# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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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 ...
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### 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 ...
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### Combinatorial Proof Problem [on hold]

I'm having trouble solving this because I'm only familiar with algebraic proofs instead of combinatorial. $$\binom{3n}{3}=n^3+6n\binom{n}{2}+3\binom{n}{3},\quad\text{for }n\ge3.$$
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### 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 $U\subset Y$ ...
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### 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 ...
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### 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, ...
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### 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 $T \subset G$ ...
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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 ... 0answers 135 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 ... 0answers 35 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 ... 3answers 455 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 ... 1answer 85 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 ... 0answers 89 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? 2answers 315 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 ... 0answers 61 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 ... 2answers 123 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 ... 0answers 51 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 ... 0answers 45 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. $$Is there a ... 0answers 42 views ### Ozsvath-Szabo orientation convention for Seifert fibred spaces I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ... 0answers 43 views ### Which of the following is true? [on hold] f(x),g(x) are defined on [-1,1], f'(0),g'(0) exist, f(0)=g(0), and f(x)\ge g(x) holds for an open interval containing 0. Then which of the following is correct: I, f(x) and g(x) have ... 0answers 59 views ### Hitting time of two dimensional continuous martingale Let (\Omega, \mathcal{F}, P) be a probability space, on which \mathcal{F}_t is filtration satisfying general conditions. W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T} is a two dimensional Brownian ... 1answer 55 views ### Limit-circle and limit-point at endpoints I was wondering if the following holds: If you have an ODE$$-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 ... 1answer 118 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 ... 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 ... 0answers 129 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 ... 1answer 121 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 ... 0answers 33 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 ... 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? ... 0answers 75 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 ... 0answers 143 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 ... 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 ...
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 ...