.tat.physik.uni-tuebingen.de/~tmueller/visual.html

- Warren

5. May 29, 2007

### cesiumfrog

The tubingen work is pretty good in this field (they've also got the advantage of great resources).

Regards "all the bells and whistles", I don't think jets are well enough understood yet, but maybe matter accretion is something that could be done soon. Hmm...

6. May 30, 2007

### Chris Hillman

Be very careful!

We are talking about ray tracing in curved spacetime, of course. If you try to go through this I think you'll find unexpected issues, particularly if you try to model the visual experience of an observer falling through the horizon. And there are other issues like frequency shifting, not to mention a reasonable simulation of an accretion disk and toroidal cloud dust if you want to include such things.

Thanks Warren--- the first site is new to me. And goody, it is about strong field lensing! (The sites I list at the current version of RelWW are concerned with the much more tractable but still very valuable, nontrivial and interesting weak-field lensing theory.) Check out the movie depicting the view of a rotating accretion disk from a vantage point near a Kerr hole at http://astro.ago.uni-lj.si/bh/pmwiki.php?n=Main.Accretion

It seems Cadez has published with de Felice, which is a good sign that this is based on careful work. (If that's not an oxymoron concerning anything run under Windows, an inherently unsafe/unstable OS--- good grief!) Still, I wouldn't trust anything unless I'd gone over it with a fine toothed comb. My experience is that whenever things are really complex, very very few authors can remember to keep all the relevant issues in play, and thus, unstated assumptions or even outright errors creep in.

Last edited: May 30, 2007
7. May 31, 2007


text:
# Basic group theory question on the index of subgroups

I'm doing some summer reading of Fraleigh's A First Course in Abstract Algebra and I came across this exercise in Section 10. I'm trying to prove the following theorem and I'm wondering if I can get some feedback on my proof. Note that $(G:H)$ refers to the index of $H$ in $G$.

Theorem: Suppose $H$ and $K$ are subgroups of the group $G$ such that $K \leq H \leq G$ and suppose that $(H:K)$ and $(G:H)$ are finite. Then $(G:K)$ is finite and $(G:K) = (G:H) (H:K)$.

Proof: First, define $\{a_iH | i = 1, 2, \cdots, r\}$ and $\{b_jK | j = 1, 2, \cdots s\}$ to be the collection of distinct left cosets of H in G and K in H, respectively. Note that since (H:K) and (G:H) are finite, we know that the size of these groups are finite and $r,s < \infty$. Then $r = (G:H)$ and $s = (H:K)$. Note that $\cup_{j=1}^s b_jK = H$, so $\{a_iH| i = 1, 2, \cdots, r\} = \{a_i(\cup_{j=1}^s b_jK)| i = 1, 2, \cdots, r\}$. Similarly we have that $\cup_{i=1}^r\left[a_i\left(\cup_{j=1}^s b_jK\right)\right] = G$. Note that for each $i$, we have $a_i\left(\cup_{j=1}^s b_jK\right)$ is distinct via assumption. Now suppose that there exists integers $1\leq m < n \leq s$ such that $a_ib_mK = a_ib_nK$. Since $a_i \in G$ and $G$ is a group, via group cancellation law we have $b_mK = b_nK$. This is a contradiction as we assumed the left cosets of $K$ are distinct. Thus for all $i,j$ we have $a_ib_jK$ are distinct. Therefore, as all are distinct and $\cup_{i=1}^r\cup_{j=1}^s a_ib_jK = G$, we see that $S = \{a_ib_jK | i=1,2,\cdots, r; j=1,2,\cdots, s\}$ is the collection of distinct left cosets of $K$ in $G$ and $(G:K) = |S|$ where $|S| = rs = (G:H)(H:K)$. Thus $(G:K)$ is finite and $(G:K)=(G:H)(H:K)$.

Any advice would be appreciated, thanks!

First, define $\{a_iH|i=1,2,\ldots,r\}$ and $\{b_jK|j=1,2,\ldots,s\}$ to be the collection of distinct left cosets of $H$ in $G$ and $K$ in $H$, respectively. Note that since $(H:K)$ and $(G:H)$ are finite, we know that the size of these groups are finite and $r,s<\infty$. Then $r=(G:H)$ and $s=(H:K)$.

I think this is a lot more throat-clearing than you need, and depending on precisely what you mean by the middle sentence might even be wrong. It's not true that if $(G:K)$ is finite the orders of $G$ and $K$ are finite (quick counter-example: $(\mathbb{Z}:2\mathbb{Z})=2$).

I don't know if you've reached quote groups yet, but if you did and you meant that $\lvert G/K\rvert=(G:K)$, well, you're right -- provided that $G/K$ exists, but it doesn't always exist (quick counter-example: There is no such thing as $S_3/S_2$).

All this is beside the point though, because the claim that $G$ and $K$ are finite could just as well have been omitted from your proof, and you're right to start with labelling the cosets of $K$ in $H$ and $H$ in $G$, and to stipulate $(G:H)=r$ and $(H:K)=s$.

Note that $\cup^s_{j=1}b_jK=H$, so $\{a_iH|i=1,2,⋯,r\}=\{a_i(\cup_{j=1}^sb_jK)|i=1,2,\ldots,r\}$. Similarly we have that $\cup^r_{i=1}[a_i(\cup_{j=1}^sb_jK)]=G$. Note that for each $i$, we have $a_i(\cup^s_{j=1}b_jK)$ is distinct via assumption. Now suppose that there exists integers $1\leq m<n\leq s$ such that $a_ib_mK=a_ib_nK$. Since $a_i\in G$ and $G$ is a group, via group cancellation law we have $b_mK=b_nK$. This is a contradiction as we assumed the left cosets of $K$ are distinct. Thus for all $i,j$ we have $a_ib_jK$ are distinct. Therefore, as all are distinct and $\cup^r_{i=1}\cup^s_{j=1}a_ib_jK=G$, we see that $S=\{a_ib_jK|i=1,2,\ldots,r;j=1,2,\ldots,s\}$ is the collection of distinct left cosets of $K$ in $G$ and $(G:K)=\lvert S\rvert$ where $\lvert S\rvert=rs=(G:H)(H:K)$. Thus $(G:K)$ is finite and $(G:K)=(G:H)(H:K)$.

