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The Annals of Probability Ann. Probab. Volume 25, Number 4 (1997), 1621-1635. Strong approximation theorems for geometrically weighted random series and their applications Abstract
Let ${X_n;n\geq 0}$ be a sequence of random variables. We consider its geometrically weighted series $\xi(\beta)=\sum_{n=0}^\infty \betaX_n... |
The Kunen inconsistency The Kunen inconsistency, the theorem showing that there can be no nontrivial elementary embedding from the universe to itself, remains a focal point of large cardinal set theory, marking a hard upper bound at the summit of the main ascent of the large cardinal hierarchy, the first outright refut... |
The Erdos-Rado sunflower lemma Contents The problem
A
sunflower (a.k.a. Delta-system) of size [math]r[/math] is a family of sets [math]A_1, A_2, \dots, A_r[/math] such that every element that belongs to more than one of the sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that Erdos-Rado... |
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Integration by PartsIntegration by Parts
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Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
Classification trees are used, as the name suggests, in solving classification problems. Here are some definitions and Matlab tips to help you dabble in this subject.
The objective of any problem of this nature is to assign an object to one of a number of specified categories or classes. Classification is a useful tech... |
Research Open Access Published: Iterative solution to singular nth-order nonlocal boundary value problems Boundary Value Problems volume 2015, Article number: 125 (2015) Article metrics
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Abstract
By using the cone theory and the Banach contraction mapping principle, we study the ex... |
Both portfolio valuation and management have many facets, but one they have in common is attribution, that is, how much each asset or collection of them is contributing to the return. Thus, in these series of posts, we are going to wax technical about the computation of asset exposure. We will show how an initially bal... |
Banach Journal of Mathematical Analysis Banach J. Math. Anal. Volume 9, Number 3 (2015), 248-260. A Hilbert space approach to approximate diagonals for locally compact quantum groups Abstract
For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only i... |
The speed of light in a material is $c = 1/\sqrt{\epsilon\mu}$. Very slow light therefore means either a very high permittivity (high $\epsilon$) or a very high permeability (high $\mu$).
High $\epsilon$ means high electric polarizability, which implies high van-der-Waals force. I highly doubt that a substance with suc... |
Is it possible to draw the path of a point on the boundary while the polygon is rolling, or writing a function of theta similar to the cycloid parametric equations:
$$ x = r (\theta - \sin(\theta)),\quad y = r (1 - \cos(\theta) $$
My main goal is to generalize the problem to write the parametric equations for the cyclo... |
The BCS wave function you write down depends on a parameter $\phi$, but the ground state energy is independent of it. This implies that $\phi$ is the (would be) Goldstone mode that governs the low energy dynamics of the system. The gradient of $\phi$is the conserved $U(1)$ current $\vec\jmath \sim \vec\nabla\phi$, and ... |
Functiones et Approximatio Commentarii Mathematici Funct. Approx. Comment. Math. Volume 49, Number 2 (2013), 229-240. Some elementary explicit bounds for two mollifications of the Moebius function Abstract
We prove that the sum $\sum_{\left\{d\le x, (d,r)=1\right.}\mu(d)/d^{1+\varepsilon}$ is bounded by $1+\varepsilon$... |
Astrid the astronaut is floating in a grid. Each time she pushes off she keeps gliding until she collides with a solid wall, marked by a thicker line. From such a wall she can propel herself either parallel or perpendicular to the wall, but always travelling directly \(\leftarrow, \rightarrow, \uparrow, \downarrow\). F... |
Homework Helper
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Ok so I've got a question after walking through the time dilation derivation that used 'light clocks' (think a beam of light bouncing back and forth between mirrors) to derive ##\delta t^\prime = \frac{\delta t}{\sqrt{1-\frac{v^2}{c^2}}}##. So my Q is could you derive the same equation if you ha... |
Porto and the nearby Douro Valley are famous for producing port wine. Wine lovers from all over the world come here to enjoy this sweet wine where it is made. The International Consortium of Port Connoisseurs (ICPC) is organizing tours to the vineyards that are upstream on the Douro River. To make visits more pleasurab... |
It is often useful to know what shape your data takes. Linear regression is one example, where you want to find out how well your data fits a straight line. Knowing that your data fits a straight line reasonably well gives you an advantage for understanding your data. For instance, you know that your data holds a linea... |
I'm trying to find solutions for the Poisson equation under Neumann conditions, and have a couple of questions. More specifically, I'm interested in the gradient of the function $\phi(x)$ in a space $\Omega \subset \mathbb{R}^d$. (note that I'm only interested in the gradient. For my problem I do not care about $\phi(x... |
So, I'm doing an extensive homework of electromagnetism and we are searching for the total electromagnetic angular momentum of the Thomson dipole. In the end, there is one integral we cannot solve. By '"we" I mean the entire class: nobody is getting how to solve it. Professor swears there are no expansions or approxima... |
Yes, such a directed graph exists, with a "quickly" computable path of length $O(\log|n-m|)$ from $m$ to $n$ for any distinct integers $m,n$.
