0
votes
0answers
11 views

smallest (sub-) sigma algebra containing a null set

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
0
votes
0answers
8 views

pushforward of universal objects along canonical morphisms of stacks

The kind of question I'm interested in has the following flavor: having two moduli stacks with one being an enhanced (i.e more data) version of the other with the natural "forgetting" map between them ...
0
votes
0answers
16 views

An Interesting Markov Chain Mixing time question (coupling of random variables)

Consider a simple markov chain called reproduction process. Let $N>0$ be fixed. At each time $t$, there are $N$ balls, in which $x$ balls are colored white, and $N-x$ balls are colored black. We ...
1
vote
0answers
9 views

How many points does 'the-most-point-contained-circle' contain at least?

Remark : This question has been asked previously on math.SE with receiving only a partial answer. Question : Letting $n\ge 2\in\mathbb N$, how can we find $f(n)$ such that the following two ...
0
votes
0answers
12 views

Relations commuting with logical equivalence

A beginner's question, but still research-level (I hope): I am looking for theorems of the form, 'Relation X commutes with logical equivalence', where X is NOT just uniform substitution. Pointers to ...
4
votes
0answers
15 views

What is the original reference for disorientations on tangle diagrams?

There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the ...
1
vote
0answers
19 views

Linear intersection number of a graph and chromatic number

A linear hypergraph is a pair $\pi=(V, L)$ where $V\neq \emptyset$ is a set and $L\subseteq {\cal P}(V)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
2
votes
0answers
38 views

Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...
0
votes
0answers
31 views

An integral with respect to the Haar measure on a unitary group

Let $A\in \mathbb{C}^n$ be a hermitian deterministic matrix and $D\in \mathbb{C}^n$ be a diagonal deterministic matrix. I need to calculate $\int_{O(n)}\det{(A-HLH')}[dH]$, where $[dH]$ is the unit ...
0
votes
1answer
29 views

Equality of two conditional expectations

I would like to show that for any random variable $X$ and $Z$ such that $X$ and $Z$ are independent and for any measurable functions $f$ and $g$, $$ \mathbb E \left[ f(g(X),Z) | g(X) \right] = ...
0
votes
0answers
18 views

Extension of canonical surjection to a place of fraction field

Let $R$ be an integrally closed domain, $K$ its field of fractions, and $m$ a maximal ideal of $R$. By Chevalley theorem, the canonical surjection $R\to R/m$ extends in at least one way to a place ...
1
vote
0answers
45 views

Dividing a n- cochain by a 1-cochain

Assume that $X$ is a path connected hausdorff topological space. Let $\alpha\in C^{1}(X,\mathbb{R})$ and $\beta\in C^{n}(X,\mathbb{R})$ be cochains in real singular cohomology. Asume that $\alpha ...
0
votes
0answers
21 views

Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries). Can we partition this union into at most $n$ rectangles? I think it's pretty ...
0
votes
0answers
52 views

How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, but I can't find the paper any more and I'm not sure if I remember it correctly. By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
4
votes
1answer
48 views

Intermediate submodels of Cohen and Mathias generic extensions

The first question is about Cohen forcing. Let $\mathbb{P}=Add(\omega, \omega),$ and let $G=(a_n: n<\omega)$ be $\mathbb{P}$-generic over $V$, where each $a_n \subseteq \omega.$ For each ...
-1
votes
0answers
30 views

What role does the quantum torus play in Noncommutative geometry

Quantum torus is the associative algebra generated over a field by quasi-commuting variables $x_i$ satisfying $x_ix_j = q_{ij}x_jx_i$ for suitable nonzero scalars $q_{ij}$. What is the role played ...
0
votes
0answers
13 views

Papers about decentralized search and cluster [on hold]

I just start an independent study about small world network and clusters and I try to find papers about decentralized search and clusters. Can anyone give me some references? Thanks!
0
votes
0answers
13 views

Is each multiplicative linear functional on $L^1(SL(2,R):SO(2,R))$ is triviall?

We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{\xi \in \hat{G}; \xi|_H=1\}$is triviall. Can we conclude that each multiplicative linear functional on $\{f\in ...
0
votes
0answers
28 views

how to draw geodesic on the ellipsoid? [on hold]

I try to simulate geodesic on the ellipsoid recently. I have two points on the ellipsoid. After solving the inverse problem, I can get the distance and two azimuths of two points. (I can obtain S12 ...
0
votes
0answers
14 views

About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing. Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph. I would like to understand what is the ...
2
votes
1answer
47 views

Is there a standard notation for off-diagonal transpose?

Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$. But is there a conventional way of notating the matrix ...
1
vote
1answer
70 views

Example of torsion in orientable manifolds?

An orientable manifold can have torsion in its integer homology. But I believe by Poincare duality the manifold must be at least 4-dimensional -- isn't that right? Anyway are there simple examples ...
1
vote
0answers
36 views

Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
2
votes
1answer
81 views

Bar Construction Model of Ring Spectrum Quotient

Suppose I am given a morphism $f:BG\to BGL_1(R)$ for $R$ some at least $E_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism ...
0
votes
0answers
127 views

For a mathematician that English is not the native language, does he/she think in english or graph or native language? [on hold]

For example, if you are a mathematician with Chinese the native language. During your research you find most of the books or papers are in English, of course when you read them, you probably will ...
0
votes
0answers
36 views

Continuous self-information

Let $I(X,Y)$ be the mutual information between two continuous random variables $X$ and $Y$. We have $I(X,Y) = H(X)-H(X|Y)$, and setting $X=Y$ leads to $I(X,X) = H(X)-H(X|X)$. If $X$ was discrete, ...
0
votes
1answer
60 views

Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
2
votes
0answers
46 views

A Krull-Schmidt Theorem for Lie groups?

