0
votes
0answers
49 views

Restriction of locally free sheaves and semi-stability on curves

Let $C$ be a stable curve and $\mathcal{F}$ be a locally free sheaf on $C$ such that the restriction of $\mathcal{F}$ to any of the irreducible component $C_i$ of $C$, $\mathcal{F}|_{C_i}$ is Gieseker ...
4
votes
1answer
104 views

Multivariable function analysis

Sorry if the terms I'm going to use is not professional enough:) This is about the complexity analysis of an algorithm. Let $\alpha$ be the largest zero root of the polynomial ...
1
vote
0answers
190 views

Analogues of the Monster for central charges different from 24

One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of ...
0
votes
0answers
30 views

the associated action on the transition functions

Let $X$ be a curve with an involution $\sigma$ generically unramified, given a $G-$bundle $E$ of rank $r$, than we ca take its pull-back, I want to describe the action of $\sigma$ on $G$. Fix a ...
1
vote
0answers
49 views

Probabilistic proof for expander existence [on hold]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders). I've been using the search ...
4
votes
0answers
88 views

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...
0
votes
0answers
23 views

rank of a Lie group over a non-archimedean local field of positive characteristic

In the case of a Lie algebra over a non-archimedean local field of positive characteristic (I have been led to believe that) it is not necessarily true that all Cartan subalgebras have the same ...
0
votes
0answers
50 views

Titchmarsh S function [on hold]

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of riemann-hypothesis gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...
2
votes
0answers
74 views

Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
1
vote
0answers
17 views

LU growth factor applied to LDL of a Positive Semidefinite matrix [on hold]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
0answers
105 views
+50

Local coordinates on (infinite dimensional) Lie groups, factorization of Rieman zeta functions

Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
0
votes
0answers
43 views

Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?

Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space. Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...
9
votes
1answer
301 views

What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
1
vote
0answers
127 views

Surgery to unlink $S^p$ and $S^q$ in $S^d$

We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres. I am looking for a surgery to unlink $S^p$ and ...
-1
votes
0answers
56 views

Difficult examples of invertible differentiable functions [on hold]

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
-4
votes
0answers
28 views

Finding kernel and image [on hold]

X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) What is the dimension of X? Find ker f and im f. Find bases for ker f and im f. Is f a bijection? My attempt: 1. I calculate dim ...
4
votes
0answers
97 views

Minimum number of real multiplications to multiply two quaternions [on hold]

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
0
votes
0answers
76 views

Can someone help me find a Mathematical documentary that aired on British television within the past 10 years about Leibniz? [on hold]

I have been searching for a documentary that aired on British television between around 2006 and 2012 which was centred around the German Mathematician, Gottfried Leibniz. All that I can remember ...
0
votes
0answers
64 views

Alternate proof for Caratheodory extension theorem

This question is on the intuition behind the Caratheodory definition of measurable sets as given in Billingsley. He motivates by saying that we "should" call a set $A$ measurable if $$P^*(A) + ...
1
vote
0answers
106 views

Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
0
votes
0answers
51 views

Contraction with a vector field and pullback bundle

While trying to understand the proof of "smooth" version of Kostant-Hochschild-Rosenberg theorem (which is due to Connes for compact smooth manifolds) I found the following argumentation: one is ...
0
votes
0answers
61 views

Elliptic Curve: Q=nP [on hold]

I have a question relating to Elliptic Curve Scalar Multiplication between two points. Given two points on that curve, Q and P, where Q=nP, is it possible to find n if we have an m such that mQ=P?
6
votes
1answer
83 views

Does a Gaussian process shrink under a contraction map

Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X_t)_{t\in T}$ defined by $X_t = \langle G, t\rangle$, where $G$ is a standard ...
3
votes
1answer
114 views

A question of terminology regarding integer partitions

I am wondering if there is a standard notation and name for the following. Let $\lambda$ be a partition $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_r\geq 1$ of $n$ into $r$ parts. Then we can ...
0
votes
0answers
110 views

Is there any computation of $K^0(X)$ and $K_0(X)$ for a singular curve $X$?

