All Questions
152,871
questions
1
vote
1
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343
views
The number of values of $f(x)/x$ when $f$ is a linearized polynomial
Consider an $\mathbb{F}_q$-linear map $f:\mathbb{F}_{q^n}\to \mathbb{F}_{q^n}$ (so $f$ is a linearized polynomial). Suppose also that $f$ is not $\mathbb{F}_{q^i}$-linear, where $i>1$.
My ...
6
votes
1
answer
357
views
Arithmetic progressions in Van der Waerden's theorem
Recall that a syndetic subset of the integers is any $S\subseteq \mathbb{Z}$ with bounded gaps, i.e. there is some $k< \omega$ so that consecutive members of $S$ have distance at most $k$. One way ...
13
votes
3
answers
4k
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Signature of a quadratic form
This may be a really dumb question, but here goes: is there any algorithm to compute the signature of a quadratic form (or a symmetric matrix, if you prefer) more efficient (asymptotically or ...
9
votes
2
answers
534
views
Generalize Wu formula to integral cohomology classes
For $\mathbb{Z}_2$ cohomology classes, we have a very useful Wu formula:
In $d$-dimensional manifold and for a $n$-cocycle in $x_n \in H^n(M^d; \mathbb{Z}_2)$, we have $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, ...
1
vote
0
answers
171
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Write {1,...,3n} as the disjoint union of arithmetic progressions of length 3 and steps 1, 2,...,n
For $n \equiv 0, 1, 2 \pmod 9$, write $\{\,1,\dots,3n\,\}$ as the disjoint union of arithmetic progressions $A_1, A_2,\dots,A_n$ of length 3, where $A_i$ has step $i$.
7
votes
3
answers
500
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On structures that are not submitted to compatibility conditions
In mathematics, it seems that 100 % of the time, we deal with objects that have a number of different structures that are "compatible" with each other. For example,
a Lie group is a manifold and a ...
2
votes
0
answers
178
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Macdonald polynomials: existence and specializations
I am reading Macdonald's Symmetric Functions and Orthogonal Polynomials lecture notes, and have several related questions:
In both the type A case (chapter 1) and the general irreducible root system ...
5
votes
0
answers
142
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Normal ideals on $[\lambda]^{<\kappa}$ concentrating on maximal cardinality
Is the following consistent? $2^\omega > \omega_2$, and there is a normal precipitous ideal $I$ on $[\omega_2]^\omega$ such that every $X \subset [\omega_2]^\omega$ of size $< 2^\omega$ is in $...
4
votes
1
answer
190
views
Quiver invariants as polynomials/algebraic curves
I'm interested in algebraic curves one can associate to gauge or string theories. Examples involve Seiberg-Witten curves or family of A-polynomials which define holomorphic Lagrangian submanifolds for ...
14
votes
3
answers
3k
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what is $\mathrm{Bun}(G)$?
I don't even know where to begin. There's a discussion of stacks and they talk about $\mathrm{Bun}(G)$. I don't know what it is, or what it's elements are or why it is important. Google and ...
2
votes
0
answers
590
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Volume of $SL(2,\mathbb{C})$ [closed]
So according to http://www-users.math.umn.edu/~garrett/m/v/SL2C.pdf
I can write the Haar measure of $SL(2,\mathbb{C})$ as
$$d\mu = \sinh^2(r) dr dk dk'$$
where $r$ runs over nonnegative real numbers ...
6
votes
1
answer
454
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Reference request: type C, D Catalan numbers
Catalan numbers are generalized to type B: https://oeis.org/A000984.
Are there some references about Catalan numbers of type C, D? Thank you very much.
3
votes
1
answer
186
views
Critical points of characters on semisimple groups
Let $G$ be a semisimple connected complex Lie group, compact real Lie group or linear algebraic group. Let $\chi$ be the character of a finite dimensional irreducible representation of $G$ (I am ...
1
vote
0
answers
124
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Can $GPK^{+}_{\infty}$ + $AC_{WF}$ prove "$ZFC$ proves that the class of ordinals is not weakly compact for definable classes?
Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set ...
0
votes
2
answers
236
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Verification of Turing-equivalent automata
Correct me if I slept in my computer science studium: If an automaton is Turing-equivalent, the Halting problem shows that there are programs we can not verify (since we can't even predict their ...
