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I have a basic question: if we use 1d string to replace 0d particle to gain insight of nature in string theory, and advanced to use 2d membranes, can we imagine that using $3$- or $n$-dimensional blocks/objects/branes as basic units in physics theory? Where is the end of this expansion?
There
can not only, there have t... |
Reference : Invariance in a class of operations related to weighted quasi-geometric means
E-prints/Working papers : First made available on ORBilu Physical, chemical, mathematical & earth Sciences : Mathematics http://hdl.handle.net/10993/36748
Invariance in a class of operations related to weighted quasi-geometric mea... |
So first let me state my homework problem:
Let $X$ be a set, let $\{A_k\}$ be a sequence of subsets of $X$, let $B = \bigcup_{n=1}^{+\infty} \bigcap_{k=n}^{+\infty} A_k$, and let $C = \bigcap_{n=1}^{+\infty} \bigcup_{k=n}^{+\infty} A_k$. Show that (a) $\liminf_k\; {\xi_A}_{_k} = \xi_B$, and $(b)$ $\limsup_k \;{\chi_A}_... |
I'm working on an introductory qm project, hope somebody has the time to help me (despite the length of this post), it will be highly appreciated.
My goal is to determine the bound states and their energies for the potential
$V_j(x) = -\frac{\hslash^2a^2}{2m}\frac{j(j+1)}{\cosh^2(ax)}$
for any positive value of j (I th... |
RGPV First Year Engineering (Set A) (Semester 1)
Basic Electricals & Electronics Engg. May 2013
Basic Electricals & Electronics Engg.
May 2013
Total marks: --
Total time: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons (2) Marks are given to the right of every question (3) Draw neat di... |
Let $X$ be a Hausdorff locally compact in $x \in X$. Show that for each open nbd $U$ of $x$ there exists an open nbd $V$ of $x$ such that $\overline{V}$ is compact and $\overline{V} \subset U$.
My work:
Since $X$ is Hausdorff and locally compact then $X$ is regular. Let $U$ be an open nbd of $x$. By assumption $X$ is l... |
Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, n \right\}$.
For example, if $n = 5$, then \begin{equation} B = \left(\begin{array}{rrrrr} 6 & 20 & 6 & 0 & 0 \\ 1 & 15 ... |
Here is a closely related pair of examples from operator theory, von Neumann's inequality and the theory of unitary dilations of contractions on Hilbert space, where things work for 1 or 2 variables but not for 3 or more.
In one variable, von Neumann's inequality says that if $T$ is an operator on a (complex) Hilbert s... |
The Annals of Mathematical Statistics Ann. Math. Statist. Volume 28, Number 3 (1957), 773-778. On the Power of Optimum Tolerance Regions when Sampling from Normal Distributions Abstract
In [1], optimum $\beta$-expectation tolerance regions were found by reducing the problem to that of solving an equivalent hypothesis t... |
We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called
surfaces, to explore a variety of other surfaces th... |
Good day all,
I'm new to probability theory and am currently working on a problem and was looking for some feedback on my work.
The question is this: One of two components is selected at random and tested. Component 1 is faulty with probability 1/5 and Component 2 is faulty with probability 1/10. What is the probabilit... |
The Annals of Statistics Ann. Statist. Volume 19, Number 4 (1991), 1813-1831. Central Limit Theorems for $L_p$ Distances of Kernel Estimators of Densities Under Random Censorship Abstract
A sequence of independent nonnegative random variables with common distribution function $F$ is censored on the right by another seq... |
Yes, there are many ways to produce a sequence of numbers that are more evenly distributed than random uniforms. In fact, there is a whole field dedicated to this question; it is the backbone of quasi-Monte Carlo (QMC). Below is a brief tour of the absolute basics.
Measuring uniformity
There are many ways to do this, b... |
If $\hat{T}(\Delta x) = e^{-\frac{i}{\hbar}\hat{p}\Delta x}$ is the spatial translation operator, then there exists a function $f$ from $\mathbb{R}$ to the ket space $V$ such that $\hat{T}(\Delta x) f(x) = f(x+\Delta x)$. Namely, the function that sends $x$ to the position eigenstate $|x\rangle$.
Similarly if $\hat{U}(... |
I concur with Aretino's answer; I just wanted to dig in to the details a bit, in the hopes it illustrates some of the options and approaches we can utilise here.