I'll first construct a digraph with this property but with each vertex having out-degree at most $8$, then show how to get from it a digraph of out-degree $2$.
Denote by $v(x)$ t... |
Search response: 6136 publications match your query. Listing starts with latest publication first: (1 - 10) MPP-2019-203 Determination of jet calibration and energy resolution in proton-proton collisions at $\sqrt{s}$ = 8 TeV using the ATLAS detector, ATLAS Collaboration,, arxiv:1910.04482 (abs), (pdf), (ps), CERN-EP-2... |
NDSolve can easily handle the PDE system $$\partial_t y = \partial_z w \quad\quad \partial_t w = \partial_z y $$ along with initial-boundary conditions $$w(t,0)=w(t,-1)=0\quad\quad w(0,z)=-sin^2(z\pi)\quad\quad y(0,z)=1$$ for $(t,z)\in[0,1]\times[-1,0]$.
Lets call this problem A.
The above PDE system can be altered as ... |
Home
Integration by PartsIntegration by Parts
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Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
Unless I'm overlooking something:
The groups must be conjugate in $\operatorname{Aut}(\mathbb{H})$.
Say we have two hyperbolic surfaces $S,\, T$ with their universal covers $\pi_S,\, \pi_T$ and a biholomorhism $\varphi \colon S \to T$. Choosing a point $s \in S$ and $\sigma \in \mathbb{H}$ above $s$, and a point $\tau ... |
Existence of sign changing solutions for some critical problems on $\mathbb R^N$
1.
Department of Mathematics, Yokohama National University, 156 Tokiwadai Hodogaya-ku - Yokohama, Japan
2.
Dipartimento di Matematica Applicata "U.Dini", Università di Pisa, Via Bonanno 25B - 56126 Pisa, Italy
3.
Dipartimento di Metodi e M... |
Difference between revisions of "Fujimura's problem"
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== Computer data ==
== Computer data ==
Revision as of 04:22, 7 March 2009
Let [math]\overline{c}^\mu_n[/math] the largest subset of the triangular grid
[math]\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3:... |
I raised a related question but hope to get some answer using the nonvanishing 2 form definition.
Let P be the real projective plane obtained by identifying antipodal points on the unit sphere of $R^3$.
How to prove that P is nonorientable by showing that any 2 form on P will vanish somewhere?
My idea is to consider th... |
Rankings are everywhere. They are sometimes useful and, at other times, contradicting. In such a case, we need to come up with a consensus ranking but… how do we
evaluate ranking consensus?
The other day I was reading about something called
rank aggregation, which is just a fancy name for combining preferences expresse... |
Astrid the astronaut is floating in a grid. Each time she pushes off she keeps gliding until she collides with a solid wall, marked by a thicker line. From such a wall she can propel herself either parallel or perpendicular to the wall, but always travelling directly \(\leftarrow, \rightarrow, \uparrow, \downarrow\). F... |
Infinitary logic
Infinitary logic is a type of logic which is used in the standard characterizations of several large cardinals, such as weakly compact cardinals and strongly compact cardinals. It also is used in alternate characterizations of other large cardinals such as supercompact cardinals and extendible cardinal... |
Beatty Sequences
Let $r$ be a positive irrational number. Set $s = 1/r,$ and define two sequences: $a_{n} = n(r + 1)$ and $b_{n} = n(s + 1)$, $n \gt 0.$
Obviously, all terms of both sequences are irrational. In particular, none of them is integer. A remarkable theorem discovered by Sam Beatty in 1926 states that, for a... |
Does anyone know of a reference for the following fact?
Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ denote the universal abelian variety. For any basepoint $[C] \in M_g$, there is a representation $\pi_1(M_g, [C]) \rightarrow... |
Divisibility Criteria
Divisibility criteria are ways of telling whether one number divides another without actually carrying the division through. Implicit in this concept is the assumption that the criteria in question affords a simpler way than the the outright division to answer the question of divisibility. Divisib... |
First of all, we should understand what the
R software is doing when no interceptis included in the model. Recall that the usual computation of $R^2$when an intercept is present is$$R^2 = \frac{\sum_i (\hat y_i - \bar y)^2}{\sum_i (y_i - \bary)^2} = 1 - \frac{\sum_i (y_i - \hat y_i)^2}{\sum_i (y_i - \bary)^2} \>.$$The ... |
This post contains the notes taken from the following paper:
Deep Learning Scaling Is Predictable, Empirically by Baidu Research.
The last years in Deep Learning have seen a rush to gigantism:
Models are becoming deeper and deeper from the 8 layers of AlexNet to the 1001-layer ResNet. Training on large dataset is way q... |
Caesium has a larger size, and the effective nuclear charge that the valence electron experiences will be far less compared to that of lithium's, right? But lithium is still considered the strongest reducing agent among all the alkali metals, and this is evidenced by its large and negative reduction potential. Why is t... |
LR Parsing OverviewThere are several different kinds of bottom-up parsing.We will discuss an approach called
LR parsing, whichincludes SLR, LALR, and LR parsers.LR means that the input is scanned left-to-right, andthat a rightmost derivation, in reverse, is constructed.SLR means "simple" LR, and LALR means "look-ahead"... |
Survivorship bias is one of the most common biases in finance, and it’s easy to fall victim to it. Let’s find out how to remain vigilant and overcome this hurdle.
“History is written by the victors”. – Winston Churchill
A cognitive bias is a consequence of subjective judgement. When it comes to finance, would you say t... |
As Kimball mentioned, this is all in Godement-Jacquet's monograph "Zeta Functions of Simple Algebras". For the case $n = 1$, this is just Tate's thesis. When the field is $\mathbb{Q}$, a good reference is Goldfeld-Hundley "Automorphic Representations and $L$-Functions for the General Linear Group".
The story is roughly... |
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... |
For my field theory class I am trying to build the Lagrangian for the following system. Consider a 2D square lattice where the nearest and next-nearest neighbor interactions are modeled by springs with spring constant $C$. Let $u_1(\mathbf{x},t)$ and $u_2(\mathbf{x},t)$ be the displacement fields in the $x_1$ and $x_2$... |
Current through a LR circuit is given by$$i=i_0 (1-e^{\frac{-tR}{L}})..1$$From here I can observe that as I tend R towards 0 no current flows through the circuit , and current will be 0 when the resistance in circuit actually becomes 0.
Though for an A.C. LR circuit I have seen my book presenting a proof where there is... |
Problem statement
Let $P$ be a probability measure on the positive real line and assume all it's
raw moments, $\mu_k = \mathbb{E}[x^k]$, $k=1,2,\dots$ exist and $\mu_k < \infty$ for all $k$. Let $\mu = \mu_1$ be the mean of $P$ and assume $\mu > 0$.
Does the following inequality hold? $$ 4 \frac{\mu_3}{\mu^3} + 6 \frac... |
Hey there. I hope you are doing well. In my last post, we had set up the required environment. Let us get started with today's post about perceptron.
A perceptron is the fundamental unit of a neural network. It is an algorithm for supervised learning (we say what is right and wrong) for binary classification (either ze... |
Consider a set of $n$ values $x_i$ for $i = 1, 2, \ldots n$. The arithmetic mean $\mu_n$ of this collection of values is defined as: $$ \mu_n = \displaystyle\frac{1}{n}\sum_{i=1}^n x_i \label{post_c34d06f4f4de2375658ed41f70177d59_mean} $$ The simplicity of equation \eqref{post_c34d06f4f4de2375658ed41f70177d59_mean} hid... |
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How has everyone fared on here? What do you think is going to be the cutoffs for these examinations?
