text stringlengths 256 16.4k |
|---|
Revista Matemática Iberoamericana Rev. Mat. Iberoamericana Volume 23, Number 2 (2007), 513-536. On a Parabolic Symmetry Problem Abstract
In this paper we prove a symmetry theorem for the Green function associated to the heat equation in a certain class of bounded domains $\Omega\subset\mathbb{R}^{n+1}$. For $T>0$, let ... |
Search
Now showing items 1-10 of 22
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
The question is:
"Show how the nonlinear regression equation $y=a(x-b)+c(x-b)^2$ can be converted to a linear regression equation solvable by the method of least squares."
if we take that \begin{align} a(x-b) - c(x-b)^2 &= ax - ab + cx^2 -2cbx - cb^2\\ & = \underbrace{-ab-cb^2}_{\beta_0} + x\underbrace{(a-2cb)}_{\beta_... |
Let's say that I want to find solutions $f\in C(\Bbb R)$ to the equation
$$ f(x+1) + f(x) = g(x) $$
for some $g\in C(\Bbb R)$. I can write $f(x+1) = (Tf)(x)$ where $T$ is the right shift operator and rewrite the equation suggestively as $$ (I+ T)f=g. $$ Formally, I can say that the solution of this equation is
$$ f= (I... |
I am testing Logistic Regression with stochastic gradient using
sklearn.linear_model.SGDClassifier. I have 2D independent variables and corresponding labels as below:
X = array([[-2.58733628, 2.26126322], [ 1.97831473, 2.03510032], [ 2.48324069, -2.17901384], ... ])Y = array([[ 1.], [ 1.], [-1.], [-1.], ...])
Here my o... |
#
1
Problem with understanding the proof of Sauer Lemma
I will replicate the proof here which is from the book "Learning from Data"
Sauer Lemma:
$B(N,K) \leq \sum_{i=0}^{k-1}{n\choose i}$
Proof:
The statement is true whenever k = 1 or N = 1 by inspection. The proof is by induction on N. Assume the statement is true for... |
The production of two high-p_T jets in the interactions of quasi-real photons in e+e- collisions at sqrt{s_ee} from 189 GeV to 209 GeV is studied with data corresponding to an integrated e+e- luminosity of 550 pb^{-1}. The jets reconstructed by the k_T cluster algorithm are defined within the pseudo-rapidity range -1 <... |
In trigonometry and geometry,
triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration). The point can then be fixed as the third point of a triangle with one kn... |
I'm trying to compare 2 types of data in programs: floating point decimals ($doubles$) and fractions (let's say $pair<long,long>$), but it doesn't really matter for the question.
So here is what I can't find nor do: I would like a exact expression, a good aproximation or a fast calculation O(1) of how many fractions ar... |
Many textbooks present expressions for superfields in $4$ dimensions. For my current project, I have to find out how things work in $2$ dimensions.
Let me summarise in short what we know about $4$d (the equations below are taken from the book by Muller-Kirsten and Wiedemann).
The most general form of a superfield is: \... |
Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh,
I would like to apply a Robin type boundary condition to the l.h.s. of the domain ($x=x_L)$, such that,
$$ \sigma_L = \left( d u_x + a u \right) \bigg|_{x=x_L} $$
where $\sigma_L$ is the boundary value; ... |
Learning Objectives
In this section students will:
Simplify rational expressions. Multiply rational expressions. Divide rational expressions. Add and subtract rational expressions. Simplify complex rational expressions.
A pastry shop has fixed costs of \($280\) per week and variable costs of \($9\) per box of pastries.... |
Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a linear map. I want to evaluate an expression of the type $$F((ax+b)^k)$$ in terms of $F(x)$ for some fixed value of $x$ (I already know $F(x)^r$ for $r=1,...,k$).
$x$ is typically small ($0<x<1$), and $k$ is typically about $15$. $a$ is always positive (about $30$) and $b... |
1976 IMO Problems/Problem 3 Problem
A box whose shape is a parallelepiped can be completely filled with cubes of side If we put in it the maximum possible number of cubes, each of volume , with the sides parallel to those of the box, then exactly percent from the volume of the box is occupied. Determine the possible di... |
Consider the product of two simple functions, say \( f(x)=(x^2+1)(x^3-3x)\). An obvious guess for the derivative of \(f\) is the product of the derivatives of the constituent functions: \( (2x)(3x^2-3)=6x^3-6x\).
