The goal of this paper is to give a quick survey of some important recent results and open problems in the area of function field arithmetic, which studies geometric analogs of arithmetic questions. We will sketch related developments and try to trace the multiple influences of works of John Tate in this context. keywords: Tate, function field, Drinfeld modules, elliptic curves, zeta values. Th...

Motivated by recent results, we study sums of the formSf(x)=∑n≤xf(⌊xn⌋), where f is an arithmetic function and ⌊⋅⌋ denotes greatest integer function. We improve error term in asymptotic formula for Sf(x) some specific cases.

This article studies the computational power of various discontinuous real computational models that are based on the classical analog recurrent neural network (ARNN). This ARNN consists of finite number of neurons; each neuron computes a polynomial net function and a sigmoid-like continuous activation function. We introduce arithmetic networks as ARNN augmented with a few simple discontinuous ...

This paper deals with a new method for implementing Binary Adders of two operands whose main characteristics are: They are built using only threshold gates; carry output, besides carry input, is used to obtain each sum bit and the total number of gates is a linear function of the number of bits of the operands (n).

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of i...