# All Questions

**0**

votes

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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**

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**0**answers

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**

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**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

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

**1**answer

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] = ...

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**0**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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

**0**answers

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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

**0**answers

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

**4**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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.