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+100

Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\...
Daniel Castro's user avatar
0 votes
0 answers
40 views

Gradient-based optimization of $n$ functions

I appreciate the willingness of everyone to assist me in advance. I am faced with a set of $n$ distinct convex optimization problems, each defined as follows: \begin{equation} \max\limits_{x \in \...
raian's user avatar
  • 1
2 votes
1 answer
158 views

Reference request: Best book to cite on a property of the family of Cauchy distributions

Kai Lai Chung once began a section of a textbook on probability by writing "Everybody knows" that $$ e^x = \sum_{n\,=\,0}^\infty \frac{x^n}{n!}. $$ (with those quotation marks). Other ...
Michael Hardy's user avatar
2 votes
0 answers
48 views

How many ways to win a game between two teams with arbitrary player skills

Suppose we have $n\geq 4$ players $p_1,\cdots,p_n$ of a game between two teams: team $A$ and team $B$ (disjoint sets, each with two or more players, so that $|A|+|B|=n$). Assume that each player $p_i$ ...
bernardorim's user avatar
7 votes
2 answers
280 views

Integral means vs infinite convex combinations

Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function. Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
Pietro Majer's user avatar
  • 56.5k
1 vote
1 answer
51 views

Equivalent condition for the Pick matrix being positive semidefinite

On the wikipedia page of the Nevanlinna-Pick theorem the following claim appears: Let $\lambda_1,\lambda_2,f(\lambda_1),f(\lambda_2)\in\mathbb{D}$. The matrix $P_{ij}:=\frac{1-f(\lambda_i)\overline{f(\...
JustSomeGuy's user avatar
0 votes
1 answer
99 views

Regular value theorem for Banach manifolds without surjectivity

It is well-known (e.g. Lang, Fundamentals of differential geometry, Prop. 2.3 in Chapter II) that the following extension of the regular value theorem holds for Banach manifolds: Let $\phi : M\...
Martin's user avatar
  • 141
3 votes
1 answer
204 views

Are there atoms in the lattice of intermediate logics?

A few days ago I stumbled upon this question on MS. The question is: Does the lattice of intermediate logics have an atom, i.e. an element that is strictly stronger than IPC but not strictly stronger ...
Navid's user avatar
  • 33
5 votes
0 answers
135 views

A puzzle with magic Egyptian tilings

Background I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
Max Muller's user avatar
  • 4,485
-3 votes
0 answers
43 views

Probability of sequence of estimated values belonging to a set

Suppose I have estimated residuals of an ARCH(1) process: $\hat{\varepsilon}_1, \dots, \hat{\varepsilon}_n$ from the sample of length $n$. On the other hand, I have "true" residuals: $\...
Grigori's user avatar
  • 17
0 votes
0 answers
34 views

Permutation for the fast computing of the $k$-th partition of $n$ in reverse lexicographic order

Let $a(n)$ be A000041 (i.e., $a(n)$ is the number of partitions of $n$ (the partition numbers)). Let $b(n)$ be A000070. Here $$ b(n) = \sum\limits_{i=0}^{n}a(i) $$ Let $c(n)$ be $k-1$ where $k$ is the ...
Notamathematician's user avatar
1 vote
0 answers
193 views

On $(k,\ell)$-sumfree sets

Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation $$x_1+\dots +x_k = y_1+\dots +y_\ell$$ in the set (for distinct $x_i$'s and $...
Sayan Dutta's user avatar
1 vote
1 answer
88 views

Characterization of Fellerian kernels

This question concerns Feller Markov kernels, similar to Vanessa's question. Terminology By 'Markov kernel' $N:E\to F$, we adopt exactly the same definition as Vanessa, with the exception that $E,F$ ...
Ano2Math5's user avatar
  • 103
1 vote
0 answers
140 views
+50

Specific type of PDE

While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices): $$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \...
Gennaro Marco Devincenzis's user avatar
-1 votes
0 answers
51 views

Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
Sean's user avatar
  • 313
0 votes
0 answers
76 views

I'm looking for the NLab page on particle species

This is just a reference request. I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently. If someone can point ...
Mozibur Ullah's user avatar
8 votes
1 answer
475 views

Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?

