text stringlengths 256 16.4k |
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This is not a solution, but it's too long for a comment.
As I wrote in the comment above, if $n$ is composite, then the statement is easily proved. Write $n=x \cdot y$ and pick $(a,b)=(x, xy-x)$. Since $$a^2+b^2=x^2(1+(y+1)^2)$$is divisible by $x^2$, it's not a prime.
This reduces the study of this problem to primes $n... |
I find this all easier to understand if I write $\phi$ as a map between two different manifolds (where the two might coincidentally be the same manifold).
Also, it all comes from abstract linear algebra: If $A_*: V \rightarrow W$ is a linear map (i.e., a pushforward), then there is a natural dual map $A^*: W^* \rightar... |
Elementary discrete dynamical systems problems Problem 1
Consider the dynamical system \begin{align*} x_{n+1} &= f(x_n) \quad \text{for $n=0,1,2,3, \ldots$ ,} \end{align*} where the function $f$ is graphed along with the diagonal $x_n = x_{n+1}$, below.
Find the equilibria of the dynamical system. Indicate them on the ... |
Let $X$ be a discrete random variable (r.v.) taking distinct values $x_1,x_2,\dots$, and let $p_i:=P(X = x_i)$. Then the entropy of $X$ is defined by the formula \begin{equation} H(X): = \sum_i p_i \log \frac1{p_i}. \tag{1}\end{equation}Note that it does not matter whatsoever in what set/space the values $x_1,x_2,\dots... |
I thought I had great intuition and mathematical understanding of the Metropolis-Hastings algorithm, until closer inspection... as I started compiling my notes, I realized I do not understand the rejection step of the algorithm.
Here is what I understood:
We have a target distribution $\pi(x)$, and we construct a trans... |
Using Path Difference in a Two Source Interference Pattern to Find the Wavelength of a Source
The diagram below shows two loudspeakers A and B a distance 0.8 m apart producing coherent sound of the same frequency and wavelength. 2 m away from the line joining the loudspeakers is parallel line along which a detector C i... |
I have a problem with solving the following question.
Let $\mathcal{P} = \{\mathbb{P}_\theta : \theta \in \Theta\}$ be a statistical family of discrete distributions with state space $\mathcal{X}$ and let $\textit{X}$ denote the corresponding random variable. Recall the definition of the $\textit{Kullback Leiber}$ dive... |
Geometric Drawings Lines
Two lines, \(a_1x+by_1+\;c_1=0\) and \(a_2x+by_2+\;c_2=0\) intersect if \(a_1b_2\neq a_2b_1\)
The point of intersection is
$$x=(b_1c_2-b_2c_1)/(a_1b_2-a_2b_1)$$
$$y=(a_1c_2-a_2c_1)/(a_1b_2-a_2b_1)$$
Line Segments Circle Bézier Curve Linear Bézier curves
A linear Bézier curve is simply a straigh... |
I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a higher, unified level. Categorical methods are general, but of course they can be applied to specific categories and there... |
$S^3=\{(x_1,x_2,x_3,x_4)\in R^4 ~|~~ x_1^2+x_2^2+x_3^2+x_4^2=4\}$ with the induced metric.$\forall p\in S^3 , X\in T_pS^3 , ||X||=2$, how to show $\alpha(t)=P\cos t+X\sin t$ is geodesic of $S^3$ ?
I find a local coordinate $$ u:(A,B,C)\rightarrow(2\cos A \cos B, 2\cos A\sin B, 2\sin A\cos C , 2\sin A\sin C) $$ then com... |
I have a very precise question concerning p. 82-83 of Stein's book "Singular integrals and differentiability properties of functions". Actually it is a calculation problem. For $f \in L^{2}(\mathbb{R}^n)$, denote by $u(x,y)$ the Poisson integral of $f$ $$u(x,y)=\int_{\mathbb{R}^n} P_{y}(t) f(x-t) dt.$$ Set $$|\nabla u(... |
Mini Research Project Time Updates Added CCSD(T) $n_i$ and dipole moments and tweaked discussion (the delay was caused by a system-wide storage upgrade on the machines which took nearly a week to complete). Preamble
This response is in no way meant to be contrary to what Geoff has already posted. I happen to enjoy thes... |
A cipher $E_k(m)$ is
malleable if there is a nontrivial binary relation $\sim$ on messages such that given $c = E_k(m)$, it is easy to find $c' = E_k(m')$ with $m \sim m'$. For example, AES-CTR is malleable because for any $m$ and $m'$ with $m' = m \oplus \delta$, it is easy to compute $$c' = c \oplus \delta = E_k(m) \... |
Consider the following equations $$ \begin{array}{cccx}\tag{1} z+1&=&\frac{1}{z}&,\\ z+2&=&\frac{1}{z^2}&,\\ \vdots &=& \vdots &,\\ z+k &=&\frac{1}{z^k}&. \end{array} $$ where $k$ is a positive integer number.