Is this correct? We can easily check, by rewriting \(f\) and doing the calculation in a way that is known t... |
Updated 07.11
We can chose the model to discuss the problem and so let us chose:
Model: Newtonian mechanics/Newtonian gravity, with the Universe filled with uniformly dense matter, interacting only gravitationally (in cosmology this called “dust matter”), and at the initial time of our spaceship journey all this matter... |
Lie algebras over rings (Lie rings) are important in group theory. For instance, to every group $G$ one can associate a Lie ring
$$L(G)=\bigoplus _{i=1}^\infty \gamma _i(G)/\gamma _{i+1}(G),$$
where $\gamma _i(G)$ is the $i$-th term of the lower central series of $G$. The addition is defined by the additive structure o... |
It would depend on damping effects being taken into account or not.
Invoking Newton's 2nd Law of motion, a differential equation for the motion of a damped harmonic oscillator can be written (including an external, sinusoidal driving force term):
$m\frac{d^2x}{dt^2}+2m\xi\omega_0\frac{dx}{dt}+m\omega_0^2x=F_0\sin\left(... |
DISCLAIMER: I've edited the question repeatedly for clarity and to target the most relevant answer.
I have the following general problem $$ \min \|h_1\cdot h_2\|^2 $$ such that $$\|g_1\wedge g_2-h_1\wedge h_2\|^2 = 0,$$ where $h_i,g_i\in \mathbb{R}^n\wedge \mathbb{R}^n$ are skew-symmetric matrices, $A\cdot B$ denotes m... |
Suppose I define the mapping torus $M_f$ in the usual way by identifying $(x, 0)$ and $(f(x), 1)$. If I have a homeomorphism $f: X \rightarrow X$ and another homeomorphism $f': X \rightarrow X$ that are homotopic, are the spaces $M_f$ and $M_{f'}$ necessarily homeomorphic?
Asking for homotopic homeomorphisms isn't enou... |
Proof for logical implication : $\exists_x A(x) \rightarrow \forall_x B(x) \implies \forall_x [A(x) \rightarrow B(x)]$
Can you please see whether the proof is fine or not?
We know that $ {\forall_x B(x) \implies \exists_x B(x)} \\ [\exists_x A(x) \rightarrow \forall_x B(x)] \implies [\exists_x A(x) \rightarrow \exists_... |
One thing you can do with tetrads is express quantities everywhere in terms of what "natural" observers would measure at each point in spacetime.
To be more concrete, consider a spacetime foliated by slices of constant timelike coordinate. At each point, one can imagine the "normal observer" whose 4-velocity is the uni... |
Difference between revisions of "LaTeX:Symbols"
m (→Operators)
(→See Also)
(25 intermediate revisions by 14 users not shown) Line 2: Line 2:
This article will provide a short list of commonly used LaTeX symbols.
This article will provide a short list of commonly used LaTeX symbols.
− − − − − − − − − − −
== Finding Othe... |
I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be...
Question: Does this notation$$\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$$refer to the Jacobian
matrix$$ J = \begin{bmatrix} \dfrac{\partial y_... |
In driven oscillator it can be explained by the following differential equation $$\ddot{x} + 2\beta \dot {x} + \omega_0 ^2x = A \cos(\omega t)$$ where the $2\beta$ is coefficient of friction, the $\omega_0$ is the frequency of simple harmonic oscillator, and $A \cos(\omega t)$ is the driven force divided by the mass of... |
The answer to your confusion really is that you can't see holes just as "absences" of electrons. You are right: If the electrons up, then the holes have to get up as well (at least if we stick to this picture of electrons being particles):
At first, maybe this answer might be interesting to look at.
To give a brief sum... |
Is there a general method to work out all irreducible complex representation of a group?
Describe all the the irreducible complex representation of the group $S_4$.
$S_4$ is the symmetric group on four letter.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professiona... |
Update1
I discovered that the bountary of a series of ellipses consists of the following three parts
Part (I): the first ellipse's
effectiveblack-segment;
Part (II): the
effectiveenvelope-points of the ellipes from the second to last second;
Part (III): the last ellipse's
effectivered-segment;
Given that there are $n$ ... |
Can't you just use the Lyapunov convexity theorem directly?
