Questions tagged [regularization]
The regularization tag has no usage guidance.
71
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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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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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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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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 ...
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2
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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 ...
- 19.4k
12
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1
answer
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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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2
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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, ...
- 1,692
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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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8
votes
3
answers
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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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3
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633
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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 ...
- 1,304
7
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2
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911
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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 ...
- 1,793
6
votes
2
answers
1k
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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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2
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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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6
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1
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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 ...
- 2,102
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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 ...
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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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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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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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$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder
I'd like to share a problem that I have been dealing with for a longer time now.
In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
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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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4
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2
answers
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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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1
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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:
$$
{\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int _{{a}}^{{b}}{\frac {...
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1
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Interesting questions for inverse parabolic problems
I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
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Proximal Operator image of convex functionals
Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator
$$
\begin{aligned}
&\Gamma_0\...
- 4,969
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2
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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
$$g(x) =- \sum_{n=1}^\...
- 1,692
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votes
2
answers
433
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Regularisation of $\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
As from the title, I am currently dealing with this sum
$\sum_{n=0}^\infty \frac{1}{(a+n^2)^p}$
in particular with $p=1/2,3/2,...$ (but once solved for $p=1/2$ one can derive wrt $a$ and find the ...
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A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
Introduction
I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...
- 1,692
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2
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Theta-function in the lower half-plane
Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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votes
1
answer
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General method for under and over determined systems?
Suppose I have a system:
$$
Ax = b
$$
where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.
I'd like a combination of a ...
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3
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Nonlinear circle fit with known radius
I have data points from a half circle and I already know the approximate radius. I want to find the circle which best fits the points using a fixed radius. How can I do this? If I solve the problem ...
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3
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1
answer
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Why to multiply the penalty by $n$ in the penalized least squares and likelihood?
In the SCAD paper by Fan and Li (2001), there exist two forms of penalized least squares as follows:
$$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$
...
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0
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The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$
Question
I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...
- 1,692
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Regularised value of cardinality of non trivial Zeta zeros:
This is a straight forward question so apologies in advance
Consider the following sums:
$$\sum_k1_{\rho_k}$$
$$\sum_k{\rho_k}$$
(i.e. first sum counts non trivial zeros of Zeta function)
I want ...
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0
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Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series
Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...
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New/useful method for summation of divergent series?
Questions
$$ 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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0
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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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2
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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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2
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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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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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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 ...
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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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1
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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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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)|...
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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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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 ...
- 9,312
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0
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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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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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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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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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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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