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Chemical Forums aims at helping chemistry students. That means many of the questions posted require use of formulae - either chemical ones (like H
2
PO
4 -
) or mathematical ones (like [itex]E = E_0 + \frac {RT}{nF}\ln Q[/itex]). You may even need to use a structural formula like
To make your questions easier to unders... |
Excerpt from textbook:
According to Archimedes' law the weight of a body of mass $m$ and density $\rho$ inside air is: $$G=mg\left(1 - \frac{\rho_v}{\rho}\right)$$ Where $\rho_v$ is the density of air.
A "weight" of mass $m_t$ and density $\rho_t$ in air weights $$G_t = m_tg\left(1-\frac{\rho_v}{\rho_t}\right)$$ If usi... |
The key to the efficiency of an antenna (whether for transmitting or receiving - the two processes are essentially reciprocal) is resonance, and impedance matching with the source / receiver. The size also matters in terms of the relationship between power and current.
A nice analysis of the impact of size of an antenn... |
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Now showing items 1-10 of 33
The ALICE Transition Radiation Detector: Construction, operation, and performance
(Elsevier, 2018-02)
The Transition Radiation Detector (TRD) was designed and built to enhance the capabilities of the ALICE detector at the Large Hadron Collider (LHC). While aimed at providing electron... |
PDE:
$$ u_t (x,t) - u_{xx} (x,t) = 0, \quad u(0,t)=0, \quad u_x(2,t)=0, \quad u(x,0)=\delta(x-x_0) $$
where $0<x_0<2$ is a given constant.
I worked out my solution by using separation of variables $$u(x,t)=X(x)T(t) \implies \frac{X''(x)}{X(x)}=\frac{T'(t)}{T(t)}=\lambda$$
For $\lambda>0$ and $\lambda=0$ we get trivial ... |
I'm trying to calculate the SO-coupling for a single p-electron ($l=1$, $s=\frac{1}{2}$) in the uncoupled representation.
This comes down to calculating these matrix elements:
$$\left\langle nlm_lsm_s\left|\hat{\vec l}.\hat{\vec s}\right|nlm_l^{'}sm_s^{'}\right\rangle$$
It can be easily found that:
$$\hat{\vec l}.\hat{... |
(Math Stack Exchange suggested that the same question I posted there be migrated here; The one at Math Stack Exchange was thus deleted. The recommendation message of migration can be found here, though the page is now deleted.)
Whenever students are asked to derive or prove some vector calculus identities such as
$$\na... |
In this particular case, I am talking about an extremely flexible human. In my story, this human catches another's head with his legs and manages to snap his neck. I know a human can't simply snap another's neck (They don't have enough strength), but could it be done if the attacker were able to pinch their opponent's ... |
I'm asked to find if the fixed-point iteration
$$x_{k+1} = g(x_k)$$
converges for the
fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$.
In this short video by Wen Shen,
it's explained how to find these fixed-points and to see if a fixed-point iteration conve... |
Assume there exist infinitely many $x$ such that: $$3x^2+3x+1 = 3p-2$$ Where $p$ is prime. Can it be shown there exist infinitely many $y$ such that: $$3y^2+3y+1=q$$ Where $q$ is prime? I believe that it cannot be shown as our assumption tells us nothing of which primes exist such that $3p-2 = 3x^2+3x+1$ and so knowing... |
I was thinking about hadrons in general Yang-Mills theories and I have some doubts that I'd like to discuss with you.
Suppose that we have a Yang-Mills theory that, like QCD, tend to bind quarks into color singlet states. So far nothing strange, even QED tend to bind electromagnetic charges to form neutral systems. The... |
I am reading Nakahara's Geometry, topology and physics and I am a bit confused about the non-coordinate basis (section 7.8). Given a coordinate basis at a point on the manifold $\frac{\partial}{\partial x^\mu}$ we can pick a linear combination of this in order to obtain a new basis $\hat{e}_\alpha=e_\alpha^\mu\frac{\pa... |
I've readed in Jech's book that the existence of the $\omega-$Erdös cardinal $\kappa(\omega)$ (that is the minimumm cardinal $\kappa$ for which $\kappa\rightarrow (\omega)^{<\omega}$) it is compatible with the axiom of constructibility $V=L$. More precisely, it can be shown that if $\kappa=\kappa(\omega)$ exists then $... |
Let $X = \mathrm{Spec}\ A$, and for each prime ideal $\mathfrak{p} \subseteq A$, let $A_{\mathfrak{p}}$ be the localization of $A$ at $\mathfrak{p}$. Now, for each open set $U \subseteq X$, define $\mathcal{O}(U)$ to be the set of functions $$s: U \rightarrow \coprod_{\mathfrak{p} \in U} A_{\mathfrak{p}}$$ such that $s... |
Measure here is Jordan Measure.
