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Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins i... |
C1.2 Godel's Incompleteness Theorem - Material for the year 2019-2020
This course presupposes knowledge of first-order predicate logic up to and including soundness and completeness theorems for a formal system of first-order predicate logic (B1.1 Logic).
16 lectures.
Assessment type:
The starting point is Gödel's math... |
Suppose we have the
inhomogeneous advection equation$$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (with boundary conditions not yet specified).
Assuming that we had no $v$, i.e. the homogeneous part of the equation, the ... |
I have an algorithm that produces a set of real outputs given real inputs. For practical purposes, let's say I have two inputs and one output, and the algorithm can be represented by the function $\phi: \Re² \rightarrow \Re$.
I need to calculate $\frac{\partial^2 \phi(u_1,u_2)}{\partial u_1\partial u_2} \Bigr|_{\bar{u_... |
This is an MCMC algorithm for uniform sampling over singular $n$ by $n$ Bernoulli matrices.
Let $H$ (for "hypercube") be the set of all 0/1 vectors of length $n$.
One step of the MCMC algorithm is as follows:
Generate an $(n-1)$ by $n$ matrix $A$, filled with 0/1 iid Bernoullisamples. This will be the first $n-1$ rows ... |
The concept of a "proliferating random walk" on a lattice is that at any time $t \in \Bbb N \cup 0$, there is some set consisting of at least one particle, each of which is on its own lattice point. When taking a time step, each particle also randomly decides whether to split into two particles, each of which moves to ... |
I am stuck with a basic understanding of the generalized (and even the ordinary version of) Gauss-Bonnet theorem. For a compact 2-dimensional Riemannian manifold $M$ with boundary $\partial M$, let $K$ be the Gaussian curvature of $M$ and $k_g$, the geodesic curvature of $\partial M$. Then
$$\int_M K\;dA+\int_{\partial... |
If tidal power plants are slowing down Earth's rotation then is it theoretically possible to build a power plant that would drain energy from Earth's angular momentum (thus slowing down it's rotation)?
What would such machine look like?
Physics Stack Exchange is a question and answer site for active researchers, academ... |
Schedule of the Workshop "Picard-Fuchs Equations and Hypergeometric Motives" Monday, March 26
10:15 - 10:50 Registration & Welcome coffee 10:50 - 11:00 Opening remarks 11:00 - 12:00 Frits Beukers: Some supercongruences of arbitrary length 12:00 - 13:45 Lunch break 13:45 - 14:45 Alexander Varchenko: Solutions of KZ diff... |
Let $(X,d)$ be a metric space, show that if $A \subset X$ is connected, $B \subset X$ with $A \subset B \subset cl(A)$ then B is connected.
My approach:
Let's assume that B is not connnected, then by definition there exist two open, disjunct subsets $U, V \subset X$ such that $ B \subset U \cup V, B \cap V \neq \emptys... |
The least common multiple of two integers, $a$ and $b$, can be calculated by the following procedure:
Decompose $a$ and $b$ into prime factors;
For every prime factor that is in $a$ but is not in $b$, put it in the $lcm$ factorization with the exponent it has in $a$;
For every prime factor that is in $b$ but is not in ... |
Use small o notation with a special care on which variable goes to infinity. What you said is basically the following.
Claim(?): Let $f(n,x)$ be a function in $n$ and $x$. Suppose $\lim \frac{f(n,x)}{x/n} = 0$. Then we have $\lim \frac{\sum_{n\leq x} f(n,x)}{\sum_{n \leq x} x/n} = 0$.
Is this claim true? It depends on ... |
I already asked a question regarding how to solve a nonlinear pde in mathematica which was answered nicely.
Actually this was a 1-dimensional form of a general 2D problem that I was trying to solve with matlab, but first I wanted to have a clue about the solution using mathematica. Now I think I have to restrict myself... |
Let \(I, J\) be connected open intervals such that \(I \cap J\) is a nonempty proper sub-interval of both \(I\) and\(J\). For instance, \(I = (0, 2)\) and \(J = (1, 3)\) form an example.
