All Questions

0
votes
0answers
71 views

independent subset problems [on hold]

I'm interested in the following which i suspect is probably a well studied problem. Given a set $N=\{1,2,...,n\}$ and $M=\{1,2,...,m\}$ consider a map $$f:N\rightarrow 2^{M}$$ (elements of $N$ to ...
2
votes
1answer
55 views

Embedded Contact Homology and Manifold Decompositions

Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them. In the symplectic ...
-1
votes
0answers
58 views

A new method of solutions for partial linear differential equations [on hold]

Recently,I read a book on partial differential equations,which says that the solution of second order linear equation of two differentiating variables and analytic coefficients can always be expressed ...
0
votes
1answer
98 views

totally disconnected sets and homeomorphisms

For every totally disconnected perfect subset S in the plane one finds a homeomorphism of the plane onto itself mapping S onto the ternary Cantor set. This is an exercise in a book by Engelking and ...
1
vote
0answers
58 views

Identity of Bernoulli polynomials

consider the Bernoulli polynomials defined by the generating function: $$\left(\prod_{i=1}^m \frac{a_i}{\left( e^{a_i}-1 \right)}\right)e^{xt}=\sum\limits_{n=0}^{\infty}B^{m}_n\left(x\vert ...
1
vote
0answers
69 views

A formal local triviality statement for smooth maps

Let $f:X\to Y$ be a smooth morphism of schemes of finite type over a field $k$, and suppose that $f(p) = q$. Let $Z = f^{-1}(q)$ be the fiber of $f$. Let $\hat{X}$ be the formal completion of $X$ at ...
0
votes
0answers
18 views

What is the nilradical of $\mathfrak{gl}_n$? [migrated]

I'm really embarrassed to ask but what is the nilradical of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$, i.e. the set of ad-nilpotent elements of $\mathfrak{gl}_n(\mathbb{C}) = ...
3
votes
2answers
85 views

The upper and lower bound of the projection of a subshift of finite type

I am thinking a problem: given a subshift of finte type of $\{0,1\}^{\mathbb{N}}$ and $2>q>1$, where $q$ is a real number. Then how can we find the largest and smallest numbers of the projection ...
4
votes
0answers
38 views

Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions. By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...
4
votes
4answers
267 views

Breaking up the free Lie algebra into Gl irreps

The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra , the free Lie algebra generated by any choice of ...
1
vote
1answer
60 views

What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence: ...
4
votes
0answers
92 views

Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s

It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...
1
vote
0answers
75 views

First passage time of a pure drift process

I am facing the following unusual problem: $Z_t$ is a pure drift process of the form $$ dZ_t = \kappa(X_t - Z_t) dt $$ where $X_t$ is another bounded process. I am interested in computing / ...
4
votes
1answer
298 views

Push-forward of locally free sheaves

Let $X, Y$ be smooth projective varieties and $f:X \times Y \to Y$ be the natural projetion map. Let $\mathcal{F}$ be a locally free sheaf on $X \times Y$. Is it true that $f_*\mathcal{F}$ is locally ...
2
votes
0answers
42 views

dual composition of binary relations

I'm not sure if this is of any interest at all, but I spent some time looking at it a couple of years ago so I'd like to ask for input on this. Given two binary relations $\rho,\,\sigma$ on a set ...
11
votes
1answer
208 views

Universal maps between topological spaces

Let $X,Y$ be topological spaces. We call a continuous map $u:X\to Y$ universal if for every continous map $f:X\to Y$ there is $x\in X$ such that $f(x) = u(x)$. If $u:X\to Y$ and $v:Y\to Z$ are ...
1
vote
0answers
143 views

Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...
0
votes
0answers
96 views

When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements: Statement $A_0$: Given any point $p\in X$, there ...
2
votes
0answers
71 views

What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it. Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi ...
2
votes
0answers
68 views

Gromov width of cotangent disk bundle

Given a symplectic manifold $(M^{2n},\omega)$, the Gromov width of $M$ is defined to be $w(M)=sup\{{\pi r^2| B^{2n}(r) \rightarrow M}\}$ My question is: what is the explicit value of $w(D^*S^n)$, ...
1
vote
1answer
79 views

Piercing of subspaces in a projective space?

The "piercing subspace" problem may be stated as follows: There are given several subspaces in a projective space, rather non-intersecting. Find an additional subspace of a prescribed dimension that ...
4
votes
1answer
137 views

Decomposing representations of finite groups of Lie type via computer

This is related to my previous question here. Let me remind you what that question asked: Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for ...
10
votes
4answers
1k views

Robotics, Cryptography, and Genetics applications of Grothendieck's work? [on hold]

I was reading about the passing of Alexander Grothendieck, and something caught my interest: Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal ...
2
votes
2answers
168 views

Bound on exponential sum with weights

Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function. I know one can bound sums of the form $$ \sum_{x \leq X} e(f(x)) $$ via for example Van der Corputs's result, provided we make ...
0
votes
0answers
17 views

Continuous wavelets for piecewise polynomial functions

For continuous wavelets, as Haar is to piecewise constant functions, what is to piecewise linear functions and is there some wavelet basis that spans piecewise cubic spline functions?
0
votes
0answers
35 views

What is the Vapnik-Chervonenkis dimension of sigmoidal functions? [migrated]

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...
1
vote
2answers
210 views

Is true that $[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }]_{4m} = 0$?