Length of the curve $$\begin{cases} x(t) = r t \cos(t)\\y(t) = r t \sin(t)\end{cases}\tag{1}\label{1}$$from $t_0$ to $t_1$ is$$s( t_0 ,\, t_1 ) = \int_{t_0}^{... |
Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). (answer) Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. If it converges, compute the limit. (answer) Ex 11.1.4 Determine whether \(\left\... |
Suppose $(X^m, g)$ is a closed Riemannian manifold of dimension $m$ with the following properties,
There is a constant $\kappa$ such that $\kappa r^m \leq Vol(B(x, r)) \leq \kappa^{-1} r^m$ for every $r \in (0,1)$.
For every $C^1$ function $f$, we have $(\int_X |f|^{\frac{2m}{m-2}})^{\frac{m-2}{m}} \leq C_S (\int_X |f|... |
I need to find tangent plane to surface $z = \frac{y^2 - 1}{x}$ that passes through $A(0,1,0)$ and $B(1,3,4)$
Normal vector of the plane I am looking for:
$$\vec{n} = \Big[\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}, -1 \Big]$$
$$\vec{n} = \Big[\frac{1 - y^2}{x^2}, \frac{2y}{x}, -1 \Big]$$
Plane equati... |
I was pretty confident that things are simple, but unfortunately I must have missed something. We can always change between between the bases for Dirac spinors, using unitary transformation, because
$$ \partial_\mu \bar\Psi \gamma_\mu \Psi=\partial_\mu \bar\Psi \underbrace{U^\dagger U }_{=1}\gamma_\mu \underbrace{U^\da... |
Defining parameters
Level: \( N \) = \( 4000 = 2^{5} \cdot 5^{3} \) Weight: \( k \) = \( 1 \) Character orbit: \([\chi]\) = 4000.ci (of order \(50\) and degree \(20\)) Character conductor: \(\operatorname{cond}(\chi)\) = \( 1000 \) Character field: \(\Q(\zeta_{50})\) Newforms: \( 0 \) Sturm bound: \(600\) Trace bound: ... |
To answer this question it is advantageous to treat a molecule as a graph and use the well known adjacency matrix from graph theory.Here is the wikipedia definition:
For a simple graph with vertex set $V$, the adjacency matrix is a square $|V| × |V|$ matrix $\mathbb{A}$ such that its element $\mathbb{A}_{ij}$ is one wh... |
I started thermodynamics mostly through independent study and basically built up my own definitions of terms that appeared to fit with what was going on. They seemed to work but my question is whether or not this is how they are actually supposed to be viewed.
Equipartition theorem. 'It is possible to show that at equi... |
Second Principle of Finite Induction Contents Theorem
Let $n_0 \in \Z$ be given.
Suppose that: $(1): \quad n_0 \in S$ $(2): \quad \forall n \ge n_0: \paren {\forall k: n_0 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: $\forall n \ge n_0: n \in S$ The second principle of finite induction is usually stated an... |
I didn't feel MO was the best place to ask this question, so apologies for this, but when I asked it at https://math.stackexchange.com/questions/2297837/why-is-this-cubic-polynomial-generic-for-cyclic-field-extensions, I didn't get enough information. I would really like to understand this example, so I will try to str... |
I am trying to simulate the phase separation of a binary mixture. If the free energy F is known as a function of the concentration $c$, the dynamical equation is:
$ \frac{\partial c(x,t)}{\partial t}=\frac{d^2}{dx^2} \frac{\delta F[c]}{\delta c} $
For the Flory-Huggins free energy we have:
$ \frac{\delta F[c]}{\delta c... |
Here is an example from Bhargav Bhatt's talk "Using DAG" at MSRI last week. Needless to say, any mistakes are mine.
Theorem. Let $X$ be a coherent (quasi-compact and quasi-separated) scheme, let $A$ be a ring complete with respect to an ideal $I\subseteq A$. Then$$ X(A) \to \varprojlim_n X(A/I^{n+1}) $$is bijective.
Be... |
To my understanding, mixed states is composed of various states with their corresponding probabilities, but what is the actual difference between maximally mixed states and maximally entangled states?
