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Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie group. Denote by $\mathfrak{g}$ the complexified Lie algebra, and denote by $K$ a maximal compact in the complexificati... |
I want to find ground state energy (as well as wavefunction) for spinless $tV$ model using Real-Space Renormalization Group (RSRG) approximation. The $tV$ model is defined as $$H=H_t+H_{int}=-t\sum_{i=1}^N (c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i) + V\sum_{i=1}^N n_i n_{i+1}$$ where $n_i$ is number operator.
The RSRG w... |
Recently I have discovered the method of constructing of GR from massless field with helicity 2 theory. It is considered here, in an article "Self-Interaction and Gauge Invariance" written by Deser S.
By the few words, the idea of the method is following. When starting from massless equations for field with helicity 2 ... |
Banach Journal of Mathematical Analysis Banach J. Math. Anal. Volume 9, Number 2 (2015), 127-133. On the spectral radius of Hadamard products of nonnegative matrices Abstract
We present some spectral radius inequalities for nonnegative matrices. Using the ideas of Audenaert, we then prove the inequality which may be re... |
Main Page The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math], and repl... |
Difference between revisions of "Demand/Dynamic User Assignment"
(using vtypes from additional file)
(8 intermediate revisions by 2 users not shown) Line 1: Line 1: + + + + + + + +
The tool '' {{SUMO}}/tools/assign/duaIterate.py '' can be used to compute the (approximate) dynamic user equilibrium.
The tool '' {{SUMO}}/... |
Main Page The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math], and repl... |
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Note by Aditya Kumar 6 years, 4 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and think... |
$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\om}{\omega} \newcommand{\... |
OK, here is my argument (sorry for the delay).
First of all, $Z$ is essentially the maximum of the absolute value of the polynomial $P(z)=\prod_j(1-z_jz)$ on the unit circumference (up to a factor of $n$, but it is not noticeable on the scale we are talking about).
Second, the maximum of the absolute value of a (trigon... |
Sequences
Definition: A Sequence $\{ a_n \}_{n=1}^{\infty}$ is an ordered list of numbers $a_1, a_2, ...$ where $a_j$ denotes the $j^{\mathrm{th}}$ term of the sequence.
One of the most famous sequences is the
Fibonacci Sequence that is recursively defined by $f_{1} = 1$, $f_{2} = 1$, and $f_{n} = f_{n-1} + f_{n-2}$ fo... |
Main Page Contents The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math],... |
No. In fact, every Lebesgue measurable function $f\colon I\to E$ is equal almost everywhere to a limit of simple Lebesgue measurable functions. As you hint at in the question, this is easy to show in the case where $E$ is separable. The general situation reduces to the separable case due to the following result. For a ... |
Suppose I had a ray of unpolarized light, and I was sitting inside the beam and looking at the electric fields oscillating, then , if I am looking at a point how would the oscillations look like? I cannot seem to understand it.
I guess you have this image in mind:
This hods just for a single photon, or elementary packe... |
Table of Contents
The Fundamental Theorem of Riemann Integral Calculus Part 1
Perhaps one of the most famous "fundamental" theorems in mathematics are the Fundamental Theorems of Calculus that are first introduced in an introductory calculus course. There are two parts of the fundamental theorem.
The first part of the ... |
The adjoint map at the level of the lie algebra is such that $$\text{ad}: \mathfrak{g} \rightarrow \text{End}(\mathfrak g),$$ taking $\lambda_a \mapsto \text{ad}_{\lambda_a}$ where $$\text{ad}_{\lambda_a}: \mathfrak{g} \rightarrow \mathfrak{g}\,\,\,\,\,\text{with}\,\,\, \lambda_b \mapsto [\lambda_a, \lambda_b].$$
So, t... |
On the tail section of the YF-22 further back in the image below, there is a red canister. What is that canister for? I suppose it is for some flight testing purposes from what I can see.
I can't pinpoint the image source.
