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The Fundamental Group of the Real Projective Plane
The goal of today's post is to prove that the fundamental group of the real projective plane, $\pi_1(\mathbb{R}P^2)$, is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. And unlike our proof that $\pi_1(S^1)\cong\mathbb{Z}$, today's proof is fairly short, thanks to the van Kamp... |
De Bruijn-Newman constant
For each real number [math]t[/math], define the entire function [math]H_t: {\mathbf C} \to {\mathbf C}[/math] by the formula
[math]\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du[/math]
where [math]\Phi[/math] is the super-exponentially decaying function
[math]\displaystyle... |
As a toy problem, I was looking at a classical second order system of the following form:
\begin{align} \ddot{x}(t) + c \dot{x}(t) + k x(t) = 0 \end{align}
Instead of the basic system above, I want to replace $x(t)$ with a Low-Pass filtered version of $x(t)$. This can change the equation to the following form:
\begin{a... |
A Rational Game
This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]. Each piece ... |
@Josh: I don't have any precise reference in mind for that, maybe in the Coddington-Levinson? but as far as I remember it is mostly for linear ODE's
Note first that reversing time $t\to -t$ is equivalent to changing $\sigma\to-\sigma$, so you only need to study one side (say $t\to+\infty$). The case $\sigma=0$ is a bor... |
Here's a typical op-amp inverting amplifier:
The input impedance is simply \$R_{in}\$, so for your requirements, \$R_{in} = 10k\Omega\$. \$R_f\$ is then whatever it needs to be to realize the desired gain. You want a gain of -10, so:
$$ -\frac{R_f}{10k\Omega} = -10 \\R_f = 10 \cdot 10k\Omega = 100k\Omega $$
Why does th... |
You're quite right that the other fundamental forces of Nature possess mediator particles, e.g. the photon for the electromagnetic force. For gravity, a
graviton particle has been postulated, and is included in the five standard string theories which are candidates for quantum gravity. From a quantum field theory persp... |
I am writing my undergrad thesis on the harmonic oscillator on a lattice. So far I have implemented the Metropolis Monte Carlo algorithm to generate trajectories $x_j$ for $0 \leq j < N$, where $N$ is the number of time lattice divisions.
I get a pretty good histogram for $|\psi(x)|^2$ and it matches the expectation fr... |
Consider 2 mathematical problems:
$$ f_1(x) = a - x \\ f_2(x) = e^x -1 $$
The condition number for a function is defined as follows:
$$ k(f) = \left| x \cdot \frac{f'}{f} \right| $$
Lets analyze conditioning first:
$$ k(f_1) = \frac{x}{x - a}, $$
which means that $f_1$ is
ill-conditioned near $x = a$;
$$ k(f_2) = \frac... |
TL;DR: Do
not just memorise thermodynamics equations! And if you have an issue with the equations $\Delta U = 0$ or $\Delta H = 0$ for an isothermal process, read the answer. The first problem
You said that an exothermic reaction corresponds to $\Delta U < 0$. This is
not true. It is defined by $\Delta H < 0$.
However,... |
Update
Since version 12, this functionality in integrated in Mathematica via the Option FitRegularization
Following on @Ajasja's answer in the spirit of this answer one can in fact provide controlled smoothing va an explicit Tichonov like penalty as follows:
ff = Function[{x, y}, basis // Evaluate];
a = ff @@ # & /@ (M... |
Difference between revisions of "Gay-Berne model"
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:<math>\frac{\chi \prime }{\alpha \prime^{2}}=1- {\left(\frac{\epsilon_{ee}}{\epsilon_{ss}}\right)} ^{\frac{1}{\mu}}.</math>
:<math>\frac{\chi \prime }{\alpha \prime^{2}}=1- {\left(\frac{\epsilon_{ee}}{\epsilon_{ss}}\right)} ^{\frac{1}{\mu}}.</math>
... |
$\pi R^2 \Delta p$ (1)
and the surface tension is:
$2\pi R \gamma$ (2)
and a sphere has two surfaces so, roughly:
$\Delta p = \frac{4 \gamma}{R}$
but if we take the limit:
$\lim_{R \to 0} \frac{4 \gamma}{R}$,
then obviously the pressure difference between the outside and inside surface of the bubble is $\infty$! How do... |
This is not really an independent answer; just an explanation of why the answer given by Levieux is optimal. (Of course it's easy to check by computer, but you might prefer to have a more human-comprehensible explanation.)