As usual, identify $\ell^\infty(G)$ with $C(\beta G)$, and work with $\beta G$ the Stone-Cech compactification. As this is a compact Hausdorff space, if $\mu$ is a regular measure on $\beta G$ then an atom of $\mu$ must be a point. So we can decompose $\mu$ as... |
The most general, integral form of Faraday's Law is (see this physics.SE question: Faraday's law for a current loop being deformed)\begin{align} \int_{C_t} (\mathbf E+\mathbf v\times\mathbf B)\cdot d\boldsymbol \ell = - \frac{d}{dt}\int_{\Sigma_t}\mathbf B\cdot d\mathbf a\end{align}Where $C_t$ is some closed curve that... |
The key to this question is to note that gunpowder doesn't technically explode -- it deflagrates. It doesn't have a super-sonic explosion, but rather a sub-sonic burn. To get the powerful kick needed to project a rifle or handgun bullet, we rely on the fact that gunpowder burns faster in a confined space. The tighter t... |
Arrange the given statements involving indices to show whether they are true or false.
\( (x^3)^4 \equiv x^7\)
\( \frac{x^6}{x^3} \equiv x^2\)
\(x^8 \div x^4 \equiv x^2\)
\(x^2 \times x^3 \equiv x^6\)
\( (x^3)^4 \equiv x^{12}\)
\( \frac{x^7}{x^3} \equiv x^4\)
\(x^8 \div x^5 \equiv x^3\)
\(x^2 \times x^3 \equiv x^5\)
Yo... |
We introduced the concept of a limit gently, approximating their values graphically and numerically. Next came the rigorous definition of the limit, along with an admittedly tedious method for evaluating them. The previous section gave us tools (which we call theorems) that allow us to compute limits with greater ease.... |
Learning Outcomes
Raise a number to a power using technology. Take the square root of a number using technology. Apply the order of operations when there is root or a power.
It can be a challenge when we first try to use technology to raise a number to a power or take a square root of a number. In this section, we will... |
I have two questions related to rotation of superfluids. Firstly, what is the main reason that superfluid cannot rotate as a whole object ? (I found that it is true in Landau's
Statistical Physics but without explanation.) Second question is the following. Suppose we have a rotating helium and then we cooling it in suc... |
I want to solve the Laplace Equation with pure Neumann B.C. using Finite Element Method:
$- \Delta u = f \ $ in $ \ \Omega $
$- \partial u/\partial n = g \ $ on $ \ \Gamma = \partial \Omega$
With weak formulation
$\int_{\Omega} \nabla v \cdot \nabla u = \int_{\Omega} v \ f + \int_{\partial \Omega} v \ g $.
To obtain a ... |
Exercise \(1\)
In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation \(y=ax^2+bx+c\), which was \(m=−\frac{b}{(2a)}\). Prove this formula using calculus.
Answer
Under Construction
Exercise \(2\)
If you are finding an absolute minimum over an interval \([a,b],\) why do ... |
Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ $$\Phi{(x,y)}=\dfrac{i}{4}H^{(1)}_{0}(k|x-y|),x\neq y$$ we denote the fundamental solution to the two-dimensional Helmholtz equation in ter... |
Is there any representation of the Lorentz group where $$U^{-1} f(x) U = f(\lambda^{-1}x)$$ other than the (0,0) representation?
If not then is it possible for a field (with a well defined polynomial basis) to behave like a scalar field under the Lorentz group?
Will such fields still be called the (0,0) representation ... |
Transverse momentum spectra and rapidity densities, dN/dy, of protons, anti-protons, and net--protons (p-pbar) from central (0-5%) Au+Au collisions at sqrt(sNN) = 200 GeV were measured with the BRAHMS experiment within the rapidity range 0 < y < 3. The proton and anti-proton dN/dy decrease from mid-rapidity to y=3. The... |
Question:
Let $K$ and $L$ be extensions of $F$. Show that $KL$ is Galois over $F$ if both $K$ and $L$ are Galois over $F$.
This question has been already asked here. But People provided
incomplete solution to the problem.
I have tried to attempt the problem:
Case $1$: Either $K\subset L$ or $L\subset K$. Then $KL$ is t... |
This is not really an answer to your question, essentially because there isn't (currently) a question in your post, but it is too long for a comment.
Your statement that
A co-ordinate transformation is linear map from a vector to itself with a change of basis.
is muddled and ultimately incorrect. Take some vector space... |
X
Search Filters
Format
Subjects
Library Location
Language
Publication Date
Click on a bar to filter by decade
Slide to change publication date range
2012, 1st ed., ISBN 9780312657758, cm.
