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The Binomial Theorem
Consider the expansion of the binomial $(1 + x)^n$ for $n \in \{ 0, 1, 2, ... \}$. When $n = 0$ we have that:(1)
When $n = 1$, $n = 2$, and $n = 3$ we get:(2)
Notice that if we list the terms of the expansion of $(1 + x)^n$ in ascending order then the coefficients of these terms match the numbers i... |
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Find the remainder when ∑k=1100k(k!)\sum_{k=1}^{100} k(k!)k=1∑100k(k!) is divided by 11.
Details:-
∙\bullet∙ k!k!k! stands for factorial of kkk, that is k!=k×(k−1)×(k−2)×...×2×1k!=k\times (k-1)\times (k-2) \times ... \times 2 \times 1k!=k×(k−1)×(k−2)×...×2×1
This... |
Suppose that we have right-angled triangles $ABC$ and $A_1B_1C_1$ with $\angle C=\angle C_1=90^\circ$ and $\angle A=\angle A_1$. The two triangles are similar.
So. we have $\displaystyle \frac{BC}{AB}=\frac{B_1C_1}{A_1B_1}$, $\displaystyle \frac{AC}{AB}=\frac{A_1C_1}{A_1B_1}$ and $\displaystyle \frac{BC}{AC}=\frac{B_1C... |
Let $G=(V,E)$ be a simple, undirected and connected graph. We say that $S\subseteq V$ is a
cutting set if $S\neq V$ and the induced subgraph on $V\setminus S$ is not connected any more.
If $S \subseteq V$ is a cutting set of $G$, is there a cutting set $S_0\subseteq S$ of $G$ such that for all $x\in S_0$ the set $S_0\s... |
When the stockprice follows a GBM, the arbitrage-free value of an EU call is given by the Black Scholes model:
\begin{align} C(S, t) &= N(d_1)S_0 - N(d_2) Ke^{-r(T - t)} \\ d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_0}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_2 &= d_1 - \sigma\s... |
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Now showing items 1-10 of 19
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 in... |
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Now showing items 1-7 of 7
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 i... |
Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such that\[f(a)=1.\]Let $H=\langle a \rangle$ be the subgroup of $G$ generated by the element $a$.
We show that $G\cong \ker(f)\times H$.To prove this isomorphism, it suffices to prove the following three conditions.
The subgroups $\ker(f)$ and $H$ are n... |
Commutative Laws of Sets
Commutative Laws of Sets
We will now look at the commutative laws between two sets. These proofs are relatively straightforward.
Theorem 1 (Commutative Law for the Union of Two Sets): If $A$ and $B$ are sets then $A \cup B = B \cup A$. Proof:Suppose that $x \in A \cup B$. Then $x \in A$ or $x \... |
Comparability and permutation graphs¶
The following methods are implemented in this module
is_comparability_MILP()
Tests whether the graph is a comparability graph (MILP)
greedy_is_comparability()
Tests whether the graph is a comparability graph (greedy algorithm)
greedy_is_comparability_with_certificate()
Tests whethe... |
Localization of blow-up points for a nonlinear nonlocal porous medium equation
1.
Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China, China
$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$
subject to homogeneous Dirichlet condition. We investigate the influence of the nonlocal so... |
Large time behavior of ODE type solutions to nonlinear diffusion equations
1.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
2.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
$ \begin{equation} \left\{ \begin{array}{ll} \partial... |
Note first that those $X_i = 0$ contribute nothing to the sum. We can exclude them by remembering that if we generate a Poisson variate $x$ with mean $\lambda$ and then a Binomial $z$ with probability parameter $p$ and size parameter equal to $x$, $z \sim \text{Poisson}(p\lambda)$; consequently the number of
nonzero el... |
The Riemann Sphere
Definition: The Riemann Sphere denoted $\mathbb{C}_{\infty} = \mathbb{C} \cup \{ \infty \}$ is the topological space adjoining the single point $\infty$ to $\mathbb{C}$.
We can readily define a very simple two chart atlas on $\mathbb{C}_{\infty}$, call it $\mathcal M = \{ (U_0, \phi_0), (U_{\infty}, ... |
Linear Independent Vectors and the Vector Space Spanned By Them
Problem 141
Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \... |
Table of Contents
Associativity and Commutativity of Binary Operations
Recall from the Unary and Binary Operations on Sets that a binary operation on a set $S$ if a function $f : S \times S \to S$ that takes every pair of elements $(x, y) \in S \times S$ (for $x, y \in S$) and maps it to an element in $S$.