Question: How to find all positive real solutions of $(1)$ when $k$ is given. Example: The only positive real ... |
I'm trying to come up with examples to better understand the definition of the cell decomposition of a topological space $X$.
The simplest example I could think of would be $X =[0, 1] \subseteq \mathbb{R}$.
This is the definition I'm working with.
If $X$ is a nonempty topological space, a
cell decomposition of $X$is a ... |
The equation for the rate constant ($\pu{s^{-1}}$) for Forster (or Resonance of dipole-dipole ) energy transfer at separation
R is $$ k_R= \alpha\frac{\kappa^2\phi}{\tau R^6}\int_0^{\infty} \frac{F(\nu)\epsilon(\nu)}{\nu^4} d\nu$$where the constant $\alpha =(9000\ln(10) )/(128\pi^5n^4N)$, n is the solution refractive i... |
Mathematician:John Lewis Selfridge
(Redirected from Mathematician:J. Selfridge)Jump to navigation Jump to search
Mathematician
Proved in $1962$ that $78 \ 557$ is a Sierpiński number of the second kind.
Nationality
American
History Born: February 17, 1927, Ketchikan, Alaska, United States Died: October 31, 2010, DeKalb... |
Line 1: Line 1: − An inequality for vector functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557902.png" />, and their derivatives, defi... |
Cars, of random length $L$, arrive at a gate. The first car parks against the gate. The other arriving cars park behind at a distance uniformly distributed on $[0,1]$.Let $N(t)$ be the number of cars parked at a distance $t$ from the gate. Find:$$ \lim_{t\to \infty} E[N(t)]/t $$
I have not been given a distribution for... |
In terms of energy, the 3 dimensional MB Distribution is giving the probability for a particle to have an energy $E \geq E + dE$ is: $$f(E) = \frac{2}{\sqrt \pi} \cdot \bigg(\frac{1}{k_BT}\bigg)^{\frac{3}{2}} \cdot e^{-\frac{E}{k_BT}} \cdot \sqrt{E} \cdot dE$$ It is said that the 1 dimensional MB Distribution, giving t... |
(
Convexity is closed under intersection) Given a family of convex set $C_i$, where $i\in I$ for some index set $I$, then $\cap_{i\in I}C_i$ is convex.
In other words, the intersection of convex sets are convex.
Proof. Suppose two points $p$ and $q$ are in $\cap_{i\in I}C_i$. That is, for each $i\in I$, $p\in C_i$ and ... |
Difference between revisions of "Unitriangular matrix group:UT(3,p)"
(→Other descriptions)
m (→In coordinate form)
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==Definition==
==Definition==
+ +
===As a group of matrices===
===As a group of matrices===
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<math>\left \{ \begin{pmatrix} ... |
I have the function $$f(x)=\frac{2x}{10+x}$$ and I am asked to find its power series representation which I found to be $$\sum_{n=0}^{\infty} (-1)^{n} *\frac{2x^{n+1}}{10^{n+1}}$$ and I found the radius of convergence to be $R=10$. All until here is clear and easy, but when I am asked to find the 1st few terms I tries ... |
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Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
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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 ... |
Determining the amount of time a process requires calls for a timer. These devices can be simple kitchen timers (not very precise) or complex systems that can measure to a fraction of a second. Accurate time measurement is essential in kinetics studies for assessing rates of chemical reactions.