Book
2. Highly resolved HSQC experiments for the fast and accurate measurement of homonuclear and heteronuclear coupling constants
... |
Model of a crystal and deriving the expected diffraction image
In single-crystal X-ray crystallography, the diffraction image is due to scattering of electrons in a crystalline sample. A convenient way of describing the electron density is to first specify crystal symmetry and mean atomic positions (coordinates), and t... |
The divisor function $d(n)$, is the number of $(a,b)\in\mathbb {N^+}^2$ such that $a\times b =n$. For example, $d(2)=2$ because $2=1\times 2=2\times 1$ and d(6)=4 because $6=1\times 6=2\times 3=3\times 2=6\times 1$.
The divisor summatory function is defined by : $$D(n)=\sum_{i=1}^n d(i)$$
This is sequence A006218 in OE... |
I want to ask MSE to confirm the correctness of the alternate solution and its mistake.
I know possible solution: https://math.stackexchange.com/a/2557094/456510
If $x,y,z\in {\mathbb R}$, Solve the system equation:
$$ \left\lbrace\begin{array}{ccccccl} x^4 & + & y^2 & + & 4 & = & 5yz \\[1mm] y^{4} & + & z^{2} & + & 4 ... |
I post this answer to check my understanding.
Imagine a wavefunction in 1 dimensions with a known energy and momentum it's wavefunction will be:
$$\Psi(x, t) = e^{i(kx-\omega t)} = e^{i(px-E t)/\hbar}$$
With some calculus and algebra you can derive the momentum operator and get this:
$$-i\hbar \partial_x \Psi = p \Psi$... |
Electronic Communications in Probability Electron. Commun. Probab. Volume 20 (2015), paper no. 53, 11 pp. Gluing lemmas and Skorohod representations Abstract
Let $(\mathcal{X},\mathcal{E})$, $(\mathcal{Y},\mathcal{F})$ and $(\mathcal{Z},\mathcal{G})$ be measurable spaces. Suppose we are given two probability measures $... |
[FFmpeg-devel] [PATCH] avfilter/vsrc_mandelbrot: avoid sqrt for epsilon calculation Ganesh Ajjanagadde gajjanag at mit.edu Tue Nov 24 15:23:04 CET 2015 On Mon, Nov 23, 2015 at 9:46 PM, Michael Niedermayer <michaelni at gmx.at> wrote:
> On Mon, Nov 23, 2015 at 05:19:52PM -0500, Ganesh Ajjanagadde wrote:
>> This rewrites... |
There are at least three interesting features of the problem as currently stated. First of all, it specifies that $x, y \ge 0$. Secondly, apart from $x,y \ge 0$, it specifies neither the domain nor the codomain. And finally, it does not ask for continuity.
We will have to narrow down the question in order to answer it,... |
While reading about polylogarithms, I came across the nice
polylogarithm ladder,
$$6\operatorname{Li}_2(x^{-1})-3\operatorname{Li}_2(x^{-2})-4\operatorname{Li}_2(x^{-3})+\operatorname{Li}_2(x^{-6}) = \frac{7\pi^2}{30}\tag{1}$$
where $x = \phi = \frac{1+\sqrt{5}}{2}$, the
golden ratio, or the root $1<x<2$ of,
$$x^n(2-x)... |
In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads:
QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$.Let $M$ be the set of all the points $\mathbf x$ such that $f(\mathbf x)=\mathbf 0$ and $N$ be the set of all the points $\mathbf x$ such that $$f_1(\mathbf x)=\cdot... |
This is an excellent question. As indicated by the MathOverflow link in the comments, there are many ways to think about torsion and torsion-freeness. At the risk of being repetitive, allow me to summarize some of these, adding my own thoughts.