Sometimes th... |
Zero Matrices
Definition An $m \times n$ matrix $A$ is a Zero Matrix if all entries in the matrix are $0$, that is $a_{ij} = 0$ for all $1 ≤ i ≤ m$ and $1 ≤ j ≤ n$, $i, j \in \mathbb{N}$.
The definition of a zero matrix is pretty self explanatory. For example, if $A$ is a $2 \times 3$ zero matrix would look like this $... |
this is a mystery to me, despite having changed computers several times, despite the website rejecting the application, the very first sequence of numbers I entered into it's search window which returned the same prompt to submit them for publication appear every time, I mean ive got hundreds of them now, and it's stil... |
The Annals of Statistics Ann. Statist. Volume 21, Number 4 (1993), 1663-1691. Incidental Versus Random Nuisance Parameters Abstract
Let $\{P_{\vartheta,\eta}:(\vartheta, \eta) \in \Theta \times H\}$, with $\Theta \subset \mathbb{R}$ and H arbitrary, be a family of mutually absolutely continuous probability measures on ... |
Enhancement of the HWZ vertex in the three scalar doublet model
Presented by Ms. Diana ROJAS
Content
We compute one-loop induced trilinear vertices with physical charged Higgs bosons $H^\pm$ and ordinary neutral gauge bosons, i.e., $H^\pm W^\mp Z$ and $H^\pm W^\mp \gamma$, in the model with two active plus one inert sc... |
The Derived Set of a Set in a Metric Space
Recall from the Adherent, Accumulation and Isolated Points in Metric Spaces page that if $(M,d)$ is a metric space and $S \subseteq M$ then a point $x \in M$ is said to be an accumulation point of $S$ if for all $r > 0$ we have that:(1)
In other words, every ball centered at $... |
I learned here that I can use
\\[<len>] to explicitly set the vertical skip space between lines, e.g.,
<len> set to
3ex. I'd like to set
<len> to, say, 2-times the normal length in this environment (e.g.,
align or
dcases) but I don't know the length parameter that determines this.
Another example: I'd like to change th... |
Agenda de la FDP Séminaire d'AnalyseLe lundi à 10h30 - Salle 1180 (Bât E2)(Tours)
Responsable :
Extinction in a finite time for solutions of a class of Parabolic Equations involving $p$-Laplacian Yves Belaud jeudi 17 octobre 2019 - 10h30 - Salle 1180 (Bât E2)(Tours) Résumé :
We study the property of extinction in a fin... |
The definition of the SI base unit "metre" [1] doesn't seem to rule out explicitly that a certain value of "length, in meters" could be attributed to a pair of ends which are rigid to each other, but not at rest to each other.
Consider, therefore, two such ends, $A$ and $B$, which both find constant but unequal ping du... |
Compact Sets in a Metric Space
Recall from the Coverings of a Set in a Metric Space page that if $(M, d)$ is a metric space and $S \subseteq M$ then a cover or covering of $S$ is a collection of subsets $\mathcal F$ in $M$ such that:(1)
Furthermore, we said that an open cover (or open covering) is simply a cover that c... |
Does the existence of a holomorphic square root for the identity function in a region $\Omega$ in $\mathbb C$ imply the existence of a holomorphic logarithm for the same function? I have no idea how to prove this.
closed as off-topic by José Carlos Santos, Parcly Taxel, Shailesh, user99914, TheSimpliFire Feb 4 '18 at 1... |
Prove that the matrix\[A=\begin{bmatrix}0 & 1\\-1& 0\end{bmatrix}\]is diagonalizable.Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal matrix.
Let\[A=\begin{bmatrix}2 & -1 & -1 \\-1 &2 &-1 \\-1 & -1 & 2\end{bm... |
We owe Paul Dirac two excellent mathematical jokes. I have amended them with a few lesser known variations.
A.