This is correct, but as a matter of clarity I think it could benefit from more words and fewer symbols. Here's how I'd have made the same argument:

Take $g\in G$. The cosets of $H$ partition $G$, so we know $g$ is in exactly one of them, call it $a_iH$. To put $g\in a_iH$ another way, we might say there's some $h\in H$ satisfying $g=a_ih$. We can repeat this exact same argument with $h$ and $K$ partitioning $H$ to get $g\in a_i(b_jK)=(a_ib_j)K$. We have now put each element of $G$ in exactly one of the cosets $\{a_ib_jK|1\leq i\leq r,1\leq j\leq s\}$. Now $$\lvert\{a_ib_jK|1\leq i\leq r,1\leq j\leq s\}\rvert=rs\text,$$ so $(G:K)=rs=(G:H)(H:K)$ and is finite.

• Thanks for the comments! And yeah, that second line was completely a typo. I meant to say that the index is finite, and not that the group size was finite, my mistake. Thanks for the clarified argument, I was wondering if there was a more precise way to describe it and that does it. As for the quote groups, I don't believe I've seen that before. – LordAlgebro Jul 13 '18 at 5:59


text:
Dengue genetic divergence generates within-serotype antigenic variation, but serotypes dominate evolutionary dynamics

Bell SM, Katzelnick L, Bedford T. 2019. eLife 8: e42496.

# Abstract

Dengue virus (DENV) exists as four genetically distinct serotypes, each of which is historically assumed to be antigenically uniform. However, recent analyses suggest that antigenic heterogeneity may exist within each serotype, but its source, extent and impact remain unclear. Here, we construct a sequence-based model to directly map antigenic change to underlying genetic divergence. We identify 49 specific substitutions and four colinear substitution clusters that robustly predict dengue antigenic relationships. We report moderate antigenic diversity within each serotype, resulting in variation in genotype-specific patterns of heterotypic cross-neutralization. We also quantify the impact of antigenic variation on real-world DENV population dynamics, and find that serotype-level antigenic fitness is a dominant driver of dengue clade turnover. These results provide a more nuanced understanding of the relationship between dengue genetic and antigenic evolution, and quantify the effect of antigenic fitness on dengue evolutionary dynamics.


text:
# How to convert an infinite summation to an integral [closed]

I need help converting this summation to an integral

$$\sum_{i=1}^\infty 1+\frac{\vert(n-im)\vert}{(n-im)}$$

I keep trying but get stuck so any help is appreciated.

## closed as off-topic by Saad, John Omielan, trancelocation, RRL, Eevee TrainerApr 17 at 0:31

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, John Omielan, trancelocation, RRL, Eevee Trainer
If this question can be reworded to fit the rules in the help center, please edit the question.

The fraction is $$+1$$ or $$-1$$ depending on whether $$i \lt \frac nm$$ or $$i \gt \frac nm$$. If $$m$$ divides $$n$$ you have a term the divides by $$0$$ so the sum is undefined. Assuming you don't fall into the undefined trap, the summand is just $$2$$ as long as $$i \lt \frac nm$$, and $$0$$ after, so the sum is just $$2\lfloor \frac nm \rfloor$$. To convert it to an integral the function is just $$2$$ from $$0$$ to $$\lfloor \frac nm \rfloor$$ and $$0$$ after.


text:
# Article: Cellular non-linear network model of microbial fuel cell

“Cellular non-linear network model of microbial fuel cell”
Biosystems, Vol. 156–157, June–July 2017, pp. 53-62, DOI 10.1016/j.biosystems.2017.04.003.

Tsompanas M-A, Adamatzky A., Ieropoulos I., Phillips N., Sirakoulis G. Ch., Greenman J.

Abstract

A cellular non-linear network (CNN) is a uniform regular array of locally connected continuous-state machines, or nodes, which update their states simultaneously in discrete time. A microbial fuel cell (MFC) is an electro-chemical reactor using the metabolism of bacteria to drive an electrical current. In a CNN model of the MFC, each node takes a vector of states which represent geometrical characteristics of the cell, like the electrodes or impermeable borders, and quantify measurable properties like bacterial population, charges produced and hydrogen ions concentrations. The model allows the study of integral reaction of the MFC, including temporal outputs, to spatial disturbances of the bacterial population and supply of nutrients. The model can also be used to evaluate inhomogeneous con gurations of bacterial populations attached on the electrode bio lms.

Avalable through arXiv


text:
# Direct solution of linearized phonon Boltzmann equation¶

This page explains how to use the direct solution of LBTE by L. Chaput, Phys. Rev. Lett. 110, 265506 (2013) (citation).

## How to use¶

As written in the following sections, this calculation requires large memory space. When running multiple temperature points, simply the memory space needed is multiplied by the number of the temperature points. Therefore it is normally recommended to specify –ts option. An example to run with the direct solution of LBTE for example/Si-PBEsol is as follows:

% phono3py --dim="2 2 2" --sym-fc -c POSCAR-unitcell
...

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="11 11 11" --fc3 --fc2 --lbte --ts=300
...

=================== End of collection of collisions ===================
- Averaging collision matrix elements by phonon degeneracy [0.031s]
- Making collision matrix symmetric (built-in) [0.000s]
----------- Thermal conductivity (W/m-k) with tetrahedron method -----------
Diagonalizing by lapacke dsyev... [0.111s]
Calculating pseudo-inv with cutoff=1.0e-08 (np.dot) [0.001s]
#  T(K)        xx         yy         zz         yz         xz         xy
300.0     111.588    111.588    111.588      0.000      0.000     -0.000
(RTA)     109.009    109.009    109.009      0.000      0.000     -0.000
----------------------------------------------------------------------------

Thermal conductivity and related properties were written into
"kappa-m111111.hdf5".
Eigenvalues of collision matrix were written into "coleigs-m111111.hdf5"
_
___ _ __   __| |
/ _ \ '_ \ / _ |
|  __/ | | | (_| |
\___|_| |_|\__,_|

## Memory usage¶

The direct solution of LBTE needs diagonalization of a large collision matrix, which requires large memory space. This is the largest limitation of using this method. The memory size needed for one collision matrix at a temperature point is $$(\text{number of irreducible grid points} \times \text{number of bands} \times 3)^2$$ for the symmetrized collision matrix.

