text stringlengths 256 16.4k |
|---|
$F$ is $m\times m$ diagonal, with real non negative elements
$D$ is $n \times m$ complex
$P$ is $n \times 1$ complex
$A$ is $m \times 1$ complex.
Minimize $\Gamma(A)$, with respect to $A$.
$$\Gamma(A) = \frac{m^2(DA-P)^H (DA-P) + (FA)^H(FA)}{A^HA}$$
It is known that both numerator and denominator of $\Gamma(A)$ are con... |
Without any loss of generality we can draw a tangent to a circle with centre $C\equiv(C_x,C_y)$ and radius $r$ and choose a point$P\equiv(P_x,P_y)$ on the tangent so that its distance from the origin is$\overline{CP}\equiv d$ with $d \ge r$.
From the sketch above we recognize that the tangency points can bedescribed as... |
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... |
I know special relativity says that traveling at high speeds (or really any speed) causes time dilation; and General relativity says that gravity also causes time dilation. I was wondering if relative time dilation (where two observers each measure the other's time to be slow) was caused not by time dilation, but inste... |
We present the first observation of exclusive $e^+e^-$ production in hadron-hadron collisions, using $p\bar{p}$ collision data at \mbox{$\sqrt{s}=1.96$ TeV} taken by the Run II Collider Detector at Fermilab, and corresponding to an integrated luminosity of \mbox{532 pb$^{-1}$}. We require the absence of any particle si... |
Defining parameters Dimensions
The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(80))\).
Total New Old Modular forms 57 15 42 Cusp forms 1 1 0 Eisenstein series 56 14 42
The following table gives the dimensions of subspaces with specified projective image type.
Decomposition of \(S_{1}^{... |
I'm reading "Microelectronics" by Millman, Grabel.
Premise:
The book developed, for a pn step-graded junction, the contact potential as $$ V_0 = V_T \text{ln} \frac{p_{p0}}{p_{n0}} = - V_T \text{ln} \frac{n_{p0}}{n_{n0}} \qquad (1) $$ where $p_{p0}$ is the hole concentration in the p-side, $p_{n0}$ is the hole concentr... |
I am trying to come to grip with some solid state theory. Bloch waves, energy eigenstates for hamiltonians with lattice periodic potential in $\mathbb R^d$, are frequently written as$$\phi_{n,k}(r)=e^{2\pi ik\cdot r}u_{n,k}(r)$$ with $u_{n,k}$ periodic in lattice vectors $R\in\mathbb R^d$. This is 1 version of Bloch's ... |
A lens has two surfaces. One of these surfaces still has refracting power, due to the difference in refractive index and angle of incidence.
Refraction on one surface
Let parallel rays start in glas. Radius sign convention results $R=-200\,mm$ since incident light is on same side as center of curvature.
Radius of curva... |
The problem is described as follows:
I need to generate a basis(matrix) in lexicographic order.
For two different basis vector $\{n_1,n_2,\cdots,n_M\}$ and $\{t_1,t_2,\cdots,t_M\}$, there must exist a certain index $1\leq k \leq M-1$ such that $n_i=t_i$ for $1\leq i \leq k-1$, while $n_k \neq t_k$. We say $\{n_1,n_2,\c... |
As far as I understand, Noether's theorem for fields works, as explained in David Tong's QFT lecture notes (page 14) for example, by saying that a transformation $\phi(x) \mapsto \phi(x) + \delta \phi (x)$ is called a symmetry if it produces a change in the Lagrangian density which can be expressed as a four divergence... |
I am self learning GR. This is a rather long post but I needed to clarify few things about the effect of general coordinate transformations on the global symmetries of metric. Any comments, insights are much appreciated.
To be concrete let's consider that $g_{\mu\nu}$ represents an axially symmetric spacetime i.e a Ker... |
The maximum height that a man can jump is a bit over 70cm for the very best athletes. This is because the only thing that matters is how high your centre of gravity rises. I will assume 80cm.