3
votes
0
answers
149
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Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
8
votes
1
answer
321
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ccc after strongly proper forcing
Let $P, Q \in V$ be such that $P$ is strongly proper and $Q$ is ccc. Does $Q$ continue to be ccc after forcing with $P$?
Since strongly proper forcings do not add new branches to $\omega_1$-trees, ...
1
vote
1
answer
577
views
Can I modify the singular values of a matrix in order to get a negative eigenvalue?
Let $A \in \mathbb{R}^{n \times n}$ be a real nonsymmetric matrix with eigenvalues $\left\{\lambda_i : i=1..n\right\}$ with positive real part $\Re(\lambda_i) > 0$ $\forall i=1..n$
Let $A=U\Sigma ...
4
votes
1
answer
136
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Convex hull of a connected subset on a complete surface of non-positive curvature
Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
17
votes
1
answer
2k
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Did Lagrange change his mind about infinitesimals?
Lagrange is famous for his attempt to found analysis algebraically using power series expansions, an approach that, as we know today, is limited to analytic functions. Lagrange is also known as the ...
1
vote
0
answers
143
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characterization of the subspace of the moduli space of curves with maximally degenerate Jacobian
Let $K$ be a field equipped with a non-Archimedean absolute value, for example $K=\mathbb{C}((t))$. An Abelian variety $A$ over $K$ is called maximally degenerate if it admits an analytic ...
109
votes
89
answers
29k
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Tweetable Mathematics
Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.
I am looking for ...
0
votes
0
answers
40
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A uniqueness result for a BVP over a semi-infinite interval
Let $q:[0,\infty) \to \mathbb{R}$, and consider the ODE
$$u''(x)=q(x)u(x) $$
with the boundary conditions $u(0)=\lim_{x \to \infty} u(x)=0$. Under what conditions on $q$ is $u \equiv 0$ the only ...
2
votes
1
answer
407
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A priori estimates for elliptic systems with bounded coefficients
Suppose we have a smooth solution to a system of PDE (for $\vec{u}$) given by $L\vec{u} = \vec{f}$ on the unit ball (no boundary conditions given). Assume that the coefficients and $\vec{f}$ are ...
3
votes
1
answer
534
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Lovasz theta and circulant graphs
Let $\theta(G)$ denote Shannon zero error capacity of graph $G$ and $\vartheta(G)$ be Lovasz upper bound for $\theta(G)$.
Let $C_{2n+1}$ denote cycle graph with $2n+1$ nodes.
We know following two ...
5
votes
0
answers
523
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What is the algebro-geometric or measure-theoretic "content" of Dhillon and Mináč's motivic Artin symbols over an arbitrary ground field?
1. Short version. In this text, Dhillon and Mináč define motivic Artin symbols. Having fixed a ground field $k$ and a smooth projective curve $Y$ over $k$ equipped with the action of a finite group $G$...
-1
votes
1
answer
169
views
Existence of a special type of maximal ideal in $C(X)$:
Does there exist any maximal ideal $M^p$ in $C(X)$ (the ring of continuous functions on a topological space $X$) such that each element of $M^p$ is a divisor of zero but $M^p≠O^p$?
4
votes
1
answer
822
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Metric on Siegel upper half space
I want to calculate the distance between two points in Siegel upper half space, but I do not know the explicit formula of the metric (which is invariant under the action of the real symplectic group). ...
1
vote
0
answers
324
views
Existence of solution for Poisson equation in Markov chain
Consider $X_n\in \mathcal{X}$ a controlled Markov chain taking value in a compact set $\mathcal{X}$ with action $a\in \mathcal{A}$, where the action set $|\mathcal{A}|$ is finite.
(In particular, we ...
3
votes
1
answer
262
views
The singularity type of a non-torus link
It is a well-know result that singular surfaces like
$$x^p+y^q=0$$
(for complex $x$ and $y$) can be associated with $(p,q)$-torus links by considering the intersection of this surface with a small ...
3
votes
1
answer
188
views
Constructing a Hamiltonian $S^1$-action on a neighborhood of a symplectic divisor
Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that ...
3
votes
2
answers
272
views
Concentration inequality of joint event over time of a submartingale
Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as
$$
P(X_t-X_0 \leq -t) \leq exp\...
5
votes
2
answers
544
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moduli space of flat connections $SU(N)$ on an elliptic curve is $\mathbb{C}P^{N-1}$
A physics paper [1] states the moduli space of flat $SU(N)$ connections on an elliptic curve $E$ is the projective space $\mathbb{C}P^{N-1}$. However, I need some clarifications:
the paper vaguely ...