First of all, obviously
each number's digits must be in descending order because if not, reorderi... |
I'm asking about definite integrals that can effortlessly be found numerically by high schoolers using software. For example,
$$\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$$
This link shows the numerical integration done by software.
This Reddit comment subst... |
Open Sets Are Everything
In today's post I want to emphasize a simple - but important - idea in topology which I think is helpful for anyone new to the subject, and that is:
Open sets are everything!
What do I mean by that? Well, for a given set $X$,
all the properties* of $X$ are HIGHLY dependent on how you define an ... |
Suppose we have a binary symmetric channel with $p=\frac{1}{3}$; that is, a communications channel in which each bit is flipped with independent probability $\frac{1}{3}$. I know that there is a code such that, in the (highly probable) event that no more than $p$ of the bits are corrupted, we can guarantee recovery of ... |
Consider a beta distribution for a given set of ratings in [0,1]. After having calculated the mean:
$$ \mu = \frac{\alpha}{\alpha+\beta} $$
Is there a way to provide a confidence interval around this mean?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis... |
Now we are going to build a polynomial of degree $$n$$ that only goes through one given point $$x_0$$, and its $$n$$ derivatives coincide in this point with the $$n$$ derivatives of the original function $$f(x)$$. This polynomial is named Taylor's polynomial of degree $$n$$ around the point $$x_0$$ of the function $$f ... |
Simplifying polynomials
“Polynomials” are expressions with two or more (poly) types of numbers (nomials). An example of a polynomial expression would be:
$(6x^2+5x+3)+(x^2+3x-2)$
To simplify this expression, we need to first consider PEMDAS. Within each set of parentheses, nothing can be reduced or combined. Also, no e... |
Let $C$ be a $[n,k]$ linear Code over $\mathbb{F}_q$ .
I want to show that each vector of $\mathbb{F}_q^{n-k} $ is written as a linear combination of $m$ columns of $H$ iff $\rho \leq m$.
I have thought the following:
$$ d(C)=\min \{ d \in \mathbb{N} | \text{ there are d linearly dependent columns of H}\} $$
So $H$ has... |
Multiple nonnegative solutions for elliptic boundary value problems involving the $p$-Laplacian
DOI: http://dx.doi.org/10.12775/TMNA.2005.036
Abstract
In this paper we present a result concerning the existence of two
nonzero nonnegative solutions for the following Dirichlet problem involving the $p$-Laplacian $$ \cases... |
I am reading about Dupire local volatility model and have a rough idea of the derivation. But I can't reconcile the local volatility surface to pricing using geometric brownian motion process. If I'm not mistaken the proces $dS=rSdt+\sigma(S;t) S dX$ given final condition $(S-K)^{+}$ will produce correct options values... |
Let $m,n \in \mathbb{Z}$ and let $x \in \mathbb{R}$. Let $[x]$ denote the floor function.
We will attempt to prove
$$\Big[\frac{x+n}{m}\Big] = \bigg[\frac{[x]+n}{m}\bigg]$$
Suppose without loss of generality that $m > 0$.
By definition of the floor function on $\frac{x+n}{m}$, we have
$$\Big[\frac{x+n}{m}\Big] \leq \fr... |
I'm a high school student, so I have no idea what a Jacobian or a manifold is, but as someone who's self-studied linear algebra and abstract algebra, I think it's pretty complex and takes a rather smart/dedicated person to pass these classes, so you're definitely smart enough to understand these concepts.
In my opinion... |
We have to evaluate the following integration $$\int_0^{\pi/2}\sin 2x\arctan(\sin x)dx.$$
In this question I thought of using integration by parts .
But stuck in that.
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 minu... |
Given a convex hexagon $ABCDEF$. All its sides are equal (it can be irregular). Furthermore, $AD = BE = CF$. How can I prove that a circle can be inscribed in this hexagon?