Work is force times distance. Weight is the force $W = gm_0 $ and when you jump a height $h$ under Earth gravity of 9.8m/s with ... |
Krishnan, R and Seshadri, TP (1990)
Structure of dipotassium galactose 1-phosphate pentahydrate. In: Acta Crystallographica, Section C: Crystal Structure Communications, 46 (12). pp. 2299-2302.
PDF
Structure_of_dipotassium_galactose_1-phosphate_pentahydrate.pdf
Restricted to Registered users only
Download (413kB) | Req... |
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 Finding Other Symbols
Here are some external resources for fin... |
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... |
According the work by Wang & Zhu, on toric Fano manifolds there exist Kaehler-Ricci solitons. If Futaki=0, there also exist CSCK metrics. But if the Futaki invariant does not vanish, what about extremal metrics? Is there some counterexample?
Here is an example where the Futaki invariant does not vanish, but with extrem... |
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:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
I recently started to study maths at university and in the analysis course we started, as usual, by looking at the axioms of $\mathbb R$ as a field. I think, I've understood the underlying intuition of these axioms quite well, but one question remained open for me:
How do we define, what we do when we add two numbers, ... |
As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\frac{1}{ad - bc} \begin{pmatrix} d & -b \\\ -c & a \end{pmatrix}$. This can be proved, for example, by writing the inverse as $ \begin{pmatrix} r & s \\\ t & u \end{pmatrix}$ and solving the resu... |
I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ (although in terms of different variables than $y$ and $x$). E.g., they might write something like
$$ x... |
A new Higgs tadpole cancellation condition reformulating the hierarchy problem Strings 2013 [talks] is underway.The first hep-ph paper today probably got to that exclusive place because the authors were excited and wanted to grab the spot. Andre de Gouvea, Jennifer Kile, and Roberto Vega-Morales of Illinois chose the t... |
Probabilists often work with Polish spaces, though it is not always very clear where this assumption is needed.
Question: What can go wrong when doing probability on non-Polish spaces?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.Sign up to join this comm... |
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \... |
The idea of a finite automaton is that each state stands for a combination of following kinds of properties.
properties of the whole string, which basically determine whether the string should be accepted or not. what are the last several letters of a string, which basically determine the transitions between the states... |
So I'm reading Dummit and Foote and they define the discriminant of $x_{1},...,x_{n}$ by $$D=\prod_{i<j}(x_{i}-x_{j})^2$$ and the discriminant of a polynomial to be the discriminant of the roots.
They say that a permutation $\sigma \in S_{n}$ is in the alternating group $A_{n}$ iff $\sigma$ fixes the product $\sqrt{D}$... |
I have $$F(z) = \phi + i\psi$$
Also, it is given that $$u = \frac{\delta \phi}{\delta x}, \ \ v = \frac{\delta\phi}{\delta y}$$ and $$u = \frac{\delta \psi}{\delta x}, \ \ v = -\frac{\delta\psi}{\delta y}$$
Now I have to prove that $$\frac{dF}{dz} = u - iv$$
But I am getting terms like $\dfrac{dx}{dz}$ and $\dfrac{dy}{... |
There is in fact a whole bunch of related algorithms. In modern context I believe "time warp" would be called sequence alignment. Depending on whether you want to match complete strings or optimal substrings one gets Needleman-Wunsch and Smith-Waterman.In your latter algorithm the costs seem to vary, that is one can at... |
Any set of positive Lebesgue measure, have a Lebesgue measurable proper subset with the same measure?
For example: if $A$ is a measurable set, with $m(A) > 0$,
I can say that $x \in A$ (since $A$ is not empty because $m(A) > 0$), then can I say that $B = A\setminus \{x\}$ is a measurable subset of $A$ and $m(B) = m(A)$... |
I am working on a problem to prove, but I do not understand it completely. Where should I use inductive method? What is the base case? And so on. Here is my problem:
A truth assignment $M$ is a function that maps propositional variables to $\{0, 1\}$ ($1$ for true and $0$ for false). We write $M\vDash x$ if $x$ is true... |
First note that in the special case $\kappa = \omega_1$, countably closed implies countably directed-closed.