$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated ...
FShrike's user avatar
  • 569
-3 votes
0 answers
120 views

Extending the proof of Maschke's Theorem from finite groups to algebras

In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
Dale's user avatar
  • 431
5 votes
1 answer
217 views

Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
a_g's user avatar
  • 497
-4 votes
0 answers
84 views

What is the Measure of the Permutations of the Real Numbers? [closed]

Since the permutations of the real numbers would form a set of cardinality $\aleph_2$, do we just say it has infinite measure? It would seem to be the case for a Lebesgue measure. Is there a standard ...
Carl Gueck's user avatar
0 votes
0 answers
67 views

Are gaps and loopy games interchangeable in the Surreal Numbers?

The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and ...
Farran Khawaja's user avatar
2 votes
0 answers
54 views

Union of two open, open-unicoherent sets whose intersection is connected

I stumbled upon the following proposition, and haven't found an error in my proof yet. By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition. ...
Calvin Wooyoung Chin's user avatar
4 votes
1 answer
251 views

A pair of non-conjugate subgroups: a simple proof

$\DeclareMathOperator\SO{SO}$Set \begin{equation} \begin{aligned} \Gamma_1 &= \left\{ I_{6}, \; \gamma_1:= \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &...
emiliocba's user avatar
  • 2,321
-2 votes
0 answers
45 views

estimating probabilities like mortality rates among specific demographics is impossible? [closed]

Russell had a similar view to this that i will summarize it in the following example: What is the probability that an English man who reaches the age of sixty will die within one year, and the first ...
Hadibinalshiab's user avatar
-1 votes
0 answers
27 views

Numerical solution 2-D Poisson equation [closed]

I have some kind of Poisson equation: $-au_{xx}-bu_{yy} = f(x,y)$, $\ a,b>0$ and trying to solve it using FDM and Iteration methods (no matter which one, each method has the same problem). $$-a\...
BaDWerewolf's user avatar
1 vote
0 answers
88 views

Mixing of geodesic flow and rate of mixing

I know the following result which is if $X:=G/\Gamma$ where $G$ be a Lie group and $\Gamma$ is a lattice in $G$. Then geodesic floe $F=(g_t)$ is exponentially mixing i.e., there exist $\gamma > 0$ ...
User1723's user avatar
  • 307
11 votes
4 answers
1k views

Six consecutive positive integers with certain shape

Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ? If they exist, one of those six integers A will be the product of 2 and a square of ...
Tong Lingling's user avatar
-6 votes
2 answers
365 views

Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]

I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
0 votes
0 answers
111 views

Projective subvarieties are closed?

I want to show that projective subvarieties of a quasi-projective variety are closed. One possible solution should be the following: Let $W \subseteq \mathbb{P}^n$ be a quasi-projective variety and $V ...
psl2Z's user avatar
  • 101
0 votes
1 answer
119 views

Vector bundles over a homotopy-equivalent fibration

I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here. Let $\pi:N\rightarrow M$ be a smooth ...
Bence Racskó's user avatar
1 vote
0 answers
141 views

Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$. Algebraically its $Spec$ is quite different from $k$. For example: it has plenty non-trivial "line-...
Alexander Chervov's user avatar
0 votes
0 answers
14 views

Proving Hopf bifurcations for 3D system

I am working with a 3D continuous system of ODEs. I have found Hopf bifurcation numerically for a certain value of parameter. However, I want prove it analytically. Is it enough to show that the ...
SHR's user avatar
  • 43
2 votes
0 answers
40 views

Godement product of lax 2-natural transformations

Which of the two obvious choices for the Godement product of lax $2$-natural transformations is ‘correct’? Specifically, recall that for natural categories, functors and natural transformations as ...
Alec Rhea's user avatar
  • 8,977
1 vote
0 answers
26 views

Can a lift satisfy Chen's relation, geometric condition but not be a rough path?

Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties: \begin{align} &X_{s,t}=X_{0,t}-X_{0,s}\\ &\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\...
user479223's user avatar
  • 1,250
4 votes
1 answer
248 views

A question related to "Locally Sidorenko" type problem

Let $F$ be a bipartite graph and $\delta_F=\delta(F)$ be a constant. Let $p\geq 0$ be a given constant. Let $W$ be a graphon with $\int W=p$ and for any $A,B\subseteq \left[0,1\right]$ with $|A|,|B|\...
bc a's user avatar
  • 41
2 votes
1 answer
100 views

Galois action on étale path torsors

TLDR: How is the Galois action on étale path torsors defined? Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...
kindasorta's user avatar
  • 1,473
1 vote
0 answers
72 views

In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?

Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements: $\lambda$ being a random large prime such as $w^\lambda > 2\times m$ $1 < n < m−1$. m is ...
user2284570's user avatar
2 votes
1 answer
71 views

What is the expected remaining life duration of a cell in the $t\to\infty$ limit?

Consider the following population model: We start with a population of a single cell at time $t=0$. Each cell divides into $k$ new cells at random times $T$ distributed according to a probability ...
Sam's user avatar
  • 281
-4 votes
0 answers
78 views

Basic reproduction number for an epidemic model [closed]

I have a system of 6 ODE representing infected subpopulations of a model of epidemic $$ \begin{cases} \dfrac{de}{dt} =\dfrac{dy_2}{dt} = \phi y_1(y_4+y_5)-p_2 y_2\\ \dfrac{dq^*}{dt} =\dfrac{dy_3}{dt} ...
HRAUNMAKASH's user avatar
1 vote
1 answer
73 views

Testing for equal characteristic polynomials using a single determinant calculation

Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals. If $p_1 \ne p_2$, then there is some positive ...
Brendan McKay's user avatar
-1 votes
0 answers
54 views

Does algebraic independence of logarithms conjecture imply L-W?

Assume that algebraic independence of logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(...
joaopa's user avatar
  • 3,655
0 votes
0 answers
78 views

Renormalization group in condensed physics and field theory

Renormalization group in field theory is differenct from the one in condensed physics in that the former satisfies a differential equation, but the latter does not. Does Renormalization group in ...
XL _At_Here_There's user avatar
3 votes
1 answer
277 views

Unique factorization of ideals in a quadratic field

"Suppose $k = \mathbb{Q}(\sqrt{d})$ is a real quadratic field ($d > 1$ a square-free integer) with fundamental unit $\varepsilon$, normalized as usual so that $\varepsilon > 1$ with respect ...
MATH Enthusiast's user avatar
5 votes
3 answers
320 views

Bar construction in commutative algebras is calculated by pushout

$\DeclareMathOperator\colim{colim}$ Also asked in MathStackexchange here This is a statement in Lurie's Higher Algebra 5.2.2.4. Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
Xiong Jiangnan's user avatar
3 votes
0 answers
140 views

What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
Alexander Chervov's user avatar
-1 votes
0 answers
80 views

Is it possible to evaluate $ \int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{1+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}? $ [closed]

People suggested me to upload this question on math overflow and many people gave the numerical results by desmos and wolfram alpha Motivation for the problem here: Is it possible to evaluate $$ \...
Sbsty 's user avatar
  • 111
3 votes
1 answer
145 views

Green's kernel estimates on finitely generated groups

I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...
kobeahibe's user avatar
  • 111
2 votes
0 answers
29 views

Is there any example of linear operator which is bounded on all Besov spaces but not on Triebel-Lizorkin spaces

Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
Liding Yao's user avatar
2 votes
0 answers
83 views

Existence of extensions of a flat projective morphism

Suppose that $S$ is a noetherian integral scheme and $U\subset S$ is an open subscheme. Let $f:X\to U$ be a flat projective morphism. I would like to know whether (or when) $f$ can be extended to a ...
Nachhauseweg's user avatar
6 votes
1 answer
136 views

Inscribing one regular polygon in another

Say that one polygon $P$ is inscribed in another one $Q$, if $P$ is contained entirely in (the interior and boundary of) $Q$ and every vertex of $P$ lies on an edge of $Q$. It's clear that a regular $...
Glen Whitney's user avatar

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