0
votes
0answers
15 views

Commutation of limits and pullbacks

i know that i asked this question already on stackexchange, but maybe it is also useful to aks this here: Suppose we have three directed sequences of $C^*$-algebras, say ...
0
votes
0answers
11 views

optimization problem, any solution?

The objective is as follows: $\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\}$ $s.t.\ \ \ Tr(\mathbf{F} \mathbf{F}^H)<p$ where $a$ ...
0
votes
0answers
22 views

A solution of a q-difference equation

Is it possible to find a solution of the $q$-difference equation $$f(q^{-1}x)-f(qx)=x(a-x)f(x),$$ with $f(0)=1$, (perhaps) in terms of basic hypergeometric series? Or in another rather explicit form? ...
-1
votes
0answers
36 views

Can first and second fundamental form be considered as differential 1-forms? [on hold]

Let $(M,g)$ be an abstract surface embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote the first and second fundamental form by $I$ and $II$, respectively and the smooth vector fields on ...
3
votes
0answers
34 views

Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$ i.e. Horn's inequalities for n matrices? Can such ...
0
votes
0answers
23 views

Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...
0
votes
0answers
68 views

Basketball shots and stopping rule

Moved over from StackExchange. You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can ...
2
votes
0answers
39 views

Is the set of $C^\infty$ sections a strong deformation retract of the space of $C^k$ sections?

For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre bundle. Let $k\in\mathbb{N}=\{0,1,\dots\}$, and let $U$ be an open subset of the total space $J^kE$ of the $k$-jet bundle. ...
0
votes
0answers
18 views

discrete dynamical system [on hold]

sir, I am doing research work in dynamical systems in the area alpha limit sets.I came across a situation where i can find some properties of alpha limit sets analogous to omega limit sets.I am ...
0
votes
0answers
71 views

Multiplicative Identity in a (SU(n)) Group

Let ${P_i}$ be members of $SU(n)$ such that for all $U$ in a subgroup [or subset] $H$: $$P_1UP_2U\cdots P_nU=I$$ where $I$ is the identity element. Are there ${P_i}$ can $H$ be enlarged to the whole ...
1
vote
0answers
21 views

Reference request: Continuity of unique maximizer of linear functional on convex set

Does anyone know reference for a theorem of the following sort: Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that $$f(w):=\operatorname{argmax}_{x\in K}w(x) $$ is ...
0
votes
0answers
35 views

Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution $\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$? ...
6
votes
0answers
41 views

Can $Ded(\kappa)$ be a supremum?

Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$. It ...
5
votes
2answers
76 views

Poisson ideals vs. ideals generated by Poisson central elements

Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$: $I = \langle (c_i) \rangle$ is generated by ...
8
votes
2answers
289 views

Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...
2
votes
0answers
40 views

Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...
0
votes
0answers
49 views

Double Markovity

Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows $$X\to W\to Y,$$ and $$X\to Y\to W.$$ How to prove that there exist functions $f$ and $g$ such that ...
3
votes
0answers
50 views

Connection between framed cobordisms and zero sets

Let $W\subseteq M\times[0,1]$ be a framed submanifold (a framed cobordism in $M$) and $2w<m-1$ where $w,m=\dim W,M$. Assume that $M$ is compact and that $W\cap M\times \{0\}$ is the zero set of a ...
-1
votes
0answers
14 views

Problem with equation structure and re-arrangement [migrated]

I am creating a program which used an equation for photo-efficiency in certain types of plants. Because of the limits of the programming (or maybe my abilities with said language). I need to break ...
0
votes
1answer
122 views

Problem of book Kunen [on hold]

Suppose $P$ is a notion of forcing in $M$ such that $\left | P \right | \leq \omega_{1}$ and $P$ is ccc. Suppose further $\Diamond$ holds in $M$. How does one show that $\Diamond$ also holds $M[G]$?
0
votes
0answers
157 views

Questions about the Collatz conjecture-also known as the “3x+1 problem” [on hold]

Let "F(k,m)" denote the following recursive function of two positive integer variables. For all k, F(k,1)=k. For all k and all m, if F(k,m) is even, then F(k,m+1)=F(k,m)/2. For all k and all m, if ...
1
vote
1answer
56 views

Contour integral around semi-circle

Can one use contour integration to evaluate $\int^{\pi}_{0} \frac{1}{1-\rho*sin(\theta)}d\theta$ for $0<\rho<1$? This would be trivial if the upper limit were $2\pi$ as we could let ...
0
votes
0answers
40 views

If the $L$-series does not vanish

I refer to this paper http://wstein.org/papers/shark/shark.pdf At the top of page 24, we are dealing with the issue where the $L$-series does not vanish for the case where $p$ is good and ordinary. ...
3
votes
1answer
61 views

Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...
3
votes
1answer
100 views

Positive definite quadratic forms on Banach spaces

This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if ...
0
votes
0answers
35 views

Probability that d-Brownian Motion ,d>3, avoids a set A

In other words, the probability that Brownian motion stays within $A^{c}$. So far I found that it is 1, for random cylinders and thorns. What about for connected and compact sets (traps)? Any other ...
0
votes
0answers
77 views

Could anyone help me with a problem regarding fundamental groups? [on hold]

Let G be a group and x be an element of G. N is the least normal subgroup of G containing x. If there is a normal, path-connected space whose fundamental group is isomorphic to G, then I have to show ...
-3
votes
0answers
47 views

Counting sets of tuples [on hold]

I am looking to count the size of certain sets created by taking the product of multivariate functions $F,Q,\ldots$, where the input arguments come from finite sets $D_1, D_2$. For example, ...
2
votes
0answers
33 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 ...
3
votes
0answers
112 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 ...
4
votes
0answers
98 views

Mukai flops and derived categories

As proved by Kawamata and by Namikawa, if $M$ and $M'$ are complex projective varieties which are related by a Mukai flop (elementary transformation) along projective spaces $W\subset M$ and ...
8
votes
2answers
436 views

Maximal ideals are prime (history thereof)

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
119 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
104 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
48 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
1answer
83 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
71 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?
3
votes
1answer
82 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
21 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
47 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
60 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
64 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 $\varphi:U\rightarrow Y$ ...
7
votes
1answer
195 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
34 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 ...
1
vote
2answers
154 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
117 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
48 views

An integral identity [on hold]

For all $n>1$ and $0<i<n$ we have the following identity? $$\int\limits_0^{\pi} \frac{\cos(nx)-\cos(i\pi)}{\cos x-\cos(i\pi/n)}dx=0.$$
5
votes
2answers
171 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
16 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 ...
3
votes
0answers
97 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) ...

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