Answer 1: No. In any model of CH, there is a countably closed, $\omega_2$-c.c. forcing that is inequivalent to $\mathbb P = \text{Add}(\omega_1,\omega_2)$.
Proof: Let $\mathbb Q$ be Jensen's partial order for ad... |
Take $p$ to be a prime and let $a_1,\dots,a_n\in\mathbb Z$ be some set of integers such that discrepancy of the set of fractional parts $$\{\frac{ma_1}p,\dots,\frac{ma_n}p\}$$ with $m\in\{1,\dots,p-1\}$ being $\mathcal D$.
Then we know there is an integer $t\in\{1,\dots,p\}$ such that the vectors $$\{\frac{ta_1}p,\dots... |
This post is a sequel to Concatenate and average NetCDFs where I introduced NCO (NetCDF operators), how to install NCO, and provided examples for the essential operations of record and ensemble concatenation and averaging of variables across multiple NetCDF files. If you are new to NCO I suggest familiarizing yourself ... |
I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke:
Teorem. Let $f : U ⊂ \mathbb{R}^n → \mathbb{R}^n$ be Lipschitz, and suppose that every matrix $A ∈ δ_{x_0}f$ is invertible. Then there are neighbourhoods ... |
Let \(A\) be a \(2\times 2\) matrix. Prove that if \(Av_1=\lambda_1v_1\) and \(Av_2=\lambda_2v_2\) for distinct reals \(\lambda_1\) and \(\lambda_2\) and nonzero vectors \(v_1\) and \(v_2\), then both columns of \(A-\lambda_1 I\) is a multiple of \(v_2\).
GD Star Rating loading...
Let \( H \) be an \( N \times N \) rea... |
On asymptotically lacunary statistical equivalent functions via ideals Authors
Ekrem Savaş - Department of Mathematics, Usak University, Usak, Turkey. Abstract
The goal of this paper is to introduce \(\mathcal{I}_\theta\)-asymptotically statistical equivalent by taking nonnegative two real-valued Lebesgue measurable fu... |
This is an improved version of the argument in Electromagnetic Unruh effect?
In the quantum vacuum particle pairs, with total energy $E_x$, can come into existence provided they annihilate within a time $t$ according to the uncertainty principle $$E_x\ t \sim \hbar.$$ If we let $t=x/c$ then we have $$E_x \sim \frac{\hb... |
Kitefoil introduction
Foil is a technology that allows a hull (propelled by a motor or in this case a sail) to
emerge totally from the water, thanks to the hydrodynamic action of the submerged surface.
In fact, the pressure of the water under the
wings, combined with the depression that forms above them, generates a fo... |
I don't know much of set theory, but is this mixed set allowed in this context?
There's an axiom of set theory that, for any two sets $A,B$ then $A \cup B$ is also a set. So it's definitely defined.
When does NFA ε-transition happen?
Try not to think of NFAs as actual machines, but instead focus on their mathematical d... |
I'm currently trying to minimize the following function: $$\min_{\mathbf x,\mathbf y,s} \|F\mathbf x-\mathbf a\|_2^2 + \|SF\mathbf y-\mathbf b\|_2^2 + \lambda \|\max(\mathbf x, \mathbf y)\|_1,$$ $$\mathbf x>\mathbf 0, \mathbf y>\mathbf 0.$$ Here $F$ and $S$ are both linear operators. $F$ is a discrete Fourier transform... |
Given the advection equation for an incompressible flow field $$\frac{\partial c}{\partial t} + \mathrm{Pe} \frac{\partial c}{\partial x} = 0$$
what would the best method be for discretizing this without introducing any numerical diffusion or oscillations? Specifically when we have step changes in boundary conditions, ... |
How to Model Optical Anisotropic Media with COMSOL Multiphysics®
On a bright evening in 1669, Professor Erasmus Bartholinus looked through a piece of an Icelandic calcite crystal he had placed onto a bench. He observed when he covered text on the bench with the stone, it appeared as a double image. The observed optical... |
Consider
$$\frac{a}{a}\pmod a,\ \ \ a\in\mathbb Z\setminus\{-1,0,1\}$$
There are two cases:
$1)$ $\frac{a}{a}$ is the notation for the real number $1$.