Let \(f\) (\(g\), resp.) be an orientation-preserving homeomorphism of the real line \(\mathbb{R}\) such that the set of points of \(... |
Using the reflection formula $$\Gamma(\theta)\Gamma(1-\theta)=\frac{\pi}{\sin \theta\pi}\quad,0<\theta<1$$ your population density is simply
$$f_{\theta}(x)=\frac{e^{-\theta x}x^{-\theta}\theta^{1-\theta}}{\Gamma(1-\theta)}\mathbf 1_{x>0}\quad,0<\theta<1$$
(This 'simplification' is obviously not needed for the given pr... |
Online auctions in which items are sold in an online fashion with little knowledge about future bids are common in the internet environment. We study here a problem in which an auctioneer would like to sell a single item, say a car. A bidder may make a bid for the item at any time but expects an immediate irrevocable d... |
I have the following question when I was going through the proof of the following theorem.
Theorem. For XOR function $f \circ XOR$, $rank(M_{f \circ XOR}) = ||\hat f ||_0$ where $M_{f \circ XOR}$ is a matrix such that $M_{f \circ XOR}(x,y) = f(x + y)$.
The proof of this theorem essentially shows that fourier coefficien... |
I saw this question in another post and I proved it differently than the others who answered. I was wondering if my proof works.
Problem
Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Suppose that $f$ is differentiable, that $f(0)=1$, and that $|f'(x)| \leq 1$ for all $x \in \mathbb{R}$. Prove that $|f(x)| \leq |x|+... |
Strange baryon and in particular multi-strange baryon production is suggested to be a useful probe in the search for quark gluon plasma formation in heavy ion collisions. We have measured the (Ω − + Ω + ) (Ξ − + Ξ + ) production ratio to be 0.8±0.4 at central rapidity and ϱ T > 1.6 GeV/c.
We report on measurements of t... |
2018-08-25 06:58
Recent developments of the CERN RD50 collaboration / Menichelli, David (U. Florence (main) ; INFN, Florence)/CERN RD50 The objective of the RD50 collaboration is to develop radiation hard semiconductor detectors for very high luminosity colliders, particularly to face the requirements of the possible u... |
why is there 0.7V instead of 1.2V on the common emitter?
if you go through Q1 mesh : Ve = 0.5+Vbe = 1.2 instead of Ve = 0+Vbe = 0.7 from Q2.
Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. It only takes a minute to ... |
As we know from complex analysis, Cauchy's integral formula states:
$f(z_o)=\frac{1}{2\pi i}\int_\gamma{\frac{f(z)}{z-z_o}dz}$ for a closed contour $\gamma$.
However there is also the result from other areas of maths that states:
$f(x_o)=\int_R{f(x)\delta(x-x_o)dx}$ for some region $R$
Given the similarity in the resul... |
This is inspired by an old Putnam problem from 2005, and a solution given by Professor Greg Martin (a Professor of Mathematics at the University of British Columbia, also a user on MO). The question is
Question (Putnam 2005): For non-negative integers $m,n$, let $f(m,n)$ denote the number of $n$-tuples $(x_1, \cdots, x... |
I) There are already several good answers. OP is asking about the momentum of the non-relativistic string with only transverse displacements, whose Lagrangian density usually is given as
$$ {\cal L}_T ~:=~\frac{\rho}{2} \dot{\eta}^2 - \frac{\tau}{2} \eta^{\prime 2} \tag{1}$$
in textbooks.
II) Let us fix notation: $\rho... |
Difference between revisions of "Stokes' Theorem"
m (Curl, not cross product. I'll be back after I study this some more. Needs work.)