These collision matrices contain real values and are supposed to be 64bit float symmetric matrices. During the diagonalization of each collision matrix with LAPACK dsyev solver, around 1.2 times more memory space is consumed in total.

When phono3py runs with –wgp option together with --lbte option, estimated memory space needed for storing collision matrix is presented. An example for example/Si-PBEsol is as follows:

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="40 40 40" --wgp --lbte
...

Memory requirements:
- Piece of collision matrix at each grid point, temp and sigma: 0.00 Gb
- Full collision matrix at each temp and sigma: 7.15 Gb
- Phonons: 0.04 Gb
- Grid point information: 0.00 Gb
- Phonon properties: 0.14 Gb

...

With –stp option, estimated memory space needed for ph-ph interaction strengths is shown.

The other difficulty compared with RTA is the workload distribution. Currently there are two ways to distribute the calculation: (1) Collision matrix is divided and the pieces are distributed into computing nodes. (2) Ph-ph interaction strengths at grid points are distributed into computing nodes. These two can not be mixed, so one of them has to be chosen. In either case, the distribution is done simply running a set of phono3py calculations over grid points and optionally band indices. The data computed on each computing node are stored in an hdf5 file. Increasing the calculation size, e.g., larger mesh numbers or larger number of atoms in the primitive cell, large files are created.

### Distribution of collision matrix¶

A full collision matrix is divided into pieces at grid points of irreducible part of Brillouin zone. Each piece is calculated independently from the other pieces. After finishing the calculations of these pieces, the full collision matrix is diagonzalized to obtain the thermal conductivity.

File size of Each piece of the collision matrix can be large. Therefore it is recommended to use –ts option to limit the number of temperature points, e.g., --ts="100 200 300 400 500, depending on the memory size installed on each computing node. To write them into files, --write-collision option must be specified, and to read them from files, --read-collision option is used. These are similarly used as –write-gamma and –read-gamma options for RTA calculation as shown in Workload distribution. --read-collision option collects the pieces and make one full collision matrix, then starts to diagonalize it. This option requires one argument to specify an index to read the collision matrix at one temperature point, e.g., the collision matrix at 200K is read with --read-collision=1 for the (pieces of) collision matrices created with --ts="100 200 300 400 500" (corresponding to 0, 1, 2, 3, 4). The temperature (e.g. 200K) is also read from the file, so it is unnecessary to specify –ts option when reading.

The summary of the procedure is as follows:

1. Running at each grid point with –gp (or –ga) option and saving the piece of the collision matrix to an hdf5 file with --write-collision option. It is probably OK to calculate and store the pieces of the collision matrices at multiple temperatures though it depends on memory size of the computer node. This calculation has to be done at all irreducible grid points.
2. Collecting and creating all necessary pieces of the collision matrix with --read-collision=num (num: index of temperature). By this one full collision matrix at the selected temperature is created and then diagonalized. An option -o num may be used together with --read-collision to distinguish the file names of the results at different temperatures.

Examples of command options are shown below using Si-PBE example. Irreducible grid point indices are obtained by –wgp option:

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --lbte --wgp

and the information is given in ir_grid_points.yaml. For distribution of collision matrix calculation (see also Workload distribution):

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --fc3 --fc2 --lbte --ts=300 --write-collision --gp="grid_point_numbers..."

To collect distributed pieces of the collision matrix:

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --fc3 --fc2 --lbte --ts=300 --read-collision=0

### Distribution of phonon-phonon interaction strengths¶

The distribution of pieces of collision matrix is straightforward and is recommended to use if the number of temperature points is small. However increasing data file size, network communication becomes to require long time to send the files from a master node to computation nodes. In this case, the distribution over ph-ph interaction strengths can be another choice. Since, without using –full-pp option, the tetrahedron method or smearing approach with –sigma-cutoff option results in the sparse ph-ph interaction strength data array, i.e., most of the elements are zero, the data size can be reduced by only storing non-zero elements. Not like the collision matrix, the ph-ph interaction strengths in phono3py are independent from temperature though it is not the case if the force constants provided are temperature dependent. Once stored, they are used to create the collision matrices at temperatures. Using --write-pp and --read-pp, they are written into and read from hdf5 files at grid points.

It is also recommended to use –write-phonon option and –read-phonon option to use identical phonon eigenvectors among the distributed nodes.

The summary of the procedure is as follows:

1. Running at each grid point with –gp (or –ga) option and storing the ph-ph interaction strengths to an hdf5 file with --write-pp option. This calculation has to be done at all irreducible grid points.
2. Running with --read-pp option and without –gp (or –ga) option. By this one full collision matrix at the selected temperature is created and then diagonalized. An option -o num may be used together with --read-collision to distinguish the file names of the results at different temperatures.

Examples of command options are shown below using Si-PBE example. Irreducible grid point indices are obtained by –wgp option:

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --lbte --wgp

and the grid point information is provided in ir_grid_points.yaml. All phonons on mesh grid points are saved by:

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --fc2 --write-phonon

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --fc3 --fc2 --lbte --ts=300 --write-pp --gp="grid_point_numbers..." --read-phonon

Here one temperature has to be specified but any one of temperatures is OK since ph-ph interaction strength computed here is assumed to be temperature independent. Then the computed ph-ph interaction strengths are read and used to compute collision matrix and lattice thermal conductivity at a temperature by

% phono3py --dim="2 2 2" --pa="0 1/2 1/2 1/2 0 1/2 1/2 1/2 0" -c POSCAR-unitcell --mesh="19 19 19" --fc3 --fc2 --lbte --ts=300 --read-pp --read-phonon

This last command is repeated at different temperatures to obtain the properties at multiple temperatures.

## Cutoff parameter of pseudo inversion¶

To achieve a pseudo inversion, a cutoff parameter is used to find null space, i.e., to select the nearly zero eigenvalues. The default cutoff value is 1e-8, and this hopefully works in many cases. But if a collision matrix is numerically not very accurate, we may have to carefully choose the value by --pinv-cutoff option. It is safer to plot the absolute values of eigenvalues in log scale to see if there is clear gap between non-zero eigenvalue and nearly-zero eigenvalues. After running the direct solution of LBTE, coleigs-mxxx.hdf5 is created. This contains the eigenvalues of the collision matrix (either symmetrized or non-symmetrized). The eigenvalues are plotted using phono3py-coleigplot in the phono3py package:

phono3py-coleigplot coleigs-mxxx.hdf5

It is assumed that only one set of eigenvalues at a temperature point is contained.