$d(A,B)=\inf\{\vert x-y:x\in A,y\in B\}$
So I can show the first direction fairly easily because it doesn't use the condition.
$\leq$
Since $S_i$ are measurable there exists poly rectangles $P_i$ such that $\mu S_i\leq\vert P_i\vert\leq \mu (S_i)+\frac{\epsilon}{2}$. Since $S_i\subseteq ... |
Interesting idea to use a grid. I doubt it will work for this question, since writing things in that grid format in a sense splits into arithmetic progressions modulo $p$, and usually saying things about the primes in a progression is harder.
However, I cannot help but mention that using a grid like that was applied br... |
Question. Is there a smooth proper scheme $X\to\operatorname{Spec}(\mathbb{Z})$ such that $X(\mathbb{Q}_v)\neq\emptyset$ for every place $v$ of $\ \mathbb{Q}$ ( including $v=\infty$), and yet $X(\mathbb{Q})=\emptyset$ ?
I believe the answer is No. The evidence is flimsy : $X$ cannot be a curve or a torsor under an abel... |
Hello, I've never ventured into char before but cfr suggested that I ask in here about a better name for the quiz package that I am getting ready to submit to ctan (tex.stackexchange.com/questions/393309/…). Is something like latex2quiz too audacious?
Also, is anyone able to answer my questions about submitting to ctan... |
If $f : \mathbb{R}^{n} \longrightarrow \mathbb{R}$ is $L$-Lipschitz (w.r.t. the $\|\cdot\|_2$ norm), it is a fact that if $x \sim N(0, I)$, then $$ \mathbb{P}( f(x) - \mathbb{E}f(x) \geq t) \leq e^{-t^2/(2L^2)} \:. $$
I'm reading the proof (from Theorem 4.3 of https://galton.uchicago.edu/~lalley/Courses/386/Concentrati... |
If $M$ is a smooth closed $n$-dimensional Riemannian manifold which is Riemannian embedded in $\mathbb R^{n+1}$, then there exists a point $p \in M$ such that the sectional curvatures at $p$ are all positive.
Can any one give me a hint for this problem? I was considering the maximum $p$ of function $|x|^2$ on $M$, then... |
Given a (Hausdorff separated) locally convex space $X$ what can we say about a proper convex function $f:X\to\mathbb{R}$ whose domain $\emptyset\neq D(f):=\{x\in X\mid f(x)<+\infty\}$ has a non-empty (topological) interior?
Recall that when $X$ is barreled the non-empty domain interior spells that the function is conti... |
I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be...
Question: Does this notation$$\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$$refer to the Jacobian
matrix$$ J = \begin{bmatrix} \dfrac{\partial y_... |
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Now showing items 1-10 of 34
Search for top squark pair production in final states with one isolated lepton, jets, and missing transverse momentum in √s = 8 TeV pp collisions with the ATLAS detector
(Springer, 2014-11)
The results of a search for top squark (stop) pair production in final states with one isolate... |
well, it is absolutely in agreement with theory. the correlation as measured by Pearson's coefficient $\rho$ is linear measure in the sense that the bounds [-1,1] are obtained only when transformations of our variables are linear, so if we have variables $X$ and $Y$ then something like $aX+bY+c$ where $a,b\in\mathbb{R^... |
Fitting a Damped Sine Wave
A damped sine wave is described by
$$ x_{(k)} = A \cdot e^{\alpha \cdot k} \cdot cos(\omega \cdot k + p)
\tag{1}$$
with frequency $\omega$ , phase p , initial amplitude A and damping constant $\alpha$ . The $x_{(k)}$ are the samples of the function at equally spaced points in time.
