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Is there a series of functions $\sum (-1)^n u_n(x)$ that is convergent for Leibniz rule (alternating series test) in some interval but it is not
uniformly convergent?
In particular the series must be (for using Leibniz rule)
positive $(u_n(x) \geq 0)$ decreasing $(u_{n+1}(x)\leq u_n(x))$ such that $\lim_{n \to \infty} ... |
Definitions
correlation coefficient $= r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2\sum_{i=1}^{n}(y_i - \bar{y})^2}}$
My Question
What is the motivation of this formula? It's supposed to measure linear relationships on bivariate data, but I don't understand why it would ... |
In the previous section, we discussed the relationship between the bulk mass of a substance and the number of atoms or molecules it contains (moles). Given the chemical formula of the substance, we were able to determine the amount of the substance (moles) from its mass, and vice versa. But what if the chemical formula... |
We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.
We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A... |
Perimeter is the distance around a two dimensional shape, or the measurement of the distance around something; the length of the boundary.
A
perimeter is a path that surrounds a two-dimensional shape. The word comes from the Greek peri (around) and meter (measure). The term may be used either for the path or its length... |
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It is given that 2cos2πn=x+1x 2\cos \dfrac{2\pi}{n} = x+\dfrac{1}{x}2cosn2π=x+x1 for n∈Nn \in \mathbb Nn∈N. If xn+1xn=anb x^{n} + \dfrac{1}{x^{n}} = an^{b} xn+xn1=anb, where aaa and bbb are real numbers, find a+ba+ba+b.
Problem Loading...
Note Loading...
Set Load... |
Check out section 2.3.2 of this paper by Chapelle and Zien. They have a nice heuristic to select a good search range for $\sigma$ of the RBF kernel and $C$ for the SVM. I quote
To determine good values of the remaining free parameters (eg, by CV),
it is important to search on the right scale. We therefore fix default
v... |
They perhaps want you to make the calculations
relatively, which means that even if you assume by looking at the first graph that the mass of the molecular chain length of $1$ (taken relatively and not as per scale) is $x$ units and thus, the mass of $10$ (again, taken relatively and not as per scale) such chains add u... |
Taylor series
A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point
$$f(a) \approx \sum\limits_{n=0}^{\infty}{\frac{f^{(n)}(a)}{n!}(x-a)^n}$$
where $f^{n}(a)$ donates the $n^{th}$ derivative of $f$ evaluated at t... |
I need help to solve each inequality
6. 3| x - 3 | < 12
7. 5| 2x + 8 | + 4 < 9
8. -2| 8x - 4 | + 1 < -15
6. \(3| x - 3 | < 12\)
First, divide both sides by \(3\) to get \(| x - 3 | < 4\). Now we have that \(x - 3 < 4\) or that \( x - 3 > -4\). By solving each one, we get \(x<7\) or \(x>-1\). This gives us \((-1, 7)\).
... |
Now we return to systems of distinct representatives.
A system of distinct representatives corresponds to a set of edges inthe corresponding bipartite graph that share no endpoints; such a collection of edges (in any graph, not just a bipartite graph) iscalled a
matching. In figure 4.5.1, a matching is shown in red. Th... |
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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 ... |
Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and we know that this $F$ must now satisfy a linear ordinary differential equation $$P_{k+1}(T)F^{(k+1)} + P_{k}(T)F^{(k)} + ... |
When dealing with phonons and specific heat of solids, it seems the really important quantity to obtain is the density of states $N(\omega)$. When we have it, we can find the internal energy as
$$U(T)=\int N(\omega) \dfrac{\hbar \omega}{e^{-\hbar\omega/k_BT}-1}d\omega,$$
and having it we can also find the specific heat... |
ECE662: Statistical Pattern Recognition and Decision Making Processes
Spring 2008, Prof. Boutin
Collectively created by the students in the class
Contents Lecture 28 Lecture notes The Last Class
Welcome to the last class of the semester! The kiwi is going to be a great resource.