Let $O$ be the intersection point of diagonals $AD$ and $BE$ (see diagram below) and set $a=OB$, $b=OD$, $d=AD=AE$, $\theta=\angle AOB=\angle DOE$.... |
I have an equation: $\ddot{x}+(\delta+\epsilon\cos{t})x=0$ known as the Mathieu equation.The $\delta-\epsilon$ parameter space of this equation looks something like
The red lines in this diagram indicate that if a point on the red lines is chosen, and the corresponding value of $(\delta,\epsilon)$ is plugged into the M... |
Consider a rectangular $(m \times n)$ matrix $\underline E_1$ with $m < n$ that has only $0$ or $1$ entries. It has exactly one $1$ entry in each row and not more than one $1$ entry in each column. Consider it being a selection of $m$ rows out of a $(n \times n)$ permutation matrix $\underline P$.
Given $\underline E_1... |
Let's say you want to estimate a quantity $\mu$, but you have only access to unbiased estimates of its logarithm, i.e., $\log\mu$. Can you obtain an unbiased estimate of $\mu$?
There used to be a quite nice blog post online for the estimator solving this problem, but it is now in a weird shape, not compiling the Latex ... |
Discretization-invariant Bayesian inversion and Besov space priors
1.
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hallstromin katu 2b) FI-00014, Finland, Finland
2.
Tampere University of Technology,Institute of Mathematics,, P.O. Box 553, 33101 Tampere
εis considered, where $U$... |
Difference between revisions of "Vopenka"
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** $gVP(κ, \mathbf{Σ_{n+1}})$ for a proper class of $κ$
** $gVP(κ, \mathbf{Σ_{n+1}})$ for a proper class of $κ$
** There is a proper class of $n$-remarkable cardinals.
** ... |
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My problem Deflection needs a solution and rating.
Note by Nishant Sharma 5 years, 3 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... |
In this question it is shown that being able to compute reciprocals (together with sums and differences) is enough to do do multiplication in a field of characteristic $\ne 2$. That made me wonder:
Can we formulate a set of field axioms that are based on the reciprocal function rather than a multiplication operation? S... |
Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$,
I want to prove by contradiction, and then find a continuous $g$ such that $g(a) = g(b) = 0$ but $\int_{a}^{b}f(x)g(x) dx \neq 0$
Proof: Sup... |
Given a finite group $G$, and a non-identity representative $g$ in a conjugacy class of prime order $p$, I'm trying to show that some nontrivial irreducible character of $G$ must have $\chi(g) \neq 0$ and $\chi(1) \neq 0 \text{ mod } p$.
The suggestion was to show that the nonexistence of such a character would imply t... |
I am trying to determine the parameters for the Nelson Siegel Svensson model and am solving a Non-Linear Optimization problem to do this.
I am trying to solve:
$$ \min_\theta{\sum{(p_i - \hat p_i)^2}}. $$
where $p_i$ are the observed dirty prices of the bonds and $\hat p_i$ are the prices that have been calculated usin... |
Firstly, in connexion with optics, Fermat's principle is always and approximation: it
defines an approximation, namely the first term in the WKB Approximation to the solution of either the quasi-time-harmonic Maxwell's equations or, in a more general setting, the Helmholtz equation. Here the WKB scale parameter is the ... |
So first, represent your system as the following in your case:
\begin{align}\frac{\partial q_k}{\partial t} - q_k \nabla \cdot F_k(\boldsymbol{q}) &= 0 & \forall k \in \lbrace 1, 2 \rbrace\end{align}
where $\boldsymbol{q} = [q_1, q_2]^T = [f, g]^T$, $F_1(\boldsymbol{q}) = q_2 \hat{e}_1$, and $F_2(\boldsymbol{q}) = q_1 ... |
In my previous two posts I showed worked solutions to problems 2.5 and 11.7 in Bulmer’s
Principles of Statistics, both of which involve the characteristics of self-fertilizing hybrid sweet peas. It turns out that problem 11.8 also involves this same topic, so why not work it as well for completeness. The problem asks u... |
In the previous STT5100 course, last week, we’ve seen how to use monte carlo simulations. The idea is that we do observe in statistics a sample \{y_1,\cdots,y_n\}, and more generally, in econometrics \{(y_1,\mathbf{x}_1),\cdots,(y_n,\mathbf{x}_n)\}. But let’s get back to statistics (without covariates) to illustrate. W... |
Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$
1.
