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: Saw you edited the question and added some comments at the end of my answer to address. Edit
There can't possibly be any factors between $\frac n2$ and $n$. Suppose $\frac n2 < a < n$ and $a \cdot b = n$. What could $b$ be? If $b=1$, then $a \cdot b = a < n$. If $b \geq 2$, then $a \cdot b > \frac n2 \cdot 2 = n$. So... |
Let $1<p_1,\dots,p_k<\infty$ such that $\sum_{i=1}^k\frac{1}{p_i}=1$.
Moreover, let $a_1,\dots,a_k\geq 0$. I have to show that $$\prod_{i=1}^k a_i\leq\max_i a_i^{p_i}$$ I want to use induction over $k$, but I am struggling with it. For $k=2$ I only got to here:
Since $p_1$ and $p_2$ are Hölder conjugates, we have $\fra... |
I'm not totally sure if this is answering your question, but let $\boldsymbol{\lambda}$ be a vector that you can think of as containing the eigenvalues of $\mathbf{X}^T\mathbf{X}$. I'm assuming that the Frobenius norm constraint you mentioned in your comment is of the form $||X||_{F}^2 \le c$. Let $\mathbf{1}$ be a vec... |
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Now showing items 1-5 of 5
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
I'm using the Limit Definition to find the derivative,
$$f'(x)=\lim_{\Delta x \to 0} {f(x+\Delta x) - f(x) \over \Delta x}$$ $$$$ Now, I want to find the derivative for the function,
$$f(x)={1 \over x+1}$$
So, here's what I did.
$$\lim_{\Delta x \to 0} {{1 \over (x+\Delta x) +1} - {1 \over x+1}\over \Delta x}$$
Now, I ... |
This question already has an answer here:
Background: This is part b of problem 12.4.3 from Arfken, Weber, Harris Math Methods for Physicists to show that $\int_0^\infty \frac{\ln^2(z)}{1+z^2}$dz$=4(1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\dots)$.
Part b of the question asks to show that this series evaluates to $\... |
I am trying to understand a modal logic countermodel, illustrating that the
QK theorem,$$ \Box (\phi(x) \wedge \forall x \phi (x)) \rightarrow \Box \forall x \phi (x), $$ fails in the alternate semantics for modal logic given by David Lewis's counterpart theory.The countermodel is the following:
It comes from Oliver Ku... |
Give it a circumstellar cloud of oxygen.
Some planetary nebulae, such as NGC 6826, appear green because of ionized oxygen.
Image in the public domain. Yes, this is a true-color image.
I see no reason why you couldn't surround the star with an extremely dense cloud of hydrogen, containing a relatively high fraction of o... |
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. Simply click the bounty award icon next to each answer to permanently award your bounty to the answerer. (You cannot award a bounty to your ow... |
Wikipedia said
The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of cha... |
I've recently found an article (referred somewhere on this site) criticizing the use of common rules of algebra on infinite series. To be honest, the video referred is one of the videos of Numberphile I liked the most. I mean, informally, to say a rule doesn't hold, I think one should find an example (in modern logic, ... |
Keywords
Quaternion matrix equations, Real representations, $\eta$-conjugates, $\eta$-conjugate transposes, Ordered units triple.
Abstract
For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* ... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
The Canonical energy momentum tensor is given by $$T_{\mu\nu} = \frac{\partial {\cal L}}{\partial (\partial^\mu \phi_s)} \partial_\nu \phi_s - g_{\mu\nu} {\cal L}. $$ A priori, there is no reason to believe that the EM tensor above is symmetric. To symmetrize it we do the following trick.
To any EM tensor we can add th... |
Analytic operator at a point $x_0$
An operator $A$, acting from one Banach space into another, that admits a representation of the form \begin{equation} A(x_0+h) - Ax_0 = \sum_{k=1}^{\infty}C_kh^k, \end{equation} where $C_k$ is a form of degree $k$ and the series converges uniformly in some ball $\|h\|<r$. An operator ... |
In mathematics, logarithmic functions is an inverse function to exponentiation. The logarithmic function is defined as
For x > 0 , a > 0, and a\(\neq\)1,
y= log
a x if and only if x = a y
Then the function is given by
f(x) = log
a x
The base of the logarithm is a. This can be read it as log base a of x. The most 2 comm... |
I know that if something is thrown up with a velocity
v, it comes down with the same velocity as long as its motion faces no resistance. So is it possible for me to fire a bullet and be killed if it drops down on me directly? Assuming there is no air resistance.