Partial Differential Equations (Continue... |
Nearly $\theta$-supercompact cardinals
The near $\theta$-supercompactness hierarchy of cardinals was introduced by Jason Schanker in [1] and [2]. The hierarchy stratifies the $\theta$-supercompactness hierarchy in the sense that every $\theta$-supercompact cardinal is nearly $\theta$-supercompact, and every nearly $2^{... |
Learning Objectives
Make sure you thoroughly understand the following essential ideas:
When arbitrary quantities of the different components of a chemical reaction system are combined, the overall system composition will not likely correspond to the equilibrium composition. As a result, a net change in composition ("a ... |
When it comes to buffer solution one of the most common equation is the Henderson-Hasselbalch approximation. An important point that must be made about this equation is it's useful only if stoichiometric or initial concentration can be substituted into the equation for equilibrium concentrations.
Origin of the Henderso... |
You are right. The $i$ indices of the occupation numbers $n_i$ can be thought of as multi-indices meaning that each $i$ encodes both the wavevector $k$ (which is quantized once boundary conditions are imposed) and the spin sign $\sigma$. In general, $i$ will enumerate all the allowed combinations of quantum numbers tha... |
I expected to get 27,6 mHBy doing what you guys said I get to a turns ratio of 2 (dividing the voltages voltages). That means the current I2rms is 2 times current I1rms equals to 10V. That means L22 = LM/2 = 15.9 mH
1. Homework StatementOk I have the following circuit and data (when the subscript is "ef" it means "rms"... |
Variable Importance in Random Forests can suffer from severe overfitting Predictive vs. interpretational overfitting
There appears to be broad consenus that random forests rarely suffer from “overfitting” which plagues many other models. (We define
overfitting as choosing a model flexibility which is too high for the d... |
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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... |
ZFR
I am math graduate student in one of the U.S. universities and I am deeply interested in Number theory, Additive combinatorics, Ramsey theory, Polynomial Method, Group and Ring theory, Finite Fields, Real and Fourier analysis, Measure Theory.
New York, NY, USA
Member for 5 years, 8 months
0 profile views
Last seen ... |
How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two numbers? Edit: changing the question into how $\omega^{\omega}$ can be shown countable, and how other countable ordinals can ... |
Suppose that $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and the center of $G$ is denoted by $Z(G)$ with its Lie algebra denoted $Z(\mathfrak{g})$. It's easy to show that $Z(\mathfrak{g})\subset\big\{X\in\mathfrak{g}\;|\;[X,Y]=0,\forall Y\in\mathfrak{g} \big\}$, my problem is that, when are they equal?
One can s... |
The Annals of Mathematical Statistics Ann. Math. Statist. Volume 31, Number 1 (1960), 222-224. Sums of Small Powers of Independent Random Variables Abstract
Let $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e. $\lim_{n \righta... |
How to solve this logarithmic equation? $8n^2 = 64n\log n$, ($\log n$ here is base 2) I have tried to convert it to $n-8\log n = 0$, but how to solve the latest?
This doesn't have any solutions using elementary functions. But, using the Lambert W function, we get: $$n = -\frac {8}{\ln 2} \operatorname{W} \left (-\frac ... |
Firstly, least squares (or sum of squared errors) is a possible loss function to use to fit your coefficients. There's nothing technically wrong about it.
However there are number of reasons why MLE is a more attractive option. In addition to those in the comments, here are two more:
Computational efficiency
Because th... |
Interested in the following function:$$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$where $\pi(n)$ is the prime counting function.When $s=2$ the sum becomes the following:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
My favourite one: $[0, 1]$ is compact, i.e. every open cover of $[0, 1]$ has a finite subcover.
Proof: Suppose for a contradiction that there is an open cover $\mathcal{U}$ which does not admit any finite subcover. Thus, either $\left[ 0, \frac{1}{2} \right]$ or $\left[ \frac{1}{2}, 1 \right]$ cannot be covered with a ... |
For my own convenience I'll work in $\infty$-categories, feel free to answer in whatever framework best suits you. My question is essentially how to show, given an $E_\infty$-ring object $R$ in an $\infty$-category $C$ and another object $M$ with a map $R\otimes M\to M$, that the pair $(R,M)$ is in fact an algebra, as ... |
I) In this answer we will only discuss the equilibrium shape. Recall that when we discussed the shape of Earth in this Phys.SE post, the gravitational quadrupole moment was important. Unlike the Earth, from a surface perspective, it is a very good approximation to assume that all the mass of the Sun sits in the center,... |
The CHOLMOD library provides a
CHOLMOD_rcond function that estimates the reciprocal condition number (in the one norm) of a symmetric positive definite matrix from its cholesky decomposition. This is defined as the ratio between the smallest and largest entries in the diagonal of the
L factor. Why is this a true fact?