Determining the Rate Law... |
For what $k\in\mathbb N$, $\sqrt{n}+\sqrt{n+k}$ is irrational? ($\forall n\in\mathbb N$)
Well, a possible but perhaps in the long run not exhaustive method, since you must find for what $k \in \mathbb{N}$ this is true; consider $$(\sqrt{n}+\sqrt{n+k})(-\sqrt{n}+\sqrt{n+k})=-n+(n+k)$$ Which of course gives $k$.
now, sin... |
Galois field finite field
A field with a finite number of elements. First considered by E. Galois [1].
The number of elements of any finite field is a power $p^n$ of a prime number $p$, which is the characteristic of this field. For any prime number $p$ and any natural number $n$ there exists a (unique up to an isomorp... |
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... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
Let $\zeta(n)$ denote the Riemann Zeta function for positive integers $n>1$ as usual by:
$$ \zeta(n)=\sum_{m=1}^{\infty}m^{-n}. $$
There are fast-converging series for $\zeta(2)$ and $\zeta(3)$, but not others. In the spirit of Apéry's
$$ {\displaystyle {\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=1}^{\infty }{\fr... |
This is an extremely interesting question, with complexity due to the extreme flexibility of the quotient maps $q: X \rightarrow X/\sim$ compared to covering spaces $p:\tilde Y \rightarrow Y$. One important difference is the way in which construction of covering space gives a relation on the 1-structure, the map $p$ ad... |
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[ The Atom and RSS feeds have done an unusually poor job of preserving the mathematical symbols in this article. It will be much more legible if you read it on my blog. ]
Lately I've been enjoying
He continues a little later:
As you can see, it is not written in th... |
The "integer square root" of anon-negative integer \$ n \$ is defined as the largest
integer not greater than \$ \sqrt{n} \$:$$ \operatorname{isqrt}(n) = \lfloor \sqrt{n} \rfloor = \max \{ k \in \Bbb N_0 \mid k^2 \le n \}$$It is for example needed in prime factorization, as an upper bound for the possible factors.
A si... |
A type $T$ is a specification. A term $t$ of type $T$ is an implementation
together with a proof of correctness.
Dependent types are more expressive than simple types found in programming languages. Via the propositions-as-types correspondence they allow us to express logical statements which comprise a specification, ... |
I suspect this has been asked here before, but I didn't find anything using Search.
Why is Newton's second law only second-order in position? For instance, could there exist higher-order masses $m_i$ with
$$F(x) = m\ddot{x} + \sum_{i=3}^{\infty} m_i x^{(i)}?$$
Are there theoretical reasons why $m_i$ must be exactly zer... |
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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 ... |
If $A_1,A_2,...$ is a sequence of subsets of a topological space. Prove
$\overline{\bigcup_{k=1}^{\infty}A_k} = \bigcup_{k=1}^{\infty}A_k \cup \bigcap_{k=1}^{\infty}\bigg(\overline{\bigcup_{l=0}^{\infty}A_{k+l}}\bigg)$
I am first trying to decipher the right hand side of the equation
Let $x\in$ RHS $\Rightarrow $ $x\in... |
Since $On⊂L⊆V$, properties of ordinals that depend on the absence of a function or other structure (i.e. $\Pi_1^{ZF}$ formulas) are preserved when going down from $V$ to $L$. Hence initial ordinals of cardinals remain initial in L. Regular ordinals remain regular in $L$. Weak limit cardinals become strong limit cardina... |
In complex numbers you have to take care with power functions.
In complex analysis you have infinity logarithms. A logarithm function $l$ in a region D is a holomorphic function in $D$ such that $\exp(l(z))=z$ for all $z \in D$. As $\exp$ is not injective in $\mathbb{C}$, a complex number can have infinite logarithms. ... |
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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 ... |
Let $V$ be a vector space and $T:V\rightarrow V$ a linear transformation with the property that $T(W)\subseteq W$ for every subspace $W$ of $V$. Prove that $T$ is scalar multiplication, i.e. there is an element $\lambda$ in the field of scalars such that $T(v)=\lambda v$ $\forall v\in V$.