## Installation of diagonalization solvers with multithreaded BLAS¶

Multithreaded BLAS is recommended to use for the calculation of the direct solution of LBTE since the diagonalization of the collision matrix is computationally demanding. A few examples of how to install multithreded BLAS libraries are presented below.

Scipy (also numpy) has an interface to LAPACK dsyev (scipy.linalg.lapack.dsyev). An MKL LAPACK linked scipy (also numpy) gives very good computing performance and is easily obtained using the anaconda package manager. In this choice, usual installation of LAPACKE is necessary for running dgesvd and zheev. When using anaconda, installing OpenBLAS is the easiest way to do. See OpenBLAS provided by conda (with multithread BLAS)

### OpenBLAS or MKL via LAPACKE¶

Using LAPACKE via python C-API is implemented. By this, phono3py can use LAPACK dsyev. This uses smaller memory space than using MKL linked scipy. Practically there are two choices, OpenBLAS and MKL. For MKL, proper installatin of the MKL package is necessary. The MKL library installed obtained from anaconda can not be used.

#### OpenBLAS¶

Use of OpenBLAS is an easy choice if the anaconda package is used. See OpenBLAS provided by conda (with multithread BLAS).

#### MKL¶

The BLAS multithread performance may be better in that in MKL. Using MKL-LAPACK via MKL-LAPACKE via python C-API is also implemented if the link is succeeded. See MKL LAPACKE (with multithread BLAS).

## Solver choice for diagonalization¶

For larger systems, diagonalization of collision matrix takes longest time and requires large memory space. Phono3py relies on LAPACK for the diagonalization and so the performance is dependent on the choice of the diagonalization solver.

Using multithreaded BLAS with many-core computing node, computing time may be well reduced and the calculation can finish in a realistic time. Currently scipy, numpy and LAPACKE can be used as the LAPACK wrapper in phono3py. Scipy and numpy distributed by anaconda are MKL linked, therefore MKL multithread BLAS is used through them. Multithreaded OpenBLAS is installed by conda and can be used via LAPACKE in phono3py. MKL LAPACK and BLAS are also able to be used via LAPACKE in phono3py with appropriate setting in setup.py.

Using --pinv-solver=[number], one of the following solver is chosen:

1. Lapacke dsyev: Smaller memory consumption than dsyevd, but slower. This is the default solver when MKL LAPACKE is integrated or scipy is not installed.
2. Lapacke dsyevd: Larger memory consumption than dsyev, but faster. This is not recommended because sometimes a wrong result is obtained.
3. Numpy’s dsyevd (linalg.eigh). This is not recommended because sometimes a wrong result is obtained.
4. Scipy’s dsyev: This is the default solver when scipy is installed and MKL LAPACKE is not integrated.
5. Scipy’s dsyevd. This is not recommended because sometimes a wrong result is obtained.

The solver choices other than --pinv-solver=1 and --pinv-solver=4` are dangerous and not recommend. They exist just for the tests.


text:
# Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems.

What is a good book to learn and practice this method?

would like:
- a book published after 2000
- not limited to a particular axiomset or set of connectives
- lots of examples.

What I especially would like is a book that teaches how to transform Natural Deduction or Sequent Calculi style proofs to axiomatic Hilbert Style proofs. (I know it is a complex subject, it depends on the axioms and is not even always possible)

At the moment I am studying Bergmann's "An introduction to many valued and fuzzy logic" that uses this style of proof just because other proof styles are either invalid or even more complicated for this type of proof

-
A good book for this topic is Kleene's "Introduction to metamathematics", but it is oldish and you may find the style difficult as an introduction. A good possibility is "Fundamentals of mathematical logic" by Peter Hinman, though the book covers much more than this topic. Or you may want to visit Peter Smith's webpage, Logic Matters. –  Andrés Caicedo Aug 12 '13 at 17:15
It might come as some use to consider thinking of each axiomatic system as having its own methods to some extent, since different Hilbert systems have different derivable rules of inference. Even when you can derive the same theorem set in principle, derivable rules of inference can change the ease with which a given theorem can get proved. E. G. for {CCpqCCqrCpr, CCNppp, CpCNpq} you can prove Cpp using the derivable rule "from C$\alpha$$\beta, C\beta$$\gamma$, infer C$\alpha$$\gamma" and making just one substitution. For {CpCqp, CCpCqrCCpqCpr, CCNpNqCqp} you need to work differently. – Doug Spoonwood Aug 13 '13 at 19:17 This question has lead me to look at L. H. Hackstaff's 1966 System of Formal Logic. A reviewer said "An excellent reference. Shows the properties of a great many formal logic systems (such as standard PC and intuitionist PC) by building them as extensions of simpler systems with more easily proven properties." Is the age of the book really a relevant consideration here? Does it affect the content or usability or general value of the book? Maybe it is, maybe I've missed something, I don't know. – Doug Spoonwood Aug 21 '13 at 5:08 If you send me your email address (mine is at my profile), I can send you a manuscript-in-progress of mine that may be of help. – Dave L. Renfro Aug 27 '13 at 18:25 "What I especially would like is a book that teaches how to transform Natural Deduction or Sequent Calculi style proofs to axiomatic Hilbert Style proofs." There exist different ways to do this, because there exist different ways to derive the rules needed for a deduction metateorem. If you use CpCqp and CCpCqrCCpqCpr to get to the rules \alpha \rightarrow C\beta$$\alpha$ and {C$\alpha$C$\beta$$\gamma, C\alpha$$\beta$} $\rightarrow$ C$\alpha$$\gamma, when changing a derivation of {C\psi$$\phi$, $\psi$} $\vdash$ $\phi$ into one of ... –  Doug Spoonwood May 13 '14 at 14:42

I've started studying L. H. Hackstaff's 1966 book Systems of Formal Logic. It looks useful for this purpose and seems to have fundamentally sound advice. Its systems don't have a rule of simultaneous substitution as a primitive rule of inference or involve axiom schema. If more systems did this it might actually make it possible to teach the axiomatic method in logic at a lower level than presently done.

