1
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0answers
350 views

Distance of two points in spherical coordinates

Today I was trying to solve a physics exercise and ran into some mathematical problems. Consider two concentric spheres $K_{1,2}$ with radii $R_{1,2}$. I wanna solve the following integral: $$E_{ij} ...
1
vote
0answers
24 views

Bounding the $L^\infty$ norm of the partials.

Suppose $\varphi \in C^{\infty}(\mathbf{R}^n)$. I would like to show that $$ \sum_{j=1}^n \| \partial_j \varphi \|_{L^{\infty}} \leq 2n\big(\|\varphi \|_{L^{\infty}}\big)^{\frac{1}{2}} ...
1
vote
0answers
121 views

Differential equation. Frobenius method

Let's consider the following differential equation: $t(1-t) x'' + (p-(p+2)tx' - px = 0$ a) Prove that t=0 is a singular regula point for that equation. Prove that if $p \notin\mathbb{Z}$ there are ...
1
vote
0answers
65 views

Trying to prove an identity about a product

I have a product formula that, given a tuple $(a_1, \dots, a_{n-1}, 0)$, computes a dimension (of a vector space) via the following formula: $$ \mathrm{dimension} = \prod_{i < j} {(a_i + \cdots + ...
1
vote
0answers
80 views

Borel and Abel resummation and zeta regularization

is ist posible to apply Borel resummation to sums of the form $$ 1-2^{s}+3^{s}-4^{s}+....=\eta(-s) $$ of course the idea is to link zeta regularization and Borel summation since $$ ...
1
vote
0answers
120 views

Approximate the solutions to the following parabolic partial-differential equations.

Use the: a)forward difference method b)backward difference method c)Cranck-Nicolson algorithm to approximate the solution of the parabolic partial differential equation: $$\frac{\partial u}{\partial ...
1
vote
0answers
36 views

An order reduction problem.

Consider a differential equation of the form: $F(t,x,\frac{dx}{dt},...,\frac{d^{n}x}{dt^{n}})=0$ and exist $\alpha$ and $m$ fixed such that: $F(\lambda t, ...
1
vote
0answers
27 views

looking for “invertibility and singularity”

Dear *friends* Many monts ago,i searched a lot the book of Robin Harte "invertibility and singularity". this book contains a lot of demonstrations that i need in my master. it focus on banach ...
1
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0answers
67 views

Singular values of $A$ and eigenvalues of $\left[\begin{smallmatrix} 0 & A \\ A^T & 0 \end{smallmatrix}\right]$.

(Roger p.418) Let $A$ be $m \times n$ matrix, $q=\min(m,n)$, and $B=\begin{bmatrix}0&A\\A^*&0\end{bmatrix}$. Let $\sigma_1 ,\ldots,\sigma_q \ge 0$. The singular values of $A$ are ...
1
vote
0answers
1k views

Cumulative Distribution Function (CDF) and Means of Central Tendency

Using the graph of the cumulative distributive function below, find the: (a) mean (b) median (c) mode (d) midrange (e) third quartile for the random variable I ...
1
vote
0answers
37 views

could t-test be used to calculate how much differences for two statistics?

We know t-test could be used to test whether two statistics has significant differences. My question is, could t-test be used to test how much differences are two statistics? For example, whether ...
1
vote
0answers
83 views

Question about proof for $f\in C^1\Rightarrow f$ is local Lipschitz continuous

I have found the following proof in this very forum: If $f:\Omega\to{\mathbb R}^m$ is continuously differentiable on the open set $\Omega\subset{\mathbb R}^d$, then for each point $p\in\Omega$ ...
1
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0answers
87 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
1
vote
0answers
121 views

Riemann sums and integral

If the Riemann sums of a function $f$, defined on an arbitrarily tagged partition $P$, are positive, then will the integral also be positive for any arbitrary tagged partition whose norm is bounded?
1
vote
0answers
79 views

On a theorem of dimension of affine algebra

In the book of commutative algebra of G.Kemper (can be found here), the author has proved the following theorem : Let $A$ be an affine algebra and let $P_{0}\subsetneq P_{1}\subsetneq ...
1
vote
0answers
54 views

Ring of the residual classes $(\Bbb Z/p\Bbb Z)^\times$? $p$-adic integer?

