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I found the following exercise in Introduction to Metric and Topological Spaces by Sutherland (Chapter 10 Question 20).
Prove that the topology on a space X is discrete iff the diagonal $\Delta=\{ (x,x) \mid x\in X\}$ is open in the topological product $X \times X$.
I believe I could prove in the $implies$ direction. I... |
INFN
- LABORATORI NAZIONALI DI FRASCATI
SEMINAR
Aula
Fisica del Nucleo, building 22
Monte Carlo Simulation for the background of the $\gamma d \rightarrow \Theta^+ \Lambda$ reaction using the g10 data.
The $\gamma d \rightarrow \Theta^+ \Lambda$ is one of the reactions under study for the g10 experiment at CLAS, with t... |
I'm trying to solve exercise 2614.1 from Demidovich's famous book of exercises on Analysis. Having solved the bit about convergence, I'm now stuck in trying to prove that a series of terms $a_n\geq 0$ is divergent if there exists $N$ such that $(1-\sqrt[n]{a_n})\frac{n}{\log{n}} \leq 1$ for all $n>N$. I've been trying ... |
If $n$ is a positive integer and $(p_1,p_2,p_3,p_4,\ldots, p_n)$ are distinct positive primes, show that the integer $(p_1\cdot p_2\cdot p_3\cdot p_4\cdots p_n)+1$ is divisible by none of these primes. How do I figure out that none of the primes divide that new integer?
Suppose $7$ is one of the primes, so $p_1\cdots p... |
The global attractor for a class of extensible beams with nonlocal weak damping
Department of Mathematics, Nanjing University, Nanjing, 210093, China
$ \begin{eqnarray*} u_{tt}+\Delta^2 u-m(\|\nabla u\|^2)\Delta u +\| u_t\|^{p}u_t+f(u) = h, \rm{in}\; \Omega\times\mathbb{R^{+}}, p\geq0 \end{eqnarray*} $
$ \Omega\subset\... |
The statement is false. Set $$f(z)=i\frac{1-rz}{1+rz}$$defined on the disk $D_{1/r}:= \{ z: |z|<1/r\}$ where $r$ is chosen to be $0<r<1$ and $1/r$ close enough to $1$. Then $f$ is holomorphic on $D_{1/r}$, hence on $D$. It maps $D_{1/r}$ onto the upper half plane, $f(0)=i$ but $f'(0)=-2ri$ so that $$|f'(0)|=2r>1$$ if $... |
I'm on 11.3.0 for macOS (64-bit).
On a recent CAS-enabled exam question a few weeks ago I was required to evaluate the following integral:
$$ \int_0^5\left(\sqrt[3]{125-x^3}\right)^2\,dx $$
In Mathematica, using the
Integrate function returns this answer:
Integrate[(125-x^3)^(2/3),{x,0,5}]
$$ 75\cdot 3^{2/3} F_1\left(\... |
Good evening! I'm actually doing an internship at the Archives Nationales of France and I encountered a situation I wanted to solve using graphs...
I. The dusty situation
We want to optimize the arrangement of books of my library according to their height in order to minimize their archive cost. The height and thicknes... |
In the last few days I thought a lot about (fully) time-constructible functions and I will present what I found out by answering Q1 and Q3. Q2 seems too hard.
Q3:
Kobayashi in his article (the reference is in the question) proved that a function $f:\mathbb{N}\rightarrow\mathbb{N}$, for which there exists an $\epsilon>0... |
Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question.
To illustrate my general question, which I postpone until the end, consider the following simple example in which I classify groups $G$ of order $3 \cdot 7$. Let $H$ and $K$ be the $7$- and $3$-Syl... |
Consider the following transition matrix
$$ T= \left[ {\begin{array}{cccc} \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{2}{6}\\ \frac{1}{3} & \frac{1}{4} & \frac{3}{5}& 0\\ 0 & \frac{1}{4} & 0& \frac{3}{6}\\ \end{array} } \right] $$
of a Markov chain process $$... |
Look at the right-most figure in diagram. The small triangle defines the all the major dimensions of the hexagon. Assuming the user measures the edge-to-edge dimension $c$, he or she can calculate the rest of the measurements. A simple right-triangle expression gives the relationship between $s/2$ and $c/2$, and is rea... |
I feel that as it was my comment, I am obliged to answer this :-).
