Questions tagged [regularization]
The regularization tag has no usage guidance.
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1+2+3+4+… and −⅛
Is there some deeper meaning to the following derivation (or rather one-parameter family of derivations) associating the divergent series $1+2+3+4+…$ with the value $-\frac 1 8$ (as opposed to the ...
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The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$
Question
I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...
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Assigning values to divergent oscillating integrals
I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ...
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Cutoff and regularization
Some variant of this question has probably been asked before on this site but my idea is to work with an explicit example.
Suppose we discretize the momentum space, so we work with:
$$\Lambda^{*} := \...
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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What's the true regularized value of product of all natural numbers?
Muñoz Garcia and Pérez-Marco - The product over all primes is $4\pi^2$ claims that the regularized value of product $\prod_{k=1}^\infty k$ is $\sqrt{2\pi}$ and of $\prod_{k=1}^\infty p_k$ over primes $...
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Regularised value of cardinality of non trivial Zeta zeros:
This is a straight forward question so apologies in advance
Consider the following sums:
$$\sum_k1_{\rho_k}$$
$$\sum_k{\rho_k}$$
(i.e. first sum counts non trivial zeros of Zeta function)
I want ...
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Derivative of Cauchy PV is equivalent to Hadamard regularization?
Let $\mathcal C$ and $\mathcal H$ denote the Cauchy principal value and Hadamard finite part. According to the Wiki:
$$
{\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int _{{a}}^{{b}}{\frac {...
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Normalizing a parameter in a regression
I am thinking about the possibility of making a parameter in my regression, let's say the $\lambda$ in a ridge regression, somehow, inside a range : $\lambda \in [0,1]$. Do you have any ideas how I ...
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Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series
Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...
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What's the regularized value of these divergent integrals: $\int_0^\infty \ln x \, dx$ and $\int_0^\infty \frac{\ln x}{x^2} \, dx$?
When playing with divergent integrals $\int_0^\infty f(x) \, dx$ and their transformations with operators $\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx$ and $\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)...
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A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
Introduction
I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...
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Divergent series summation beyond natural boundaries
I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
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Value of divergent sum $\sum_{n=0}^\infty (-1)^n n^n$
I'm hoping to find a reasonable value to assign to the divergent series $\sum_{n=0}^\infty (-1)^n n^n$ and $\sum_{n=0}^\infty (-1)^n (xn)^n$. For the first one, I have obtained something around 0.71, ...
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Hypermodulus and what mathematical objects have it
When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ...
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Improving regularity of the boundary of a convex set in Riemannian manifolds
Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$...
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Does this method analytically continue gap series series?
I was looking for ways to continue gap series, and it seemed to be that they could be continued outside of the boundary by simply turning
$$f(x)= \sum_{n=0}^\infty x^{n^k}$$
into
$$g(x) =- \sum_{n=1}^\...
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How is this expression for the regularization of integrals of monomials, given in a paper, justified? How strong is argument in favor?
In this answer by Carlo Beenakker he cites the following regularization formula:
$$\int_0^\infty x^p\,dx\mathrel{"="}\frac{(-1)^{p+1}}{(p+1)(p+2)},\;\;p=0,1,2,\dotsc,$$
citing Tafazoli - Calculation ...
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What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?
I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs.
So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
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Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
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A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
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Regularization of the area under hyperbola
So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
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A set of divergent integrals that I think, equal to $-\gamma$
So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-...
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Can we meaningfully ascribe values to these divergent integrals?
My gut feeling is that
$\int_0^\infty (1-\frac1{x^2})dx=0$
$\int_0^\infty (x-\frac2{x^3})dx=0$
$\int_0^\infty (x^2-\frac6{x^4})dx=0,$
etc, and in general,
$\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$
...
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On modified Euler product
Consider the modified Euler product as follows:
$$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$
Here $c$ is a constant
My questions are
Is there a compact representation for this ...
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Is the pseudoinverse the same as least squares with regularization?
