1
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0answers
48 views

Identifying modes in circular data

I have some circular data (ie, angles in $[0, 2\pi)$), and I need to analyze it in terms of its modes. I hope this is the right place to ask for help solving this. The main thing I need to be able to ...
1
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0answers
53 views

Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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0answers
82 views

Does this quadratic form represent 1?

I am stuck on the following question in Lam's quadratic forms for a few days now. Let $a,b,c$ be three elements of a field $F$ such that $0 \neq a^2+b^2 \neq c^2$. Show that the quadratic form ...
1
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0answers
75 views

Finding an invariant polynomial under a matrix action

I asked this question as a mathematica question: http://mathematica.stackexchange.com/questions/41689/finding-a-certain-invariant-polynomial-using-matrix-coordinates but maybe it will get more ...
1
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0answers
41 views

Vector bundle on a transversal intersection

Consider a curve $X$ (over a field $k$), which is an union of two smooth and connected curves $C$ and $D$ meeting at exactly one point $P \in X$ transversally. Let $E$ be a vector bundle on $X$ such ...
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0answers
35 views

Do we need to pay attention to the codomain of a differentiable function?

I came across the following definitions: We call $M\subset \mathbb R^N$ $m$-dimensional $C^k$-submanifold of $\mathbb R^N$ if for all $a\in M$ there is an open neighborhood $U$ of $0$ in $\mathbb ...
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0answers
78 views

Questions about isotypic subspaces and co-isotypic subspaces of $V$.

Let $G$ be a finite group and $\chi: G \to \mathbb{C}^*$ a character. Let $(\pi,V)$ be a representation of $G$. The $(G, \chi)$-isotypic subspace of $V$ is defined by $$ V^{\chi}=\{v \in V: ...
1
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0answers
108 views

Why are Mobius map conformal at infinity?

Why are Mobius map conformal at infinity? I think I'm missing a subtlety! So we know an analytic function, $f$, is conformal at $z$ iff $f'(z) \neq 0$ But we see that the derivative of Mobius ...
1
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0answers
35 views

Convert function form to variate polynomials form

Let $f:{\Bbb Z}\longrightarrow{\Bbb Z}$ be a function, example $f(0)=3,\ f(1)=2,\ f(2)=0,\ f(3)=1$. Now that was "in decimal". We can think of it "in binary" format: $f(00)=11,\ f(01)=10,\ f(10)=00,\ ...
1
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0answers
83 views

How to minimise an objective function which is not a direct function of the decision variable?

I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my ...
1
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0answers
55 views

Use equational proofs to solve the problem

Use equational proof to solve the problem. $ \vdash A \lor (B \rightarrow A) \equiv B \rightarrow A $ These are the axioms and theorems.
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0answers
52 views

Wedge product of a one- form and a Kähler form

Let $x$, $y$, $v$, $w$ be coordinates on $R^{4}$ and $g$ be the Riemannian metric whose matrix with respect to these coordinates is $$g=\left ( \begin{array} {cccc} 1 & 0 & -kx & 0\\ 0 ...
1
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0answers
70 views

Preimage of a sequence of cauchy

Let $X,Y$ metric spaces. If $f:X\to{Y}$ is continuous, and $\{y_n\}$ is a Cauchy sequence in $Y$. Then, my question is $\{f^{-1}(y_n)\}$ is a Cauchy sequence in $X$? I´m sorry, in a second ...
1
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0answers
48 views

Splitting a number to a sum of perfect powers

Given a number $N$, is it possible to determine whether the number can be splitted to a sum of $c$ perfect powers: $N={a_1}^q+{a_2}^w+.....{a_c}^t$
1
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0answers
25 views

Does $f_t=\frac{z}{t}$ converges to $f(z)=z$ uniformly on $\mathbb{\hat C}$ as $t\nearrow 1$?

