Most active questions
215 questions from the last 7 days
-2
votes
0
answers
82
views
Can this prime pyramid reveal deeper insights into prime distribution? Has someone seen this pattern before? [closed]
Definition
The Burz Prime-Number Pyramid is a triangular arrangement of consecutive integers, structured such that the length of each row corresponds to a prime number, except the first row which ...
- 1
0
votes
0
answers
91
views
Integral form of linking number
I am reading the paper "Gapless Floquet topology" by Cardoso et al and the following section got me confused.
I understand that now $F$ lives in a space where a 2 dimensional subspace is ...
- 101
1
vote
0
answers
86
views
Specific regularity in bipartite graphs
Let $G(A,B)$ be a bipartite graph with $|A| = |B| = n$, where $n$ is sufficiently large. The average degree of $G$ is $d = \frac{e(A,B)}{n}$, where $e(A,B)$ denotes the number of edges between sets $A$...
- 359
4
votes
1
answer
65
views
Mapping properties of the Schrödinger semigroup
The Schrödinger semigroup $e^{t(-\Delta +V(x))}$ for Kato class potentials is fairly well-understood. A classical reference is the AMS paper "Schrödinger Semigroups" by Barry Simon. I was ...
- 1,045
2
votes
1
answer
71
views
Coradical filtration for comodules is exhaustive
It is a standard fact in the theory of coalgebras and comodules that, given a coalgebra $C$ and a comodule $M$ over $C$, the coradical filtration
$$M_n := \Delta^{-1}(M\otimes C_0 + M_{n-1}\otimes C)$$...
- 518
2
votes
0
answers
90
views
Diamonds on supercompact $\kappa$ after a $\kappa$-c.c. forcing
Let $\kappa$ be supercompact. Then the (supercompact) Laver diamond holds at $\kappa$: There is $f:\kappa\to V_\kappa$ such that for all $\lambda\geq \kappa$ and $x\in H(\lambda^+)$ there is $j:V\to M$...
- 61
6
votes
0
answers
93
views
Generic representations of $\mathrm{GL}_n(\mathbb{R})$
Let $F$ be a local field of characteristic $0$, $G=\mathrm{GL}_n(F)$.
When $F$ is $p$-adic, Bernstein and Zelevinsky classified the irreducible generic representations. The statement is:
Let $\delta_{...
- 709
2
votes
0
answers
80
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
-5
votes
0
answers
83
views
Every smooth function contains a bijection [closed]
Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
4
votes
0
answers
92
views
Transferring $A_\infty$-structure from a module to its homology
Given an $A_\infty$-module $M$, which is a graded module $M=\bigoplus_{k\in\mathbb{Z}}M_k$ with morphisms $m_n^M\colon A^{\otimes(n-1)}\otimes M\rightarrow M$ of degree 2-n satisfying the $A_\infty$ ...
- 213
1
vote
0
answers
76
views
Can one decompose quasi finite morphism as a composition of an open immersion and a finite morphism?
Can one decompose quasi-finite separated morphism of schemes as a composition of an open immersion and a finite morphism?
I am willing to assume that all the involved schemes are Noetherian.
- 2,649
-2
votes
0
answers
72
views
There is a typo in Stall's textbook on Set Theory: unable to prove the trichotomy of sets (m ∈ n or m = n, or n ∈ m) [migrated]
Here is the textbook, chapter 7, page 300. This lemma seems very of important, and I've spend about 8 hours trying to figure it out, but I'm unable to prove even the weaker version of the lemma (only ...
- 105
1
vote
0
answers
72
views
Hasse principle for Brauer groups of fields of transcendence degree 2
In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
- 41
0
votes
0
answers
73
views
Can a generalized root formula exist for polynomials with finite degrees? [closed]
Let $p\in\mathbb{Z}[x]$, set $d = \textbf{Deg}(p)$, and write $$p(x) = \sum_{i=0}^{d}{a_ix^i}$$ for some sequence $\{a_0,a_1,a_2,...., a_{d-1}\}$. Is there a mapping $\mathcal{F}$ so that $$\mathcal{F}...
3
votes
0
answers
92
views
+100
Can I get a spherical coordinate from a real cocycle?
