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If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold itself from relatively basic information about the gradient flow diffeomorphisms of $f$. To sketch briefly: for each pair of... |
Redirected from Twin Prime Conjecture
There are an infinite number of primes psuch that p+ 2 is also prime.
Such a pair of prime numbers is called a twin prime. The conjecture has been researched by many number theorists. The majority of mathematicians believe that the conjecture is true, based on numerical evidence an... |
I'm trying to evaluate these integrals using Convergence Theorems, but I'm not really sure how to go about it. Here are the integrals:
For $\phi$ bounded and continuous, $\psi\in L^1(m)$, $\lim_{n\to\infty}\int_{\mathbb R}\phi(x/n)\psi(x)dm(x)$ For $\phi$ continuous and compactly supported, $\psi\in L^1(m)$, $\lim_{n\t... |
TODO¶
Listed are plans/directions the project is going to do in the next stage.
Demanding¶ lower search granularity to sector tree. merge CONST and VAR tokens. different symbol weight: Math token > math variable > sub/sup-script.
\( \begin{equation} \left\{ \label{eq154} \begin{array}{ll} \text{Score} &= \sum_t \text{s... |
Given that $(ab)^2=(bc)^4=(ca)^x=abc$ Then what is the value of $x$?
$2(\log a+\log b)=4(\log b+\log c)=x(\log c+\log a)=\log a+\log b+\log c$
Then I am lost, any other easier way to solve?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields... |
I know this may seem like a homework problem at first, but please bear with me...
In this problem, we have two masses sliding without friction in a horizontal and vertical track, connected by a rigid massless link. Choosing $\theta$ as the single generalized coordinate, we can derive a single equation of motion, e.g. v... |
If an electron has spin and volume than the point on the surface is rotating at constant speed according to Planck constant defined angular momentum.If this electron accelerates then this point on the surface has to add its rotating velocity to the velocity of translation of the electron but this sum must not reach the... |
Defining parameters
Level: \( N \) = \( 15 = 3 \cdot 5 \) Weight: \( k \) = \( 3 \) Nonzero newspaces: \( 3 \) Newforms: \( 4 \) Sturm bound: \(48\) Trace bound: \(3\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(15))\).
Total New Old Modular forms 24 12 12 Cusp forms 8 8... |
I know a very similar question has been asked on this site already: Reciprocal lattices. The top answer states:
...The reciprocal lattice is simply the dual of the original lattice. And the dual lattice has a simple visual algorithm.
Given a lattice $L$, for each unit cell of $L$ find the point corresponding to that ce... |
Search
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 ... |
Help:Color Color
{\color{Blue}{x^2}}+{\color{Orange}{2x}}-{\color{LimeGreen}{1}}
x_{1,2}=\frac{{\color{Blue}{-b}}\pm\sqrt{\color{Red}{b^2-4ac}}}{\color{Green}{2a}}
There are several alternate notations styles
{\color{Blue}x^2}+{\color{Orange}2x}-{\color{LimeGreen}1}works with both texvc and MathJax
\color{Blue}x^2\colo... |
Im trying to maximize the probability of a particular outcome occurring subject to a constraint. In particular
$$\max \prod_{i \leq n} 1 - (1 - x_i)^{y_i} \;\;\; \text{ s.t. } \;\;\; i \in \mathbb{N}^+,\; 0 \leq x_i \leq 1,\; x_1\cdots x_n = z, \forall n \in \mathbb{N}^+$$
where $y_i \in \mathbb{N}^+$ and $0 \leq z \le... |
Consider the following question in classical mechanics
Are Newton's Second Law, Hamilton's Principle and Lagrange Equations equivalent for particles and system of particles?
If
Yes, where can I find a complete proof? Are there certain conditions for this equivalence?
If
No, which one is the most general one?
I couldn't... |
Since sufficiency of fixed dimension only occurs in exponential families (Darmois-Pitman-Koopman lemma), apart from distributions with varying support like the Uniform, let us consider an exponential family with parameter $\theta$ and density [against a fixed dominating measure]$$f_\theta(x)=\exp\left\{\sum_{i=1}^k a_i... |
Complex Numbers
Problem 1-1
Let z = 2 - 3 i where i is the imaginary unit. Evaluate z z* , where z* is the conjugate of z , and write the answer in standard form.
