0
votes
0answers
16 views

Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...
0
votes
0answers
5 views

$GL(2, \mathbb{Z})$ modular form

Recall that an ordinary modular form (of weight $k$) is a holomorphic function on the upper half plane $\mathbb{H}^+$ satisfying $$ f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{k} \, f(\tau) . ...
2
votes
0answers
70 views

Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
2
votes
1answer
43 views

An upper bound on the number of sets of parallel lines covering points in a finite plane?

Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a ...
0
votes
0answers
8 views

Role of the trivial group in decidability of the theory of a class of groups

Let K be a class of groups, and K* the class of all non-trivial groups from K. Clearly, if Th(K*) is decidable then Th(K) is decidable, too. Is it true that Th(K*) is decidable if and only if Th(K) is ...
-1
votes
0answers
17 views

Parallel transport along a geodesic and the related Jacobi field

Crossposted from: http://math.stackexchange.com/questions/1255018/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found mentioned in ...
1
vote
0answers
10 views

Linear algebra formulation for colored node graph isomorphism

(Please see a few paragraphs below by what I mean by “colored node graph isomorphism”.) Some basic definitions for completeness: Given two graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ the graph ...
2
votes
0answers
49 views

Residue for the generating function of the Euler totient function

Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series $$ f(z) := \sum_{n=1}^{\infty} \varphi(n) z^n $$ Since $0\le \varphi(n)\le n$, I believe this gives a ...
0
votes
0answers
9 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
1
vote
0answers
38 views

Atiyah's vector bundles over an elliptic curve

I'm reading through Atiyah's paper that classifies vector bundles over an elliptic curve, and I'm a little confused about one of his proofs. Lemma 15(i) states that if $E \in \mathcal{E}(r,d)$ is a ...
0
votes
0answers
28 views

Factoring a polynomial in a specific manner

Let $f(x,y,z) = ax^d + b y^k z^{d-k} + c$, where $a,b,c \in \mathbb{C}$ and $abc \ne 0$, and $d \geq 3$. Let $d'$ denote the smallest non-negative integer $l$ such that there does not exist a positive ...
1
vote
1answer
304 views

Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$. For $m$ large, let ...
0
votes
0answers
24 views

Backward Uniqueness for the wave equation

Does the wave equation $u_{tt} - \Delta u = 0$ have any backward uniqueness results that are similar to the ones for the heat equation (see for example Theorem 11 page 64 in Evans)? If not, are there ...
1
vote
1answer
119 views

Counting prime powers $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is: $$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...
-1
votes
1answer
28 views

Algebraic Groups of Type H_3 and H_4

By coincidence i stumbled over this page http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/abstracts.html , which was installed for a workshop on algebraic groups in 2012. In the ...
0
votes
0answers
29 views

Generalization of Schur polynomials

I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of ...
3
votes
1answer
58 views

Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces. Let $B_1, \ldots, B_k \in B(\mathcal{H}).$ Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via ...
2
votes
1answer
344 views

Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism: $$g_Y \colon Y \times_X Y \cong Y ...
4
votes
1answer
189 views
+50

Continuous maps to fat geometric realizations of simplicial spaces

The nLab page on partitions of unity mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel ...
-2
votes
0answers
58 views

Specific examples or applications of homotopy coherent diagrams [on hold]

A homotopy coherent diagram is a special case of a functor between higher categories where the source category is an ordinary category. Homotopy coherence can be precise in a topological category. In ...
-2
votes
1answer
31 views

Proof of the measure representation lemma

Please can someone tel me where i can find the proof of this : Thank you
44
votes
4answers
1k views

Is the set AA+A always at least as large as A+A?

Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...
-5
votes
0answers
14 views

0-1 Integer Problem, On Constructing A General Case Algorithm [on hold]

Maximise: $8x_1+11x_2+6x_3+4x_4$ subject to: $5x_1+7x_2+4x_3+3x_4<=14$ $x_j$ is an element of $\{0,1\} j=1,...4$ superficial to solve even with pen and paper but what algorithm would be used to ...
2
votes
2answers
53 views

How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?