It has sections on the axiomatic method which says things like:

1. Know (the major ones/some of the major ones) the derivable rules of inference of the logical systems by heart.
2. Know the axioms of the logical system by heart. Otherwise you'll have to refer to the axioms constantly when trying to construct a proof.
3. Know the rules of inference by heart.
4. Know the problem.
5. Know the evidence.

Unlike many other books I have seen it also has plenty of examples axiomatic proofs (with the abbreviations for wffs adopted by the text).

As an example of how to prove something using just one variable at a time substitution. Suppose our first thesis is that of suffixing or reverse hypothetical syllogism.

Our language here has only has variables {a, b, c, ..., f} and subscripted variables if necessary. Our meta-language will happen with {p, ..., z} so we can talk about rules of inference.

1 C Cab C Cca Ccb

If we used condensed detachment would tell us that the next most general result of this system will be C CdCab C d C Cca Ccb. Can we prove this using only single variable at a time substitution and detachment?

1 has form Cxy, and so does Cab. So, since we only have one thesis, it seems to try to make or find some form of CCabCCcaCcb such that we can detach something. How might we do this? Well the antecedent of 1 is Cab and the antecedent of Cab is a. So, we need to substitute something for a such that a has form Cxy. Would Cab work?

2? C CCabb C CcCab Ccb.

We want to have "b" in one have the form CCcaCcb. But, the rule of uniform substitution happen tells us we have to substitute for all propositional variables. So, now if we used 2 to substitute something, we'd have to substitute the b in the first Cab. But, then we'd have to make even more substitutions. Do we have a simpler path possible here?

Well, maybe we could adopt a procedural technique to vary substitutions such that when substituting for variables, more variables appear than in the original wff. So, let's try substituting a with Cad in 1. Then we obtain 2:

Now we want to have some form of 1 become the antecedent. So we want something like CCbaCbd. Oops, I didn't follow my own technique. I mean 2 b/CCeaCed yields 3

Now we want CCabCCcaCcb to match 3 (this exercise, surprising to me, I find kind of hard, because I know the form CCqrCCpqCpr by heart, but CCabCCcaCcb I don't know by heart). So, 1 b/d yields

Now 4 c/e yields 5

5 C Cad C Cea Cea

Now we can detach 6 from 3 and 5:

6 C CcCad C c C Cea Ced We want the following:

G C CdCab C d C Cca Ccb

So, 6 c/d yields

7? C CdCad C d C Cea Cdb

No, that won't work since substitutions have to happen uniformly. b does not appear in 6. So, if we substitute d for b first, we can still apply substitutions later. So, 6 d/b yields

7 C CcCab C c C Cea Ceb Now 7 c/d yields

8 C CdCab C d C Cea Ceb Now 8 e/c yields

9 C CdCab C d C Cca Ccb

This proof isn't unique since I might have made other substitutions first. Let's look at 3 again:

The Detachment Theorem gives us a tip or hint that this from a wff of the type CCadCCeaCed, another wff of the type CcCad, we may infer CcCeaCed. Do we have a more general rule? Let's back up to 1

1 C Cab C Cca Ccb

But this time let's substitute b with CCeaCed first. We then obtain

10 C CaCCeaCed C Cca CcCCeaCed.

Along with The Detachment Theorem this gives us a clue about more general rules of inference than 3 would. So, we have

CpCCqpCqr => CC s p C s CCqpCqr (s does not appear on the left-side of the statement of the rule)

CpCCqpCqr, Csp => CsCCqpCpr

CpCCqpCqr, Csp, s => CCqpCpr

CpCCqpCqr, Csp, s, Cqp => Cpr and

CpCCqpCqr, Csp, s, Cqp, p => r

-
Nice book it is from 1958 so older than i wanted and other than Doug describes it is in (normal) infix notation still it is the only book i know that comes close to what i wanted. –  Willemien Aug 27 '13 at 10:39

Are you looking for a book on Hilbert axiomatic systems in particular, or just a book on proof theory more generally? There are many different kinds of proof systems, apart from the axiomatic Hilbert-style systems. In particular, proof theory benefits most from the sequent calculus presentation. If you're only interested in Hilbert systems, then I don't know of any resources that focus exclusively on that. If you're just interested in proof theory, there's an array of books that would suit most of your needs.

The thing about proof theory books is that, most of the time, they focus mostly on two logics: classical and intuitionistic. Those, it seems, have been the most influential in developing the course of proof theory. Still, to understand proof theory for other logics, you need to understand some of the basics for proof theory in classical and intuitionistic logic first. Negri and von Plato's Structural Proof Theory is a good place to start. Takeuti's Proof Theory develops proof theory from a different angle, focusing on arithmetic and consistency proofs. For something a little faster paced, look at Troelstra and Schwichtenberg's Basic Proof Theory. All of these are written post-2000.

If you've got some understanding of sequent calculi, then the book you're looking for is definitely Greg Restall's An Introduction to Substructural Logics. This is a fantastic book (one of my favorites actually), and I keep going back to it again and again. Only the first 7 chapters talk about proof theory, but he talks about proof systems with fewer connectives, nonclassical proof systems, modal proof systems, etc., and gives plenty of good examples. The cool thing about the substructural viewpoint is that it's an amazingly elegant way to characterize a lot of nonclassical and many-valued logics in one fell swoop. Again though, to fully appreciate it, you should be sure you know some sequent calculus before reading this book.

-
No I am looking especiallyfor a book on hilbert style axiomatic proofs (which is good for either classical , intuitionistic and relevant logics) –  Willemien Aug 12 '13 at 19:37
Ah, alright then. Well, all these proof systems are good for any of those logics as well, it's just Hilbert systems are usually hard to manipulate. An easy natural deduction or sequent proof can very easily become a long and tedious Hilbert proof. I'd be interested to see if there is a book focusing exclusively on techniques for dealing with Hilbert systems though. –  Alex Kocurek Aug 12 '13 at 20:04
Before I made this post and saw your question math.stackexchange.com/questions/374937/… I tried to make a Answer but realised it was all more complex then I assumed and my answer was far from complete (so never published) and that led to this question –  Willemien Aug 13 '13 at 8:16

Try Cori & Lascar "Mathematical Logic" or check out a course book of some University.