First of all, birational equivalence is really a geometric notion. As far as I know, there is no analogue for groups, rings or fields and therefore the cryptographic relevance is limited. It becomes relevant when speaking of geometric objects: for examp... |
What new approach did Yitang Zhang try & what did the experts miss in the first place?
Yes it is a good question as to why (say) FI did not hit upon such a result, as the two major glue components, dispersion a la BFI, and beating the square-root barrier akin as Friedlander/Iwaniec, are due to them. As Zhang puts it, l... |
After some thought, I think the answer is in fact NO, even for IND-1-CCA
* and even for Shoup's OAEP+.
RSA-OAEP/OAEP+ work by taking a message $m$, producing a padding $p(m,r)$ and then encrypting this, so $c = f(p(m,r))$ where $f$ is RSA encryption, and
$f(u) = u^e \pmod{N}$ is deterministic. In fact, the whole point ... |
closed as off-topic by Mark McClure, happy fish, bbgodfrey, Michael E2, yohbs May 10 '17 at 3:54
This question appears to be off-topic. The users who voted to close gave these specific reasons:
"This question cannot be answered without additional information. Questions on problems in code must describe the specific pro... |
Update: The MathJax Plugin for TiddlyWiki has a new home: https://github.com/guyru/tiddlywiki-mathjax Some time ago I came across MathJax, a nifty, Javascript based engine for displaying TeX and LaTeX equations. It works by “translating” the equation to MathML or HTML+CSS, so it works on all modern browsers. The result... |
253 23 Homework Statement An object is undergoing circular motion in horizontal plane at fixed radius## r = 0.12m## Radial acceleration is ##2+2t ##m/s Calculate arc length the object swept through the first 2 seconds. Homework Equations -
From what I understand,
##a_{r} = v_{tan}^2 /r## ##a_{r} = (r\omega)^2 /r## ##a_... |
Congratulations Aleksander from GdyniaBilingual High School No3, Poland for your excellent solution tothe Cocked Hat problem. As you will see, the solution hinges onsimplification of an algebraic expression and solving a quadraticequation.
Here is Aleksander'ssolution.
First we will rearrange the expression from implic... |
2019-09-04 12:06
Soft QCD and Central Exclusive Production at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The LHCb detector, owing to its unique acceptance coverage $(2 < \eta < 5)$ and a precise track and vertex reconstruction, is a universal tool allowing the study of various aspects of electroweak an... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
Problem: Let $\phi(x)$ be the normal probability density function (pdf), and $\Phi(x)$ the normal cumulative distribution (cdf). I'm interested in the asymptotic behavior of the following integral
$I(a,d)=\int_{-\infty}^{\infty}dx\left[\Phi\left(x/a\right)\right]^{d}\phi(x)$
in the limit that $a \rightarrow \infty $, a... |
We study the evolution of convex hypersurfaces with initial at a rate equal to <i>H</i> 鈥<i>f</i> along its outer normal, where <i>H</i> is the inverse of harmonic mean curvature of is a smooth, closed, and uniformly convex hypersurface. We find a <i>胃</i>* > 0 and a sufficient condition about the anisotropic function ... |
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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... |
A parabola is a U-shaped plane curve where any point is at an equal distance from a fixed point (known as the focus) and from a fixed straight line which is known as the directrix. Parabola is an integral part of conic section topic and all its concepts parabola are covered here which include the following:
Definition ... |
An important theorem in Diophantine approximation is the theorem of Liouville:
Liouville TheoremIf x is a algebraic number of degree $n$ over the rational numbers then there exists a constant c(x) > 0 such that: $\left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $p,q\in N^*$ where $q>0$.
Th... |
$\newcommand{\al}{\alpha}\newcommand{\de}{\delta}\newcommand{\De}{\Delta}\newcommand{\ep}{\varepsilon}\newcommand{\ga}{\gamma}\newcommand{\Ga}{\Gamma}\newcommand{\la}{\lambda}\newcommand{\Si}{\Sigma}\newcommand{\thh}{\theta}\newcommand{\R}{\mathbb{R}}\newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\PP}{\operatorn... |
In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-Singer index theorem. I cannot find any clue how one can derive that formula from APS index theorem.