There are two questions: How to prove that in general $[\hat{A}(\mathbb HP^m)]_{4m} = 0$ It is possible to verify it for low values of $m$. How to prove that in general $[\frac{\hat{A}(\mathbb ...
1
vote
1answer
165 views

Smoothness and smoothness over formal neighborhood

Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$. We assume that the map on the level of fomal neighborhoods $X_{x}\rightarrow Y_{y}$ is formally smooth, can we find ...
0
votes
0answers
49 views

Complex conjugate orbifold of C^n [on hold]

I have a very simple question: what is the result of identifying each point $(z_1,\ldots,z_n) \in \mathbb{C}^n$ with $(z_1^\ast,\ldots,z_n^\ast)$? Is it just $\mathbb{C}^{n-1} \times \mathbb{H}$, ...
0
votes
0answers
29 views

Prove that the subset sum problem with fixed size and number reusability is NP complete

I'm trying to solve the following problem: There are B lists of unspecified size containing integers. Pick a number from each list so that the sum of all the picks is exactly A. Prove that this ...
1
vote
1answer
73 views

Whitehead's second Lemma and invariants of exterior square

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
9
votes
2answers
236 views

Is there any relationship between the topologies of the clique complex and the independence complex?

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...
-3
votes
0answers
81 views

Does the sequence tan(n) diverge? [on hold]

Does the sequence tan(n) diverge or inverge? Also, how can I prove it?
3
votes
2answers
162 views

Cohomology of SL(2,R) with coefficients given by linear action

Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication. What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$? And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a ...
-6
votes
0answers
39 views

Question about integrals [on hold]

In an electric circuit , suppose E is the electromotive force in volts t seconds and E = cos ( ln ( t ) ) . Determine the mean value of E t = 1 to t = e ^ pi .
0
votes
0answers
18 views

measures of global stability [on hold]

Local stability of a point attractor or stable state can be checked from Lyapunov exponent. Likewise, what are the measures which can be used to tell about global stability in complex systems.
0
votes
0answers
44 views

$\Omega(G)$ is homeomorphism to $\Omega(T)$ [on hold]

Is it possible that the end set of graph $G$ is homeomorphism to the set of end for every spanning tree of $G$ where $G$ has a cycle. In other hand, for given graph $G$ and spanning tree $T$ of $G$. ...
-1
votes
0answers
55 views

Decomposition of separable metric space with certain topological property [on hold]

Is there any information about when separable metric space with certain topological property can be decomposed into finetely many zero-dimensional subspaces with the same property? I am mainly ...
1
vote
0answers
67 views

Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?

Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...
1
vote
0answers
111 views

Questions on prime integral ideal congruences

Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...
9
votes
2answers
360 views

Can a parent and child node have the same type in a well-founded digraph tree?

$\newcommand\toward{\rightharpoonup}$It would help me to understand something in a current research project if someone could provide an example of directed graph $\langle G,\toward\rangle$ with the ...
3
votes
1answer
83 views

Stable rank of finite rings

Has any finite ring (not necessarily commutative) always stable rank 2 ? How do you prove that or does it follow from something ? May be this question is trivial but I'm not familiar with K-theory.
0
votes
0answers
92 views

Decision method for a partial mapping: very strange [on hold]

Consider this definition: A decision method for a partial mapping $F$ from $A$ to $B$ is a method which, if applied to an element $a$ of $A$, will give the value $F(a)$ if $a$ is in the domain of $F$ ...
4
votes
0answers
46 views

Best constant for a trace inequality

Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla ...
7
votes
4answers
199 views

4-regular graph with every edge lying in a unique 4-cycle

What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle? Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one ...
2
votes
1answer
141 views

not Gauss sum with the same magnitude

Gauss sum is a sum of $p$ roots of unity with magnitude $\sqrt{p}$. Does another sum with such property exist? More exactly. Let $p$ be a prime number. $\zeta^p=1,\;\zeta\ne 1$. Causs sum: ...
0
votes
0answers
103 views

Unitary operators on Hilbert spaces [on hold]

Consider the set $U(H)$, of unitary operators on the real separable Hilbert space $H$. Fix an orthonormal basis $\{e_i\}$ of $H$. Is the subset $S$ of $U(H)$ corresponding to finite dimensional ...
6
votes
1answer
167 views

Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general. But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...
25
votes
1answer
547 views

“Nyldon words”: understanding a class of words factorizing the free monoid increasingly

BACKGROUND. Let me first introduce some classical definitions, which appear, e.g., in §5 of Lothaire's Combinatorics on Words, in §5.1 of Reutenauer's Free Lie algebras, and in §6.1 of Victor ...
0
votes
1answer
149 views

A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...

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