All Questions

68 views

Quotient of a vector space by a linear finite group action

Let the cyclic group $\mathbb{Z}_n$ act on $\mathbb{C}^n$ (or on $\mathbb{R}^n$, I'm interested in both) by permuting coordinates. What does the quotient $Q$ look like? Is there some nice way to ...
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Algebra working with quadratic and linear equations [on hold]

Hi guys I'm writing my SATthis Saturday, while practicing I came accross this: Question snapshot I tried thinking as far as I can but couldn't figure out how to work it out. I would truly ...
280 views

Is Lehmer's polynomial solvable?

The degree 10 polynomial $$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$ given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is ...
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Problem related to Frobenius coin problem

Lets say a linear form $ax+by$ represents $n$ if $ax+by=n$ for some $x,y≥0$. Call a pair $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if for some $r,s,t,u>1$ with each of ...
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Can we get infinitely often better approximations for algebraic numbers from continued fractions with algebraic partial coefficients?

I am wondering how good approximations to algebraic numbers one can get via convergents of continued fractions with algebraic partial coefficients coefficients. Let $\alpha$ be algebraic integer ...
81 views

Regarding a new divergence function of two probability distributions

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, ...
184 views

Convergence rate of Fagin's 0-1 law for first-order properties of random graphs

Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the ...
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History of Mathematical Notation

I would like to see a simple example which shows how mathematical notation were evolve in time and space. Say, consider the formula $$(x+2)^2=x^2+4{\cdot}x+4.$$ If I understand correctly, Franciscus ...
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concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...
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Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...
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In a fibration, where does the generic object live?

In Bart Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, 1999. isbn: 9780444508539, the author writes, p. 326: Sometimes, for ...
458 views

Is the moduli space of graphs simply connected?

The moduli space of graphs $MG_n$ is the quotient of Culler-Vogtmann's outer space $X_n$ by the action of $\mathrm{Out}(F_n)$. It can be thought of as the space of metric graphs homotopy equivalent to ...
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Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...
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Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$? There are some results of the ...
267 views

The density of integers represented by a binary form

Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of ...
440 views

Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...
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“The Two Sheriffs” puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler. Two sheriffs in neighboring towns are on the track of a killer, in a case involving eight ...
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hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...
11k views

nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...
66k views

Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion ...
107 views

Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
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$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
7k views

Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
7k views

Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
809 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 ...
12k views

What are some correct results discovered with incorrect (or no) proofs?

Many famous results were discovered through non-rigorous proofs, with correct proofs being found only later and with greater difficulty. One that is well known is Euler's 1737 proof that ...
484 views

Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There is no surface in $R^3$ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
232 views

Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...
528 views

Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...
46k views

Which are the best mathematics journals, and what are the differences between them?

Suppose you have a draft paper that you think is pretty good, and people tell you that you should submit it to a top journal. How do you work out where to send it to? Coming up with a shortlist isn't ...
27k views

Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
111 views

Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation $$\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3$$ in ...