In the following, I... |
I recently came across this in a textbook (NCERT class 12 , chapter: wave optics , pg:367 , example 10.4(d)) of mine while studying the Young's double slit experiment. It says a condition for the formation of interference pattern is$$\frac{s}{S} < \frac{\lambda}{d}$$Where $s$ is the size of ...
The accepted answer is c... |
entropy is the measure of surprise
That's informal, short and non-quantitative, but correct within that. In the case of a Random Number Generator, we must make that: entropy is the measure of surprise in the outputs of the RNG, for one skilled person (with arbitrarily large computing power) knowing the RNG design inclu... |
I am interested in ring-theoretic properties of rings of modular forms. Consider the ring $R$ of integral modular forms for some level, say $\Gamma_1(n)$ -- and to be gentle, let's invert $n$. Algebro-geometrically, this can be defined as the sections of powers of the line bundle $\lambda$ on the compactified or uncomp... |
I have a question about a reflecting Brownian motion and its boundary local time.
Bass and Hsu studied the existence of Reflecting Brownian motion and boundary local time on a bounded Lipschitz domain in 1991. Although I don't state the definition of boundary local time here, I will briefly explain what it is. Roughly ... |
Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution
1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2.
School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, Chi... |
Hello guys! I was wondering if you knew some books/articles that have a good introduction to convexity in the context of variational calculus (functional analysis). I was reading Young's "calculus of variations and optimal control theory" but I'm not that far into the book and I don't know if skipping chapters is a goo... |
Intuitively, I would expect the Taylor expansion around $x_0$ of a polynomial in $(x-x_0)$ to be identical to the polynomial. However, I cannot seem to show that/whether this is the case:
For a finite power series $f(x) = \sum_{i=0}^{n} a_i (x-x_0)^i$ the $k$-th derivative is given by $$\frac{d^k f(x)}{dx^k} = \sum_{j=... |
We have used UTE sequence to obtain the subject-specific susceptibility distribution, which was then used to simulate motion-induced B
0 change at two head positions. A Fourier-based dipole-approximation method was used to map susceptibility to B 0. We have evaluated the simulation results against the measured B 0 at t... |
Here's a geometric argument, but it isn't as slick as some of the Calculus-based ones.
Consider the unit circle about $O$, through $R$ and $S$, with $\theta = \angle ROS$. The perpendicular from $S$ to $\overline{OR}$ has length $\sin\theta$, while the perpendicular from $R$ up to $T$ on the extension of $\overline{OS}... |
I'm having some trouble by trying to solve Euler equations by using the Frobenius method. For example, I'm asked to solve the Euler differential equation
$$ x^2y'' + xy' - y = 0 $$
using a power series solution.
I start by assuming there is at least one solution with the form $ y = x^\sigma\sum{a_nx^n} $.
First, I divi... |
This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that increases to $\mathcal{F}$.
Is there a way to quantify the "rate of convergence" of $\mathcal{F}_n \uparrow \mathcal{F}?... |
In quantum field theory Feynman has invented a diagrammatic method to encode various terms in the Taylor decomposition of integrals of the following form below which I will write in a baby version as finite dimensional integral rather than path integral (and using "imaginary time"):$$Z(j_1,\dots,j_n):=\frac{\int_{\math... |
Let $M$ denote a smooth $n$-dimensional manifold.
(a) Let $\phi$ denote a smooth $n$ form which is nowhere zero. Show that every $x_{0} \in M$ has a neighborhood on which we can find smooth local coordinates $x^{1}, ...x^{n}$ such that : $\phi=dx^{1}\wedge...\wedge dx^{n}$
(b) Let $\psi$ denote a closed smooth $n-1$ fo... |
I have series $\sum_{n=2}^{\infty} \frac{n+1}{n^3-1} $
I think that comparison test would give the easiest solution but i'm not sure how to apply it. I know that i need to find larger sum to prove convergence or smaller to prove divergence, so
$\frac{1}{n} $ would give us larger sum (correct me if i am wrong), and we k... |
I'm trying to prove the following:
If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that $\sum_{n=1}^\infty b_n^2<\infty$, then $\sum_{n=1}^\infty a_n^2 <\infty$.