I wondered whether there is an analogue of the Krull-Schmidt theorem for real Lie-groups. More precisely, what conditions do you have to impose on a finite dimensional Lie group $G$ so that for each ...
11
votes
4answers
479 views

Categorical proof subgroups of free groups are free?

This is a crossport of this question from MSE. Is there a categorical proof that subgroups of free groups are free? How about the result that subgroups of free abelian groups are free abelian? ...
-1
votes
0answers
31 views

Iterative Calculation? [on hold]

Apologies, as I do not know how to phrase this question in the correct terms; however, I will try my best. I have an equation that looks like this: D = A - B - C However, C = ( [A - B] * X ) Is ...
11
votes
0answers
227 views

Why is “The Higman Rope Trick” thus named?

I'm studiyng Higman's Embedding Theorem, and a fundamental part of the proof is the following lemma: If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a ...
2
votes
0answers
53 views

Besse p134 Riemann tensor in dimension 4

Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that $$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$ because we are in dimension ...
-1
votes
0answers
38 views

Improper integral calculation - limit at infinity [on hold]

Will you please help me prove the following limit is zero ? $ lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} cos(x\cdot u) du $ Thanks in advance
5
votes
0answers
76 views

Alexander polynomial in branched covers

Suppose I am given a homology sphere as a double branched cover over a link (of determinant one). Let a knot in this space be given as a lift of an arc with endpoints on the link. Is there a way to ...
2
votes
0answers
34 views

Decidability of first order theory of subclasses of posets

Is the first order theory of finite posets known to be undecidable? Does anyone know a survey about such results?
0
votes
0answers
83 views

p-divisibility of the connected component of the Picard group

Let $X$ to be a smooth projective variety over a field of positive characteristic $p>0$, then can one claim $Pic^0(X)$ is p-divisible.
3
votes
3answers
240 views

Find an integrable, positive, unbounded, analytic function

Is there a standard example of a function $f \in L^1( \mathbb R)$ which is analytic, positive, integrable but not bounded? An example which comes immediately to mind is to take the series of narrower ...
2
votes
1answer
74 views

Schur covering group for S4

It is well-known that the symmetric group S4 has two Schur covering groups, S4-tilde and S4-hat. There are explicit presentations for both groups, and we know that S4-hat is isomorphic to GL(2,3). ...
2
votes
0answers
81 views

Hilbert vs Chow in nice cases

I'm trying to understand the relationship between the Hilbert schemes and Chow varieties in situations where everything is simple. Suppose that $X$ is a smooth projective variety over $\mathbb C$, ...
0
votes
0answers
22 views

Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $R$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e. ...
1
vote
0answers
21 views

Progressive measurability and functional composition

Suppose we have a progressively measurable process $X$ taking values in $\mathbb{R}^d$. What are sufficient conditions on a function $f( x, t, \omega ) \colon \mathbb{R}^d \times [0,\infty) \times ...
-4
votes
0answers
33 views

Whats the formula to work out the minimum monthly payment of a loan? [on hold]

I'm a developer, and i'm building a snowball debt calculator. I want a formula to work out what the minimum monthly repayment would be on a debt with a given interest. And I really want to get the ...
1
vote
1answer
91 views

Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...
0
votes
0answers
20 views

Integrability - conditions of lax pairs

I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation: $$ \partial_t U - \partial_x V + [U,V]=0 $$ where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ ...
0
votes
0answers
22 views

Upper bound a function

The problem is of finding the maximum of the following function (in terms of i) $\ f_i = \frac{(2m-i) \cdot i}{2 b} \ln(b) - \tfrac{1}{2} i \ln(i) +O(i) $ providing $\ 0<b \leq m, 0<i \leq m ...
3
votes
0answers
50 views

Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the ...
1
vote
0answers
51 views

Help in understanding “Local well-posedness for the Maxwell-Schrodinger system”

Is there someone who knows the following paper "Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada. I'm trying to study it but I've some doubts. In particular I'm not ...
9
votes
1answer
303 views

Can ZFC prove it cannot derive an inconsistency in $n$ steps?

Let $Con(\mathtt{ZFC}, n)$ denote the statement "$\mathtt{ZFC}$ cannot prove the contradiction within $n$ steps (or better within $n$ symbols) within a given proof system (say a natural deduction to ...
9
votes
0answers
226 views

Who first talked about “holes” in homology?

The question Why do the homology groups capture holes in a space better than the homotopy groups? and many others here use the idea that homology counts the ``holes'' in a space. The comments on this ...
0
votes
1answer
22 views

distances-based dispersion measuring approach

Is there any known approach or method to measure the dispersion of a set depending on the distances between its points (i.e.: without calculating the average or the mean) ? thanks.

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