All Questions
5,076 questions with no upvoted or accepted answers
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45
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Sensitivity of optimization solutions?
I've been working with the following optimization problem:
$$ \max \int \left(\frac{1}{2}\left\| x \right\|^2 + \tilde{u}(x)\right) \,ds(x) + \int \left(\frac{1}{2}\left\| y \right\|^2 + \tilde{v}(y) ...
- 383
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114
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The best error term for the second moment
Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
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64
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Relationship between the vortex filament equation and the transport equation
Let us consider the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$.
How is the Cauchy problem for the ...
- 277
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0
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395
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Topological entropy of logistic map $f(x) = \mu x (1-x)$, $f:[0,1] \to [0,1]$ for $\mu \in (1,3)$
As stated in the question, I want to find the topological entropy of the logistic map on the interval $[0,1]$ for a "nice" range of the parameter $\mu$, namely $\mu \in (1,3)$. I think the fact that $...
- 281
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63
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Coarea-like formula for BV functions (not their derivative)
Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Unfortunately, the formula
$$f = \int_{\mathbb R} \...
- 839
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2k
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The collected works of John von Neumann
Might there be an online collection of John von Neumann's collected works in pdf format? I'm particularly interested in his approach to applied mathematics(ex. shockwaves, hydrodynamics).
Note: I ...
- 3,871
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112
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Formal definition of episodic Markov Decision process?
David Silver, in his lecture 4 from his Youtube lectures, speaks about episodic Markov Decision Processes (MDPs) and Monte-Carlo Policy Evaluation.
I could not find a formal definition of episodic ...
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122
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References for the extension of Euler's phi function to number rings
Can anyone post a self-contained reference concerning the extension of the Euler phi function to number rings and its basic properties (reminiscent of those that the classic Euler phi function has)? ...
- 123
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56
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linear functions/hyperplanes vs. convex functions/convex sets in Hilbert space
The simplest Hahn-Banach extension theorem in Hilbert space $X$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the ...
- 1,461
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87
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Necessary Conditions for a Graph not possible to Rainbow Color?
Suppose we have a $t$-uniform hypergraph ($t \ge 3$) $G$, and have $t$ colors available. A question in my research is equivalent to asking what the necessary and sufficient conditions are on $G$ for ...
- 103
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94
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Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients
Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:
$$(i)\ 1 + 80q^3 ...
- 3,209
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140
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Reference for convergence to a Poisson Point Process
Edited after comment by Ofer Zeitouni
I have a sequence of discrete time stochastic processes $\big((S_n(i))_{i \geq 1}\big)_{n\geq 1}$ such that for every $n$, $i$,
\begin{equation}S_n(i)=\sum_{j=...
- 66
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63
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Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
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424
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Bounding the total variation distance between two measures from a given set
I have a distance on the space of probability measures on $[0,2]$. It is defined as such for two probability measures $\mu_1$ and $\mu_2$ :
$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \...
- 31
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215
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Resolvent estimate of compact perturbation of self-adjoint operator
Let $T$ be a selfadjoint operator on Hilbert space $H$. Then we know that there is a resolvent estimation $$\left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \...
- 31
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132
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Final step in Coppersmith?
In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
- 13.9k
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139
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(Semi-)Riemannian geometry for working PDE analysts
What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)?
The closest thing I know to this, are two books by ...
user60665
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448
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Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713
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759
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On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
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59
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Existence and uniqueness for semilinear parabolic problem using fixed point approach
Where can I find a proof of existence and uniqueness of solutions for a semilinear parabolic problem
$$u_t -\Delta u +f(t,x,u,\nabla u) =0$$
which is based on a fixed point approach?
user60665
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403
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Mathematical Problems of General Relativity II
In the introduction of D. Christodoulou's book "Mathematical Problems of General Relativity I", he refers a few times to the second volume. My question is does it exists? Has it been (or will it be) ...
- 133
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65
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Elliptic Dirichlet problems with measure boundary data
Can you point out any references on the Dirichlet problem for divergence-type elliptic operators with a Radon measure as boundary data?
user60665
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116
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Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
user57432
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225
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Convergent sequences in projective varieties
It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-...
- 1,252
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387
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Is there a known shorter axiomatization of NF than this?
Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...
- 11.3k
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267
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Publ Math Debrecen 1949
I am trying to find a paper (by Egervary) in Publ Math Debrecen, 1949 - I am having trouble finding an online version (the journal has an online presence but it does not go back that far). Any help ...
- 96.4k
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115
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What is known about the lower bound for the integers $n$ for which $n$ minus the first $k$ odd primes are $k$ composite numbers?
