All Questions
46,938
questions with no upvoted or accepted answers
3
votes
0
answers
247
views
Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?
In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...
3
votes
0
answers
144
views
On an inequality for the arithmetic function counting the number of primes $\lfloor n^c\rfloor$ in the spirit of Ramanujan's prime counting inequality
In page 3 of [1] (please see if you need it the book by Berndt) Axler refers an inequality that involves the prime-counting function $\pi(x)$ and that was deduced by Ramanujan. I'm curious to know if ...
3
votes
0
answers
199
views
The Fourier transform of a compactly supported smooth function on Lie groups over $\mathbb{Q}_S$, where $S$ contains finitely many primes and $\infty$
Let $G$ be a semisimple Lie group defined over global Field $\mathbb{Q}$. Let $S$ be a set of finitely many non-Archimedean places including Archimedean places. Let $P_{0}=M_{0}A_{0}N_{0}$ be the ...
3
votes
0
answers
141
views
How can one construct this dendrite?
In the early 1970s Pelczynski noticed that the only surjective isometries on $C(K)$ for the following compact Hausdorff space $K$ are $\pm Id$. I believe this was the first such example.
Quoting from ...
3
votes
0
answers
101
views
When is the least-area surface unique?
Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...
3
votes
0
answers
157
views
What are the common critical points of all Hitchin fibrations?
Let $C$ be an complex curve of genus at least two. Fix some rank $n$ and degree $d$, and consider the moduli $M_{Dol}(C, r, d)$ of stable Higgs bundles of rank $n$ and degree $d$, and the Hitchin ...
3
votes
0
answers
95
views
Number of real roots of a degree 10 polynomial with several parameters
Consider the real polynomial $x^2 + (x^2+1)(ax^8 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + g)$ where $a < 0$, $g > 0$ and there are no terms of degree $1$ or $7$ in the second bracket. Can this ...
3
votes
0
answers
185
views
What is the initial semiring category with a (commutative) semiring?
Recall that
The biinitial monoidal category with a monoid is given by the augmented simplex category together with the monoid $([0],\sigma^{0}_{0},\delta^{0}_{0})$ there.
The biinitial symmetric ...
3
votes
0
answers
75
views
A question about simple and finitely-generated objects in Grothendieck categories
Let $\mathcal{E}$ be a Grothendieck category and consider the following conditions:
(LFG) $\mathcal{E}$ is locally finitely generated (that is, the finitely generated objects of $\mathcal{E}$ generate ...
3
votes
0
answers
200
views
On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
3
votes
0
answers
40
views
Non-existence of local generators for Sobolev tangent subundles
Let $U\subset\mathbb R^n$ be a bounded open set, a rank $r\le n$ measurable tangent subbundle $\mathcal V$ on $U$ is a map to the Grassmannian $\mathcal V:U\to Gr(r,\mathbb R^n)$ which is only defined ...
3
votes
0
answers
78
views
Why are we interested in proper holomorphic embeddings?
Dealing with an $n$-dimensional Stein manifold $X$, a problem which captured the attention was to find proper holomorphic embeddings
$$
f\colon X\hookrightarrow\Bbb C^N
$$
for some $N$. Of course the ...
3
votes
0
answers
105
views
Next smooth number
I want to find the next $n \in \mathbb{N}$ such that
$$s < n = \prod_{p_i \in \mathbb{P}_B} {p_i}^{a_i}$$
Where $\mathbb{P}_B$ is the set of primes not greater than $B$
I know that we can generate ...
3
votes
0
answers
135
views
Matrix equation and spherical harmonics
I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$),
$$
\eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi)
$$
Similar to the ...
3
votes
0
answers
83
views
Calculations regarding the weight one Morphic cohomology
According to the paper "A theory of algebraic cocycles" (Friedlander and Lawson 1992; this is a summary of the original paper), theorem 8, gives the morphic cohomology of weight one. More ...
3
votes
0
answers
150
views
When is the Fermat Catalan surface a rational surface?
Related to Fermat Catalan conjecture and scholar.google.com didn't return any results.