Throughout, we let $M$ be a smooth manifold, $\nabla$ a connection on $TM$,... |
Let \(a_1,a_2,\ldots\) be an infinite sequence of positive real numbers such that \(\sum_{n=1}^\infty a_n\) converges. Prove that for every positive constant \(c\), there exists an infinite sequence \(i_1<i_2<i_3<\cdots\) of positive integers such that \(| i_n-cn^3| =O(n^2)\) and \(\sum_{n=1}^\infty \left( a_{i_n} (a_1... |
Current browse context:
math.AC
Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Commutative Algebra Title: On graphs related to co-maximal ideals of a commutative ring
(Submitted on 1 Jun 2011)
Abstract: This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of ... |
Given the first $n$ primes, we can label the $k$th prime as $p_k$. So, what is the least common multiple(LCM) of {$p_1 - 1$, $p_2 - 1$, $p_3 - 1$, ..., $p_n-1$}? In other words, if we subtract $1$ from each of the first $n$ primes, and wish to find the LCM of these new values, can we find a lower bounds for this LCM?
B... |
Let $f:[0;1]\to \mathbb{R}$ be a continuous function satisfying
$$f\left(\frac{x}{2}\right) + f\left(\frac{x+1}{2}\right)=3f(x).$$
How to show that $f\equiv0$?
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 si... |
Equivalence of Definitions of Isolated Point Contents Theorem Let $T = \left({S, \tau}\right)$ be a topological space.
Let $H \subseteq S$ be a subset of $S$.
$x \in H$ is an
isolated point of $H$ if and only if: $\exists U \in \tau: U \cap H = \left\{{x}\right\}$ Proof Definition 1 implies Definition 2
Let $x$ be an i... |
Data on the mean multiplicity of strange hadrons produced in minimum bias proton--proton and central nucleus--nucleus collisions at momenta between 2.8 and 400 GeV/c per nucleon have been compiled. The multiplicities for nucleon--nucleon interactions were constructed. The ratios of strange particle multiplicity to part... |
Definition:Symmetric Mapping Definition
Let $\R$ be the field of real numbers.
Let $\F$ be a subfield of $\R$.
Let $V$ be a vector space over $\F$
Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping.
Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is s... |
NLO Higgs+jet production at large transverse momenta including top quark mass effects Abstract
Here, we present a next-to-leading order calculation of
H+jet in gluon fusion including the effect of a finite top quark mass $$m_t$$ at large transverse momenta. Using the recently published two-loop amplitudes in the high e... |
The
amsmath package provides a handful of options for displaying equations. You can choose the layout that better suits your document, even if the equations are really long, or if you have to include several equations in the same line.
Contents
The standard LaTeX tools for equations may lack some flexibility, causing o... |
In Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why $\epsilon_0$ is countable.
Thanks.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related f... |
Ground state degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system.
I just want to find a potential $V(\mathbf{r})$, not necessary the central potential, such that Schrodinger equation in d-dimensional (no internal degrees o... |
I had a teacher pose this interesting question yesterday:
Suppose you're running a high-energy scattering experiment at the LHC. For concreteness, let's suppose it's a 2 to 2 scattering event which involves electrons and/or muons.
The theorist uses QFT to compute some cross-section which comes from the amplitude $$ \ma... |
I read in my textbook that if we multiply a chemical reaction by some factor(let's say $b$) its new equilibrium constant becomes $K^b$.But I don't understand why this happens..What is the difference ...
While I was reading about the usefulness of the quantity $\Delta H$, I found that it can be used to calculate the how... |
I am reading a paper and there is the following theorem:
Let $n$ be a fixed integer, and $n >1$.
Denote divisibility in $\mathbb{Z}[\frac{1}{n}]$ by $|_n$, thus for all $x, y \in \mathbb{Z}$ $$x |_n y \leftrightarrow \exists q, f \in \mathbb{Z}: y=xqn^{-f}$$ Then the positive existential theory of $(\mathbb{Z}; +, |_n)... |
Suppose you are given a single unit square, and you would like to completely cover the surface of a cube by cutting up the square and pasting it onto the cube's surface.
Q1. What is the largest cube that can be covered by a $1 \times 1$ square when cut into at most $k$ pieces?
The case $k=1$ has been studied, probably ... |
Using a silicon vertex detector, we measure the charged particle pseudorapidity distribution over the range 1.5 to 5.5 using data collected from PbarP collisions at root s = 630 GeV. With a data sample of 3 million events, we deduce a result with an overall normalization uncertainty of 5%, and typical bin to bin errors... |
$\sqrt{x} = \frac{x}{\sqrt{x}}$
Seeing the above equation may look completely logical, or not. When i saw it a few days ago, i thought it was wrong. When i understood that it was correct, i thought it was the most beautiful thing ever. I’m not 100% sure why this intrigued me so much, but it just looks great.