Given a linear system $Ax=b$, the pseudoinverse of $A$ is found as the matrix $A^+$ such that $x=A^+ b$ where $x$ solves the least squares problem $\min \| Ax - b \|^2 $ and $x \perp \mathcal{N}(A)$. ...
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The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$
Background: I'm facing the computation of the zeta regularization of the infinite product given by
$$\prod_{m=-\infty}^\infty (km+u)$$
for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. ...
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Sparse signal recovery (nonlinear case)
Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
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More or less universal formula for regularization of divergent integrals?
Is there a simple formula that would produce the regularized value for the most common divergent integrals?
I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-...
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Theta-function in the lower half-plane
Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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Is regularization of infinite sums by analytic continuation unique?
There are ill-posed summations that we can assign values to, take for concreteness,
$$ S = \sum_{k=0}^\infty k $$
to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...
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Less fundamental applications of Zeta regularization:
As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect.
Are there less fundamental applications of zeta function regularization? By "less ...
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Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$? [closed]
The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates:
$$S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...
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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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Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?
I wonder if there is any sensible generalization of regularization which would be able to ascribe finite values to $\int_0^\infty \tan x \,dx$ and $\int_{-\infty}^0 \psi(x)dx$?
Perticularly, since $\...
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Interesting questions for inverse parabolic problems
I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
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Proximal Operator image of convex functionals
Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator
$$
\begin{aligned}
&\Gamma_0\...
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How to derive the solution of Tikhonov Regularization via SVD [closed]
The solution to Tikhonov Regularization is
$$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$
where $\sigma^2_{min}$ is the minimum of the singular values of $A$.
Then we apply $SVD$ to $A$ such that,
$$A=U\Sigma ...
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New/useful method for summation of divergent series?
Questions
$$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$
Also obeys (see background for argument):
$$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
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Why we cannot speak about the main or natural regularization?
Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it".
But in practice if we consider some ...
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Regularized linear vs. RKHS-regression
I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two.
Given input-output pairs $(x_i,y_i)...
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What is the generalized sum of the following series? $\sum _{x=1}^{\infty } \sqrt{s^2 x^2-1}$ [closed]
I tried Mathematica, various regularization methods, including Borel, with no result.
On Math.SE the question was attacked with claims that divergent series cannot have a sum, so I decided to ask at ...
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Regularization on divergent series [closed]
I want to compute $$ \sum ^{\infty }_{n=1}\dfrac {\left( 2n+k-2\right) !\zeta \left( 2n\right) \left( -1\right) ^{n-1}}{\left( 2\pi \right) ^{2n}}. $$
This series is surely not convergent for any ...
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Why to multiply the penalty by $n$ in the penalized least squares and likelihood?
In the SCAD paper by Fan and Li (2001), there exist two forms of penalized least squares as follows:
$$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$
...
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When will the upper regularization of a bounded function not defined?
Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...
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Existence of analysis regularization solution
I am interested in the optimization problem known as "analysis regularization":
$$ {\rm argmin}_{x \in \mathbb{R}^{p}}\frac{1}{2}\|y - Ax\|_2^2 + \lambda \|D^T x\|_1,$$
where $y \in \mathbb{R}^n$, $...
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Is there optimal sparse and minimum energy solution?
I am interested to know if anyone worked on the the inverse solution of the problem defined below, which means I want to find $\tilde{y}$.
$M_{\delta}^{\alpha}[\tilde{y}]=\|A\tilde{y}-\tilde{b}\|_{...
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Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
As from the title, I am currently dealing with this sum
$\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
in particular with $p=1/2,3/2,...$ (but once solved for $p=1/2$ one can derive wrt $a$ and find the ...
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Difference between Sobolev norm and L2 norm for regularization?
I am interested to find a inverse solution of the problem defined below, which means I want to find $\tilde{y}$.
$M_{\delta}^{\alpha}[\tilde{y}]=\|A\tilde{y}-\tilde{b}\|_{L_2}^2 + \alpha\|\tilde{y}\|...
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