Square root of the Laplacian: we want $\Delta$ to be $D^2$ for some first order differential operator (for example, because it is easier to solve first order partial differential equations than second order PD... |
The term "distributed ray tracing" was originally coined by Robert Cook in this 1984 paper. His observation was that in order to perform anti-aliasing in a ray-tracer, the renderer needs to perform spatial upsampling - that is, to take more samples (i.e. shoot more rays) than the number of pixels in the image and combi... |
Suppose that a random variable has a lower and an upper bound [0,1]. How to compute the variance of such a variable?
You can prove Popoviciu's inequality as follows. Use the notation $m=\inf X$ and $M=\sup X$. Define a function $g$ by $$ g(t)=\mathbb{E}\left[\left(X-t\right)^2\right] \, . $$ Computing the derivative $g... |
Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces
1.
School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330032, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
J. Funct. Anal., 87(1989), 359-... |
The aim of this test case is to validate the following functions:
The simulation results of SimScale were compared to the analytical results derived from [Roark]. The mesh used was created locally consisting of quadratic hexahedral elements and uploaded to the SimScale platform.
The bimetallic strip has a length of l=1... |
Note that the zero vector in the vector space $C[-\pi, \pi]$ is the zero function\[\theta(x):=0.\]
Let us consider a linear combination\[a_1\cos(x)+a_2\sin(x)=\theta(x)=0 \tag{*}.\]If this linear combination has only the zero solution $a_1=a_2=0$, then the set $\{\cos(x), \sin(x)\}$ is linearly independent.
The equalit... |
There are two outgoing links in this question of mine here, but only an incoming link is shown in the "Linked" box.
EDIT
Also an incoming link from this question Number of ways, powers of $2$ sum up specific values doesn't appear in the "Linked" section of this question What's the non-trivial root of $\lim \limits_{n\t... |
Yes, you should taking into account the scale of the output $y$ and should also take into account the scale of the covariates in $X$.
Let $X \in \mathbb{R}^{n \times p}$ be the design matrix, whose rows are vectors with each entry being a covariate that together seek to
explain the response $y \in \mathbb{R}^n$. Each e... |
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Now showing items 1-9 of 9
Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 ... |
Properties of Vector Spaces
Properties of Vector Spaces
We will now look at some important properties of vector spaces and provide what may seem like trivial proofs. Please review the Vector Spaces page first nevertheless.
Theorem 1: If $V$ is a vector space and $x \in V$ then there exists only one additive identity $0... |
Put $\alpha=\sqrt{2+\sqrt{2}}$. Then we have $\alpha^2=2+\sqrt{2}$. Taking square of $\alpha^2-2=\sqrt{2}$, we obtain $\alpha^4-4\alpha^2+4=2$. Hence $\alpha$ is a root of the polynomial\[f(x)=x^4-4x+2.\]By the Eisenstein’s criteria, $f(x)$ is an irreducible polynomial over $\Q$.
There are four roots of $f(x)$:\[\pm \s... |
I asked this on MathStackExchange and was instructed it would be better here.
I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on $\mathbb{P}^{2}$ of rank $N$ and second Chern class ... |
The Matrix Form of the Chain Rule for Compositions of Differentiable Functions from Rn to Rm
Recall from The Chain Rule for Compositions of Differentiable Functions from Rn to Rm page that if $S \subseteq \mathbb{R}^n$ is open, $\mathbb{a} \in S$, $\mathbf{g} : S \to \mathbb{R}^p$, and if $\mathbf{f}$ is another functi... |
Almost six years ago, Michael Hardy raised the issue of the "partitions" tag being used for some very different concepts, and subsequently edited its tag wiki excerpt to at least be clear about what the various concepts covered were. Two months ago the tag wiki was changed to be solely about in...
I have seen some post... |
I derived demand, given a Cobb-Douglas utility function but I am not really sure if I did it correctly. I am especially struggling with the sum signs and the subscripts of $i$ & $j$. It would be really great if someone could check. I want to maximize utility for 2 goods, here $j$ and $i$.
$\ u(x_i)=\prod_{i=1}^n x^a_i ... |
Let $C$ be the event that a randomly chosen person has lung cancer. Let $S$ be the event of a person being a smoker.Suppose that 10% of the population has lung cancer and 20% of the population are smokers. Also, suppose that we know that 70% of all people who have lung cancer are smokers.
