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80 views

Regularization by mean curvature flow

I have a $C^{1,\alpha}$ surface defined as the graph of some function $\varphi : B \to \Bbb{R}_+$ ($B$ is a ball). This surface has positive and bounded mean curvature in the weak sense (since the ...
18 views

A possible condition on a set of vectors to be uniquely determined by their gramian?

Suppose you are given a gramian matrix $G=W^TW\in R^{k\times k}$, $W\in R^{d\times k},\;d>k$. The given gramian $G$ determines, up to a unitary transformation, the columns of $W$, i.e. an ...
138 views

Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$...
159 views

A question about some notation involving the exclamation mark [closed]

What does the symbol ‘!’ signify? Is it $\text{argmin}$? For example, $\| A x - y \| = \min!$.
339 views

On increasing the penalty term in convex optimization with regularization

Given the two strictly convex (unique solution) optimization problems as: $$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$ where $X\in\mathbf{S}_{++}^{n}$ (...
302 views

double integral and Hadamard finite part

Given the divergent integral $$\int _{0}^{\infty}dx \int_{0}^{\infty}dy \frac{x^{2}y+1}{1+x+y}$$ how can I apply Hadamard's finite part to give a finite meaning to it ? It is just made by ...
220 views

General method for under and over determined systems?

Suppose I have a system: $$Ax = b$$ where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$. I'd like a combination of a ...
196 views

Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system $$x=A^{-1}b$$ This inverse problem is basically a least square problem with a rank-1 update.  x=[uv^{T}...
491 views

What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?

Introduction: This is slightly edited and generalised version of a question I asked on the Physics Stack Exchange website. This question has a twin brother asked here on MO, only now we consider ...
1k views

Choosing the order of Tikhonov regularization of an inverse problem

This question is migrated from math.stackexchange. Let me first describe the problem I am trying to solve and then the question I have. I greatly appreciate anyone who can shine some light on it. ...
2k views

Understanding zeta function regularization

I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced zeta function regularization in a compelling way. The goal is to define the determinant of a positive self-...