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
$-\epsilon \Delta u+V(x)u-\epsilon k(\Delta(|u|^2))u=g(u), \quad u>0,x \in \mathbb R^N,$
where g has superlinear growth at infinit... |
Constructing the Tensor Product of Modules The Basic Idea
Today we talk tensor products. Specifically this post covers the construction of the tensor product between two modules over a ring. But before jumping in, I think now's a good time to ask, "What are tensor products good for?" Here's a simple example where such ... |
Mass, Weight and Density
Category : Railways
Mass, Weight and Density
Mass
The mass (m) of a body of matter is quantitative measure of its inertia i.e., its resistance to a change in the state of rest or motion of the body, when a force is applied.
Inertia is the property of a mass which resists change from its states ... |
Difference between revisions of "State Feedback"
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<li><p>A ''linear quadratic regulator'' minimizes the cost function
<li><p>A ''linear quadratic regulator'' minimizes the cost function
<center><math>
<center><math>
−
\tilde J = \int_0^... |
Link to arXiv Preprint.
Abstract
Batch normalization is often used in an attempt to stabilize and accelerate training in deep neural networks. In many cases it indeed decreases the number of parameter updates required to reduce the training error. However, it also reduces robustness to small input perturbations and noi... |
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recog... |
This note addresses the typical applied problem of estimating from data how a target “conversion rate” function varies with some available scalar score function — e.g., estimating conversion rates from some marketing campaign as a function of a targeting model score. The idea centers around estimating the integral of t... |
A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.
bidibangboom Posts: 34 Joined: May 10th, 2019, 6:38 pm
triple poster
Code: Select all
#C A... |
This is essentially an addition to the list of @4tnemele
I'd like to add some earlier work to this list, namely Discrete Gauge Theory.
Discrete gauge theory in 2+1 dimensions arises by breaking a gauge symmetry with gauge group $G$ to some lower
discrete subgroup $H$, via a Higgs mechanism. The force carriers ('photons... |
This is essentially an addition to the list of @4tnemele
I'd like to add some earlier work to this list, namely Discrete Gauge Theory.
Discrete gauge theory in 2+1 dimensions arises by breaking a gauge symmetry with gauge group $G$ to some lower
discrete subgroup $H$, via a Higgs mechanism. The force carriers ('photons... |
I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align} (x-z)^2+y^2 &\text{ is a perfect square,}\\ x^2+y^2 &\text{ is a perfect square,}\\ (x+z)^2+y^2 &\text{ is a perfect square.} \end{align} I failed tying to find such triple numerically. I tried to look at the ... |
Let $X$ be an algebraic stack. In Perfect complexes on algebraic stacks (4.1) a
perfect complex $P \in \mathsf{D}_{\mathrm qc}(X)$ is defined to be a complex such that, for any smooth morphism $\operatorname{Spec}(A) \to X$ (where $A$ is a commutative ring), the complex $\mathbf{R}\Gamma(X,P_{|\operatorname{Spec}(A)})$... |
The possibility of recursive self-improvement is often brought up as a reason to expect that an intelligence explosion is likely to result in a singleton - a single dominant agent controlling everything. Once a sufficiently general artificial intelligence can make improvements to itself, it begins to acquire a compound... |
I’m attempting to write a C program to gather values from a coupled spring system: There is a wall, connected to a mass $m_1$ by a spring, then this mass is connected to a second mass $m_2$ by another spring.
The values I require are the positions of the two masses and their velocities.
Using initial values for positio... |
I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The question is as follows:
Let X and Y follow a bivariate normal distribution with means (3, 2), variances(1, 4) and covar... |
Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation:$$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$However, I can always represent a 4-vector $p^\mu$ using left and right handed spinor indices, writing$$ \tag{2} p_{\alpha \dot{\alp... |
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* [[Siril:Tutorial_sequence|Work on a sequence of converted images]]
* [[Siril:Tutorial_sequence|Work on a sequence of converted images]]
* [[Siril:Tutorial_preprocessing|Pre-processing images]]
* [[Siril:Tutorial_preprocessing|Pre-processing images]]
−
*... |
I will set up a naive static/short run model to examine the case (so this post may be a bit long - I will try to dispense with some algebraic steps). I will use convenient functional forms, which are nevertheless consistent with usual assumptions.