Suppose we have two Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$. A quantum state on $\mathcal{H}_A$ is a normali... |
I'll start with Earth
Earth is hurling through space at a speed of approximately $29.78 km/s$ If the sun were to disappear, the Earth would move in a straight line until the sun reappears. Since there are $259,200 seconds$ in three days that gives Earth the time to travel $29.78 km/s \times 259,200 s = 7,718,976 km$ Th... |
Answer
$$d = 0.958 \space g/mL$$
Work Step by Step
$$V = 2.18 \space L \times \frac{1000 \space mL}{1 \space L} = 2180 \space mL$$ $$d = \frac{m}{V} = \frac{2088 \space g}{2180 \space mL} \approx 0.958 \space g/mL $$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim ... |
"""Author: John VolkDate: 10/10/2016"""from __future__ import print_functionfrom sympy.parsing.sympy_parser import (parse_expr, standard_transformations, implicit_multiplication,\ implicit_application)import numpy as npimport sympyimport re
Python, regex, and SymPy to automate custom text conversions to LaTeX¶ This pos... |
The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places
(for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a well-defined adjoint for an antilinear operator? Suppose we have an antilinear operator $A$ such that $$ A(c_1|\psi_1\rangle+c_... |
We present the first observation of exclusive $e^+e^-$ production in hadron-hadron collisions, using $p\bar{p}$ collision data at \mbox{$\sqrt{s}=1.96$ TeV} taken by the Run II Collider Detector at Fermilab, and corresponding to an integrated luminosity of \mbox{532 pb$^{-1}$}. We require the absence of any particle si... |
Also it was stated there that maxwell's equations are invariant under Lorentz transformation but not under Galilean transformation?
Please provide me with some explanation regarding this.
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minu... |
When light is red shifted from distant galaxies, the photons have lost energy. When dark energy pushes objects apart, those objects have gained energy from a larger gravitational potential. Is the amount of energy that dark energy applies to push objects apart equal to the amount of energy lost because light from dista... |
The spacing in the following output looks off to me. In particular, the integral symbol has not grown to accommodate the height of the integrand, the spacing in the fraction seems large, and the enclosing brackets do not rise high enough in the matrix. The issue remains whether
mathtools is loaded or not, but since I u... |
(Edited as suggested)
I have following code involving connected nodes:
\documentclass{article}\usepackage{tikz}\tikzstyle{every picture}+=[remember picture]\begin{document}\begin{equation} P(t) = \tikz[baseline]{\node[fill=blue!50, anchor=base] (t1) {$ \epsilon_{0}\chi^{(1)}E(t) $};} + \tikz[baseline]{\node[fill=red!50... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
I’m working on a number theory proof that has been giving me some trouble for a while. I will explain the problem and the attempts I’ve made.
Let $x\in \mathbb{R}$ and $d \in \mathbb{Z}$ where both $x, d > 0$ (i.e. positive values). Prove that the number of integers, say k, that are $\leq $ $x$ and divisible by $d$ is ... |
I am interested in showing continuity/boundedness of the weak solution to the following problem pde:
\begin{align*} 0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\ 0 &= \mathbf{\nabla} \cdot \mathbf{q} && \quad x\in \Omega,\\ 0 &= u && \quad x\in \partial \Omega_D,\\ g &= \mathbf{q}\cdot \mathbf{\eta} &&\qua... |
Planck's law
Until stars were formed a few hundred million years after the Big Bang (BB), the brightness of the Universe was extremely homogeneous and given by a near-perfect blackbody Planck spectrum with a temperature of $T = T_0(1+z)$, where $T_0=2.725\,\mathrm{K}$ is the current temperature of the CMB, and $z$ is t... |
I have a matrix $P \in M_n(\mathbb N)$, where
$$ P = \begin{bmatrix} 0 & P_{12} & \ldots & P_{1n}\\ P_{21} & 0 & \ldots & P_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ P_{n1} & P_{n2} & \ldots & 0 \end{bmatrix}$$
with $P_{ii} = 0$ for all $i \in \{1,2,\dots,n\}$. I need to find matrices $A, B \in M_n(\mathbb N)$ that sa... |
Each equation gives information about the body's location on each axis in Cartesian coordinate system (A is some constant and $t$ is time). We know that $\sin^2(x)+\cos^2(x)=1$ (Pythagora's theorem applied to unit circle which gives us the radius of unit circle). This is the answer given for this problem:
Each velocity... |
I know that the integral $\int_0^1 \frac{(x+1)^n-1}{x} dx,$ for $n \in \mathbb{Z}^+$, can be evaluated by expanding the numerator with the binomial theorem and integrating term by term. You get the nice expression $$\int_0^1 \frac{(x+1)^n-1}{x} dx = \sum_{k=1}^n \binom{n}{k} \frac{1}{k}.$$ My question is this:
Is there... |
We are here with you hands in hands to facilitate your learning & don't appreciate the idea of copying or replicating solutions. Read More>>
MTH101 Calculus And Analytical Geometry GDB Solution & Discussion
For a functiona pointand a positive numberFind
Moreover find a number such that
Note:
Please follow the following... |
Homework Helper
1,020 0
Hi,
I'm having some trouble with solving this indefinite integral.
