# All Questions

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### family of polynomials with square discriminant

The title pretty much sums it up: do people know of nice parametrized families of polynomials (with integer coefficients) with square discriminant. I should say that one such family consists of ...
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### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...
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### Convergence of local stable manifolds

This question is a kind of local version of a previous post (MO224171). Let $\ f_n \$ be a sequence of Morse functions on $\mathbb{R}^d$, converging, in the $C^\infty$-topology, to a limit Morse ...
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### Information and intuition packed in the Chern character for coherent sheaves

even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting. Let us consider a smooth projective algebraic variety ...
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### “The” natural double complex associated to a principal $G$-bundle?

Disclaimer: Part of the purpose of this question is to make sure i'm not terribly wrong about some of these constructions. Let $\pi: P \to M$ be a principal $G$-bundle. We have the associated ...
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### Goldbach for certain classes of $n$

Asked on MSE without response here. $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$. The Wiki article on the Goldbach conjecture states that In 1975, ...
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### How to calculate error for 3D-trilateration

I'm developing a 3D positioning system that uses four anchor nodes of known location to position a fifth node of unknown location. I'm calculating the position by using trilateration, MATLAB code ...
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### Euler Characteristic of simple sheaves

Let $X$ be a projective curve over a field $K$ (any characteristic). Let $\mathcal{F}$ be a coherent simple sheaf (In the sense, that $\mathcal{F}$ doesn't have non-trivial subsheaves). What is the ...
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### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$. I am interested in the following questions: How ...
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### On the number of divisors in a given range

Given $\alpha\in\Bbb N$, can there be more than $(\log N)^4$ divisors (composites allowed) of $N$ in $\big[\frac\alpha2,\alpha\big]$ when $\sqrt N\in\big[\frac\alpha2,\alpha\big]$?
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### number of maximal subgroups of the symmetric group

What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere... EDIT I am actually more interested in the number of conjugacy ...
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### Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty$ for all $\lambda \in [-c,c]$. Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...
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### Quaternions: ellipse effect

I would be interested in an explanation of the "six ellipse effect" produced by the pseudocode below (I also wonder how close are these to being actual ellipses). Note the code is somewhat similar to ...
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### Is there a function with a fixed point but does not satisfy the Banach contraction principle? [on hold]

This is my question Is there a function which does not satisfy the Banach contraction principle, but has a fixed point? thank
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### When is an erratum necessary?

A typo, a spelling error etc., in a published article, is definitely not enough for issuing an erratum. If a mistake destroys a main result, then an erratum is definitely necessary, and the proof ...
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### Examples of Noetherian overkill

I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite ...
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### Joyal's construction of the spectrum of a commutative ring

I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well. I know this is a lot to ask, but basically, I ...
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### Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...
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### Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows: $$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$ Upper bounds for $\phi$ can be simply derived from ending the product early, e.g. ...
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### dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
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### Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
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### Grothendieck - sheaves as meter sticks

I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it. ...
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### Mathematician trying to learn string theory

I'm a mathematician. I want to be able to read recent ArXiv postings on high energy physics theory (String theory) (and perhaps be able to do research). I want to understand compactifications, ...
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### Should I not cite an arxiv.org paper?

I want to cite a paper which is on arxiv.org but is not published or reviewed anywhere, and no publication or review seems to be in the pipeline. Would citing this arxiv.org paper be bad? Should I ...
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### Compact embedding and fractional Sobolev spaces in unbounded domain

It is known that there exists a compactness results involving fractional sobolev spaces in bounded domain. What about unbounded domain? More precisely, Under which conditions, we can extend the ...
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### What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...
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### Name for the variety of preimages of a finite morphism

If $f:X\to Y$ is a finite morphism of degree $d$ between two varieties, you get a closed subset of the symmetric product $X^{(d)}$ (or perhaps rather the Hilbert scheme $X^{[d]}$), defined as the ...
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### rational numbers and triangular numbers

This question is an offshoot of Ratio of triangular numbers. Suppose $ka(a+1)=nb(b+1)$, where $k,n >1$ are relative prime integers, and $a,b \geq 0$ are integers. Which $k,n$ pairs have no solution ...
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### Algorithm: In every vertex whose distance from $v_i$ is not greater that $d_i$ place $r_i$ objects

You are given a tree with $N$ $(1 \le N \le 10^5)$ vertices and $N - 1$ edges. Weight of edge won't exceed 200. Design an algorithm to do $Q$ $(1 \le Q \le 10^5)$ operations of two types as fast as ...
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### Bound that random walk stays within with constant probability?