There are ... |
Closure Properties
Once you have a small collection of non-context-free languages you can often use closure properties of $\mathrm{CFL}$ like this:
Assume $L \in \mathrm{CFL}$. Then, by closure property X (together with Y), $L' \in \mathrm{CFL}$. This contradicts $L' \notin \mathrm{CFL}$ which we know to hold, therefor... |
Given an instance I of an optimization constraint satisfaction problem (CSP), finding solutions with value at least the expected value of a random solution is easy. We wonder how good such solutions can be. Namely, we initiate the study of ratio \(\rho _E(I) =(\mathrm {E}_X[v(I, X)] -\mathrm {wor}(I))/(\mathrm {opt}(I)... |
I am thinking of the property in probability of inequality. In particular, we assume \begin{equation} P[\zeta>a]\leq b, \end{equation} where $a>0$, $b>0$ and $\zeta\in R$ is a random variable.
Now we would like to consider whether the inequality of $P[\zeta^2>a^2]\leq b$ holds.
In fact, for differential and monotonic t... |
Dieter Lüst and two co-authors from Monkberg (Munich) managed to post the first hep-th paper today at 19:00:02 (a two-second lag is longer than usual, the timing contest wasn't too competitive):
In particular, they claim that whenever there are particles whose spin is two or higher, they have to be massive and there ha... |
Secans and Cosecans
I is amazing how easily old stuff gets lost, when it is used no longer.
I bet most mathematicians don’t know how to spot the secans or the cotangens in the unit circle. Neither did I, when the question came up on the German version of geopardy (Wer wird Millionär). Of course, we can look that up in ... |
Yahtzee Waiting Times
I recently was asked about waiting times in the game of Yahtzee. If you do not know the game it suffices to say that you throw 5 dice, and one of the goals is to get 5 dice with the same number, a Yahtzee. I wrote about waiting times a long time ago. But let me repeat the main points.
The trick is... |
Consider a particle of mass $𝑚$ in an infinite square well of width $𝐿$. The wave function of the particle at $𝑡 = 0$ is $$ \psi (x,0)=Ax^2(x^2-L^2), \quad 0\leq x \leq L$$
a.) What is $\psi(x,t)$ for $ t \geq 0 $?
b.) At some time $t >0$, what is the probability of measuring the particle to have energy $16\pi ^2\hb... |
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... |
How to Perform Various Rotor Analyses in the COMSOL® Software
Vibration in rotating machinery is very sensitive to the geometric, structural, and inertial properties of the various rotating and stationary components interacting with each other. These properties include the location of the mounted components and their i... |
So the knapsack problem has an integer programming formulation as follows,
$$ \max_x v\cdot x\\s.t \\x_i \in \{0,1\}\\w\cdot x \leq C$$
Now consider the second integer program which might be a variation of the knapsack integer program.
$$ \max_x v\cdot x\\s.t \\x_i \in \{0,L_i\}\\ x_i \leq R \cdot \delta_i\\ \delta_i \... |
Defining parameters
Level: \( N \) = \( 100 = 2^{2} \cdot 5^{2} \) Weight: \( k \) = \( 1 \) Nonzero newspaces: \( 1 \) Newforms: \( 1 \) Sturm bound: \(600\) Trace bound: \(0\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(100))\).
Total New Old Modular forms 74 25 49 Cus... |
Skills to Develop
In this section, we strive to understand the ideas generated by the following important questions:
What is a graphical justification for why \( \frac { d } { d x } \left[ a ^ { x } \right] = a ^ { x } \ln ( a )\)? What do the graphs of \(y = \sin ( x ) \) and\( y = \cos ( x )\) suggest as formulas for... |
I have a question about Sobolev spaces.
In the following, we assume $d \ge 2$.Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$.
Note that $D$ is not necessary bounded. $H^{1}(D)$ denotes first order $L^2$-Sobolev space on $D$ with Neumann boundary condition. I am interest... |
Our new book (NAT)
Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15
uses mainly cubical, rather than simplicial, sets. The reasons are explained in the Introduction: in strict cubical higher categories we can easily express
algebraic inverse... |
If $\sum_{m=0}^{\infty} b_m$ is
conditionally convergent, then is $\sum_{m=0}^{\infty} m^2b_m$ divergent? JUSTIFY
An example of conditionally convergent series is $\sum_{m=0}^{\infty} (-1)^m/\sqrt{m+1}$ and multiplying by $m^2$ it is divergent.