[tex] \int {\sqrt {\frac{{6\cos ^2 x + \sin x\cos (2x) + \sin x}}{{2 - \sin x}}} } dx [/tex]
I was able to lose the sin(x) and get a cos(x) out of the square root by doing this:
[tex] \int {\sqrt {\frac{{6\cos ^2 x + \sin x(2\cos... |
Another way we often think about numbers is as abstract quantities that can be measured: length, area, and volume are all examples.
In a measurement model, you have to pick a
basic unit. The basic unit is a quantity — length, area, or volume — that you assign to the number one. You can then assign numbers to other quan... |
Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ denote the square root of the metric $g$, implicitly defined by the formula $\hat g_{ai} \hat g_{bj} \delta^{ab} = g_{ij}$,... |
The least count of the watch used for the measurement of time period is $0.01$ s
This information is just telling you to round off to the second decimal place, as you correctly did.
The sample mean is $\mu = 0.56$ and the sample standard deviation is $\sigma = 0.02$. The answer the text is referring to is
$$\frac \sigm... |
With regard to the density parameter derived from Friedmann Equations which is:
$$ Age = D_H\int_{z}^{\infty}\frac{1}{(1+z)\sqrt{\Omega_R(1+z)^4 + \Omega_M(1+z)^3 + \Omega_K(1+z)^2 + \Omega_L(1+z)^{(3(1+w))}}} dz $$
(setting $z$ to $0$ will provide the current age of the universe)
where $D_H=$ Hubble Distance, $z$ = Re... |
Since I just finished optimizing a lot of them in a software, DifferentialEquations.jl, I decided to just lay out a comparison of the main Order 4/5 methods. The Fehlberg method was left out because it's commonly known to be less efficient than the DP5 method.
Backstories Dormand-Prince 4/5
The Dormand-Prince method wa... |
Let $(M^{2n},\omega)$ be a symplectic manifold with an integral symplectic form $\omega$. Due to the work of M.Gromov and D.Tischler (M.Gromov "A topological technique for the construction of solutions of differential equations and inequalities", D.Tischler "Closed 2-forms and an embedding theorem for symplectic manifo... |
We assume that $G\in G(n,p),p=\frac{\ln n +\ln \ln n +c(n)}{n}$. Then the following fact is well known:
\begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\rightarrow \infty) \\ 0 & (c(n)\rightarrow - \infty) \\ e^{-e^{-c}} & (c(n)\rightarrow c) \end{cases} \end{eqnarray}
I want to know resu... |
Physicists tend to be a bit casual about sign conventions when it seems to be obvious. So let's attempt to be completely rigourous.
The key step is getting the flight time $t$ since the range is just $v\cos\theta\, t$. We do this using the SUVAT equation:
$$ v = u + at $$
We'll use the usual conventions that up and rig... |
The Schwarzchild metric is for the gravitational field of an object of mass $M$ with no electric charge and no angular momentum. The metric is
$$ {ds}^{2} = \frac{dr^2}{1 - \frac{r_\mathrm{s}}{r}} - c^2dt^2\left(1-\frac{r_\mathrm{s}}{r}\right) + r^2 \left(d\theta^2 + \sin^2\theta \ d\varphi^2\right) $$
with $r_s = \fra... |
Suppose that $K$ is a finite extension of $\mathbb Q$, say of degree $n$. By the primitive element theorem, $K=\mathbb Q(\alpha)$. Then $\alpha$ has $n$ conjugates and we correspondingly get $n$ embeddings of $K$ into $\mathbb C$. But I believe that all these embeddings need to be have the same images in $\mathbb C$ (a... |
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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 ... |
Ever since day one of of my Mathematical Logic course, this fact has really bothered me. I cannot wrap my head around how an empty set is a subset of every possible set. Could someone kindly explain how this is true? Any help is appreciated!
If you're comfortable with proof by contrapositive, then you may prefer to pro... |
In statistics, the
Breusch–Godfrey test, named after Trevor S. Breusch and Leslie G. Godfrey, [1] [2] is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial dependence that has not been i... |
Update November 16th: Oui, the constants are constant now!
