# All Questions

**2**

votes

**0**answers

25 views

### Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element:
The square of $a$ equals Muger's "squared dimension" of $X$, an ...

**-4**

votes

**0**answers

77 views

### Algebraic Curves: Exercise 2.17 (William Fulton) [on hold]

Let $V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2$, e $\overline{X}, \overline {Y}$ the residues of $X,Y$ in $A(V)$ its coordinate ring; let $z= \dfrac{\overline{Y}}{\overline{X}}\in K(V)$. Find the pole sets ...

**0**

votes

**0**answers

107 views

### Numbers summing to distinct integers

We want to find $r$ positive integers $\{a_i\}_{i=1}^r$ such that of atmost $(2s+1)^r$ values obtained from $$\sum_{i=1}s_ia_i$$ where $s_i\in\{-s,-s+1,\dots,0,\dots,s-1,s\}$ with $s\leq r$, we insist ...

**0**

votes

**0**answers

16 views

### Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of ...

**-2**

votes

**1**answer

124 views

### Direct image of structural sheaf [on hold]

I am sorry if my question is not of high level!!
Let $\pi:X\rightarrow Y$ be a double cover where $X$ and $Y$ are projective smooth curves.
Is it true that $R^1\pi_*\mathcal O_X=0$ ? Why ?
Thanks ...

**4**

votes

**1**answer

225 views

### Formal group law over $\mathbb{F}_p$

Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...

**2**

votes

**1**answer

49 views

### Enumerating matrices function of ranks

Is there an expression/approximate expression for number of real matrices $M\in\{0,1\}^{n\times n}$ of rank $r\leq n$?

**4**

votes

**2**answers

237 views

### how to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N
$$\sum_{i = 1}^{N} N \bmod i$$
It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...

**0**

votes

**0**answers

98 views

### Weyl group representation

Let $G$ be a reductive p-adic group.
Let $W$ be a weyl group. if $w$, and $w_o \in W$.
I want to know in which case we have $w w_o w^{-1}= w_o$ ?
in case if $w_o(\theta)=\theta $ where $\theta$ is a ...

**0**

votes

**0**answers

17 views

### Statistics of strongly connected components in random directed graphs

I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for.
...

**-3**

votes

**0**answers

57 views

### The most general splitting of a field extension [on hold]

This question has been posted here on math.stackexchange, but I felt it was maybe better to post it here.
Take $L/K$ an extension of the field $K$. I have questions on how we can "split" the ...

**0**

votes

**0**answers

36 views

### format of grading Witt Lie Algebra

Let $W(n,m)$ be generalized Jacobson-Witt algebra over a field of characteristic p>3, according to the grading of $W(n,m)$ , we know that it inherit the grading from $A(n,m)$ as follows: ...

**1**

vote

**0**answers

43 views

### Question regarding a theorem of Erdos and Renyi on $B_2(g)$ sequence

Let $S \subset \mathbb{N}$. We say $S$ is of type $B_2(g)$ if the number of representation
of the form $n = s_1 + s_2 \ (s_1 \leq s_2)$ is bounded by $g$ for every $n \in \mathbb{N}$.
Let $S(n)$ be ...

**0**

votes

**0**answers

69 views

### endomorphisms algebra of a real representation [on hold]

Let $G$ be a finite group. Given a real irredcible representation of $G$, we know that its endomorphisms algebra is a division algbra and hence is the real, complex or quaternion algebra. Is there a ...

**-8**

votes

**0**answers

67 views

### Polinominal equations [on hold]

Explain why it is possible that polynomial has no real solutions. Use reasoning
to expand your explanation to find the general characteristics of polynomials that have no
real solutions

**1**

vote

**0**answers

39 views

### solution to a parabolic PDE

I'm reading a paper where the following parabolic PDE is considered:
$u_t(x,t)=u_{xx}(x,t)+b(x)u_x(x,t)+\lambda(x)u(x,t)$, with boundary conditions
$u_x(0,t)=qu(0,t) \text{ and } u(1,t)=\int_0^1 ...

**5**

votes

**1**answer

78 views

### Find subset of collection of sets whose intersection has minimum average value

Let $a_1,\ldots,a_n>0$, and let $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ (all non-empty).
For any $I\subseteq\{1,\ldots,d\}$, define $S(I)=\bigcap_{i\in I} S_i$.
Given some $1\leq s < d$, consider ...

**1**

vote

**0**answers

40 views

### A cohomology associated with a codimension one foliation(2)

What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension $*$?
Moreover what is the description of this cohomology for ...

**1**

vote

**1**answer

181 views

### Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv −1 \mod p$. Is there a possibility to say ...

**-1**

votes

**1**answer

62 views

### How to compute the Expectation of the random variable using Taylor Series expansion [on hold]

I don't know how to solve the following expression:
$ = nm^2 E \bigg[\frac{ \exp(\theta) {(\log(R))}^2}{N(R,x)}\bigg] \hskip 5 pt Eq(4) $ which I have explained below. $R$ follows Poisson ...

**3**

votes

**1**answer

59 views

### Are there 2-connected regular graphs whose maximum matching leaves 3 vertices uncovered?

I'd like to use Corollary 5 of a paper by Hell & Kirkpatrick on graph packings to obtain an NP-hardness result. They want a 2-vertex-connected graph $F$ such that every matching in $F$ leaves at ...

**3**

votes

**1**answer

86 views

### A multinomial-type sum over compositions of an integer

I find myself needing to compute (or asymptotically estimate) the following sum over the $2^{S-1}$ compositions of $S$. I am hoping an expert in combinatorics (I am a computer scientist) will ...

