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3
votes
1answer
79 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 ...
0
votes
0answers
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 ...
5
votes
1answer
137 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 $a \neq 0$...
2
votes
1answer
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! $.
2
votes
0answers
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 $X\in\mathbf{S}_{++}^{n}$ (...
2
votes
2answers
297 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 ...
2
votes
1answer
216 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 ...
0
votes
0answers
189 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}...
5
votes
1answer
486 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 ...
2
votes
2answers
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. ...
21
votes
3answers
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-...
2
votes
1answer
355 views

Regularization of Zygmund functions

Dear community. I would like to derive a "good" estimate on $\frac{d}{dt}f_\epsilon(t)$, where $f_\epsilon$ is a regularization of a Zygmund-continuous function $f$, i.e. $|f(x-\tau)+f(x+\tau)-2f(x)|...
3
votes
3answers
877 views

Nonlinear circle fit with known radius

I have data points from a half circle and I already know the approximate radius. I want to find the circle which best fits the points using a fixed radius. How can I do this? If I solve the problem ...
2
votes
1answer
340 views

What is the regularity of the argument of a complex function?

Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
5
votes
2answers
676 views

What's the correct notion of determinant of a bilinear pairing?

By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...
13
votes
2answers
2k views

Zeta-function regularization of determinants and traces

The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form. Let A be an operator (on an infinite-dimensional ...