Does $f_t=\frac{z}{t}$ converges to $f(z)=z$ uniformly on $\mathbb{\hat C}$ as $t\nearrow 1$? I know it doesn't converge uniformly on $\mathbb{C}$. But I am always confused about $\mathbb{\hat C}$. ...
1
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0answers
53 views

Let X be a sequentially compact metric space. Prove that X is disconnected iff…

Let $X$ be a sequentially compact metric space. Prove that $x$ is disconnected iff there exist nonempty subsets $A$ and $B$ of $X$ and $\epsilon > 0$, with $A \cap B=\emptyset$, $A \cup B = X$, and ...
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0answers
29 views

There are 10 non distinguishable balls and one of these has different weight

There are 10 non distinguishable balls and one of these has different weight(one does not know whether it is weigh more or less than others). One can use scales 3 times to compare their weight. You ...
1
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0answers
53 views

ODE - Why Bernoulli

I forgot what the original exercise looked like (you can get it be by substituting u back again, but it is not relevant) but in class, by using the $u$ substitution $u = y+1$ and $dy=du$, the ...
1
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0answers
21 views

Prime Decomposition in Two Dimensional Manifold

There is Prime Decomposition in three dimensional manifold, so I want to ask is there Prime Decomposition in two dimensional manifold. I think no because $\mathbb T^2$#$\mathbb RP^2$=$\mathbb ...
1
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0answers
32 views

Finite element method boundary estimate

I just want to recall a formula. In finite element let $u$ be the exact solution, $u^h$ be the approximation. If $\Omega$ is the region, is there a formula $$||u-u^h||_{H^{\frac{1}{2}}(\Omega)}\leq ...
1
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0answers
60 views

Properties of eigenpairs of Sturm-Liouville Problems

I'm trying to prove the following and am struggling. I've tried to prove this in a similar method as done in proving this for matrices, but don't believe this is correct. Help is appreciated, as ...
1
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0answers
26 views

What is the probability density function of $e^{\sigma a}b$

I am trying to figure out the probability density function of $u = e^{\sigma a} b$, where $\sigma > 0$ and $(a, b) \sim N(0, \Sigma)$ $$ \Sigma = \begin{pmatrix} 1 & \psi \\ \psi & 1 ...
1
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0answers
182 views

How to prove orbit periodicity in some conservative systems?

Good afternoon, I have some trouble proving periodicity with a conservative system. The problems is that I don't know how to make a formal demonstration although I can see it quite true. I have the ...
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0answers
48 views

How to solve the following equality

Is it possible to solve the following equation analytically for $\beta$: $$y'(A+\beta B)^{-1}y = \alpha,$$ where $A$ and $B$ are both positive-semidefinite and symmetric matrices (essentially, some ...
1
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0answers
145 views

Question on stochastic process

let $(\Omega, \mathcal{F},\pi)$ be a probability space with $\sigma$-algebra $\mathcal{F}$ and measure $\pi$. Let $$X:[0,+\infty)\times \Omega\rightarrow \mathbb{R}$$ a family of random variables ...
1
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0answers
64 views

Quotient rule for the Jacobian

Is there an analog to the quotient rule that can be applied to the calculation of the Jacobian? Example: Can the jacobian of a quotient of two functions be decomposed into some series of linear ...
1
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0answers
38 views

Estimate distance between approximated posterior and true posterior

I'm working on a paper about using graphical models to do some prediction tasks with known observations. Since the model is complicated, finding the maximum a posteriori on the true posterior ...
1
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0answers
38 views

Equations for Intersection of Plane and circle

I am having a problem in getting the related equations for an intersection between a point on a plane and the edge of a circle in 3D space. Any suggestions? Or also a tangent plane and a point on a ...
1
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0answers
59 views

maximization over a max operator

My question is as follows \begin{equation} \max_W\max_{D_k}D_kW \end{equation} where \begin{equation} D_k = ((x_{i}^{1}-x_{j}^{1})^{2},(x_{i}^{2}-x_{j}^{2})^{2},...(x_{i}^{n}-x_{j}^{n})^{2}) ...
1
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0answers
39 views

Should I find the angle between a normal vector and a line to find the angle between the line and a plane?

I'm asked to find the angle between a line and a plane. I've calculated the normal vector of the plane to be $[7,4,-5]$ and the direction vector of the line to be $[3,-2,4]$. Is it right to find the ...
1
vote
0answers
48 views

Balls and their cardinality

Let metric space $X$ and fixed point $a\in X$ be given. I search for not restrictive assumption under what all balls in metric space $X$ with center in $a$ have the cardinality equal the carinality ...
1
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0answers
42 views

Eigenvalues of $B$ and $-B$

For a real symmetric matrix $B$, if we assume $QBQ^{T}=-B$ for some orthogonal matrix $Q$, then we know that the eigenvalues of $B$ and $-B$ are the same. My question is whether matrices of this ...
1
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0answers
34 views