Aviation Stack Exchange is a question and answer site for aircraft pilots, mechanics, and enthusi... |
Continuity at a Point
Definition: Let $f : A \to \mathbb{R}$ be a function and let $c \in A$. We say that $f$ is continuous at the point $c$ if $\forall \epsilon > 0$ $\exists \delta > 0$ such that $\forall x \in A$ with $\mid x - c \mid < \delta$ then $\mid f(x) - f(c) \mid < \epsilon$. Another definition of continuit... |
Pointwise Cauchy Sequences of Functions
Definition: A sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $X$ is said to be Pointwise Cauchy on $X$ if for all $\epsilon > 0$ and for all $x \in X$ there exists an $N \in \mathbb{N}$ such that if $m, n \geq N$ then $\mid f_m(x) - f_n(x) \mid < \epsilon$.
In... |
Main Page Contents The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math],... |
Group Homomorphisms From Group of Order 21 to Group of Order 49
Problem 346
Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.Suppose that $G$ does not have a normal subgroup of order $3$.Then determine all group homomorphisms from $G$ to $K$.
Let $e$ be the identity element of the gro... |
Description
This data set contains approximately 140,000 shear-stabilized jammed packings, as described in [1].
These packings contain (N = 16... 4096) particles with harmonic interactions, under a confining pressure p=10^{-7}\...10^{-2}\). Ensemble sizes range from 10 (N=4096) to 5000 (N=16). Particle interactions The... |
Question:-Use Chebychev's inequality to show that for any $k>1$, $e^{k+1}\ge k^2$
Chebychev's inequality states that for any random variable $X$ with finite mean, and for $k>0$ $$P(|X-\mu|\le k\sigma)\ge 1-\frac {1}{k^2}$$ Or $$P(|X-\mu|\ge k\sigma)\le \frac {1}{k^2}$$ A general for of the inequality is given in the bo... |
Illinois Journal of Mathematics Illinois J. Math. Volume 54, Number 2 (2010), 621-647. Spectral multipliers for Schrödinger operators Abstract
We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ ... |
The point is that sometimes, different models (for the same data) can lead to likelihood functions which differ by a multiplicative constant, but the information content must clearly be the same. An example:
We model $n$ independent Bernoulli experiments, leading to data $X_1, \dots, X_n$, each with a Bernoulli distrib... |
Assume on the contrary that each matrix equation $A\mathbf{x}=\mathbf{e}_i$ has a solution.Let $\mathbf{b}_i\in \R^n$ be a solution of $A\mathbf{x}=\mathbf{e}_i$ for each $i=1, \dots, n$.That is, we have\[A\mathbf{b}_i=\mathbf{e}_i.\]Let $B=[\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n]$ be the $n\times n$ matrix wh... |
How does he do the algebra? (page 134 Rudin, chapter 6 ,theorem 6.20)
$\left| \frac {F(t)- F(s)}{t-s} -f(x_o) \right| = \left| \frac{1}{t-s} \int_s^t[f(u) - f(x_o)]du \right|< \epsilon $
also, how does he conclude that $F'(x_o) = f(x_0)$
: here is the theorem(and the proof by rudin)
Let $f \in \Re$ on $[a,b]$. For $ a ... |
Let $A_1, A_2, A_3, \,\ldots$ be sets such that $A_1 \cap A_2 \cap \cdots \cap A_n \ne \emptyset$ holds for all $n$.
Must it be that $\bigcap_{n = 1}^{\infty}A_n \ne \emptyset$?
I answered no. Here is my "proof".
Define $A_n = \{n+1, n+2, \,\ldots\}$.
Then $A_1 \cap A_2 \cap \cdots \cap A_n = \{n+1, n+2, \,\ldots\}$.
F... |
I'm having trouble with the understanding on this last problem from my homework set.
Two cars playing demolition derby are moving towards each other with constant velocity, as measured by Marvin, who is a stationary observer. As measured by Marvin, a green car on the left is moving rightward at 0.80 c and an orange car... |
Motivating examples:
Let $V$ be a
real vector spacewith Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$ The LHS being the $W^{s,2}$-type Sobolev space and the RHS a weighted $L^2$ space on the dual.