With $x_{(... |
Difference between revisions of "Algebra and Algebraic Geometry Seminar Spring 2018"
(→John Lesieutre)
(→John Lesieutre)
Line 212: Line 212:
===John Lesieutre===
===John Lesieutre===
−
Some higher-dimensional cases of the Kawaguchi-Silverman conjecture
+
Some higher-dimensional cases of the Kawaguchi-Silverman conjectu... |
Find if the series converges or diverges, $a_n=\sum_{n=1}^\infty\frac{1}{1+\ln (n)}$. Comparing it with another series $b_n=\frac{1}{\ln(n)}$. Dividing both the series and taking limits, we get $\lim_{n\to\infty}\frac{\ln(n)}{1+\ln(n)}$. Since it is the $\infty/\infty$ form, applying H'opitals rule, we get, $\lim_{n\to... |
I'm currently working through the book
Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized Lorentz force via a velocity-dependent potential.
I understand the generalized force that they derive from a Lagrangian of ... |
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Now showing items 1-6 of 6
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
Consider two random variables $X$ and $Y$ and their joint cumulative probability distribution function $F$. I'm attempting to show that $P\{a_1<X<a_2\mathrm{,}\, b_1<Y<b_2\} = F(a_2, b_2)+F(a_1,b_1)-F(a_2,b_1)-F(a_1,b_2)$ by simplifying the left hand side but I am not quite getting there.
My attempt $$P\{a_1<X<a_2\math... |
First, if $a=b$ then$$b_1=a=a_1$$Now, let $a=1$ and $b=2$. Then$$a_1=1.5 \ \text{ and }b_1=\sqrt{2} \approx 1.414$$Hence $a \ne b$ and $b_1 \leq a_1$. There's a problem with what you wrote isn't it ?
It might be $a_1 \leq b_1$ iff $a=b$. Indeed, suppose$$a_1 \leq b_1$$Then using the fact that $x \mapsto x^2$ is increas... |
I am reading a book on Skyrmions, and I am at the part where the interaction of skyrmions with electrons is discussed. The chapter speaks of Spin-Transfer Torque (STT) and makes the following statement about magnetic metals:
In magnetic metals, the outer-shell electrons of magnetic atoms participate in the formation of... |
I have been studying about the SU(2) symmetry in Heisenberg Hamiltonian with a paper 'SU(2) gauge symmetry of the large U limit of the Hubbard model' written by Ian Affleck et al(Phys. Rev. B 38, 745 – Published 1 July 1988). In the paper, they represent the spin in terms of fermionic operators with constraint that num... |
Last Updated: May 5, 2019
When we solve a partial differential equation with a source term, we must pay attention to how to treat it numerically for better convergence. The
is available in OpenFOAM so that we can handle a linearized source term in numerically stable way. SemiImplicitSource fvOption Keywords Source term... |
Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as
$$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} + v^1 \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix} +v^2 \begin{pmatrix} 0&-i \\ i&0 \end{pmatrix} + v^3 \begin{pmatrix} 1&0\... |
$\newcommand{\supp}{\operatorname{supp}}$ We are given with distributions $f,g \in D'(\Bbb R)$. If $\supp f\subset (-\infty,a)$ and $\supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution, where $a$ and $b$ are real numbers.
What I know:
$\langle f*g\rangle=\langle f(x)g(x),\phi(x+y)\rangle$
[Abo... |
The total wavefunction of an electron $\psi(\vec{r},s)$ can always be written as $$\psi(\vec{r},s)=\phi(\vec{r})\zeta_{s,m_s}$$ where $\phi(\vec{r})$ is the space part and $\zeta_{s,m_s}$ is the spin part of the total wavefunction $\psi(\vec{r},s)$. In my notation, $s=1/2, m_s=\pm 1/2$.
Question 1 Is the above statemen... |
Introduction Energy and Power Basic Operations Practice Problems Transformation of signals defined piecewise Periodic signals Even and Odd Signals Commonly encountered signals
Table of Contents
What is a signal?
The word 'signal' has been used in different contexts in the English language and it has several different m... |
Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$.Let's denote the width of a sample by $h$ where $$h\rightarrow0$$Now, for finding the area under the curve between the bounds $a ~\& ~b $ we can a...
@Ultradark You can try d... |
I think while doing the calculations, I have found the answer. @GK gave an answer but I think there is a subtle mistake in that approach because both q and $\tau$ are coupled transformations of x and t. Thus the total derivative is not equal to the partial one. So this is my solution.
