# All Questions

**-4**

votes

**0**answers

28 views

### Vector application [on hold]

A rope is hung at both ends from a horizontal beam, and a weight m is suspended from it. The left part of the rope exerts a force G at P, while the right part of the rope exerts a force H. Find the ...

**15**

votes

**1**answer

302 views

### A nice subcategory of the category of measurable spaces

Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties?
The real line equipped with the Lebesgue $\sigma$-algebra is nice.
Any ...

**2**

votes

**0**answers

58 views

### clustering permutations by shared subsequences [on hold]

I have a question, stimulated by some biology, about comparing sets of permutations.
The problem
Let's think of genes on a bacterial chromosome as beads on a string - atomic, unique objects, with ...

**3**

votes

**1**answer

88 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 ...

**-5**

votes

**0**answers

38 views

### Maple: In Matrices A x B = C, how do I find matrix A given B and C [on hold]

I have matrix A, B, and C which are all 8x8 matrices in Maple.
in the equation A x B = C, when B and C are known, how do I find matrix A?
I know how to do it by hand, but I don't know the maple ...

**19**

votes

**0**answers

315 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 ...

**-4**

votes

**0**answers

44 views

### A set containing more than half elements of a group [on hold]

I wish to prove the exercise which states that for a set $A$ containing more than half elements of a group $G$, every element of $G$ is a product of two elements of $A$.
My attempt:
By Lagrange ...

**2**

votes

**0**answers

53 views

### Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?

**1**

vote

**1**answer

131 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 ...

**-6**

votes

**0**answers

35 views

### Arrange numbers? [on hold]

hello the question i would like to ask is very difficult as english is not my native language so here it goes...I need a program or exel chart that arrange numbers in a group in order to get the ...

**-5**

votes

**0**answers

31 views

### How to calculate how much more wins to get a certain winrate? [on hold]

I have two values, wins and total games played.
To calculate the win rate I use the normal «formula»:
wins/totalGamesPlayed*100;
But let's say I have 21 wins ...

**1**

vote

**1**answer

235 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

**0**answers

61 views

### Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$.
So $G\in\mathcal{G}$ can be written ...

**2**

votes

**0**answers

136 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

**2**answers

83 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

**1**answer

52 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 ...

**2**

votes

**1**answer

114 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

**1**answer

92 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 ...

**0**

votes

**0**answers

55 views

### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...

**3**

votes

**0**answers

76 views

### extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$.
Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$.
If we have a $\mathbb{Z}_{\ell}$ local ...

**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

**1**answer

65 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,
$n=a_1+\dots+a_k$, ...

**0**

votes

**1**answer

233 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 ...

**8**

votes

**3**answers

283 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 ...

**2**

votes

**1**answer

49 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 ...

**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

24 views

### Common Point of Intersection of n-dimensional ellipsoids [on hold]

Suppose we have two ellipses in 2-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have 4 points of intersection. Can we say that in ...

**2**

votes

**0**answers

70 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

**1**answer

120 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 ...

**-6**

votes

**0**answers

25 views

### Trouble with an assignment [on hold]

Can anyone please be kind enough to help me with this.
On one shelf there are 5 hardcover books and 6 paperbacks and on the other shelf there are 7 hardcover and 4 paperback. From the first shelf ...

**7**

votes

**2**answers

259 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} ...

**6**

votes

**0**answers

101 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 ...

**-2**

votes

**0**answers

21 views

### AQA A Level Normal Distribution [on hold]

The question goes like:
A wholesaler decides to grade such oranges by weight. He decided that the smallest 30% should be graded as small, largest 20% as large and in between as medium.
The ...

**6**

votes

**2**answers

320 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 ...

**1**

vote

**0**answers

84 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 ...

**27**

votes

**5**answers

2k views

### What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...

**1**

vote

**0**answers

44 views

### moduli space of curves under prescribed tangency conditons

We consider an irreducible component of the Hilbert Scheme of curves in $\mathbb P^2$. Denote it as $\mathcal D.$ We fix a line $L$ and a point $A\in L.$ Denote $\mathcal D_0$ as the subscheme of ...

**0**

votes

**0**answers

19 views

### Markov Modulated Markov Chain

Consider a discrete time Markov chain $X_t$ on some finite state space $\mathcal{S}$ with transition matrix $P$. Now consider a process $Y_t$ also on $\mathcal{S}$, which conditioned on $X_{t}=s$ ...

**1**

vote

**0**answers

62 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**

vote

**0**answers

55 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

**1**answer

653 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 ...

**0**

votes

**0**answers

27 views

### How do we prove that a specific kernel is positive definite (case of logarithm)? [on hold]

I have a problem proving that some specific kernels are positive definite. In general, I can find the answer quickly enough but here I have a specific case involving a logartihm :
$K : ...

**1**

vote

**1**answer

94 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 $f$. ...

**2**

votes

**1**answer

127 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

71 views

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

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

55 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

**0**answers

34 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 ...

**2**

votes

**1**answer

119 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

65 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](:= ...

**1**

vote

**0**answers

82 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 ...