Note by Tarun Yadav 3 years, 10 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to t... |
If elementary particles (specifically, those with mass, such as the electron or other leptons) are pointlike particles, wouldn't that mean they are naked singularities?
But these particles have spin- wouldn't that make them naked
ring singularities, thus giving them an observed radius, making them non-pointlike?
If I r... |
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
Here is a nice one ${}{}{}{}{}{}{}{}{}$, can you find the equation?
Here is a bijection from $(-\pi/2,\pi/2)$ to $\mathbb{R}$: $$ f(x)=\tan x. $$ You can play with this function and solve your problem.
$g(x)=\frac 1{1+e^x}$ gives a bije... |
In one problem I'm working on, I came across the computation of the reduced matrix element for the component of the rank 2 tensor
$$\langle j'\|[J^{(1)}\times J^{(1)}]^{(2)}_{(1)}\|j\rangle$$
I'm having a bit of trouble computing the respective tensor:
$$[J^{(1)}\times J^{(1)}]^{(2)}$$
I tried the following. Given two ... |
There are special integrals such as the logarithmic integral and exponential integrals. I want to know if there are primitives for such integrals. If not, why not?
This last link should clarify some of the ideas used :
the only new term appearing during an integration (i.e. that was not in the integrand) is a linear co... |
Let $X$ be a real random variable and $f$ the density belonging to $X$. Let $a\neq0$, and $b \in \mathbb R$, while $Y:=aX+b$. Show:
i) The density of $Y$ wrt the lebesgue measure exists and is $g(x):=\frac{1}{|a|}f(\frac{x-b}{a}), x\in\mathbb R$
ii) Find the distribution of $Y$ if $X$ ~ $\mathcal{N}(\mu,\sigma^{2})$
My... |
Question:
In the early morning hours of June 14, 2002, the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated 2002 MN, remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was 73,600 miles... |
Given the following
\[K = \min_{x} \left\{ c^T{x} + \sum_{k=1}^{M} p_kq_k^Ty_k ~|~ Ax = b; T_kx + W_ky_k = h_k; y_k \geq 0, x \geq 0 \,\,\forall ~k=1,2, \ldots M \right\} \]
where \((T_k, W_k, h_k, y_k)\) occurs with probability \(p_k > 0 ~\forall ~k = 1,2, \ldots, M\)
Now, I want to prove the following claim:
For \(M ... |
Given a differentiable function $\kappa(s)$, $s \in I$, show that the parametrized plane curve having $k(s)=k$ as curvature is given by:
$\alpha(s)=(\int cos(\theta(s))ds+a,\int sin(\theta(s))ds+b)$
where
$\theta(s) = \int k(s)ds + \gamma$
And that the curve is determined up to translation of the vector (a,b) and a rot... |
Motivation : Here is a motivation as to why this problem is so important.
Let $f(t)$ be an audio signal. We can safely asume it to be bandlimited to 0-20kHz as we cannot hear anything above that. Capture this signal in digital computer with appropriate sampling frequency and denote it as $f[n]$.
Now take Discrete Hilbe... |
Geometric Meaning of the Geometric Mean
The
geometric mean of two positive numbers $a\;$ and $b\;$ is the (positive) number g whose square equals the product $ab\;$:
$g^{2} = ab\;$.
While it is possible to (at least partially) adapt the definition to handle negative numbers, I do not believe this is ever done.
The geom... |
I find myself needing to compute (or asymptotically estimate) the following sum over the $2^{S-1}$ compositions of $S$. I am hoping an expert in combinatorics (I am a computer scientist) will recognise this summation.