Consider $S^1$ a
circlewith haar m... |
Expressing a Linear Functional as a Linear Combination of Other Linear Functionals
If $X$ is a linear space and $\varphi, \psi_1, \psi_2, ..., \psi_n$ are linear functionals, we would like to obtain a criterion for when $\varphi$ is a linear combination of $\psi_1, \psi_2, ..., \psi_n$.
The theorem below gives us such ... |
1. UNNECESSARY PROBABILITIES.
The next two sections of this note analyze the "guess which is larger" and "two envelope" problems using standard tools of decision theory (2). This approach, although straightforward, appears to be new. In particular, it identifies a set of decision procedures for the two envelope problem... |
From this, we see that the matrix $A$ is nonsingular if and only if the $(3, 3)$-entry $4a^2+a+1$ is not zero.By the quadratic formula, we see that\[a=\frac{-1\pm \sqrt{-15}}{8}\]are solutions of $4a^2+a+1=0$.
Note that these are not real numbers. Thus, for any real number $a$, we have $4a^2+a+1\neq 0$.
Hence, we can d... |
Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\]The set $W$ is called the center of $V$.
We must show that $W$ satisfies the three criteria for vector subspaces.Nam... |
Wave energy converters in coastal structures Introduction
Fig 1: Construction of a coastal structure.
Coastal works along European coasts are composed of very diverse structures. Many coastal structures are ageing and facing problems of stability, sustainability and erosion. Moreover climate change and especially sea l... |
Looking for an example of a group homomorphism to better grasp the concept. Something along the lines of defining a simple group such as $X = \{1,0\}$ and $G = (X, \circ)$, and $A = \{1,2,3\}$ is an arbitrary set. Then taking that (or preferrably a better example) and demonstrating explicitly how the group homomorphism... |
My issue is about the proper development of the action of the momentum operator $P^{\mu}$ - the generator of spacetime translations - on multi-particle states. I'm a little clueless on this, so I'm going to build up my question without developing it at all. Hopefully somebody can fill in the holes.
Consider a scalar qu... |
Table of Content What is Radiation?
The process of the transfer of heat from one place to another place without heating the intervening medium is called radiation. The term radiation used here is another word for electromagnetic waves. These waves are formed due to the superposition of electric and magnetic fields perp... |
I want to simplify the expression $\tan(\arctan(a) - \arctan(b))$ to $\tfrac{a-b}{1+a\cdot b}$ (if $b \neq - \tfrac{1}{a}$). Right now, the output of
FullSimplify[Tan[ArcTan[a] - ArcTan[b]], {a > 0, b > 0, a < Pi/2, b < Pi/2}]
is the same as the input. The assumptions are to assure that $b \neq -\tfrac{1}{a}$ and to gi... |
Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that\[b^m=a.\]Add to solve later
Since $m$ and $n$ are relatively prime integers, there exists integers $s, t$ such that\[sm+tn=1.\]
Then we have\... |
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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 ... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Why does the following DFA have (to have) the state $b_4$?
Shouldn't states $b_1,b_2,b_3$ already cover "exactly two 1s"? Wouldn't state $b_4$ mean "more than two 1s", even if it doesn't trigger an accept state?
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of... |
I'm top-editing this since it answers the question directly.
The sinc series is fundamentally a $C/x$, so you can extract as many absolutely convergent series out of it as you want, but what is left over is still only conditionally convergent. Also, you can rescale $x$ and it is still a $C/x$ series.
Saying you have a ... |
Fourier transformations:
$$\phi(\vec{k}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{r\text{ space}} \psi(\vec{r}) e^{-i \mathbf{k} \cdot \mathbf{r}} d^3r$$
for momentum space and
$$\psi(\vec{r}) = \left( \frac{1}{\sqrt{2 \pi}} \right)^3 \int_{k\text{ space}} \phi(\vec{k}) e^{i \mathbf{k} \cdot \mathbf{r}} d^3k$$
f... |
Class Member of Class Builder
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Theorem
Let $A$ be a class.
Let $x$ be a set.