Then determine the probability... |
Invariant Subspaces
Definition: Let $V$ be a vector space over the field $\mathbb{F}$, and let $T$ be a linear operator from $V$ to $V$, that is $T \in \mathcal L (V)$. A subspace $U$ of $V$ is said to be Invariant Under $T$ if for all $u \in U$ we have that $T(u) \in U$.
Alternatively we can say that the subspace $U$ ... |
Dear All
In the classical refutation method, one searches for a proof of $\Gamma, \lnot A \vdash \bot$ instead of $\Gamma \vdash A$. The method works, i.e. is complete and correct, since it is for example easily seen that both sequents are interderivable (*).
In a Robinson resolution method based on the refutation meth... |
When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition could be derived.
In the rest of this post $\bar{f}$ is such that $\bar{f}'=f$ and $t_{j}^{f}$ is such that $t_{j}\leq t_... |
Let $1\leq p\leq +\infty$, $0<s<1$ and $\Omega\subseteq \mathbb{R}^n$ an open set. The
fractional Sobolev space $W^{s,p}(\Omega)$ is defined to be
$$ W^{s,p}(\Omega) = \left\{ u\in L^p(\Omega) : \frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p} + s}} \in L^p(\Omega\times\Omega) \right\} $$
equipped with the norm
$$ \|u\|_{W^{s,p}(... |
There is a vast range of problems that fall under the broad umbrella of making sequential decisions under uncertainty. While there is widespread acceptance of basic modeling frameworks for deterministic versions of these problems from the fields of math programming and optimal control, sequential stochastic problems ar... |
Just 2 simple questions I´m struggling with. Hope you can help.
Suppose that the model $y=\beta X +\epsilon$ with $\epsilon \sim \text{ Normal}(o,\sigma^2I_n)$ has a prior $\beta \sim \text{ Normal}( \beta_0, k(X^TX)^{-1})$
I want two things:
1.) I want to show that for the density of $\beta$ that
$$p(\beta) \propto \e... |
Table of Contents
Oscillation and Continuity of a Bounded Function at a Point
Recall from the Oscillation of a Bounded Function at a Point page that if $f$ is a bounded function on $[a, b]$ and $x \in [a, b]$ then the oscillation of $f$ at $x$ is defined to be:(1)
Earlier, on the Oscillation of a Bounded Function on a ... |
I'm trying to implement threshold RSA operations, starting with decryption based on
Peeters, R., Nikova, S., & Preneel, B. (2008). Practical RSA Threshold Decryption for Things That Think. Retrieved from http://www.cosic.esat.kuleuven.be/publications/article-1178.pdf
and running into problems where it seems I would hav... |
I'm assuming here that the $x_{\lambda}$ are real numbers. (Complex numbers would be fine too -- that doesn't matter. This was written before the edit mentioning topological vector spaces in general, and I haven't thought about it at that level of generality.)
In the first part of this answer, we'll see how to define c... |
In this paper, Peter Acquaah asserts that an important difference between odd perfect and even perfect numbers is that:
(A)The greatest component of an odd perfect number $N$ is less than $\sqrt{N}$.
(B)The greatest component of an even perfect number $M$ is greater than $\sqrt{M}$. ATTEMPT TO PROVE STATEMENT (B)
Let $... |
Equations of Planes in Three Dimensional Space
We will now look at equations of planes in $\mathbb{R}^3$. There are three forms of planes that we will look at.
Definition: An equation in the form $Ax + By + Cz + D = 0$ represents the Standard Form Equation of a plane in $\mathbb{R}^3$.
For example, the equation $2x + 3... |
But if you don't want to have a Google account: Chrome is really good. Much faster than FF (I can't run FF on either of the laptops here) and more reliable (it restores your previous session if it crashes with 100% certainty).
And Chrome has a Personal Blocklist extension which does what you want.
: )
Of course you alr... |
I have a question about a proof in Rosenberg and Schochet's paper "the Künneth theorem and the Universal Coefficient Theorem for Kasparov's generalized K-functor", proposition 2.6. First of all, the setting:
Def.: Let $N$ be the bootstrap class of $C^*$-algebras, it's the smallest full subcategory of the seperable nucl... |
This is related to Dirac's theorem.
For any finite, simple, undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimal degree of all vertices.
Are there positive integers $n,c\in\mathbb{N}$ with the following property?