... |
Divergence and curl are two measurements of vector fields that are very useful in a variety of applications. Both are most easily understood by thinking of the vector field as representing a flow of a liquid or gas; that is, each vector in the vector field should be interpreted as a velocity vector. Roughly speaking, d... |
Prove that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+...+ \frac{1}{n^2} \ge 1$, for all natural numbers $n$. I tried to use mathematical inducion but failed and I tried to figure out a short formula for the sum but I couldnt find any.
Without integrals. If $n\ge4$ $$ \sum_{k=n}^{n^2}\frac1k>\sum_{k=n}^{2n-1}\frac1k+\sum... |
I'm adding another answer since you've substantially changed the question. I feel we're getting out of the realm of appropriate stackexchange etiquette here with so much discussion and changing thing around. Maybe a mod can suggest how best to proceed. Anyways, here's my answer.
I agree whole-heartedly with your first ... |
Well, for the longest coherence time ever, I'm finding this Science from 2013 entitled Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28, which indicates qubits that lasted for over 39 minutes; these, however, only had an 81% fidelity rate. (This is for qubits used in computat... |
In Miles Reid's Undergraduate Commutative Algebra he defines a ring $B$ to be finite as an $A$ - algebra if it is finite as an $A$ - module. Now what I don't understand is suppose we look at the polynomial ring $k[x_1,\ldots,x_n]$ where $k$ is a field. Then as a $k$ - algebra it is finitely generated. Is this the same ... |
Let
$E_4(z)= - \frac{B_4}{8}+ \sum_{n=1}^\infty \sigma_3(n) q^n$
and
$E_6(z)= - \frac{B_6}{12}+ \sum_{n=1}^\infty \sigma_5(n) q^n$
How does one show they are algebraically independant over $\mathbb{C}$ ?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.Sign u... |
We have seen how to compute certain areas by using integration; some volumes may also be computed by evaluating an integral. Generally, the volumes that we can compute this way have cross-sections that are easy to describe.
Example 8.3.1 Find the volume of a pyramid with a square base that is 20 meters tall and 20 mete... |
As you might guess, a first order linear differential equation has the form $\ds \dot y + p(t)y = f(t)$. Not only is this closely related in form to the first order homogeneous linear equation, we can use what we know about solving homogeneous equations to solve the general linear equation.
Suppose that $y_1(t)$ and $y... |
"Gold and silver ornaments are precious".
The following notations are used $:$
$G(x):x$ is a gold ornament
$S(x):x$ is a silver ornament
$P(x):x$ is precious
Options are $:$
$\forall x(P(x) \implies (G(x) \wedge S(x)))$ $\forall x((G(x) \wedge S(x)) \implies P(x))$ $\exists x((G(x) \wedge S(x)) \implies P(x))$ $\forall... |
Consider the helium atom with two electrons, but ignore coupling of angular momenta, relativistic effects, etc.
The spin state of the system is a combination of the triplet states and the singlet state. I will denote a linear combination of the three triplet states as $\lvert\chi_+\rangle$ (because it's symmetrical und... |
I try to find the local minimum of a function $$\Pi(\vec{d})$$ where $$\vec{d} = (d_1,d_2,...,d_n)$$ should satisfy $$d_i>0$$ for $i \in [1,n]$.
