2
votes
0answers
40 views

Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...
5
votes
0answers
172 views

Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where ...
10
votes
2answers
536 views

Maximal ideals are prime (history answer please!)

Please can someone tell me the history of the simple argument that any maximal ideal of a commutative ring or distributive lattice is prime? (It is understood that we have found the maximal one using ...
3
votes
1answer
130 views

Vector Laplace Beltrami operator of the Gauss map

Consider an abstract surface $(M,g)$ embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote by $N:M \to \mathbb{R}^3$ the Gauss map (normal field) of the surface. Write the Laplace Beltrami ...
3
votes
2answers
131 views

Bochner's theorem for measures of positive type

Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying ...
0
votes
0answers
52 views

Rank 2^n quadratic form with non trivial invariant e_n

I advise you to have a look this question of mine first Rank four quadratic Form with non trivial discriminant in I(k) From quadratic form theory its well known that for a field $k$ and the ...
2
votes
2answers
153 views

Examples of cubic Julia sets

I'm looking for information on explicit cubic filled Julia sets with interesting properties such as: Having two different finite attractors (such as $f(z)=z^3-1.5z$) Being disconnected with ...
3
votes
3answers
88 views

graded generalization of the Moyal–Weyl product

Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?
5
votes
1answer
96 views

Clifford algebras for quadratic modules over ringed spaces

What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ...
0
votes
1answer
25 views

Bivariate skew-normal distribution [on hold]

I'm using the Gaussian distribution as a weight function for solving pde's. I'm interested in skewing the function. For one-dimensional problems, it was easy to derive the resulting skewed Gaussian ...
4
votes
0answers
62 views

First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
-2
votes
0answers
66 views

Calculating of the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$? [on hold]

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?
-1
votes
0answers
69 views

Can a linear map on a finite-dimensional subspace be extended to the whole space “trivially”? [on hold]

I have a question concerning the extension of continuous linear maps. Let $X$ be a normed vector space and let $U$ be a finite-dimensional subspace of $X$. Furthermore, let $\varphi:U\rightarrow Y$ ...
8
votes
1answer
244 views

Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$. Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$: $$R_q = M_q^T (M_q ...
0
votes
0answers
48 views

asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question. Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf Namely theorem 5. Now, feel ...
2
votes
2answers
204 views

Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...
1
vote
0answers
128 views

On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces, $$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
-2
votes
0answers
53 views

An integral identity [on hold]

For all $n>1$ and $0<i<n$ we have the following identity? $$\int\limits_0^{\pi} \frac{\cos(nx)-\cos(i\pi)}{\cos x-\cos(i\pi/n)}dx=0.$$
6
votes
2answers
197 views

String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...
1
vote
0answers
19 views

Gelfand Kirillov dimension of induced modules

How does the Gelfand-Kirillov dimension of induced modules behave? In patricular, if $S$ is an Ore extension of $R$, how is the GK-dimension of an $R$-module say $N$ related to the GK dimension of $N ...
4
votes
1answer
141 views

Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...
13
votes
4answers
346 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
1
vote
0answers
45 views

A Density Problem

Let $ \mathscr{D}=\mathscr{D}(\mathbb{R}^n - {0}) $ be the space of all smooth functions with compact support in $ \mathbb{R}^n - {0} $ topologized by the standard Schwartz topology and let $ ...
-2
votes
1answer
32 views

Real Analysis : Uniform Convergence of sequences on the real line [on hold]

I want a sequence of continuous functions whose limit function is continuous but the convergence is not uniform. I have an example : fn(x)=x/n ; The limit function f(x)=0 is continuous but the ...
6
votes
1answer
204 views

Cases where the number field case and the function field (with positive characteristic) are different

In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the ...
0
votes
1answer
238 views

Yang-Mills equations are not elliptic [on hold]

How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic? Alternatively, how does one calculate the principal symbol of the Yang-Mills equations? Can ...
3
votes
0answers
64 views

Subplanes of Finite Projective Planes

If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub ...
5
votes
1answer
206 views

Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
0
votes
1answer
22 views

Constructing a transition matrix of a time-homogeneous, finite Markov chain with full support stationary distribution

is there a way to construct a transition matrix of a time-homogeneous, finite Markov chain such that the stationary distribution always has full support (this is equivalent to all states of the chain ...
1
vote
0answers
46 views

Borel class of a set of measures

Let $K$ be a compact Hausdorff space and consider $X=Ball(M(K))$, the unit ball of the space of regular Borel measures on $K$. Endow $X$ with the weak-$*$ topology $\sigma(M(K),C(K))$, regarding ...
-1
votes
0answers
125 views

Why do I study a lot but still don't understand the material too well? [on hold]

I had a linear algebra test yesterday and I studied a week in advance for it. I did all the assigned homework questions, past exam questions, problem set questions, but I still did poorly on the test. ...
11
votes
0answers
260 views

Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
9
votes
3answers
798 views

When did coordinate plane “as we know it” come into play?

This is a historical question that needs some background to make sense. Let me start with the longer version of the question: When did negative numbers, algebra and coordinate plane come together? ...
2
votes
1answer
131 views

The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$. One important example of left distributive algebras arises ...
14
votes
2answers
428 views

Why polarization of abelian varieties?

Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange. I could never understand the intuition behind polarization of abelian ...
11
votes
2answers
358 views

Asymptotics for algebraic numbers of height less than one

The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? The rather ...
1
vote
2answers
140 views

Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
9
votes
1answer
157 views

Algebraic $K$-theory of algebras in symmetric spectra: reference

I want to use the technicalities of structured ring spectra for the first time in my life, and I am not really familiar with the relevant literature. I am looking for a reference that defines ...
-3
votes
0answers
33 views

Laplace transform for integral of product of two functions [on hold]

Consider the following integral of product of two function $$ \int_0 ^t f(s)g(s-t)ds $$ i want to know the laplace transform of above term w.r.t t. if $g(s-t)$ is replaced by $g(t-s)$, there is a ...
4
votes
0answers
131 views

Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of: $$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$ are on the ...
3
votes
2answers
61 views

Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e. $$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$ It is known that for arbitrary $A_{n\times n}$ $$\|T\circ ...
2
votes
1answer
146 views

Scheme vs $A$-scheme morphisms

Let $A=S^{-1}\mathbb Z$ be a localization of a multiplicative set $S\subset \mathbb Z$. Question 1: Let $X$ be an arbitrary $A$-scheme, and view $X_A=X\times_{\mathbb Z} A$ as an $A$-scheme via the ...
0
votes
0answers
25 views

Inapproximability of logarithmic factor of independent set

The hardness result derived using PCP theorem for Independent set suggests that there exists some absolute constant $\epsilon_0$ such that for $0< \epsilon < \epsilon_0$, it is hard to ...
-4
votes
0answers
24 views

Help with simple rotation on an x,y plane [migrated]

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
-1
votes
0answers
78 views

a question about holomorphic vector field on complex manifold [on hold]

Let $(M,J)$ be a complex manifold, we know that every holomorphic vector field $X$ can be written as follow $$X=-J(Z)+\sqrt{-1}Z,$$ where $Z$ is a real vector field on $M$. My question is that if ...
1
vote
0answers
132 views

Hodge modules and Deligne-Beilinson cohomology of function fields

Let $K$ be a function field over complex numbers i.e. the fraction field of a complex variety. Then one can define the Deligne-Beilinson cohomology and mixed Hodge modules for $K$ as the direct limit ...
0
votes
0answers
56 views

Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition

there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the ...
2
votes
1answer
96 views

Existence of an infinite finitely generated $p$-group with nontrivial intersection of nontrivial subgroups

Is there an infinite finitely generated (non-cyclic) $p$-group $G$ such that the intersection of all nontrivial subgroups of $G$ is a nontrivial subgroup?
-2
votes
0answers
33 views

Laplace transform of non-central chi- square distribution [closed]

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
0
votes
1answer
66 views

Reference for the result that the systol map from Teichmuller space to curve complex is coarsely Lipschitz

Let $\mathcal{T}(S)$ denotes the Teichmuller space of a finite type surface $S$ equipped with Teichmuller metric and $\mathcal{C}(S)$ denotes the curve complex. Define a map ...

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