Detailed Solution.
Problem 1-2
Evaluate and write in standard form \( \dfrac{1-i}{2-i} \) , where i is the imaginary unit.
Detailed Solution.
Quadratic Equat... |
Tokyo Journal of Mathematics Tokyo J. Math. Volume 34, Number 2 (2011), 383-406. Nested Subclasses of the Class of $\alpha$-selfdecomposable Distributions Abstract
A probability distribution $\mu$ on $\mathbf{R}^d$ is selfdecomposable if its characteristic function $\widehat\mu(z), z\in\mathbf{R}^d$, satisfies that for... |
So you have found out $X \in \mathbf{NP}$. Unfortunately, as Juho hints at in their answer, there is no "list" you can systematically go over to try and investigate $X$ further. This is mainly because separation results in complexity theory are (yet) few and far between, so most of the things you may try will not expli... |
I am currently reading through my electrodynamics lecture notes and I can't understand a calculation.
$$ E = \frac{1}{2} \int \mathrm{d}^3x \, \phi(\mathbf{x}) \rho(\mathbf{x}) = - \frac{\epsilon_0}{8\pi} \int \mathrm{d}^3x \, \phi(\mathbf{x}) \Delta \phi(\mathbf{x}) \\ = \frac{\epsilon_0}{8\pi} \int \mathrm{d}^3x \, \... |
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... |
Regression (OLS) - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)
$z$ test for the diff... |
Difference between revisions of "Group cohomology of elementary abelian group of prime-square order"
(→Over an abelian group)
(→Over an abelian group)
Line 35: Line 35:
The homology groups with coefficients in an abelian group <math>M</math> are given as follows:
The homology groups with coefficients in an abelian grou... |
How does a gauge covariant derivative in a non-abelian field theory act on various quantities which are not valued in the algebra, and why? In particular, how does it act on a scalar valued function $f(x)$ and a matrix-valued function $\chi(x)$ which is not valued in the [representation of the] algebra of the gauge sym... |
Your proposal is a type of perpetual motion device which is popularly referred to as a "magnetic motor" (NOT an "electric motor", which is a real thing). Every one of these magnetic motors is impossible because they all presuppose that it is in some way possible to have one permanent magnet "sneak up" on another and th... |
Regression (OLS) - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)
$z$ test for the diff... |
Frequency Domain View of Upsampling
Why Interpolator needs a LPF after Upsampling
Outline Background Introduction Derivation Example Conclusion Background
$ {f}_{s} $ = sampling frequency (number of samples/second) Hz
$ {T}_{s} $ = sampling period (number of seconds/sample) seconds $ {f}_{s} = {\frac{1}{{T}_{s}}} $ Sam... |
Say we have used the TFIDF transform to encode documents into continuous-valued features.
How would we now use this as input to a Naive Bayes classifier?
Bernoulli naive-bayes is out, because our features aren't binary anymore.
Seems like we can't use Multinomial naive-bayes either, because the values are continuous ra... |
$$\sum_{i=1}^n \frac1{4i-1}$$
I know I have to integrate the function but from what to find lower and upper bound.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
It sound... |
Bathe's
Finite Element Procedures shows the "nonlinear strain stiffness matrix" for a 2D truss element as$$\frac {^tP} {L_0 + \Delta L}\left[ \begin{array}{ccc}1 & 0 & -1 & 0 \\0 & 1 & 0 & -1 \\-1 & 0 & 1 & 0 \\0 & -1 & 0 & 1 \\ \end{array} \right]$$
But other sources, such as this http://people.duke.edu/~hpgavin/cee42... |
I start with three independent random variables, $X_1, X_2, X_3$. They are each normally distributed with:
$$X_i \sim N(\mu_i, \sigma^2), i = 1, 2, 3.$$
I then have three transformations,
$$\eqalign{ Y_1 &= -X_1/\sqrt{2} + X_2/\sqrt{2} \cr Y_2 &= -X_1/\sqrt{3} - X_2/\sqrt{3} + X_3/\sqrt{3} \cr Y_3 &= X_1/\sqrt{6} + X_2... |
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)
$z$ test for the difference between two proportio... |
"""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 Planck function that describes the blackbody emission is a function of temperature
and wavelength:$$B_\lambda(T)=\frac{2hc^2}{\lambda^5}\cdot\frac1{e^{hc/\lambda k_BT}-1}$$Due to the temperature dependence, blackbodies at different temperatures have different emissions. The graph below, from Wikipedia shows the dra... |
Show that $$\int_0^\pi{d\theta\over(a-b\cos\theta)^2}={a\pi\over (a^2-b^2)^{3\over 2}}$$ where $a>b>0$. I'm not sure how to simplify this integral or evaluate it. Any solutions or hints are greatly appreciated.