(→References: Category)
(11 intermediate revisions by 2 users not shown) Line 1: Line 1: −
'''Stokes' Theorem'''
+
'''Stokes' Theorem''' that the
+ + + + + +
integral of the [[curl]] of ... |
I'm reading the notes here and have a doubt on page 2 ("Least squares objective" section). The probability of a word $j$ occurring in the context of word $i$ is $$Q_{ij}=\frac{\exp(u_j^Tv_i)}{\sum_{w=1}^W\exp(u_w^Tv_i)}$$
The notes read:
Training proceeds in an on-line, stochastic fashion, but the implied global cross-... |
The following apparently elementary question came out of a somewhat naive attempt to prove that every distribution $u\in \mathscr D'(\mathbb R^2)$ with $\partial_1 u=\partial_2 u =0$ is a constant function (this can be reduced to $\mathscr C^1$-functions by convolution with an approximate identity and for $\mathscr C^1... |
Brauchart, Johann S and Hesse, Kerstin (2007)
Numerical integration over spheres of arbitrary dimension. Constructive Approximation, 25 (1). pp. 41-71. ISSN 0176-4276 Abstract
In this paper, we study the worst-case error (of numerical integration) on the unit sphere $\\mathbb{S}^{d}$, $d\\geq 2$, for all functions in t... |
Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces
Independent scholar
We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $ L^r_v L^\infty_x $ spaces where $ r \in (1,\infty] $ by usin... |
$\mathbf{Geometrical \ approach}:$ Each point on the surface of a sphere is at an equal distance (equal to radius) from the center this results in the minimum surface area for a given volume. This can be proved analytically by comparing the surface area of a sphere with that of any other geometrical shape for a given v... |
Prove the identity:
$$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$
I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$ but got stuck along the way.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. I... |
I have a set of data $y_i (z_i)$ with errors $\Delta y_i$
data = {{0.015, 34.1114},{0.0277, 35.705},{0.048948, 36.7316},{0.0651, 37.3067},{0.100915, 38.4567},{0.159, 39.4164},{0.248508, 40.2722},{0.455, 42.3239},{0.655, 42.3151},{0.75, 43.243},{0.84, 43.5143},{0.961, 44.2642},{1.188, 44.6076},{1.34, 45.0675},{1.414, 44... |
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called
separable if it can be decomposed in the following way:
$$ M=\sum_{i=1}^{k} \:\: \lambda_i E_i\otimes F_i $$
where $k\le n^2$, $\lambda_i>... |
The canonical counterexample is when you throw a football, but you don't get a good spiral. The football is freely flying through the air so there is no torque, but the $\omega$ and $L$ are not colinear.
To explain this example and the general case I will go into the math now. The moment of inertia tensor can be writte... |
In geometry, the notion of a
connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most e... |
I'm trying to solve the following congruence:
$71x-1 \equiv 0 \pmod{59367} $
Given that $59367=771 \times 77$, I have previously solved that:
$71x \equiv 1 \pmod{771}$ such that $x=-76$
$71x \equiv 1 \pmod{77}$ such that $x=-13$
I'm trying to use the Chinese Remainder Theorem, but seem to be getting the wrong answer, i... |
Because you are looking only at the so-called
global part, i.e. the part of the gauge transformation which resembles a group action.
Recall that the vector bosons transform as$$A_\mu \to g A_\mu g^{-1} - (\partial_\mu g) g^{-1}$$where the first part is the
global part of the gauge transformation, which tells you that $... |
In The Feynman Lectures on Physics, Volume I 39-2 The pressure of a gas, the following is presented:
If $v$ is the velocity of an atom, and $v_{x}$ is the $x$-component of $v$, then $mv_{x}$ is the $x$-component of momentum
in; but we also have an equal component of momentum 0utand so the total momentum delivered to th... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
Learning Outcomes
Use Order of Operations in Statistics Formulas.
We have already encountered the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In this sections we will give some additional examples where order of operations must be used properly to evaluate statist... |
This question is concerned with the long-standing problem confusing so many people which is: how is it that we can view $z$ and $\bar{z}$ as independent variables. To be more precise, I understand all the formal manipulations using the chain rule and the trick with $x = \frac{z+\bar{z}}{2}$ and $y=\frac{z-\bar{z}}{2\ma... |
For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is $$\lim\limits_{n\to\infty}Y_{n,p}=\mathbb{E}[|B_H(1)|^p] \ a.s.\text{ and in } L^1.$$ The Ergodic Theorem of Birkhoff says:
Let $(\Omega,\mathcal{... |
Let $$\phi_k(x)=\sum_{1\le n \le x \\(n,x)=1} n^k$$ What's the asymptotic behavior of $$\sum_{n=1}^x\phi_k(n)?$$
The possible routes Route 1 (For someone who wants some practice with Abel Summations): There should be an approach which is an analog to the techniques shown here: sum of the divisor functions and I think t... |
So I got a gift for a friend based on
Mathematica code.
