All Questions

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Stability of moment representation of a scalar real-valued function

Let $f \in C([0,1],\mathbb R)$ be a continuous function. Define the moments of $f$ by \begin{align*} m_i(f) := \int_0^1 x^i f(x) dx, \end{align*} which yields a sequence of real numbers. We then ...
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Finding equivalent matrix combination

I have a program I've written that is solving some problems with some matrix-vector math, but I have a feature I want to add and while I've found a work around an analytic solution would be superior. ...
43 views

Smoothness and Cohen Macaulay

One always get the idea (almost a slogan in Alg. Geom.) that Cohen-Macaulay varieties will have some (mild) singularities and Gorenstein can be smooth. I found a smooth scheme that by construction ...
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Interpolation with double second differences [on hold]

My question is about an interpolation method used in an astronomy book that I would like to understand, and that can be found here: ...
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mutual information problem [on hold]

In mutual information we have: if $x$ and $y$ are independent then $p(x,y)=p(x)p(y)$ and then $I(X;Y)=0$. Do If $I (X;Y) = 0$ when $x$ and $y$ are independent?
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Fourier transformation [on hold]

I need an example s.t. The function f belongs to L^2 but not in L^1, and the fourier transformation is in L^1? If you have any hint that will be perfect.
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When does a “universal” quot scheme exist?

Suppose $M$ is a moduli space of semistable sheaves on a projective variety $X$. Let $v$ be some the discrete invariants. I would like to form a space $\mathcal Q(v) \rightarrow M$, where the fiber ...
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Existence and summability of cumulant [on hold]

I posted this question on math stackexchange, but no one answered. So I am seeking help here. 1) Is the statement "the $r$-th order moment exists" equivalent to "the $r$-th order cumulant exists"? ...
16 views

mutual information entropy problem [on hold]

In mutual information we have: if $x$ and $y$ are independent then $p(x,y)=p(x)p(y)$ and then $I(X;Y)=0$. Do If $Y (X;Y) = 0$ when $x$ and $y$ are not necessarily independent?
66 views

Doob Martingale: Where is the catch?

I am working on a research problem in uncertainty propagation that involves sums of possibly dependent random variables with bounded sets of support. I am attempting to use the method of bounded ...
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Non-normality of limit of random variables

I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that ...
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A general theory for boundary value problems

One can study the characterization of a linear differential operator $T$ from scalar product $(f,g)=\int_{a}^{b}f(t)g(t)dt$ and the theory of adjoint operators solving $Tf=g$ by finding a right ...
32 views

Power-spectrum of quasi-periodic functions

From Scholarpedia: Quasiperiodic oscillation is an oscillation that can be described by a quasiperiodic function, i.e., a function $F$ of real variable $t$ such that ...
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Investigate the $\inf$ of the sequence $(\sqrt[n]{|\sin{n}|})_{n=1}^{\infty}$ [on hold]

According to this MathStackExchange post years ago we have $\displaystyle\lim_{n\to\infty}(\sqrt[n]{|\sin{n}|})=1$. So the $\limsup$, the $\liminf$, and the $\sup$ of this sequence are clearly 1. But ...
31 views

For v = (x, y, z) let a, b, c denote the angles between v and the respective x, y, z axes. Show that cos^2(a) + cos^2(b) + cos^2(c) = 1 [on hold]

I am unsure how to approach this problem, as I have not yet learned many of the trig identities for working in 3 dimensions. The only thing I can think of is if A, B, and C have to add up to 180 (I am ...
17 views

Bounding function of norms in constrained vector space

$v$ is a vector of length $n$, where $v_1 = 1$ and very element $v_i \in [0,1]$ $w = \| v \|_1^1$ $x = \| v \|_2^2$ $y = \| v \|_3^3$ $z = \| v \|_4^4$ Can you recommend a strategy for achieving a ...