closed as off-topic by Travis Willse, C. Dubussy, imranfat, heropup, Martin R Apr 21 '16 at 21:25
This questi... |
I happen to have revised our calculus syllabus for first year biology majors about one year ago (in a French university, for that matter). I benefited a lot from my wife's experience as a math-friendly biologist.
The main point of the course is to get students able to deal with
quantitative models. For example, my wife... |
Prove that doesn't exist $N\in\mathbb{N}$ with property: for all primes $p>N$ exist $n\in\{3, 4,\ldots, N\}$ such that $n, n-1, n-2$ are quadratic residues modulo $p$.
By Dirichlet's theorem, there exists $p>N$ such that each prime $l\leq N$,with the exception of $l=3$, satisfies $(l/p) = (l/3)$. I claim thatthis $p$ i... |
For the pressure exerted by compressed gas, its stated in my textbook: "The result is that at high pressures the molecules of a gas are so compressed that their volume becomes a significant fraction of the total volume of the gas. Since this reduces the volume available for molecular motion, collision occur more freque... |
I asked a student I tutor what $\sqrt{x^2}$ was so I could show him why the solution is $|x|$ instead of just $x$. He ended up changing the problem to $(x^2)^{1/2}$ and then multiplied the exponents to get $x^{2/2}=x^1=x$. Now I'm well aware why the answer to the problem I gave him is $|x|$ and explained it, and the an... |
Two hours before the seminar, the auditorium was already half-full. By 2:40, it was full. The web chat room for the event was
!iframe> A Dutch variation of this blog post is available. Astrophysicist Mario Livio has joined the community of people who spread the rumor about a bump that will be presented on Tuesday betwe... |
Note first that the first $=$ (equals) in $\frac{dl(\theta)}{d\theta} = 0 = −\frac{1}{2\sigma^2}(0−2X^TY + X^TX \theta)$ should be interpreted as a "is set to", that is, we set $\frac{dl(\theta)}{d\theta} = 0$. Given that (apparently) $\frac{dl(\theta)}{d\theta} = −\frac{1}{2\sigma^2}(0−2X^TY + X^TX \theta)$, $\frac{dl... |
As I am working on a problem with 3 linear equations with 2 unknowns I discover when I use any two of the equations it seems I always find a solution ok. But when I plug it into the third equation with the same two variables , the third may or may not cause a contradiction depending if it is a solution and I am OK with... |
I don't have much to add to mweiss' nice answer in terms of teaching suggestions. But I would like to add my own point of view on what the abuse of notation $\mathbf{r}=\mathbf{r}(s)=\mathbf{r}(t)$
means and where it comes from historically.
The first person to do this abuse of notation (implicitly) seems to have been ... |
I recently came across this in a textbook (NCERT class 12 , chapter: wave optics , pg:367 , example 10.4(d)) of mine while studying the Young's double slit experiment. It says a condition for the formation of interference pattern is$$\frac{s}{S} < \frac{\lambda}{d}$$Where $s$ is the size of ...
The accepted answer is c... |
I am learning the Riemann Surfaces and I have a question when I read the appendix A.1.5. of textbook A course Complex analysis and Riemann surfaces by W.Schlag.
One way to define the degree of the non-constant and analytic map $f$ between Riemann surfaces is that $$\deg(f) := \sum_{p\in f^{-1}(q)}(b_{p}(f)+1), $$ where... |
Edit: I'm leaving the old post below, but before I want to write the proof as suggested by Bruce from his book, which uses the ideas in a more efficient way.
