**4**

votes

**0**answers

57 views

### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**0**

votes

**1**answer

82 views

### Are constructive characterisations of k-regular (simple) graphs known?

By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some ...

**4**

votes

**1**answer

105 views

### Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors?
In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr ...

**5**

votes

**2**answers

48 views

### convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...

**-3**

votes

**1**answer

36 views

### upper bound for a convex fractional function [on hold]

Consider the following convex fractional function
$$f\left( {\bf{x}} \right) = \frac{1}{{1- {\bf{x}}}}$$
where ${1- {\bf{x}}} > 0$. Is it possible to obtain a linear or quadratic upper bound ...

**-1**

votes

**0**answers

112 views

### Is $(X_G, d_G)$ , compact manifold?

Let compact topological group $G$ acts on $(X,d)$ . We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$:
$$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$
It is clear that ...

**12**

votes

**2**answers

340 views

### Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.
Probable primes are the union of the primes and base two pseudoprimes.
This definition is much ...

**7**

votes

**1**answer

405 views

### Sums of unique squares

Let $\mathbb{N}$ denote the positive integers and let $S = \{n^2: n\in \mathbb{N}\}$. For any positive integer $k$ we define $$\text{sq}(k) = |\{F\subseteq S: F\neq \emptyset, F\text{ is finite and } ...

**-1**

votes

**0**answers

31 views

### Why do there is a unique continuous homomorphism? [migrated]

Is this a right place to ask help for an exercise?
Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ...

**6**

votes

**0**answers

163 views

### On an unpublished result of Magidor

In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and ...

**1**

vote

**1**answer

117 views

### Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that
$$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$

**2**

votes

**2**answers

132 views

### Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 ...

**2**

votes

**0**answers

25 views

### Strong solution to an SDE with a discontinuous diffusion term

I am having an SDE for which I would be in trouble if there were no strong solution.
The SDE is -
$ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$
where $b_1$ and $b_2$ are two ...

**-3**

votes

**1**answer

103 views

### Encyclopedia of Mathematics?(non-Alphabetical) [on hold]

Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level.
And what's the difference between say, ...

**2**

votes

**0**answers

52 views

### Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...

**21**

votes

**2**answers

896 views

### What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...

**7**

votes

**2**answers

171 views

### Repeats of all binary strings of length k

The question seems like it should be known, but I was not able to find it anywhere.
How many binary strings of length $n$ are required so that for every $k$ positions in these strings, all $2^k$ ...

**2**

votes

**0**answers

81 views

### An equality of discriminant and resultant divisors

Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant
$$\prod_{\alpha\in\Phi}d\alpha$$
on $\mathfrak ...

**9**

votes

**1**answer

105 views

### What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdos-Renyi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Alber model
...

**1**

vote

**0**answers

63 views

### Closed form answer to a naive integral [on hold]

Let a and b be positive real numbers. How to find a closed form answer to the integral
$$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$
If it is not possible to find a ...

**-4**

votes

**0**answers

38 views

### Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$
are linearly independent for $(x,y) \in [0,1]^2$.
Is the set of functions, $g_1,\ldots,g_n$, defined by
$$
g_i(x) = \int_{y\in [0,1] } ...

**-4**

votes

**0**answers

26 views

### Discrete time equivalent to ODE [on hold]

I'm reading a paper in which it is noted that
$$\frac{dv(t)}{dt} = f(t) - \varepsilon v(t)$$
has the discrete time equivalent
$$v(t+1) = v(t)\exp(-\varepsilon) + \frac{f(t)}{\varepsilon}[1 - ...

**1**

vote

**0**answers

50 views

### Perturbed Chebyshev polynomials

It is well-known that the Chebyshev polynomials of the first kind satisfy the recurrence relation
$$
\begin{cases}
T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\
T_{0}(x)=1, \ \ T_{1}(x)=x \\
...

**7**

votes

**2**answers

82 views

### Generate polyhedra by collapsing vertices of a polyhedron

I am looking for basic information about the following idea:
(I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle.
(II) Consider a three-dimensional cube. By collapsing a ...

**2**

votes

**0**answers

45 views

### Lattice Flatness Measure

I am looking for the definition of a flatness measure in lattice theory.
More generally, I am looking at finite-height lattices and I want to measure their complexity, with a perfectly flat lattice ...