$q$ and $\tau$ both can be express... |
Cartesian Form Polar or Exponential Form Euler's Identities Conjugate Operations on two complex numbers nth power and nth roots of a complex number Functions of a complex variable Complex functions of a real variable Magnitude and Phase Plot Examples and References History Why use complex numbers?
We will use the lette... |
The unit of illumination is the lux, lumens per square meter.
What is the minimum lux required for reading? How many lux does the Sun provide at distance D?
What is the minimum lux required for reading?
You can plug all sorts of numbers into this depending on how good your eyes are, how big the print is, and how close ... |
This question arises from the one asked here about a bound on moment generating functions (MGFs).
Suppose $X$ is a bounded zero-mean random variable taking on values in $[-\sigma, \sigma]$ and let $G(t) = E[e^{tX}]$ be its MGF. From a bound used in a proof of Hoeffding's Inequality, we have that $$G(t) = E[e^{tX}] \leq... |
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January 2011 , Volume 18
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Abstract:
We present a novel class of functions that can describe the stable and unstable manifolds of the Hénon map. We propose an algorithm to construct these functions by using... |
The number of storks and the number of human babies delivered are positively correlated (Matthews, 2000). This is a classic example of a spurious correlation which has a causal explanation: a third variable, say economic development, is likely to cause both an increase in storks and an increase in the number of human b... |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphe... |
Please assume that this graph is a highly magnified section of the derivative of some function, say $F(x)$. Let's denote the derivative by $f(x)$.Let's denote the width of a sample by $h$ where $$h\rightarrow0$$Now, for finding the area under the curve between the bounds $a ~\& ~b $ we can a...
@Ultradark You can try d... |
I'll skip irrationals, but can anyone give an interpretation of the imaginary number $\sqrt{-x}$? or of the complex number $a + \sqrt{-x}$?
That is, can you form a sentence that makes sense, which points to the square root of a negative number?
I am explicitly not just looking for a way to complex number operations
Sen... |
Is there an Intercept theorem (from Thales, but don’t mistake it with the Thales theorem in a circle) in hyperbolic geometry?
Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, AC are respectively parallel to SB, SD, BD; and SB > SA, SD > SC. Then, we have SB/SA = SD/SC = BD/AD
(see wiki illustra... |
Please, help me make equivalent transformations with this formula (A∨C→B)(A→C)(¬B→¬A∧C)(¬A→(C→B))(B→¬C→¬A). Thanks.
You cannot "mix" in this way different "conventions" regarding symbols.
If you want to use
propositional connectives (like : $\lor, \land$) instead of boolean operators (like : $\cdot, +$) you have to rew... |
How to Find a Fast Floating-Point atan2 Approximation ContextOver a short period of time, I came across nearly identical approximations of the two parameter arctangent function, atan2, developed by different companies, in different countries, and even in different decades. Fascinated with how the coefficients used in t... |
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2014
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Microelectronics Reliability, ISSN 0026-2714, 05/2018, Volume 84, pp. 1 - 6
This paper discusses the reliability performance of Wolfspeed GaN/AlGaN high... |
Given an $n$-vector $y$ (responses) and a design matrix $X$, I wish to fit them with a simple linear regression model $$y=X\beta+e,$$ where $e\sim\mathcal{N}(0, \sigma^2I)$. Then, we have $$y\sim\mathcal{N}(X\beta, \sigma^2I).$$ Then the maximum likelihood estimations (MLE) of $\beta$ and $\sigma^2$ are just $$\hat\bet... |
I would like to fit a GLM to the rate underlying a Poisson process, for data with variable exposure (period of measurement) - and the question is about aggregating/grouping the data before fitting or not. So the model is
$$\mu = \exp(\beta_0+\beta_1 X_1+\beta_2 X_2)$$
$$ Y_i \sim Poisson(\mu_i t_i) $$
So $\mu_i$ is the... |
CryptoDB Changmin Lee Affiliation: Seoul National University, Republic of Korea Publications Year Venue Title
2019
JOFC
Cryptanalysis of the CLT13 Multilinear Map
In this paper, we describe a polynomial time cryptanalysis of the (approximate) multilinear map proposed by Coron, Lepoint, and Tibouchi in Crypto13 (CLT13).... |
Unitary elements of a Banach space have been defined in this paper as follows:
Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be (geometrically) unitary if $A'=\text{ span }S_{a}$.
Here, $A'$ is the dual space of $A$.
We note that this property is true for uni... |
Is it true for $n=1$?
