1
vote
0answers
66 views

Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...
10
votes
1answer
606 views

Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of $$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...
1
vote
0answers
219 views

Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$, $$\mathsf{F(p)}=a, ...
2
votes
1answer
91 views

amalgamation of locally finite groups

It is well known that in category of groups there are Push-outs so it is possible to realize amalgamation in some kind of most free way. My question is what about category of locally free groups? I ...
0
votes
0answers
26 views

Binary Operator and Vector Combination Problem [on hold]

Given a set of vectors, a set of binary operators, and a solution constraint. Is there a mathematical way to combine the vectors and operators to attain the solution constraint? The vectors are all ...
0
votes
1answer
104 views

Why does this vector bundle on the surface sit in this exact sequence?

Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E_0\subset E_1\cdots\subset E_s=E$ be the Harder-Narasimhan filtration. Suppose $E_1$ is $\mu$-stable and rank ...
0
votes
0answers
41 views

Determining primitive ideal space of C*-algebra

How do I in general determine a primitive ideal space of C*-algebra? Is there any standard way of determining it? Thanks.
3
votes
1answer
134 views

Hodge-Tate weights of induced representation

Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$ and let $E$ be another such extension, such that all the $\mathbb{Q}_p$ embeddings $K \to \bar{\mathbb{Q}}_p$ are ...
5
votes
3answers
378 views

is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. We know ...
2
votes
0answers
75 views

How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i ...
0
votes
0answers
20 views

product of two multivariate normal densities for the same vector, if one is only specified for a subset [migrated]

A random vector x with n elements has a multivariate-normal density f(x). Another distribution is known for m linear combinations of elements of x. The linear combinations are given in the form ...
1
vote
1answer
81 views

If $f$ is separately holomorphic on $\Omega$ then $f\in\mathcal{C}^0(\bar\Omega)\Leftrightarrow f\in L^1(\Omega)$

Let $\Omega\subseteq\Bbb C^2$ be open bounded (and connected), $f:\Omega\to\Bbb C$ separately holomorphic (i.e. $f$ is holomorphic in each variable when the other is fixed). Hartogs theorem is not ...
3
votes
0answers
69 views

Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and g is radially symmetric, the function $ (0, \infty )\ni t \mapsto g ...
9
votes
3answers
277 views

“The Two Sheriffs” puzzle -2: threshold for security

I've already asked a question “The Two Sheriffs” puzzle with wrong assumption. Yoav Kallus in his amazing answer using Fano plane showed that the problem has a solution in the case of seven suspects. ...
5
votes
2answers
219 views

Stabilisers of group actions

Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$. Suppose there exists some point $x \in X$ whose ...
1
vote
0answers
30 views

Edge-disjoint path-systems in infinite digraphs

Let $ D=(V,A) $ be a directed graph without backward-infinite paths and let $ \{ s_i \}_{i<\lambda},U \subset V $ where $ \lambda $ is some cardinal. Assume that for all $ u\in U $ there is a ...
1
vote
0answers
100 views

identity component of a formal group

Let $G=\operatorname{Spf} A$ be a formal group, the it is stated that the identity component $G^\circ$ (defined as $\operatorname{Spf} A_{\operatorname{fm}}$ for some open maximal ideal ...
4
votes
3answers
147 views

Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$. I am looking for the expectation ...
0
votes
0answers
46 views

Zero divisors and boundary elements of $A^{-1}$ [on hold]

I need to understand basic properties of (a) zero divisors in Banach algebras or rings, and (b) spectral properties of elements of the boundary of the set of invertibles in a complex unital Banach ...
4
votes
0answers
101 views

Nash's proof of De Giorgi-Nash-Moser theorem

I saw this question, but I think the answer didn't fully address what I want to know about it: Nash's paper on parabolic equations. It says almost everything developed later in elliptic and ...
0
votes
0answers
65 views

$K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Under what conditions does the following exact sequence split? $1\to E(R)\to Gl(R)\to K_1(R)\to 1$.
4
votes
0answers
50 views

An identity of complicated combinations of gamma functions (related to hypergeometric functions)

Can somebody help me in proving the following equation? \begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) ...
4
votes
1answer
126 views

Raising coefficients of a power series to some power

Suppose you are given a power series $P=\sum_{i=0}^\infty{a_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form $$ ...
3
votes
1answer
103 views

Is this characterization of (-1)-eigenspaces of the Weyl group of $E_6$ known?

I recently needed to know which circles $S$ in a maximal torus $T^6$ of the compact exceptional group $E_6$ yield one-dimensional subspaces $\mathfrak s$ of the Lie algebra $\mathfrak t^6$ that are ...
-4
votes
0answers
54 views

What is this formula Name? Can anybody teach me guide me to understand this? [on hold]

Formula If XT(t) at any time t relative to Timeframe T, then almost surely, there exists positive integers h and k such that every price belonging to the set [XT(t) – k , XT(t)+ k] is h(T) recurrent. ...
3
votes
1answer
70 views

When is the direct product of two graph cores itself a core?

