相关论文: A proof of the multi-component $q$-Baker--Forreste…
Following Lang and Trotter we describe a probabilistic model that predicts the distribution of primes $p$ with given Frobenius traces at $p$ for two fixed elliptic curves over $\mathbb{Q}$. In addition, we propose explicit Euler product…
Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,\ldots,d_{k}],$ with all partial quotients…
In this paper, we are interested in the $\beta$-ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form $|x|^{p}$ with $p \geq 2$. Since this potential is not of class $\mathcal{C}^{3}$ when $p \in (2,3]$, most of…
In this paper we prove a $K$-homology index theorem for the Toeplitz operators obtained from the multishifts of the Bergman space on several classes of egg-like domains. This generalizes our theorem with Douglas and Yu on the unit ball.
We expand upon the notion of bottlenecking introduced in our earlier work, characterizing a spectrum of graphs and showing that this naturally extends to a concept of coarse bottlenecking. We show how the notion of bottlenecking provides a…
On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii)…
We consider the minimal free resolution of a generic set of n+1 forms (not necessarily of the same degree) in a polynomial ring of n variables. The Hilbert function for such an ideal is known, thanks to a result of Stanley and of Watanabe.…
Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator of a finite continued fraction b/d = [d1,d2,...,dk], with all partial quotients d1,d2,...,dk being bounded by an absolute constant…
Let $k\in \mathbb Z_+$ and $(X, \mathcal B(X), \mu)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots, T_k \colon X\to X$. Let $P_1,\ldots, P_k\in\mathbb Z[\rm n]$ be…
The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of…
Boij-S\"oderberg theory concerns resolutions of graded modules over a polynomial ring over a field. Specifically Boij-S\"oderberg theory gives a description of the cone of Betti diagrams for Cohen-Macaulay modules. Eisenbud and Schreyer…
The Ramanujan $_1\psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised…
Pusey, Barrett and Rudolph (PBR) have recently given a completely novel argument that restricts the class of possible models for quantum phenomena (arXiv:1111.3328). In these notes the assumptions used by PBR are considerably weakened, to…
A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X,L), the cohomologies of L over the GIT quotient X // G equal the invariant part of…
An arithmetic function $f$ is called a sieve function of range $Q$, if it is the convolution product of the constantly $1$ function and $g$ such that $g(q)\ll_{\varepsilon} q^{\varepsilon}$, $\forall\varepsilon>0$, for $q\leq Q$, and…
Given a pair of modular forms with complex multiplication by distinct imaginary quadratic fields, the four dimensional Galois representation associated to their Rankin--Selberg convolution is induced from a character over an imaginary…
The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are…
Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the…
We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha…
J.C.Lagarias (2000) conjectured that if $\mu$ is a complex measure on p-dimensional Euclidean space with a uniformly discrete support and its spectrum (Fourier transform) is also a measure with a uniformly discrete support, then the support…