My attempt: I gather that for ... |
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math.CA
Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Classical Analysis and ODEs Title: Lower semicontinuity via W^{1,q}-quasiconvexity
(Submitted on 14 Jun 2011 (v1), last revised 22 Dec 2012 (this version, v7))
Abstract: We isolate a general condition, that... |
I am currently reading Topological Classification and Stability of Fermi Surfaces by Y. X. Zhao and Z. D. Wang (PRL
110, 240404 (2013)).
They remark that the Green's function (along the complex frequency axis) can be viewed as a mapping $S^p \to \mathrm{GL}(N,\mathbb{C})$, where $p$ is the co-dimension of the Fermi sur... |
Can we divide two vector quantities? For eg., Pressure( a scalar) equals force (a vector) divided by area (a vector).
No, in general you cannot divide one vector by another. It is possible to prove that no vector multiplication on three dimensions will be well-behaved enough to have division as we understand it. (This ... |
I wanna clarify some issues about renormalization in the $\bar{MS}$ scheme that I glossed over when I first learnt about this stuff. I am following http://arxiv.org/abs/1411.7853 section 3.1. The gluon part of the QCD Lagrangian is considered and the renormalized coupling and gluon field are written
$$g=\bar{\mu}^{\eps... |
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Is the usual \(\leq\) ordering on the set \(\mathbb{R}\) of real numbers a total order?
So, yes.
1. Reflexivity holds
2. For any \\( a, b, c \in \tt{R} \\) \\( a \le b \\) and \\( b \le c \\) implies \\( a \le c \\)
3. For any \\( a... |
I would like to check if two subpopulations of my data have the same parameters in a model. Model 1 is based on subpopulation 1 and Model 2 is based on subpopulation 2.
Model 1: $y=x^\alpha + \gamma +\varepsilon$
Model 2: $y=x^\beta+ \theta +\varepsilon$
The parameters of the two models are estimated with Nonlinear Lea... |
Please correct me if I got you wrong, but let me a bit elaborate your question and provide possible answers for two different cases.
Say, you have a descrete HMM model, i.e., transition probabilities $A_{ij} = P(q_{t+1}=S_i~|~q_t=S_j)$, and emission probabilities $B_{i}(k) = P(o_t = V_k~|~q_t=S_i)$, where $q$'s denote ... |
Let $X_1, X_2, ..., X_n$ be a random sample from a distribution who's PDF is given by $f(x; \theta)=(\theta+1)x^{\theta} $ for $0 \leq x \leq 1$ or $0$ otherwise.
Find the method-of-moments estimator for $\theta$.
So I have done the following:
$\mathbb{E}(X)=\int_{-\infty}^{\infty} xf(x) dx = (\theta +1)\int_{0}^{1}x^{... |
Let $1\leq d$ be an integer.
Consider the $d$-dimensional moment curve $\mu\colon \mathbb R\to \mathbb R^d$ given by $t\mapsto (t,t^2,\dots, t^d)$. Given a finite subset $S\subset \mathbb R$ of cardinality $\geq d+1$, the $d$-dimensional cyclic polytope $C(d,S)$ is the convex hull of $\mu(S)$ in $\mathbb R^d$.
It is we... |
Let it be given that $A$ is a real valued $m$ by $n$ matrix with entries $p_j\left(\frac{i}{m-1}\right)_{i,j=0}^{m-1,n-1}$, where $p_j$ is a Legendre polynomial. Show that $$\frac{\|{x}\|_2}{2} \leq \sqrt{\frac{2}{m}}\|{Ax}\|_2\leq 3\frac{\|{x}\|_2}{2}$$ when $m\geq Cn^2$ for $C\in\mathbb{R}$ and for all $x\in\mathbb{C... |
Simple pursuit
Zombies moving towards you will always catch you, but due to their lack of intelligence, your survival time increases exponentially with your relative speed. In $k=O(1)$ dimensional space ($k=2$ in the problem), the expected survival time is $d⋅(1/Θ(μd^k))^{(1+1/v)(1±o(1))/(k-1)}$ if $v$ is bounded below... |
It rather depends on what you mean by "get". In general you can't obtain population quantities from sample information. However, you can often obtain estimates, though in this case the estimates may not be very good.