Define the Fermat Catalan surface
$$ S_{m,n,k}: x^m+y^n=z^k$$
Where $\frac1m+\frac1n+\frac1k < 1$.
Q1 When is ...
3
votes
0
answers
76
views
connected Hopf algebra of infinite Gelfand-Kirillov dimension but of finite dimensional primitive space
I would like to know some examples of connected Hopf algebras which has infinite Gelfand-Kirillov dimension but with primitive space finite dimensional. Any commments are welcome!
3
votes
0
answers
188
views
Base-change theorems for stable $\infty$-categories
Omitting some technicalities, the base-change theorem for quasicoherent sheaves says that if we have the following diagram of (derived) schemes
$\require{AMScd}$
\begin{CD}
X \times_S Y @>\pi_2>&...
3
votes
0
answers
213
views
Enumerating multi-core binary partitions
An integer partition $\lambda$ of $n$ is called a binary partition provided that its parts are powers of $2$ (dyadic). Example: Let $n=3$. The binary partitions are $\lambda=(2,1)$ and $\lambda=(1,1)$ ...
3
votes
0
answers
185
views
Deformation to a normal cone for a holomorphically symplectic manifold
Let $X$ be a subvariety in $M$.
"Deformation to the normal cone"
is a holomorphic deformation of a neighbourhood
of $X$ in $M$ over the disk such that its central fiber
is the total space ...
3
votes
0
answers
100
views
Character formula for real representations
For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. The real representations of a real semisimple Lie algebra are classified using their ...
3
votes
0
answers
845
views
Reconstructing an analytic ring from its module category
When reading Lectures on Analytic Geometry, I found that in the data of an analytic ring, the underlying ring seems unnatural and the module category should be the soul, but in fact one needs the ...
3
votes
0
answers
103
views
Efficient computation of "higher order" Jacobi symbols
Let $N> 1$, and suppose $a \in (\mathbb{Z}/N\mathbb{Z})^{\times}$. There are efficient algorithms to compute the Jacobi symbol $\left(\frac{a}{N}\right)$ using quadratic reciprocity. Fix $k \geq 1$....
3
votes
0
answers
180
views
The Brauer group and the second Galois cohomology group
I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...
3
votes
1
answer
380
views
Minimum upper bound for sum of the entries of the inverse covariance matrix
Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$
and let $\mathbf{K}$ be the following $n \times n$ covariance matrix
$$\mathbf{K} = \...
3
votes
0
answers
116
views
intersection of two height 2 primes must contain a non-zero prime?
I saw in some contexts the following statement, which I do not have a reference for this:
"Kaplansky asked if in a Noetherian domain the intersection
of two height 2 primes must contain a non-...
3
votes
0
answers
168
views
Probabilistic behavior of greedy point selection in the plane
Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
3
votes
0
answers
104
views
Periodicity of algebraic $K$-theory in high enough degrees with finite coefficients
Given this it seems that higher algebraic $K$-theory and the etale one coincide in high enough degrees. The etale $K$-theory with finite coefficients is also Bott inverted $K$-theory, so it should be ...
3
votes
0
answers
243
views
Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
3
votes
0
answers
215
views
Confused about the definition of the Kahn-Priddy map
The Kahn-Priddy map is defined in various papers as follows:
Define $i:\mathbb {RP}^{n-1}\rightarrow O(n)$ by taking $\ell\in \mathbb {RP}^{n-1}$ to the reflection across $\ell^\perp.$ Define $j:O(n)\...
3
votes
0
answers
230
views
Is there a closed form of $ \displaystyle \sum_{k=0}^{\infty}{\frac{\phi^{xk}}{k!_F}}$
where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...
3
votes
0
answers
171
views
The group of fixed points of an involution of a Weyl group
Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$.
Let $W=W(R)$ denote its Weyl group.
Let $S\subset R$ be a basis of $R$ (a system of simple roots).
Let $D=D(R,S)$ denote the ...
