0
votes
0answers
4 views

Decomposition of semi simple local systems

I found A question similar to this, but the answer wasn't clear to me and I'm not supposed to ask for further clarification in the answer section. Let $L$ a semi simple local system defined over an ...
0
votes
0answers
5 views

Optimal covering

Let consider a problem of optimal covering of Hamming space. So we have Hamming space $\{0,1\}^n$ and some integer $r$. We want to find a set $A \subseteq \{0,1\}^n$ such that any point from ...
-5
votes
0answers
16 views

Summation of Geometric Series

Im really desperate please help!!! how can you show that a. the sum oscillates between the two values a and b for the summation of geometric series {a*r^(n-1)}` provided that this is divergent? ...
0
votes
0answers
4 views

Finiteness properties for graph of groups decompositions

My curiosity was raised by the following question and the huge variety of comments and suggestions it attracted. I wondered if a converse statement might be equally interesting. Let $G$ be a finitely ...
0
votes
0answers
15 views

Interchange summation and differentiation

I asked this question already on math.stackexchange, but did not receive any answers see here Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that ...
0
votes
0answers
15 views

Covering finite groups by kernels

Let $G$ be a finite group. When does there exist a finite group $H$ such that every $h\in H$ is in the kernel of some epimorphism $G\to H$? This is well-known to be true for $G$ abelian, for example ...
-3
votes
0answers
22 views

Summation of geometric series divergence [on hold]

The summation of some geometric series a*r^(n-1) is divergent. But what i don't understand is this: If the summation of a geometric series is divergent, then one of its sum is: a. the sum oscillates ...
1
vote
0answers
34 views

Octonions product: inversion in the right and identity in the left

Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena ...
1
vote
0answers
28 views

Asymptotic expansion square root matrix

I am looking for an asymptotic expansion for $\underline\gamma$ which is the "square root" matrix of a symmetric $p\times p$ matrix $\gamma$. Here $\underline\gamma$ is assumed to be symmetric, e.g. ...
0
votes
0answers
32 views

Regular embeddings of reductive groups

A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...
1
vote
1answer
66 views

Is anything like $\phi(n)<\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?

Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions? Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime ...
3
votes
2answers
74 views

Complete sets of incompatible totally ordered down-set in a partially ordered set

Let $(P,\leq)$ be a partially ordered set. A down-set is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ...
0
votes
0answers
33 views

Calculate the intersection numbers by a plane section [on hold]

This question is from the chapter A of Reid's note: Chapters on algebraic surfaces Let X = X$_d$ $\subset$ P$^3$ be a nonsingular surface of degree d and suppose that X has a plane section P ...
0
votes
0answers
23 views

A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...
4
votes
0answers
59 views

Flooding a cycle digraph via chip-firing: $n^{k-1} + n^{k-2} + \cdots + 1$ bound (a Norway 1998-99 problem generalized)

Let $k > 1$ and $n$ be positive integers. Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Let $D$ be a digraph which has exactly $k$ vertices $v_0$, $v_1$, ..., $v_{k-1}$ and exactly $k$ arcs $v_0 ...
-1
votes
0answers
14 views

probability distribution [on hold]

X is a continous random variable of normal distribution for the length of the rulers produced in a factory. Given X has mode of 15 cm and standard deviation of 1 cm. A ruler is randomly selected from ...
-1
votes
0answers
28 views

How to show Well Founded Induction false? [on hold]

The abstract reduction system ({a,b,c,d},→) where the → is defined as: http://i.stack.imgur.com/TS0Ud.png Let Q be a monadic predicate on {a,b,c,d} such that Q(a) = Q(b) = false and Q(c) = Q(d) = ...
1
vote
2answers
92 views

If d(“G/H”) < d(G) = 2, must H contain a primitive element?

Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...
-2
votes
0answers
30 views

Underlying Set in Model Theory [migrated]

In model theory a structure has an underlying set. In addition to the interpreted relations, are there (implicit) assumptions made about possible operations on this set? For example, is it assumed to ...
3
votes
0answers
62 views

Blowig-up a point in the singular locus

Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...
0
votes
0answers
19 views

Practical Way to Detect a Markov Chain is Regular Given the Transition Matrix [migrated]

I understand that a Markov Chain is reducible if, given its transition matrix $P$, there exists $n$ such that every element of $P^n$ is greater than 0. However, I am wondering that if there is an ...
4
votes
0answers
72 views

Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, ...
9
votes
1answer
159 views

When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?

Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...
-2
votes
0answers
28 views

Backpropagation KL-divergence for training “one level neural network” [on hold]

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
44 views

Strict partition of size n [on hold]

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 ...
-1
votes
0answers
35 views

Analytic formula to evaluate the exact value of solid angle subtended by an ellipse at any arbitrary point lying on the vertical axis [on hold]

`Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center. Standard equation ...
0
votes
0answers
51 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
60 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
42 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 ...
0
votes
0answers
29 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
19 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 ...
0
votes
1answer
72 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 $H$. In other ...
8
votes
2answers
323 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 ...
3
votes
1answer
93 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 ...
6
votes
0answers
134 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 ...
-1
votes
0answers
143 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
1answer
78 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 ...
0
votes
1answer
81 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 ...
1
vote
1answer
123 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
123 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 ...
4
votes
1answer
64 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, ...
0
votes
0answers
53 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
46 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 ...
-1
votes
0answers
28 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". ...
-2
votes
0answers
46 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?
-4
votes
0answers
50 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 ? The first part of the question is about smooth ...
0
votes
0answers
60 views

Rolling map as a diffeomorphism?

Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map) $$(\Phi\gamma)(t) = ...
0
votes
0answers
48 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 $\{ ...
0
votes
0answers
128 views

Multiplication Map, Is it invariant?

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 $Z$, let $E=\pi^*F$ and fix $x\in X$. The involution $i$ ...
2
votes
1answer
67 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 $\left(B^T B\right)^{-1}$ ...

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