FIRMS There are $i=1,...,n$ identical, price taking firms. In the short ... |
Nonlinear Eigenvalue Problems for Even Functionals 2007 (English)In: Applicable Analysis: an international journal, ISSN 1563-504X (electronic) 0003-6811 (paper), Vol. 86, no 7, p. 829-849Article in journal (Refereed) Published Abstract [en]
Let $H$ be a Hilbert space and let $g\in C^1(H,\mathbb R)$ be an even Fréchet ... |
There's a 59.5125% chance of survival.
Naively, we might have thought there'd be a 55% chance of survival as 55% of the roll results are good. But the 20 is a slightly better result than the 1 is a bad one, so that pushes up the probability a bit. Let's see how.
The approach
The simplest way to tackle this is to look a... |
My problem is that I am about to write a longer mathematical text, and it will be filled with integrals. Integrals tend to be filled with pesky fractions, square roots and what not.
Personally I feel like LaTeX is spacing things "wrongly" I prefer to have more space in my fractions, and a tad more space after the squar... |
Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
We show that an arbitrary factor map $\pi :X \to Y$ on an irreducible subshift of finite type is a composition of a finite-to-one factor code a... |
Question: How to classify/characterize the phase structure of (quantum) gauge theory?
Gauge Theory (say with a gauge group $G_g$) is a powerful quantum field theoretic(QFT) tool to describe many-body quantum nature (because QFT naturally serves for understanding many-body quantum problem with (quasi-)particle creation/... |
As far as I know, absorbing of the positive coefficient of $i\epsilon$ in a propagator seems to be a trivial operation without even the need of justification.
In Peskin page 286, he did this:$$k^0\rightarrow k^0(1+i\epsilon)$$$$(k^2-m^2)\rightarrow (k^2-m^2+i\epsilon)$$
In M. Srednicki's Quantum Field Theory, page 51,
... |
Introduction
In this post, I review the convergence proofs of gradient algorithms. Our main reference is: Leon Bottou, Online learning and stochastic approximations. I rewrite the proofs described in Bottou's paper but with more details about the points which are subtle to me. I tried to write the proofs as clear as po... |
Hi, Can someone provide me some self reading material for Condensed matter theory? I've done QFT previously for which I could happily read Peskin supplemented with David Tong. Can you please suggest some references along those lines? Thanks
@skullpatrol The second one was in my MSc and covered considerably less than my... |
Let $X$ be an uncountable set. We define the cocountable topology $\tau$ as the set of all subsets $U\subseteq X$ such that either $U=\emptyset$ or $X\setminus U$ is countable.
I am interested in the conditions under which a set $E\subseteq X$ will be dense in $X$, and I honestly don't know where to start.
I have alrea... |
I'm having trouble with this problem in Ahlfors' Complex Analysis (page 238):
If a vertex of the polygon is allowed to be at $\infty$, what modification does the formula undergo? If in this context $\beta_k = 1$, what is the polygon like?
The formula he is referring to is (probably) $$F(w) = C \int_0^w \prod_{k=1}^n ( ... |
The Tensor Product, Demystified
Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Mathematicians do this all the time:
When you have two integers, you can find their greatest common divisor or least common multiple. When you have some sets, you can form... |
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Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Yes. Let's improve the wording here: that is not just an expression, it's the definition of a dimensionless variable that eases the understanding of the solution of the transient diffusion is a semi-infinite medium. Because there is no clear characteristic length as in a finite diffusion path problem, Buckingham π theo... |
NOTE: This answer is community wiki: If you feel there is a need for
correction -please update the answer right here.
1. Frequency response: For any system - the (discrete time or continuous time) Fourier Transform of the impulse response is same as the frequency response of it. You can replace $s$ by $j\omega$ to work... |
I'm trying to find the roots of $x^3 -2$, I know that one of the roots are $\sqrt[3] 2$ and $\sqrt[3] {2}e^{\frac{2\pi}{3}i}$ but I don't why. The first one is easy to find, but the another two roots?
