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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 ...
user440858's user avatar
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 ...
Andy's user avatar
  • 369
13 votes
3 answers
4k views

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 ...
Igor Rivin's user avatar
  • 95.5k
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 $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, ...
Xiao-Gang Wen's user avatar
1 vote
0 answers
171 views

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 $i$.
Willliam D. Weakley's user avatar
7 votes
3 answers
500 views

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 ...
Phil-W's user avatar
  • 975
2 votes
0 answers
178 views

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 ...
Roger Van Peski's user avatar
5 votes
0 answers
142 views

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 $...
Monroe Eskew's user avatar
  • 18.1k
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 ...
Caims's user avatar
  • 243
14 votes
3 answers
3k views

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 ...
john mangual's user avatar
  • 22.6k
2 votes
0 answers
590 views

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 ...
Alireza Behtash's user avatar
6 votes
1 answer
454 views

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.
Jianrong Li's user avatar
  • 6,101
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 ...
Gro-Tsen's user avatar
  • 29.8k
1 vote
0 answers
124 views

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 ...
Thomas Benjamin's user avatar
0 votes
2 answers
236 views

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 ...
Hauke Reddmann's user avatar
3 votes
0 answers
149 views

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 ...
Guangbo Xu's user avatar
  • 1,197
8 votes
1 answer
321 views

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, ...
tci's user avatar
  • 662
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 ...
Astor's user avatar
  • 323
4 votes
1 answer
136 views

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 ...
aglearner's user avatar
  • 14k
17 votes
1 answer
2k views

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 ...
Mikhail Katz's user avatar
  • 15.1k
1 vote
0 answers
143 views

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 ...
Dima Sustretov's user avatar
109 votes
89 answers
29k views

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 views

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 ...
user1337's user avatar
  • 463
2 votes
1 answer
407 views

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 ...
Vamsi's user avatar
  • 3,323
3 votes
1 answer
534 views

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 ...
Turbo's user avatar
  • 13.7k
5 votes
0 answers
523 views

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 $G$...
Tyler Foster's user avatar
-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 $M^p≠O^p$?
S.B's user avatar
  • 5
4 votes
1 answer
822 views

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). ...
Zhiwei Zheng's user avatar
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 ...
Sung-En Chiu's user avatar
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 ...
cduston's user avatar
  • 145
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 ...
aglearner's user avatar
  • 14k
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\...
Sung-En Chiu's user avatar
5 votes
2 answers
544 views

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 ...
john mangual's user avatar
  • 22.6k
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 ...
Tomo's user avatar
  • 1,167
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 ...
Mikhail Bondarko's user avatar
2 votes
0 answers
116 views

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 ...
Rodrigo Torres's user avatar
18 votes
1 answer
1k views

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 ...
Benighted's user avatar
  • 1,701
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 ...
Rdrr's user avatar
  • 881
4 votes
0 answers
127 views

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}\...
Henry.L's user avatar
  • 7,951
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 ...
Syu Gau's user avatar
  • 415
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 ${\mathbb E}(X_i) = 0$, ${\...
user108997's user avatar
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 ...
Amir Sagiv's user avatar
  • 3,544
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 ...
Xiao-Gang Wen's user avatar
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 \...
Alexi Quevedo S.'s user avatar
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 $B$ -- ...
Barbot's user avatar
  • 143
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=...
Dr. Pi's user avatar
  • 2,939
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 ...
Mike Shulman's user avatar
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)
swati setia's user avatar
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 ...
user349424's user avatar
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 ...

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