Assume that $\|p-q\|<1$, with $p,q\in A$, a unital C$^*$-algebra. Let $x=pq+(1-p)(1-q)$. Then, as $2p-1$ is a unitary, $$\|1-x\|=\|(2p-1)(p-q)\|=\|p-q\|<1.$$So $x... |
I have this funny equation
$$ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^3 u}{\partial x^3}, \qquad x \in [0,1], \qquad t \in (0,T] $$ with initial conditions $u(x,0) = \sin(2\pi x)$, $\frac{\partial u}{\partial t}(x,0) = 0$ and periodic boundary conditions: $u(0,t) = u(1,t)$, $\frac{\partial u}{\partial x}(0,t... |
Welcome to the LHCb web Organisation
We are a relatively new group on the LHCb experiment, and as such we are quite small. However, we are rapidly growing, and this section will keep track of who is working on what. Please feel free to edit your section if I have missed anything out. %$\mu$%
Nigel Watson: Head of the L... |
In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field.
$$grad(f) = \vec{\nabla}(f) = \left< \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2} , ... , \frac{\parti... |
This is standard theory. Try
Birrell, N. D., & Davies, P. C. W. (1982). Quantum Fields in Curved Space. Cambridge: Cambridge University Press.
Bog standard Curved space QFT text. Don't remember how much is said specifically about spinors though.
Brill, D., & Wheeler, J. (1957). Interaction of Neutrinos and Gravitationa... |
Set is Subset of Union/Set of Sets
Jump to navigation Jump to search
Theorem
Let $\mathbb S$ be a set of sets.
Then: $\displaystyle \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$ Proof
Let $T$ be any element of $\mathbb S$.
We wish to show that $T \subseteq S$.
Let $x \in T$.
Then:
\(\displaystyle x\) \(\in\) ... |
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... |
Let $(A,m)$ be a commutative noetherian local ring such that $m$ is principal, say $m=(t)$. Let $(\hat A,\hat m)$ be its $m$-adic completion. Let $A\subset B\subset\hat A$ be any intermediate subring such that $n=tB$ is a maximal ideal of $B$.
The question is: Is it true that the localisation $B_n$ is contained in $\ha... |
BICEP2 near the South Pole might have found a gem In the morning, Sam Telfer asked Matt Strassler, Adam Falkowski, and myself about the new buzz related to the B-modes. It turns out that he was ahead of us. But now, all of us know what is supposed to happen soon, and it is exciting. Update, Monday 4 pm:The rumor was 10... |
Focus Questions
The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts mo- tivated by these questions and be able to write precise, coherent answers to these questions. What is an identity? How do we verify an identity?
Consid... |
I have a question about the definition of a Markov process and a Markov family by Karatzas/Shreve in the book "Brownian Motion and Stochastic Calculus" (cf. p. 74). Let me first recall their definitions:
Markov process: Let $d$ be a positive integer and $\mu$ a probability measure on $(\mathbb{R}^d, \mathcal{B}(\mathbb... |
Let $\gamma : [0,1] \to \mathbb R^2$ be a finite $C^2$-curve in the plane which does not intersect itself. Let $p(z)$ be a second-degree polynomial in $z \in \mathbb R^2$. Can we construct a Riemannian metric $g$ along $\gamma$ such that
$\gamma$ is a geodesic of $g$, $\gamma$ has length 1, and $p(z)$ is the 2-jet of $... |
Suppose there are $N$ subjects under study, with subject $i$ contribution $n_i$ observations, for $i =1,...,N$. And let $y_{ij}$ denote a response variable for subject $i$ at observation $j$. Let $x_{ij}$ denote a $p\times 1$ vector of predictors, and let $z_{ij}$ denote a $q\times 1$ vector of predictors. In general, ... |
Asymptotic large time behavior of singular solutions of the fast diffusion equation
1.
