2
votes
1answer
734 views

Three term recurrence relation.

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$: $$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a_{j-1} - ...
2
votes
0answers
118 views

When does order matter when decomposing a boundedly generated group

A group $G$ is said to be boundedly generated if (it is finitely generated and) there exists a finite family of cyclic subgroups (not necessarily normal or distinct) $\lbrace C_i \rbrace_{i =1, ...
0
votes
1answer
848 views

Finding the lowest cost set of disjoint paths using all nodes in a directed graph?

I have a directed graph with edges connecting nodes representing costs. I wish to find the set of paths which -go from node 'start' to node 'end' -are node-disjoint (except for the start and end ...
7
votes
2answers
659 views

What is a twisted modular operad?

I find Getzler and Kapranov's article Modular Operads difficult to understand. Can anyone explain what a (twisted) modular operad is conceptually, or what the underlying idea behind the concept of a ...
4
votes
0answers
263 views

What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?

Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
0
votes
1answer
365 views

$q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ?

Definition: Let $(V,\Omega)$ be a symplectic vector space, we define $\perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast})$ by $\perp(\omega)=i_{X_{\Omega}}(\omega)$ here if ...
0
votes
3answers
543 views

Convex Combination of 2 hermitian matrices

Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices ...
4
votes
1answer
1k views

Ask some matrix eigenvalue inequalities.

Let $ \begin{bmatrix} A& B \\\\ B^* &C \end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$. Are the following plausible inequalities true? I have seen a lot of ...
2
votes
2answers
285 views

Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$ polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. Define $\mathcal{A}$ to be ...
10
votes
3answers
2k views

ubiquitous quantum cohomology

Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...
1
vote
1answer
391 views

Boundary of star-shaped domain

Assume that $M\subset R^n$, $n\ge 3$, is a boundary of an open bounded set $D$ containing $0$, which is starlike w.r.t. 0, meaning that each ray $[0,x]$ from $x\in M$ to $0$ meets $M$ only once. Is ...
6
votes
2answers
721 views

Reference Request: Steinberg's 1975 paper “On a paper of Pittie”(retrieved)

I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference ...
9
votes
1answer
683 views

Is there a mathematical explanation for the Aharonov-Casher effect?

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
3
votes
2answers
379 views

What collections of convex sets result in non-trivial uses of Helly's theorem?

Consider Helly Theorem, taken from notes by Igor Pak: Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then ...
1
vote
1answer
151 views

Recommendations for binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\ ...
3
votes
2answers
607 views

Does equality of Hodge star and symplectic star imply Kähler structure?

Question The question asked is: On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does ...
0
votes
1answer
72 views

rational rotation vector and closed curves

On $\mathbb{T}^n$ with a Riemannian metric, the stable norm is defined as $$\Vert h\Vert=\inf \sum |r_i| \cdot \mathrm{length}(\sigma_i),$$ where $h\in H_1(\mathbb{T}^n,\mathbb R)$ and $\sum_i ...
7
votes
2answers
229 views

“generic” in elementary submodels

Suppose you have a Suslin tree $T$ and you have a countable elementary submodel $M$ containing the usual "enough stuff" (including $T$). A comment in Todorcevic's Partition Problems in Topology ...
0
votes
2answers
364 views

Solution to differential equation

a) How to solve, or at least to prove the existence of a solution to differential equation for given initial condition $y(s)=y_0>0$ and $y'(s)=y_1$, $s<0$, $$y''+(2-n)\coth(t) ...
7
votes
1answer
563 views

What does the t in t-category stand for?

To my knowledge the notion of a t-category was first introduced Beilinson, Bernstein and Deligne's Faiseaux Pervers. But while they explain the name "perverse sheaf", they don't give any indication ...
1
vote
2answers
207 views

Ascending chain condition on ideals of free products

In my previous question: M Shahryari (mathoverflow.net/users/29488), Normal Subgroups of Free Products, Normal Subgroups of Free Products (version: 2012-11-28), I asked if a group $A$ has max-n ...
9
votes
3answers
2k views

Showing a matrix is negative definite [formerly Showing a sum is always positive]

For each $d$, I have a matrix $M$ with values $$ M_{ij} = \begin{cases} \frac{4ij}{d} - \binom{2d}{d} & i \neq j & \\\\ \frac{4i^2}{d} - \binom{2d}{d} - ...
2
votes
1answer
288 views

Graphical calculus in braided G crossed fusion categories: Explanation request and a question

I am trying to understand the equivalence between the 2 category of braided G crossed categories and the 2 category of braided categories containing Rep(G) as a symmetric category. The references in ...
3
votes
2answers
1k views

The double of a smooth manifold with boundary?

$\def\mc#1{\mathcal#1}\def\seq#1{\langle#1\rangle}\def\bbR{\mathbb R}\def\gt{>}\def\dom{{\rm dom\ }}$In some instances, I have seen an appeal to the concept of "the double" of a smooth manifold ...
6
votes
2answers
727 views

reverse mathematics strength of “Lipschitz functions are somewhere differentiable”

What is the reverse mathematics strength of "For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ? ...
2
votes
2answers
5k views

Maximum likelihood estimator for Power-law with Exponential cutoff

Hi, for fitting empirical data to power-law I am aware of the work by Clauset et al. (http://arxiv.org/abs/0706.1062) and how to use maximum likelihood estimation. There exists also a simple maximum ...
3
votes
2answers
439 views

how can i solve a boundary value numerically on an infinite interval ??

let be the differential equation $ -y''(x)+x^{4}y(x)-E_{n}y(x)=0 $ with the boundary conditions $ y(0)=0=y(\infty) $ how could i use the shooting method or other numerical method to solve this ...
5
votes
3answers
2k views

Bertrand theorem - central forces

Here is a version of Bertrand theorem. Let us consider a force $F(r)$ which depends only on the distance to a given point. If all trajectories which remain bounded are closed, then either $F(r)=ar$ ...
14
votes
2answers
1k views

Frobenius splitting and derived Cartier isomorphism

Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results. 1. If $X$ is ...
6
votes
4answers
642 views

How many finite simple groups of order $p+1$?