4
votes
0
answers
166
views
Why is flatness needed for the Segre classes of a family of cones to be equal in the Chow ring of the base
Let $X$ be an algebraic scheme and $\mathscr C$ a cone on $X\times\mathbf A^1$ and $C_t$ denote the restriction of $\mathscr C$ to $X\times\{t\}$ ($t=0,1$, or whatever). The claim in Fulton's ...
15
votes
1
answer
513
views
What are the advantages of various "models" for the motivic stable homotopy category
People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...
2
votes
0
answers
116
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About existential graphs
I would like to read articles about Peirce's existential graphs. I already have the works of Brady and Trimble on it, and I would like to know if there exists a geometric approach to them, or a ...
18
votes
1
answer
1k
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Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory?
I asked this on PhysicsSE, but I think it also fits here as it's related to algebro-geometric connections to string and gauge theory.
I'm thinking about the beautiful story of "geometrical ...
6
votes
1
answer
352
views
Semi-Simplicity of Mod-$\ell$ Galois Representations
I am currently studying mod-$\ell$ Galois representations of $CM$ elliptic curves. More precisely, let $\ell$ be a prime, $K$ a number field and $E/K$ a $CM$ elliptic curve, where all endomorphisms of ...
4
votes
0
answers
127
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Is there an example that both Berry-Essen bound and DKW bound are attained?
The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
7
votes
1
answer
267
views
generalized elements in monoidal categories
In a category $\mathcal{C}$, a generalized element of an object $A$ means a morphism to $A$. It follows from Yoneda lemma that the object $A$ is determined by the collection of generalized of elements ...
1
vote
2
answers
804
views
Improved bounds in the Berry-Esseen theorem
One consequence of the Berry-Esseen approach to the Central Limit theorem, states the following. Let $X_1,\ldots ,X_n$ be real valued random variables, all independent, with ${\mathbb E}(X_i) = 0$, ${\...
10
votes
2
answers
473
views
Proving a system is nonintegrable /not solvable with Inverse Scattering Transform
Question:
Given a PDE, is there a general method to show that it is not solvable using the inverse scattering transform?
Specifically, for the perturbed 1D NLS or the 2D cubic NLS, where was it ...
7
votes
1
answer
900
views
Associativity of Steenrod's cup-i product
In the paper Products of Cocycles and Extensions of Mappings,
Steenrod introduced the cup-i product (and Steenrod square). I would like to ask if Steenrod's cup-i product associative or not? The paper ...
6
votes
1
answer
125
views
the sigma-p property
An operator ideal $\mathcal{I}$ possesses the $\sum_{p}$-property, say $1<p<\infty$, if for arbitrary collections of Banach spaces $E_{m}, F_{n} (m,n=1,2,\ldots)$ the following holds:
if $T\in \...
14
votes
1
answer
409
views
References for reasoning about the spectrum of a convex body?
By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
3
votes
1
answer
339
views
Squarefree values of polynomials at prime arguments
This is a reference request.
Assume that $f_1,\ldots,f_r \in \mathbb{Z}[t]$ are non-zero linear polynomial.
Letting $\mu$ be the M\"{o}bius function, is there any work on
$$ \sum_{p\leq x} \prod_{i=...
3
votes
0
answers
159
views
Exponentials of profunctors
Suppose $f:B\to A$ is an exponentiable functor, so that pullback $f^\ast$ has a right adjoint $\Pi_f$. Then $f\times \mathbf{2} : B\times \mathbf{2} \to A\times \mathbf{2}$ is also exponentiable, and ...
1
vote
1
answer
244
views
Lattice Sieving
What are some good references for Lattice Sieving in Number Field Sieve? Could someone suggest some research papers in this area?(Theoretical and Computational Perspective)
2
votes
0
answers
512
views
Orbifold Vector bundle from parabolic vector bundle
I am reading the paper 'Parabolic Bundles as Orbifold bundles' by Indranil Biswas. In Sect. 3 of the paper, one starts with a Parabolic vector bundle and constructs a corresponding orbifold vector ...
7
votes
3
answers
5k
views
Was Cauchy prescient?
Cauchy proved a sum theorem for series of continuous functions in 1821, and published another article on the subject in 1853.
Michael Segre, writing in Archive for History of Exact Sciences, claimed ...