Question edited in view of the comments below
By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.
My ...
user57432
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44
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A name for algorithms that perform well in an asymptotic sense, when inputs are random
Is there a term for an algorithm that performs "well" (say, within a constant factor of optimality) in an asymptotic sense when a large number of random inputs are provided? For example, say I had an ...
- 4,069
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165
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Probability that the perturbed convex hull is larger than the original one
I am wondering if any convex geometers/probabilists have looked at the following question:
Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each ...
- 133
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152
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Reference request concerning the generalized Jacobian Conjecture
On page 287, A. van den Essen says:
Furthermore one can show that it suffices to prove JC for all $n \geq 2$ and for all $F$'s of the form:
$F=(l_1,\ldots,l_r,x_{r+1}+M_{r+1},\ldots,x_n+M_n)$ ...
- 2,837
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72
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Invariant measures for a renewal process driven by Interarrival times bounded away from zero
Good morning, I apologize in advance if my question sounds too basic but after some research I was unable to come up with satisfactory answers to my doubts.
I am currently studying a model which ...
- 277
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142
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Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.3k
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46
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linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
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0
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121
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Barnes's double $\Gamma$-function, $\gamma_{h}(0) = -\frac{1}{12}$?
I apologize if I may have gotten the name of the function incorrectly. The function $\gamma_h(x)$ is defined in Appendix A of the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/...
- 425
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35
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Iterating the Voter protocol
Assume you have an array of length $n$ filled with the numbers $1,2,...,n$. (Actually, it only matters that all numbers are different.) This corresponds to a Dirac delta distribution for the number ...
- 4,829
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84
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Name of an equivalence identity on sums of weighted binomial coefficients
In my research, I have encountered the following equivalence identity:
For $a,b,n\in \mathbb{N}$, the equivalence
$$
\left({\sum_{0\leq w < b}n^w}\right)^a \substack{\equiv\\n^b} \sum_{0\leq w ...
- 103
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0
answers
140
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Finite group and cyclic cover
Suppose the finite group $N$ surjects to finite group
$F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and
surject to $F$.
But is this ...
- 1,755
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89
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Hausdorff methods of summation
From the book of Boss "Classical and modern methods in summability":
"The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
- 109
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80
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Not exactly directed percolation
Is the following problem known/well-studies? I'm looking for references or a name that I can look up.
I start with $N$ cell, each one divides into two cells, each one of the new cells either dies ...
- 371
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65
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Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at ...
- 623
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257
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Hercules and the Hydra with time constraints
The game of Hercules vs. the Hydra can be put in terms of a single number in hereditarily-factorized form. For example, if the Hydra is $2^{19^3} \cdot 5^{11^7}$, Hercules must choose between two ...
- 2,302
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74
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Transformation or correspondence between language and real number
As we know, formal language can be regarded as a set of strings of alphabet, and real number can be regarded as sequence generated by set of integers, for example, denominators of the simple continued ...
- 3,104
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98
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Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
- 10.4k
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94
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Name and theory of multi-valued functions $F:\mathbb{N}^k \rightarrow \mathbb{N}^l$
In computability theory there are considered mostly single-valued functions $f:\mathbb{N}^k \rightarrow \mathbb{N}$. (Let $\mathbb{N}$ be a placeholder for $\mathbb{N}$ or any initial segment $[0,n]$ ...
- 9,730
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0
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167
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Negative $q$-binomial series: reference request
There seems to be a result for formal series (I hope this is right) for all integer $r\ge 0$
$$
\sum_{n\ge 0} (-x)^n\ {{n+r}\choose{r}}_{q} = (1+x)^{-1}(1+qx)^{-1}\dots
(1+q^{r}x)^{-1}
$$
where the $q$...
- 1,143
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0
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283
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A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular
A paper I'm reading in representation theory states the following result:
Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
- 6,180
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195
views
Automorphisms of weighted quiver
I am reading this paper strongly primitiv species with potentials I : mutations.
In page 6, they give the definition of weighted quiver: a weighted quiver is a pair $(Q,d)$, where $Q$ is a loop-...
- 775
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0
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234
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whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?
I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...
- 3,104
0
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0
answers
124
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Reference for the Hardy maximal function on the torus
I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
- 3,425
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0
answers
86
views
Generalized concept of subgraph (input/output graphs)
The usual definition of a vertex-induced subgraph goes like this:
A vertex-induced subgraph1 is a subset of the vertices of a graph $G$
together with any edges with both endpoints in this subset....
- 9,730