0
votes
2answers
35 views

A question about open subsets of Hilbert space whose complements are compact sets

Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. Is $C$ ...
0
votes
0answers
28 views

When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...
11
votes
6answers
1k views

Polynomial contact structures on $RP^3$

Let us consider polynomial contact structures on $\mathbb RP^3$, i.e. contact structures on $\mathbb R^3$ defined by a form $w=Pdx+Qdy+Rdz,\ P,Q,R\in \mathbb R[x,y,z]\ $ in an affine part and then ...
-1
votes
0answers
4 views

Backpropagation KL-divergence for training “one level neural network”

Hi I hope that some one could help me. I have a L matrix (25x1000) (I have 10000 works each work is represent by 25 bits) I map each word to one of 5 classes {vary negative,negative, ...
-3
votes
0answers
22 views

Strict partition of size n

I want to know how to calculate how many strict partitions of X are with size n. For example there are 22 partitions of number 8, and there are 6 strict partitions of 8 (partitions with distinct ...
6
votes
1answer
142 views

Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of ...
4
votes
2answers
362 views

Based loop groups as stacks?

I have been stuck for some time, thinking about the following question. Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension ...
0
votes
0answers
16 views

Relating overlapping simplicial complexes

Let $X$ be a simplicial complex and let $A,B\subset X$ be subcomplexes such that $C=A\cap B$ is a non-empty simplicial complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$, ...
0
votes
1answer
32 views

Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E (i.e. line bundles). Let $$ n_1:= ...
0
votes
0answers
24 views

On rank of matrix splittings

Given $M\in\Bbb Z_{\geq0}^{n\times n}$ with highest entry $b$, do there exist non-negative $A,B$ of rank atleast $c\cdot r$,atmost $d\cdot r$ with fixed $c,d>0$ such that $M=A-B$? Posted: ...
3
votes
1answer
64 views

Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
5
votes
2answers
400 views

Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to \left\[G\right]$, where $\left\[G\right]$ is the ...
5
votes
0answers
87 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0}^{n\times n}$ with highest entry $b$, is there a nice method to characterize non-negative matrices $Q$ such that $$\mathsf{rank}(M+Q)\leq s\cdot\mathsf{rank}(Q)$$with ...
8
votes
2answers
222 views

Integrals of representations over geodesics

Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...
7
votes
2answers
201 views

Residual finiteness: why do we care? [on hold]

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ...
0
votes
0answers
23 views

Integral geometry and curvature of surfaces

I'll stick to 2-surfaces in $\mathbb{R}^3$ for simplicity. Higher dimensions generalizations welcome. In classical integral geometry, we may obtain up to a scaling factor, for example, the surface ...
1
vote
1answer
281 views

Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$. For $m$ large, let ...
1
vote
1answer
218 views

Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism: $$g_Y \colon Y \times_X Y \cong Y ...
0
votes
0answers
28 views

Find all integers x, y, and z such that 1/x + 1/y = 1/z [migrated]

Characterize all positive integers x, y, and z such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. For example, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$
0
votes
0answers
15 views

contiguity and infinite complexes

Two simplicial maps $f: K^{(n)}\to L$, $g: K\to L$ are called contiguous if for every simplex $\tau\in K^{(n)}$ and it carrier $\sigma$ we have that $f(\tau)\cup g(\sigma)$ is a simplex in $L$. It ...
8
votes
2answers
263 views

$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$

Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...
3
votes
1answer
42 views

Product of binary Boolean operators

I asked this question a day ago on math.stackoverflow but figured it could have an interest here. I'm interested in the set $\mathcal{P}_N$ of boolean functions of boolean variables $p_1, p_2, ...
-1
votes
0answers
115 views

About a problem requiring long calculations [on hold]

Given six points: $$P₁=(129/256,65/256)=(x₁,y₁)$$ $$P₂=(132097/262144,66561/262144)=(x₂,y₂)$$ $$P₃=(2164277249/4294967296,1090535425/4294967296)=(x₃,y₃)$$ $$P₄=(δ_{q-1}-2α_{q-1},β_{q-1})=(x₄,y₄)$$ ...
5
votes
0answers
100 views

Polynomial growth without Gromov's theorem

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies ...
8
votes
1answer
167 views

Piecewise linear (PL) structures on $\mathbf R^4$

One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...
1
vote
1answer
90 views

How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?

Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and $\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of points of $M$, where $d\mu|_p$ fails to be ...
0
votes
1answer
260 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
51
votes
30answers
5k views

books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
0
votes
1answer
52 views

Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...
0
votes
1answer
91 views

Function field of the Jacobian of genus 2 curve over $\mathbb{F}_q$

I have been trying to build the function field of the jacobian of a genus 2 smooth curve over a finite field, but I am having problems making it explicit, I need to work with another curve with points ...
2
votes
0answers
23 views

Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...
6
votes
4answers
251 views

Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows: Pick $k$ distinct numbers out of numbers ...
7
votes
3answers
176 views

How do small central extensions drop the dimension of a faithful representation?

Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied... I am interested in the phenomenon ...
0
votes
0answers
41 views

Characterization of the stable manifold [on hold]

Assume we study a (finite dimensional) differential system $$ x'(t)=f(x(t)), \quad x(t) \in \mathbb R^n, $$ for a smooth function $f$ and such that $0$ is an equilibrium point. Thus, we have existence ...
6
votes
4answers
591 views

Multiplicity one conjecture

I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for ...
1
vote
1answer
55 views

Upper bounds on elements of a matrix

During my research I have come across matrices this type $$C=B\left(B^T B\right)^{-1}B^T\ ,$$ where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
2
votes
1answer
75 views

Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory ...
9
votes
1answer
414 views

Can you write $\mathbb R^2$ as a disjoint union of two totally disconnected sets?

Can you write $\mathbb R^2$ as a disjoint union of two totally disconnected sets?
0
votes
0answers
42 views

Pro-constructible subset of scheme intersects very dense subsets?

Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$? If $Y$ is just constructible, this is true. ...
0
votes
0answers
72 views

Rank two vector bundle without theta divisor?

Let $\pi:X\rightarrow Z$ a double cover of an elliptic curve with genus $g\geq 3$. Choose a general rank 2 and degree 1 vector bundle $F$ on $Z$, then $E=\pi^* F(p)$ ($p$ a ramification point) ...
6
votes
5answers
543 views

Applications of Chevalley Restriction Theorem

Let $G$ be a simple linear algebraic group (over $\mathbb{C}$, say) and $\mathfrak{g}$ be its Lie algebra, $\mathfrak{t}\subset \mathfrak{g}$ the Lie algebra of a maximal torus in $G$ and $W$ the ...
0
votes
0answers
57 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
11
votes
3answers
761 views

Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let $f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
2
votes
2answers
236 views

Do you know of any asymmetric, nonparametric measure of dependence?

A measure of dependence is a way to assign a number (usually normalized between 0 and 1) to a couple of random variable, such that $\delta(X,Y)=0$ if and only of $X$ and $Y$ are independent, and ...
-1
votes
0answers
24 views

Combinatorial optimization problem [on hold]

Suppose I have a population, divided into 6 known classes. I get a feasible solution when I select 2 elements from each class (so, 12 in total). For every feasible solution, I can compute a "cost". ...
37
votes
18answers
5k views

What are some deep theorems, and why are they considered deep?

All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
-2
votes
0answers
39 views

Good upper bound for an alternanting series [on hold]

Someone know a good upper bound for the partial sums of $S=\sum(-1)^{n+1}\sqrt{n}$? I mean how fast is the growth of this sum?
11
votes
2answers
664 views

Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic. According to the axioms of Kolmogorov, probability theory is formulated with a (normalized) probability measure ...
-4
votes
0answers
41 views

Can a non-compact manifold be embedded? [on hold]

Can a non-compact smooth manifold be embedded into another smooth manifold? Moreover, Can we get a diffeomorphism between tow non-compact manifolds and what is the type of diff. group action if yes?

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