If you have them, you can readily calculate the parameters from the
population mean and median; if $\ti... |
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, ... |
Applet: Lotka-Volterra model, visualized as functions of time
Illustration of the solution to the predator-prey system \begin{align*} \diff{r}{t} &= \alpha r - \beta r p\\ \diff{p}{t} &= - \gamma p + \delta r p\\ r(t_0) &= r_0\\ p(t_0) &= p_0 \end{align*} for the population sizes $r(t)$ of prey and $p(t)$ of predators ... |
we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $
wtih $Ric\ge-(n-1)K$, for all '$B$' of radius $R$ and volume $V$, $F\in C^{\infty}_0(B)$, $q=n/(n-2)$.
My question is whether this inequality was esta... |
Apologies for my inability to share intuition, a frequently subjective issue... I have learned a lot by reading the Steuernagel group numerical flows and topological features of such flows, in practice. For a recent discussion/proof of the zeros, singularities,and negative probability density features, hence your sourc... |
Since the dawn of quantum mechanics,
entanglement has been a central notion of the theory [18], a very important body of work being dedicated to understanding, classifying, measuring, and characterizing this very important property of quantum states. Once it has been recognized that it is computationally hard to decide... |
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Production of $K*(892)^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$ =7 TeV
(Springer, 2012-10)
The production of K*(892)$^0$ and $\phi$(1020) in pp collisions at $\sqrt{s}$=7 TeV was measured by the ALICE experiment at the LHC. The yields and the transverse momentum spectra $d^2 ... |
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Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
I found the following problem on a comprehensive exam:
Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a continuous function and consider the function $F: \mathbb{R}^2 \to \mathbb{R}$ given by $$F(x,y) = \int_{D_{x,y}} f(u,v)\,du\,dv, \qquad D_{x,y} = \left\{(u,v) \in \mathbb{R}^2 \,\middle|\, u^2 + v^2 \leq x^2 + y^2 \right\... |
This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is thus an edited version with the relevant details included.
Given a random vector $X \in \mathbb{R}^k$, with a known pdf gi... |
Difference between revisions of "Fujimura's problem"
Line 3: Line 3:
:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n \}</math>
:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n \}</math>
−
which contains no equilateral triangles <math>(a+r,b,c), (a,b+r,c), (a,b,c+r)</math> with <math>r > 0</math>; call su... |
I have a question of a more mathematical nature on the mathSE (Symmetric Direct Product Distributive?) that received a good answer, but I think an answer more oriented to chemists would be a useful resource here.
I'm trying to determine the symmetry of the second overtone band of the degenerate $\Pi_u$ bend of $\ce{CO2... |
This is a perfect storm of notational dissonance between QM and QFT. Your statement
I had read that the symmetry is spontaneously broken if $A \left|\psi \right>_n^{(1)}\neq 0 $ and symmetric if $A \left|\psi \right>_n^{(1)}= 0. $
is inapposite and misconstrued—justly paradoxical. I suspect labels on symmetric phase ve... |
Main Page The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math], and repl... |
The Annals of Statistics Ann. Statist. Volume 5, Number 4 (1977), 646-657. Upper Bounds on Asymptotic Variances of $M$-Estimators of Location Abstract
If $X_1, \cdots, X_n$ is a random sample from $F(x - \theta)$, where $F$ is an unknown member of a specified class $\mathscr{F}$ of approximately normal symmetric distri... |
Could anyone help with this problem? Thanks
A joint density function is given as follows:
$$f(x,y) =\begin{cases}{} 0.125\cdot (x+y+1) \ \ \text{for} -1<x<1, 0<y<2 \\ 0, \text{otherwise} \end{cases}$$
Calculate $P(X>Y)$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and p... |
Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or at least a category with weak equivalences, such that its $\infty$-categorical localization is the homotopy limit of th... |
We are given a set of grayscale image patches obtained from images after edge detection. Each patch is 10x10 pixels with intensity varying between 0 and 255 for each pixel. This set may contain a very few (maybe one or two) or a large number of patches. We want to represent the set by a single 10x10 pixel patch or “fea... |
Modulo Multiplication on Reduced Residue System is Closed Theorem
Let $m \in \Z_{> 0}$ be a (strictly) positive integer.