Whenever $G=(V,E)$ is connected and $\delta(G)\geq n$, there is a matching $M\subseteq E$ such tha... |
How do I estimate the residual $\varepsilon_{t}$ of a Seasonal ARIMA model $\hat{Y}_t=\hat{\phi}{Y}_{t-1}+\hat{\Phi}{Y}_{t-12}+\varepsilon_{t}$?
If the MSE is 0.114, what does it mean?
You can calculate $\varepsilon_{t}$ as follows. $$\varepsilon_{t} = {Y}_t - \hat{Y}_t $$
Mean squared Error can be calculated as
$$MSE ... |
Continuity of Functions of Several Variables
Recall that a function of a single variable $y = f(x)$ is continuous at $c \in D(f)$ if $\lim_{x \to c} f(x) = f(c)$. We will now extend the concept of continuity of a function of a single variable to a function of several variables.
Definition: A two variable real-valued fu... |
Proofs Regarding The Supremum or Infimum of a Bounded Set
We will now look at some proofs regarding the supremum/infimum of a bounded set.
Before we do though, let's first recall that for a bounded set $A$, to prove that $\sup A = u$ for some $u \in \mathbb{R}$ we must show that:
1)$u$ is an upper bound to the set $A$.... |
The Lp(E) Normed Linear Space
The Lp(E) Normed Linear Space
Definition: Let $E$ be a Lebesgue measurable set and let $1 \leq p < \infty$. Then the $L^p(E)$ Space is the set $L^p(E) = \{ f \: \mathrm{measurable} : \int_E |f|^p < \infty$ with the norm $\| \cdot \|_p : L^p(E) \to [0, \infty)$ defined for each $f \in L^p(E... |
We study the relations between Adams operation on a lambda-ring and the power structure on it, introduced by S. Gusein-Zade, I. Luengo and A. Melle-Hernandez. We give the explicit equations expressing them by each other. An interpretation of the formula of E. Getzler for the equivariant Euler characteristics of configu... |
The Alternating Series Test
So far we have looked at the following tests to determine if a series was convergent or divergent:
The Integral Test for Positive Series The p-Series Test The Comparison Test for Positive Series The Limit Comparison Test for Positive Series The Ratio Test for Positive Series The Root Test fo... |
I am interested in finding a formula for the inertia matrix of a rigid body about its center of mass. This particular rigid body is composed of other rigid bodies with known inertia matrices about their center of masses. An example is more cubes and some spheres welded together in some shape. Supposing the mass and cen... |
Difference between revisions of "De Bruijn-Newman constant"
(→Threads)
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It is known that <math>\xi</math> is an entire function of order one ([T1986, Theorem 2.12]). Hence by the fundamental solution for the heat equation, the <math>H_t<... |
Since then I introduced constrained CSMC, which is like vanilla CSMC but with some of the classes forbidden as part of the instance specification. Constrained CSMC was designed with the goal of simplifying the CSBM reduction, but it does slightly more. It actually allows me to define a reduction from average constraine... |
The Set of Real-Valued Continuous Functions on a Compact Metric Space X, C(X)
We will soon look at a very important theorem known as The Arzelà–Ascoli Theorem but we will first need to define an important type of metric space. We first define the sets for which our metric space will be over.
Definition: Let $(X, d)$ be... |
@mickep I'm pretty sure that malicious actors knew about this long before I checked it. My own server gets scanned by about 200 different people for vulnerabilities every day and I'm not even running anything with a lot of traffic.
@JosephWright @barbarabeeton @PauloCereda I thought we could create a golfing TeX extens... |
It is not that $h(S_1)=h(S_2)$ is true; in fact $h(S_1)=a$ and $h(S_2)=c$, as explained in the example at pag. 80, where $h$ is the minhash function associated to the permutation $\{abcde\}\mapsto \{beadc\}$ and $S_1$ resp. $S_2$ are given in figure 3.3. What is true is that the
probability of having $h(S_1)=h(S_2)$, f... |
Traveling wave solutions of a generalized curvature flow equation in the plane
1.