Institute of Mathematics, Academia Sinica, Taipei, Taiwan
2.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China
$u_t=Δ u^m$
$({\mathbb R}^n\setminus\{0\})×(0, ∞)$
$0<m<\frac{n-2}{n}$
$n≥3$
... |
Recently, I read the definition of oxidation state on Wikipedia. It read that a 100% ionic bond is impossible. So what does a 75% ionic and 25% covalent bond mean at all?
A "100% ionic" bond would be a bond whose bonding electron(s) were
never in the vicinity of the cation, but rather always in unperturbed valence orbi... |
Tuned Mass Dampers
A Tuned Mass Damper (TMD) is a mechanical device designed to add damping to a structure for a certain range of exciting frequencies. The extra damping will reduce the movement of the structure to an acceptable level.
A tuned mass damper contains a mass that is able to oscillate in the same direction ... |
Let $C_n = \{1, x, \dots, x^{n-1}\}$ and define $\varphi_r : C_n \to C_n$ as $\varphi_r(x^s) = x^{rs}$.
Theorem: $\varphi_r$ is bijective $\iff$ $\gcd(r,n)=1$.
This is the proof given in a set of notes I'm using:
$\varphi_r$ is bijective $\iff \varphi_r$ is surjective $\iff$ $\{\varphi(x)^t : 0 \leq t < n\} = C_n \iff ... |
Working in the set theory ZFC, all reasoning about classes is strictly informal. A class is informally taken to consist of all sets satisfying some condition. For example, the 'universe' is the collection of sets $x$ which satisfy $x=x$. More formally, two formulae $\phi$ and $\psi$ (with one free variable) refer to th... |
I read in a book about optical fibers that the different spectral components of a light pulse transmitted in the fiber propagate with different velocities due to a wavelength dependent refractive index. Can someone explain that? Why is that silica refractive index depends on the wavelength/frequency of the wave?
The fu... |
I got a task, I don't quite know how to solve.
I've got the following Hamiltonian: $$ \hat H = \frac{B}{\hbar^2}\hat{\mathbf S}_1\cdot \hat{\mathbf S}_2+\frac{C}{\hbar}\left(\hat S_{1z}+\hat S_{2z}\right), \qquad \hat{\mathbf S}_{j=1,2}=(\hat S_{jx},\hat S_{jy},\hat S_{jz}). $$ ($B,c$ constants)
My task is to calculate... |
The solution is quite nice, and simply relies on the fact that $\lim_{n\to 0^+} x^x = 1$, hence for $n$ large enough, we can approximate the integral with $\int_0^{1/n} x\, dx$ instead.There’s an easy generalization of this problem:\begin{align*}\lim_{n\to \infty} n^{k+1} \int_0^{1/n} x^{x+k} \, dx = 1/(k + 1).\end{ali... |
We were introduced to hyperbolic functions previously, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.
Derivatives and Integrals of the Hyperbolic Functions
Recall that the hyperbolic sine and hyperbolic cos... |
EDIT: As i mentioned in my original question, i do not have the background to fully understand @Timaeus answer (which was very detailed indeed). I would appreciate if someone could give a more 'classical physics' answer ,even not so detailed ,in order to clarify some things my self a little bit more. In addition i woul... |
In single-variable calculus, differentiation and integration are thought of as inverse operations. For instance, to integrate a function \(f (x)\) it is necessary to find the antiderivative of \(f\) , that is, another function \(F(x)\) whose derivative is \(f (x)\). Is there a similar way of defining integration of rea... |
Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. (That is, $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$.) Let $\sigma(x)$ denote the
sum of the divisors of $x \in \mathbb{N}$.
Define $$D(n^2) := 2n^2 - \sigma(n^2)$$ to be the deficiency of the non-Euler part $n^2$.
CLAIM
$\gcd(n^2, D(n^2)) \neq 1$.
M... |
Negative of Subring is Negative of Ring Theorem
Let $\struct {R, +, \circ}$ be an Ring.
For each $x \in R$ let $-x$ denote the ring negative of $x$ in $R$.
Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$.
For each $x \in S$ let $\mathbin \sim x$ denote the ring negative of $x$ in $S$.