Rose[x_, theta_] := Module[ {phi = (Pi/2)Exp[-theta/(8 Pi)], X = 1 - (1/2)((5/4)(1 - Mod[3.6 theta, 2 Pi]/Pi)^2 - 1/4)^2}, y = 1.95653 x^2(1.27689 x - 1)^2 Sin[phi]; r = X(x Sin[phi] + y Cos[phi]); {r Sin[theta], r Cos[theta], X (x Cos[phi] - y Sin[phi]), EdgeForm... |
MY ATTEMPT AT PROVING THE LIMIT
By the definition of a limit, we know that:
$\displaystyle \lim_{x \to a} g(x) = 0$
Means that for every:
$\displaystyle \epsilon_2 > 0$,theres is a $\displaystyle \delta_1 > 0$ such that:
$\displaystyle 0 < |x - a| < \delta_1$
implies:
$\displaystyle |g(x) - 0| = |g(x)| < \epsilon_2$
Th... |
"""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... |
I cannot plot this function as I am getting errors. I am not sure what I am doing wrong.
Seems to work when $$n \rightarrow 1,$$ but when $$n \rightarrow 2,$$ I get a whole bunch of errors and nothing is plotted.
Here is the code:
$$ \text{Plot}\left[\frac{n e^{-t (\lambda +\mu )} I_1\left(2 t \sqrt{\lambda \mu }\right... |
The path-groupoid $\mathcal{P}_1(X)$ of a (smooth) topological space $X$ is a refinement of the fundamental groupoid $\Pi_1(X)$ whose morphisms are given by (piecewise smooth) paths in $X$ up to thin-homotopy (rather than full homotopy). Thin-homotopies are basically homotopies sweeping zero area. (Note that there's a ... |
Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is an infinite-dimensional manifold; moreover, it is the total space of a smooth principal fiber bundle with structure group... |
Discussion area to prepare for the Final Exam On problem 2. I am breaking the curve $ \gamma $ up into two piece wise curves $ \gamma_1 $ and $ \gamma_2 $ that meet when the curve $ \gamma $ crosses the negative real axis at the point $ z_0 $. I then am taking the principle branch of log as an analytic function to eval... |
“Unperformed measurements have no results.” —Asher Peres
With two looming paper deadlines, two rambunctious kids, an undergrad class, program committee work, faculty recruiting, and an imminent trip to Capitol Hill to answer congressional staffers’ questions about quantum computing (and for good measure, to give talks ... |
I would like to know whether I have correctly proved the following statement and have correctly extrapolated out a general situation.
We're asked two things:
a) Prove there is no rational number solution to $x^2-3x+1=0$
b) The problem (a) suggests a more general problem. State and outline a proof of this.
a) Proof: We ... |
I would like to know if the Hirzebruch-Riemann-Roch theorem exists for bundles over Riemann surfaces with a boundary. I am asking this because the Hirzebruch-Riemann-Roch theorem is used in the following paper (https://arxiv.org/pdf/0707.2786v2.pdf) on page 10 to compute the index of the following differential operator... |
Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The cofactor matrix of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$.
Thus, it is immediate that $A \in \operatorname{SO}_n$ if and only if $$ (**) \operatorname{cof} A =A,\det A =1$$
However, if $n \neq 2$ the condition on the de... |
Central limit theorems are a set of weak-convergence results in probability theory. Intuitively, they all express the fact that any sum of many independent identically distributed random variables is approximately normally distributed. These results explain the ubiquity of the normal distribution.
The most important an... |
This question already has an answer here:
Given $X_1,\ldots,X_n$, where $X_i\sim U(-\theta,\theta)$, what the MLE for $\theta$? Apparently the answer is $\max\{|X_1|,\dots,|X_n|\}$ but I can't figure out why.
The density function is $$f(x,\theta) = \begin{cases} \frac{1}{2\theta}, & x\in[-\theta,\theta] \\ 0, & \text{e... |
Caesium has a larger size, and the effective nuclear charge that the valence electron experiences will be far less compared to that of lithium's, right? But lithium is still considered the strongest reducing agent among all the alkali metals, and this is evidenced by its large and negative reduction potential. Why is t... |
M4: Geometry - Material for the year 2019-2020
15 lectures
The course is an introduction to some elementary ideas in the geometry of euclidean space through vectors. One focus of the course is the use of co-ordinates and an appreciation of the invariance of geometry under an orthogonal change of variable. This leads in... |
Prove the following.