I need help
Thank you
Mathematics Stack Exchange is a question and answer site for people studying math at any level an... |
Suppose $x$ and $y$ are integers. Given $2xy+14y=-53-13x$, what does $xy$ equal? The answer is $-15$, but how do I get that? I feel like I should be able to find this.
A "general" way of attacking these types of problems is with a method that could be called "completing the rectangle", analogous to completing the squar... |
In Minkowski space even-dim (say $d+1$ D) spacetime dimension, we can write fermion-field theory as the Lagrangian:
$$\mathcal{L}=\bar{\psi} (i\not \partial-m)\psi+ \bar{\psi} \phi_1 \psi+\bar{\psi} (i \gamma^5)\phi_2 \psi$$
with Yukawa coupling to scalar term (with $\phi_1$) and pseudoscalar term (with $\phi_2$). Here... |
This is a question that I asked in the mathematics section, but I believe it may get more attention here. I am working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of commuting matrices. The g... |
Treasury futures are actually really complicated... There are complete books dedicated to this topic (e.g., The Treasury Bond Basis) and really good sell-side research papers ("Understanding Treasury Bond Futures" by Salomon Brothers) that I highly recommend.You're actually very much on the right track, but I'll try to... |
Difference between revisions of "ORD is Mahlo"
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If there is a pseudo [[uplifting]] (proof in that article) cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a [[reflecting]] cardinal and consequently also a transitive model of ... |
Difference between revisions of "De Bruijn-Newman constant"
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where
where
−
:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s)</math>
+
:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} ... |
The conclusion merely counts dimensions of vector spaces. However, it is not generally true.
The most basic properties of matrix multiplication show that the linear transformation represented by the matrix $\mathbb{H}=X(X^\prime X)^{-}X^\prime$ satisfies
$$\mathbb{H}^2 = \left(X(X^\prime X)^{-}X^\prime\right)^2=X(X^\pr... |
The Annals of Mathematical Statistics Ann. Math. Statist. Volume 43, Number 5 (1972), 1412-1427. Likelihood Ratio Tests for Sequential $k$-Decision Problems Abstract
Sequential tests of separated hypotheses concerning the parameter $\theta$ of a Koopman-Darmois family are studied from the point of view of minimizing ex... |
Queries ping a certain computer server at random times, on average $\lambda$ arriving per second. The server can respond to one per second and those that can’t be serviced immediately are queued up. What is the average wait time per query? Clearly if $\lambda \ll 1$, the average wait time is zero. But if $\lambda > 1$,... |
Trig Basics
Despite its reputation as being "high math", trigonometry can be very simple. At least its use on the SAT or ACT is simple—and that's all we're worried about here.
To use trig, you need two things:
a triangle with a right angle (AKA, a right triangle) one of the other parts labeled (either an angle or a leg... |
Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity
1.
Institute of Contemporary Mathematics, Henan University, School of Mathematics and Information Science, Henan University, Kaifeng 475004, China
2.
School of Mathematics and Information Science, Henan University, Kaifen... |
I am trying to represent myself quotient groups and I'm having trouble seeing what the kernel of a homomorphism : $\Phi: G \rightarrow G/H$ is (be it a ring homomorphism or a group homomorphism).
I understand that $\mathbb{Z}/n\mathbb{Z}$ is a quotient group that is cyclic and that the kernel of the application: $\Pi :... |
This is false for $n\geq 4$. Consider the Grassmannian $\mathrm{Gr}(2,n)$ of all two-dimensional subspaces of $\mathbb{R}^n$, and recall that $\mathrm{Gr}(2,n)$ is a compact manifold of dimension $2n-4$. For each $\varphi\in\mathrm{GL}_n(\mathbb{Q})$, let$$S_\varphi = \{A\in \mathrm{Gr}(2,n) \mid \varphi u=v \text{ for... |
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form $$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$ where $X$ is a semimartingale on a manifold $M$ and $\Theta$ is a predictable process above $X$ taking values in the second order tangent bundle $\tau M$ (the bundle ... |
I've recently read Yan LeCuns comment on 1x1 convolutions:
In Convolutional Nets, there is no such thing as "fully-connected layers". There are only convolution layers with 1x1 convolution kernels and a full connection table.