Let $P \left({A}\right)$ denote the formula $P\left({x}\right)$ with all free instances of $x$ replaced with instances of $A$.
Then: $A \in \left\{{x : P \left({x}\right)}\right\} \iff \left({\exists x: x = A \... |
Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term
Università di Cagliari, Dipartimento di Matematica e Informatica, V.le Merello 92,09123 Cagliari, Italy
$\left\{ \begin{array}{l}\begin{split}u_t- \lambda \triangle u_t=& k(t) \text{div}(g(| \nabla u|^2) \nabla u) +f(t,u,| \nabla u| ) \quad {... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed i... |
Given a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$
where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the constraint (2), then there exists a constant $\lambda$ such that $y=y(x)$ is also an extreme of the functional $$\int_a^b [F(x,y,y')+\... |
In this section,
sdr means complete sdr.It is easy to see that not every collection of sets has an sdr. Forexample,$$A_1=\{a,b\}, A_2=\{a,b\}, A_3=\{a,b\}.$$The problem is clear: there are only two possible representatives, soa set of three distinct representatives cannot be found. This exampleis a bit more general tha... |
14.1. Generative Adversarial Networks¶
Throughout most of this book, we’ve talked about how to make predictions. In some form or another, we used deep neural networks learned mappings from data points to labels. This kind of learning is called discriminative learning, as in, we’d like to be able to discriminate between... |
Suppose I have a high-dimensional vector space $X$, a subspace $V \subset X$, and a collection of $n$ vectors $\{x_i\}_{i=1}^n \subset X$.
My question is: How can I choose a small collection $k < n$ of the vectors $x_i$ so that the span of this smaller collection "well-approximates" the subspace $V$?
The notion of "wel... |
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Skills to Develop
Identify the age of materials that can be approximately determined using radiocarbon dating.
When we speak of the element Carbon, we most often refer to the most naturally abundant stable isotope
12C. Although 12C is definitely essential to life, its unstable sister isotope 14C has become of extreme i... |
Your presumption in the first paragraph is incorrect. The fact that more particles collide with enough energy to make the activation energy with an increase in temperature means that the
rate of reaction increases as temperature goes up, but that doesn't affect the equilibrium of the reaction.
Instead, the effect on te... |
First, I propose to rewrite your original system (by multiplyingnumerator and denominator by $x\,y$) as\begin{align} x' &= \frac{d_2 r_1\,x + (r_1 a_{22} - r_2 a_{12})\,x\,y}{a_{12}a_{21}\, x\, y - (d_1 + a_{11}\,x)(d_2 + a_{22}\,y)}, \tag{1a}\\ y' &= \frac{d_1 r_2\,y + (r_2 a_{11} - r_1 a_{21})\,x\,y}{a_{12}a_{21}\,x\... |
Lahiri's A First Book on Quantum Field Theory states on problem 4.24 that from the Pauli-Lubansky vector $$W_\mu=-\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}P^\nu J^{\lambda\rho}$$ one can prove that for eigenstates of momentum and angular momentum $W^\mu W_\mu=-m^2s(s+1)$ where $m$ is the mass of the particle and $s$ is t... |
This is a slightly more advanced version of another question here.
Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by which I mean a non-trivial commutative ring with unit such that the zero ideal is prime. Let $\textbf{Fld}$ be the category ... |
Let $\xi$ be a positive parameter. We say a positive integer $n$ is $\xi$-smooth, or friable, if for all primes $p$ dividing $n$, we have $p \leq \xi$. Let $T(X, \xi)$ denote the set of $\xi$-smooth numbers up to $X$, and $\Psi(X,\xi) = \# T(X,\xi)$. Then it is known that for all $a > 0$, we have $\Psi(X,X^{1/a}) \sim ... |
New Impedance Boundary Conditions for Acoustics Simulations
When developing a new product or functionality, the first step is typically to understand the functional properties in isolation. To achieve reliable and accurate predictions via mathematical modeling, it is essential that the critical components, test setup, ... |
Interactions between ions, dipoles, and induced dipoles account for many properties of molecules - deviations from ideal gas behavior in the vapor state, and the condensation of gases to the liquid or solid states. In general, stronger interactions allow the solid and liquid states to persist to higher temperatures. Ho... |
This site is going to need TeX markup to express mathematical symbols and formulas well.
