2
votes
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
3answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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 ...

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