Although it's just a bunch of conventions, I have been sort of excited about the systems of units – and the SI units in particular – for more than three decades.
The most recent blog post, one from July 2017, announced plans to redefine the fundamental SI units... |
Integration Exercises on indefinite and definite integration of basic algebraic and trigonometric functions.
This is level 1 ? Use the ^ key to type in a power or index and use the forward slash / to type a fraction. Press the right arrow key to end the power or fraction. Click the Help tab above for more.
Each of your... |
Defining parameters
Level: \( N \) = \( 6041 = 7 \cdot 863 \) Weight: \( k \) = \( 2 \) Nonzero newspaces: \( 8 \) Sturm bound: \(5958144\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(6041))\).
Total New Old Modular forms 1494708 1431773 62935 Cusp forms 1484365 1423161 ... |
Current browse context:
astro-ph.HE
Change to browse by: Bookmark(what is this?) Astrophysics > High Energy Astrophysical Phenomena Title: Optical, J, and K light curves of XTE J1118+480 = KV UMa: the mass of the black hole and the spectrum of the non-stellar component
(Submitted on 12 Sep 2019)
Abstract: Optical, J, a... |
Standard treatments of the Buckingham Pi Theorem seem to imply that given a dimensionless function $f$ of variables $q_1, q_2, \dots, q_n$ with associated dimension matrix having rank $r$, there exists a function $\phi$ of $\nu = n-r$ variables and pi groups $\pi_1, \pi_2, \dots, \pi_\nu$ such that $$ f(q_1, q_2, \dots... |
The answer is
yes if $R$ is any atomic domain, e.g., $R$ is a Noetherian domain.
Claim 1. Let $R$ be any integral domain. The set $S = S_R$ is saturated in the sense that if $ab \in S_R$ for some $a,b \in R$, then $a \in S_R$.
Proof. Let $x \in R$ and let $I = Ra + Rx$. As $Ib$ is principal, so is $I$.
Lemma. Let $R$ b... |
Define the problem $W$:
Input:A multi-set of numbers $S$, and a number $t$.
Question:What is the smallest subset $s \subseteq S$ so that $\sum_{k \in s} k = t$, if there is one? (If not, return
none.)
I am trying to find some polytime equivalent decision problem $D$ and provide a polytime algorithm for the non-decision... |
I am trying to better understand Mercer's Theorem, by applying it to some specific kernels.
Background
Let $D \subset \mathbb{R}^N$ be a closed, bounded subset. We associate a function $K: D \times D \rightarrow \mathbb{R}$ with an operator (the Hilbert-Schmidt Integral Operator) $T_K: L^2(D) \rightarrow L^2(D)$ being ... |
Learning Outcomes
Compare two fractions Compare two numbers given in different forms
In this section, we will go over techniques to compare two numbers. These numbers could be presented as fractions, decimals or percents and may not be in the same form. For example, when we look at a histogram, we can compute the fract... |
In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an $n$-dimensional vector space over $GF(x)$ (which I was using to determine the number ... |
How would I prove that $\mathbb{N}$ has no limit points? Why does $\mathbb{N}$ have no limit points?
I have tried proving with integers but clearly this is not the same.
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 mi... |
We are still waiting for a good solution for Problem 2014-15.
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For a (simple) graph \(G\), let \(o(G)\) be the number of odd-sized sets of pairwise non-adjacent vertices and let \(e(G)\) be the number of even-sized sets of pairwise non-adjacent vertices. Prove that if we can delete \(k\) vertic... |
Planets orbit around stars, satellites orbit around planets, even stars orbit each other. So the question is: Why don't galaxies orbit each other in general, as it's rarely observed? Is it considered that 'dark energy' is responsible for this phenomenon?
There are plenty of satellite galaxies orbiting larger galaxies. ... |
Counting Plane Graphs with Exponential Speed-Up Abstract
We show that one can count the number of crossing-free geometric graphs on a given planar point set exponentially faster than enumerating them. More precisely, given a set
P of n points in general position in the plane, we can compute pg( P), the number of crossi... |
To begin our study, we will look at subspaces \(U\) of \(V\) that have special properties under an operator \(T\) in \(\mathcal{L}(V,V)\).
