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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 ...
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 ...
5 votes
1 answer
284 views

The spectral radius of a binary matrix - polynomial growth?

(This is a follow-up to The spectral radius of a binary matrix) Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$. QUESTION. Is there a $\delta\in\bigl(0,\frac12\...
Nikita Sidorov's user avatar
5 votes
1 answer
484 views

Graduate-level reference on temporal point processes

I am looking for a modern, graduate level, rigorous book on temporal point processes which also treats self-correcting point processes, and self-exciting point processes. It would be even more ...
geo.wolfer's user avatar
7 votes
1 answer
2k views

Determinant of real Wishart matrix

Suppose $A$ is a real $N \times P$ matrix, $P \geq N$, with entries drawn independently according to $A_{ij} \sim \mathcal{N}(0,1)$. Then $W = A \, A^\top$ is a member of the real Wishart ensemble. ...
Latrace's user avatar
  • 73
5 votes
0 answers
113 views

A general form of a maximal totally isotropic subspace in the split octonion algebra

Let $\mathbb O'$ be the split octonion algebra over $\mathbb R$. For each nonzero divisor of zero $x\in \mathbb O'$ $\mbox{($x \neq 0, N(x)=0$)}$ the kernel of the left multiplication by $x$, $Ker ...
Yourij1's user avatar
  • 175
7 votes
0 answers
195 views

What are (the motivations for Dominic Joyce's definition of) smooth functions for manifolds with corners?

In this article, Joyce defines a new kind of smooth map of manifolds with corners. The standard requirement would be that a smooth map is smooth in every chart. He calls such maps weakly smooth. In ...
Manuel Bärenz's user avatar
2 votes
1 answer
277 views

Constructible sets II (Grothendieck rings)

Here is my second question on constructible sets, now on Grothendieck rings. Let $K_0(Sch_k)$ be the Grothendieck ring of schemes over $k$. I have read that if $S$ is a constructible set in a ...
THC's user avatar
  • 4,313
3 votes
0 answers
142 views

Algorithm for holonomic sequence

A sequence $(a_n)_{n\geq 0}$ of complex numbers is called polynomially recursive (P-recursive) or holonomic if there exists a number $r$ and rational functions $P_1(n), \ldots, P_r(n)$ such that $a_n =...
user108968's user avatar
16 votes
1 answer
3k views

How to construct a basis for the dual space of an infinite dimensional vector space?

Let $V$ be an infinite-dimensional vector space over a field $K$. Then it is known that $\dim V < \dim V^*$. More precisely, by a result attributed to Kaplansky and Erdos, we have $\dim V^* = |K|^{\...
spin's user avatar
  • 2,781
2 votes
0 answers
197 views

Constructible sets, I (Morphisms)

I have a number of questions on constructible sets. The first one is on morphisms: suppose $X$ and $Y$ are constructible sets, respectively in projective spaces $\mathbf{P}_1$ and $\mathbf{P}_2$ over ...
THC's user avatar
  • 4,313
6 votes
0 answers
454 views

On the average number of subgroups per conjugacy class

At some point in the future, I hope to do some work on estimates for the number of conjugacy classes of subgroups of a finite group (by an estimate here, I mean an upper bound). Assuming, for the ...
the_fox's user avatar
  • 347
1 vote
3 answers
1k views

How to be a Great mathematician in prison/without a master? [closed]

Is it possible to be a great mathematician in our home with a laptop+poor internet+electronic books+some books+a little food +a little money or not? without having a constant job without studying P.H....
4 votes
0 answers
377 views

On modified Bessel solutions to complex ODE's using Kummer's series

I am trying to reduce the following ODE to Bessel's ODE form and hence solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$ I tried to solve it via the standard method, i.e.,...
Spoilt Milk's user avatar
4 votes
1 answer
353 views

Which monoids can be realized as the monoid of ideals of a commutative monoid?

Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an ideal of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) ...
Salvo Tringali's user avatar
2 votes
0 answers
60 views

When are solutions of the Schrödinger equation radial?

Let $S$ be a nonnegative self-adjoint operator on a complex Hilbert space $X$. (For example, $X$ consists of functions on $\mathbb R^d$; it could be $L^2(\mathbb R^d), \dot{H}^2(\mathbb R^d)$, etc.)...
abcd's user avatar
  • 223
1 vote
1 answer
152 views

terminology problem related to finitely generated objects

If $x$ is an object of a category $C$, one usually says that $x$ is if finite presentation (or compact) if for any direct filtered system ($y_i$) in $C$, the canonical map $$f : \text{colim}_i \, \...
Alain Bruguieres's user avatar
7 votes
4 answers
3k views

Generating Random Curves with Fixed Length and Endpoint Distance

Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given? ...
Manfred Weis's user avatar
  • 12.6k
15 votes
2 answers
899 views

What are the tangent $\infty$-categories to $\mathrm{Top}^\mathrm{op}$ and $E_\infty$-$\mathrm{Ring}^\mathrm{op}$?