Then the expression is equivalent to $1$ modulo $a$.
E.g., see this, where the expression on the LHS would not even exist if what would be meant by $99$ in the denomina... |
For "attenuation coefficient" as it applies to electromagnetic theory and telecommunications see propagation constant. For the "mass attenuation coefficient", see the article mass attenuation coefficient.
The
attenuation coefficient is a quantity that characterizes how easily a material or medium can be penetrated by a... |
Electroproduction of the omega meson was investigated in the p(e,e'p)omega reaction. The measurement was performed at a 4-momentum transfer Q2 ~ 0.5 GeV2. Angular distributions of the virtual photon-proton center-of-momentum cross sections have been extracted over the full angular range. These distributions exhibit a s... |
Your loss function would not work because it incentivizes setting $\theta_1$ to any finite value and $\theta_0$ to $-\infty$.
Let's call $r(x,y)=\frac{1}{m}\sum_{i=1}^m {h_\theta\left(x^{(i)}\right)} -y$ the
residual for $h$.
Your goal is to make $r$
as close to zero as possible, not just minimize it. A high negative v... |
Using this theorem:
Let $(a_k)$ be a sequence in $\Bbb R$ and let $x_0\in \Bbb R$:
The power series: $$\sum_{k=0}^\infty a_k(x-x_0)^k \qquad \text{and} \qquad \sum_{k=0}^\infty ka_k(x-x_0)^{k-1} $$ converge uniformly in $[x_0-r,x_0+r]$ for all $r\in (0,R)$ where $R$ is the radius of convergence of $\sum_{k=0}^\infty a_... |
Suppose after some calculations I arrive at an equation of the form: \begin{equation} [\nabla^2 +V(\mathbf{r})]\psi(\mathbf{r})=k^2\psi(\mathbf{r}) \tag{1} \end{equation} As you can see here the Laplacian has the same sign as the eigenvalue. The eigenvalue $k^2$ will then be used to determine an energy defined as $E=-\... |
I am studying how to get the normalization factor algebraically in the exited states of the harmonic oscillator using the beautiful ladder operators technique.
I am stuck at the point where is stated that the product of the ladder operators yields the energy levels (in $\widehat{a}^\dagger\widehat{a}$ case) and the ene... |
Difference between revisions of "Image Dimensions"
(Created oage, added disclaimer)
(Added a bit of description of the properties dialog)
Line 1: Line 1:
<div style="background-color:#DDFFDD; border:thin solid green; padding:1em">
<div style="background-color:#DDFFDD; border:thin solid green; padding:1em">
'''Disclaime... |
Why do we choose these numbers and not for e.g $e$ and $\pi$ for success and failure in an experiment?
What is the logic and how badly would it affect my calculations if I choose other values instead of 0 and 1?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data an... |
This is a post regarding the paper Efficient Preconditioning for the $p$-Version Finite Element Method in Two Dimensions. In Lemma 3.3, the basis required is the bubble (or interior) polynomials, the set of edge functions orthogonal to the interiors, and linear (of bilinear) functions.
The edge functions orthogonality ... |
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... |
Hey guys! I built the voltage multiplier with alternating square wave from a 555 timer as a source (which is measured 4.5V by my multimeter) but the voltage multiplier doesn't seem to work. I tried first making a voltage doubler and it showed 9V (which is correct I suppose) but when I try a quadrupler for example and t... |
This article will be permanently flagged as inappropriate and made unaccessible to everyone.
Are you certain this article is inappropriate?
Excessive Violence
Sexual Content
Political / Social
Email Address:
Article Id:
WHEBN0000098663
Reproduction Date:
Orbital elements are the parameters required to uniquely identify... |
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... |
Here is a possible suggestion how such an approach might work. The mechanism follows from rather general considerations and does not require quantum mechanics. So I'll use a more general mathematical notation.