Let $X$ be a projective curve over an algebraic closed field $k$ which characteristic zero. Define $K^0(X)$ as the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ as the Grothendieck ...
1
vote
0answers
72 views

Syzygies in integral domains

Let $f$ be a homogeneous irreducible element in a graded commutative Noetherian ring. What is the possible set of elements $f_1$ and $f_2$ such that $f=f_1g_1+ f_2g_2$? Even in very particular cases ...
2
votes
2answers
89 views

Variation of Hodge structures associated to a hermitian symmetric domain

Let $D$ be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures $(h_s)_{s\in D}$ on a vector space $V$ satisfying specific conditions which depend on $D$ such ...
5
votes
1answer
107 views

Bounding the number of lattice points inside an $n$-dimensional ellipsoid

I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse. That is, given an $n$-dimensional ellipsoid ...
3
votes
4answers
194 views

Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$

I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click). In equation (27) the authors, apparently, used the following ...
18
votes
1answer
337 views

Can topological cyclic homology compute Picard groups?

Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism $$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$ where $Pic(\mathcal{O}_K)$ is the ...
5
votes
1answer
165 views

An unusual metric reconstruction problem

$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have ...
-3
votes
0answers
41 views

Calculate Inverse Fourier Transform [on hold]

How to calculate the Inverse Fourier Transform of the following functions: $\dfrac{1}{-1+2\pi i x}$ $\dfrac{1}{(2\pi ix)^2-2 \pi ix +1}$ I don't know how to evaluate the integrals.
1
vote
1answer
62 views

Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R \times \mathbb R/\mathbb Z \to \mathbb R$, $f = f(s,t)$, is the unknown ...
3
votes
1answer
106 views

Regarding left-to-right minima

Let $\rho$ be a permutation on $[1,n]$ and $l_i$ be the number of left-to-right minima in $\rho_{i\ldots n}$, I know that for a random permutation $E[l_1] = H_n$ (the $n$-th Harmonic number) but is ...
18
votes
2answers
1k views

An unfair marriage lemma

I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem: Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into ...
4
votes
2answers
147 views

Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community. Setup: Let ...
5
votes
0answers
76 views

Embedded spheres in the K3 surface

Using the Seiberg-Witten theory, we know that every (smoothly) embedded $S^{2}$ in $K3$ with trivial normal bundle is null-homologous. We know we have a lot of interesting knotted $S^{2}$ inside ...
0
votes
1answer
119 views

Gluing locally free sheaves on curves

Let $C$ be a quasi-projective curve, $C_i$ for $i=1,...,r$ are the irreducible components of $C$. Assume that $C_i$ is non-singular and $F_i$ locally free sheaf on $C_i$ of the same rank for all $i$. ...
-1
votes
0answers
72 views

Excercise of commutative rings, S. Balcerzyk and T. Józefiak [on hold]

Let $R=k[[x_1,...,x_n,y_1,...,y_n]]/(x_i y_j-x_j y_i)$, $i,j=1,\ldots,n$, where $k$ is a field. Prove that (a) $R$ is a domain. (b) $\dim R=n+1$. (c) $R$ is Cohen-Macaulay. (d) the type of $R$ ...
-1
votes
0answers
38 views

Cohen-Macaulay rings and Normal rings [migrated]

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
-1
votes
0answers
24 views

Geodesic parameterization under conformal mapping [on hold]

Under a conformal deformation of the euclidean metric, say: $\hat{g}_{ij}=e^{\phi}\delta_{ij}$, where $\phi$ depends on the radial coordinate alone, I am struggling to see the following fact: "With ...
2
votes
0answers
44 views

l1 Quadratic Programming [on hold]

Within a SQP- algorithm it can happen that the constraints of the quadratic sub- problems are infeasible. In order to overcome this infeasibilities, a l1 penalty method can be used according to ...
1
vote
1answer
102 views

Singularities in minimal surfaces [on hold]

There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times ...
1
vote
1answer
86 views

Construction of curves and morphisms

Given any triple of positive integers $(g',g,d)$ with $2g'-2\geq d(2g-2)$. Does there always exist curves $C_{g'},C_g$ of genus $g',g$ with a degree $d$ morphism $f\colon C_{g'}\to C_g$? If we fix ...
5
votes
0answers
133 views

How to prove that a projective module is not free?

Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...
0
votes
0answers
12 views

Simple RK4 measure of a force in 2nd order ODE [migrated]

Consider that I am solving a second order ODE using RK2/RK4. The ODE represents simple equations of motion: Equations of motion I am trying to solve: \begin{align} \frac{dx}{dt} &= v \\[.3em] ...
0
votes
0answers
22 views

Degree $2$ nilpotent matrices with non-zero product [migrated]

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and ...
2
votes
1answer
96 views

Are lax functor categories into a cartesian closed 2-category cartesian closed?

Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F ...
6
votes
1answer
231 views

Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
0
votes
0answers
68 views

$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are: Is $\inf\{i\in \mathbb N \cup \{0\}\cup ...

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