Definition \(\PageIndex{1}\): invariant subspace
Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with \(\dim(V)\ge 1\), and let \(T\in \mathcal{L}(V,V)\) be an op... |
For $1 \leq i \leq n$ and $1 \leq j \leq k$:
Let $a_{ij}$ be the $j$th bit of string $a_i$, and let $b_{ij}$ be the $j$th bit of string $b_i$.
Define $x \oplus y$ as $x$ xor $y$, and define $x \parallel y$ as the concatenation of $x$ and $y$.
Then $A = \parallel_{j=1}^{k}\bigoplus_{i=1}^{n} a_{ij}$ and $B = \parallel_{... |
C3.8 Analytic Number Theory - Material for the year 2019-2020
Basic ideas of complex analysis. Elementary number theory. Some familiarity with Fourier series will be helpful but not essential.
16 lectures
Assessment type:
The aim of this course is to study the prime numbers using the famous Riemann $\zeta$-function. In... |
This question is from Lang's Algebra Chapter VI Exercise Q8
Let $f(x)=x^4+ax^2+b$ be an irreducible polynomial over $\mathbb{Q}$, with roots $\pm\alpha$, $\pm\beta$ and splitting field $K$.
I have shown that the Galois Group is either $\mathbb{Z_{4}}$ or $\mathbb{Z_{2}}\times\mathbb{Z_{2}}$ or $D_{8}$
The second part o... |
Suppose $K/\mathbb{Q}\_p$ is a finite extension with residue field $k$. Fix a uniformizer $\pi\in K$ and choose a coherent sequence $(\pi^{1/p^n})$ of $p$-power roots of $\pi$, and let $K_\infty/K$ be the extension of $K$ generated by these roots. The field $K_\infty$ is arithmetically pro-finite and so we can consider... |
This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the number of $p_2$ not exceeding x.
My question is whether I have done anything illegal in adapting this method. The numbers (... |
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed as a quotient of the tensor algebra $T\left(L\right)$ of $L$. Let $U\left(L\right)$ denote the universal enveloping algeb... |
The simplest forcing to add a dominating function is Hechler forcing $\newcommand{\D}{\mathbb{D}}\D$. In set-theoretic circles, conditions in $\D$ are pairs $(s,f)$ where $s$ is a finite sequence of natural numbers and $\newcommand{\N}{\mathbb{N}}f:\N\to\N$, extension is defined by $(s,f) \leq_{\D} (t,g)$ if $t \supset... |
Since the total mass-energy for the neutrino presumably does not change when a neutrino changes lepton flavor, though the mass is different, what compensates for the gain or loss of mass? Does the propagation speed of the neutrino change?
There are a couple of misconceptions here.
The flavor states are not mass states.... |
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Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, ISSN 0370-2693, 06/2015, Volume 746, pp. 178 - 185
A sample of 1.69×10 7 fully reconstru... |
Degree $n$ : $28$ Transitive number $t$ : $34$ Group : $C_{14}\times D_7$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,26,22,18,13,10,6,2,25,21,17,14,9,5)(3,8,12,16,20,23,28,4,7,11,15,19,24,27), (1,11,25,8,22,4,17,27,13,23,9,19,6,16)(2,12,26,7,21,3,18,28,14,24,10,20,5,15) $|\Aut(F/K)|... |
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In statistics and probability theory, the standard deviation (SD) (represented by the Greek letter sigma, σ) measures the amount of variation or dispersion from the average.[1]
A low standard deviation indicates that the data points tend to be very close to the mean (also called expected value); a high standa... |
Here’s a recipe for finding the coordinates of your position after $n$ steps along the spiral.
It’s simpler to number the positions on the spiral starting at $0$: position $0$ is $\langle 0,0\rangle$, the origin, position $1$ is $\langle 1,0\rangle$, position $2$ is $\langle 1,-1\rangle$, and so on. Using $R,D,L$, and ... |
Consider the Sturm-Liouville problem
$$y''(x) + \lambda x^2 y=0, \ y(0)=0,\ y(1)=0$$
The analytical solution is given by
$$\lambda_n=4\alpha_n^2, \ y_n(x)=\sqrt{x}J_\frac14(\alpha_nx^2)$$
where $\alpha_n$is the $n$-th zero of the Bessel function of the first kind of order 1/4. Thus, $\lambda_1\approx30.93$ and $y_1(x) ... |
I am attempting to model the steady state behavior of a cylinder using the finite volume method (FVM) subjected to a variety of boundary conditions in Matlab. First off, I am treating the cylinder as being axisymmetric so I am only determine the temperature profile in the r-Z plane. I set up a 2D grid so that the r coo... |
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∏j=1nj≢0(mod ∑j=1nj) \large \prod_{j=1}^n j \not \equiv 0 \quad \left ( \text{mod} \ \sum_{j=1}^n j \right) j=1∏nj≡0⎝⎛mod j=1∑nj⎠⎞
For how many integers nnn satisfying 1≤n≤1801 \le n \le 1801≤n≤180 is the non-congruence above fulfilled?
