All Questions
152,885
questions
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 ...
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
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}\...
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 ...
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\...
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 ...
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. ...
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 ...
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 ...
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 ...
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 =...
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|^{\...
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 ...
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 ...
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} \, .$$
I tried to solve it via the standard method, i.e.,...
4
votes
1
answer
353
views
Which monoids can be realized as the monoid of ideals of a commutative monoid?
Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an ideal of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) ...
2
votes
0
answers
60
views
When are solutions of the Schrödinger equation radial?
Let $S$ be a nonnegative self-adjoint operator on a complex Hilbert space $X$. (For example, $X$ consists of functions on $\mathbb R^d$; it could be $L^2(\mathbb R^d), \dot{H}^2(\mathbb R^d)$, etc.)...
1
vote
1
answer
152
views
terminology problem related to finitely generated objects
If $x$ is an object of a category $C$, one usually says that $x$ is if finite presentation (or compact) if for any direct filtered system ($y_i$) in $C$, the canonical map
$$f : \text{colim}_i \, \...
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?
...
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 $\...
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 ...
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)\...
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 ...
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 ...
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 $\alpha$ ...
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 ...
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-...
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<...
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 ...
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.$ ...
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 ...
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 ...
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 ...
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 ...
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. ...
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{...
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 ...
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 $\...
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 \...