Let $B_{S,k}$ denote the set of compositions that have exactly $k$ parts, and define $T_{k}$ as,
$$T_{... |
Communications in Mathematical Analysis Commun. Math. Anal. Volume 14, Number 2 (2013), 143-162. Asymptotic Behavior and Stability of Solutions to Barotropic Vorticity Equation on a Sphere Abstract
Orthogonal projectors on the subspace $\mathbf{H}_{n}$ of homogeneous spherical polynomials of degree $n$ and on the subsp... |
Mahlo cardinal A cardinal $\kappa$ is Mahlo if and only if it is inaccessible and the regular cardinals below $\kappa$ form a stationary subset of $\kappa$. Equivalently, $\kappa$ is Mahlo if it is regular and the inaccessible cardinals below $\kappa$ are stationary. Every Mahlo cardinal $\kappa$ is inaccessible, and i... |
Hello! I'm struggling with this math problem:
The beginning of coordinate system moved to point \(\displaystyle O'(-1,2)\) and then rotated it over angle \(\displaystyle \alpha\) that \(\displaystyle tg\alpha = \frac{5}{12}\). Designate the point \(\displaystyle M\)coordinates in the old coordinate system if it coordin... |
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AMC I is just around the corner guys.....
Note by Xuming Liang 5 years, 8 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution... |
The following equation, which relates the pH of an aqueous solution of an acid to the acid dissociation constant of the acid, is known as the Henderson-Hasselbach equation.
\[ pH = pk_A + \log_{10} \dfrac{[\text{conjugate base}]}{[\text{weak acid}]}\]
The Henderson-Hasselbach equation is derived from the definition of ... |
Home
Integration by PartsIntegration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
Deep Neural Networks (DNNs) are notorious for requiring less feature engineering than Machine Learning algorithms. For example convolutional networks learn by themselves the right convolution kernels to apply on an image. No need of carefully handcrafted kernels.
However a common point to all kinds of neural networks i... |
Equilateral Triangle on Three Lines What is this about? Problem
Given three straight lines (denoted below by two points $AB,$ $CD,$ $EF$).
Construct an equilateral triangle with vertices one per line.
Construction
Choose an arbitrary point on one of the line, say $X$ on $EF.$ Rotate $CD$ around $X$ $60^{\circ}$ into $C... |
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∑i=1n2(i)=a2\sum_{i=1}^{n^2} (i) = a^2i=1∑n2(i)=a2
(n,a)∈N(n,a)\in N(n,a)∈N , NNN denotes natural number.
Does there exist only one such aaa to satisfy the above conditions?
Note by Akash Shukla 3 years, 4 months ago
Easy Math Editor
This discussion board is a pl... |
I'm trying to get the AdS solution to the circular wilson loop. The standard AdS metric is:
$ds^2 = \frac{L^2}{z^2}(\eta_{\mu \nu} dx^{\mu} dx^{\nu} + dz^2)$
If I take the circle of radius R at the x1,x2 plane I can choose polar coordinates:
$x_1 = R cos(\theta)$, $x_2 = R sin(\theta)$
$ds^2 = \frac{L^2}{z^2}(-dt^2 + d... |
Given an isolated $N$-particle system with only two body interaction, that is $$H=\sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}+\sum_{i<j}V(\mathbf{r}_i-\mathbf{r}_j)$$
In the thermodynamic limit, that is $N\gg 1$ and $N/V=$constant, it seems that not all two body interaction can make system approach thermal equilibrium automa... |
Astrid the astronaut is floating in a grid. Each time she pushes off she keeps gliding until she collides with a solid wall, marked by a thicker line. From such a wall she can propel herself either parallel or perpendicular to the wall, but always travelling directly \(\leftarrow, \rightarrow, \uparrow, \downarrow\). F... |
Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)
Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, Algeria
$ x'(t)=-f(x(t))+\in... |
I want to count in how many ways three dice can sum to a given number without brute forcing it. In fact, I would like to do it using generating functions and without having to expand out the product.
To do this, I have thought of making a differential equation out of $y=(x+x^2+x^3+x^4+x^5+x^6)^3$ and then solving the e... |
I'd like to offer a heuristic partial answer to a more general question... which relates to some of the discussion in the comments.