I know that if something is thrown up with a velocity
Yes... |
1. We can think about what $A_{5}$ actually is, that is, all even permutations of 5 symbols. If you think about it, what element in $A_{5}$ could possibly have order 60? For example if we look at something like the permutation $(1 2 3)$ which can be seen as $(13)(12)$ what is the order of this? Knowing that the order o... |
"half sliced unit disk"
Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty sketch, this is a unit disk cut at every 45 degree.
Mathematics Stack Exchange is a question and answer site for people stu... |
Agglomerative clustering methods that rely on a multiple sequence alignment and a matrix of pairwise distances can be computationally infeasible for large DNA and amino acid datasets. Alternative k-mer based clustering methods involve enumerating all
k-letter words in a sequence through a sliding window of length k. Th... |
Given an ellipse with foci $F_1, F_2$ and a point $P$. Let $T_1, T_2$ the points of tangency on the ellipse determined by the tangent lines through $P$. Show that $\widehat {T_1 P F_1} = \widehat {T_2 P F_2}$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professiona... |
According to the first and second law for a closed system containing different chemicals we have
\begin{align} &\delta Q - \delta W = dU = T dS - p dV +\sum_i \mu_i d N_i\\ &\Rightarrow\;\delta W - p dV + \sum_i \mu_i dN_i = \delta Q - T d S \le 0\qquad\because\text{2nd law}\\ &\Rightarrow\;\delta W - p dV + dG\bigr|_{... |
Hi I'm having trouble with this homework question on wave mechanics. The question is as follows:
"State the boundary condition which must be met at a point where the string of question $2$ is fixed.
Hence find the real standing wave solutions to the wave equation, and determine the allowed oscillation frequencies, when... |
I have a problem with this exercise because I really don't know how to proceed. It's related with the "S-matrix".
In class we saw this example:
Consider the spherically symmetric potential:
$$V(r)=\begin{cases} -V_{0}, & 0\leq r\leq a\\ 0, & r>a \end{cases} $$
We know that for $r>a$ and $l=0$ we have
$$U_>''(r)+k^{2}U_... |
If $Y_n$ is limited in probability and $X_n \to 0$ in probability then $X_nY_n \to 0$ in probability.
Attempt: I would go like:
for every $\epsilon$, $\mathbb P(|X_nY_n|\leq \epsilon) = \mathbb P(|Y_n|\leq \frac{ \epsilon}{ |X_n|})\geq\mathbb P(|Y_n|\leq C_{\delta}) \geq 1-\delta \to 1$ but the truth is that $\frac{ \e... |
A Statistical Background A.1 Basic statistical terms
Note that all the following statistical terms apply only to
numerical variables, except the distribution which can exist for both numerical and categorical variables. A.1.1 Mean
The
mean is the most commonly reported measure of center. It is commonly called the avera... |
Gravitational force $\vec{F_g}$ is trying to rotate the cylinder around point A.
Lever arm has length $r_c= r\,\sin\theta$ Torque $\tau$ is $$\tau = F_g \, r_c = m \, g\, r \, \sin\theta$$Torque causes angular acceleration $\alpha$ of the cylinder. We are interested in acceleration $\vec{a}$ of point C.Absolute value o... |
It is plain, from looking at the question geometrically, that
the expected distance between two independent, uniform, random points within a convex set is going to be a little less than half its diameter. (It should be less because it's relatively rare for the two points to be located within extreme areas like corners ... |
Difference between revisions of "Probability Seminar"
(→February 14, TBA)
(→April 11, Eviatar Proccia, Texas A&M)
Line 59: Line 59:
== April 4, TBA ==
== April 4, TBA ==
−
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar
+
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar ], [http://www.m... |
Katok/ Hasselblatt, "Modern Theory of Dynamical Systems", p. 156:
Every probability distribution $p=(p_0,\ldots,p_{N-1})$ where $0\leq p_i\leq 1$ for $i=0,\ldots,N-1$ and $\sum_{i=0}^{N-1}p_i=1$, determines the product measure $\mu_p$ on the space $$ \Omega_n=\{\omega=(\ldots,\omega_{-1},\omega_0,\omega_1,\ldots)|\omeg... |
Let $f:\mathbb{R}\to\mathbb{R}$ be a function satisfying $|f(x+y)-f(x-y)-y|$$\le y^2$
for all $x,y\in\mathbb{R}$. Show that $f(x)={x\over2}+c$, where $c$ is a constant.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a min... |
How do I solve this system of equations?
$$\begin{cases} 7(a+b)=b-a \\4(3a+2b)=b-8\end{cases}$$
Progress
I tried both substitution and elimination, but when I set $a$ or $b$ free on one side, I keep getting $a$ or $b$ also on the other side.