Let $\{A_n \}_{n \in \mathbb{N}}$ and $\{ B_n\}_{n \in \mathbb{N}}$ be sequences of sets with $$ A_1 \subset A_2 \subset A_3 \dots \subset A_n \dots $$ $$ B_1 \subset B_2 \subset B_3 \dots \subset B_n \dots $$ then $\left( \bigcup_{n=1}^{\infty} A_n\right ) \cap \left ( \bigcup_{n=1}^{\infty} B_n\r... |
I've come across this problem and managed to get the right answer, but there remains a mystery that I wasn't quite able to solve: the minus sign (or a lack thereof)! Here's the problem and my solution:
A man walks across a bridge and when he's $40 \%$ of the way through, he spots a train incoming towards him at $40$ mp... |
The answers currently posted are ignoring a few important details so I'm going to give my own.I may rehash some things already said.To make everything absolutely clear I write here a complete derivation of the forced damped oscillator with emphasis on the role of the $Q$ factor.
Basic equations
Consider the equation of... |
An investigator wishes to produce a combined analysis of several datasets. In some datasets there are paired observations for treatment A and B. In others there are unpaired A and/or B data. I am looking for a reference for an adaptation of the t-test, or for a likelihood ratio test, for such partially paired data. I a... |
Difference between revisions of "Image Dimensions"
(Added some speculative content for the next section)
m (Text replace - "{{Category|NewTerminology}}" to "{{NewTerminology}}")
(7 intermediate revisions by 4 users not shown) Line 1: Line 1: +
<div style="background-color:#DDFFDD; border:thin solid green; padding:1em">... |
Congratulations on deriving the exponential law for yourself, one learns a great deal about science working like this. Now to your last question:
If I had a group of atoms that have an 'average lifetime' of say 5 seconds, after 5 seconds has elapsed, what is the 'average lifetime' of the remaining atoms? I don't think ... |
Definition:Antireflexive Relation Contents Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
$\mathcal R$ is
antireflexive if and only if: $\forall x \in S: \tuple {x, x} \notin \mathcal R$ Also known as
Some sources use the term
irreflexive.
However, as
irreflexive is also found in other sources t... |
The Annals of Applied Probability Ann. Appl. Probab. Volume 23, Number 5 (2013), 1879-1912. On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime Abstract
We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the $M/M/n$ queue in the Ha... |
These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
Exercise 5.1.4: Find eigenvalues and eigenf... |
In Chapter 7.5 in Peskin and Schroeder, the authors define the physical charge in eq. (7.76), $$\text{(physical charge)}=\sqrt{Z_3}\cdot\text{(bare charge)}$$ where $\dfrac{1}{1-\Pi(0)}\equiv Z_3$. Here, $$\Pi^{\mu\nu}(q)=(q^2g^{\mu\nu}-q^\mu q^\nu)\Pi(q^2)$$ $i\Pi^{\mu\nu}(q)$ being the sum of all 1PI insertions into ... |
Notice:
If you happen to see a question you know the answer to, please do chime in and help your fellow community members. We encourage our fourm members to be more involved, jump in and help out your fellow researchers with their questions. GATK forum is a community forum and helping each other with using GATK tools a... |
Definition:Multiplication/Modulo Multiplication/Definition 2 Definition
Let $m \in \Z$ be an integer.
Let $\Z_m$ be the set of integers modulo $m$:
$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$ The operation of multiplication modulo $m$ is defined on $\Z_m$ as: Also denoted as
Although the operation of multiplication mod... |
Shift the triangle to the origin by A -> A - A = 0; B -> B - A; C -> C - A.
Points in the plane of the shifted triangle can be expressed with {B - A, C - A} as a basis, in other words you have a linear expression for the translated point P in the form $\alpha$ (B - A) + $\beta$ (C - A).
For the given $x_4$ and $y_4$, t... |
“Unperformed measurements have no results.” —Asher Peres
With two looming paper deadlines, two rambunctious kids, an undergrad class, program committee work, faculty recruiting, and an imminent trip to Capitol Hill to answer congressional staffers’ questions about quantum computing (and for good measure, to give talks ... |
A friend of mine posed this brain teaser to me recently:
What's the length of shortest bit sequence that's never been sent over the Internet?