I had to generalize the definition to make my argument work, so that $p$ is a $\textit{survivor of order}$ $m$ if there is a set of primes $\{q_1, \dots, q_m\}$ such that for every $k \le ... |
Home
Integration by PartsIntegration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
The Annals of Statistics Ann. Statist. Volume 22, Number 1 (1994), 1-20. The Order of the Remainder in Derivatives of Composition and Inverse Operators for $p$-Variation Norms Abstract
Many statisticians have adopted compact differentiability since Reeds showed in 1976 that it holds (while Frechet differentiability fai... |
Proceedings of the Centre for Mathematics and its Applications Proc. Centre Math. Appl. The AMSI-ANU workshop on spectral theory and harmonic analysis. Andrew Hassell, Alan McIntosh and Robert Taggart, eds. Proceedings of the Centre for Mathematics and its Applications, v. 44. (Canberra AUS: Centre for Mathematics and ... |
On the LUQ decomposition
The algorithm implemented in
luq (see reference given below) computes bases for the left/right null spaces of a sparse matrix $A$. Unfortunately, as far as I can tell, there seems to be no thorough discussion of this particular algorithm in the literature. In place of a reference, let us clarif... |
I am conducting a linear regression: $Y=\alpha+\beta\times X+\epsilon$, $\epsilon\sim N(0,\sigma^2)$. It turned out that the confidence interval for the predicted mean $Y$ was really small (figure 1), that it was hardly differntiable from the predicted mean.
On the other hand, we can see that the observed $Y$s actually... |
I want to learn how to construct spaces of quantum states of systems.
As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is the Hamiltonian of the harmonic oscillator with the quadratic term: $\hat{H}=\hat{H}_{0}+\hat{H}_{1}$, where
$\hat{H}_{0}... |
Play around with different values in the matrix to see how the linear transformation it represents affects the image. Notice how the sign of the determinant (positive or negative) reflects the
orientation of the image (whether it appears "mirrored" or not). The arrows denote eigenvectors corresponding to eigenvalues of... |
Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones?
Or, more formally...
Suppose $v_1,v_2,\ldots,v_k \in \mathbb C^m$. Take $n \ge m$ as small as possible such that if we consider $\mathbb C^m$ as a subsp... |
There is the standard argument, using the definition of the inner product; that $\langle f|A|g\rangle =\langle g|A|f\rangle ^{*}$ for a Hermitian operator $A$, given any wave vectors $|f\rangle,~ |g\rangle$.
Also consider the following:
Consider the infinite dimensional one dimensional position space, with a column vec... |
Research Open Access Published: Boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear sensitivity and logistic source Boundary Value Problems volume 2019, Article number: 120 (2019) Article metrics
280 Accesses
Abstract
In this paper, we deal with the following quasilinear attraction–repuls... |
The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary $$ \sup_{t \in \mathbb{R}} |F(t) - G(t)| \ll \frac{1}{T} + \int_{-T}^{T} \bigg | \frac{f(t) - g(t)}{t} \bigg | dt $$ provided that one of t... |
It is essentially impossible to answer the general question of "how does multilinearity come up naturally in physics?" because of the myriad of possible examples that make up the total answer. Instead, let me describe a situation that very loudly cries out for the use of tensor products of two vectors.
Consider the pro... |
Differential and Integral Equations Differential Integral Equations Volume 32, Number 7/8 (2019), 423-454. Structure of conformal metrics on $\mathbb{R}^n$ with constant $Q$-curvature Abstract
In this article, we study the nonlocal equation $$ (-\Delta)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb R$}, \quad\int_... |
I wonder whether 'deepness' is subjective or not. The Compactness Theorem of first-order logic has several proofs.
Theorem. (Gödel-Maltsev) Given a language $L$ and a set $S$ of first-order sentences in that language, if every finite subset of $S$ has a model, then $S$ has model.
(A
language is just a set of relational... |
Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's
Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=1}^n$ is a real valued martingale difference with respect to the filtration $(\mathcal F_i)_{i=1}^n$ (that is, $(S_j:=\s... |
Research Open Access Published: Global existence and exponential stability for a nonlinear Timoshenko system with delay Boundary Value Problems volume 2015, Article number: 206 (2015) Article metrics
1303 Accesses
3 Citations
Abstract
This paper is concerned with a nonlinear Timoshenko system modeling clamped thin elas... |
In a previous post, I described how one can compute the point ${\bf S}$ on a line $r$ which is closest to a given point ${\bf X}$. There I assumed the line $r$ passed through two distinct points ${\bf P}$ and ${\bf Q}$ (see figure 1). In this post I will describe how one can use that technique to compute the closest po... |
I'm trying to find the expectation value of the spin in the $y$ direction after applying a $\pi/2$-$\pi/2$ sequence of pulses. In Slichter's Principles of Magnetic Resonance, he does a $\pi/2$-$\pi$ pulse which makes sense because that will get you the largest free induction signal you could get. I understand everythin... |
Sodium has a volume expansion coefficient of $15 * 10^{-5} K^{-1}$. Calculate the percentage change in the fermi energy as the temperature is raised from $T = 0K$ to $T = 300K$.