3
votes
0
answers
238
views
Counting twin primes with a sieve-like algorithm
The sequence A002822, denoted as $S$, represents all the twin primes except $\{3, 5\}$. Other than that exception, $k$ and $k+2$ are twin primes iff $(k+1)/6\in S$. Let $S(N)$ be the subset of $S$ ...
3
votes
0
answers
164
views
$(-n-1)$-connected spectra vs. reduced excisive functors from $n$-truncated pointed spaces
It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of ...
3
votes
0
answers
159
views
Justification for the definition of equivariant curvature
Let $G$ be a compact Lie group which act on a smooth manifold $M$.
Let $\mathbb{C}[\mathfrak{g}] \otimes \mathcal{A}$ be the algebra of polynomial maps from $\mathfrak{g}$ to $\mathcal{A}(M),$ we ...
3
votes
0
answers
292
views
Simultaneous Galois closure
For a finite separable extension $L/K$ of fields, there exists a Galois closure, which is a finite field extension $\tilde L/L/K$ where $\tilde L/K$ is Galois. (given by the compositum of $\sigma L$, ...
3
votes
0
answers
193
views
The regularity of solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary
While doing my research, I encountered the following problem as:
is there any regularity result for solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary? For ...
3
votes
0
answers
199
views
Dynamical degree and spectral radius
Let $X$ be a smooth, projective surface over an algebraically closed field $k$ of characteristic zero, and let $f \in \mathrm{Bir}(X)$ a birational map.
Let's denote $f_{\ast} : \mathrm{NS}(X) \...
3
votes
0
answers
69
views
Analytical solution of 2nd order nonlinear ODE ($y''+(a+by^2)y'+cy = 0$)
I encountered the following ode in the attempt to solve the problem of nonlinear van der pol equation. I have tried for a long time to give it a solution but failed.
$y''+(a+by^2)y'+cy = 0$
where $a$, ...
3
votes
0
answers
256
views
Rational mapping from affine space to projective space
This problem is highly related to this one. Over there I learnt about the Segal's problem regarding the rational mapping spaces. There seems to be a generalization of the result of the Segal for ...
3
votes
0
answers
249
views
What are quantum extremal surfaces from a mathematical viewpoint?
It is said that they are surfaces which locally maximize area and bulk entanglement entropy. It would be great if I could receive some introductory material on it and some prerequisites to understand ...
3
votes
1
answer
200
views
Smooth cut-off in homogeneous Besov space
Given a Littlewood-Paley decomposition
$$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$
where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported ...
3
votes
1
answer
182
views
Existence of a quasi-open (a.k.a semi-open) map into a Cantor cube
Recall that a topological space is extremally disconnected if the closure of any open set is open.
A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For ...
3
votes
0
answers
158
views
On analogues of Weber's formula
Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that
$$
\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
3
votes
0
answers
297
views
Articles of Casnati on algebraic varieties
I am attempting to track down online copies of the following two algebraic geometry articles.
Is there some repository where these might be found? If necessary I could use the first few pages of each ...
3
votes
0
answers
78
views
Coxeter polynomials of graphs
Let $Q$ be a finite connected and directed graph with $n$ points.
Assume $Q$ is acyclic as a directed graph.
Let $C=C_Q$ be the Cartan matrix of $Q$, that is the matrix with entries $c_{i,j}$ being ...
3
votes
0
answers
128
views
What is the smallest sequence $a_k$ with nondecreasing $\frac{\sigma(a_k)-H_{a_k}}{\exp(H_{a_k})\log(H_{a_k})}$?
This is inspired by the Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH , an answer and some comments there.
For $n\geqslant2$ denote
$$
L(n):=\frac{\sigma(n)-H_n}{\exp(...
3
votes
0
answers
96
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
3
votes
0
answers
49
views
Conditions of parameters to have bounded solution of Dynkin's equation in exit problem
Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
3
votes
0
answers
106
views
Local expression of a quasi projective variety under finite morphism
I am studying the article Around the Chevalley-Weil Theorem by Zannier, Turchet and Corvaja and I am stuck in the following point: let $V$ and $W$ be two complex quasi-projective varieties and $\pi\...