Imagine I want to know the time evolution $f(x,t)$ when I only have knowledge of the initial condition $f(x,0)... |
Set Difference is Subset
Jump to navigation Jump to search
Theorem $S \setminus T \subseteq S$
\(\displaystyle x \in S \setminus T\) \(\leadsto\) \(\displaystyle x \in S \land x \notin T\) Definition of Set Difference \(\displaystyle \) \(\leadsto\) \(\displaystyle x \in S\) Rule of Simplification
The result follows fr... |
Learning Objectives
Draw and interpret scatter diagrams. Use a graphing utility to find the line of best fit. Distinguish between linear and nonlinear relations. Fit a regression line to a set of data and use the linear model to make predictions.
A professor is attempting to identify trends among final exam scores. His... |
When does $\int_Af(x,y)dA$ represent a surface area geometrically, and when does it represent a volume? In my lecture notes I'm told it represent the volume underneath the surface $z=f(x,y)$, but I've found examples online computing double integrals to find the surface area of a surface. Thanks a bundle
$\iint_S f \ope... |
Let $(X_1, d_1), (X_2, d_2)$ be metric spaces and define $(X_1 \times X_2, d)$, $(X_1 \times X_2, p)$ where $d[(x_1, x_2),(y_1, y_2)] = d_1(x_1, y_1) + d_2(x_2,y_2)$ and $p[(x_1, x_2),(y_1, y_2)] = \max\{d_1(x_1,y_1), d_2(x_2,y_2)\}$
My attempt:
Notice that $p[(x_1, x_2),(y_1, y_2)] \leq d[(x_1, x_2),(y_1, y_2)]$, that... |
Are the Milnor's seven dimensional exotic spheres parallelizable?
A much more general result is true.
Theorem: Let $\Sigma$ be a homotopy sphere and $f: S^n \to \Sigma $ be a homotopy equivalence. Then $f^{\ast} T \Sigma \cong T S^n$.
It says that exotic spheres cannot be distinguished by looking at the tangent bundle.... |
"Definition 5 Let X be a nonempty set. By a partician P of X we mean a set of nonempty subsets of X such that:
(a) If A, B$\in$P and A$\neq$B, then A$\bigcap$B=$\emptyset$ (b) $\bigcup \limits_{C \in P}C$ = X"
Source: Set Theory by You-Feng. Lin and Shwu-Yeng T. Lin
"3.7 Definition Let A be a set; by
a partition of A w... |
My question revolves around finding a function based on its derivative of the type below :
Problem :The limit below represents the derivative of some real-valued function $f$ at some real-number $a$. State such an $f$ and $a$ in this case.
$$\lim_{h \ \to \ 0} \frac{\sqrt{9+h}-3}{h}$$
Now this type of problem is slight... |
Transfinite Induction/Principle 1 Theorem
Let $A$ denote a class.
Suppose that: For all elements $x$ of $\operatorname{On}$, if $x$ is a subset of $A$, then $x$ is an element of $A$. Then $\operatorname{On} \subseteq A$. Proof This page is beyond the scope of ZFC, and should not be used in anything other than the theor... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
I have the following velocity vector (in spherical polars): \begin{equation} \textbf{v} = u \hat{\textbf{r}} + v_{\phi}\hat{\boldsymbol\phi} \end{equation} Where $u(r) = u$ and $v_{\phi} (r) = v_{\phi}$. $\theta =0$ in this system.
I am now required to prove the following: \begin{eqnarray} \rho(\textbf{v}\cdot\nabla)\t... |
The XTT^2 ALSV(FD) Specification
Author: Grzegorz J. Nalepa, based on the work with Antoni Ligęza
Version: Draft 2008Q3 Introduction to Attributive Logics
Attributive logics constitute a simple yet widely-used tool for knowledge specification and development of rule-based systems.
In fact in a large variety of applicat... |
The boiling points of group 15 elements increase on going down the group (or, as size increases) but the same is not true for the melting points. The melting points increase from $\ce{N}$ to $\ce{As}$ and then decrease from $\ce{As}$ to $\ce{Bi}$. Why is this the case?
A lower melting point is an indication of a lesser... |
I can turn-the-crank and show that $\frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0$ etc, but what would be a strategy to proving the general statement for spin representations that $j\otimes s =\bigoplus_{l=|s-j|}^{|s+j|} l$.