Department of Mathematics, Tongji University, Shanghai 200092, China
n, $x)$, where for a simple plane curve $\Gamma$ and for any $P \in \Gamma, k$ denotes the curvature of $\Gamma$ at $P$, ndenotes the unit normal vector at $P$ and $V$ d... |
$Z_1, Z_2, .., Z_{100}$ are independent identical distributed random variables with expected value $E(Z_i)=0$ and variance $Var(Z_i)=1$
Calculate the probability for the event $\sum_{i=1}^{100}Z_{i} \in \left(-10,10\right )$ approximatively.
Hint: We have that $\Phi(1) = 0.8413$ where $\Phi$ is the cumulative distribut... |
Stripes, spots, or a mix of both appear on the skin of many animals — from tigers to beetles to whale sharks. These patterns are typically unique to individual creatures, and biologists often use them for identification. While distinct patterns may seem random, they obey certain rules that suggest a common underlying d... |
Current browse context:
astro-ph.GA
Change to browse by: References & Citations Bookmark(what is this?) Astrophysics > Astrophysics of Galaxies Title: Dynamical Histories of the Crater II and Hercules Dwarf Galaxies
(Submitted on 3 Jan 2019)
Abstract: We investigate the possibility that the dwarf galaxies Crater II and... |
The symbol \(\sigma _{\widehat
p}\) is also used to be smaller than the population standard deviation of individual scores. Please answer the questions: feedback current community blog chat Cross Validated Cross to 360) makes the SE decrease by a factor of 3. For unweighted data, $\omega_i = 1/n$, giving $\sum_{i=1}^n ... |
(
Note: This post focuses on a single simple example, however I'm asking about the error in general in my logic).
Consider the infinite potential well "particle in a box" system described by
$$V(x)=\begin{cases}0&\text{if }0<x<L\\\infty&\text{otherwise}\end{cases}.$$
It's fairly easy to find the wavefunctions $\psi_n(x... |
I'd like to quote, with kind permission of the original author, the text of an article about LaTeX and MathML from access2science. The aim of the website is to provide "articles and links on accessibility of science, technology, engineering, and math (STEM). Its purpose is to provide practical information to people wit... |
We know heavier objects fall faster when dropped at certain height. I was wondering if I am going downhill on my mountain bike without any peddling, will I travel faster or slower because I am fat?
Heavier objects do not fall faster per se. But for heavy objects the influence of the air resistance will be smaller, if t... |
Convexity properties of graphs¶
This class gathers the algorithms related to convexity in a graph. It implements the following methods:
ConvexityProperties.hull()
Return the convex hull of a set of vertices
ConvexityProperties.hull_number()
Compute the hull number of a graph and a corresponding generating set
AUTHORS:
... |
The condition implies that $BA$ commutes with $AB$ and hence they are simultaneously triangularisable over $\mathbb C$.
Let $AB$ and $BA$ be simultaneously triangularised. Since $AB$ and $BA$ in general have identical spectra, if $\lambda_1,\ldots,\lambda_n$ are the eigenvalues of $BA$ along its diagonal, then the entr... |
To send content items to your account,please confirm that you agree to abide by our usage policies.If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.Find out more about sending content to .
To send content items to your Kindle, first ensure no-rep... |
BARC talk by Till Miltzow Tuesday, 18 June, Till Miltzow, assistant professor, Utrecht, will give a talk "Smoothe Analysis of the Art Gallery Problem" Title: Smoothe Analysis of the Art Gallery Problem Abstract:In the Art Gallery Problem we are given a polygon P \subset [0,L]^2 on n vertices and a number k. We want to ... |
I am trying to follow a derivation in this paper from Wald
Specifically, at the end of the paper, just under Eq. 4.7 there is given the equation
$$ Q/m \leq 2 B_0 m$$
where $Q$ is the system charge, $m$ the mass and $B_0$ the magnetic field. This equation in in geometrized units such that $c=G=1$ (as defined at the sta... |
Diagonal Matrices of Linear Operators Examples 1
Recall from the Diagonal Matrices of Linear Operators page that if $V$ is a finite-dimensional vector space and $T \in \mathcal L (V)$, then $T$ is said to be diagonalizable if there exists a basis $B_V$ such that $\mathcal M (T, B_V)$ is a diagonal matrix.
