Most active questions
215 questions from the last 7 days
2
votes
0
answers
66
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545
1
vote
1
answer
41
views
Relationships between two stochastic matrices
Consider two $n \times n$ stochastic matrices $A$ and $B$. If for any two probability vectors $x$, $y$ in $R^n$, we have $xA=yA$ implies $xB=yB$, what can we say about the relationship of $A$ and $B$?
- 11
-1
votes
0
answers
58
views
Solving special multivariable limits by Euclidean geometry
General Problem: Inspect function $L(a)$ for $a\in \mathbb{R}^{n}$, given that:
$$L(a)=\lim_{x\to 0_{+}}\frac{x^{a}}{F(x)}$$
Notation legends:
$x=(x_1,\ldots,x_n),\ a=(a_1,...,a_n),\ x^a=x_{1}^{a_1}\...
- 231
1
vote
0
answers
59
views
Minimum number of sets for a union-closed sets conjecture counterexample
Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
- 1,000
3
votes
0
answers
48
views
An algebraic group $G$ over $L^+$ such that $G_{\mathbb{R}}$ is compact for almost all embeddings
Does there exist a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is ...
- 105
1
vote
1
answer
55
views
Proving bound on expectation of likelihood ratio involving mixtures
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 13
0
votes
0
answers
46
views
Fractal dimension using wavelets [closed]
I'm trying to estimate the fractal dimension of a function.
I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation
$$
\alpha = -...
1
vote
0
answers
111
views
Zariski Connectedness Theorem: From Analytic & Topological Viewpoint
Let $p:Y \to X$ be a proper, flat (see later why) surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ ...
- 6,038
-1
votes
0
answers
45
views
Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
-3
votes
0
answers
75
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
1
vote
0
answers
56
views
Quantitative multivariate CLT from quantitative CLT of linear combinations
Suppose $Z_1, \ldots, Z_k$ are random variables with mean $0$ and variance $1$ that are "approximately jointly Gaussian" in the sense that for any scalars $c_1, \ldots, c_k$, we have that $\...
- 111
1
vote
0
answers
43
views
Could this closed-form expression for the integral of the Riemann $\xi$-function along the critical line provide new insights?
The Riemann $\xi$ and $\Xi$-functions are respectively defined as:
\begin{align}
\xi(s) &= \frac{s\,(s-1)}{2}\, \pi^{-s/2} \,\Gamma\left(\frac{s}{2}\,\right) \zeta(s) \qquad s \in \mathbb{C} \\
\...
- 21
0
votes
0
answers
49
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,038
0
votes
0
answers
47
views
Maximize mixing in a 12 person dinner party [closed]
Is this question well posed? If not, can you improve it? If so, what is the solution?
I am holding a dinner party for 12 people. Their names are A, B,...L. The seats are numbered: 1, 2, ... 12. The ...
- 101
0
votes
0
answers
85
views
Is there a symbolic computation program that can deal with differential forms, Stokes theorems, the Hodge * operator, etc
I am looking at a messy series of computations of integrals of differential forms on manifolds with boundary, involving repeated application of Stokes' theorem, and also involving the Hodge * operator....
- 1
0
votes
0
answers
67
views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported.
I am wondering if we can ...
- 91
2
votes
0
answers
34
views
Maximum number of connected components of surfaces in three dimensions, what is known?
Part of Hilbert's 16th problem is:
It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the ...
- 21
-5
votes
0
answers
45
views
New Framework for Handling Undefined Mathematical Operations: Seeking Comments and Suggestions [closed]
The framework addresses undefined mathematical operations like zero raised to the first power (0^1) and division by zero (1/0) by introducing definitions that include infinity as the inverse of zero. ...
- 1
-1
votes
0
answers
33
views
Tightest decreasing majorant
I had asked this question here but received no answer.
Let $O$ be an operator that maps sequences to sequences such that the elements of the sequence $O(a)$ are given by
$$\bigl(O(a)\bigr)_n ~{}={}~ \...