Mathematics Stack Exchange is a question and answer site for people studying m... |
Lasing characteristics of heavily doped single-crystal Fe:ZnSe Abstract
Characteristics of a Fe:ZnSe laser are studied at room temperature. The laser active elements are heavily doped single crystals with the \(\hbox {Fe}^{2+}\) ion concentration \(n=0.64\times 10^{19}-5.7\times 10^{19}\hbox {cm}^{-3}\), grown from mel... |
The
Frobenius norm of a matrix is the square root of the sum of the squares of its elements; it provides a rough measure of the “size” of the matrix.
Wikipedia gives three ways of expressing the Frobenius norm of a matrix whose elements are real numbers:
\|A\|_{\mathrm{F}}\equiv \sqrt{\sum_{i=1}^{m} \sum_{j=1}^{n}\left... |
I am currently working on a research project for a pairs trading strategy and would like to know the correct positions to take when a signal has been triggered.
Say we are using this equation to generate signals: \begin{equation} z = y - \beta x \end{equation} $\mu_z$ = 0, $\sigma_z$ = 0.5, and $\beta$ = 1, for simplic... |
Supercells¶
Supercell is a widely used crystallographic concept
1. A general theoretical background is offered in the expandable section below for the readers unfamiliar with the topic. The rest of this documentation page explains how supercell generation is implemented in the Materials Designer interface. Theoretical ... |
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Now showing items 11-20 of 76
Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisi... |
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Now showing items 1-10 of 55
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... |
Prove that $ {(x+y)}^n-x^n-y^n$ is divisible by $ xy(x+y) \times (x^2+xy+y^2)$ if $ n$ is an odd number not divisible by $ 3$ . Prove that $ {(x+y)}^n-x^n-y^n$ is divisible by $ xy(x+y) \times {(x^2+xy+y^2)}^2$ if $ n \equiv \pmod{6}1$ Solution
1.Considering the given expression as a polynomial in $ y$ , let us put $ y... |
I am looking to describe all functions of type $f:\mathbb{R} \rightarrow \mathbb{R}$ as a vector space. Is this possible? What is a basis? Can I write any function of this type as $$g(x) = \int^{\infty}_{-\infty}a(k)sin(kx) + b(k)cos(kx)dk$$ for $a(k), b(k) \in \mathbb{R}$, and $k, \in \mathbb{R}$. I am assuming that $... |
Ah, the smell of a new seismic interpretation project. All those traces, all that geology — perhaps unseen by humans or indeed any multicellular organism at all since the Triassic. The temptation is to Just Start Interpreting, why, you could have a map by lunchtime tomorrow! But wait. There are some things to do first.... |
Probability Seminar Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:... |
Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision infinite_geometric_series [2016/07/22 15:19]
nikolaj
infinite_geometric_series [2016/07/22 15:20] (current)
nikolaj
Line 25: Line 25: == q-Integral == == q-Integral == - For a function $f$, the q-... |
User:Nikita2/sandbox A mapping $\varphi:D\to D'$ possesses Luzin's $\mathcal N$-property if the image of every set of measure zero is a set of measure zero. A mapping $\varphi$ possesses Luzin's $\mathcal N{}^{-1}$-property if the preimage of every set of measure zero is a set of measure zero. Briefly\begin{equation*}\... |
Define the function $f:\mathbb{N}\to\mathbb{R}$ as
$$ f(n)=e^\gamma n \log \log n, $$
where $e$ is Euler's constant and $\gamma$ is the Euler-Mascheroni constant. Then Robin's criterion states that for all positive integers $n>5040$, the statement
$$ \sigma(n) < f(n), $$
where $\sigma$ denotes the divisor sum function,... |
This is the first in a series of articles in which I want to transmit what I learned (or what I think I learned) from the books, papers and lectures of Alexander Stepanov.