I use the
FindMinimum function in
Mathematica; i.e., use something like $$\mathtt{FindMinimum}[\{\Pi(\vec{d}),d_i>0\},\vec{d}].$$ But I don't know how to apply the constraints... |
Here's one way of interpreting this statement (this is essentially an elaboration of the comment by use Learning is a mess). The basic idea behind Wilsonian renormalization is that renormalization can be regarded as the process by which we understand how the theory
effectively behaves at different scales, like momentum... |
Contents
Intersecting a ray with a sphere is probably the simplest form of ray-geometry intersection test which is the reason why so many raytracers show images of spheres. It also has the advantage (because of its simplicity) to be very fast. However, to get it working reliably, they are always a few subtitles which a... |
No such liquid, safe or otherwise, can exist. Evaporation is a strictly endothermic process in all cases.The change in state from liquid to gas is marked by the individual particles gaining enough translational kinetic energy to overcome the mutual attractions present in the liquid phase to "fly free" in the gas phase.... |
I am trying to show that for a convex set $A$ and $s,r>0$ positive real numbers we have $rA+sA = (r+s)A$.
Clearly $(r+s)A$ is contained in $rA+sA$ but I am having trouble showing the other inclusion.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in rela... |
I have a function $f:\mathbb{C}\to\mathbb{C}$ that is analytic on $A = \{s\mid \mathfrak{R}(s)\geq1\}$ and does not vanish on $A$.
Intuitively, I would expect this to mean that $f$ has an analytic logarithm on $A$, but I am not sure if that is true.
I know that it has a holomorphic logarithm on $\{s\mid\mathfrak{R}(s)>... |
Given a general 2nd degree homogeneous differential equation $y'' +p(x)y' +q(x)y=0$. Change the independent variable from $x$ to $z=z(x)$. Show that the above given homogeneous 2nd order differential equation can be transformed into an equation with constant coefficients if and only if $(q' + 2pq)/q^{3/2}$ is constant.... |
Informally, two functions $f$ and $g$ are
inverses ifeach reverses, or undoes, the other. More precisely:
Definition 9.1.1 Two functions $f$ and $g$ are inverses if for all $x$ in the domain of $g$, $f(g(x))=x$, and for all $x$ in the domain of $f$, $g(f(x))=x$.
Example 9.1.2 $f=x^3$ and $g=x^{1/3}$ are inverses, since... |
Visitors sometimes feel bored with our web blog because of too many boring stuffs which not often appear in their casual work/study. I just want to post such a simple tutorial for beginners and if you are experienced in CTF's pwn then just skip it. Enjoy!
Chinese Remainder Theorem =========================
Suppose are ... |
In Gali chapter 2 we have the following constraint to the classical monetary model
$P_t C_t + Q_t B_t \leq B_{t-1} + W_t N_t-T_t$.
Then it seems that this is treated as an equality. Therefore my question is: are we assumig that $\partial U/\partial B_t >0$? So households would achieve the optimun at the equality.
Anoth... |
Wikidot uses a markup language called $\LaTeX$ (pronounced "lay-tech") along with jsMath to generate properly typeset mathematics. $\LaTeX$ requires memorizing (or looking up) "commands" for creating certain kinds of characters. Since you'll be posting to the forum and creating wiki pages with serious mathematical cont... |
Let $G=(V,E)$ be a directed graph, $\omega : E \rightarrow R$ a weight function, and $s,t \in V$ a pair of different nodes. It's given that $G$ doesn't have a negative cycle. Moreover, 10 of its edges are colored in red (let's say that the rest are colored in blue).
I want to find an efficient algorithm that find the s... |
Using randomized approach we can guarantee that the equality problem has O(1) complexity (in communication). With other definitions of of equality (not strictly equal), is there a general approach to design a randomized algorithm with bounded error? Or how to determine if there is no way such an algorithm can be design... |
This is what we call in the pitch-detection biz, the "
octave problem".
First of all, I would change the AMDF to ASDF. And I would not reduce the window size as the lag increases. (Also, I am changing notation to what I consider to be more conventional. "$x[n]$" is a discrete-time signal.)
The Average Squared Differenc... |
I could understand the derivation of the "bulk-to-boundary" propagators ($K$) for scalar fields in $AdS$ but the iterative definition of the "bulk-to-bulk" propagators is not clear to me.
On is using the notation that $K^{\Delta_i}(z,x;x')$ is the bulk-to-boundary propagator i.e it solves $(\Box -m^2)K^{\Delta_i}(z,x;x... |
So my question is: is Pauli-repulsion a phenomenon that has also not
yet been explained in terms of any of the three other forces that we
know of?