The context here is functional analysis homework, in the subject ... |
We have a block matrix:
$$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$
Here $A$, $B$ and $C$ are all permutation matrices
of varying sizes, raised to a power. For example, all of the block matrices take on forms such as:
$$ \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\... |
1,764 69
Hi PF!
Given the ODE system ##x'(t) = A(t) x(t)## where ##x## is a vector and ##A## a square matrix periodic, so that ##A(t) = A(T+t)##, would the following be a good way to solve the system's stability: fix ##t^*##. Then $$ \int \frac{1}{x} \, dx = \int A(t^*) \, dt \implies\\ x(t) = x(0)\exp\left( A(t^*)t \r... |
A well known fact in probability is that a uniform random variable on $[0,1]$ can be used to simulate any other probability distribution on $\mathbb{R}$.
A standard way of doing this is to define, given $\mu$ a probability on $\mathbb{R}$, the random variable $F(u,\mu) = \min\lbrace x \in \mathbb{R}: \mu((-\infty,x]) \... |
The Annals of Statistics Ann. Statist. Volume 26, Number 3 (1998), 1011-1027. Estimation of the truncation probability in the random truncation model Abstract
Under random truncation, a pair of independent random variables $X$ and $Y$ is observable only if $X$ is larger than $Y$. The resulting model is the conditional ... |
Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$.
A
Frostman measure on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r_0>0$ with $\mu(B_r(x)) \leq C r^t$ for all $x\in X$ and all $0<r\leq r_0$. Let's call the supremum of such $t$ the Frostman-Exponent $Fro... |
This is a neat little trick that I don’t
think I ever shoe-horned into a publication. It’s essentially a means to leverage side information in the form of NLP corpora to improve multi-class classification.
In short, the trick is: multiply output confidence scores by a co-occurrence matrix. This helps your model predict... |
The problem is NP-complete. Here is a reduction from 3SAT.
Given an instance of 3SAT with variables $v_1,\ldots,v_n$ and clauses $c_1,\ldots, c_m$, construct numbers as follows. We represent a number as the form $\lambda_1p^{\mu_1}+\lambda_2p^{\mu_2}+\cdots$ where $p$ is a very large number (we will estimate $p$ later)... |
In this vignettes, an application of a locally and a globally efficient adaptive sample determination to a confirmatory randomized clinical trial is illustrated.
This trial evaluated whether oral adjuvant chemotherapy with tegaful and uracil (UFT) and leucovorin (LV) reduces the recurrence after resection of liver meta... |
We know that for a point particle, the action is
$$ S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) -m^2e(\lambda)\right] , $$
with signature convention $(-,+,+,+)$. It was mentioned on some website as a I googled that $e$... |
November 5th, 2017, 11:23 PM
# 1
Senior Member
Joined: Sep 2012
Posts: 201
Thanks: 1
Legendre reccurence relation
I am having a slight issue with generating function of Legendre polynomials and shifting the sum of the generating function.
So here is an example:
I need to derive the recurrence relation $lP_l(x)=(2l-1)xP... |
Answer
One nautical mile is 1.1 statute miles.
Work Step by Step
We can convert the angle to radians: $\theta = 01' = (\frac{1}{60})^{\circ}(\frac{\pi~rad}{180^{\circ}}) = 0.00029~rad$ The can find the arc length $S$: $S = \theta ~r$ $S = (0.00029~rad)(3963~mi)$ $S = 1.1~mi$ One nautical mile is 1.1 statute miles. |
Let $F$ be a field and $R$ a commutative ring where $R$ is not the zero ring. Suppose $\varphi : F \rightarrow R$ is a ring homomorphism. Show that $\varphi$ is injective.
Any help would be appreciated. I feel pretty lost on this.
Mathematics Stack Exchange is a question and answer site for people studying math at any ... |
The main reason to prefer the colon notation $t : T$ to the membership relation $t \in T$ is that the membership relation can be misleading because
types are not (just) collections.
[
Supplemental: I should note that historically type theory was written using $\in$. Martin-Löf's conception of type was meant to capture ... |
Exercise :
Calculate a Maximum Likelihood Estimator for the model $X_1,\dots, X_n \; \sim U(-\theta,\theta)$.