These are the lessons that Alex gives us, and I want to show them in this series:
Specify our algorithms correctly Programming must be based on a so... |
How many solutions do for the following equation exists?
\begin{align} \varphi(n)=\pi(n), n\in\mathbb{N}, \end{align}
where $\varphi(n)$ is Euler's totient function and $\pi(n)$ is the prime-counting function.
Thank you very much
Mathematics Stack Exchange is a question and answer site for people studying math at any l... |
This question is somewhat naive. Please see the following proof before reading the question itself.
Let $L$ be a vector space over a field $\Bbb{F}.$ Assume there is an associative multiplication on $L$ so that $L$ is an associative ring, and denote this multiplication simply as $xy$ for $x,y\in L$. Furthermore, suppos... |
This is a long question, so I thank anyone willing to help me sort it out. That said, the material is elementary (to anyone who's taken real analysis) and I'm just long winded. The catalyst of the question is the following theorem and proof, which I repeat here verbatim:
Compact subsets of metric spaces are closed.
\be... |
Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?
Since you've used the tag
Functional analysis, you might be interested in non-metrisability of certain topologies ubiquitous in analysis: Two questions on the Zariski topology on $\mathbb{R}$
Some other examples can... |
Math question on Newton's method and detecting actual zeros - Printable Version
+- HP Forums (
https://www.hpmuseum.org/forum)
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/forum-3.html)
+--- Forum: General Forum (
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+--- Thread: Math question on Newton's method and detecting actual zeros (
/thr... |
In general, an algebraic closure of a field $K$ is denoted by $\overline{K}$. Typical examples arising in number theory are $K=\mathbb{Q}$, $K=\mathbb{F}_p(t)$, $K=\mathbb{Q}_p$. Usually one needs the axiom of choice in order to prove the existence of algebraic closures. There are (at least) two exceptions: For $K=\mat... |
On a radial positive solution to a nonlocal elliptic equation
DOI: http://dx.doi.org/10.12775/TMNA.2003.017
Abstract
The existence of a solution to the Dirichlet boundary value problem for
nonlinear Poisson equations with the nonlocal nonlinear term $$ -\Delta u=f\bigg(u,\int (g\circ u)\bigg),\quad u\vert\partial U=0, ... |
Denote by $X := [\mathbb{N}]^\infty$ the set of infinite subsets of $\mathbb{N}$. Recall that the Ellentuck topology is a topology on $X$ generated by sets of the form $\{A\text{ infinite} \mid s\text{ is an initial segment of }A\text{ and }A\setminus s\subset B\}$ for some finite $s\subset \mathbb{N}$ and (infinite) $... |
Introduction Energy and Power Basic Operations Practice Problems Transformation of signals defined piecewise Even and Odd Signals Commonly encountered signals
A variety of operations can be carried out on signals to obtain new signals. These operations can be classified into two categories - operations that are perform... |
Stochastic 3D Modeling of Three-Phase Microstructures for Predicting Transport Properties: A Case Study 143 Downloads Abstract
We compare two conceptually different stochastic microstructure models, i.e., a graph-based model and a pluri-Gaussian model, that have been introduced to model the transport properties of thre... |
Define $f: \ell^2(\mathbb{N}) \to \Bbb C$ by: $$f(x) = \sum_{n \geq 1}\frac{x_n}{n^2},$$where $x = (x_n)_{n \geq 1} \in \ell^2(\mathbb{N})$. It is pretty clear that $f$ is linear. Also, since $\ell^2(\mathbb{N}) \subset \ell^\infty(\mathbb{N})$ and $\|x\|_\infty \leq \|x\|_2$, we have: $$\begin{align}|f(x)| &= \left|\s... |
Remark: Throughout this post, I will be assuming the Axiom of Choice. This won't always be necessary (in fact, in most of the instances where we'll use it, we could've instead used weaker choice principles), but it makes things far less messy to prove. Let me know if you're interested in a more choice-free version (or ... |
The other answers so far focus on how to transform the integral so that you don't have a singularity. But that's not always viable so it's important to know how to deal with a singularity when one exists.