I'm looking at finite simple groups of order $p+1$ where $p$ is a prime number. But they don't seem to fall into any classification - have these all been determined? Is the number of them even ...
10
votes
2answers
6k views

Mathematics for machine learning

I would like to know what mathematics topics are the most important to learn before actually studying the theory on neural networks. I ask that because I will start to learn about neural networks and ...
6
votes
3answers
2k views

Maximal ideal in polynomial ring

Is it true that the intersection of a maximal ideal in $A[x]$ with $A$ is a maximal ideal in $A$? Let's say A is Noetherian. I would be surprised if it isn't true but somehow I can't seem to show it. ...
0
votes
1answer
121 views

Quotient of Lie rings and quotient of Lie groups!

Is there any correspondence between Lie rings and Lie groups such that if one proves for a Lie algebra $g$ with ideal $I$ that $g/I$ is a Lie algebra, then the same result holds automatically for the ...
1
vote
1answer
155 views

question of topos and site

Let $T, P$ be two topoi, and $f:T \longrightarrow P$. Does there exist two site $S_{T}, S_{P}$ and a morphism $g: S_{T} \longrightarrow S_{P}$ such that $f$ is induced by $g$ ?
8
votes
2answers
463 views

When do functors induce monadic adjunctions of presheaf categories

For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor ...
2
votes
1answer
80 views

Isometries between Hilbertian homogeneous finite dimensional operator spaces

We know that if $i:R_n\rightarrow C_n$ is an isometry then for any $n$-dimensional operator space E, there is a factorization $i=uv$ with $v:R_n\rightarrow E$, $u:E\rightarrow C_n$ such that ...
2
votes
2answers
312 views

A certain type of Quadratic Constrained Quadratic Programming Problem (QCQP)

Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP) \begin{equation} \min_{z}~z^{H}z \\\ ~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0 \end{equation} I am ...
2
votes
0answers
219 views

A question about fiberbundles in algebraic geometry

I hope this question is not too simple: Let $F,E,B$ complex algebraic varieties such that there exists a fiber bundle $$F\to E\to B$$ where all morphisms are assumed to be algebraic. Question: If ...
6
votes
0answers
222 views

Lovász function of the Möbius ladder

Quantum motivation Noncontextuality inequalities (and in particular Bell inequalities) can be mapped into graphs, in such a way that its relevant properties can be calculated via some simple ...
2
votes
0answers
159 views

what does the decomposition theorem say for a Lefschetz pencil?

The setting is the following: let $X$ be a smooth projective variety (say over $\mathbb{C}$), $D$ a simple normal crossings divisor on $X$ and $(H_t)_{t \in \mathbb{P}^1}$ a Lefschetz pencil on $X$ ...
5
votes
2answers
616 views

Reference request - CDGA vs. sAlg in char. 0

Hello, Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...
5
votes
0answers
81 views

Symmetric product and seminormality (curve case)

Let $d$ be a positive integer. Is the $d$-symmetric product of a nodal curve over a field seminormal?
2
votes
0answers
174 views

How to find quotients of infinite triangle groups or von Dyck groups?

I need the following information about the quotients of infinite triangle (or von Dyck) groups. (1) Let $G(l,m,n)$ defined as $S^l$ =$T^m$ = $(ST)^n$ = $E$ is the hyperbolic ($1/l+1/m+1/n<1$) ...
15
votes
2answers
900 views

Interpretation of elements of H^1 in sheaf cohomology.

Given a variety V and a locally free (coherent) sheaf $\mathcal{F}$ of rank 1 (equivalently a line bundle $L$), I can do a Cech cohomology on it. Then $H^0(V; \mathcal{F})$ are just global sections. ...
4
votes
2answers
607 views

Simple uses for the Entropy bound on the volume of a Hamming ball

I'm a teaching assistant in an introductory course of Information Theory. I intend to prove the following well-known fact that easily proven using elementary information theoretic consideration: ...
8
votes
2answers
769 views

ad-nilpotent degree of a nilpotent Lie Algebra

Let $\mathfrak{g}$ be a Lie Algebra (finite dimensional, over $\mathbb{C}$). Engel's theorem tells us that if there exists a $m\in \mathbb{N}$ such that $ad(x)^m = 0$, $\forall x\in \mathfrak{g}$, ...
3
votes
1answer
195 views

Twisting an object P by an H-Torsor I

I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization. The Statement Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism ...
8
votes
1answer
539 views

Creating Models of $ZFC$

I know of only 2 main techniques to create a model of $ZFC$. The first one is creating a model which is an extension of $V$: this is forcing. The second technique is that of inner model theory and ...
7
votes
3answers
954 views

Checking whether the image of a smooth map is a manifold

I have a specific problem, but would also like to know how to tackle the general case. I will first state the genral question. Let $M$ be an embedded submanifold of $\mathbb{R}^n$ and let $F: ...
5
votes
0answers
167 views

Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong ...

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