$\def\ket#1{|#1\rangle} \let\up=\uparrow \let\dn=\downarrow \def\PD#1#2{{\partial#1\over\partial#2}}$There is no repulsion and no unexplained force. I would also add that PE... |
The idea for estimating the mean is roughly as follows:
For any $f(x)$ that gives outputs in the reals, define a rescaled $F(x)$ that gives outputs in the range 0 to 1. We aim to estimate the mean of $F(x)$.
Define a unitary $U_a$ whose operation is $$U_a:|0\rangle|0\rangle\mapsto\frac{1}{2^{n/2}}\sum_x|x\rangle(\sqrt{... |
High Dimensional LDA Posted on
This note is for Cai, T. T., & Zhang, L. (n.d.). High dimensional linear discriminant analysis: Optimality, adaptive algorithm and missing data. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 0(0).
Introduction
Suppose $Z$ is drawn with equal probability fro... |
This will be a talk for the CUNY Logic Workshop, September 16, 2016, at the CUNY Graduate Center, Room 6417, 2-3:30 pm.
Abstract. In analogy with the ancient views on potential as opposed to actual infinity, set-theoretic potentialism is the philosophical position holding that the universe of set theory is never fully ... |
I'm trying to solve a full-vectorial wave equation for an arbitrarily shaped wave guide, by using
NDSolve and perfectly matched layer (PML) conditions.
The PML conditions can be stated as a coordinate transformation of the form $\text{$\partial $/$\partial $x $\to $ $\alpha $x(x) $\partial $/$\partial $x}$ and so on. A... |
This question is a continuation of the discussion at How can a linear operator on DFT vector produce the same vector using only half of the DFT vector?, where the higher level motivation is better explained, in case there is interest.
Question:
How to take the positive frequencies from an hermitian symmetric DFT vector... |
Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible to get a bound on the coefficients of $g(x)$ in terms of $M$ i.e. if $g(x)=\sum_{i=0}^mb_ix^i$ then does there exist some... |
Answer
91.9 Hz
Work Step by Step
We find the resonant frequency: $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\\ f = \frac{1}{2\pi}\sqrt{\frac{2.5 \times 10^3}{7.5\times10^{-3}}}=\fbox{91.9 Hz} $
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hour... |
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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 ... |
I am having some trouble knowing how to correctly start a problem of finding the Fourier Coefficients using complex exponential form. The problem is given below:
$$g_1(t)=\begin{cases} 1,~~\qquad t<1\\2-t,\quad t\ge 1\end{cases}$$
and
$$g_2(t)=\begin{cases} 1+2t, \qquad -0.5<t\le 0.5\\\tfrac{(7-2t)}{3},~~~\,\qquad0.5<t... |
I've been trying to prove it for a while, but can't seem to get anywhere.
$$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$
Could someone please provide a valid proof?
I am not allowed to work on both sides of the equation.
Work so far:
RS:
$$ \begin{align} & \frac{\sin^2\theta}{\cos^2... |
Definition:Continuous Mapping (Metric Space) Contents Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in ... |
This is a pretty interesting problem. First, define the right-hand side.
f[θ_, μ_] := Sin[θ]/(μ + Cos[θ]);
If $|\mu|>1$, there are only the proper equilibria you noted at $\theta=n\pi$:
Plot[f[θ, 1.5], {θ, 0, 4 π}, AxesLabel -> {"θ", "θ'"}]
However if $|\mu|<1$ the denominator might equal zero, resulting in the followi... |
What makes it occur? How do the protons and nucleus know that they have to lose mass to produce energy...? And is the mass of a compressed Spring more than an uncompressed one?? does a body which has a greater energy has more mass than the one which has a less energy?
"How do the protons and nucleus know that they have... |
Sub Gaussian Posted on
This post is based on Wainwright (2019).
A random variable $X$ with mean $\mu=\E X$ is sub-gaussian if there is a positive number $\sigma$ such that $$ \E [e^{\lambda(X-\mu)}]\le e^{\lambda^2\sigma^2/2}\;\forall \lambda\in \IR\,. $$
Several equivalent characterizations of sub-Gaussian variables.
