# All Questions

**0**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**4**answers

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

**1**answer

60 views

### What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence:
...

**4**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**4**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**4**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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) = ...