It's easy to prove the formula if you just look at the individual basis vectors of the tensor prod... |
The book EXACT SPACE-TIMESIN EINSTEIN’SGENERAL RELATIVITY by Podolsky and Griffiths has a section on Taub-Nut space-time metrics and there is defines the singularity made in the Taub metric as quasi-regular singularity $$ \mathrm{d} s^{2}=-f(r)\left(\mathrm{d} t+4 l \sin ^{2} \frac{1}{2} \theta \mathrm{d} \phi\right)^{... |
The production of J/ψ mesons in proton-proton collisions at a centre-of-mass energy of $ \sqrt{s}=13 $ TeV is studied with the LHCb detector. Cross-section measurements are performed as a function of the transverse momentum p$_{T}$ and the rapidity y of the J/ψ meson in the region p$_{T}$ < 14 GeV/c and 2.0 < y < 4.5, ... |
Consider the flow $\vec u=(uy,0,0)$ between two plates $y=0$ and $y=1$ (chosen out of simplicity). I want to find the stress tensor of such a flow given by: $$\sigma_{ij}=-\delta_{ij}p+\eta \left( \frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_k}\right)$$ Although this seems quite easy (e.g. use Navie... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
This question appears somewhat similar to other questions asking about why wavelength affects diffraction (a concept which I'm still not 100% sure on...) however my query is different and not answered that I can find. (The focus of my question is to what degree slit size affects diffraction in terms of the wavelength, ... |
Critical superlinear Ambrosetti-Prodi problems
DOI: http://dx.doi.org/10.12775/TMNA.1999.022
Abstract
We consider the existence of multiple solutions for problem (1.1) below
with either $\lambda\neq \lambda$ or $\lambda=\lambda_1$, where $\lambda_k$, $k=1,2,\ldots$ are eigenvalues of $(-\Delta, H^1_0(\Omega))$. The loc... |
The approach is wrong because you don't say which random variable satisfies which binomial distribution. The answer is that the number $X$ of bits that differ between the two strings satisfies $B(n, \frac{1}{2})$. Hence the probability is$$P(X=d)=\binom{n}{d}(1/2)^d (1/2)^{n-d} = \frac{1}{2^n}\binom{n}{d} $$
Edit As yo... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
I'm a newbie reading quantum mechanics from "Inroduction to Quantum Meachanics" by Griffiths and in the early pages of the book the author defines:
$$\langle x\rangle =\int_{-\infty}^{\infty} x|\Psi(x,t)|\,dx = \int_{-\infty}^{\infty} \Psi^* (x)\Psi \,dx,$$
$$\langle v\rangle = \frac{d}{dt}\left(\langle x\rangle\right)... |
I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how they are unique from each other in any physical sense (other than their difference in transformations). Is there any phys... |
I know how to combine two observed values with different standard deviations into one mean value:
$$ \left(\frac{\sigma_b^2}{\sigma_a^2+\sigma_b^2}\right)x_a + \left(\frac{\sigma_a^2}{\sigma_b^2 + \sigma_a^2}\right)x_b = \mu $$
With a new standard deviation of:
$$ \frac{1}{\sigma_{new}} = \frac{1}{\sigma_a} + \frac{1}{... |
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... |
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... |
A cool generalization of Rédei's method furnishes the following lower bound:
Theorem-lower bound.Let $U\subset \mathbb F_p$ have cardinality $2p$. Then $U$ is contained in the union of two lines or it determines at least $\lfloor\frac{p+4}{3}\rfloor$ rich directions.
The key idea is to consider the
y-derivative of the ... |
A good theoretical analysis of with and without replacement schemas in the context of iterative algorithms based on random draws (which are how many discriminative Deep Neural Networks (DNNs) are trained against) can be found here
In short, it turns out that sampling
without replacement, leads to faster convergence tha... |
Learning Outcomes
Solve linear equations for the variable.