Minimum Number of piles

Suppose we have N stones. They have the same size and weight, but they might have different strength.The $i$-th stone can hold at most $x_i$ stones on its top (we'll call xi the strength of the ...
1
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0answers
21 views

Describing a point in im $\Bbb P_A^n/I \rightarrow$ Spec $A$ as those $p$ for which $\emptyset \neq V(I) \subset$ Proj $\kappa(p)[x_0,…,x_n]$

Let $Z$ be a closed subset of $\Bbb P_A^n$ cut out by functions $f_i$. Consider the map $\pi: \Bbb P_A^n \rightarrow \operatorname{Spec} A$. I am trying to see why the points $p\in \pi (Z)$ are ...
1
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0answers
43 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
1
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0answers
53 views

Definition of Manifold's Orientation

I am reading the book of manifold. And I find there are many definitions about one object, such as orientation, Euler character and degree of map. I am confused with the conception of orientation. ...
1
vote
0answers
52 views

A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
1
vote
0answers
27 views

What is the dot product of two randomly generated 0-mean unit-vectors?

Given pairs of random 0-mean unit vectors, what kind of distribution is generated by their dot products? Judging from a number of results I've generated myself, the distributions appear to be ...
1
vote
0answers
30 views

Obtaining estimates for $|I(f)|$

Let $R >0$ and $R\neq 1$, set $I(f)=\int_{|z|=R}\frac{Log z}{z^2+1}\,dz.$ By obtaining estimates on $|I(f)|$ proves $I(f)\to 0$ as $R \to 0$ and also as $R\to \infty$. Using $z=Re^{i\theta}$ ...
1
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0answers
125 views

Is this a correct way to prove what the derivative of a polynomial function is?

After trying a polynomial long division problem with a lot of wondering how to go about answering it I proceeded by most likely overcomplicating things but the equation derived seems to work at ...
1
vote
0answers
21 views

Linear Gaussian system, covariance of the normalisation constant

If we have the following multivariate Gaussian distributions: $$p(x) = N(x|\mu_x,\Sigma_x)$$ $$p(y|x) = N(y|Ax + b, \Sigma_y)$$ Now how can you deduce p(y) ? p(y) is called the normalisation ...
1
vote
0answers
120 views

About $f(s)=\sum_{a^2+b^2>0} \frac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}=0$ and the Extended Riemann Hypothesis.

Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$). Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$ where the sum runs over all positive integers $a,b$ ...
1
vote
0answers
182 views

Algebra trick question, multiple choice, choosing one suggests a subgroup of $S_4$ has order 6!? - is it a trick question or am I silly

First: Let $G=S_4$ - the group of all permutations of {1,2,3,4}. I must prove $N=\{1,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$ is a subgroup of $G$ - this is easy, each transposition is its own inverse, ...
1
vote
0answers
64 views

How does SVD work?

Trying to find information, and, no-one seems to know the answers. I have a time-series, represented by $T = [0, 1, 1, 0, \ldots, n]$ the time series is then transformed into the Spectral results: ...
1
vote
0answers
39 views

Does this Gamma posterior make sense?

quick question about the form of a posterior distribution. Suppose that $\theta \sim Gamma(a, b)$ and that, given $\theta$, $Y$ has CDF $$F(Y\mid\theta) = 1 - e^{-\theta(e^y - 1)},\quad ...
1
vote
0answers
32 views

Litterature: Convex Geometry using differential forms

I've been looking for papers or books that discuss/use differential forms in convex geometry. I have been very unsuccessful in my search and was wondering if anybody here had come across such ...
1
vote
0answers
132 views

Rings, annihilators and (maximal) ideals

Let $R$ be a unital, associative, non-commutative ring. If $P$ is an ideal of $R$, what is the annihilator of quotients $R/PR$ and of $R/P$? Does something change if $P$ is supposed to be a maximal ...
1
vote
0answers
59 views

Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...
1
vote
0answers
47 views

Continuity domain for momentum operator

The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. ...
1
vote
0answers
37 views

proving that the series $ 1+2^{s}+3^{s}++ $ is divergent but borel summable

suppose that in the sense of distribution $ \int_{0}^{\infty}dxx^{n}T(x,s) =n^{s} $ for some distribution $ T(x) $ i do not know :( so if we apply borel generalized resummation $$ ...

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