-
I looked at amazon to the index of this book and it looks like the book has only a very few pages on proofs and proofconstruction themselves (pages 193-200) or am i mistaken? –  Willemien Aug 12 '13 at 9:32
@Willemien No, you are not mistaken. This book does not cover the material you are asking for in any depth. In fact, this is one of the points of their approach. –  Andrés Caicedo Aug 12 '13 at 17:04

A classic presentation of logic via an axiomatic presentation -- rather than, say, via a natural deduction system -- is Elliot Mendelson's Introduction to Mathematical Logic (5th edition 2009). This is the book I've always recommended to students wanting to see an old-school axiomatic system being put to work.

The first edition of Mendelson's book is almost 50 years old, however. For a modern book which is axiom-based you could try the admirable Christopher C. Leary’s A Friendly Introduction to Mathematical Logic (Prentice Hall, 2000).

-
@Peter_Smith , thanks for your sugestions, but both books are more about proofs and what it all means(mathematical logic) than about how to make proofs and how to prove theorems, thinking about what I would like: it would be something like "Logic primer " by Allan and Hand, mitpress.mit.edu/books/logic-primer but then for Hilbert style axiomatic logic. also maybe I should let go of the publication date limit. –  Willemien Aug 23 '13 at 10:21


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# ICPAQGP-2010

5-10 December 2010
6th International Conference on Physics and Astrophysics of Quark Gluon Plasma (ICPAQGP 2010)
Home > Timetable > Contribution details

Thermal photons in QGP and non-ideal effects

Content: We investigate the thermal photon production-rates using one dimensional boost-invariant second order relativistic hydrodynamics to find proper time evolution of the energy density and the temperature. The effect of bulk-viscosity and non-ideal equation of state are taken into account in a manner consistent with recent lattice QCD estimates. It is shown that the \textit{non-ideal} gas equation of state i.e $\varepsilon-3\,P\,\neq 0$ behaviour of the expanding plasma, which is important near the phase-transition point, can significantly slow down the hydrodynamic expansion and thereby increase the photon production-rates. Inclusion of the bulk viscosity may also have similar effect on the hydrodynamic evolution. However the effect of bulk viscosity is shown to be significantly lower than the \textit{non-ideal} gas equation of state. We also analyze the interesting phenomenon of bulk viscosity induced cavitation making the hydrodynamical description invalid. It is shown that ignoring the cavitation phenomenon can lead to a very significant over estimation of the photon flux. It is argued that this feature could be relevant in studying signature of cavitation in relativistic heavy ion collisions.
Id: 144
Place:
Room: Main Auditorium
Starting date:
 --not yet scheduled--
Duration: 00'
Primary Authors: Mr. V., Sreekanth (Physical Research Laboratory)
Co-Authors: Prof. BHATT, Jitesh (Physical Research Laboratory)
Mr. MISHRA, Hiranmaya (Physical Research Laboratory)
Presenters: Mr. V., Sreekanth


text:
# Differentiating x^y + y^x + (lnx)^x etc

1. Apr 26, 2004

### pig

How can this be done?

I don't even know how I would begin.. How would you differentiate stuff like x^(y^(x^y))? Where y is a function of x, not a constant of course..

2. Apr 26, 2004

### arildno

The example:
x^(y^(x^y))=x^(z(x))

d/dx x^(z(x))=z(x)x^(z(x)-1)+x^(z(x))*ln(x)*z'(x)..

3. Apr 26, 2004

### pig

I don't understand this :(

d/dx x^(z(x))=z(x)x^(z(x)-1)+...

Where does the + come from?

I know I should use the chain rule somehow but I can't seem to figure out how.. I'm having problems with differentiating both x^(f(x)) and f(x)^x..

4. Apr 26, 2004

### pig

Hmm if I do this:

f(x) = x^y
ln f(x) = ln x^y
ln f(x) = ylnx
(ln f(x))' = (ylnx)'
f'(x)/f(x) = y'lnx + y/x
f'(x) = f(x)*(y'lnx + y/x)

(x^y)' = x^y*(y'lnx + y/x)

Is this right?

5. Apr 26, 2004

### pig

(x^y)' = x^y*(y'lnx + y/x)

(x^y)' = yx^(y-1)+x^y*lnx*y'

I think I understand what you wrote after all.. Thanks arildno :)

Last edited: Apr 26, 2004
6. Apr 26, 2004

### NateTG

Use the chain rule:

$$\frac{d}{dx} f(g(x))=g'(x)f'(g(x))$$

So
$$\frac{d}{dx} x^{(y^{(x^y)})}= \frac{d}{dx} e^{(\ln x \times y^{(x^y)})} = \frac{d}{dx} (\ln x \times y^{(x^y)}) \times x^{(y^{(x^y)})}$$
$$=(\frac{y^{(x^y)}}{x} + \ln x \times \frac{d}{dx} y^{(x^y)}) \times x^{(y^{(x^y)})}$$
$$=(\frac{y^{(x^y)}}{x} + \ln x \times \frac{d}{dx} e^{(y \times x^y)}) \times x^{(y^{(x^y)})}$$
$$=(\frac{y^{(x^y)}}{x} + \ln x \times \frac{d}{dx} {(y \times x^y)} \times y^{(x^y)}) \times x^{(y^{(x^y)})}$$
$$=(\frac{y^{(x^y)}}{x} + \ln x \times (y' x^y + y \frac{d}{dx} (x^y)) \times y^{(x^y)}) \times x^{(y^{(x^y)})}$$
$$=(\frac{y^{(x^y)}}{x} + \ln x \times (y' x^y + y \frac{d}{dx} (y \ln x ) \times x^y) \times y^{(x^y)}) \times x^{(y^{(x^y)})}$$

$$=(\frac{y^{(x^y)}}{x} + \ln x \times (y' x^y + y (\frac{y}{x} + xy' ) \times x^y) \times y^{(x^y)}) \times x^{(y^{(x^y)})}$$
Obviously, some regrouping is necessary. I somehow doubt that that's the same as what arnildo had.