... |
It will come as no surprise that we can also do triple integrals—integrals over a three-dimensional region. The simplest application allows us to compute volumes in an alternate way.
To approximate a volume in three dimensions, we can divide the three-dimensional region into small rectangular boxes, each $\Delta x\time... |
See section 4.4 to review some basic terminology about graphs.
A graph $G$ consists of a pair $(V,E)$, where $V$ is the set of vertices and $E$ the set of edges. We write $V(G)$ for the vertices of $G$ and $E(G)$ for the edges of $G$ when necessary to avoid ambiguity, as when more than one graph is under discussion.
If... |
I would like to numerically solve a system of differential equations that describes the dynamics of $N$ coupled oscillatory units (Kuramoto model) via their phase variables $\phi_i$:
$\frac{\partial\phi_i}{\partial t} = \omega_i + \frac{K}{N}\sum_{j=1}^N \sin(\phi_j-\phi_i)$
A simple implementation with NDSolve works n... |
Setup
Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements of the group ring $\mathbb Z[\Lambda]$ of $\Lambda$ as linear combinations of elements of the form $e^\lambda$, $\lambda... |
We show here how any interpretation of $\operatorname{Tr} A$ when $A : V \to V$ is an isomorphism can be extended to an interpretation of the trace of an arbitrary endomorphism.
As described elsewhere, if you view $A : V \to V$ as a vector field on $V$ in the canonical way then the trace of $A$ is the same as its diver... |
So, Bob is given the following state (also called the
maximally-mixed state):
$\rho = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{bmatrix}$
As you noticed, one nice feature of density matrices is they enable us to capture the uncertainty of a... |
The second qubit of a two-qubit system in the Bell state
$$|\beta_{01}\rangle= \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$$ is sent through an error channel which introduces a bit flip error with probability $p$. I want to calculate the state of the system after the second qubit exits the error channel.
I know how to do... |
A
directed graph,also called a digraph,is a graph in which the edges have a direction. Thisis usually indicated with an arrow on the edge; more formally, if $v$and $w$ are vertices, an edge is an unordered pair $\{v,w\}$, while adirected edge, called an arc,is an ordered pair $(v,w)$ or $(w,v)$. The arc $(v,w)$ is draw... |
So Im having some trouble in solving projective module problems. I will give an example which extends to the other problems I also wasn´t also to solve.
Let A be a ring, $P$ a left $A$-module. We call a dual basis for $P$ to the family $\{(x_i,f_i)\}_{i \in I}$ where $(x_i,f_i)\in P \times P^*$ for every $i\in I$ such ... |
Since my answer is long, here is a loose heuristic summary answer: A permutation $\pi$ such that $|\pi(i+1) - \pi(i)|$ is large for many values of $i$ occurs less frequently than permutations where such values are small.
The first results on this might be from Bandt and Shiha's 2005 paper Order patterns in time series,... |
Suppose an object moves so that its speed, or more properly velocity, is given by $\ds v(t)=-t^2+5t$, as shown in figure 7.3.1. Let's examine the motion of this object carefully. We know that the velocity is the derivative of position, so position is given by $\ds s(t)=-t^3/3+5t^2/2+C$. Let's suppose that at time $t=0$... |
Methods for Enforcing Inequality Constraints
How do you find the shortest overland distance between two points across a lake? Such obstacles and bounds on solutions are often called inequality constraints. Requirements for nonnegativity of gaps between objects in contact mechanics, species concentrations in chemistry, ... |
The continuous time Hilbert transform is $$\hat x(t) := x(t) + j\left( p.v. \left\{\frac{1}{t\pi} \ast x \right\} (t)\right)$$
where $$ \theta(t) = \tan^{-1}\left( \frac{ p.v. \left\{\frac{1}{t\pi} \ast x \right\} (t)}{x(t)} \right)$$ and $$ \omega(t) = \theta'(t) $$
and since $\theta(t)$ is smooth, you can actually ca... |
Image of Subset under Relation is Subset of Image/Corollary 3 Corollary of Image of Subset under Relation is Subset of Image
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $C, D \subseteq T$. Then: $C \subseteq D \implies f^{-1} \sqbrk C \subseteq f^{-1} \sqbrk D$ This can be expressed in t... |
We have seen how integration can be used to find an area between a curve and the $x$-axis. With very little change we can find some areas between curves; indeed, the area between a curve and the $x$-axis may be interpreted as the area between the curve and a second "curve'' with equation $y=0$. In the simplest of cases... |
We start by considering equations in which only the first derivative of the function appears.