Given an $\infty$-category $\mathcal{C}$ with finite limits and finite colimits, there are two ways to make it stable. One is to pass to the pointed objects and take the category of $\Omega$ objects $\...
Tim Campion's user avatar
  • 60.6k
0 votes
0 answers
298 views

Proof of Krull's intersection theorem with Taylor expansion

I asked this question last year in MSE, but I didn't get an answer. I took a commutative algebra course last semester (using Kaplansky's book), and I learned about Krull's intersection theorem. In ...
Seewoo Lee's user avatar
  • 1,911
1 vote
0 answers
109 views

How do I justify these nontheorems in the absence of the Existence Property for $PA$

Let $\Pi$ be the provability predicate for $PA$. I want to conclude that $PA\nvdash\exists x(\alpha(x)\wedge\lnot\Pi\ulcorner \alpha(\overset{.}{x}) \urcorner)$ and $PA\nvdash\exists x(\lnot \alpha(x)\...
Frode Alfson Bjørdal's user avatar
2 votes
1 answer
218 views

How does the existence property fail in $PA$?

Who or what is a good reference that explains how the (numerical) existence property fails for $PA$? Alternatively, what is a good example? It e.g. is clear that the disjunction property must fail ...
Frode Alfson Bjørdal's user avatar
3 votes
1 answer
59 views

Star-shapeness of polynomial tracts containing a single zero

In The Shape of Level Curves (link to article on JSTOR), George Piranian constructs a polynomial $p$ with $n$ distinct zeros, such that the set $\{z:|p(z)|<\epsilon\}$ has $n$ components (each of ...
Trevor J Richards's user avatar
4 votes
1 answer
379 views

Symmetrizing with respect to Galois Group: Trace and Norm

In invariant theory the Reynold's Operator gives rise to an element invariant for that group. For a Galois extension $K/F$ with $K=F[\alpha]$ the trace of $\alpha$ is an element of $K$. If $\alpha$ ...
P Vanchinathan's user avatar
1 vote
1 answer
82 views

Clarification on margin bound uniform w.r.t. the margin parameter

Theorem 4.5. in the book "Foundations of Machine Learning" by Mohri et al: http://prlab.tudelft.nl/sites/default/files/Foundations_of_Machine_Learning.pdf derives a generalization bound to hold ...
axk's user avatar
  • 517
2 votes
0 answers
285 views

An approach for Singular Hermitian-Einstein metric

Motivation: If we extend Hitchin-Kobayashi correspondence along holomorphic fibre space such that each vector bundle $E_s$ of fibers $X_s$ be stable then for finding canonical twisted Hermitian-...
user avatar
1 vote
1 answer
98 views

Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<...
Sammyy Delbrin's user avatar
12 votes
3 answers
949 views

Fixed point set of smooth circle action

Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
Jens Reinhold's user avatar
2 votes
0 answers
115 views

Some operations on categories - nomenclature question

Suppose that we have a "diagram of categories", i.e. a map from some small category $J$, viewed as a 2-category (with trivial two-morphisms) to the category of categories, $j\mapsto \mathcal{C}_j.$ ...
Dmitry Vaintrob's user avatar
7 votes
1 answer
271 views

A Converse of the Skorokhod Embedding Theorem

I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds: Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
Probabilist's user avatar
0 votes
2 answers
612 views

References on KKT Conditions applied to Calculus of Variations

I am looking for research references for KKT conditions and Slack Variables applied to variational calculus problems in things like Control Theory and such. For example, naively I might try solving a ...
user avatar
6 votes
1 answer
710 views

Relation - Anabelian geometry and Tate conjecture

A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture. I would like to know what is the relation between Anabelian algebraic geometry and Tate ...
tttbase's user avatar
  • 1,700
5 votes
1 answer
179 views

The uniqueness of a $K$-fixed vector in a spinor representation

Consider $G=SO(2n)$ and $K=U(n)$. $(G,K)$ is a symmetric pair. I'm interested in (zonal) spherical functions on $G/K$ which are matrix elements with respect to $K$-fixed vectors in irreducible ...
Ninnat Dangniam's user avatar
3 votes
1 answer
738 views

Gauss' Circle Problem at $\left ( \frac{1}{2}, \frac{1}{2} \right ) $

GCP (Gauss' Circle Problem) asks for a closed form for the number of square-lattice points inside a circle, centered at the origin, of radius $r$. Let's denote by $N(r)$ the number of these points. ...
user3141592's user avatar
6 votes
1 answer
772 views

Yoneda extensions in exact categories and their derived categories

If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism $$\mathrm{Ext}_\mathcal{A}^i(X,Y) \simeq \mathrm{...
Arkandias's user avatar
  • 951
3 votes
1 answer
285 views

Counting Bipartitions

Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$. Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of ...
Steven Spallone's user avatar
14 votes
1 answer
676 views

$\mathbb{Z}$-module structure of the subring generated by an algebraic number

Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
user108921's user avatar
22 votes
4 answers
2k views

Anti-large cardinal principles

I'm interested in axioms that prevent the existence of large cardinals. More precisely: (Informal definition) $\Phi$ is an anti-large-cardinal axiom iff $V \models \Phi \Rightarrow V \not \models \...
Neil Barton's user avatar

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