7. Apr 27, 2004

### arildno

Well, I didn't bother to compute z'(x).
Here's the simplest way to compute x^(z(x)):

F(x,z)=x^(z), H(x)=F(x,z(x)).

dH/dx=d^(p)F/dx+d^(p)F/dz*z'(x), where d^(p)/dx is the partial derivative with respect to x.


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# How do you find the distance between (-sqrt2,-sqrt2), (sqrt2, 6sqrt2)?

Jun 13, 2017

abs((-sqrt2,-sqrt2):(sqrt2,6sqrt2))=color(red)(sqrt(106)

#### Explanation:

The easiest way to do this is to observe that the given points are equivalent to $\left(- 1 , - 1\right)$ and $\left(1 , 6\right)$ scaled by a factor of $\sqrt{2}$

The distance between the given points will be the same as the distance between $\left(- 1 , - 1\right)$ and $\left(1 , 6\right)$ multiplied by $\sqrt{2}$

The horizontal distance between $\left(- 1 , - 1\right)$ and $\left(1 , 6\right)$ is $1 - \left(- 1\right) = 2$

The vertical distance between $\left(- 1 , - 1\right)$ and $\left(1 , 6\right)$ is $6 - \left(- 1\right) = 7$

The diagonal distance between $\left(- 1 , - 1\right)$ and $\left(1 , 6\right)$ is given by the Pythagorean Theorem as
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{2}^{2} + {7}^{2}} = \sqrt{4 + 49} = \sqrt{53}$

and the corresponding diagonal distance between $\left(- \sqrt{2} , - \sqrt{2}\right)$ and $\left(\sqrt{2} , 6 \sqrt{2}\right)$ is
$\textcolor{w h i t e}{\text{XXX}} \sqrt{2} d = \sqrt{2} \cdot \sqrt{53} = \sqrt{106}$


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# Complex conjugates

#### Everything You Need in One Place

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##### Examples
###### Lessons
1. Find the conjugate of
$\blacksquare \;$ $3+4i$
$\blacksquare \;$ $4-\sqrt6i$
$\blacksquare \;$ $-\sqrt{10}+4i$
$\blacksquare \;$ $i-5$
1. Given that $z=5+6i$, determine $\overline{z}+\overline{z}$
1. Given that $w=i-14$, determine $\overline{w}+w$
1. If $z=15+2i$, $w=28i-7$, determine $w+\overline{z}$
###### Topic Notes
We have seen how conjugates had helped us simplify perplexing problems in the trigonometric identities chapter and the radical functions chapter. What if I tell you that the same principles can also be applied to complex numbers to make our lives better! In this section, we will introduce complex conjugates, as well as how to perform addition and subtraction on them.


text:
Free Version
Easy

# Ionic and Molecular Compounds

CHEM-Z11XRX

Which of the following is an ionic compound?

A

${ H }_{ 2 }{ O }_{ 2 }$

B

${ Na }_{ 2 }{ O }_{ 2 }$

C

${ C }_{ 2 }{ H }_{ 2 }$

D

${ B }_{ 2 }{ H }_{ 6 }$


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What is Symplectic Area?

In classical Mechanics, momentum and position can be paired together to form a symplectic manifold. If you have the simple harmonic oscillator with energy $H = (k/2)x^2 + (m/2)\dot{x}^2$. In this case, the orbits are ellipses. How is the vector field determined by the (symplectic) gradient, then?

Also, does anyone know an interpretation for the area inside a closed curve in phase space?

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The symplectic area contained in a closed curved, that is the boundary of map of a disc, is the "action along the curve". $$\int_\sigma \omega = \int_\sigma d\lambda = \int_{\partial \sigma} \lambda = \int_0^{2\pi} \lambda_{\gamma(t)}(\dot \gamma(t)) dt,$$ where $\sigma$ is a smooth map from the disc to $M$, and $\gamma = \partial \sigma$. In all cases, the pullback of the 2-form $\omega$ by $\sigma$ is exact since the disc is contractible, so there exists a primitive $\lambda$, on the disc, and you apply Stokes' theorem.

[I apologize for the lengthy answer]

Let me try to elaborate a little bit on a not too complicate but not that simple example to see where the symplectic form makes sense. Let us consider a point on the sphere $S^2$, let $$TS^2 = \{ (x,v) \in S^2 \times {\bf R}^3 \mid x \cdot v = 0 \}$$ Let $$L : TS^2 - S^2 \to {\bf R} \quad \mbox{with} \quad L(x,v) = \Vert v \Vert$$ be the "length function" as lagrangian. And you look for the variational problem $$\delta \int L(x(t),\dot x(t))\ dt = \delta \int \Vert \dot x(t) \Vert\ dt = 0.$$ I don't put the limits of the integral on purpose, it would lead to a too long discussion. Since the lagrangian is homogeneous of degree 1 in $v$, we have the Euler identity $$L(x,v) = \frac{\partial L(x,v)}{\partial v}(v)$$ And the nature of the partial derivative involved above is a map from $TS^2-S^2$ to the cotangent $T^*S^2$ $$\forall v \in T_xS^2 - \{0\}, \quad \frac{\partial L(x,v)}{\partial v} = \frac{\bar v}{\Vert v \Vert} \in T^*_xS^2$$ where the bar denotes the transposed, that is $\bar v w = v \cdot w$. Let's call this map $P$ $$P : TS^2 - S^2 \to T^*S^2 \quad \mbox{with} \quad P(x,v) = \left(x,\frac{\partial L(x,v)}{\partial v}\right) = \left(x, \frac{\bar v}{\Vert v \Vert}\right).$$ Now let $\lambda = pdx$ the Liouville form on $T^*S^2$, its pullback by $P$, integrated along the curve $\gamma = [t \mapsto (x(t),\dot x(t))]$ is exactly the action $$\int \Vert \dot x(t) \Vert \ dt = \int_\gamma P^*(\lambda) = \int_{P \circ \gamma} \lambda.$$ Now, let $\tilde \gamma = P \circ \gamma$, this is a path in the image $Y$ of $P$, which is the unit-cotangent bundle $$Y = {\rm Im}(P) = \{ (x, \bar u) \in T^*S^2 \mid \bar u u = 1 \}$$ And the variational condition becomes then $$\delta \int_{\tilde \gamma} \lambda = \int d\lambda\left(\delta\tilde\gamma(t), \frac{d\tilde \gamma}{dt}\right)\ dt = 0.$$ But $\varpi = d\lambda$ is a 2-form on $Y \simeq US^2 \simeq SO(3)$ which is of odd dimension, actually $3= 2\times 2 -1$. Now, $\varpi$ has a kernel of dimension 1, and $\gamma$ is a solution of the variational problem if and only if $$\frac{d\tilde \gamma}{dt} \in \ker \varpi_{\tilde \gamma(t)}$$ In this case, the kernel is given explicitly by $$\frac{dx}{dt} = \alpha u \quad \mbox{and} \quad \frac{du}{dt}= -\alpha x.$$ The quotient space ${\cal S} = Y/\ker\varpi$, the space of solutions of the variational problem, is then equivalent to the sphere $S^2$, thanks to the (SO(3)-moment map) $$\pi : (x,u) \mapsto x \times u.$$ By construction this space inherits a symplectic form $\omega$ such that $$\pi^*(\omega) = \varpi.$$ And $({\cal S}, \omega)$ is the space of oriented non parametrized geodesics of the sphere $S^2$ (which by chance is also a sphere $S^2$). Finally what do we get? A space $Y \simeq US^2 \simeq SO(3)$ made of couples $(x,u)$ or matrices $y=[x\ u \ x \times u]$, a 1-form $\lambda$, the "action-form" (actually called the "Cartan 1-form"), a characteristic distribution $y \mapsto \ker(d\lambda)$ whose leaves are the pre-images of the point of the sphere $S^2$ by the moment map $\mu : (x,u) \mapsto x \times u$, and the image of $\mu$ is a symplectic manifolds for the projection $\omega$ of $d\lambda$. Note that in this case $\omega$, proportional to the standard area-form, is closed but not exact.