A graph homomorphism $f$ is a function $f : V(X) \to V(Y)$ such that if $uv \in E(X)$, then $f(u)f(v) \in E(Y)$. If such an $f$ exists, write $X \to Y$. $X$ and $Y$ are hom-equivalent if $X \to Y$ and ...
5
votes
1answer
264 views

Primes isolated by large gaps to either side

Say that the $n$-th prime $p_n$ is isolated to degree $k$ (my notation) if the prime gap to either side is larger than $\log p_n$ to the $k$-th power: \begin{eqnarray*} p_n - p_{n-1} & > & ...
3
votes
0answers
105 views

A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.) ...
1
vote
1answer
108 views

Generating finite groups using subgroups

For finite groups $L \leq K$ we define $d(L,K)$ to be the least $n \in \mathbb{N}$ for which there exist $a_1, \dots, a_n \in K$ such that $\langle L, a_1 \dots, a_n \rangle = K$. Is there some $m ...
3
votes
3answers
199 views

Reference request about the representations of the group $PSL_2(\mathbb{F}_q)$

Is there a review/exposition of the representation theory of $PSL_2(\mathbb{F}_q)$ ? Like an enumeration of its irreducible representations and their dimensions as a function of $q$?
0
votes
0answers
82 views

A question on complexity notation

I am considering writing ''$\Pi^{n}_{i_{0},...,i_{n-1}}$-comprehension'' as abbreviation for ''$\Pi^{0}_{i_{0}}$-comprehension plus ... plus $\Pi^{n-1}_{i_{n-1}}$-comprehension'' in the context of an ...
1
vote
0answers
58 views

Outer measure preserving bijection

Suppose X is a Sierpinski set (So X is uncountable and every null subset of X is countable). Let f be a bijection on X. Must/Does there exist a non null subset Y of X such that for every subset W of ...
-2
votes
0answers
30 views

Independent and Dependent Variables [on hold]

Hi guys i have a question regarding independent and dependent variables. Provide an example that shows the variance of the sum of two random variables is not necessarily equal to the sum of their ...
4
votes
1answer
200 views

Self-contained book on Ricci Flow/Geometric Analysis

Can someone please tell me whether there is any self-contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self-contained I mean it does not assume that ...
1
vote
2answers
73 views

Integrability at $z$ of the 2-form $ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $

Given $g\in\mathcal{C}^1(\bar\Delta)$, and $z\in\Delta$, how can i prove that the 2-form $$ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $$ is integrable in $z$? At ...
0
votes
0answers
17 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Initially posted on math.stackexchange, was recommended that this is a more relevant forum: Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters ...
2
votes
0answers
160 views

An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
0
votes
0answers
22 views

Calculating age with decreasing year values [migrated]

This is my first question on mathoverflow.net, with everything this entails. This question is about the perceived duration of every year as one ages. We will call $Y_0$ the perceived duration of our ...
0
votes
0answers
47 views

Good covering of a (singular) curve in a complex surface

Let $W$ be a $2$-dimensional complex manifold and $C\subset W$ a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection ...
4
votes
2answers
142 views

Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$

In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...
1
vote
0answers
25 views

Biggest volume parallelotope inside the union of two parallelotopes

Given a parallelotope $P$ symmetric around the origin, and a vector $v$, such that $(P+v)∩(P−v)$ is not empty, is there a simple way to obtain a parallelotope $Q⊂(P+v)∪(P−v)$, symmetric around the ...
1
vote
0answers
28 views

Mappings between adaptive networks and Markov processes

Are there any known mappings between adaptive networks models (i.e. graph model representations of networks where the internal vertex dynamics and connectivity topology can change subject to specific ...
1
vote
0answers
46 views

boundedness of a sequence $ \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ implies that its temporal derivative is bounded as well [on hold]

Hi I have the next claim which I would like to find a proof of it. I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in ...
7
votes
2answers
188 views

A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. It is also known that the first moment exists for each of them, ...
-2
votes
4answers
294 views

Studying topology: which first, algebraic or differential? [on hold]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...
3
votes
0answers
111 views

Are all minimal totally separated spaces compact?

Let us call a space $(X,\tau)$ totally separated if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq ...
2
votes
1answer
110 views

Minimal totally separated spaces

Let us call a space $(X,\tau)$ totally separated (ts) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with ...
-1
votes
0answers
32 views

Is there any theorem which guarantees the existence of an eigenvalue for a non-normal matrix in the vicinity of its perturbed matrix? [on hold]

Let $A=(a_{ij})$ be a non-normal square matrix of order $n$ such that $a_{ji}=1/a_{ij}$ if $a_{ij}\neq 0$ and $0$ otherwise. If $B$ is the perturbed matrix obtained from $A$ such that $B$ also ...
1
vote
0answers
56 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
6
votes
2answers
189 views

A variant of random walk

Standard random walk assumes a sequence of iid RVs $\{X_i\}_{i\geq 0}$ and studied the distribution of $S_n=\sum_{i=0}^n X_i$. Here, I am wondering whether there is some work on $T_n=\sum_{i=0}^n ...

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