**11**

votes

**1**answer

181 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 ...

**5**

votes

**2**answers

111 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 ...

**0**

votes

**0**answers

15 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 ...

**1**

vote

**0**answers

38 views

### Repeats of all binary strings of length k

The question seems like it should be known. However, 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 $2^k$ ...

**4**

votes

**1**answer

29 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 ...

**9**

votes

**1**answer

586 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,
...

**0**

votes

**1**answer

62 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 ...

**33**

votes

**1**answer

734 views

### Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...

**4**

votes

**0**answers

49 views

### Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an ...

**0**

votes

**1**answer

65 views

### Solutions of an nonlinear evolution problem

We consider the following continuous-time nonlinear evolution problem
\begin{equation}
\begin{cases} \dot{y}(t)=Ay(t)+F(y(t),u(t)),\quad t\geq0\\y(0)=f\in\mathcal{X}\end{cases}
\end{equation}
where ...

**7**

votes

**2**answers

258 views

### For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ ...

**6**

votes

**1**answer

165 views

### Consistency of the nonrigidity of $P(\omega_1)/NS$

Is it consistent with ZFC that there exists an automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$ which is not the identity?

**2**

votes

**0**answers

142 views

### Rank of the Jacobian of twists of hyperelliptic curves

Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...

**0**

votes

**1**answer

61 views

### Convergence to equilibrium via gradient descent

J. B. Rosen proved that in concave games of n players (which assumes that Cartesian product of strategy profiles is convex) if the game satisfies the condition of diagonally strictly concave then ...

**84**

votes

**90**answers

11k views

### What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...

**-1**

votes

**0**answers

153 views

### Hom-set with functions as arguments? [closed]

In Awodey's "Category Theory" (p48) and elsewhere a Hom-set Hom(A,B) is defined as the set of functions between the objects A ...

**14**

votes

**0**answers

586 views

### What is the best lower bound for 3-sunflowers?

A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = Z $ for all $i \neq j$ for some fixed $Z$. A well-known conjecture of Erdos and Rado says that there is a constant $C_t$ such ...

**42**

votes

**4**answers

3k views

### If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...

**8**

votes

**7**answers

15k views

### Notation for the all-ones vector [closed]

What's the most common way of writing the all-ones vector, that is, the vector, when projected onto each standard basis vector of a given vector space, having length one? The zero vector is frequently ...

**2**

votes

**0**answers

18 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

35 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 ...

**-4**

votes

**0**answers

26 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] } ...

**0**

votes

**0**answers

34 views

### Closed form answer to a naive integral

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 ...

**-3**

votes

**0**answers

32 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 $A = PQ^T$?

**-4**

votes

**0**answers

21 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

24 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 ...

**0**

votes

**0**answers

27 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 \\
...

**1**

vote

**1**answer

60 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 ...

**8**

votes

**1**answer

58 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 + ...

**5**

votes

**1**answer

124 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 ...

**6**

votes

**1**answer

233 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?

**2**

votes

**1**answer

41 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 ...

**10**

votes

**1**answer

278 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 ...

**2**

votes

**2**answers

65 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

**1**answer

74 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} - ...

**5**

votes

**1**answer

87 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 ...

**-1**

votes

**0**answers

48 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: ...

**20**

votes

**8**answers

1k views

### Advanced Differential Geometry Textbook

I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual ...

**1**

vote

**0**answers

25 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 ...

**7**

votes

**4**answers

315 views

### Lower bounding the multiplicative order of 2 modulo p

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.
Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?
...

**2**

votes

**1**answer

46 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?

**0**

votes

**1**answer

54 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) ...

**0**

votes

**0**answers

35 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 ...

**-4**

votes

**0**answers

23 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

**-4**

votes

**0**answers

40 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 ...

**52**

votes

**41**answers

6k views

### Important formulas in Combinatorics

Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...

**3**

votes

**1**answer

39 views

### Maximum and minimum diameter of categorical graph product

Let $G_i$ be connected finite simple undirected graphs with diameter $d_i$ for $i=1,2$. Assume that $G_1\times G_2$ is connected. (Here $G_1\times G_2$ denotes the categorical product.)
In terms of ...

**2**

votes

**0**answers

43 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 ...

**-4**

votes

**0**answers

84 views

### Why should “small” P be preferred? [on hold]

In contrast, of course, is the approach of finding an NP language of super-polynomial complexity.
But why the overwhelming, obvious yet implicit favoritism?
Has it anything to do with our ...

**-4**

votes

**0**answers

21 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