We saw that i... |
Table of Contents
Lebesgue Integrability of the Absolute Value of a Function
Recall from the Lebesgue Integrability of the Positive and Negative Parts of a Function page that if $f$ is a Lebesgue integrable function on $I$ then the positive and negative parts of $f$, $f^+$ and $f^-$, are both Lebesgue integrable on $I$... |
It looks like your program is using an approximation based on$q \approx w = w_s*RH$ with an approximation of Clausius-Clapeyron to find $w_s$. Looking at a few values of RH,T and P, your approximation is pretty close (+/- 5%) to an analytic answer. Based on the output you quoted it looks like you are providing incorrec... |
Why negative interest rates might not work Matthew Martin9/04/2014 01:58:00 PM
Tweetable
1so that we can eliminate liquidity traps by implementing negative interest rates. Here's why I'm uncertain whether negative interest rate policy is expansionary.
To start, let's go through the standard logic: the fisher equation r... |
In recent years, processing and exploration of time series has experienced a noticeable interest. Growing volumes of data and needs of efficient processing pushed the research in new directions, including hardware based ...
The aim of this dissertation is to investigate the geometry of resolutions of quotientsingularit... |
ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree inform... |
Table of Contents
The Pythagorean Theorem for Inner Product Spaces Examples 1
Recall from The Pythagorean Theorem for Inner Product Spaces page that if $V$ is an inner product space and if $u, v \in V$ are such that $u$ and $v$ are orthogonal to each other, that is, $<u, v> = 0$, then:(1)
We will now look at some examp... |
Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$ f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy, $$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I shall ignore), and assume that $g\in L^q$ for all $q\in[1,2d/(d+2)]$. By the Hardy-Littlewood-Sobolev inequality (or any... |
Table of Contents
Basic Theorems Regarding Connected and Disconnected Metric Spaces
Recall from the Connected and Disconnected Metric Spaces page that a metric space $(M, d)$ is said to be disconnected if there exists $A, B \subseteq M$, $A, B \neq \emptyset$ where $A \cap B = \emptyset$ and:(1)
We say that $(M, d)$ is... |
Preprints (rote Reihe) des Fachbereich Mathematik Refine Year of publication Document Type Preprint (62) (remove)
306
In this paper we study the space-time asymptotic behavior of the solutions and derivatives to th incompressible Navier-Stokes equations. Using moment estimates we obtain that strong solutions to the Nav... |
Without loss of generality, all the $a_i$'s and $b_i$'s are nonzero. Let $\tilde d$ denote the difference between the left- and right-hand sides of the conjectured inequality $(*)$, which then of course can be rewritten as $\tilde d\ge0$. In the previous version of my answer, I rewrote $\tilde d$ in new variables, $x_i... |
(a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ is invertible, then $ABA^{-1}=B$. (d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. (e) If... |
Table of Contents
The Ratio Test for Positive Series of Real Numbers
We will now develop yet another important test for determining the convergence or divergence of a series. This test is known as the ratio test for positive series.
Theorem 1: Let $(a_n)_{n=1}^{\infty}$ be a positive sequence of real numbers and let $\... |
Question #032b6 1 Answer Answer:
The net force is the vector sum of all the forces acting on an object.
Explanation:
Whenever a number of forces act on an object, and if the vector sum of all the forces is not balanced, then we have a resultant force. This is called
net force. A net force is capable of accelerating a m... |
You should be able to find a proof of this fact in any undergraduate stochastic processes books. Durrett's book Essentials of Stochastic Processes has a good proof of this.
I'll give an outline of how to prove it. Suppose that the Markov chain starts at $X_0=x$. Let $0 = R_0 < R_1 < R_2 < \ldots$ be the sequence of ret... |
Basic Theorems Regarding Compact Sets in a Metric Space
Basic Theorems Regarding Compact Sets in a Metric Space
Recall from the Compact Sets in a Metric Space page that if $(M, d)$ is a metric space then a set $S \subseteq M$ is said to be compact in $M$ if for every open covering of $S$ there exists a finite subcoveri... |
My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion:
As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic process $\{X_t\}_{t \in [0,T]}$, we can solve a (boundary?) value problem, whose PDE has parameters are given by the dynamics o... |
I am a bit unsure if my calculations are correct, but on my scribbling paper it seemed to work out.
P1 and P2 lie on a circle around M.
This allows us to measure the distance of the two points by just taking the radius (which is half the line length) of the circle and constructing two rectangular triangles. The accepte... |
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