# Questions tagged [regularization]

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### What intuitive meaning “determinant” of a divergency (divergent integral, series, germ, pole or a singularity) can have?

I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
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138 views

### Did anyone ever propose the distinction between “divergent to infinity” as opposed to “divergent but with finite average”?

There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...
1answer
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### A question on assigning finite values to divergent sums involving expression of primes

We know the following: $$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$ This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$. ...
0answers
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### Regularization of the area under hyperbola

So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
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### What is the regularized numerocity of prime numbers?

Basically, I want to find this sum: $$\lim_{s\to0}\sum_{n=1}^\infty \frac{p(n)}{n^s}$$ where $p(n)$ is the membership function of the set of primes. That is, $p(n)=1$ if $n$ is prime and $0$ otherwise....
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### Assigning negative integer moments to random variables with Hadamard regularization

Let $X\sim F_X$ denote a continuous random variable that admits a density $f_X$ with support $\mathcal S=\operatorname{supp}(X)\ni 0$ and assume $f_X(0)>0$. I am interested in defining a ...
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223 views

### A set of divergent integrals that I think, equal to $-\gamma$

So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-...
2answers
187 views

### Can we meaningfully ascribe values to these divergent integrals?

My gut feeling is that $\int_0^\infty (1-\frac1{x^2})dx=0$ $\int_0^\infty (x-\frac2{x^3})dx=0$ $\int_0^\infty (x^2-\frac6{x^4})dx=0,$ etc, and in general, $\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$ ...
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863 views

### On modified Euler product

I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it. Consider the modified Euler product as ...
4answers
963 views

### Is the pseudoinverse the same as least squares with regularization?

Given a linear system $Ax=b$, the pseudoinverse of $A$ is found as the matrix $A^+$ such that $x=A^+ b$ where $x$ solves the least squares problem $\min \| Ax - b \|^2$ and $x \perp \mathcal{N}(A)$. ...
1answer
161 views

### The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$

Background: I'm facing the computation of the zeta regularization of the infinite product given by $$\prod_{m=-\infty}^\infty (km+u)$$ for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. ...
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### Sparse signal recovery (nonlinear case)

Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
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196 views

### More or less universal formula for regularization of divergent integrals?

Is there a simple formula that would produce the regularized value for the most common divergent integrals? I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-...
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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 ...
3answers
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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 ...
2answers
501 views

### 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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### 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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257 views

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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/...
2answers
652 views

### 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 ...
1answer
241 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 ...
1answer
193 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 $a \neq 0$...
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208 views

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

What does the symbol ‘!’ signify? Is it $\text{argmin}$? For example, $\| A x - y \| = \min!$.
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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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493 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 ...
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630 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 ...
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791 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}...
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862 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 ...
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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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3k 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-...
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