Solution :
The distribution function $f(x)$ for the given Uniform model is :
$$f(x) = \begin{cases} 1/2\theta, \; \; -\theta \leq x \leq \theta \\ 0 \quad \; \; , \quad\text{elsewhere} \end{cases}$$
Thus, we ca... |
$\displaystyle\sum_{k=1}^\infty (2k)!/k!(k+1)!$
Let $a_k = (2k)!/k!(k+1)!$
$\lvert a_{k+1}/a_k\rvert \to 4$ as $k \to \infty$
Thus the series is divergent. Can someone double check ... my gut says it is convergent.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profes... |
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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 set $Q$ of $n$ points, we want to find the subset $S_\max \subset Q$ of $k$ elements that maximize the total distance between them.
$$S_\max = \max_S \sum_{\substack{ i,j\in S\\ i \neq j}} d(x_i,x_j)$$
where in my case $x_i$ is a boolean vector and the distance considered is the Manhattan/Hamming distance.
Is t... |
This question already has an answer here:
I know $ \omega ^ 2 $ is countable, but I'm unable to find a bijection from $ \omega * \omega \rightarrow \omega $
This should be simple, but I'm very stuck.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in rela... |
N Saradha
Articles written in Proceedings – Mathematical Sciences
Volume 100 Issue 2 August 1990 pp 107-132
Under certain assumptions, it is shown that eq. (2) has only finitely many solutions in integers
Volume 127 Issue 4 September 2017 pp 565-584 Research Article
Let $F(X, Y) = \sum^{s}_{i=0}a_{i}X^{r_i}Y^{r−r_i} \i... |
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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... |
I am having issues in integrating the numerical solution obtained with FEM method and NDSolve, over a boundary. Specifically, i am integrating a specific combination of the gradient over a boundary obtaining errors which are much larger than the specified tolerances. I would like to have a way to avoid the errors and t... |
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\ a_j & b_j & c_j \\ a_k & b_k & c_k \end{matrix}\right)\right|^p \right)^{\frac{1}{p}} \leq c_p \left( \sum_{i=1}^n |a_i|^p \right)^{\frac{1}{p}} \left( \sum_{1\leq j... |
If you want the expected value, one answer is $n E[S_{(m)}]$, where $S_{(m)}$ is the $m$th order statistic of a sample of $n$ gamma$(k,1)$ random variables. While this expression may not have a simple closed form, you may be able to get a decent-sized approximate answer from the literature on moments of order statistic... |
Let $x$ be a nonnegative real number and denote $[x]$ as the greatest integer less than or equal to $x$. We will attempt to prove that $\big[\sqrt{x}\big] = \big[\sqrt{[x]}\big]$.
First suppose that $x$ is a perfect square. Then the equation trivially holds.
Assuming that $x$ is not a perfect square, we have $\sqrt{x} ... |
The most frequently used evaluation metric of survival models is the concordance index (c index, c statistic). It is a measure of rank correlation between predicted risk scores $\hat{f}$ and observed time points $y$ that is closely related to Kendall’s τ. It is defined as the ratio of correctly ordered (concordant) pai... |
Laser Beam Expanders
This is a supplementary section of the Laser Optics Resource Guide.