I have already started using TeX in my markup.
The way that Math Overflow does it is to surround the TeX with dollar signs, using the jsMath library.
I suggest we stick with that convention, so that when we have support for it, our... |
To small numerical factors, the fusion reaction rate is $r_{AB} \propto n_A n_B <\sigma_{AB}\, v>$, where $<\sigma_{AB}\, v>$ is the (temperature-dependent) "reactivity" for the reaction, formed from the averaging the cross-section over an appropriate Maxwellian velocity distribution, and $n$ are the number densities o... |
Product Rule for Derivatives Theorem
Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable.
Let $\map f x = \map j x \, \map k x$. Then: $\map {f'} \xi = \map j \xi \, \map {k'} \xi + \map {j'} \xi \, \map k \xi$ $\forall x \in I: \map {f'} x = \map j x \, \map {k'} x + \map {j'} x \, \map k x$... |
Answer
56.55 $ mm ^ 2 $
Work Step by Step
We know that the area is equal to: $$A=\pi r^2 \times \frac{\theta}{2\pi \ radians}$$ We know that the area is equal to: $$A=r^2 \times \frac{\theta}{2}$$ $$A=12^2 \times \frac{\pi/4}{2}=56.55$$
You can help us out by revising, improving and updating this answer.Update this ans... |
Let assume that running a
single DES key search operation requires 1 key schedule, encryption, and 1 comparison that now we omit.
for key in 2^56
k = key_schedule(key) //note to distingush the key schedule has a funtion
test (E(k,m), c)
Your 3DES requires3 encryptions, only 2 schedule runs if we want to attack brute-fo... |
$\ce{NH3}$ is a weak base so I would have expected $\ce{NH4+}$ to be a strong acid. I can't find a good explanation anywhere and am very confused. Since only a small proportion of $\ce{NH3}$ molecules turn into $\ce{NH4+}$ molecules, I would have expected a large amount of $\ce{NH4+}$ molecules to become $\ce{NH3}$ mol... |
Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}\mathrm dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) $$ which is the Fourier Transform of the cube of Bessel function. The Bessel function $J_0$ is given by $$ J_0(x)... |
Contents Procedural Texturing
Texturing in computer graphics is a very common technique used to add details to the surface of an object. The principle is similar to wallpaper in a way. The only thing you can do with objects is to render them as diffuse object for example using a solid color for the entire object. The a... |
October 5th, 2017, 04:38 PM
# 31
Senior Member
Joined: Sep 2016
From: USA
Posts: 666
Thanks: 437
Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
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$f(x) = x$ maps continuously to the entire real line. Therefore $xf(x) = x^2$ must also map to the entire real line.
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Hmm, my points just went from ~111,000 to 77,000. Did I miss something?
Note: I'm not actually too awfully concerned about points, just wondering.
Note by Cody Johnson 5 years, 2 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Ch... |
Zurek 2001 is a review article on decoherence in quantum mechanics. Equation 5.36 on p. 24 gives an estimate of the decoherence time, which I'll paraphrase as follows:
$ \frac{t_D}{t_R} = \left(\frac{\lambda_T}{x-x'}\right)^2 $ .
My attempt to interpret the meaning of the variables is:
$t_D$ is the decoherence time
$t_... |
I am working on my master thesis on acoustic pattern recognition and have a question to the calculation of cepstral coefficients.I understand the step with a filter bank in MFCC calculation, it is ...
I've seen different equations that calculate cepstrum from power spectrum, but the equations are not consistent. Some p... |
Please help to solve this question:
Instability of a difference scheme under small perturbations does not exclude the possibility that in special cases the scheme converges towards the correct function, if no errors are permitted in the data or the computation. In particular let $f(x)=e^{\alpha x}$ with a complex const... |
I am reading mixed opinions comparing bayesian A/B testing and frequentist ones.