It is a common task in algebra to solve an equation for a variable. The goal will be to get the variable on one side of the equation all by itself and have the other side of the equation just be a number. The process will involve identifying the operations that... |
A Fourth-Order-Centered Finite-Volume Scheme for Regular Hexagonal Grids
Related Articles Dust Lofting and Ingestion by Supercell Storms. Seigel, Robert B.; van den Heever, Susan C. // Journal of the Atmospheric Sciences;May2012, Vol. 69 Issue 5, p1453
Recent research pertaining to aerosol impacts on cloud microphysics... |
Article
Keywords: ${\omega }$-covering set; ${\mathcal E}$; hereditarily nonparadoxical set
Summary: We prove the following theorems: There exists an ${\omega }$-covering with the property $s_0$. Under $\mathop {\mathrm cov}\nolimits ({\mathcal N}) = $ there exists $X$ such that $ \forall _{B \in {\mathcal B}or} [B\cap... |
Suppose I take the $0$-skeleton $X_0$ to have a single inhabitant $base$.
Let $S_1$ have a single point $base'$, with attaching map $f(base', -)=base$. If I construct the pushout to the 1-skeleton $X_1$, as below, $$ \require{AMScd} \begin{CD} S_1 \times \mathbb{S}^0 @>{f}>> X_0;\\ @VVV @VVV \\ \mathbf{1} @>>> X_1; \en... |
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... |
My best guess is that the series $$ \sum_{i=1}^n i(n-(i-1)) $$ becomes $$ 2 \Bigg[ n + 2(n-1) + ... + \frac{n}{2} \bigg(n-\bigg(\frac{n}{2}-1\bigg)\Bigg)\Bigg] $$ So the highest term is $n^2$ and there are $n$ terms. Does that mean its $O(n^3)$? That seems high. Intuitively, it seems like it should be closer to $O(n^2)... |
Given a real symmetric matrix $M$, ostensibly of "low rank", efficiently find an expression $M = \sum \alpha_i u_i u_i^T$ using the number of terms rank($M$).
A 2011 StackOverflow Question Dense Cholesky Update in Python asked about doing low rank updates to Cholesky decompositions, but was answered largely with ways t... |
I am trying to show the asymptotic expansion for $$\sum_{n\le x}\frac1{\sqrt n}=2\sqrt x+\zeta(1/2)+O(x^{-1/2}).$$
(The exact identity of the zeta term is not important, it need only be some $c$.) To that end, I am attempting to prove the following slightly stronger theorem, which is supported by numerical evidence:
Le... |
To solve the Poisson equation for the Newton Potential, say $\phi$, one can use the divergence theorem, such that $$\int_U \nabla^2 \phi \sqrt{g}~ \mathrm dV= \int_{\partial U} \langle \nabla \phi,n\rangle \sqrt{g_\Sigma}~ \mathrm d\Sigma=1,$$ where $\sqrt{g_\Sigma}$ represents the induced metric on the border.
Conside... |
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... |
In econometrics, fixed effects binary choice models are important tools for panel data analysis. Our package provides an approach suggested by Stammann, Heiss, and McFadden (2016) to estimate logit and probit panel data models of the following form:
\[ y_{it} = \mathbf{1}\left[\mathbf{x}_{it}\boldsymbol{\beta} + \alpha... |
The Numerical Aperture (NA) (for fiber optics) is usually used to denote the acceptance cone for a multi-mode fiber.
Does NA also describe the expansion of light emitted from the end of a fiber?
I have a 1 mm core 0.22 NA PMMA fiber, I would like to collimate the light once it's reached a 1.5" diameter.
$$ n\sin(\theta... |
Regression (OLS) - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)
$z$ test for the diff... |
For certain vector fields, the amount of work required to move a particle from one point to another is dependent only on its initial and final positions, not on the path it takes. Gravitational and electric fields are examples of such vector fields. This section will discuss the properties of these vector fields.
Defin... |
just a few remarks:
note: i use the standard abbreviation $(a,b)$ for the GCD of $a$ and $b$
a. it is possible to approach a problem
in the wrong way. occasionally this may lead to startling insight, but usually it leads to a waste of time, and can even be demotivating. here, OP's introduction of:
$$(a,b) = ax+by$$sugg... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.