Laser beam expanders increase the diameter of a collimated input beam to a larger collimated output beam. Beam expanders are used in applications such as laser scanning, interferometry, and remote sensing. Contemporary laser beam e... |
11 0
In free electron 3D box model, we can calculate the density of state on the Fermi surface g([tex]\epsilon[/tex]f) easily, but how about the level spacing near the Fermi surface? I think this level spacing [tex]\Delta[/tex]E should satisfy [tex]\Delta[/tex]E=d/g([tex]\epsilon[/tex]f) where d is the degree of degene... |
Annals of Functional Analysis Ann. Funct. Anal. Volume 4, Number 1 (2013), 138-148. Coupled coincidence point theorems for nonlinear contractions under c-distance in cone metric spaces Abstract
In this paper, among others, we prove the following results:\\ $(1)$ Let $(X,d)$ be a complete cone metric space partially ord... |
I was talking about the following tentative argument. The 2-category of distributors (also called profunctors) $\mathrm{Dist}$ has (small) categories for objects. For $C,D:\mathrm{Dist}$ the 2-category of morphisms is defined as $$\mathrm{Dist}(C,D):=[D^{op} \times C; Set]$$and denoted as $C \nrightarrow D$ (the middle... |
Let $E$ be an elliptic curve defined over $\mathbb{Q}$.The
canonical height of a rational point $P\in E(\mathbb{Q})$ is computed by writing the $x$-coordinate $x(nP)=A_n(P)/D_n(P)$ as a fraction in lowest terms and setting $$ \hat h(P) = \lim_{n\to\infty} \frac{1}{n^2}\log \max\bigl\{|A_n(P)|,|D_n(P)|\bigr\}. $$( Note.... |
The way I know of to bound generalization error by Rademacher complexity is Theorem 2.4 in this lecture notes, http://ttic.uchicago.edu/~tewari/lectures/lecture9.pdf. Here the quantity on the LHS that Rademacher complexity is trying to upperbound is given as, $L_{\phi}(\hat{f}_{\phi}^*)-\min_{f \in F} L_{\phi}(f)$ wher... |
The solution is easy by employing $\ gf\bmod gh\, =\, g(f\bmod h)\ \ $ [mod Distributive Law]
$$\begin{align}f(x)\!-\!f(a)\bmod (x\!-\!a)(x\!-\!b) &= (x\!-\!a)\left(\dfrac{f(x)\!-\!f(a)}{x\!-\!a}\bmod x\!-\!b\right)\\&= (x\!-\!a)\left(\dfrac{f(b)\!-\!f(a)}{b\!-\!a}\right)\ \ {\rm if}\ \ a\neq b\\&= (x\!-\!a)\,\ f'(a)\q... |
EDIT
I think I have to correct myself. I believe now that the integral is an analytic function of k and that the cut coming from Mathematica's antiderivative is spurious and so are the complications in performing the integral.
The argument is that the integrand is analytic in k and hence is expandable into a convergent... |
I'm not that familiar with stellar formation, so you might be right. According to this website:http://www.josleys.com/show_gallery.php?galid=313the earth is malleable because of its liquid core and tectonic plates on the surface (quite interestingly, according to that website, if the...
Thanks for your help. I apprecia... |
This question already has an answer here:
There is a set with $n$ elements. Why is the maximum number of subsets that can be formed out of it $2^n$?
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 minute to sign up.Sign ... |
(a) To find a basis for the plane $x-y+z = 0$, you could solve this equation in terms of $x$ and $z$: $y= x+z$. Then the set of vectors of your plane could be described as:
$$V = \left\{ (x, x+z, z) \ \vert \ x,z \in \mathbb{R} \right\} \ .$$
From this description it's easy to find a basis for your plane $V$: it will h... |
$$\mathrm{molarity} = \frac{\text{amount of solute}}{\text{volume of solution}} $$ and amount of substance is based on quantity (larger mass means larger amount), so how come it is an intensive property. Shouldn't it be an extensive property?
Concentration is an intensive property. The value of the property does not ch... |
I would like to know how to find an adjoint of an operator $T$ on a Hilbert space.
I tried to find out on my own but it's not solid. Here is what I did:
I picked a concrete example. Let $H=\ell^2$ and let $R: H \to H$ be the right shift operator. Let $e_1 = (1,0,0,\dots), e_2=(0,1,0,0,\dots)$ etc. Then I used the defin... |
I have been trying to derive the Einstein equation from the Einstein-Hilbert action $$ S[g_{\mu \nu}] = \frac{1}{16 \pi} \int_M \text{d}^4x \sqrt{-g}R $$ The standard derivation states that the variation $\delta S =0$ when we vary the metric components. In this derivation, we use the fact that $\delta (\sqrt{-g}R)= \de... |
Randomized learning of the second-moment matrix of a smooth function
1.
Institute of Electrical Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
2.
Department of Electrical Engineering, Colorado School of Mines, Denver, CO 80401, USA
3.