Let $\Z'_m$ be the reduced residue system modulo $m$:
$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Then $S$ is closed, in the sense that: $\forall a, b \in \Z'_m: a \times_m b \in \Z'_m$ Proof
L... |
Electronic Journal of Statistics Electron. J. Statist. Volume 10, Number 2 (2016), 3894-3944. Robustness in sparse high-dimensional linear models: Relative efficiency and robust approximate message passing Abstract
Understanding efficiency in high dimensional linear models is a longstanding problem of interest. Classic... |
You not only can, but also must treat symbols for units by the ordinary rules of algebra, since unit symbols are mathematical entities and not abbreviations.
The
value of a quantity is expressed as the product of a number and a unit. That number is called the numerical value of the quantity expressed in this unit.
This... |
Short answer
I think the formula for the expected successes is this:
\begin{align}E &= n \cdot \frac{3d - t - 2e + 1}{e-1}, &\text{where } & 1 ≤ t ≤ e ≤ d\end{align}
While the variance could be this (not tested):
\begin{align} V = n \cdot \left(\frac{d-t+1}{d-1} - \frac{(e-t)^2-(d-e+1)^2}{(d-1)^2}\right)\end{align}Here... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
On the shape of planar Brownian paths
We establish a formula describing the shape of the convex hull of sample paths in the case of planar Brownian motion : viz. the average number of edges joining paths’ points separated by a time-lapse \Delta \tau in [ \Delta \tau _1, \Delta \tau_2 ] is equal to 2\log (\Delta \tau_2 ... |
Difference between revisions of "Group cohomology of elementary abelian group of prime-square order"
(→Over an abelian group)
(→Over the integers)
(27 intermediate revisions by the same user not shown) Line 1: Line 1: + + + + +
Suppose <math>p</math> is a [[prime number]]. We are interested in the [[elementary abelian ... |
Let $\mathscr A$ be an algebra of sets. Let $\Sigma$ be the smallest sigma algebra (also the smallest monotone class) containing $\mathscr A$. Let $A_0 \in \Sigma$. Then $A_0 \cap \Sigma$ is a sigma algebra and is also the smallest sigma algebra (or monotone class) containing the algebra $\mathscr A \cap A_0$.
To prove... |
Definition:Particular Negative Contents Definition
A
particular negative is a categorical statement of the form: Some $S$ is not $P$
where $S$ and $P$ are predicates.
In the language of predicate logic, this can be expressed as: $\exists x: \map S x \land \neg \map P x$ Its meaning can be amplified in natural language ... |
Is there a proper proof of the following property:
Let $p$ be a prime number. The number of invertible elements in $\mathbb{Z}/p^n\mathbb{Z}$ is $(p-1)p^{n-1}$.
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 s... |
Find the limit of the sequence:
$$a_{n + 1} = \int_{0}^{a_n}(1 + \frac{1}{4} \cos^{2n + 1} t)dt,$$
such that $a_0 \in (0, 2 \pi)$
That was one of the tasks in the Olympiad.
Here is my approach.
First, I wanted to simplify the integral:
$\int_{0}^{a_n}(1 + \frac{1}{4} \cos^{2n + 1} t)dt = \int_{0}^{a_n}(dt) + \frac{1}{4... |
Fix $r>0$. For $h>0$ let $u_h\in W^{1,2}(B_h,B_r)$, where $B_h$ and $B_r$ are the ball of radius $h$ and $r$ centered at the origin in $\mathbb R^n$, respectively.
I want to approximate the $u_h$ with Lipschitz function $f_h^\lambda\in C^{0,1}(B_h,B_r)\cap W^{1,2}(B_h,B_r)$. There is a theorem which says (https://onlin... |
The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162. Free version here. Thanks to Aaron Meyerowitz's answer to ... |
We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$$
Suppose $\displaystyle\sum^\infty_{n=1}a_n$ does converge, so by the comparsion test the given series also converge.
Suppose $\displaystyle\sum^\infty_{n=1}a... |
The idea behind Stokes' theorem
Green's theorem states that, given a continuously differentiable two-dimensional vector field $\dlvf$, the integral of the “microscopic circulation” of $\dlvf$ over the region $\dlr$ inside a simple closed curve $\dlc$ is equal to the total circulation of $\dlvf$ around $\dlc$, as sugges... |
This has been stuck in my head and although I've found quite some info, I can't get to the final answer. I'm probably overthinking something, so I hope you can point it out or help me out otherwise.
