# All Questions

**0**

votes

**0**answers

7 views

### Is there for every word a Hall set such that it factors to a few small Hall words?

Is it true that for any word $w$ there is a Hall set $H$ such that for the unique factorization of $w$ in $H$ as $w=x_1\ldots x_k$ it holds that (where the first two are just the conditions for unique ...

**0**

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**0**answers

22 views

### cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:
1) is $X(\mathbb{Q})$ an empty set ?
2) is $X(\mathbb{Q})$ a finite (non empty) set ...

**0**

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**0**answers

4 views

### Laplace equation between circles

I need to solve the simple Laplace equation $$\nabla^2f(r,\theta)=0$$
with boundary conditions:
$$f(a,\theta)=g(\theta)$$
$$\lim_{A\rightarrow\infty}f(A,\theta)=1$$
where $a$ is a fixed real radius. ...

**0**

votes

**1**answer

39 views

### Alexeev's projective torus embeddings

I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry.
In chapter 2 in order to obtain an affine toric variety he takes ...

**0**

votes

**0**answers

15 views

### Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where ...

**5**

votes

**1**answer

45 views

### Dual of Banach-valued $L^p$

Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb ...

**3**

votes

**1**answer

53 views

### Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated,
For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...

**0**

votes

**0**answers

13 views

### Separating Two Groups of Data using Fisher's Linear Discriminant

I found an article (starting on page 8) that gives a neat method for finding the line/plane/hyperplane that maximizes the separation between two groups of data points in n-dimensions. It uses Fisher's ...

**4**

votes

**0**answers

31 views

### Orders in Central Simple Algebras. Applications

It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ...

**0**

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**0**answers

15 views

### Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$

Let $\mathcal T = D^b(\mathbb P^1)$, the bounded derived category of $\mathbb P^1$ over a field $k$. It is well known that $\mathcal T$ admits a strong and full exceptional sequence $\{\mathcal O, ...

**0**

votes

**1**answer

29 views

### Genus of Covering Space of 3-Manifold

Let $M_g$ and $M_h$ be closed orientable 3-manifolds of genus $g$ and $h$ respectively and suppose that $M_g$ is an $n$-sheeted cover of $M_h$. Is there a formula that would allow us to compute $g$ if ...

**5**

votes

**0**answers

92 views

### A question on compact sets

Let $K\subset \mathbb{R}^N$ be a compact set. We say
$K$ is "good" if the following property holds:
Given a set of open neighborhoods $\{x\in U_x\subset \mathbb{R}^N\}_{x\in K}$ there exists a finite ...

**0**

votes

**0**answers

26 views

### Completeness of spaces $\Lambda(\varphi, p)$

Definition 1. Let $f$ be measurable function on a mesurable subset $E\subset \mathbb R^n$. Non-increasing rearrangement of $f$ is a function $f^\ast(x)=\inf\{s>0: \operatorname{mes} E[|f| > s] ...

**4**

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**0**answers

36 views

### Is a finitely generated subgroup of a free profinite group virtually a retract?

Let $F$ be a nonabelian finitely generated free profinite group, and let $H \leq F$ be a finitely generated closed subgroup. Must there be some open subgroup $H \leq U \leq F$, and a closed normal ...

**4**

votes

**0**answers

34 views

### Closed geodesics avoiding points in hyperbolic surfaces

Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...

**0**

votes

**0**answers

13 views

### How can I run Normal Boundary Intersection (NBI) method with matlab or any other software? [on hold]

I have a constraint optimization problem. I want to convert it to a multi-objective mathematical programming problem (so-called Multi-objective optimization , vector optimization). I need to run ...

**0**

votes

**0**answers

24 views

### Limiting Ratio of Solutions to Ordinary Differential Equations

I'm trying to find the limit of the ratio of two functions
$ \lim_{t \rightarrow \infty} \frac{f(t)}{g(t)} $ but only have the initial conditions and the differential equations they solve, but the ...

**1**

vote

**0**answers

38 views

### On the Heat Kernel for Generalized Laplacians on non-compact manifolds

In my setup $M$ is a complete non-compact Riemannian manifold with Ricci curvature bounded from below; and $D_L$ is a generalized Laplacian for $M$. This means that $D_L$ is a linear second-order ...

**3**

votes

**0**answers

51 views

### Examples of Symplectic Questions Solved by ``Mirror Symmetry Translation'' to Complex Questions

According to the proponents of homological mirror symmetry, when a complex and symplectic manifold are mirror symmetric, we can take difficult questions about the symplectic space and transfer them ...

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6 views

### Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$.
Let $1<p<\infty$.
Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

**0**

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**0**answers

26 views

### BPP with expected polynomial time

Suppose we change the definition of BPP to require the TM to run in expected polynomial time (such as in ZPP).
Will the resulting class be equal to BPP, or larger?
Thanks.

**0**

votes

**0**answers

12 views

### Optimization of a multilinear function over a product of hypersimplices

Let $P = \Delta_1 \times \cdots \times \Delta_N$ be the Cartesian product of $N$ hypersimplices. Let $f : P \to \mathbb{R}$ be a multilinear function of $N$ variables, ie $x_i \mapsto f(x_1, \ldots, ...

**2**

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**0**answers

59 views

### Is an anticanonical Weil divisor in $\mathbb{Q}$-Gorestein variety Calabi-Yau?

Let $P$ be a normal, $\mathbb{Q}$-Gorestein variety with terminal singularities. Let $X \subseteq P$ be a normal, irreducible Weil divisor such that $X \sim_{\mathbb{Q}} - K_P$, that is ...

**3**

votes

**0**answers

166 views

### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
...

**1**

vote

**0**answers

31 views

### Geometric automorphism of free group respect to nonorientable suface

An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...

**7**

votes

**1**answer

157 views

### singularize the least inaccessible?

Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while ...

**4**

votes

**1**answer

44 views

### Linear intersection number and vertex covering number

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

**0**

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**0**answers

13 views

### Algebra textbook recommandation for person already familiar with algebra [migrated]

Can you guys please recommend me a good textbook on algebra, given that I am already familiar with lots of algebra stuff and I want to revisit, and deepen my knowledge?
(Little backstory: I started ...

**-1**

votes

**0**answers

36 views

### smallest (sub-) sigma algebra containing a null set [on hold]

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 ...

**1**

vote

**0**answers

33 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 ...

**3**

votes

**0**answers

53 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

28 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

41 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 ...

**6**

votes

**1**answer

58 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 ...

**3**

votes

**0**answers

63 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 ...

**2**

votes

**0**answers

90 views

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

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**-1**

votes

**2**answers

83 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

**0**answers

24 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

64 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 ...

**2**

votes

**0**answers

35 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

**0**answers

70 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} - ...

**-1**

votes

**0**answers

41 views

### What role does the quantum torus play in Noncommutative geometry [on hold]

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

**0**answers

58 views

### Papers about decentralized search and cluster

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!
EDIT (David ...

**0**

votes

**0**answers

18 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

**0**answers

30 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

17 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 ...

**3**

votes

**1**answer

80 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

105 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

**0**answers

50 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 ...

**4**

votes

**1**answer

159 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 ...