All Questions
152,891
questions
5
votes
1
answer
289
views
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 ...
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^...
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 $...
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}}...
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/{\...
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 ...
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 ...
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 ...
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 ...
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-...
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 ...
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 ...
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(...
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)....
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 \...
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^{...
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 ...
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 ...
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 ...
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 ...
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 ...
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(...
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 ...
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 ...
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}...
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 ...
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 ...
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 ...
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 ...
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:...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
5
votes
0
answers
386
views
Parallelizable spheres are H-spaces
Adams's paper On the Non-Existence of Elements of Hopf Invariant One famously includes the following diagram of implications in the introduction:
Implications in the Hopf Invariant One problem
I ...
4
votes
0
answers
94
views
Topological hyperfields
I am trying to generalize the notion of reorientation class of an oriented matroid to the context of matroids over hyperfields (compare Baker and Bowler, 2016). I have already got some results in this ...
8
votes
0
answers
379
views
A way to twist a manifold
This question is inspired by an answer to the question Manifolds with polynomial transition maps.
What I need from that answer is this. Suppose given, on a smooth $n$-manifold $M$ with some charts $(...
4
votes
1
answer
200
views
Smooth intertwining operators
Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}_{\geq 2}$.
Then $V$ is uniquely ...
1
vote
0
answers
89
views
Proving positivity of high degree homogenous polynomials in 4 variables
I have several homogenous polynomials of degrees 6, 8,.... They are in 4 different variables A,B,C,D.
I need to show that they are positive for any positive value of A,B,C,D.
Looking at a plot of ...
1
vote
1
answer
307
views
On Riemann zeta function and Dirac delta function/distribution
Let $$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)!...
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 ...
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, ...
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}} =...
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
$Y=U\Lambda U^T$...
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 &...
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 ...