5
votes
1answer
179 views

What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$: $\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...
9
votes
13answers
1k views

Obscure Names in Mathematics [on hold]

I recently stumbled over the "Happy Ending Problem" (cf e.g. http://en.wikipedia.org/wiki/Happy_Ending_problem), which made we wonder, if there are other conjectures, theorems or problems, whose names ...
1
vote
0answers
52 views

Uniqueness of the maximum derivative of a rational function

This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function $$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$ with all $a_i\geq 0$ and ...
1
vote
2answers
82 views

Deformation quantization of a closed Riemann surface with genus >1

Quantization of of an elliptic curve can be done in different ways. In C^*-algebraic version, one can start with the C^*-algebra ...
-2
votes
0answers
44 views

Proving how many divisors of a prime factorization (including 1 and n) there are [on hold]

I'm trying to figure out this problem but I'm not sure where to start. Could anyone explain to me the question a bit more in depth or give a few hints? The problem is, Let n in Z+ with prime ...
0
votes
0answers
19 views

Comparing the inverse of a diagonally dominant matrix [migrated]

I have a $d \times d$ matrix $A$ whose entries are bounded (C1): $I - \epsilon X \preceq A \preceq I + \epsilon X$, where $I$ is the identity matrix and $X = 11^\top - I$ is the matrix with ones ...
-4
votes
0answers
26 views

How to know which make which surjective and which Tor is correct to represent them surjective for codordism [on hold]

when A connect B through cobordism A --- cobordism ----B from view of function, when define surjective there exist a function g to make f surjective such that g ...
-2
votes
0answers
92 views

Is the configuration space of infinite sphere contractible? [on hold]

Let $\Sigma$ be suspension. Let $S^0$ be $0$-sphere. Let $\Sigma^\infty S^0$ be the union $\cup_n \Sigma^n S^0$ with respect to the inclusion $\Sigma^kS^0\subset \Sigma^{k+1}S^0$. Let ...
19
votes
1answer
423 views

Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$ with $i>0$. Does there always exist a variety $Y$ and a ...
-1
votes
0answers
16 views

state of art pseudo-boolean optimization solver [on hold]

I am actually constructing engineer application based on pseudo-boolean optimization. I want to ask what is the current status (how many variables, interaction parameters) the solver could generally ...
0
votes
0answers
166 views

A problem of a hacked article [on hold]

I am surprised by the fact that a journal published an article that I have in arxiv for a few months. The date of publication is after the date that I have in arxiv. The submission date in the ...
3
votes
2answers
275 views

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [on hold]

In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$ However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...
2
votes
3answers
317 views

How did the summation operation come into use? [on hold]

So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...
1
vote
1answer
138 views

Quotients and radicals

Let $I, J$ be ideals in a commutative ring with identity $R$. Define the quotient ideal $(I : J)$ by $$(I : J)=\{x\in R : xJ\subseteq I\}.$$ Define the radical $r(A)$, of an ideal $A$ of $R$ by ...
-3
votes
0answers
27 views

Operations research and Linear Programming [on hold]

I am working on a linear programming maximization problem and need help in understanding how to reformulate this problem so that it has only two functional constraints and all variables have ...
-1
votes
0answers
69 views

Combinatorial Proof Problem [on hold]

I'm having trouble solving this because I'm only familiar with algebraic proofs instead of combinatorial. $$\binom{3n}{3}=n^3+6n\binom{n}{2}+3\binom{n}{3},\quad\text{for }n\ge3.$$
4
votes
0answers
54 views

The name for the quotient property

I asked this question on math@stackoverflow and was suggested to ask it here as well. We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$ ...
-1
votes
0answers
48 views

How to prove that the vertices of a convex hull are only defined by some specific subsets, in this particular case [on hold]

Let $\mathbb{V}$ a vector space of dimension $2^N$, where each vector (of size $N$) is a combination of $0$ and $1$. ex: for $N=2$, $\mathbb{V}$={[0 0],[1 0],[0 1],[1 1]}. Consider (in ...
13
votes
2answers
467 views

Teaching the fundamental group via everyday examples

This is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys. What stories, ...
2
votes
1answer
118 views

Various definitions of the Bruhat decomposition of the affine Grassmannian

Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...
3
votes
0answers
57 views

A problem on chains of squares — can one find an easy combinatorial proof?

Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection $$ A = \left\{ \left[ \frac{i}{n},\frac{i + 1}{n} \right] \times ...
0
votes
0answers
132 views

Publishing in mathematics [on hold]

I apologize if mathoverflow is not the right place for this question, but I guess it is the only place where I can get an answer. The question is the following: is publishing a paper in mathematics ...
-1
votes
0answers
34 views

Conformal map from a sector of unit disk onto upper half plane [on hold]

How do we construct a conformal map from $\{z=x+iy,x>1/2,|x+iy|<1\}$ onto the upper half plane? My idea is first create a sector sending one of the two intersection points to infinity.Any help ...
3
votes
3answers
443 views

What should be considered a finite size of an infinite dimensional space? [on hold]

I've got a map between two infinite dimensional spaces, $f: A\to B$, where $A$ seems "larger" than $B$. For the sake of conversation let's assume that $A$ is the set of smooth maps $\mathbb R^3\to ...
3
votes
1answer
72 views

Ising model: probability of a long path of minus under plus boundary conditions

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions. Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...
-2
votes
0answers
86 views

The field of rational functions on a smooth projective absolutely irreducible curve over a finite field [on hold]

We mean a variety (over "k") of dimension 1 by the curve in the expression "The field of rational functions on a smooth projective absolutely irreducible curve over a finite field k", don't we?
9
votes
2answers
282 views

Splitting integers 1, 2, 3, … n to avoid least possible sum

For each positive integer n, partition the integers 1, 2, 3, … 2n into two sets of n integers each. Let g(n) be the least integer such that there is such a partition in neither of whose parts there is ...
2
votes
0answers
60 views

What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
7
votes
0answers
61 views

Which real Pin groups agree?

In the Lie theory notes on my website it is claimed (Example 7.3.3.5) that $\mathrm{Pin}(4,0)$ and $\mathrm{Pin}(0,4)$ are not isomorphic. As Nigel Ray pointed out to me, this claim is not quite ...
0
votes
0answers
48 views

Generalized weight space

In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space: If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
0
votes
0answers
34 views

Green's function of the Ornstein-Uhlenbeck operator

Consider $\mathbb R^d$ with the Gaussian measure $d\gamma(x) = e^{\frac{1}{4}|x|^2}\,dx$. The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a ...
1
vote
0answers
38 views

Ozsvath-Szabo orientation convention for Seifert fibred spaces

I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...
0
votes
0answers
42 views

Which of the following is true? [on hold]

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct: I, $f(x)$ and $g(x)$ have ...
1
vote
0answers
51 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
2
votes
1answer
49 views

Limit-circle and limit-point at endpoints

I was wondering if the following holds: If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
2
votes
1answer
113 views

A question about running MMP with scaling

Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective. Suppose $K_X + \Delta$ is not nef (over $U$) and there ...
0
votes
0answers
24 views

question about Baer sum of extensions

Let $E_1$ and $E_2$ be extensions of $\mu_p$ by $\mathbb{Z}/{p\mathbb{Z}}$. Assume that $E$ contains $E_1$ and $E_2$ both, and $E_1 \cap E_2 = \mathbb{Z}/{p\mathbb{Z}}$. Then, does $E$ contain their ...
1
vote
0answers
119 views

Is there any progress on Problem 13 (from Schoen and Yau)?

This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks: Let $M_1$ and $M_2$ ...
3
votes
1answer
115 views

Real points of zero-dimensional real algebraic varieties

There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?). Here is a closely related ...
1
vote
0answers
30 views

Classification properties of fusion rings

Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...
-3
votes
0answers
32 views

Does $(n + 2)$ have a multiplicative inverse mod $(n - 1)$ over $GF(5)$? [on hold]

I have been stuck on understanding this for hours. The reason I am confused is that I thought over $GF(k)$, only constants have inverses. Also, how would one go about applying EGDC to figure this out? ...
-2
votes
0answers
74 views

Closure in Hilbertspace [on hold]

I know that i asked this question already on stackexchange.com (http://math.stackexchange.com/questions/983377/closure-in-a-hilbertspace) Define for a pure contraction $S$ (remember: $\|S\|\leq1$ and ...
8
votes
0answers
131 views

Mixed Hodge structure on configuration spaces

Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed ...
0
votes
0answers
43 views

Estimation of growth rate of spectral radius

I have following problem: Let the spectral radius of $S=(a_{ij})_{n\times n}$ be $\lambda>1$, where each $a_{i,j}$ is a positive integer, then we have that $$\lim_{k\to ...
3
votes
1answer
141 views

Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
1
vote
0answers
26 views

How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below. If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...
2
votes
0answers
51 views

Number of maximal chains in Bruhat order

Is there a formula for the number of maximal chains between two permutation in the (strong) Bruhat order?
4
votes
0answers
68 views

Picard scheme of varieties over imperfect fields

Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
3
votes
0answers
36 views

Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
1
vote
2answers
136 views

Calculating Exterior Distance from Measurements of Inner Geometry

Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ...

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