The Setting
I am currently working on a project in Topological Data Analysis (TDA), where I aim to construct a density-robust spherical coordinate associated with a dataset $X$, sampled from a ...
- 71
3
votes
0
answers
89
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 73
5
votes
0
answers
62
views
Underlying noncommutative topologies of noncommutative complex varieties
Let $X$ be a (separated) complex algebraic variety. Then we can view its analytification $\newcommand\topo{\text{top}}X^{\topo}$ as a locally compact Hausdorff space. I wonder whether the same ...
- 2,856
-3
votes
0
answers
64
views
Can both conditions about vertex degrees hold true in a planar graph? [closed]
I am working on a problem about planar graphs and trying to understand if two statements can both be true at the same time.
The problem states that for any planar graph with at least 3 or more ...
- 1
2
votes
0
answers
116
views
Induced homology map zero implies zero in cobordism?
I had asked this in math stackexchange, but got no reply. Hence, I'm asking here.
[I'm no expert in (co)bordism theory, and I've been struggling with it for the past few weeks. Any good references on ...
- 21
2
votes
0
answers
65
views
Classification of centralizers of elements of finite simple groups of Lie type
I am currently studying the twisted Ree finite simple groups given by $^2G_2(3^{2n+1})$ and I was wondering if there is a reference for the classification of centralizers of elements in this family of ...
- 71
-4
votes
0
answers
62
views
Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...
1
vote
0
answers
70
views
To partition a triangle into $n$ convex pieces with sum of number of sides over all pieces maximized
This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.
Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over ...
- 5,979
5
votes
0
answers
97
views
Is the pullback of differential forms on a compact manifold smooth tame as a map of Fréchet manifolds?
In Hamilton's paper on the Nash-Moser inverse function theorem he shows that if $M$ is a smooth compact manifold and $V\to M$ a smooth vector bundle then its smooth sections $\Gamma(V)$ equipped with ...
- 51
5
votes
0
answers
101
views
Query about extender embeddings
This seems as though it should be a result which is possible to prove but I was just wondering if I have it right and also if there is a source for it.
Suppose that $j:V_{\alpha} \rightarrow V_{\beta}$...
- 2,125
0
votes
0
answers
89
views
+100
Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 537
0
votes
0
answers
100
views
Algebraic relations for $\Gamma$ function
Let $N$ and $n$ be positive integers with $\mathrm{GCD}(n,N)\ne1$. I want to prove the following claim:
$\Gamma\left(\frac nN\right)$, $\pi$ and the $\Gamma\left(\frac uN\right)$ ($u\in[1,N-1]$, $\...
- 3,998
2
votes
0
answers
85
views
Is there a natural topology for subsets of a fixed topological space?
This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?
The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a ...
- 719
1
vote
0
answers
63
views
Non metrizable uniform spaces
Bourbaki's book on general topology states that a uniform space is metrizable if it is Hausdorff and the filter of entourages of the uniformity has a countable basis. However, he doesn't provide an ...
- 111
1
vote
1
answer
49
views
Graph classes which have small edge k-cuts
I am interested in graph classes that have the following property: There exists a function $f(k)$ such that for every graph $G$ in the class, for every choice of $k$ vertices $v_1, \ldots, v_k$ in the ...
- 526
-3
votes
0
answers
73
views
Is this a conclusion group for a new fundamental geometry problem? [closed]
Let σ(n) represent all possible values of the types of different lengths of segments connected to each other among n points in the definition,such as in the plane,σ(3)=(1,2,3),σ(3)min=1,in the ...
1
vote
0
answers
66
views
Construction of the smallest nucleus above a prenucleus: what does this proof tell us?
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is implicit:
...
- 32.5k
0
votes
0
answers
78
views
Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
4
votes
0
answers
57
views
Positivity of elementary symmetric polynomials under linear fractional transformations
The general question
For $1\leq k\leq n$, let $$e_k(a_1,\dots,a_n):=\sum_{j_1<\dots<j_k}a_{j_1}\cdots a_{j_k}$$ be the $k$-th elementary symmetric polynomial.
Let $a_1,\dots,a_n<1$ and $e_1(...