Now you can ask the same question as previously: "What does mean the area include in a disc $\sigma : D^2 \to {\cal S}$?"

Consider the pullback by $\sigma$ of the $S^1$-principal bundle $\pi : Y \to {\cal S}$, this is a principal bundle on $D^2$, but $D^2$ is contractible, so this fiber bundle is trivial, thus it admits a smooth section, that is a lift $\tilde \sigma : D^2 \to Y$, that is $\pi \circ \tilde \sigma = \sigma$. Now, $$\int_\sigma \omega = \int_{\pi\circ\tilde\sigma} \omega = \int_{\tilde\sigma} \pi^*(\sigma) = \int_{\tilde\sigma} d\lambda = \int_{\tilde\gamma} \lambda \quad \mbox{with} \quad \tilde\gamma = \partial\tilde\sigma.$$ Let us write $\tilde \gamma(s) = (x_s,\bar u_s) \in Y$, and let us assume that the parameter $s$ runs over $[0,2\pi]$ to describe $\tilde \gamma = \partial \tilde \sigma$, then $$\int_\sigma \omega = \int_{\tilde\gamma} \lambda = \int_0^{2\pi} \bar u_s \frac{dx_s}{ds} \ ds.$$ And this is the action of the unit vector $s \mapsto u_s$ distribution along the curve $s \mapsto x_s$. And let us remember that the vector $x_s \times u_s$ describes a geodesic of the sphere $S^2$ for all $s$, and $s$ is not the time parameter of this geodesic.

Note 1. that this construction can be applied to any homogeneous lagrangian, and for non-homogeneous lagrangian, first we homogenize them and after we apply this construction.

Bibliography Jean-Marie Souriau, "Structure des Systèmes Dynamiques", Dunod ed., Paris 1970

-
 Is that primitive known as the "Lagrangian"? (In mechanics sense.) – John Mangual Jan 22 2011 at 16:36 It is related, I am afraid that giving a precise answer to your question could lead us too far :-) But in your simple case if $\omega = dp \wedge dq$ (let's say $q=x$ and $p = \dot x$), ${\bf R}^2$ being simply connected there is no major problem, you can chose once and for all $\lambda = p dq$, a standard choice. Then the action is $$\int_0^{2\pi} p(t)\frac{dq(t)}{dt} dt$$, where $\gamma : t \mapsto (p(t), q(t)$. If you choose another primitive, the difference is the differential of a real function which doesn't contribute to the integral on the loop. – Patrick I-Z Jan 22 2011 at 17:22 Wonderfully informative answer! No apologies necessary; I (and I am sure, others) appreciate the tutorial. :-) – Joseph O'Rourke Jan 24 2011 at 23:15

For one interpretation of the area inside a curve in phase space, see Arnold's Mathematical Methods of Classical Mechanics page 20. In case you do not have a copy of the book, he defines a function $S: (E_0 - \epsilon, E_0 + \epsilon) \to \mathbb{R}$ which gives the area inside the curve associated to an energy level $E$ (assuming this is well defined). The problem on page 20 asserts that $T = \frac{dS}{dE}$ where $T$ is the period of motion along the curve.

Note: The relevant page is available on google books.

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You can view $\mathbb{R}^{2n}$ as a quotient of the real Heisenberg group $\mathcal{H}^{2n+1}$ modulo its center. For a closed loop $\alpha$ in $\mathbb{R}^{2n}$ and a point in $\mathcal{H}^{2n+1}$ over $\alpha(0)$ there's unique lift $\tilde{\alpha}$ of $\alpha$ to $\mathcal{H}^{2n+1}$ going through this point. The symplectic area enclosed by $\alpha$ expresses the signed distance from $\tilde{\alpha}(0)$ to $\tilde{\alpha}(1)$ with respect to a left invariant Riemannian metric on $\mathcal{H}^{2n+1}$.

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 Which is also the "action". Actually this is always true, there exists always above a symplectic manifold (or not symplectic, or not a manifold) a principal bundle with group the torus of periods of the closed 2-form, that is the quotient of the real line by the group of periods (can be not a Lie group), equipped with a connection 1-form with curvature the 2-form, and any loop can be lifted horizontally (in the kernel of the connexion), such that the area enclosed is equal to the "length" of the path along the fiber of the lift. – Patrick I-Z Jan 23 2011 at 18:47