**-1**

votes

**0**answers

115 views

### Ext of Skyscraper sheaf [on hold]

Let $X$ be projective curve over the complex number field; and let $\mathbb C_p$ be the skyscraper sheaf whose fiber aver $p\in X$ is $\mathbb C$ and $0$ otherwise.
How could we prove that ...

**0**

votes

**0**answers

38 views

### Supremum of positve kernel

Let $A(x,y)\geq 0$ $\forall x,y$ be a positive kernel of a bounded, positive operator $A$.
How does one prove that
$$\sup_{x,y}A(x,y)=\sup_x A(x,x)?$$

**1**

vote

**0**answers

23 views

### Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil

Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...

**0**

votes

**1**answer

25 views

### Reorder rows and columns to find as close to block diagonal structure in a non-symmetric sparse matrix [on hold]

I have a sparse matrix with no apparent structure and am wanting to reorder the rows and columns in such a way that the matrix becomes as close to block diagonal as possible. I am using R and have ...

**0**

votes

**0**answers

125 views

### Twisting sheaf of Serre

I'm sorry if my question is rather trivial, but I can't figure it out.. Given $A$ a ring and $P=Proj(A[X_0,\cdots,X_n])$, I know that $\oplus_n H^0(P,\mathcal{O}(n))=A[X_0,\cdots,X_n]$. This equality ...

**11**

votes

**3**answers

270 views

### Sets of points containing permutations - a Ramsey-type question

The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with ...

**1**

vote

**0**answers

15 views

### Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?

**1**

vote

**0**answers

36 views

### connectedness of coincidence set

Consider the following obstacle problem in the whole domain $\mathbb{R}^n$
min{$\Delta u$, $u$-$\phi$}=0
with prescribed boundary value $\lim_{|x|\rightarrow\infty}u(x)=0$ and $\phi$ (can be assumed ...

**0**

votes

**0**answers

109 views

### Under which conditions the inclusion of a sub-simplicial set of the nerve of a category is a Joyal equivalence?

Let $i: X \to N\mathcal C$ be a monomorphism in the category of simplicial sets, with $C$ a category and $NC$ its nerve. I am looking for sufficient conditions (and not too difficult to check) under ...

**0**

votes

**1**answer

55 views

### Discrete Taylor's Formula in n dimensions [on hold]

I am searching for discrete form of Taylor's formula in n dimensions. Please share the appropriate resources.

**1**

vote

**0**answers

143 views

### Polynomial existence over finite field

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote ...

**1**

vote

**1**answer

98 views

### Combinatorics problem involving counting the number of certain substrings

I'm not sure if this question is suited for MO, but it does seem quite challenging to me, and is required for a research problem in chemistry I'm working on. I did try getting help from elsewhere ...

**1**

vote

**0**answers

22 views

### A curious example envolving moment's convergence

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...

**0**

votes

**0**answers

39 views

### An embedding of modules by tensor product over a Noetherian domain

I have a problem on Ring theory. I would like to prove or disprove the following statement:
Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...

**5**

votes

**0**answers

159 views

### Conjugation of the quotient of $SL(n,\mathbb{C})$ by a finite subgroup

EDITED Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$.
Let $H\subset G$ be a finite subgroup.
Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety.
...

**0**

votes

**0**answers

118 views

### Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...

**0**

votes

**0**answers

30 views

### Helmholtz boundary value problem in 2D [on hold]

I want to solve the Helmholtz equation in 2D with constant nonhomogeneities: $$\nabla^2w-\lambda w=C$$ and with Dirichlet boundary conditions such that $$w(0,0)=0$$ ...

**5**

votes

**0**answers

86 views

### Examples of Brody hyperbolic affine varieties which are not Kobayashi hyperbolic

Let $X$ be a complex space.
We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$.
We say that $X$ is Kobayashi hyperbolic if the Kobayashi ...

**-4**

votes

**0**answers

37 views

### Prove that a Graph is connected using eigen values $\lambda$ [on hold]

Prove that for a graph is connected if and only if $\lambda_{max}$ > $\lambda_{1}$
Prove that for a $d$-regular graph $\lambda_{\max} = \lambda_1 =
\cdots = \lambda_{k-1}$ if and only if the graph ...

**0**

votes

**0**answers

39 views

### Asymmetry of functions defined on the $n$-th roots of the unity

Let $\mathcal{A} = \{V : \mathbb{U}_n \rightarrow \mathbb{C}\}$ where $\mathbb{U}_n$ is the group of the complex $n$-th roots of the unity. This group naturally acts on $\mathcal{A}$: for any $a \in ...

**1**

vote

**0**answers

86 views

### A Lie algebra assiciated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

**1**

vote

**1**answer

64 views

### A cohomology associated with a codimension one foliation

Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:
$$\phi:\Omega^{i}(M)\to ...

**7**

votes

**0**answers

275 views

### Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in ...

**3**

votes

**1**answer

69 views

### Homogeneous polynomial vector fields tangent to the unit sphere

This question has something to do with that one.
Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v_1,\ldots,v_n)$ whose components $v_j$ are homogeneous of ...

**3**

votes

**2**answers

456 views

### Co-Hausdorffification

Given a topological space $(X,\tau)$ we can define the "$T_2$-ification" of $X$ by setting $T_2(X,\tau) = X/\simeq$ where $x\simeq y$ in $X$ if and only if for every open neighborhood of $x$ has ...

**3**

votes

**0**answers

52 views

### Connected sum of chiral manifolds

Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two ...

**-1**

votes

**1**answer

153 views

### Parabolic subgroup

I have a question about root set corresponding to $P_θ ∩ M_Ω$
where $θ$ is a subset of simple roots, $Ω=θ∪{α}$ where $α$ is a simple root and not in $θ$, $P_θ$ is a parabolic subgroup corresponding ...

**1**

vote

**1**answer

103 views

### Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor?

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...