It's a too-rarely-understood fact that ConvNets don't need to have a fixed-size input. You can... |
I am assuming you are working over a number field $K$. Then, $f(x,y)$ describes a conic section. If you homogenize it by adding suitable powers of $z$ to every monomial, you get a ternary quadratic form, which can always be diagonalized by suitable change of variables to obtain something of the form $Q(x,y,z) = ax^2 + ... |
I have a linear system of equations of size mxm, where m is large. However, the variables that I'm interested in are just the first n variables (n is small compared to m). Is there a way I can approximate the solution for the first m values without having to solve the entire system? If so, would this approximation be f... |
Feynman diagrams provide a very compact and intuitive way of representing interactions between particles. These diagrams can be included into LaTeX documents thanks to a few packages. One of the older packages is
feynmf which uses MetaPost in order to generate the diagrams. More recently, a new package called Ti
kZ-Fey... |
ProblemFind the Fourier transformation of $u(x) = \frac{1}{1+x^2}$
I want $\int_\mathbb R e^{-itx} \frac{1}{1+x^2} dx$. Let $f(z) = e^{-itz} \frac{1}{1+z^2}$, $z \in \mathbb C$, let's integrate this over the semi-circunference of radius $R$ and the line $[-R, R]$.
First, the integral on the circumference vanishes (with... |
Let $G_1$, $G_2$ be arbitrary groups, and $H$ be any group with homomorphisms $\theta_1:G_1\rightarrow H$, $\theta_2:G_1\rightarrow H$. Show that there exist a group $G$ and homomorphisms $\beta_1:G_1\rightarrow G$, $\beta_2:G_2\rightarrow G$, and a unique homomorphism $\theta:G\rightarrow H$ such that $\beta_1\theta =... |
Let us consider monic polynomials of $n^{\text{th}}$ degree $p(t) = t^n + a_{n-1} t^{n-1} + \dots + a_1 t + a_0$ with real coefficients. We know the roots are continuous functions of the coefficients. Let us define a map $F: \mathbb R^n \to \mathbb R_+^n$ by \begin{align*} F: (a_{n-1}, \dots, a_0) \mapsto (r_1, \dots, ... |
Let $a_1, a_2, \ldots$ be an infinite set of non-negative samples taken from a distribution $P_0(a)$, and write
$$\tag{1} \label{problem} S = 1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \ldots. $$ Notice that if the $a_i$ were all the same, $S$ would be a regular geometric series, with value $S = \frac{1}{1-a}$. How will the int... |
DOI:10.1128/AEM.01277-08 ABSTRACT
Network models offer computationally efficient tools for estimating the variability of single-cell lag phases. Currently, optical methods for estimating the variability of single-cell lag phases use single-cell inocula and are technically challenging. A Bayesian network model incorpora... |
Let us define a predicate
$$T(i,k_1,...,k_n)\in\{True, False\}$$Where $T(i,k_1,...,k_n)$ means "using the $i$ first values of $S$, we can find $n$ disjoint subsets with sums $k_1, ... k_n$". The answer you are looking for is $T(|S|,K,..,K)$ where $K$ appears $n$ times. We have the following recurrence formula : $$T(i,k... |
The Wiener–Khinchin-Einstein theorem states that the auto-correlation $(r_{xx}(\tau))$ and spectral density $(S(f))$ are Fourier duals, i.e.
$$r_{xx}(\tau) = \int^{+\infty}_{-\infty} S(f) \exp\left( 2\pi i \tau f \right)\:df$$
This relationship has several assumptions:
-The process must be stationary, i.e the spatial c... |
Let $(X,d)$ a metric space, $\alpha >0$ (fixed ) and $T: X \rightarrow X$ a map such that exist $n \in N$, where :
$$ d(T^n x , T^n y) \leq \alpha^n d(x,y), \forall x,y \in X$$
Define $h(x,y) = [d^2 (x,y) + \frac{1}{\alpha^2} d(Tx,ty)+...+ \frac{1}{\alpha^{2(n-1)}} d(T^{n-1}x,T^{n-1}y)]^{1/2}$. I want to prove that $h ... |
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