Definition 19.1.1 A
first order differential equation is an equation ofthe form$F(t, y, \dot{y})=0$.A solution of a first order differential equation is afunction $f(t)$ that makes $\ds F(t,f(t),f'(t))=0$ for every value of $t... |
The question Should chat have TeX support was first asked3½ years ago, and it has been asked at intervals since then. I am not asking that question. I
know that chat should have TeX support and so does everyone else. My questions are, why hasn't this been done and when will it be done? The lack of MathJax support in ch... |
If $E \subset \mathbb{R}^n$ and $x \in E$, we have that $ \operatorname{dist}(x,\partial E) = \sup \{ \gamma \geq 0 : B(x, \gamma) \subset E \}$. What characteristics of $\mathbb{R}^n$ make this true? In other words, what restrictions would we have to impose on a general metric space (or topological space) for this to ... |
For reasons that I find very mysterious, Unicode has the full range of Greek in sans serif bold, upright and italic, but it doesn't cover sans serif Greek in medium weight; to wit, there are
MATHEMATICAL SANS-SERIF BOLD CAPITAL GAMMA (U+1D758)
MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL GAMMA (U+1D792)
and the other Gr... |
Rayleigh Scattering of Isolated Species
( Species == ions, atoms, molecules )
See Polarization for the low wavenumber (frequency / speed of light ) approximation used for Rayleigh scattering.
Scattering is due to the polarization of species. The polarization can be summed from the behavior of individual resonances and ... |
As mentioned in the comments by Roland, this term is not common and is first used by the authors of the mentioned paper (also the package mentioned by Roland in comments - STING).
From this link you can find the definition of the Last Heavy Atom which is possibly the most distal non-hydrogen (N, C, O, S) atom in the am... |
There are reasons that any modern example is likely to resemble the status of Legendre's constant. Most (but not all) interesting numbers admit a polynomial-time algorithm to compute their digits. In fact, there is an interesting semi-review by Borwein and Borwein that shows that most of the usual numbers in calculus (... |
What does the number in brackets mean in these two examples?
$$21(1)\ \mathrm{cm^{-1}}$$
and
$$1.0(3)\times10^{-7}$$
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What d... |
An original contribution in this work was the proposition of a contrast gain control mechanism (Section 2.3
)
via
the differential equation
$$\frac{\displaystyle {dV}}{\displaystyle {dt}}(x,y,t)=I_\mathrm{OPL}(x,y,t) -g_\mathrm{A}(x,y,t) V(x,y,t)$$
(22)
with
$$g_\mathrm{A}(x,y,t) = G_{\sigma_\mathrm{A}} \stackrel{x,y}{... |
This "decay" of later values is a direct consequence of the episodic formula for the objective function for REINFORCE:
$$J(\theta) = v_{\pi_\theta}(s_0)$$
That is, the expected return from the first state of the episode. This is equation 13.4 in the book edition that you linked in the question.
In other words, if there... |
Apart from the formal result about #P-hardness, there's something worth touching on, about the nature of strong simulation itself. I'll comment first on strong simulation, and then specifically on the quantum case.1. Strong simulation even of classical randomised computation is hardStrong simulation is a very powerful ... |
Bach, Marc A and Parameswaran, Pattiyil and Jemmis, Eluvathingal D and Rosenthal, Uwe (2007)
Bimetallic Complexes of Metallacyclopentynes: cis versus trans and Planarity versus Nonplanarity. In: Organometallics, 26 (9). pp. 2149-2156.
PDF
Bimetallic-45.pdf
Restricted to Registered users only
Download (342kB) | Request ... |
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