# All Questions

**2**

votes

**0**answers

95 views

### Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor on Kahler variety $X$. I am looking for a proof that such moduli space exists?
The log ...

**1**

vote

**0**answers

56 views

### Ideal structure of group $C^*$-agebras [on hold]

What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ?
If G to be the group of integers $Z$ , then $C^*$($Z$)= C($T$) so because ideal structure of $ ...

**5**

votes

**1**answer

209 views

### Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold.
Given a smooth $(n-1)$-dimensional smooth ...

**6**

votes

**0**answers

93 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...

**1**

vote

**0**answers

76 views

### Optimization on Binomial coefficients

Suppose we are given integers $1\leq r\leq N$, we want to study the following
$$
\max_{m_0+m_1=N,m_0,m_1\geq 1}\max_j \tbinom {m_0}{j}\tbinom {m_1}{r-j}.
$$
$N$ is very large, for instance $N\geq ...

**0**

votes

**0**answers

41 views

### Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space over a field of char $0$. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected cocommutative Hopf algebra and in ...

**1**

vote

**1**answer

161 views

### Weight 12 cusp forms for $\Gamma_0(p)$

Let $S_k$ be the space of weight 12 cusp forms of $\Gamma_0(p)$, ($p$ prime), then Sage tells that $\dim S_k^{\text{new}}=\dim S_k-2$. Thus the old forms spans a 2-dimensional subspace. One of the ...

**-1**

votes

**0**answers

68 views

### A simple question about ordinary diffential equations of first order [on hold]

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation:
$$ F(x,y,y',...,y^{(n)})=0$$
Then $F(x,y,y^{(1)})=0$ defines a an ODE of order one. In "basic standard texts", for purposes ...

**-3**

votes

**1**answer

50 views

### A question on matrix polynomial [on hold]

Suppose
${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...

**10**

votes

**2**answers

446 views

### Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows.
Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...

**0**

votes

**1**answer

229 views

### If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all.
Question
If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$?
Heuristic
From the Descartes spoof, with quasi-Euler ...

**1**

vote

**1**answer

222 views

### Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...

**0**

votes

**0**answers

73 views

### A quantity measuring weak compactness

Let us fix some notations. If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $d(A,B)=\inf\{\|a-b\|\mid a\in A,b\in B\},\widehat{d}(A,B)=\sup\{d(a,B)\mid a\in A\}.$ Let $A$ be a ...

**13**

votes

**1**answer

408 views

### Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.

**0**

votes

**0**answers

44 views

### continuous vs discrete random walk [on hold]

For 1D random walk in discrete case the probability $P_N(X)$ of finding walker at position $X$ after $N$ steps has a binomial distribution, moreover when $N+X$ is odd then probability is 0.
Let's ...

**8**

votes

**2**answers

286 views

### What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...

**2**

votes

**1**answer

104 views

### Can tests for the convergence and divergence of series be used to create undecidable sentences?

Let f(k) be a recursive function which maps the set of positive integers into itself. Let T be a formalized theory which is axiomatizable and contains Peano's Arithmetic as a sub-theory. For example, ...

**8**

votes

**2**answers

87 views

### How many uniquely colored degree two vertices in 3-coloring of subcubic graph?

Is there a graph with maximum degree three that has 3 degree two vertices that must get the same (resp. different) color in every 3-coloring of the graph?
I'm interested in any similar results as ...

**1**

vote

**2**answers

75 views

### Relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$

Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?

**-4**

votes

**0**answers

97 views

### How to prove that the only integer solutions to ${r}^{3}/\left({r-1}\right)$ are $r\in \left\{{0,2}\right\}$ [on hold]

This seams simple, how to prove that the only integer solutions to ${r}^{3}/\left({r-1}\right)$ are $r\in \left\{{0,2}\right\}$. If $r$ is odd then we have an odd/even where there is always at least ...

**-3**

votes

**0**answers

26 views

### Statistics, probability [on hold]

Eight of the 40 newly built cars are selected at random to be checked for steering defects. Suppose 10 0f the cars have such defects. What is the probability that all 8 of the selected cars have ...

**1**

vote

**0**answers

32 views

### Limits on a parameter $\alpha$ to get positive definite matrix

Given positive semi-definite $n\times n$ matrices $B_k$, how would I go about getting the limits on $\alpha_k$ such that the expression
\begin{equation}
\mathbb{I}-\sum_{k=1}^{m-1}\alpha_kB_k
...

**-1**

votes

**0**answers

78 views

### The duality between an reductive group and an endoscopic group [on hold]

Can an integral of an reductive group G on a Calabi-Yau surface to be dual to the integral of it's endoscopic group E on the mirror surface?

**0**

votes

**1**answer

96 views

### Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$
Of course if $f\in H^s(\mathbb{R}^3)$ ...

**0**

votes

**1**answer

118 views

### Frankl's union-closed sets conjecture for infinite families

This question is motivated by Frankl's union-closet sets conjecture.
Let $X$ be a non-empty set. We say that a family ${\cal A} \subseteq {\cal P}(X)$ is union-closed if $\emptyset\notin{\cal A}$ and ...