- 349
3
votes
0
answers
36
views
Proving a proposition about the provability logic GL without using its completeness theorem
Let $GL$ be the provability logic containing the axioms $K := \Box (\varphi\to \psi)\to (\Box \varphi\to \Box \psi)$ and $L := \Box(\Box \varphi \to \varphi)\to \Box \varphi$, along with the ...
0
votes
0
answers
41
views
Markov chain on the real line: Numerical methods for evaluating the stationary distribution
Consider a Markov chain on the real line with transition probabilities
$$
p(x_0,x)=\mathbf 1_{\{x\geq x_0+\alpha\,\cup\,x\leq x_0-\beta\}}\phi(x)+\delta(x-x_0)\left(\Phi(x_0+\alpha)-\Phi(x_0-\beta)\...
0
votes
0
answers
64
views
Copy and repeat or copy and sum integer coefficients
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor.
$$
Let $T(n,k)$ be an integer coefficients with row length $f(n)$ (number of zeros in the binary expansion of $n$ plus $2$ for $n>0$ with $f(0)=1$...
- 4,938
3
votes
0
answers
28
views
Given a metric space $X$, is there a natural way to view the quasi-isometry group $QI(X)$ as a topological group?
Given a metric space $(X,d)$, we define $QI(X)$ as the set of quasi-isometries $f : X \to X$, modulo the equivalence relation
$$
f \sim g \ \ \ \ \text{ if and only if } \ \ \ \sup_{x \in X} \ d(f(x)...
- 639
-3
votes
0
answers
33
views
Bayesian Inference for Parameters Estimation in ARMA Model [closed]
In the usual sense, Maximum Likelihood Estimation is the common method for Parameter estimation in ARMA(p,q) model.
If I am looking to estimate parameters for ARMA(p,q) with Bayesian Inference, how ...
- 1
0
votes
0
answers
53
views
Special determinant formula
Consider two column vectors $\textbf{a}$ and $\textbf{b}$ of length $k$ and $m$ respectively, $km$ variables denoted $y_{i,j}$ (i=1 to k, j=1 to m), and a quadratic form $\textbf{y}^{T}\mathbb{M}\...
- 419
1
vote
0
answers
29
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
vote
0
answers
38
views
Reference request - Fourier multiplier of vector valued function
I would like to understand the concept of multiplier for vector valued functions and find appropriate references for the multiplier theorems out there.
For instance say that we would like to express $\...
- 111
0
votes
0
answers
48
views
Translation Invariants of Polynomials
The function $f(k)$ is a (numerical) polynomial in $\mathbb{Q}[k]$, and the set
$ S_f = \{ f_d : d \in \mathbb{N} \} $
is a set associated with $ f $, where $f_d(k)=f(k+d)$.
I am interested in finding ...
- 89
3
votes
0
answers
39
views
Obstruction theory for specializing perfect complexes?
I'm considering a problem around the moduli of perfect complexes.
Let's consider a $X$ smooth proper over Spec($R$), $R$ is a DVR, $char=0$, equipped with a perfect complex $\mathcal F$ on $X_{K(R)}$.
...
- 31
-1
votes
0
answers
42
views
Homomorphism from field of hyperreals to field of reals?
I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?)
Assuming that ...
- 55
3
votes
0
answers
42
views
When does a lax monad morphism induce a functor between categories of algebras that preserves reflexive coequalisers?
Let $S$ be a monad on a category $\mathbf C$. If $\mathbf C$ is cocomplete, and the category of algebras $\mathbf C^S$ admits reflexive coequalisers, then $\mathbf C^S$ is cocomplete. Thus, it is ...
- 10.7k
1
vote
0
answers
34
views
Vertex coloring of the Rado graph
Is there a reference for the following fact about the Rado graph (the random countable graph) which came up in an answer to this question?
If the vertices of the Rado graph $G=(V,E)$ are colored with ...