The trapezoid rule is a poor choice for this case, because it uses the values of the function at the endpoints, whi... |
Monday-Sept. 30 Tuesday-Oct. 1 Wednesday-Oct. 2 9:00-9:30 Registration/Coffee Coffee Coffee 9:30 - 10:30 Lehalleur Wendt Sechin 10:30-11:00 Coffee Coffee Coffee 11:00 - 12:00 Yakerson Dotto Xu 12:00-14:00 Lunch Lunch Lunch 14:00 - 15:00 Drew Ozornova Jin 15:00-16:00 Coffee Coffee Coffee 16:00 - 17:00 D’Addezio Wang Bac... |
Large Eddy Simulation ( LES) is one of the most promising methods for computing industry-relevant turbulent flows. It is used to predict unsteady flow behaviors with lower computational cost as compared to Direct Numerical Simulation (DNS).
The essential idea of LES dates back to
Joseph Smagorinsky (1963) in meteorolog... |
In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism.
Let $\xi^a$ be a vector field on the $n$-dimensional spacetime manifold $M$. Physical fields are collectively represented by $\phi$, which could be the m... |
Fix a sequence $a_n={n+2\choose 2}$ of triangular numbers with the initial condition $a_0=1$,such that
$1,3,6,10,15,21,\dots$
given by
$F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$
Now if we consider the following generating function
$G(x)=-\frac{\frac{1}{x^5}\Big(1+\frac{1}{x}\Big)\Big(1-\frac{1}{x^2}\Big)... |
Your mistake was forgetting that when you square a non-$0$ equality, the result satisfies two possible equations. In particular, both $x = y$ and $x = -y$ gives that $x^2 = y^2$. From what you describe, you did the following:
$$\sqrt{3}\sin(x) = 2 - \cos(x) \tag{1}\label{eq1}$$
then square both sides to get
$$3\sin^2(x... |
For an entangled pure state, the Schmidt decomposition is such that there are at least two non-zero Schmidt coefficients. Tracing out one subsystem implies that the other subsystem is mixed.
Explicitly, we have
$$\psi = \sum_k\sqrt{\lambda_k}\vert k\rangle_A \vert k\rangle_B$$
$$\rho = \vert\psi\rangle\langle\psi\vert ... |
I think that the source is Richard Dedekind's Was sind und was sollen die Zahlen ? (Vieweg: Braunschweig, 1888).
I quote from the English translation : THE NATURE AND MEANING OF NUMBERS, (The Open Court Publishing Co., 1901) :
98. Definition [page 37]. If $n$ is any number, then will we denote by $Z_n$ the system [set]... |
We will examine situations where quantities are increasing exponentially. This situation is known as $\textit{exponential growth modelling}$, and occur frequently in our real life around us.
Population of species, people, bacteria and investment usually $\textit{growth}$ in an exponential way.
Growth is exponential whe... |
Here is a report of the Ray Tracer written by myself Christopher Chedeau. I've taken the file format and most of the examples from the Ray Tracer of our friends Maxime Mouial and Clément Bœsch. The source is available on Github.
It is powered by Open Source technologies: glMatrix, CodeMirror, CoffeeScript, Twitter Boot... |
You mean that functions are formally defined as
sets of ordered pairs. Thus, if $I=\{0,1,2\}$ and $X_0=X_1=X_2=\{a,b,c\}$, one member of $\prod\limits_{i\in I}X_i$ is the function $\{\langle 0,a\rangle,\langle 1,b\rangle,\langle 2,a\rangle\}$. You're asking, I take it, how to square this with the idea that a member of ... |
ABC is equilateral triangle inscribed in circle with Radius R. D point on the circle. I want to proof that: $DA^2+DB^2+DC^2=6R^2$ (in three ways- the proof can be in anyway not just geometry)
closed as off-topic by Namaste, Qwerty, Daniel W. Farlow, MathOverview, Watson Dec 29 '16 at 21:38
This question appears to be o... |
Simplest Calculation of Half-band Filter Coefficients
Half-band filters are lowpass FIR filters with cut-off frequency of one-quarter of sampling frequency f
s and odd symmetry about f s/4 [1]*. And it so happens that almost half of the coefficients are zero. The passband and stopband bandwiths are equal, making these ... |
Let $G,G_1$ and $G_2$ are three abelian groups with group homomorphisms $\phi_i:G\to G_i$. This gives $k$-algebra homomorphisms $k[\phi_i]:k[G]\to k[G_i]$. So we can consider $k[G_i]'s$ as $k[G]$-module via the homomorphisms $k[\phi_i]$. We can consider the tensor product $k[G_1]\otimes_{k[G]}k[G_2]$ and this will be a... |
There are many properties of circle geometry with semi circles, such as equal arcs on circles of equal radii subtend equal angles at the centres equal angles at the centre stand on equal chords the angles at the centre is twice an angle at the circumference subtended ay the same arc the perpendicular from the centre of... |
*All sequences considered are non-negative.