In a recent question we raised the theorem: for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique ...
1
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0answers
168 views

Creating polyhedra with playing cards

In here, George Hart gives some exciting examples on how to create polyhedra by cutting playing cards and sliding them inside one another. I was wondering if such an approach can be generalized to ...
1
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0answers
69 views

A probability problem on periodicity of Markov Chain

Assume for a Markov Chain with period $d$, $\{C_0, C_1, \dots, C_{d−1}\}$ be the equivalence classes induced by $∼$ $(i$~$j$ means all the paths from $i$ to $j$ is of length $0$ mod $d$ )and numbered ...
1
vote
0answers
82 views

3d Implicit Trigonometry help?

I'm trying to understand implicit 3D trigonometry, specifically with this equation: $$\sin(y)+\cos(z)=\cos(x)$$ Can someone please explain to me what is going on with this equation? I really can't ...
1
vote
0answers
846 views

Matlab code for fixed point iteration

I want to write in Matlab a function that appreciates the fixed point iteration for a system of equations. The idea is: $\begin{bmatrix} x{_{1}}^{m+1}\\ x{_{2}}^{m+1} \end{bmatrix}= ...
1
vote
0answers
36 views

Optimal paths between two closed lines

Imagine I have an outer shape and an inner shape (that may overlap like the following picture) Is there any algorithm or mathematical property I can use to find a third set of points which will be ...
1
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0answers
26 views

How to understand the infraconnected set

I begin to study some p-adic analysis. I find it is hard to understand the infraconnected set. Who can give me some examples to show it? Is it relate with the connected set in the topology? I also ...
1
vote
0answers
78 views

Calculation of conditional joint probability given certain conditionals for data which aren't independent

The context of this problem is the estimation of the distribution of a parameter $v$ given sets of data $A$ and $B$, where $A$ and $B$ are not independent. Suppose I know $P(v | A)$ and $P(v | B)$. ...
1
vote
0answers
117 views

Approximation or solution to infinite series involving modified Bessel Functions

I'm looking for an analytical solution or approximation to the following infinite series. $$\sum_{i=0}^\infty t^i I_i(z)\;,\;t\neq0,$$ with $I_i(z)$ as the modified Bessel function of the first ...
1
vote
0answers
34 views

Functions defined using expectation values of random sequences

How does one show the following: Let $h:\mathbb{R}^2 \rightarrow \mathbb{R}$ a non-negative and continuous function. If $\{X_n \}$ and $\{Y_n \}$ are random sequences such that $\sup_n ...
1
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0answers
96 views

Conjugate prior where prior is normal, applying Bayes theorem

So X_1,...,X_n forms a random sample from a normal dist with unknown mean µ and variance σ^2. So we have the prior distribution with mean 0 and variance σ^2. We then need to show that if n is large ...
1
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0answers
25 views

How did we arrive at this form of Markov's Inequality in this proof?

In the book I am reading (complexity and cryptography by Talbot and Welsh, chapter 4), there is a proposition on $\textbf{ZPP}$($ \textbf{ZPP} = \textbf{RP} \cap \textbf{coRP}$-proposition $4.13$), ...
1
vote
0answers
87 views

Let $X$ be an integrable RV with median $m$. Show that $m=\text{argmin}_a\mathbb{E}|X-a|$

I have tried to look for mistakes in these calculations until I turned blue, but still clueless. If anyone could help me pointing out the errors, I would be really grateful. My apology for the long ...
1
vote
0answers
73 views

Greedy algorithm for Nephite coinage system

I have a question about problem 2 of this homework set and its solution. The task is to show whether the greedy algorithm works or not for the Nephite coinage system from the Book of Mormon (with ...
1
vote
0answers
139 views

Confusion related to optimization of log(det(X)) function

I have this confusion related to optimization of the log(det(X)) function. I didn't get how it implicitly maintains the constraint of X being positive definite. For eg if I have a matrix ...
1
vote
0answers
25 views

Problem on reflections - clarification needed

Suppose that $R_1,...,R_p$, $S_1, . . . , S_q$ are all reflections across planes that contain $\bf0$. Show that if $R_1 · · · R_p = S_1 · · · S_q$ (denoting composition of reflections) then $(−1)^p = ...
1
vote
0answers
38 views

System of second order lineal differential equations

I have the following system: $x'' = \alpha^2 y - x $ $y'' = x- y $ I have no idea how can I start. Please give me some hint.
1
vote
0answers
56 views

About relation between conditional variance and covariance determiant of gaussian vector

$\newcommand{\Var}{\operatorname{Var}}\newcommand{\Cov}{\operatorname{Cov}}$Could you help me to check that for every Gaussian vector $(Z_1,Z_2,\ldots, Z_n)$, we have $\det\Cov(Z_1,\ldots, Z_n)$ $= ...
1
vote
0answers
50 views

If two plotted meshes of nonlinear equations in a system intersect, does that mean there are always infinite solutions?