For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$. What is the bound, in terms of $n$, such that if the ...
I am having a problem with the definition of the space $W^{-k,p}$. I use Adams's definition $$W^{-k,p} = \left\{T \in D'(\Omega) \ \middle| \sum \limits_{0 \leq |i| \leq k} (-1)^{|i|} \int_{\Omega} ... 2answers 415 views ### Did Bishop, Heyting or Brouwer take partial functions seriously? The partial μ-recursive functions which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one ... 0answers 43 views ### Pull back of a semistable vector bundle to a product is semistable? Let X be a smooth projective surface over \mathbb{C}. Let L be an ample line bundle on X. Let F be a \mu_L semistable rank 2 vector bundle on X (semistability in the sense of ... 0answers 42 views ### Modifying tensor to be positive definite everywhere Consider a (0,2)-tensor. It is known that it is positive definite somewhere and it is negative definite otherwise. Is there a theory how to "make" that tensor positive definite everywhere, while ... 0answers 75 views ### infinite loop-suspension of classifying space of symmetric groups [on hold] Let X be a manifold and p an odd prime. By the Brown representation theorem,$$ H^n(X;\mathbb{Z}/p)=[X; K(\mathbb{Z}/p,n)]. $$Let \Sigma_p be the p-th symmetric group and B\Sigma_p be the ... 1answer 37 views ### Intuition about Skorohod integral I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ... 2answers 658 views ### Is there any pattern to the continued fraction of \sqrt[3]{2}? Is there any pattern to the continued fraction of \sqrt[3]{2} ? Wolfram Alpha returns for cube root of 2: \sqrt[3]{2}= [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ... 0answers 27 views ### heat equation in 2D with absorbing and reflecting boundary conditions [on hold] could you please help me with solving the following problem$$u_{xx}+u_{yy}=u_t, \quad t>0,x∈(−∞;∞),y>0u(x_0,y_0,0)=δ(x−x_0,y−y_0)u_y(x,0,t)=0u(x_t,y_t,t)=0$$The ... 0answers 119 views ### Homological stability for orthogonal groups In Vogtmann's paper "Spherical posets and homological stability for O_{n,n}" it is shown that for all fields different than the field F_2 with two elements the homology groups of the orthogonal ... 0answers 127 views ### An abstract zero-sum problem I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ... 1answer 187 views ### Practical advantages of univalent foundations I'm interested in the machine translation of mathematics from informal to formal (a la Ganesalingam in The Language of Mathematics). As a first step, I am designing a computer language for expressing ... 2answers 460 views ### Rational points on the “quintic circle” x^5 + y^5 = 7 I suspect that the curve x^5 + y^5=7 has no \mathbb Q points, and a brief computer search verifies this hypothesis for denominators up to 10^4. What techniques can be used to show that there are ... 2answers 261 views ### When is the closed unit ball in a smaller Banach space closed in a larger Banach space? Recently I saw an interesting lemma: For any s>0, the closed unit ball in H^s is also closed in the L^2 norm. That is, suppose u_j\in H^s and \|u_j\|_{H^s}\le 1. Suppose u_j\to u in ... 0answers 81 views ### probability of a quadratic form being nonnegative at a random point I am looking for good and explicit lower and upper bounds on the probability that x^TGx\ge 0, where G is a symmetric matrix with zero trace and x is a vector whose components are independent ... 2answers 146 views ### Computer Algebra Systems that support variable sized matrices I'm familiar with sympy, the matlab symbolic package, reduce, and have tried out a few other computer algebra systems. However, as far as I can tell, none of them seem to be able to do algebra on ... 0answers 51 views ### Lower semi-continuity of the Hellinger-Fisher-Rao distance I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance$$ ...
Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has: and in Theorem 16 of Chapter 5 proves that: $p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...