$\sum\limits_{i=1}^1(6i-3) = 6\cdot 1 - 3 = 3 = 3\cdot 1^2$ Yes, it is true for $n=1$.
Now, supposing that it is true that $\sum\limits_{i=1}^n(6i-3)=3n^2$ for
some $n\geq 1$, we are curious whether or not it follows that it will necessarily also be true for $n+1$
I.e. we
want to show that $\sum\l... |
Let $S$ be a non-empty set. Consider the statement$$\forall A\subseteq S \quad \forall B\subseteq S\quad\exists D\subseteq S \ (D\neq \emptyset \ \wedge \ D\cap (A\cup B)=\emptyset) $$The negation is$$\exists A \subseteq S \quad \exists B\subseteq S \quad \forall D \subseteq S \ (D=\emptyset \ \vee \ D\cap (A \cup B)\n... |
Funny problem. There is a number of ways it can be treated, starting from elementary (working with real & imaginary parts) & proceeding to more advanced (including some more geometric approaches).
Your questions 1 & 2 read pretty much the same to me, & the only sensible answer I can provide is '
because working w/ z & ... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red marbles found in a sample of 3. If we sample
without replacement, what is a maximum likelihood estimator for $\theta$ (the... |
It is safe to say that there will always be intermolecular forces at play. At the time where you will consider these you should already have a good idea about the molecules involved in your system.
Based on the composition and molecular structures you can make certain assumptions. In a molecule it is straight forward t... |
Let's assume that $f\colon\mathbb R\to\mathbb R$ is continuous and hence Lebesgue measurable. Then, the Lebesgue integral $\int_{(0,\infty)}f(x)\,d\lambda(x)$ makes sense (but of course can be equal to $\pm\infty$). Given such $f$, how can we find this Lebesgue integral? Will it always be the case that $$\int_{(0,\inft... |
Mathematics - Functional Analysis and Mathematics - Metric Geometry
Abstract
The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)-$separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subseteq S_X$ an... |
OWN 1.25d Is the even part of $x(t) = \cos \left( 4 \pi t\right) \ u(t)$ periodic? If it is, what is the time period? OWN 1.25e Is the even part of $x(t) = \sin \left( 4 \pi t\right) \ u(t)$ periodic? If it is, what is the time period?
OWN 1.26b Is the discrete-time signal $x[n] = \cos \left(\frac{n}{8}\right)$ periodi... |
Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$.
I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all $n\in\mathbb Z$.
First attempt. I have tried to prove that: $$f\left(\frac{\pi}n\right)=\cos\left(\frac{\pi}n\right),\quad\t... |
The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions
1.
Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés, Spain
2.
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom, United Kingdom
nonlinear eigenv... |
Hilbert transform Partial Function
definiendum $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ definiendum $H(f):=y\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(x)}{y-x}\,\mathrm dx$ Discussion
$(H(f))=-f$
The Hilbert transform commutes with the Fourier transform up to a simple factor and is a... |
I am looking at the problem of a coupled oscillator, whereby we have 3 springs connected between two walls in the following way: wall, then spring (k), then mass (m), then spring (2k), then mass (m), then spring (k), then wall.
I have calculated the characteristic frequencies (I think) by using the fact that we will ha... |
Many popular indices such as the S&P 500, Nasdaq 100 and the Wilshire 5000 are value weighted - a company’s weight in the index is proportional to its market-capitalization (price shares outstanding). This post explores alternative weighting methods and their implications.
The Nasdaq
Given previous posts on the Nasdaq,... |
Hi I am trying to prove this $$ I:=\int_0^{\pi/4}\log\left(\tan\left(x\right)\right)\, \frac{\cos\left(2x\right)}{1+\alpha^{2}\sin^{2}\left(2x\right)}\,{\rm d}x =-\,\frac{\pi}{4\alpha}\,\text{arcsinh}\left(\alpha\right),\qquad \alpha^2<1. $$ What an amazing result this is! I tried to write $$ I=\int_0^{\pi/4} \log \sin... |
From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action
$$I(g)=\frac{1}{4\pi}\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega)$$
i.e. it changes under a twist of framing on $M$ b... |
I cannot claim to be an expert on AQFT, but the parts that I'm familiar with rely on local fields quite a bit.
