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On Polynomial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers. $D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace. $A_1,\cdots,A_m$ are $n\times n$ Hermitians. We are interested in the ...
gondolf's user avatar
  • 1,487
4 votes
1 answer
360 views

Examples of norm forms where the numbers represented can be readily described

In case of impatience: the question here is a request for examples, especially degree six or seven where the norm form might represent some prime$p,$ then some $q^2$ but not $q,$ then some $r^3$ but ...
Will Jagy's user avatar
  • 25.3k
6 votes
1 answer
425 views

Is there a classification of pointed nodal genus 1 curves?

Any pointed nodal (ie, proper semistable with a specified rational point lying in the smooth locus) curve of arithmetic genus 1 over a field $k$ must be irreducible and has precisely 1 node, which ...
stupid_question_bot's user avatar
3 votes
0 answers
146 views

Question about Nash functions

I am reading Kollár's recent survey on Nash's work in algebraic geometry. I am trying to understand why the retraction $\pi:U_M\to M$ introduced in Discussion 7 is a Nash map. Kollár applies Claim 8.4 ...
GH from MO's user avatar
  • 98.2k
19 votes
4 answers
1k views

The number of commuting m-tuples is divisible by order of group: Improvements?

The number of commuting pairs of elements in finite group G is equal to the product $k(G)*|G|$ (see MO271757 ) where $k(G)$ is the number of conjugacy classes. Thus it is is divisible by $|G|$ (the ...
Alexander Chervov's user avatar
15 votes
2 answers
903 views

What homotopy classes can attaching an $E_n$-cell kill?

Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k(...
Jonathan Beardsley's user avatar
12 votes
0 answers
413 views

What does deformation theory have to do with Serre duality?

The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the ...
Jonathan Wise's user avatar
9 votes
1 answer
250 views

Decomposition of Henstock-Kurzweil-integrable functions

Let $f:[a,b]\to\mathbb R$ be a Henstock-Kurzweil-integrable function (short: HK-integrable). Can $f$ always be written as a sum of a Lebesgue-integrable function and a function which has a ...
sranthrop's user avatar
  • 231
2 votes
0 answers
379 views

Matrix optimization of a random quadratic form

I am interested in maximizing a quadratic form which looks like $$f(\Sigma) = E(\operatorname{trace}(SJ)) = E(1^{\top} S 1)$$ where $J$ is a matrix of $1$'s, $S= \Sigma_{mm} - \Sigma_{mo} \Sigma_{oo}...
tony's user avatar
  • 21
11 votes
1 answer
798 views

Large Complex Structure Limit of Calabi-Yau family and uniqueness of limit

Let $\mathcal X$ be a smooth complex manifold of dimension $n+1$. We say $\mathcal X \to ∆$ is a large complex structure limit if and only if it’s maximal unipotent degeneration . $T: H^n(\mathcal ...
user avatar
5 votes
2 answers
296 views

Hypotheses for exponent pairs

The theory of exponent pairs provides bounds for $$\sum_{N<n<2N} e(f(n)),$$ where f behaves like a monomial. Precise formulations of this are in Graham and Kolesnik (GK) which seems to be what ...
George Shakan's user avatar
2 votes
1 answer
202 views

Existence of polynomial p with real coefficients such that p(n) is prime if and only if n is palindrome

Does there exist any polynomial $p(x) \in \mathbb{R}[x]$ such that $p(n)$ is a prime number if and only if $n$ is a palindrome number ? ($n$ must be a positive palindrome number to give $p(n)$ a prime ...
Aditya Guha Roy's user avatar
4 votes
2 answers
145 views

Which necklaces require maximal cuts?

Given an unclasped necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. A fair division means that each person is given the ...
EgoKilla's user avatar
  • 143
10 votes
1 answer
348 views

limits of stable theories

Say that a complete theory $T$ is a limit of stable theories if for every $\phi \in T$ there is a stable completion of $\{\phi\}$. (Equivalently, $T$ is the ultraproduct of stable theories.) Question:...
Danielle Ulrich's user avatar
12 votes
2 answers
572 views

Bounding weight multiplicities by number of certain Coxeter elements

This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras). Let's say $G$ is a simple algebraic group ...
Jingren Chi's user avatar
5 votes
0 answers
138 views

Local family index theorem, but with Chern class?

