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Are there infinitely many primes N congruent to 1 mod 8 with h(-N) and h(-2N) both powers of 2?

The title says it all. For N less than 1000, if I've looked up the tables correctly, when the condition holds N can only be 17, 41, 73, 113, or 257. Motivation: Let N be an odd positive integer, f ...
paul Monsky's user avatar
  • 5,412
2 votes
0 answers
164 views

Lower bound for the sum of cosines between singular vectors of diagonally dominant matrices

Let $A \in \mathbb{R}^{n \times n}$ be a nonsymmetric diagonally dominant matrix with $a_{ij} < 0$ $\forall i \ne j$ and $a_{ii}>0$. Let the singular value decomposition of $A$ be $A=U \Sigma V^...
Astor's user avatar
  • 323
9 votes
0 answers
146 views

A characterization of Moishezon manifolds via sections of $L^k$ with $k\to \infty$

Let $X$ be a smooth compact complex manifold of dimension $n$. Suppose $L$ is a line bundle on $X$ such that $dim(H^0(X,L^k))>c\cdot k^n$ for $c>0$ and $k>>0$. Question. Is it true that $...
aglearner's user avatar
  • 14k
4 votes
1 answer
180 views

Cardinality of a set of pairwise non-order-isomorphic ultrafilters on $\omega$

It is well known that there are $2^{2^{\aleph_0}}$ many non-principal ultrafilters on $\omega$. Is there a set ${\frak U}$ of non-principal ultrafilters on $\omega$ with $|{\frak U}| = 2^{2^{\aleph_0}}...
Dominic van der Zypen's user avatar
2 votes
0 answers
145 views

Interchanging limit and infinite product in Euler product for Dedekind function s=1

For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation $$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
Job Rauch's user avatar
2 votes
0 answers
82 views

Linearly Isometric Banach lattices

Do there exist two Banach lattices which are linearly isometric but not Banach-lattice isometric? This is so basic that you would expect a textbook or monograph to have answered this, but I have not ...
Fred Dashiell's user avatar
1 vote
0 answers
228 views

Geometric interpretation of k-th power of first n natural numbers and summation using Pick's theorem

I want to know is there any interesting properties of this approach or generalization to find $S_k(n)=1^k+2^k+3^k+\cdots+n^k$ by using Pick's Theorem $S=i+\tfrac{b}{2}-1$, where $i$-number of ...
serg_1's user avatar
  • 19
1 vote
1 answer
223 views

OEIS Sequence A002846 and properties of matrix inverses

Sequence A002846 in https://oeis.org/A002846 (OEIS) gives, for each positive integer $n$, the number $a(n)$ of ways of transforming a set of $n$ indistinguishable objects into $n$ singletons via a ...
Helmut's user avatar
  • 169
7 votes
1 answer
1k views

Property between trace class and Hilbert-Schmidt

Consider the following condition on a bounded operator $T$ on a Hilbert space: $\ \ \ \ \ $(A) there exists an orthonormal basis $(e_j)$ with $\sum_j\parallel Te_j\parallel<\infty$. We have the ...
user avatar
7 votes
3 answers
907 views

Notable examples of syntactic proofs whose existence is guaranteed by completeness, but having been found later than a semantic proof?

Question. What are examples (preferably documented and explicitly commented on from this perspective in the literature, preferably in an article dedicated to this aspect alone) of the following well-...
3 votes
2 answers
128 views

$T_2$-spaces with order-isomorphic topologies

Suppose $X\neq \emptyset$ is a set. Let $\tau_1, \tau_2$ be Hausdorff topologies on $X$ with the property that the partially ordered sets $(\tau_1,\subseteq)$ and $(\tau_2,\subseteq)$ are order-...
Dominic van der Zypen's user avatar
1 vote
1 answer
238 views

Is there an example with Area $0<F(f)<\infty$ for some transcendental entire function

It seems that there may be example of a transcendental entire function with finite (but positive) planar area of the Fatou set in Eremenko-Lyubich class. However, I can't not find it in the ...
yaoxiao's user avatar
  • 1,664
7 votes
1 answer
325 views

Internal hom in $(\infty,2)$-categories

Let $X,Y$ be two $(\infty,2)$-categories, viewed as two fibrant objects in $\mathrm{Fun}(\Delta^{op},\mathrm{Set}_\Delta)$ with the complete Segal model structure (one uses the Joyal model structure ...
Xin Jin's user avatar
  • 337
1 vote
2 answers
207 views

Characterisation of a poset

Let $X$ be a finite set ordered by $R$, where $R$ is a transitive, reflexive, and antisymmetric relation on $X$. We define, for all $x\in X$, $C_R(x)=(m_R(x),M_R(x))\in \mathbb N^2$, such that $m_R(...
jcdornano's user avatar
  • 469
9 votes
0 answers
520 views

Mysterious relationship between central charges of conformal field theories and the Beraha numbers

Background: Conformal field theories (CFTs) in two dimensions are partially characterized by a so-called central charge (characterizing the central extension of the Virasoro algebra which defines it)....
Ruben Verresen's user avatar
2 votes
1 answer
128 views

Separability of a subring and of a pre-image

Let $A \subseteq B \subseteq C$ be commutative rings. A known result, which can be found in De Meyer and Ingraham's book, says that separability of $A \subseteq C$ implies separability of $B \...
user237522's user avatar
  • 2,783
8 votes
0 answers
464 views

Hard Lefschetz property and Moishezon manifolds

Cohomology ring of a closed manifold $M^{2n}$ is said to satisfy hard Lefschetz property if there exists an element $a\in H^2(M,\mathbb R)$ such that multiplication by $a^k$ yields an isomorphism $$H^{...
Yury Ustinovskiy's user avatar
1 vote
1 answer
129 views

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

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