First, a clarification. In your question, I think you may be conflating two ideas: local fields ($\phi(x)$, $F^{\mu\nu}(x)$, $\bar{\psi}\psi(x)$, etc) and unobservable local fields ($A_\mu(x)$, $g_{\mu\nu}(x)$... |
I am trying to understand how a perturbation of a skew symmetric matrix by another skew symmetric matrix affects the dominant eigenvector corresponding to $\lambda=0$.
Specifically, let $S$ be an $n \times n$ real-valued skew symmetric matrix , with $n$ being odd. Assume $S$ has eigenvectors $(e_1,...,e_n)$ and corresp... |
We have
$$\sum_{k = 1}^n \cos \sqrt{k} = \frac{\cos 1 + \cos \sqrt{n}}{2} + \int_1^n \cos \sqrt{t}\,dt - \int_1^n \bigl(\lbrace t\rbrace - \tfrac{1}{2}\bigr)\frac{\sin \sqrt{t}}{2\sqrt{t}}\,dt.$$
The first term is bounded, and the second integral is $O(\sqrt{n})$ as one sees from the boundedness of $\sin \sqrt{t}$ and ... |
Let $\mu_n$ be a sequence of positive radon measures on $\mathbb{R}^n$ weakly converging (as dual of continuous compactly supported functions) to a measure $\mu$.
Assume that $f_n(z)$ is a sequence of positive, compacly supported functions such that: they are uniformily supported in a ball, i.e.: for every $n$ their su... |
Question
Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to ... |
I have the following system of equations that involve some rather complicated exponents:
$$ \left\{\frac{\mathit{a} \sigma ^{41/34}}{\mathit{m}^{8/119} \mathcal{M}^{28/17}}=1,\frac{\mathit{b}\mathit{q}\mathit{m}^{2/3} \mathcal{M}^3}{\sqrt{\sigma }}=1,\frac{\mathit{f} \mathit{c} \mathit{m}^{436/357}}{\sigma ^{48/17} \ma... |
The
cumulative probability distribution function $F_X(x)$ tells us how muchprobability mass there is to the left of $x$ or at $x$ for each $x$on the real line. (The choice of notation, though almost universally usedis truly dreadful for use in a classroom setting! How on earth does one read out aloud $F_X(x)$ or $P\{X\... |
https://doi.org/10.1351/goldbook.C01032
The chemical potential of a substance
Green Book, 2nd ed., p. 49 [Terms] [Book] PAC, 1994, PAC, 1996, Bin a mixture of substances B, C... is related to the Gibbs energy \(G\) of the mixture by: \[\mu _{\text{B}}=(\frac{\partial G}{\partial n_{\text{B}}})_{T,p,n_{\text{C}\neq \tex... |
First of all, a topological space that is locally homeomorphic to a Euclidean space is locally compact, cf. this, and locally connected as those properties are local and preserved by homeomorphisms (and they hold for Euclidian spaces). Then
$M$ second-countable $\Longrightarrow\ M$ paracompact with countably many conne... |
I'm studying Linear Algebra for the second time, using Hoffmann & Kunze. Currently I'm trying to prove the following theorem:
Theorem 7. If $A$ is an $n \times n$ matrix, then $A$ is row-equivalent to the $n \times n$ identity matrix if and only if the system of equations $A\vec{x} = \vec{0}$ has only the trivial solut... |
Input Convex Neural Network (ICNN) is a neural network that has convexity in a subset of its input. In this post, I give a brief review of the model in the context of continuous action reinforcemen learning.
Motivation
Many inference problems are non-convex, especially when their objective functions are represented by ... |
In science we learn about single, double, and sometimes triple bonds. From a quick search I have found up to sextuple bonds.
Is there a maximum bond order? If yes/no, what causes this?
Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. It ... |
Implementing Impractical Digital Filters
This blog discusses a problematic situation that can arise when
we try to implement certain digital filters. Occasionally in the
literature of DSP we encounter
impractical
digital IIR filter block diagrams, and by impractical I mean block
diagrams that cannot be implemented. Thi... |
Nothing would happen for a while, assuming we're talking about Sun just disappearing overnight. You can even demonstrably witness what happens with 12 hours of no sunlight in the Equator (or during polar night in the Arctic, for that matter). The primary reason for this is the vast amount of water.