... |
19 Mar 2018 Using a Bernoulli VAE on Real-Valued Observations
The Bernoulli observation VAE is supposed is used when one’s observed samples $x \in \sset{0, 1}^n$ are vectors of binary elements. However, I have, on occasion, seen people (and even papers) that apply Bernoulli observation VAEs to real-valued samples $x \i... |
I'm considering to study some high-dimensional Navier-Stokes equations. One problem is to do write the viscous equation for vorticity, helicity and other conserved quantities. I think it might be better if it is possible to work with differential form and exterior calculus? Is there any reference that I may find somewh... |
This is a fairly tricky concept, and I see research physicists stumbling with (analogues of) this problem with a somewhat alarming frequency. When you say
the nucleus is seemingly spherical
what that really means is that the electrons interact with a spherical object (essentially a point Coulomb charge at the nuclear c... |
There is much misconception in the question, but I'll hazard an answer. I am not going to give any links (no point linking to Wikipedia, it's
just there), but I'll highlight the important terms in bold. If you want to research more, search for these. Any good general astronomy course textbook will cover these topics, t... |
Frequent Links ISO 31-11
Its definitions include the following:
[2] Contents Mathematical logic
Sign Example Name Meaning and verbal equivalent Remarks ∧ p ∧ q conjunction sign p and q ∨ p ∨ q disjunction sign p or q (or both) ¬ ¬ p negation sign negation of p; not p; non p ⇒ p ⇒ q implication sign if p then q; p impli... |
I see definitions (in Wikipedia, for example) about inner product spaces over arbitrary fields. But I don't understand how positivity makes any sense for fields which are not ordered? Am I missing something?
Clarification: Let me elaborate, to make clear what I already know.An inner product $g$ is a $\mathbb F$-sesquil... |
Suppose I begin with the time-independent Schrodinger equation $$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$ ordinarily we specify the function $V$ and then solve for a set of eigenfunctions and eigenvalues. And just to be slightly more general, we do the same thing with Sturm-Liouville e... |
I have found many sources (c.f. Schwartz's QFT book section 10.4) that try to obtain the non-relativistic limit of the Dirac equation by first "squaring it" so that it looks somewhat like the Klein-Gordon equation.
Kleiner eqn 6.113 for examples shows that in the chiral basis, this "squared" Dirac equation becomes
$\le... |
Demonstrate this.
But it is given in the OP as "p". The expectation takes the largest value for p = 1/2, because as post #6 shows the expected number of games played...
Let n = \frac{N-1}{2} \geq 0Then our expectation value goes like this:E = \sum_{i=0}^{2i -1 < N} \sum_{j=0}^{2j -1 < N} { {i+j} \choose i } p^i...
Thus... |
The discovery of TON 618 have created a new black hole species (already fingerprinted by M87 or even IC1101 cores): the ultramassive black holes with masses greater than $10^{10}M_\odot$. As said in the previous answer, in classical settings, there is no upper limit of the mass of black holes (I am not so sure if you g... |
It seems that you are having difficulty somewhere with converting the informal short sketch into a fully detailed sketch.
So let's try to see in detail how a (non-empty) set being r.e. (recursively enumerable) implies that it is the range of some total recursive function.
Let $A\subseteq \mathbb{N}$ be a non-empty r.e.... |
I was reading about fibre optic cables and it was mentioned that ,the individual "light pipes" are coated with a material whose refractive index is less than that of that of the glass. My question is why a material with smaller $\mu$ ? According to sin $\theta$=1/$\mu$, if I decrease $\mu$ then $\theta$,which is the cr... |
so here's something that might be a neat topic for discussion, if anyone ever comes around and wants to talk: recall the spectra $X(k)$, which are the Thom spectra of the maps $\Omega SU(k)\to \Omega SU\simeq BU$. This is pretty much all I talk about these days... sorry.
Well, here's a neat fact, if $E$ is complex orie... |
If you want rigorous proof, the following lemma is often useful resp. more handy than the definitions.
If $c = \lim_{n\to\infty} \frac{f(n)}{g(n)}$ exists, then
$c=0 \qquad \ \,\iff f \in o(g)$,
$c \in (0,\infty) \iff f \in \Theta(g)$ and
$c=\infty \quad \ \ \ \iff f \in \omega(g)$.