Departments of Statistics and Computer Science, Ru... |
The square root of 2 is 1.41421356237... Multiply this successively by 1, by 2, by 3, and so on, writing down each result without its fractional part: 1 2 4 5 7 8 9 11 12... Beneath this, make a list of the numbers that are missing from the firstRead More →
Dear Uncle Colin How does $\sqrt{9 - \sqrt{17}} = \frac{\sqrt{... |
Let $H_{n}$ be the $n$th harmonic number. In Lagarias's paper "An Elementary Problem Equivalent to the Riemann Hypothesis," he shows that the statement $$\sum_{d\mid n}d\leq H_{n}+\exp(H_{n})\log(H_{n})\tag{1.1}$$ for every positive integer $n$, with equality if and only if $n=1$, is equivalent to the Riemann hypothesi... |
The ideal encryption scheme $E$ would be one that, for every ciphertext $C=E(K, M)$, if the key remains secret for the adversary, the probability of identifying $M$ is
negligible. Since that is not possible in practice, the second most reasonable approach is to define constraints strong enough to satisfy some definitio... |
Apologies if this is obvious -- I'm very new to
Mathematica.
I'm trying to minimize the solution to an ODE with respect to a variable. The following code generates the solution to the ODE,
sol=DSolve[ {(1/2) * σ^2 * k''[q] + μ*k'[q] - λ*k[q] == 0, k'[0] == -mc, k'[b] == me}, k, q]
but when I try minimizing using the
Mi... |
Your public key contains two numbers. First it is a number n, which is called the Modulus and are computed through $p \cdot q = n$.The second number is e, which is the public exponent and are used to encrypt your message m. The number e is choosen that it have the following properties:
\begin{equation}1 < e < \phi(n) =... |
I am interested in the following problem which seems like an extension of the Kruskal-Katona Theorem.
Let $A_k \subseteq \{0,1\}^n$ be a subset of the hypercube such that every element in $A$ has exactly $k$ ones. For any element $x \in \{0,1\}^n$ let $N_l(x)$ be the set of elements obtained by flipping one of the 1's ... |
Given n random points on a circle, find, with proof, the probability that the convex polygon formed by these points does not contain the center of the circle. at first i thought it easiest to find probability that the center IS enclosed and then take 1 minus that . the probability would be that at least 1 point lie on ... |
The general theorem is: for all odd, distinct primes $p, q$, the following holds: $$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$
I've discovered the following proof for the case $q=3$: Consider the Möbius transformation $f(x) = \frac{1}{1-x}$, defined on $F_{p} \cup {\inft... |
Publications by Caroline Terquem
MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY
482 (2019) 530-549
MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY
476 (2018) 5032-5056
MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY
464 (2017) 2429-2440
MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY
464 (2017) 924-932 CoRoT 2... |
There is a huge variety of feasible approaches. Which is best suited depends on
what you are trying to show, how much detail you want or need.
If the algorithm is a widely known one which you use as a subroutine, you often remain at a higher level. If the algorithm is the main object under investigation, you probably w... |
What is a formal definition of a irrational number? Usually, we say that it is a number that it is not rational. Is it enough?
Uncle Google and auntie Wikipedia are your friends. Wikipedia correctly states:
In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers
In a way,... |
Found this on Complexity Zoo warning expired certificate check NP Over The Complex Numbers.
[BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum number of additions, subtractions, and multiplications needed to construct $n$, starting from 1. If for every sequence ${n_k}... |
452 0 1. Homework Statement
A piece of wire is bent to form a circle with radius r. It has a steady current I flowing through it in a counterclockwise direction as seen from the top (looking in the negative z direction).
What is B_z(0), the z component of B at the center (i.e., x = y = z = 0) of the loop?
Express your ... |
2019-09-04 12:06
Soft QCD and Central Exclusive Production at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The LHCb detector, owing to its unique acceptance coverage $(2 < \eta < 5)$ and a precise track and vertex reconstruction, is a universal tool allowing the study of various aspects of electroweak an... |
In an occasion, I'd like to use Fubini's Theorem to swap the order of integration of a countour integral with an integral with respect to a measure (show that $\int_{\gamma} \int_{\Omega} f(x,z) d\mu(x) dz = \int_{\Omega} \int_{\gamma} f(x,z) dz d\mu(x)$). For that, I need to take two measure spaces, $(\Omega, \mathcal... |
Solving the complex equation $z^2=1+2\,i$ using
Solve[z^2 == 1 + 2 I]
returns $\left\{\left\{z\to -\sqrt{1+2\,i}\right\},\left\{z\to\sqrt{1+2\,i}\right\}\right\}$, but how do I force Mathematica to always output on the form $a+b\,i$, $a,b\in\mathbb{R}$? Or, if there is no output form from
Solve to do this, to convert/t... |
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 ... |
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