I am trying to wrap my head around this, as I have no formal education on it. I have a system of two nonlinear equations, with two variables. If I plot them in Matlab/Octave, and get two ...
1
vote
0answers
22 views

Terminology: interior subunits of polytope compounds

Edit: trying a better time of day. I'm trying to learn some geometry on my own and am getting drowned in terminology. The structures I'm looking to identify are those created by the overlap of two ...
1
vote
0answers
71 views

Derivative of a vector with scalar product in denominator

I'm struggling with a partial derivative of the following form: \begin{equation} \frac{\partial}{\partial \vec{x}} \frac{\vec{x}}{\vec{e}_3^T\,\vec{x}}, \end{equation} where $\vec{x} \in ...
1
vote
0answers
185 views

A basic probability question on uniform distribution

Suppose that independent trials, each of which is equally likely to have any of $m$ possible outcomes, are performed until the same outcome occurs $k$ consecutive times. If $N$ denotes the number of ...
1
vote
0answers
41 views

Proving Weierstrass sine representation mass bound: $\inf_n \sup_{|w| \geq m} |s_n(w)| \leq 4^{1-m}$

If one defines $$ s_n(w):=\frac{\sin\pi w}{\pi w}\prod_{j=1}^n\left( 1-\frac{w^2}{j^2}\right)^{-1}=\prod_{j=n+1}^\infty\left( 1-\frac{w^2}{j^2}\right),$$ how may one show that, for every $m \geq 0,$ ...
1
vote
0answers
144 views

Using MATLAB to do spectral analysis for a periodic signal

As is said, I was asked to do spectral analysis for a periodic signal, i.e. plot the magnitude of its Fourier transform against frequencies. I know that for periodic signals, the spectral is just ...
1
vote
0answers
95 views

Discrete probability distribution for a multi-stage experiment

Say we choose $n$ experiments to perform (with replacement) from the experiment types $1\dots C$, where an experiment of type $i$ is chosen with probability $q_i$. Experiments of type $i$ have ...
1
vote
0answers
174 views

pdf of the minimum of two dependent random variables

I am struggling at calculating the pdf of the minimum of two random variables as follows: $v = \min(v_1 + k_1 \cdot v_3, v_2 + k_2 \cdot v_3)$ where: $v$: continuous random variable I want to ...
1
vote
0answers
46 views

A question about the solutions of a diophantine equation

I would like to know if it's possible to find the solution of the following equation: $$x^k+y^h=z^{kh}$$ in which: $$\{x,y,z\}\subset\mathbb{N}$$ given $k$ and $h$ with: $$\{k,h\}\subset\mathbb{N}$$
1
vote
0answers
111 views

Why are complete lattice homomorphisms defined that way?

Given a function $f : X \rightarrow Y$ and a set $A \subseteq X$, there's at least two possible ways of interpreting the direct image $f(A),$ as explained here. In the notation of that question, write ...
1
vote
0answers
20 views

Bivariate Quasi-interpolants

Consider the bivariate spline quasi-interpolant $S2$ defined on a bounded rectangle $R$ with simple interior knots. It is known that, if $f \in C^k(R)$, then $||f-S_2f||_{\infty}$ behaves like ...
1
vote
0answers
184 views

Fourier transform of convolution in a finite range

Can anyone help me evaluate the Fourier transform of of the following function, $t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $f(t) = \int_{t_0}^t ...
1
vote
0answers
32 views

If Lie(H) preserves a subspace, must H also preserve that subspace?

Assume $H \subset G$ is a closed connected subgroup of a linear algebraic group over an arbitrary field (both assumed to be smooth). Assume $G$ acts linearly on the (finite dimensional) vector space ...
1
vote
0answers
46 views

Compute the joint probability of two “ordering” events

Let $\{o_1, o_2, o_3, o_4\}$ be a set of objects, each one associated with a score $s(o_i)$. This score is uncertain, and thus described by means of a probability density function $f_i$. When a total ...
1
vote
0answers
48 views

Getting an inverse function

I have a cubic function $N_3(x) = a x^3 + b x^2 + c x + d$ which guaranteed is non-negative in each point on interval $x \in [0,1]$. I building an other function $N_4(x) = \int_0^x{N_3(t)dt}$. Sure ...
1
vote
0answers
795 views

Represent non-periodic functions in a Fourier Series like function

I have this question of whether it is possible to represent non-periodic functions in a form just like you would represent a periodic function through a Fourier series. I understand this question ...
1
vote
0answers
86 views

Simplifying expression

I am looking for a way to simplify this expression: $$ \sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over ...

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