**-3**

votes

**0**answers

40 views

### Decomposition of orthogonal matrix into 2 orthogonal matrices [on hold]

Is there anyway to find a decomposition of orthogonal matrix $A$ into 2 orthogonal matrices $P$ and $Q$ such that $A = PQ^T$?

**14**

votes

**2**answers

395 views

### Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html.
At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb ...

**3**

votes

**0**answers

41 views

### Adjunctions of uniformly locally connected spaces

A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and ...

**5**

votes

**1**answer

132 views

### Optimisation of betting strategy

Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game:
We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we ...

**-5**

votes

**1**answer

84 views

### is this formula correct [on hold]

I found this formula in a book:
for cylindrical coordinates where: $x=r\cos\theta$ and $y=r\sin\theta$, then:
$$\dfrac{\partial}{\partial x} = \cos\theta \dfrac{\partial}{\partial r} - ...

**4**

votes

**1**answer

59 views

### Is every pair of writable reals one-tape-ITTM-computable?

I've been reading this paper, in which authors prove that not all ITTM-computable functions $\Bbb R\rightarrow\Bbb R$ are 1-tape-computable, but if we put some restriction on the output of the ...

**1**

vote

**1**answer

78 views

### Projective dimension of a quotient ring

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$,
$Z$ an indeterminate.
The first comment in this question says that, if $A$ is noetherian, then
$pd_{B\otimes_A B}(B) ...

**7**

votes

**1**answer

275 views

### About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive ...

**-1**

votes

**0**answers

56 views

### Interpolating Product of two Polynomials

Consider we have two non-constant polynomials $A(x)$ and $B(x)$. We define the polynomials over field $\mathbb{Z}_p$, for a large prime number $p$.
We define Polynomial $A(x)$ as follows: ...

**3**

votes

**1**answer

62 views

### Computer algebra system for Weyl algebra computations

Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra?

**5**

votes

**3**answers

134 views

### Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...

**-5**

votes

**0**answers

29 views

### Topology Proof about a open ball [on hold]

Show that B(a,r1) < B(a, r2) r1
show that the open ball with radius r1 is a subset of the of the open ball with radius r2

**1**

vote

**0**answers

47 views

### Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...

**-5**

votes

**0**answers

41 views

### Developping the gradient [on hold]

We know from the tensor calculus that: $\vec\nabla(a\cdot b) = b\vec\nabla a + a \vec\nabla b $ , where $a$ and $b$ are two scalar functions.
But in the case where for example $a$ is a scalar ...

**10**

votes

**1**answer

87 views

### Commuting ODE's implies existence of nonzero vanishing two variable polynomial?

Write $\partial := d/dt$, fix $m, n > 0$, and let$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ...

**11**

votes

**1**answer

639 views

### A problem in elementary geometry

Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle:
On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise,
...

**7**

votes

**1**answer

118 views

### cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra
(1)
$$
H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...

**-4**

votes

**0**answers

25 views

### Find limit of polynomial in detail [on hold]

I want to know what is the limit n->infinity for
(n + a)^ b/ n^b
Please provide a detailed answer. Regards

**7**

votes

**1**answer

261 views

### Irreducible cubics modulo primes

Is there a small finite (perhaps of cardinality two or three) collection of cubic polynomials $p_1, \dotsc, p_k \in \mathbb{Z}[x]$ such that for every prime $p$ at least one of these is irreducible?

**13**

votes

**1**answer

326 views

### Integral cohomology ring of K(Z,3)

Computing the cohomology of Eilenberg Maclane spaces is a feasible but difficult problem in algebraic topology. The general answer is quite complicated (see the MO answer and the reference therein ...

**3**

votes

**0**answers

66 views

### Unibranch partial normalization

In a paper I recently read something about the "unibranch partial normalization" of a curve.
Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it ...

**3**

votes

**0**answers

54 views

### Perturbations to a vector field

I ran into some problems while working through a proof of the Poincare-Hopf theorem that essentially boiled down to the following question: given a smooth vector field $V$ on a (compact Riemannian) ...

**2**

votes

**0**answers

22 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

**2**

votes

**1**answer

96 views

### Odds of residue being small

Given $\mathsf{c\geq1}$, what is the probability that if you choose $\mathsf{A,B,\alpha\in\Bbb N}$ such that $\mathsf{A,B<\alpha<AB}$ holds we will have both ...

**0**

votes

**1**answer

80 views

### Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...