- 13.4k
-1
votes
0
answers
42
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
-4
votes
0
answers
25
views
Instrumental Variable model prove inequality holds [closed]
very stuck on this proof for my homework, professor didn't really teach this concept as it was supposed to be "self-learning", so I'm not really sure where to start.
homework problem
1
vote
0
answers
26
views
On $N$-partition of some common subsets $\Omega\subset\mathbb R^d$
Let $\Omega\subset\mathbb R^d$ be compact and convex, and denote by $\ell$ the normalised Lebesgue measure such that $\ell(\Omega)=1$. Let $N$ be an arbitrary but fixed integer.
In this post we set $d=...
- 1,399
-1
votes
0
answers
23
views
A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 49
0
votes
0
answers
52
views
A question of irreducibility of certain affine algebraic sets
Let $K$ denote an algebraically closed field of characteristic zero, and let $p_1(T), \dots, p_m(T)$ denote $m$ irreducible polynomials in $K[x_1, \dots, x_n][T]$ of degree at least $1$. Set
$$ S= \{ (...
- 6,224
2
votes
0
answers
35
views
On the solution sets to the ODE $(a^2-1)Q''(a) + \left(3a-\frac{d-1}{a} \right)Q'(a) + \frac{3}{4}Q(a)=0$
Looking for self-similar, radial solutions to a certain nonlinear wave equation in $\mathbb{R}^{d+1}$ leads to studying the following linear ODE in the self-similar variable $a\in[0,1)$:
$$(a^2-1)Q''(...
- 890
-2
votes
0
answers
27
views
Convergence of $ \vec{x_{n+1}} = A^{-1} f(\vec{x_{n}},\vec{y_{n}}), \:\:\: \vec{y_{n+1}} = A^{-1} g(\vec{x_{n}},\vec{y_{n}}) $
I have two systems
$$ A \vec{x} = f(\vec{x}, \vec{y}), \:\:\: A \vec{y} = g(\vec{x}, \vec{y}) $$
Both have the same constant, square, invertible matrix $A$. I implemented an iterative algorithm with ...
- 97
1
vote
0
answers
31
views
Primary invariants on MAGMA for a graded ring
I have asked this question on mathstacks, but a collegue of mine recommended me to post it here.
I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
- 11
0
votes
1
answer
37
views
Action of point stabilizers in finite doubly transitive groups
Suppose that $(H,X)$ is a finite faithful doubly transitive permutation group (where $H$ acts on the set $X$). Moreover, suppose that $H$ also acts doubly transitively (and faithfully) on a set $Y$, ...
- 4,547
-1
votes
0
answers
26
views
Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\mathbf{x})$
Let $f(\mathbf{x})=f(x_1,x_2,\dotsc,x_N)$ with $N>2$ be a real and continuous function and $f(\mathbf{x})\ge f_0$ for any $\mathbf{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\dotsc,x_N$ be the i.i.d. ...
- 375
-1
votes
0
answers
18
views
What is the expected value of the set when N elements are chosen from the same probability distribution?
Suppose we have a parameter that follows some probability distribution $f(x)$. When simulating an $N$-body with that parameter as an attribute, how should values of the parameter be chosen?
Let each ...
- 1
0
votes
0
answers
38
views
Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
- 455
2
votes
0
answers
50
views
Convergence of finite-difference method for Cauchy-Riemann equations
Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$.
We will assume the following: we are ...
- 312
0
votes
0
answers
42
views
Ranges of operators acting on Fréchet Spaces
My question is in the spirit of Reference request: Baire's theorem for operator ranges. It mentioned that :
Finite intersections and sums of operator ranges are operator ranges.
Images and pre-...
- 101
1
vote
0
answers
49
views
Square-integral involving Brownian bridge
Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of
$$
\int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt?
$$
As noted I am interested ...
- 2,217
3
votes
0
answers
54
views
Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?
This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested:
Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
- 2,830
1
vote
0
answers
41
views
Can conditional distributions with respect to a sufficient sub-$\sigma$-algebra be represented by a single Markov kernel?
Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be ...
- 111
0
votes
0
answers
35
views
Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467