Basically my understanding is the following; consider the case of A/B testing with binary outcomes (https://www.evanmiller.org/bayesian-ab-testing.html#mjx-eqn-binary_ab_pr_alpha_b):
With frequentist approach, you basically calculate the $... |
The capillary forces referred to the fact that surface tension causes liquid to rise or penetrate into area (volume), otherwise it will not be there. It can be shown that the height that the liquid raised in a tube due to the surface tension is \[h = \frac{2\sigma\cos(\beta)}{g\Delta\rho r}\tag{73}\] Where \(\Delta\rho... |
Sometime ago we came across invariant quantities under twisting of all affine algebra. (See the appendix E of http://arxiv.org/abs/hep-th/0403076 .) Choose the convention so that the longest root has length $\sqrt{2}$, and the co-root of a root $\alpha$ is $\alpha^*=2\alpha/\alpha^2$. The lowest negative root $\alpha_0... |
Consider the Heston model given by the following set of stochastic differential equations: $$\frac{dS_{t}}{S_{t}}=\mu_{t}dt+\sqrt{V_{t}}dW_{t}, S_{0}>0,$$ $$dV_{t}=\kappa(\theta-V_{t})dt+\xi\sqrt{V_{t}}dZ_{t}, V_{0}=v_{0}>0,$$ $$d<W,Z>_{t}=\rho dt$$ where $W_{t}$ and $Z_{t}$ are two brownian motion, $\kappa,\theta,\xi>... |
Entropy is a notoriously tricky subject. There is a famous anecdote of John von Neumann telling Claude Shannon, the father of information theory, to use the word “entropy” for the concept he had just invented, because “nobody knows what entropy really is, so in a debate you will always have the advantage“.
Entropy mean... |
I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$
Insulator (Neumann BC) Electrode set at some potential (Dirichlet BC). Additionally I have some electrodes $E_l$ which are set to have a fixed current $I_l$, so on these th... |
J. D. Hamkins and R. Yang, “Satisfaction is not absolute,” to appear in the Review of Symbolic Logic, pp. 1-34, 2014.
@ARTICLE{HamkinsYang:SatisfactionIsNotAbsolute,
author = {Joel David Hamkins and Ruizhi Yang},
title = {Satisfaction is not absolute},
journal = {to appear in the Review of Symbolic Logic},
year = {2014... |
The following is a theorem given in Bella Bollobas's Modern graph theory(Springer 2002) Page-73
If $G$ is nontrivial (that is, has at least two vertices), then the parameters $\delta(G)$, $\lambda(G)$ and $\kappa(G)$ satisfy the following inequality: $\kappa(G) \le \lambda(G) \le \delta(G)$, where $\delta(G)$ is the mi... |
When I try Find the volume of the region $R$ lying below the plane $z = 3-2y$ and above the paraboloid $z = x^2 + y^2$
Solving the 2 equations together yields the cylinder $x^2 + (y+1)^2 = 4 $ How do I get the volume then???
Mathematics Stack Exchange is a question and answer site for people studying math at any level ... |
Denote the following:
$S_i$ to be the sum of the digits in column $i$. $D_i$ to be the digit in column $i$ of the sum. Thus $D_i = S_i \mod 10$ $C_i$ to be the carry over of $S_i$.
Given that all the numbers are even, we know that $N,X,E \in \{0,2,4,6,8\}$. Looking at $S_1=N+N+X+E \le 8+8+6+4=26 \implies C_1 \le 2$.
Le... |
I am looking at Kemna and Vorst's paper:
A PRICING METHOD FOR OPTIONS BASED ON AVERAGE ASSET VALUES. see http://www.javaquant.net/papers/Kemna-Vorst.pdf
Let $\text{d}S_t = S_tr\text{d}t + S_t\sigma\text{d}W_t$. Let $t_0 \leq t \leq T$, define $A(t)=\frac{1}{T-t_0}\int^T_{t_0}S_\tau\text{d}\tau$.
The Asian option has pa... |
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D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
The question asks: if $\sum a_n$ converges, $\{b_n\}$ is monotonic and bounded, prove that $\sum a_nb_n$ converges.
