11
votes
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
90answers
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
0answers
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
0answers
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
4answers
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
7answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
8answers
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
0answers
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
4answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
41answers
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
1answer
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
0answers
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
0answers
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
0answers
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

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