My conclusion is that it diverges. But what will the "justify" be? |
1. State whether the following augmented matrices are in RREF and compute their solution sets.
$$\left(\begin{array}{rrrrr|r}1 &0 &0 &0 &3 &1 \\ 0 &1 &0 &0 &1 &2 \\ 0 &0 &1 &0 &1 &3 \\ 0 &0 &0 &1 &2 &0\end{array}\right),$$
$$\left(\begin{array}{rrrrrr|r}1 &1 &0 &1 &0 &1 &0 \\ 0 &0 &1 &2 &0 &2 &0 \\ 0 &0 &0 &0 &1 &3 &0 ... |
By means of $\varepsilon$-$\delta$, I am looking for some ideas to prove (a beginner math class) that limit does
not exist.For instance, consider the function\begin{equation}f(x)=\begin{cases}x,&x>1\\3-x,&x\leq1,\end{cases}\end{equation}show that $\lim_{x\to1}f(x)$ does not exist.
By means of $\varepsilon$-$\delta$, I ... |
I was studying variational methods in theoretical physics and I got stuck with a few simple questions. I have possible answers but I cannot see clearly and rigorously if they are correct.
Suppose we have an action $S$ that depends on two fields: an antisymmetric tensor field $T_{\mu \nu}$ and the spacetime metric $g_{\... |
Prince Rupert's Cube
Jump to navigation Jump to search
Prince Rupert's Cube
Let $C$ be a unit cube.
$\dfrac {3 \sqrt 2} 4 = 1 \cdotp 06066 \, 0$ Proof Source of Name
This entry was named for Prince Rupert of the Rhine.
The correct answer was determined by Pieter Nieuwland.
This provides a solution of $\sqrt 6 - \sqrt 2... |
In more than two dimensions we use a similar definition, based on the fact that all eigenvalues of the coefficient matrix have the same sign (for an elliptic equation), have different signs (hyperbolic) or one of them is zero (parabolic). This has to do with the behavior along the characteristics, as discussed below.
L... |
Let us reformulate OP's question as follows:
Give a proof that a local coordinate transformation $x^{\mu} \to y^{\rho}=y^{\rho}(x)$ between two local coordinate systems (on a 3+1 dimensional Lorentzian manifold) must be affine if the metric $g_{\mu\nu}$ in both coordinate systems happen to be on constant flat Minkowski... |
You have two classes of points. Instead of managing them in two sets, one just assigns each point in the first class the value $-1$ and in the second class the value $+1$. So in fact you have point-value pairs $(x_i,y_i)$. To classify future points in a consistent way you now want to construct a function $f(x)$ that ha... |
For the 2018 version of our NMR Mandhala (working towards higher homogeneity along the cylindrical axis), I did a close reading of Soltner and Blumler’s 2010 article “Dipolar Halbach Magnet Stacks Made from Identically Shaped Permanent Magnets for Magnetic Resonance”. Below are some particularly useful findings.
Useful... |
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... |
Using basic circuit analysis techniques we can find the voltage gain of this basic integrator as follows:
\$i_1=\frac{v_I}{R_I}\quad\text{and}\quad i_2=-C(\frac{dv_O}{dt})\\\text{since:}\quad i_1=i_2 \ \rightarrow \ \frac{v_I}{R_I}=-C(\frac{dv_O}{dt})\$
From this we can derive the output voltage to be:\$-\frac{1}{RC}\i... |
Problem - need help for part (ii)
Let $\vec{F} = y \vec{i} -x \vec{j} + z \vec{k}$ and let the surface $S$ be the part of the paraboloid $ z = 4 - x^2 - y^2$ with $z \geq 0 $, oriented with $\vec{n}$ upwards. Calculate the flux integral $\int_S \vec{F} \cdot d\vec{S}$ using
i) Cartesian coordinates
ii) cylindrical coor... |
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... |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut... |
I have been reading a recent paper. In it, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes. To simplify the situation, let us suppose that the liquid between the electrodes is argon liquid.
The system has a "slab"... |
OBD Reasoner
The OBD reasoner uses definitions of transitive relations, relation hierarchies, and relation compositions to infer implicit information. These inferences are added to the OBD Phenoscape database. This section documents the inherited code in Perl and embedded SQL, that extracts implicit inferences from the... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.