Suppose we are given sequences $a_n,b_n$ such that $\sum a_n=1$ and $b_n$ is a sequence going to infinity. I am searching for any sequence $c_n$ with the properties that
(1) $c_n\to \infty$
(2) $c_n\le b_n$
(3) $c_n\cdot\max\lbrace a_k:b_n\le k\le 2b_n\rbrace\to 0$
Does such ... |
Let $x, y \in \mathbb{R}$ be arbitrary, and without loss of generality, assume $y < x$.
The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(t) = \cos(t)$ is continuous and differentiable in $\mathbb{R}$, so it is obviously continuous in $[y, x]$ and differentiable in $(y, x)$.
By the Mean Value Theorem, there exi... |
Let $X$ and $Y$ be normed spaces, either both real or both complex. Let $f \colon X \to Y$ be a linear operator. Then $f$ is said to be
bounded if there exists a real number $r > 0$ such that $$ \lVert f(x) \rVert_Y \leq r \lVert x \rVert_X \mbox{ for every } x \in X. $$If there is no such $r$, then $f$ is said to be u... |
Consider the following integral,
$$I(n)=\int_0^{\pi/2}\cos^nx\cos(nx)dx $$
I tried taking one $\cos x$ out and then integrating by parts. I also tried integrating by parts using $\cos(nx)$ as the other function but didn't get any relation between $I(n)$ and $I(n-1)$.
In both the ways, $\cos(nx)$ was creating a problem ... |
Let's be perfectly clear about the assumptions. I assume that "randomly selecting 4 points" A,B,C,D in the unit square means selecting uniformly and independently at random their coordinates $x_A,y_A,x_B...$ in the $[0,1]$ range; and that one of the 6 possible pairings between points to form the segments is also chosen... |
Nevertheless, I wonder whether there are conditions for existence of a complement $M$ to the kernel $N$ of a bounded linear operator $T:V\to Q$. That is, under which conditions there is a closed subspace $M\subset V$ such that $N \oplus M = V$?
In my particular case, the operator $T\colon V \to Q$ fulfills these equiva... |
Suppose you have a vector function of space: $\vec{E}(x,y,z)=Ex(x,y,z)\hat{x}+Ey(x,y,z)\hat{y}+Ez(x,y,z)\hat{z}$.
Suppose now you want to rotate the whole vector function by using a unitary rotation matrix $\mathbf{R}(\theta)$ but you still want to describe it in terms of the original coordinates. That is, a bunch of l... |
Following appendix A of
"Ergoregions in Magnetised Black Hole Spacetimes" by G. W. Gibbons, A. H. Mujtaba and C. N. Pope, starting from the Lagrangian
$$\mathcal{L} = \hat{R} - \hat{F}_{\mu\nu}\hat{F}^{\mu\nu},$$ metric and field
$$\mathrm{d}\hat{s}^2 = e^{2\phi} \mathrm{d}s^2 + e^{-2\phi}(\mathrm{d}z + 2\mathcal{A})^2... |
The details about what an RSA key is made up of are explained succinctly here.
Is it possible to reduce the amount of data that's usually packaged with the (private) key and then derive it later?
Cryptography Stack Exchange is a question and answer site for software developers, mathematicians and others interested in c... |
June 17th, 2019, 01:49 PM
# 1
Senior Member
Joined: Oct 2015
From: Greece
Posts: 137
Thanks: 8
How to solve this simple Laplace integral?
Check this Image. I tried to post it as an image but it takes the entire space of the post...
It seems I have forgotten a lot about solving integrals... But I really need to remember... |
When can we do a linear regression without fixed or random effects and when do we need to use those in the regression analysis? I have tried studying a lot but have got only a vague idea. I would be extremely thankful if anyone can explain me with examples.