Let $\pi:X\to B$ be a proper submersion with spin fibers, and $E\to X$ a Hermitian vector bundle with a unitary connection $\nabla$. Then the local family index theorem for spin Dirac operator twisted ...
Ho Man-Ho's user avatar
  • 1,087
2 votes
1 answer
358 views

Counting cosets in the Quotient of Weyl groups

Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its ...
MIQ's user avatar
  • 83
5 votes
0 answers
270 views

Constructive treatment of Jacobson rings

Which result is closest to the classical General Hilbert's Nullstellensatz: Finite type algebras over Jacobson rings are Jacobson. and constructively true at the same time? And where can I find a ...
Jakob Werner's user avatar
  • 1,093
2 votes
0 answers
77 views

Separability of the ring extension $A \subset A[T]/(h_1,\ldots,h_n)$

Is there a generalization of Wang's known result, Corollary 8, which says the following: $A \subset B=A[T]/(h(T))=A[w]=B$ is a separable ring extension if and only if $h'(w)$, the formal derivative of ...
user237522's user avatar
  • 2,783
1 vote
0 answers
82 views

Gradient descent with gradient evaluated at transformed coordinates

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a given function and let us consider the unconstrained problem, $$\min_{x\in\mathbb{R}^n}f(x)$$ The standard iterative method for this is the gradient descent ...
Samrat Mukhopadhyay's user avatar
0 votes
1 answer
294 views

Asymptotic behaviour of fixed points in permutations

For any $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
Dominic van der Zypen's user avatar
5 votes
0 answers
386 views

Parallelizable spheres are H-spaces

Adams's paper On the Non-Existence of Elements of Hopf Invariant One famously includes the following diagram of implications in the introduction: Implications in the Hopf Invariant One problem I ...
Reuben Stern's user avatar
4 votes
0 answers
94 views

Topological hyperfields

I am trying to generalize the notion of reorientation class of an oriented matroid to the context of matroids over hyperfields (compare Baker and Bowler, 2016). I have already got some results in this ...
snaleimath's user avatar
8 votes
0 answers
379 views

A way to twist a manifold

This question is inspired by an answer to the question Manifolds with polynomial transition maps. What I need from that answer is this. Suppose given, on a smooth $n$-manifold $M$ with some charts $(...
მამუკა ჯიბლაძე's user avatar
4 votes
1 answer
200 views

Smooth intertwining operators

Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}_{\geq 2}$. Then $V$ is uniquely ...
MathStudent's user avatar
1 vote
0 answers
89 views

Proving positivity of high degree homogenous polynomials in 4 variables

I have several homogenous polynomials of degrees 6, 8,.... They are in 4 different variables A,B,C,D. I need to show that they are positive for any positive value of A,B,C,D. Looking at a plot of ...
Michael Roberts's user avatar
1 vote
1 answer
307 views

On Riemann zeta function and Dirac delta function/distribution

Let $$I_{N} = \int_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N}(2\pi x)dx = \frac{2N-1}{2N} \int_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N-2}(2\pi x)dx $$ therefore (I think) $$ I_N = \frac{(2N-1)!...
C Marius's user avatar
  • 207
11 votes
2 answers
1k views

Does this product have analytic continuation?

The product $$ F(s)=\prod_{p}\frac1{(1-p^{-s})^p}, $$ converges for $\mathrm{Re}(s)>2$, when $p$ runs over all primes. Does it admit analytic continuation beyond the line $\mathrm{Re}(s)=2$? Any ...
user avatar
2 votes
0 answers
116 views

explicit formulae of heat kernel on graphs

I have just discovered this article about heat kernels on graphs. It has been written by a respected theoretical physicist, but seemingly never made it into a peer-reviewed journal. On the other hand, ...
Delio Mugnolo's user avatar
5 votes
1 answer
183 views

How do we classify all possible extensions of the Fibonacci recursion to the complex plane?

Take the straight forward Fibonacci equation $$F_0 = F_1 = 1$$ $$F_{n-2} + F_{n-1} = F_n$$ Let's consider a holomorphic function $F: \mathbb{C} \to \mathbb{C}$ such that $$F(z)\Big{|}_{\mathbb{N}} =...
user avatar
2 votes
0 answers
441 views

Eigendecomposition of the Hadamard product of a rank one symmetric matrix and a positive definite symmetric matrix

Is it possible to say anything about the eigenvalues and eigenvectors of a matrix $X = Y \circ xx^T$ where $Y$ is a positive definite symmetric matrix with known eigen-decomposition $Y=U\Lambda U^T$...
Tzonathan's user avatar
1 vote
0 answers
95 views

Factoring in discrete Heisenberg group $H_3(\mathbb{Z})$

Let $H_3(\mathbb{Z})$ be the discrete Heisenberg group generated by $x=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix},\ \ y=\begin{pmatrix} 1 & 0 &...
user avatar
4 votes
0 answers
82 views

$\mathrm{\Gamma}$ functor of Barratt-Eccles in simplicial context

In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$. Proposition 6.2 states if ...
Victor TC's user avatar
  • 795
2 votes
0 answers
202 views

What is the maximum weighted earth-movers distance between two permutations?

