# All Questions

**-2**

votes

**0**answers

7 views

### Trigonometric substitution [migrated]

Been out of touch with trigonometry for some time now. Need help proving this expression.
Sin2x/2 = 1/2(1-Cosx)
Any help will be appreciated. Thanks.

**1**

vote

**1**answer

124 views

### Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...

**4**

votes

**1**answer

229 views

### Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$

I was talking to a friend and the following set $S$ came up.
Let $f$ be some real valued function tending to infinity.
Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...

**3**

votes

**1**answer

92 views

### Matching in the Boolean Algebra

We consider the Boolean Algebra of the set $[n]=\{1,\dots,n\}$ and the matching $\psi$ from the $m+1$ subsets of $[n]$ into the $m$ subsets of $[n]$ for $(n+1)/2\leq m+1\leq n$, which is used to show ...

**0**

votes

**0**answers

106 views

### Where are there defined objects between gerbes and bundle gerbes?

Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism.
Does this exist in the literature?

**3**

votes

**1**answer

115 views

### Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that
$\|A\| < x$ and ...

**16**

votes

**1**answer

974 views

### Unexpected $\sqrt{3}$

A somewhat lengthy calculation, involving integrals, reveals that the probability an $n\times n$ Hermitian matrix, drawn from the Gaussian unitary ensemble, is positive definite, decays as ...

**4**

votes

**0**answers

45 views

### Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...

**6**

votes

**3**answers

467 views

### Links between Geometric Group Theory and Number Theory

Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.

**1**

vote

**4**answers

207 views

### Homology of infinite intersection

If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...

**2**

votes

**3**answers

301 views

### Variations to Cayley's Embedding Theorem for Groups

Early in a course in Algebra the result that every group can be embedded as a subgroup
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher ...

**0**

votes

**0**answers

3 views

### Abstract Algebra [migrated]

Show that a group that has only a finite number of subgroups must be a finite group.(Fraleigh, A First Course in abstract Algebra-7th Edition,pg.67) I could not show properly so I need help. Thank ...

**2**

votes

**2**answers

215 views

### Morita equivalence via Kan extension

Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}_yf=F\colon \widehat{\mathcal C}\leftrightarrows ...

**6**

votes

**3**answers

158 views

### Existence of nonergodic polygonal billiard

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one.
A standard conjecture is that a ...

**-1**

votes

**0**answers

32 views

### Maximum chi-square distance between norm vectors [closed]

What is the maximum possible chi-square distance between two normalized vectors? The representation of chi-square distance is below.
$d(x,y) = \sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$

**1**

vote

**1**answer

115 views

### Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum_{i=0}^n |z_i|^2)).$$
What is the analogous formula for a Kaehler ...

**2**

votes

**0**answers

51 views

### DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...

**-5**

votes

**0**answers

90 views

### The problem of Reimann zeta function [closed]

$\zeta(2)=\sum_0^\infty 1/n^{2}<\pi^2/6=1.644934<2$
From the popular knowledge
$\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$
but
$\int_0^\infty x/(e^x-1)dx=\int_0^\infty ...

**1**

vote

**0**answers

65 views

### K-theory of ringed spaces (including henselian and formal schemes); excision and Mayer-Vietoris

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. ...

**-5**

votes

**0**answers

61 views

### Prove determinant of nxn matrix is (a+(n-1)b)(a-b)^(n-1)? [closed]

Prove det(mat) is (a+(n-1)b)(a-b)^n-1 where matrix is nxn matrix with a's on diagonal and all other elements b, off diagonal?
For example, suppose matrix with diagonal composed solely of a's. All ...

**4**

votes

**0**answers

107 views

### $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows
$$
(A,B) \cdot (M_1, \ldots, M_m) ...

**3**

votes

**1**answer

215 views

### Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...

**3**

votes

**1**answer

702 views

### Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?

The usual cohomology theory on schemes uses injective or flasque resolutions of quasi-coherent sheaves. Hence it uses Axiom of Choice.
However, if the base scheme is a noetherian separated scheme, the ...

**-3**

votes

**0**answers

105 views

### The problem of Riemann zeta function [closed]

$\zeta(2)=\sum_0^\infty 1/n^{2}<1.8$
From the popular knowledge
$\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$
but
$\int_0^\infty x/(e^x-1)dx=\int_0^\infty xe^{-x}/(1-e^{-x})dx$
$=\int_0^\infty ...

**3**

votes

**1**answer

234 views

### Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.

**0**

votes

**0**answers

183 views

### Recreating the wheel

I recently finished my Phd in pure maths and I am looking for open problems in my research area, functional analysis. Without going into the details, I stumbled onto an interesting problem and I ...

**0**

votes

**0**answers

54 views

### Cover one finite subset of integers by another one

Let $A$, $B$ be two finite subsets of integers. We denote by $C(A, B)$ the minimum number of shifts of $A$ to cover $B$. More formally, it can be written as
$$
C(A, B)=\min\{|S|: S\subseteq ...

**6**

votes

**1**answer

411 views

### Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$.
Place $K$ on an inclined plane, and let it roll down the plane,
under some reasonable assumptions of friction between $K$ and
the plane, ...

**0**

votes

**0**answers

41 views

### additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by
$$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...

**-2**

votes

**0**answers

42 views

### n-th prime in first order arithmetics [closed]

Recently I have thought about formalizing Turing machine in first order arithmetics, step by step, starting from the most basic things. But I quickly struck a problem - to continue, I need to find a ...

**1**

vote

**0**answers

184 views

### A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version:
Randomly assign (with replacement) $N$ balls to $M$ urns. ...

**-1**

votes

**0**answers

82 views

### Why integer should have finite many digits? [closed]

for example, if we take the real part of pi 3.1415926... and write it from right to left
like ...6295141, we can get a number, but this number is not a integer, why ? why it is not a integer?
Can we ...

**1**

vote

**1**answer

26 views

### a class of directed hypergraphs

I am interested in a certain class of directed hypergraphs, more precisely in the class of those hypergraphs each of whose hyperedges contain an even number of nodes (not necessarily the same even ...

**6**

votes

**0**answers

72 views

### When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request.
In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...

**9**

votes

**2**answers

323 views

### What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...

**3**

votes

**1**answer

236 views

### What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech.
Does anyone know when Hechler forcing was first used in a publication?

**-2**

votes

**0**answers

34 views

### Simplifying Trig Equation with Identifities [closed]

I have an equation I have been given to solve, I know how to start but I do not know what to do after I use the Trig Identities. Any help?
Here is what I was given
(cos(A + B) + cos(A - B))
/
...

**4**

votes

**2**answers

202 views

### Can finitely generated subgroups of limit groups be detected in free group quotients?

In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant ...

**4**

votes

**1**answer

214 views

### Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...

**3**

votes

**1**answer

84 views

### Extremal graph theory for directed graphs

In extremal graph theory, there are results such as
$$t(C_4,G)\geq t(K_2,G)^4,$$
where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and ...

**4**

votes

**1**answer

118 views

### What is the early history of the concepts of probabilistic independence and conditional probability/expectation?

In the 1738 second edition of The Doctrine of Chances, de Moivre writes,
Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards ...

**3**

votes

**1**answer

98 views

### Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...

**0**

votes

**0**answers

28 views

### Grassmann algebra morphism with universal property

I'm pretty sure that the following doesn't work, but nevertheless i wanted to ask, maybe this is a kind of well-known construction i've never heard of:
Let $\Lambda(\mathbb{R}^n)$ be a finite ...

**1**

vote

**0**answers

80 views

### Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the
Fourier transform of a random variable $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

**1**

vote

**1**answer

103 views

### A comparison principle for parabolic equation

(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...

**3**

votes

**1**answer

91 views

### Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...

**-1**

votes

**0**answers

11 views

### regu tools l_curve regularization stanford ee 263 [migrated]

I am trying to solve one of the famous stanford EE263 problems, which gives me matrix A representing blurring of an image and y, representing the blurred image. For that I have been trying to use ...

**4**

votes

**4**answers

857 views

### Advice for number theory library

Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I ...

**3**

votes

**0**answers

86 views

### Infinite series of determinants

I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...

**1**

vote

**0**answers

71 views

### $p$-groups with $\Omega_1(G)\leq\Phi(G)$ [on hold]

Let $G$ be a finite $p$-group with $\Omega_1(G)\leq\Phi(G)$.
What do we have information about this group?