With this, you should be able to ord... |
Search
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 ... |
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in terms of the integral of the products, and forget about composition! Why i... |
Here is a somewhat different way from Johan's of looking at thisproblem. At each stage of the walk, choose a number $x$uniformly from $[0,1]$ and then walk either a distance $x$ to theright or $1-x$ to the left. This does not affect the probabilityof becoming negative since there is still a uniform probabilityof taking... |
Theory Definition. A hierarchy is a single-rooted tree.
Let \(T=(V,E)\) be a connected, acyclic directed graph that expresses a hierarchical structure. The set of vertices \(V\) represents the entities that are being organized, and the set of edges \(E\) maps from a given entity to its respective superior.
A vertex is ... |
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This is a question from one of my statistics courses at uni, ... |
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... |
What did you try ? To prove that $A$ is a subring of $K$, you need to prove that:
(i) $1 \in A$ (note: it is easy to forget this one, so I put it first!), which means that $v(1) \geq 0$. We could easily prove this one using fact 1. of your question: namely, $v(1 \times 1) = v(1) + v(1)$, what could $v(1)$ be ? (ii) for... |
Case 1: all roots are distinct
In this case, assume $f$ has total of $n$ real roots. I can take any consecutive pair of roots $a,b$, $a < b$ and say that since $f(a) = f(b) = 0$, using Rolle's theorem, $\exists c \in \mathbb{R}$ such that $f'(c) = 0, a < c < b$. Then I end up with $n-1$ real roots in $f'$. How do I sho... |
Topic: Discrete Fourier Transform
(This problem clarifies how zero-padding a signal changes its DFT.)
Question
Let x[n] be a signal with duration N. More precisely, assume that $ x[n]=0 $ for $ n> N-1 $ and for $ n<0 $.
Let y[n] be the zero-padding of x[n] to length M>N:
$ y[n]= \left\{ \begin{array}{ll} x[n], & 0\leq ... |
Next we consider series with both positive and negative terms, but in a regular pattern: they alternate, as in the
alternating harmonic series for example:
\[ \sum_{n=1}^\infty {(-1)^{n-1}\over n}= {1\over1}+{-1\over2}+{1\over3}+{-1\over4}+\cdots= {1\over1}-{1\over2}+{1\over3}-{1\over4}+\cdots. \]
In this series the si... |
I am studying the dual simplex method from Lieberman - 10e. An approach called dual simplex method was described that is "applied" on the "primal table" itself, i.e., We do not convert it into its dual problem. For the Primal problem, I know how to find an initial basis using Big-M method or Two phase method. Such a ba... |
What is a function? Informally, it is a process, or an assignment, from an input set to an output set. It is not just the process or assignment that forms a function, but specifying the input and output is part of what it is. Formally, a function is a triple $(A,B,f)$ where $A,B$ are sets and $f\subseteq A\times B$ is ... |
The first observation of top quark production in proton-nucleus collisions is reported using proton-lead data collected by the CMS experiment at the CERN LHC at a nucleon-nucleon center-of-mass energy of $\sqrt{s_\mathrm{NN}} =$ 8.16 TeV. The measurement is performed using events with exactly one isolated electron or m... |
Square roots are typically computed by root finding; if you want to find $\sqrt{x}$, you would use a root finding algorithm on
$$g(t) = t^2 - x$$
One such algorithm would be, for example, Newton's Method. Wikipedia has an entire page describing algorithms for root finding, and some more for computing square roots.
The ... |
There is a very simple method to simulate from the Gaussian copula which is based on the definitions of the multivariate normal distribution and the Gauss copula.
I'll start by providing the required definition and properties of the multivariate normal distribution, followed by the Gaussian copula, and then I'll provid... |
In rudimentary thermodynamics, an equation of state is any equation relating state variables such as pressure, volume, temperature, energy, and so on. In cosmology and fundamental physics, the term is most often used for equations describing the relationship between the energy density \(\rho\) and pressure \(p\):
\[ p ... |
Hipster dynamics¶
This week I started seeing references all over the internet to this paper:
The Hipster Effect: When Anticonformists All Look The Same. It essentially describes a simple mathematical model which models conformity and non-conformity among a mutually interacting population, and finds some interesting res... |
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