This question is in reference to this other question, Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
2
votes
2answers
256 views

Other Variant of Schur Polynomials/Functions

We know that an important variant of the Schur Polynomial is the shifted Schur Polynomials that was developed by Okounkov & Olshanski. The question here is: is there any other variant of Schur ...
-4
votes
0answers
48 views

Have there been attemps to manage the pool of worldwide mathematics students? [on hold]

(Foreword: I am well aware that MO is not a blog, and not for argumentative questions. Therefore I have phrased my question in a rather specifically answerable form. If moderators still feel it is ...
-3
votes
0answers
35 views

prove that the sequence a_n = [n((2)^(1/2))] + [n((3)^(1/2))] contains infinitely many even and odd integers [on hold]

im not sure how to proceed.i have done a few problems similar to this but i can't solve this one. thanks for any help.( by the way this is a problem from croatia team selection test )
18
votes
2answers
449 views

The word problem for fundamental groups of smooth projective varieties

While attending a very nice talk on the geometric group theory of fundamental groups of Kahler manifolds by Pierre Py last weekend, I realized that I don't know the answer to the following question. ...
-4
votes
0answers
27 views

Caledonian college level2 [on hold]

assume we have a stick of one meter length. we put 999 ants on the stick, at arbitrary positions and arbitrarily facing either left or right. At a certain time, all ants start moving with the same ...
7
votes
0answers
467 views

Commutator subgroup of a surface group

Let $\Sigma_{g,n}$ denote a compact orientable genus $g$ surface with $n$ boundary components. Assume that $g \geq 1$ and fix a basepoint $p \in \Sigma_{g,n}$. Define $S \subset ...
2
votes
0answers
20 views

A tensor equation related to an invariant of a diffeomorphism

Let $M$ be an $n$-dimensional differentiable manifold, $f : U \rightarrow V$ a diffeomorphism between open neighbourhoods $U$, $V$ of $M$ with $f(x)=x$ for some $x \in U$, and let $R$, $S$, $T$ be ...
1
vote
0answers
14 views

random odes adapted solution

Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode $$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$ Where $b$ is a bounded continuous function (not ...
10
votes
2answers
299 views

For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
0
votes
1answer
288 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
2
votes
0answers
38 views

Permutation-invariant matrix representation

The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...
1
vote
0answers
61 views

Complexity :: Integer Programming :: Non-Poly Example [on hold]

When learning about computational complexity I find that when discussing the NP-Complete problems authors always give examples of such problems that can in fact be solved in poly time. I understand ...
3
votes
1answer
231 views

Can states on commutative Banach algebras be understood as probability measures?

Suppose $\mathcal{A}$ is a commutative Banach algebra (over $\mathbb{R}$) or commutative Banach *-algebra (over $\mathbb{C}$). Is there always a measurable space $(\Omega,\mathcal{F})$ such that there ...
4
votes
3answers
245 views

When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure?

Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure. Is it always possible to construct a smooth structure on $M$ ...
5
votes
1answer
203 views
+50

Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
4
votes
1answer
61 views

Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question. Is ...
11
votes
2answers
504 views

Mysterious identity between numbers of odd/even meander systems

Definitions: An upper arch system of order $n$ is a subset of the plane consisting of $n$ non-intersecting closed semicircles in the upper half-plane whose endpoints belong to the set $\{(k,0)\mid ...
-4
votes
0answers
24 views

Something about iterated logarithm [on hold]

that's my first question there. So, can you explain, why iterated log well-defined with base more than e^(1/e). I considered a f = w^(1/w), and prove that f(max) = e^(1/e), so if I prove that log* ...
1
vote
3answers
140 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
0
votes
0answers
44 views

On covering by smooth numbers

Denote $P(y)=\mathsf{greatest}\mbox{ }\mathsf{prime}\mbox{ }\mathsf{factor}\mbox{ }\mathsf{of}\mbox{ }y$. Denote $S(x,y)=\{n<x: P(n)<y\}$. Denote $S_t(x,y)=\sum_{i=1}^tS(x,y)$ as $t$-fold ...
-2
votes
0answers
66 views

Why are algebraic cycles rational? [migrated]

Let $X_{/\mathbb{C}}$ be a projective non-singular variety of dimension $n$ and $Z \subset X$ be an irreductible subvariety of dimension $p$. Denote by $\mathrm{H}_{\mathrm{dR}}^i(X,\mathbb{C})$ the ...
1
vote
1answer
138 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
11
votes
2answers
553 views

Sum and product estimate over integers, rationals, and reals

My question is the following: is finding a lower bound for $|A+A\cdot A|$ (as a function of $|A|$) where $A$ is any finite subset of the positive integers equivalent to finding the same lower bound ...
8
votes
4answers
443 views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...
5
votes
3answers
270 views

Introductory texts to mathematics [on hold]

I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would ...
1
vote
1answer
107 views

Can we find structure constants of Lie Algebra for Lie Symmetry of ODE without solving determining equations?

Let's consider (for example) one scalar ODE. We are searching for Lie Symmetries of it. There is well-known result, that we can find size of Symmetry Group without solving determining equations. ...

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