We can never know for sure because we don't have a comprehensive list of all the data. But what can we say probabilistically? Restating it like so:
At what value for X is there a... |
Let $Z_i \sim \mathcal{N}(0,1)$ be independent normal distributions. Consider the following correlated variables, defined by $$ X_1 = \frac{Z_1 + Z_2}{\sqrt{2}},\;\;\;X_2= \frac{Z_2 + Z_3}{\sqrt{2}},\;\;\;X_3= \frac{Z_3 + Z_4}{\sqrt{2}},\ldots$$
Thus each $X_i$ by itself is also a standard normal distribution but is co... |
We can find the arc length of a curve by cutting it up into tiny pieces and adding up the length of each of the pieces. If the pieces are small and the curve is differentiable then each piece will be approximately linear.
We can use the distance formula to find the length of each piece:
\[ L = \sqrt{ \left(\Delta{x}\ri... |
$\newcommand{\vec}[1]{\mathbf{#1}} \newcommand{\dd}{\mathrm{d}}$I'm reading Landau and Lifshitz' book on non-relativistic quantum mechanics and I have some doubts about a passage in the chapter about elastic scattering. I have the French edition of 1966 so I cannot quote precisely, but it should be in §125, from around... |
Kyle Kanos's answer looks to be very full, but I thought I'd add my own experience. The split-step Fourier method (SSFM) is extremely easy to get running and fiddle with; you can prototype it in a few lines of Mathematica and it is, extremely stable numerically. It involves imparting only unitary operators on your data... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
It has "been known" since 1908 that for any such function, this holds for all but countably many real numbers $c$, even when we additionally require all the sequences to approach $c$ from the same side.
In May 1908 William Henry Young presented several results for general functions from $\mathbb R$ to ${\mathbb R},$ in... |
LaTeX:Symbols
LaTeX About - Getting Started - Diagrams - Symbols - Downloads - Basics - Math - Examples - Pictures - Layout - Commands - Packages - Help
This article will provide a short list of commonly used LaTeX symbols.
Contents Common Symbols Operators Relations Finding Other Symbols
Here are some external resourc... |
I am looking for a proof, a hint or an idea to the following problem:
Is the unique solution $x\in (0,2\pi)$ of
$$ x\sin(x) + \cos(x) = 1 $$
which is equivalent to
$$ 2\arctan(x) = x$$
a rational multiple of $\pi$. I.e. is $\frac{x}{\pi} \in \mathbb{Q}$?
I believe that this is not true. This idea is based on the numeri... |
Let $X$ be a compact metric Borel space. Suppose $\mu_{n}(A)\rightarrow\mu(A)$ for all $\mu-$continuity sets $A$ (sets with zero boundary measure), where $\mu_{n}$ is a sequence of probability measures. (some people call it weak other weak* convergence)
If $E$ is a measurable set such that $\mu(E)>0$ and the Cesaro ave... |
This is essentially an addition to the list of @4tnemele
I'd like to add some earlier work to this list, namely Discrete Gauge Theory.
Discrete gauge theory in 2+1 dimensions arises by breaking a gauge symmetry with gauge group $G$ to some lower
discrete subgroup $H$, via a Higgs mechanism. The force carriers ('photons... |
Exam-Style Question on Logarithms A mathematics exam-style question with a worked solution that can be revealed gradually
Question id: 410. This question is similar to one that appeared on an IB AA Standard paper (specimen) for 2021. The use of a calculator is allowed.
(a) Show that \( \log_4 (\sin 2x +2) = \log_2 \sqr... |
Difference between revisions of "LaTeX:Symbols"
m (→Dots)
(→Operators)
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=== Operators ===
=== Operators ===
+ + + + +
=== Relations ===
=== Relations ===
Revision as of 21:23, 4 December 2017
LaTeX About - Getting Started - Diagrams - Symbols - Downloads - Basics - Math - Examples - Pictures - Layout - C... |
For the proof of Lemma 1 we need some auxiliary results on the eigenvalues of \({\mathbf {M}}_{22}-m_2^2{\mathbf {J}}_{C(K,2)}\) for a symmetric invariant design when \(K\ge 4\).