You may use this List of ... |
Interesting recursive functions -
et R={i∣∃j:f(j)=i} be the set of distinct values that...can sum1 explain clearly ???wat is d meaning of dis
R={i∣∃j:f(j)=i} be the set of distinct values that f takes
R={i∣∃j:f(j)=i} be the set of distinct values that f takes
Someone please explain why is it written "takes" ? R contain... |
You only need to consider the case $\mathfrak{h}_{s}^\ast(A) \lt \infty$, but you need to be a bit careful in choosing the outer approximations since swapping $\inf$ and $\sup$ certainly isn't allowed without some thinking. If you knew that you can always take the same set $E$ in the $\inf$ (which I will show in \eqref... |
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So guys today INCHO And INAO Are over . Tomorrow is INPHO .
How were your exams? your expectations
Note by Prakhar Bindal 2 years, 8 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to t... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
Let $M=\left (\omega\mathbb{I}-A\right )\left(\omega^{*}\mathbb{I}-A^{\dagger}\right)$ be a Hermitian matrix of size $n\times n$ where $A$ is a real non symmetric matrix and $\omega=a+\mathrm{i}b$. $A^{\dagger}$ represents the conjugate transpose of $A$.
I want to compute $\det[M]^{-\frac{1}{2}}$.
I know that for a rea... |
There are several and almost similar inequalities in MSE that some of them can be proved in long page. some of these questions listed below:
For $abc=1$ prove that $\sum\limits_{cyc}\frac{a}{a^{11}+1}\leq\frac{3}{2}.$ For positive $a$, $b$, $c$ with $abc=1$, show $\sum_{cyc} \left(\frac{a}{a^7+1}\right)^7\leq \sum_{cyc... |
Let \(G\) be a group. A topology on \(G\) is said to be a group topology if the map \(\mu: G \times G \to G\) defined by \(\mu(g, h) = g^{-1}h\) is continuous with respect to this topology where \(G \times G\) is equipped with the product topology. A group equipped with a group topology is called a topological group. W... |
Using some suggestions from the other commenters:
The alternating group, $A_4$, has the set $H=\{I,(12)(34),(13)(24),(14)(23)\}\cong V_4$ as a subgroup. If $f\in S_4\supseteq A_4$ is a permutation, then $f^{-1}[(12)(34)]f$ has the effect of swapping $f(1)$ with $f(2)$ and $f(3)$ with $f(4)$. One of these is $1$, and de... |
I'm trying to answer this question and you are supposed to use the multiplication rule to solve it:
A deck of 52 playing cards is randomly divided into four piles of 13 cards each. Compute the probability that each pile has exactly 1 ace.
I started off by defining following 4 events: $A_{1}, A_{2}, A_{3}$ and $A_{4}$ w... |
I would give an economical argument:
For every field $F$ of cardinality $q=p^d$ we have $x^q -x=0$ for all $x \in F$.(this is easy)
For every $P \in \mathbb{F}_p[X]$ irreducible of degree $d$ we have $P \mid X^{q}-X$
Indeed, work in the field $F = \mathbb{F}_p[X]/P(X)$. The polynomial $P(X)$ has a root in $F$ which is ... |
There is problem with ppx values they are ambiguous. It may be w/w, w/v, v/v, n/n.
Salt water has density significantly different to $\pu{1 g/ml}$, so $\pu{1 ppt(parts per thausand) }$ may mean $\pu{1000 mg/L}$ or $\pu{1000 mg/kg}$, with the recalculation factor of the solution density.
The former ($\pu{ppt w/v as 1000... |
As far as I know, Integer Linear Programming(ILP) problem is NP-complete. According to the following paper, Binary Linear Programming problem(BLP) can be solved in Polynomial time. http://dx.doi.org/10.4236/ajor.2016.61001
I'm not familiar with the convex Quadratic Problem mentioned in the paper. However, I know that I... |
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