All Questions
152,874
questions
1
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1
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129
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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 ...
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
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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
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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
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0
answers
413
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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
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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
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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
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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
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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
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0
answers
82
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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
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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 $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
5
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0
answers
386
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Parallelizable spheres are H-spaces
Adams's paper On the Non-Existence of Elements of Hopf Invariant One famously includes the following diagram of implications in the introduction:
Implications in the Hopf Invariant One problem
I ...
4
votes
0
answers
94
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Topological hyperfields
I am trying to generalize the notion of reorientation class of an oriented matroid to the context of matroids over hyperfields (compare Baker and Bowler, 2016). I have already got some results in this ...
8
votes
0
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379
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A way to twist a manifold
This question is inspired by an answer to the question Manifolds with polynomial transition maps.
What I need from that answer is this. Suppose given, on a smooth $n$-manifold $M$ with some charts $(...
4
votes
1
answer
200
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Smooth intertwining operators
Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}_{\geq 2}$.
Then $V$ is uniquely ...
1
vote
0
answers
89
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Proving positivity of high degree homogenous polynomials in 4 variables
I have several homogenous polynomials of degrees 6, 8,.... They are in 4 different variables A,B,C,D.
I need to show that they are positive for any positive value of A,B,C,D.
Looking at a plot of ...
1
vote
1
answer
307
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On Riemann zeta function and Dirac delta function/distribution
Let $$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
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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 ...
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 ...
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?
...
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 $n$?
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(...
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+...
10
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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 ...
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).
...
9
votes
0
answers
171
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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 ...
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 ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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\...