Lemma 3
Let
$$\begin{aligned} \lambda _{\varvec{1}}= 1+2(K-2)m_{2} +\textstyle {\frac{1}{2}}(K-2)(K-3)m_{4} - \textstyle {\frac{1}{2}}K(K-1)m... |
If the solution of $Ax=b$ is unstable, the matrix is very ill-conditioned (i.e., has a very large condition number), and (paraphrasing Lanczos) no amount of mathematical trickery can make it stable. The best you can hope for is to solve a
different problem that is a) stable and b) gives you a solution that is sufficien... |
Smoothing effects for some derivative nonlinear Schrödinger equations
1.
Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162
2.
Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacana
3.
Department of Applied Mathematics, Scie... |
There is a "standard" way to consider normed spaces over arbitrary fields but these are not well-behaved in the case of scalars in finite fields. If you want to work with norms on vector spaces over fields in general, then you have to use the concept of valuation.
Valued field:Let $K$ be a field with valuation $|\cdot|... |
As with the sine, we do not know anything about derivatives that allows us to compute the derivatives of the exponential and logarithmic functions without going back to basics. Let's do a little work with the definition again:
\[\eqalign{ {d\over dx}a^x&=\lim_{\Delta x\to 0} {a^{x+\Delta x}-a^x\over \Delta x}\cr& =\lim... |
Let $\Gamma = \langle S \mid R \rangle$ be a finitely generated group, with the neutral element $e \not \in S= S^{-1}$.
Let $\ell : \Gamma \to \mathbb{N}$ be the world length related to $S$.
For any $g \in \Gamma$ and for any $s \in S$, let $p_s(g)$ be the number of geodesic paths from $g$ to $e$ beginning by the edge ... |
How do I write by proof, the ground state of the toric code (by Kitaev) Hamiltonian $ H=-\sum_{v}A(v)-\sum_{p}B(p) $ where $A(v)=\sigma_{v,1}^{x}\sigma_{v,2}^{x}\sigma_{v,3}^{x}\sigma_{v,4}^{x}$ and plaquette term $B(p)=\sigma_{p,1}^{z}\sigma_{p,2}^{z}\sigma_{p,3}^{z}\sigma_{p,4}^{z} $ ? Here $v$ are indices of vertice... |
When evaluating the integral below in python using scipy.quad I get the following warning:
UserWarning: The maximum number of subdivisions (50) has been achieved. If increasing the limit yields no improvement it is advised to analyze the integrand in order to determine the difficulties. If the position of a local diffi... |
Interested in the following function:$$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$where $\pi(n)$ is the prime counting function.When $s=2$ the sum becomes the following:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
also, if you are in the US, the next time anything important publishing-related comes up, you can let your representatives know that you care about this and that you think the existing situation is appalling
@heather well, there's a spectrum
so, there's things like New Journal of Physics and Physical Review X
which are... |
Mertens' third theorem is just the exponentiated version of the second theorem (without the bounds that Mertens proved for his second theorem):
\begin{align}-\ln\Biggl(\ln n\prod_{p\leqslant n}\biggl(1 - \frac{1}{p}\biggr)\Biggr)&= -\ln \ln n - \sum_{p\leqslant n} \ln \biggl(1 - \frac{1}{p}\biggr)\\&= \Biggl(\sum_{p\le... |
There are three types of neutrinos known today. When detecting them, how can we tell which type we are detecting?
Neutrino flavor is defined as agreeing with the flavor of the charged lepton participating in the interaction, so that the neutrino in the reaction $$ \nu + A \to \mu + X \,, $$ is defined to be a muon neut... |
1. The problem statement, all variables and given/known data
Electron of is in a 1-D potential well of depth $20eV$ width $d=0.2 nm$ in his ground state $N=1$. What is the energy of the ground state? Write the normalized wavefunction of the ground state. What is the probability, to find the particle outside the well?
2... |
TL;DR Initially published crystal structure of $\ce{[NEt4]2[InCl5]}$ [1] according to the further investigations [3], is not valid. The $\ce{InCl5^2-}$ ion does not have $C_\mathrm{4v}$ symmetry, and VSEPR theory pretty much explains formation of numerous slightly distorted trigonal bipyramidal $\ce{InCl5^2-}$-containi... |
In single-variable calculus, the functions that one encounters are functions of a variable (usually \(x\) or \(t\)) that varies over some subset of the real number line (which we denote by \(\mathbb{R}\)). For such a function, say, \(y = f(x)\), the \(\textbf{graph}\) of the function \(f\) consists of the points \((x, ... |
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