All Questions
152,891
questions
3
votes
1
answer
186
views
+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(\...
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 \...
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 ...
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 $p_i$ ...
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 ...
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(\...
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\...
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 ...
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 ...
-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: $\...
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 ...
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 $...
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 $E,F$ ...
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} \...
-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, ...
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 ...
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 ...
-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 ...
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 ...
-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 ...
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 ...
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.
...
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\\
&...
-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 ...
-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\...
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 $\gamma > 0$ ...
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 ...
-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 ...
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 ...
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-...
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 ...
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 ...
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|^\...
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|\...
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},\...
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 ...
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 ...
-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} ...
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 ...
-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(...
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 ...
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 ...
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{...
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 ...
-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
$$
\...
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 ...
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_{...
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 ...
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 $...