# Questions tagged [regularization]

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### Theta-function in the lower half-plane

Standard theta function $$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$ has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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### Is regularization of infinite sums by analytic continuation unique?

There are ill-posed summations that we can assign values to, take for concreteness, $$S = \sum_{k=0}^\infty k$$ to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...
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### Less fundamental applications of Zeta regularization:

As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect. Are there less fundamental applications of zeta function regularization? By "less ...
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### Generalised limits via derivatives of integrals?

Assuming that $f$ is a continuous function, we have that $$f(x) = \frac{d}{dx}\int f(t)\,dt.$$ Assuming instead that $f$ has a removable singularity at $x=a$, and is otherwise continuous, we have ...
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### Regularizing the sum of all primes

In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes? $$\sum_{p \text{ prime}} p$$ Neither of these questions obtained a ...
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### Why we cannot speak about the main or natural regularization?

Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it". But in practice if we consider some ...
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### Assigning values to divergent improper integrals

I am investigating a question about possibility of assigning values to divergent improper integrals the same way as we do regularization on divergent series. The solution to the problem seems to be ...
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### Significance of Tikhonov matrix

I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
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### Comparing sizes of sets of natural numbers

It seems natural to consider $\lim_{q \rightarrow 1^-} \sum_{n \in S} q^n - \sum_{n \in T} q^n$, when it exists, as a way of comparing the sizes of two sets $S,T \subseteq {\bf N}$ that have the same ...
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### 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 ...
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### 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 $a \neq 0$...
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### A question about some notation involving the exclamation mark [closed]

What does the symbol ‘!’ signify? Is it $\text{argmin}$? For example, $\| A x - y \| = \min!$.
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### 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 $X\in\mathbf{S}_{++}^{n}$ (...
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### 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 ...
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### 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 ...
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### 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}...
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### 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 ...
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### 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. ...
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### 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-...