My attempt at the solution is below:
Volume expansion coefficient is given by $\alpha _v = \frac{1}{V}\left( \frac{\partial V}{\partial T} \ri... |
o.m.'s conclusions are correct (if you like my answer you should up vote his too).
Theoretically Yes. Realistically Possibly.
Plugging in realistic efficiencies shows that certain sized craft could do this in theory. However, solar power isn't very potent and we don't have a good mechanism for turning solar power into ... |
Some Properties of Common Tangents What is this about? Problem
Let $HE$ and $FG$ be two internal tangents of non-overlapping circles $(A)$ and $(C),$ as shown below.
Let $FH$ intersect the second time $(A)$ in $I$ and $(C)$ in $J.$ Define $L$ to be the intersection of $EH$ and $FG,$ $K$ the intersection of $EI$ and $GJ... |
Why does there exist a non-split sequence with the condition that $\mathrm{pd} M = \infty$?
Remarks. I am reading
wherein on p. 476 there is the
Theorem. Let $\Lambda$ be a left artinian ring and assume that each indecomposable finitely generated left $\Lambda$-module has finite projective dimension or finite injective... |
Most electronic components dissipate heat whenever a current flows through them. The amount of heat depends on the power, device characteristics, and circuit design. Besides the components, the resistance of the electrical connections, copper traces, and vias contribute to some heat, and power losses.
To avoid failures... |
Cardinality Cardinality is the bare notion of size, we measure "how many elements" are in a given set. Finite sets are simple that way, a set $A$ is finite it there is a natural number $k$ such that $A=\{a_1,\ldots,a_k\}$. That is to say that there exists a bijection between $A$ and $\{0,\ldots,k-1\}$. We generalize th... |
This question is related to this Helicity states.
Suppose we have $k=[\omega,0,0,\omega]$. In Weinberg's book The Quantum Theory of Fields: Volume I he defines the state$|k,\sigma\rangle$ as an eigenstate of the operator $J_{3}$ that is
\begin{equation} J_{3}|k,\sigma\rangle=\sigma|k,\sigma\rangle \end{equation} where ... |
Lephenixnoir af424d1baa update documentation after writing the wiki 3 miesięcy temu config 4 miesięcy temu include/TeX 3 miesięcy temu src 3 miesięcy temu .gitignore 3 miesięcy temu Makefile 4 miesięcy temu README.md 3 miesięcy temu TODO.md 3 miesięcy temu configure 3 miesięcy temu font5x7.bmp 4 miesięcy temu font8x9.b... |
Suppose I have data points $(x_1^1, x_2^1), (x_1^2, x_2^2), (x_1^3, x_2^3), \ldots$ in $\mathbf{R}^2$ that fall in one of two classes, $y^i=0$ or $y^i=1$. I can find a linear separator for these points using logistic regression by finding $b_0, b_1, b_2$ that maximize $$P\left(y\;|\; (x_1, x_2)\right) = \frac{1}{1+e^{-... |
ReferencesShahvooz, B., S. Boy, T. Baseheart. Flexural Strengthening of Four 76-Year-old T-Beams with Various Fiber-Reinforced Polymer Systems: Testing and Analysis. ACI Structural Journal , 99 (2002), No. 5, 681-691.Sheikh, S., D. De Rose, J. Mardukhi. Retrofitting of Concrete Structures for Shear and Flexure with Fib... |
If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ g_{\mu\nu}(x^\alpha) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \\ 0 & 0 &-1 & 0 \\ 0 & 0 & 0 &-1 \end{array} \righ... |
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