# All Questions

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### Quantiles moments and Convergence

QUESTION: Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
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### Different approaches to the multiverse of sets

There are some different approaches to the multiverse of sets, in particular: 1) The approach by Woodin, 2) The approach by Sy Friedman, ..., 3) The approach by Hamkins. I wonder to know if ...
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### The Alexander-Conway polynomial: from knots to braids?

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...
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### Spectral Graph Theory

Let G be an undirected graph, then Laplacian Matrix(L(G)) = Degree Matrix (D(G)) - Adjacency Matrix (A(G)). What is the relationship between laplacian and adjacency spectrum of undirected graphs?
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### Learning representation theory of real reductive lie groups

I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject ...
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### Laplacian matrix of a graph with negative weights

I am trying to calculate the Laplacian and Adjacency matrix of a graphs for positive and negative weights. If a graph be simple with only non-negative weight it is easier. But in my graph I have some ...
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### Powers of $2$ and the products of initial odd primes

NOTATION: $O_x$ -- the product of all odd primes $\le x$. E.g. $O_7=3\cdot 5\cdot 7 = 105$. QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\$ the only solutions of the ...
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### Lie group about the quantum harmonic oscillator [on hold]

We konw that in quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ is annihilation and creation operator, $H$ is the Hamiltonian operator. ...
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### Deterministic shifts

We consider (topological) dynamical systems $(\Omega, S)$, where $S$ is the shift $(Sx)_n=x_{n+1}$, and $\Omega$ is a compact, shift invariant subspace of $[0,1]^{\mathbb Z}$. I call $(\Omega, S)$ ...
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### Quadratic twist of an elliptic curve given by non-Weierstrass model

Suppose $f(x)$ is a polynomial of degree 4 with integer coefficients and nonzero discriminant. Let $C$ be the hyperelliptic curve of genus 1 defined by $y^2=f(x)$. If we assume that $C$ has a rational ...
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### Find the Flaw in the Argument [on hold]

I have got a question. The problem is to find the composite numbers $C$ $(n)$ and $C$ $(n(n+1))$ such that we get $C$ $(n(n+1))$ $-$ $C$ $(n)$ $=$ $n$. My argument is simply this. The inequality in ...
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### Number of lattice points inside a parallelogram defined by a vector on the “rhombic plane”

I will give here a specific case and try to explain the question the best I can: The rhombic plane is the plane with the x-axis the same as the Cartesian plane but the y-axis tilted at $60$ degrees. ...
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### What is the maximal ideal of $z[t,t^{-1}]\otimes Q$? [migrated]

I know the $z[t,t^{-1}]$ is a localization of $z[t]$.But I do not know the maximal ideal of $z[t,t^{-1}]\otimes Q$? Many thanks!
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### Homotopy equivalent type of a knot complement [on hold]

Let $S$ denote the bounded complement of a tame knot in $S^3$,then $S$ is homotopy equivalent to a finite 2-dimensional simplicial complex $K$ [Milnor's paper "infinite cyclic covering"],I do not ...
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### Fermat surface known to have very few rational integer solutions

The motivation for this question is the Selmer curve, given by $$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$ One can show that this curve has no rational integer solutions, despite having a solution ...
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### How can I solve a cubic equation in a finite field with characteristic 2?

I need to solve the usual cubic equation x^3 + ax^2 + bx + c = 0 over a finite field GF(2^n). This is to avoid doing a brute-force Chien search in a BCH decoder. I read in a paper about an easy way ...
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### Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...
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### Unoriented bordism and homology, reference?

The following has undoubtedly been known to the experts for years, but I only noticed it the other day. Can anyone give a reference? One can prove Thom's theorem to the effect that every mod $2$ ...
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### Surjectivity of $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$

Let $R$ be a Noetherian ring and let $M$ is finitely generated $R$-module.Suppose $p$ is a minimal prime in $\text{Supp}_RM$. Then $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$ that $f(m)=m /1$ is ...
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### Moment map of CP^1 as rational normal curve

I am little confused about some basic symplectic geometry about Hamiltonian actions on sphere. I appreciate your comments. Consider sphere $S^2 = \mathbb{C}P^1$ with its standard symplectic (Kaehler) ...
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### Elliptic curves with square conductor

Is there a characterization of elliptic curves over $\mathbb Q$ whose conductor is a square? Does this property have a geometric meaning?
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### An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product ...
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### Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
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### When are completely positive maps monic/epic?

In the category of C*-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is ...
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### Intuitive explanation for Hardy-Littlewood maximal function

I came across the Hardy-Littlewood maximal function in an analysis course. Could someone help me intuitively understood what the purpose of this map is, and why it is useful? Thank you. Regards ...
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### Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most ...
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### $C^\infty$ approximations of $f(r) = |r|^{m-1}r$ [migrated]

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
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### What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?
What is the current status of representations of $GL_n(F)$ (and other algebraic groups)? When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in ...