What is the maximum weighted earth-mover's distance (as defined in Sun et. al. 2010) between two permutations in $\mathfrak S_n$ where the transposition $(i, i+1)$ has cost given by weight $w_i$. In ...
Max Flander's user avatar
5 votes
2 answers
224 views

Properties of right rejective subcategories

I am reading this paper finiteness of representation dimension, on page 1012 there is a place I don't understand: Why $\mathcal{C}(, \mathbb{F}(X)) \rightarrow [\mathcal{C'}](,X)$ is an isomorphism? ...
Xiaosong Peng's user avatar
3 votes
1 answer
173 views

Rate of convergence of weakly null sequences

If $x_n$ is a normalized, weakly-null sequence in a Banach space, and $\epsilon_n\to 0$, does there exists a non-zero functional $f$ such that $|f(x_n)|<\epsilon_n$ for all $n$?
Markus's user avatar
  • 1,361
10 votes
1 answer
476 views

The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$

I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$: Is the double cover of $Sp_6(...
Shawn's user avatar
  • 453
3 votes
1 answer
342 views

Polynomials splitting into linear factors modulo certain primes

Given integers $a,b$, we say a polynomial $f(x) \in \mathbb{Z}[x]$ is an $(a,b)$-filter, if $f(x)$ splits completely into linear factors modulo an odd prime $p$ only if $p=a \pmod b$. For example $x^2+...
Marco's user avatar
  • 537
10 votes
0 answers
406 views

Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?

Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
David Roberts's user avatar
  • 33.8k
7 votes
0 answers
544 views

What is the geometric intuition for the sheaf-theoretic terms "soft", "fine", and "flabby"?

The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here). ...
ಠ_ಠ's user avatar
  • 5,933
9 votes
0 answers
171 views

How many components are there in the space of "generic" planar N-gons? (Mnev's revenge)

Call an ordered $N$-tuple of points in the Euclidean plane ${\mathbb R} ^2$ "in general position" if no three points of the points in the set are collinear. As a function of $N$ how many components ...
Richard Montgomery's user avatar
3 votes
1 answer
325 views

Jacobians of curves with maximal Picard number

What can be said about a complex curve $C$, if its jacobian $J(C)$ has the maximal Picard number? It is natural to expect that for a general curve of given genus its Jacobian has Picard rank 1 (isn't ...
 V. Rogov's user avatar
  • 1,115
2 votes
1 answer
209 views

When will the value of automorphic function $f(x)$ satisify an algebraic equation?

When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic? If the question is too ...
XL _At_Here_There's user avatar
2 votes
2 answers
533 views

Names of convex geometry journals

What are the names of the research journals that focus on convex geometry? I know of "Advances in Geometry" and "Discrete & Computational Geometry" but no others. Context of question: I am a ...
math4's user avatar
  • 155
1 vote
2 answers
851 views

Find all edges not covered by a shortest path in an all-pairs shortest path over a subset of vertices

Given a graph $G(V,E)$ where every edge $e\in E$ has some positive weight $c(e)$. The graph can be directed/undirected/mixed. The graph is assumed to be strongly connected. Moreover, we define a ...
Joris Kinable's user avatar
2 votes
0 answers
153 views

Sums of nonzero triangular numbers

OEIS A002097 gives numbers that are not the sum of 3 nonzero triangular numbers. They are just seven numbers: 1, 2, 4, 6, 11, 20, 29. The site says that the seven numbers are all without references. ...
P.-S. Park's user avatar
4 votes
0 answers
181 views

Distributions over permutation groups $\mathcal{S}_n$

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
Henry.L's user avatar
  • 7,951
1 vote
1 answer
152 views

automorphisms of a finite $p$-group

If $G=\langle a, b : a^{p^{n}}= 1= b^{p^{n+1}}, [b, a]= b^{p} \rangle$, such that $n\geq 2$ and $p$ is an odd prime number, then how can I define a non-inner automorphism of $G$? Is it possible to ...
banoo's user avatar
  • 21
6 votes
1 answer
418 views

Complete residue system modulo n (permutation of numbers 0 to n-1) such that

I have a task: Find all $n\ \epsilon \ N, \ n > 1$ for which a permutation $a_1,\ a_2,\ ...,\ an\ $ of numbers $ 0,1, ..., n - 1$ exists such that $a_1,\ a_1+a_2,\ ...,\ a_1+a_2+\ ...\ +an\ $ form ...
vixenn's user avatar
  • 63
6 votes
3 answers
486 views

Proof of the Schauder Lemma

Schauder's Lemma in functional analysis states the following: Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\...
Dominic Wynter's user avatar

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