1
vote
0
answers
96
views
Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
- 111
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
-5
votes
0
answers
83
views
Research without context is no research? [closed]
I was wondering what is research in mathematics. Can it be considered research only the research in open problems or it can be considered research also the finding of new formulas, new classes of ...
1
vote
0
answers
109
views
The value of the Hauptmodul at CM point
Let $J$ be a classical normalized $j$-invariant (that is, J=j-744).
Then, it is a classical result that $J(\tau)$ is an algebraic integer if $\tau$ is an imaginary quadratic number (that is, $a\tau^2+...
- 111
1
vote
0
answers
53
views
Description of all biholomorphic maps from annulus [duplicate]
Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected?
In ...
- 11
1
vote
0
answers
63
views
Reference request: Proof theory in $W_1^1$
Buss defined $V_2^1$ as a second-order bounded arithmetic corresponding to $\mathsf{PSPACE}$.
Later, Skelley introduced $W_1^1$, a third-order bounded arithmetic of $\mathsf{PSPACE}$.
Since the ...
- 11
0
votes
1
answer
77
views
Newton method for polynomials with random starting points
I know that this question exists, but unfortunately it doesn't cover my issue sufficiently.
Assume that we have a polynomial $p(x)$ of degree $n$ with real coefficients, we can assume that all its ...
- 1,171
0
votes
1
answer
37
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 $B$?
- 1
2
votes
0
answers
52
views
Can we bound the degree of a one dimensional smooth compact leaf of a holomorphic foliation in terms of its genus?
Let $X$ be a smooth projective variety over the complex numbers with a fixed ample line bundle $H$. Suppose that $\cal F$ is a foliation in curves over $X$ (which may be singular).
Can you find a ...
- 388
-1
votes
0
answers
44
views
How to prove the following theorem by distribution function and series
Let $g$ be nonnegative and measurable function in $\Omega$ and $\mu_{g}$ be its distribution function, i.e.,
$$
\mu_{g}(t)=\left|\left\{ x\in \Omega:g(x)>t\right\}\right|,\; t>0.
$$
Let $\eta>...
- 1
3
votes
0
answers
50
views
Martingale problem for the Wiener process
Consider $\Omega \triangleq \mathbf{C}([0,T];\mathbb{R})$, $\mathbf{F} \triangleq \mathbf{B}(\mathbf{C}[0,T];\mathbb{R})$ (Borelian $\sigma$-algebra) and $\mathbf{F}_t \triangleq \sigma \left \{ W_s, \...
- 395
1
vote
0
answers
80
views
The definition of Hodge bundles with metric
A system of Hodge bundles is a direct sum of holomorphic vector bundles $E = \oplus_{p+q=n} E^{p,q}$ with a morphism $\theta : E^{p,q} \rightarrow E^{p-1,q+1} \otimes \Omega_X^1$ such that $\theta^2 = ...
- 11
3
votes
0
answers
47
views
Semisimple elements and fixed points
The following statement seems to be well-known:
Let $X$ be a variety on which an affine algebraic group $H$ acts with
finitely many orbits and let $s \in H$ be semisimple. Then $H_s = \{h
\in H \mid ...
- 53
-3
votes
0
answers
48
views
Do the domains of the two square roots of a positive (unbounded) operator coincide? [closed]
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
- 415
3
votes
0
answers
81
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
2
votes
0
answers
55
views
+50
Local Lipschitz continuity of signature map $S:C^{1-\text{var}}([0,T],\mathbb{R}^d) \to \mathcal{H}$
Just came across the claim that the signature map (between path space and tensor space) is locally Lipschitz continuous with respect to the $1-$variation norm (see section A.2.1).
More specifically, ...
- 161
2
votes
0
answers
61
views
Wieferich primes and identities for the Euler quotients of $2^n+1$ and $\frac{2^n+1}{3}$
Let $n>1$ be odd integer.
Define the Euler quotient $a(n)=\frac{2^{\varphi(n)}-1 \bmod n^2}{n}$.
Number $n$ with $a(n)=0$ is Wieferich number and if it is prime
it is Wieferich prime.
It is open ...
- 25.4k