**1**

vote

**1**answer

91 views

### A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...

**8**

votes

**3**answers

479 views

### Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product ...

**7**

votes

**2**answers

250 views

### Even parking functions and spanning trees of complete bipartite graphs

Set $\mathbb{N} := \{0,1,2,\ldots\}$. A parking function of length $n$ is a sequence $(\alpha_1,\ldots,\alpha_n) \in \mathbb{N}^n$ whose weakly increasing rearrangement $\alpha_{i_1} \leq \alpha_{i_2} ...

**2**

votes

**0**answers

88 views

### DG natural transformation Serre functors

This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer.
Let $X$ be a smooth projective variety ...

**0**

votes

**0**answers

23 views

### How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?

Say we have a simple HPN with 2 continuous places $A$ and $B$ and one transition. We want a transition not only add and substract $N$ marks from $A$ and add $M$ to $B$ but use mathematical function ...

**-5**

votes

**0**answers

52 views

### How many ways are there to order the numbers from 1 to 25 so that no primes occur consecutively? [on hold]

I just had this on an exam and was wondering if I answered it correctly. My solution was to first order the 16 composite numbers in this range, giving 17 spots to insert the 9 primes. So we have ...

**-5**

votes

**0**answers

28 views

### Algebra math word problem to be solved using elimination or substitution method [on hold]

A two-digit number is such that the sum of its digits is 1/4 of the number. When the digits of the number are reversed and the number is subtracted from the original number, the result obtained is ...

**2**

votes

**1**answer

118 views

### Some very weak statements on choice

This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Consider the statements
$(\text{S}1)$ For any infinite set $X$ there ...

**1**

vote

**1**answer

75 views

### Dirac bundle and spinor bundle

What is the difference between Dirac bundle and spinor bundle?Morever, every spinor bundle is Dirac bundle, is it true?

**1**

vote

**0**answers

48 views

### Pairing for non-uniformizable Anderson T-motives

Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ...

**2**

votes

**1**answer

111 views

### Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?

Consider the statement
For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.
Does this imply the ${\sf AC}$?

**1**

vote

**2**answers

118 views

### question about literature in the field of Ramsey's theory [on hold]

i am searching for an on- line paper or a book, or maybe just a paper or a book which consists a proof of finite Ramsey's theorem for sets (not for graphs). i need a combinatorial proof which is not ...

**-2**

votes

**0**answers

28 views

### The Heisenberg uniquness pairs invariants by translation and rotaion? [on hold]

why the Heisenberg uniquness pairs invariants by translation and rotaion ?

**3**

votes

**0**answers

49 views

### Unitizations of Banach algebras and matrix norms

Consider the short exact sequence of Banach algebras $0\rightarrow A\rightarrow A^+\rightarrow\mathbb{C}\rightarrow 0$ where $A$ is a Banach algebra without unit and $A^+$ denotes the unitization of ...

**6**

votes

**2**answers

107 views

### Symplectic Reduction on infinite dimensional manifolds

Let $X$ be a compact, oriented Riemann manifold. Let $\pi_{P}: P \rightarrow X$ be a principal $G$-bundle over $X$, for a compact Lie group $G$. Let $(M, \omega)$ be a symplectic manifold endowed with ...

**2**

votes

**0**answers

87 views

### Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$

Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...

**3**

votes

**1**answer

38 views

### Inverse Laplace transform of a hypergeometric function

This is a repost from Math Stack-exchange where I did not manage to get an answer.
http://math.stackexchange.com/questions/1491027/inverse-laplace-transform-of-a-hypergeometric-function
I managed to ...

**12**

votes

**1**answer

1k views

### What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two
$$
\zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1
$$
...

**1**

vote

**0**answers

36 views

### What's the advantage of majorization-minimization (MM) algorithm [on hold]

The majorization-minimization (MM) algorithm is a framework for convex and nonconvex optimization. When applied to nonconvex optimization, the MM algorithm solves a sequence of convex problems to ...

**0**

votes

**3**answers

158 views

### Divergence of general random series and a special case

Is there any sufficient condition in terms of moments under which
$$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent
I am given that $\sum_n E[X_n]$ diverges. Actually, I am ...

**2**

votes

**2**answers

150 views

### odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: ...

**1**

vote

**0**answers

25 views

### Bit complexity versus arithmetic complexity of polynomial multiplication

Given degree $d_1$ and $d_2$ polynomials in $\Bbb Z[x]$ with coefficient sizes of bits $b_1$ and $b_2$ respectively
(1) what is the bit complexity of multiplying the two polynomials?
(2) What is ...

**3**

votes

**0**answers

182 views

### Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme

I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes?
At this point I am still ...

**4**

votes

**1**answer

94 views

### Glueing together functions defined on the leaves of a foliation

Even though my question can be asked for very general types of foliations, I am interesetd only in its answer for Poisson manifolds, which are what I am currently studying.
Consider a Poisson ...

**12**

votes

**1**answer

162 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...