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We develop the foundations of a general framework for producing optimal upper and lower bounds on the sum $\sum_p a_p$ over primes $p$, where $(a_n)_{x/2<n\le x}$ is an arbitrary non-negative sequence satisfying Type I and Type II…

In this work we use intersection of different pseudo-orbits obtained by interval extensions to reduce the bounds of the exact solution provided by the toolbox Intlab. The method is applied on the logistic map.

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on $\Omega\times \mathbb R_t$ (with $\Omega$ open set in $\mathbb R^n$) by $$ L_j = \frac{\partial}{\partial t_j} + i \frac {\partial…

Sawin recently gave an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb{F}_{q}(T)$ and proved their existence by exhibiting the coefficients as trace functions of specific perverse sheaves. However,…

The paper investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of…

We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends…

We consider chiral fermionic conformal field theories constructed from classical error-correcting codes and provide a systematic way of computing their elliptic genera. We exploit the $\mathrm{U}(1)$ current of the $\mathcal{N}=2$…

This paper provides norm-based generalization bounds for the Transformer architecture that do not depend on the input sequence length. We employ a covering number based approach to prove our bounds. We use three novel covering number bounds…

In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and…

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this…

We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical…

We construct compact descriptions of function fields and number fields.

Due to wide applications of binary sequences with low correlation to communications, various constructions of such sequences have been proposed in literature. However, most of the known constructions via finite fields make use of the…

By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect…

We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of $L$-functions defined over number fields. Specifically, we give a…

Inference in expressive probabilistic models is generally intractable, which makes them difficult to learn and limits their applicability. Sum-product networks are a class of deep models where, surprisingly, inference remains tractable even…

We study a general class of multiplicative functions by relating "short averages" to its "long average". More precisely, we estimate asymptotically the variance of such a class of functions in short intervals using Fourier analysis and…

We prove three more general supercongruences between truncated hypergeometric series and $p$-adic Gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez-Villegas and proved by E. Mortenson…

Bent functions are of great importance in both mathematics and information science. The $\mathcal{P}\mathcal{S}$ class of bent functions was introduced by Dillon in 1974, but functions belonging to this class that can be explicitly…

Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic…