# All Questions

**2**

votes

**0**answers

69 views

### A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.
The Künneth-Theorem which I ...

**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 ...

**6**

votes

**3**answers

195 views

### Numerical invariants for a graph or its complement that are bounded by some constant

I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but ...

**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 ...

**2**

votes

**1**answer

48 views

### Properties of bipartite graphs

For a connected bipartite graph $G$ are the two following properties equivalent:
1)Every minimal cycle in $G$ has length 4, that is every cycle of length strictly greater than 4 can be divided in ...

**5**

votes

**2**answers

150 views

### When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of ...

**3**

votes

**0**answers

139 views

### A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form
$$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...

**1**

vote

**0**answers

53 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 ...

**0**

votes

**0**answers

33 views

### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n^2$ solitary?

(Note: A similar question was asked in MSE two months ago.)
Let $\sigma(x)$ be the sum of the divisors of the natural number $x$, and denote the abundancy index $\sigma(x)/x$ by $I(x)$.
Here is my ...

**1**

vote

**1**answer

129 views

### l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic.
If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...

**0**

votes

**1**answer

62 views

### writing an integer as particular summation [on hold]

I think my question is an elementary question. Thanks for any help or comment.
Is there any formula for the number of writting a natural number $n$ in a summation as follows,
$n=a_1+\dots+a_k$, ...

**-4**

votes

**1**answer

51 views

### Win/Lose ratios and selection [on hold]

Imagine a following scenario:
You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...

**1**

vote

**0**answers

59 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 $ ...

**2**

votes

**0**answers

99 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 ...

**0**

votes

**0**answers

42 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

**0**answers

81 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 ...

**6**

votes

**0**answers

99 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 ...

**3**

votes

**1**answer

85 views

### Is the module action $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map:
$$\gamma: M\times M^*\to M^*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...

**11**

votes

**0**answers

325 views

+200

### Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are:
Definition 1 ("naive"): Let $X$ be a (real) ...

**-3**

votes

**1**answer

53 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 ...

**-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 ...

**0**

votes

**0**answers

45 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 ...

**15**

votes

**6**answers

549 views

### What (fun) results in graph theory should undergraduates learn?

I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...

**-2**

votes

**0**answers

51 views

+50

### 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

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 ...

**-4**

votes

**0**answers

105 views

### I am looking for a general solution for when $n$ and a rational function $f (n)$ are both integers [on hold]

I am looking for a general solution for when $n$ and a rational function $f \left({n}\right)$ are both integers. One example is below.
This seams simple, how to prove that the only integer solutions ...

**-2**

votes

**0**answers

29 views

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

why the Heisenberg uniquness pairs invariants by translation and rotaion ?

**-1**

votes

**0**answers

71 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 ...

**-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 ...

**1**

vote

**0**answers

77 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 ...

**1**

vote

**0**answers

50 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 $H_1(E)$ ...

**1**

vote

**1**answer

231 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 ...

**2**

votes

**1**answer

109 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

**0**answers

27 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 ...

**2**

votes

**0**answers

90 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 ...

**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 ...

**1**

vote

**1**answer

162 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 ...

**2**

votes

**2**answers

103 views

### Number of facets of a polyhedron when a vertex is removed

My question, informally: I have a bounded polyhedron in R^n with k facets, and I want to remove a vertex of this problem. How many facets does the remaining polyhedron have at most?
More formally: ...

**0**

votes

**0**answers

30 views

**7**

votes

**2**answers

342 views

### Asymptotics of product of Euler's totient function (A001088)?

Conjecture:
\begin{align}
\lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...}
\end{align}
The numerical result from 100000 terms is:
My questions ...

**4**

votes

**2**answers

132 views

### No normal coordinates on general Finsler manifolds

I recently read a footnote in Chern's article stating that a non-Riemmanian Finsler manifold does not possess normal coordinates.
As I'm still new to non-Riemmanian Finsler geometry I don't see why ...

**1**

vote

**0**answers

131 views

### $L^\infty$ bound on solutions of linear parabolic equations

We work on a closed Riemannian manifold $M$. Let $u$ and $v$ be the non-negative weak solutions of
$$au_t - 2d\,\Delta au = cv - f$$
$$bv_t - d\,\Delta bv = f$$
$$u(0)=u_0, \quad v(0)=v_0$$
where $f$ ...

**1**

vote

**0**answers

37 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 ...

**5**

votes

**2**answers

512 views

### mod 5 partition identity proof

I am looking for a proof that:
$$\prod_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{x^i}{\prod_\limits{j=1}^i (1-x^{5j})}$$
The left hand side expands into:
...

**1**

vote

**2**answers

80 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

**3**answers

169 views

### Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$.
...

**-5**

votes

**1**answer

47 views

### My question is about Constructions [on hold]

If one angle of an triangle is 60 degrees and two of it's sides are equal then will it be an equilateral triangle?

**16**

votes

**0**answers

265 views

+50

### Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.
In the ...

**0**

votes

**1**answer

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

**10**

votes

**1**answer

289 views

+50

### Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...