Here is a standard linear panel data model: $$ y_{it}=X_{it}\d... |
One recovers Einstein's equations by considering the bosonic closed string theory with a zero B-field, a constant dilaton $\phi$ and by taking the limit $\alpha' \rightarrow 0$, $g_s \rightarrow +\infty$ such that $g_s \sqrt{\alpha'}$ is some constant $l_p$. Here $\alpha'$ is the Regge's slope i.e. the inverse of the s... |
June 18th, 2019, 01:53 AM
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Function of two variables and differentiability
I read the following three statements for a real valued two variable function 'f' which are true:
1. I know that when the function(of two variables) is differentiable, then it implies th... |
Colloquia/Fall18 Contents 1 Mathematics Colloquium 1.1 Spring 2018 1.2 Spring Abstracts 1.2.1 January 29 Li Chao (Columbia) 1.2.2 February 2 Thomas Fai (Harvard) 1.2.3 February 5 Alex Lubotzky (Hebrew University) 1.2.4 February 6 Alex Lubotzky (Hebrew University) 1.2.5 February 9 Wes Pegden (CMU) 1.2.6 March 2 Aaron Be... |
While reading up on quantile regression I found a really nice hack described in Bayesian Quantile Regression Methods (Lancaster & Jae Jun, 2010). It is called
Jeffreys’ substitution posterior for the median, first described by Harold Jeffreys in his Theory of Probability, and is a non-parametric method for approximatin... |
For the wave equation $u_{tt} =c^2u_{xx}$ subject to the Robin boundary conditions $$a_0u(0, t)-b_0u_x (0, t) = 0$$ $$a_1u(1, t)+b_1u_x (1, t) = 0$$ for constants $a_0, b_0, a_1, b_1$ with $b_0\neq0$ and $b_1\neq 0$. Calculate the derivative $E'(t)$ of the energy given in the example.
How should $E(t)$ be modified in o... |
For proving the quadratic reciprocity, Gauss sums are very useful. However this seems an ad-hoc construction. Is this useful in a wider context? What are some other uses for Gauss sums?
Gauss sums are not an ad-hoc construction! I know two ways to motivate the definition, one of which requires that you know a little Ga... |
Article Title Keywords
Unitarily invariant norms, the Frobenius norm, Singular values
Abstract
Let $A$ and $B$ be complex square matrices. Some inequalities between $\mid A \mid + \mid B \mid$ and $\mid A^{*} \mid + \mid B^{*} \mid$ are established. Applications of these inequalities are also given. For example, in the... |
Recently I read an interesting paper: A Contextual-Bandit Approach to Personalized News Article Recommendation. I noticed its dataset is available, so I thought to play with it. Here, I share a background theory and basic intermediate experimetnal results.
Suppose you are running a local news company that earns most of... |
For giving sequence $ A_1,A_2,A_3 ……A_n$ , find number of triples $ i,j,k$ such that $ 1<=i<j<=k<=N$ and $ A_i + A_{i+1} + … A_{j-1} = A_{j} + A_{j+1} ….. A_{k}$ .Where $ +$ is bitwise xor operation .
I tried to solve it using dynamic programming somewhat similar to https://www.geeksforgeeks.org/count-number-of-subsets... |
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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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Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
I come from the world of precise satellite orbit determination, so my references also come from there. I realize this means most of the equations in my references relate to orbits and not general measurements, however, I do think you'll find them interesting.
I think a good reference is Fundamentals of Astrodynamics an... |
Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it has a deep connection with the theory of binary quadratic forms as shown in this. By this question, $1, \omega = \fra... |
We often use Ricker wavelets to model seismic, for example when making a synthetic seismogram with which to help tie a well. One simple way to guesstimate the peak or central frequency of the wavelet that will model a particlar seismic section is to count the peaks per unit time in the seismic. But this tends to overes... |
The number of of the cells touched by the line is (following Byron's answer) $N (|X_1-X_2| + |Y_1-Y_2| +1) $ . Calling $X =|X_1 - X_2|$ and $Y = |Y_1 - X_2|$ and leaving apart the $N$ factor, this is approximately equivalent, for large $N$, to $d_M=X+Y$, the Manhattan distance.
Hence, I'll attack the following problem:... |
Logical AND: Use the linear constraints $y_1 \ge x_1 + x_2 - 1$, $y_1 \le x_1$, $y_1 \le x_2$, $0 \le y_1 \le 1$, where $y_1$ is constrained to be an integer. This enforces the desired relationship. (Pretty neat that you can do it with just linear inequalities, huh?)
Logical OR: Use the linear constraints $y_2 \le x_1 ... |
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