My proof goes as follows:
Let $\varepsilon>0$, and let $S_k$ denote $k$-th partial sum of $\sum a_nb_n$. Now, $\{b_n\}$ is monotonic and bounded, so it converges, say to $b$.
Since $\{b_n\}$ converges, we... |
I am trying to design an efficient algorithm that retrieves the ith to jth largest elements in an array. For example, if the following array is the input:
[10, 14, 18, 3, 21, 25, 27] i = 2 j = 4
The following array is returned:
[25, 21, 18]
18 is the 4th largest element in the array and 25 is the 2nd smallest element i... |
MathJax TeX Test PageThe moment of inertia of a point particle is $\boxed{mr^2}$. $r$ is the distance from the axis of rotation. For future reference, it is important to note that moments of inertia always add.
Rod rotated about End
MathJax TeX Test PageBecause moments of inertia always add, we can think of a road as a... |
The other person is likely thinking about the concept of escape velocity. There are people who claim no man-made spaceship could ever have gone into space because they cannot reach the escape velocity for the earth. What they fail to acknowledge is that the escape velocity for the earth is not a constant, it depends on... |
Let's say we have an object in 3D space located at
P(1,1,1) and if we decided we wanted to rotate this object within any of the planes
XY,
XZ,
YZ along any of the 3 axis where this is a basic 3D Cartesian system in a Right-Hand System the rotation matrices $R_n$ for
P will look like these:
$$ R_x \space|| \space R_y \s... |
Electronic Communications in Probability Electron. Commun. Probab. Volume 23 (2018), paper no. 73, 14 pp. Approximation of a generalized continuous-state branching process with interaction Abstract
In this work, we consider a continuous–time branching process with interaction where the birth and death rates are non lin... |
Does Smeaton's coefficient, k, have a modern value or it is dependent of the air density?
Why is the accepted value of k so high?
In various texts about the Wright brothers (see 1 and 2) one can read about Smeaton's coefficient that troubled them a lot and that they finally discovered the parameter had a much lower val... |
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Let ωn=eiπ/n\omega_n=e^{i\pi/n}ωn=eiπ/n. The value of ∣∏j=12014(∏i=1j(ωj2i−1−1))∣\left|\prod_{j=1}^{2014}\left(\prod_{i=1}^j\left(\omega_j^{2i-1}-1\right)\right)\right|∣∣∣∣∣j=1∏2014(i=1∏j(ωj2i−1−1))∣∣∣∣∣ can be expressed as aba^bab where a,ba,ba,b are positive in... |
Hi, Can someone provide me some self reading material for Condensed matter theory? I've done QFT previously for which I could happily read Peskin supplemented with David Tong. Can you please suggest some references along those lines? Thanks
@skullpatrol The second one was in my MSc and covered considerably less than my... |
A local system is a sheaf of finite dimensional vector spaces that is locally isomorphic to the constant sheaf $k^n$. If $\gamma: [0,1] \to X$ is a continuous path in $X$, $\gamma^{-1}(L)$ is again local system on $[0,1]$ but one shows that any locally constant sheaf on $[0,1]$ is actually constant. So the fibers at 0 ... |
Hello,Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)?Also, does it have torsion? It must either have torsion or not be...
Hello!I have learned Ri... |
\( \newcommand{\norm}[1]{\left \lVert #1 \right \rVert} \)
\( \newcommand{\Real}{\mathbb{R}}\)
Hello! So this semester has been a fairly busy one for me and so I have not made much time to get anything new written.. until now!
Lately I’ve been having some real fun with optimization, especially convex optimization and s... |
Maxwell's equations are the set of four equations by James Clerk Maxwell that describe the behavior of both the electric and magnetic fields. Maxwell's equations provided the basis for the unification of electric field and magnetic field, the electromagnetic description of light, and ultimately, Albert Einstein's theor... |
I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that
$$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,$$
where $\|\cdot\|_\infty$ denotes the operator norm, and $\hat{A_i}$ denotes the omission of $A_i$.
E.g. if all product... |
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