I'm looking for the minimum spot size of:
A 1 µm laser of quality $M^2 = 1.5$ Leaving the system through a 40cm diameter ... |
In almost all Quantum Field Theories textbooks the same approach to quantization is presented as the first example: one considers the scalar real Klein-Gordon field $\phi$ and just write it as
$$\phi(x) =\int_{} \dfrac{d^3 p}{(2\pi^3)\sqrt{2\omega_p}} (a(p) e^{-i p_\mu x^\mu}+a^\dagger(p) e^{i p_\mu x^\mu})$$
being $a(... |
Definition:Probability Density Function Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $\Omega_X = \Img X$, the image of $X$.
Then the probability density function of $X$ is the mapping $f_X: \R \to \c... |
Electronic Journal of Probability Electron. J. Probab. Volume 23 (2018), paper no. 20, 27 pp. Evolution systems of measures and semigroup properties on evolving manifolds Abstract
An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ o... |
Main Page The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math], and repl... |
Difference between revisions of "Quasirandomness"
(→Introduction: further cleanup of vandalism)
(→Introduction)
Line 1: Line 1: − +
..././../.:.
− + − + − + − + − + − + − + − + − + − + − + − +
==A possible definition of quasirandom subsets of <math>[3]^n</math>==
==A possible definition of quasirandom subsets of <math>... |
Segment 7 Calculation Problems
1. Prove the result of "mechanical way". <math> \begin{align} \Delta^2 &= < (x-a)^2 > \\ &= < x^2 -2ax + a^2 > \\ \frac{d{\Delta^2}}{da} &= 0 \\ 2 <a - x> &= 0\\ 2(a - <x>) &= 0\\ a &= <x> \end{align} </math>
2. Thought process while solving the problem:
It is easier to construct a piecew... |
So I have a graph and need to find shortest path between two points in it. I need
1 to do it it using bidirectional search. The bidirectional search should be goal-directed, i.e. A*.
So let $l(u,v)$ be length of the (oriented) edge $u,v$, $\pi_f(v)$ the potential of vertex $v$ in forward search and $\pi_r(v)$ potential... |
In my textbook (Chemistry Part - I for Class XI published by NCERT), there is an equation for the energy of an electron in an energy state: $$E_n = -R_\mathrm H\left(\frac{1}{n^2}\right)$$ and there is a paragraph below it with the following text:
where $R_\mathrm H$ is called
Rydberg constantand its value is $2.18\tim... |
The Suzuki coupling reaction (also called Suzuki-Miyaura coupling reactions; Ref.1) is the coupling of an aryl or vinyl boronic acid with an aryl or vinyl halide or triflate using a palladium(0) catalyst similar to Heck reaction and Negishi reactions in mechanistic aspects. In particular, Negishi reaction uses organozi... |
I have a question about understanding the proof of Theorem 4.11 in the paper
A Potential Theory for Monotone Multivalued Operators (accessible here). The authors claim to construct a convex functional and I'm not sure I follow their argument.
My specific question is at the end, but I provide some background from the pa... |
The process $X$ is not gaussian and its increments are not independent.
Note first that $X$ is a Brownian martingale, hence a Brownian motion with a change of time, thus, it is distributed like $(\beta_{\langle X\rangle_t})$, where $\beta$ is a Brownian motion independent of $X$. For example, $X_1$ has the distribution... |
We can estimate this using the Polya-Vinagradov method. We get a main term, which comes from the fact that two elements of $\mathbb F_p$ that sum to something greater than $p$ are more likely to sum to something a little bit greater than $p$ than a lot, and an error term. The formula is:
$$ \frac{ i p}{2\pi} + O( \sqrt... |
(Updated)
I have looked the draft of Ch4 of the book "Abelian Varieties" by Gerard van der Geer and Ben Moonen. It looks like in order to see the group scheme structure on G/H, one should consider the fppf quotient. It is eaiser to see the group scheme structure on the fppf quotient. And one can prove that the fppf quo... |
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