# All Questions

**1**

vote

**1**answer

226 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 ...

**6**

votes

**0**answers

96 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 ...

**8**

votes

**3**answers

480 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 ...

**0**

votes

**1**answer

231 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

**0**answers

80 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 ...

**13**

votes

**1**answer

413 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.

**2**

votes

**1**answer

106 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, ...

**1**

vote

**2**answers

79 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 $C^\ast(E)$?

**1**

vote

**0**answers

51 views

### Computing abelianizations of some explict finite subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$

I have been attempting to find some abelianizations of some subgroups of $GL_2(\mathbb{Z}/p^n \mathbb{Z})$. I have been using brute force for the most part but I get very messy results. Here is an ...

**8**

votes

**1**answer

91 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

**0**answers

58 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 $ ...

**-1**

votes

**0**answers

70 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 ...

**0**

votes

**1**answer

119 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 ...

**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 ...

**0**

votes

**0**answers

75 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 ...

**7**

votes

**2**answers

252 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} ...

**1**

vote

**0**answers

18 views

### Equivalence of first order quasilinear PDE to linear PDE [migrated]

Given a system of nonlinear PDE of the special form:
$\sum_{i=1}^n A_i(x, \phi) - \frac{\partial \phi_j}{\partial x_i} = B_j(x,\phi) $ $(1)$
with $(j=1,...,m)$ and $x \in R^n,\phi \in R^m$. If we ...

**1**

vote

**1**answer

92 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 $f$. ...

**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 ...

**2**

votes

**1**answer

64 views

### Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= ...

**0**

votes

**0**answers

16 views

### Connected sum of two “same” Kleins bottles [migrated]

If I have two surfaces of Klein's bottle, K, given by edge words [a b- a- b-] and [a- b a b] what space do I get when I identify same edges. In other words, i have two same Klein's bottles that are ...

**0**

votes

**1**answer

98 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 $f\in H^s(\mathbb{R}^3)$ ...

**2**

votes

**1**answer

114 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 ${\sf AC}$?

**2**

votes

**1**answer

124 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**

votes

**0**answers

79 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?

**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?

**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 ...

**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
$$
\zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1
$$
...

**-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 ...

**2**

votes

**0**answers

89 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 ...

**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 ...

**8**

votes

**3**answers

281 views

### Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters

Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...

**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
...

**0**

votes

**1**answer

632 views

### Hard maths on viXra? [closed]

A few years ago a nice paper surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic ...

**6**

votes

**2**answers

310 views

### First Galois cohomology of Weil restriction of $\mathbb{G}_m$

Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}_m)$ the Weil restriction of $\mathbb{G}_m$, from $L$ to $K$. I want to show that its first Galois cohomology ...

**12**

votes

**1**answer

164 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 ...

**0**

votes

**3**answers

159 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 ...

**-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 ...

**9**

votes

**1**answer

197 views

### Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...

**3**

votes

**2**answers

175 views

### How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...

**2**

votes

**2**answers

152 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: ...

**2**

votes

**0**answers

89 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

**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 ...

**1**

vote

**0**answers

49 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 $H_1(E)$ ...

**-5**

votes

**0**answers

29 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 ...

**3**

votes

**0**answers

51 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 ...

**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 ...

**3**

votes

**0**answers

96 views

### Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...

**3**

votes

**1**answer

39 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 ...