We can, however, giv... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Differences
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Line 13: Line 13: Generally, material physics of finite (i... |
Find if the series converges or diverges, $a_n=\sum_{n=1}^\infty\frac{1}{1+\ln (n)}$. Comparing it with another series $b_n=\frac{1}{\ln(n)}$. Dividing both the series and taking limits, we get $\lim_{n\to\infty}\frac{\ln(n)}{1+\ln(n)}$. Since it is the $\infty/\infty$ form, applying H'opitals rule, we get, $\lim_{n\to... |
I am asked to verify that the sequence $\left(\frac{1}{6n^2+1}\right)$ converges to $0$:
$$\lim \frac{1}{6n^2+1}=0.$$
Here is my work:
$$\left|\frac{1}{6n^2+1}-0\right|<\epsilon$$
$\frac{1}{6n^2+1}<\epsilon$, since $\frac{1}{6n^2+1}$ is positive
$$\frac{1}{\epsilon}<6n^2+1$$
$$\frac{1}{\epsilon}-1<6n^2$$
$$\frac{1}{6\e... |
I have the following differential equation $$y''-xy=0\,.$$ By using $y=\sum\limits_{n=0}^{\infty} a_n x^n$, I show that $$(n+2)(n+1)a_{n+2}=a_{n-1}$$ for $n=1,2,...$, and $a_2=0$. (I differentiate the sum and by plugging it into the differential equation which is zero I get that result.) My question is how can I show t... |
In Carr and Madan (2005), the authors give sufficient conditions for a set of call prices to arise as integrals of a risk-neutral probability distribution (See Breeden and Litzenberger (1978)), and therefore be free of static arbitrage (via the Fundamental Theorem of Asset Pricing)
These conditions are:
Call spreads ar... |
I have seen some examples of metrics on $\mathbb R$ which make it incomplete, like $d(x,y)=|e^{-x}-e^{-y}|$ and $d(x,y)=\arctan x - \arctan y$ where $1,2,3,\ldots$ is a Cauchy sequence which doesn't converge. I am curious about what other examples there are. Is it always infinity which turns out to be 'missing' or are ... |
The normal ordening is a way to say: ''we throw away the zero-point energy'' (since it becomes infinity and wa say we only look at energy-differences), or to put it in the words of A. Zee: ''Create before you annihalite''.
The chronological ordening comes in when you calculate the Feynman propagator (also called the Gr... |
Consider the following three phenomena.
Stein's paradox: given some data from multivariate normal distribution in $\mathbb R^n, \: n\ge 3$, sample mean is not a very good estimator of the true mean. One can obtain an estimation with lower mean squared error if one shrinks all the coordinates of the sample mean towards ... |
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E. D. Reese, J. E. Carlstrom, J. J. Mohr, G. P. Holder (U.Chicago), M. K. Joy (NASA/MSFC), L. Grego (SAO), S. Patel (U.Alabama), W. L. Holzapfel (U.C.Berkeley)
With analysis of Sunyaev-Zel'dovich effect and x-ray data it
is possible to determine the distances to galaxy clusters.
Cosmi... |
I know that the definition of an infinite series is the limit as $n \to \infty$ of its partial sums.
$$\underbrace{\sum_{n=0}^{\infty} \ \ a_n}_\text{Infinite Series}\ \stackrel{\text{def}}{=} \ \lim_{n \to \infty} \ \underbrace{\sum_{i=0}^{n} \ a_i}_\text{Partial Sums}$$
But then how do you define a finite series?
If ... |
Say that a function $\,f: \mathbb{R}^{n\times n} \to \mathbb{R}$ is given in an element-wise form as
$$f(A) = \sum_{i,j} A_{ij} F_{ij}.$$
When is this function convex on the whole set $\mathbb{R}^{n\times n}$?
While trying to find an answer to this question I've stumbled upon a definition of quadratic forms based on ma... |
This problem is closely related to Hook Length Formula.
The formula gives a way how to compute the number of fillings of a matrix $m\times n$ with distinct numbers $1, 2, \ldots, mn$ in such a way that all rows and all columns are increasing. Whenever you have such a matrix, you can translate it into a process of $m$ z... |
The beginning of a measure theory book I am read introduces Cartesian products. It contains the following statements that confuse me:
If $\{X_\alpha\}_{\alpha \in A}$ is an indexed family of sets, their Cartesian product $\prod_{\alpha \in A}X_\alpha$ is the set of all maps $f: A \rightarrow \bigcup_{\alpha\in A}X_\alp... |
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