Stochastic analysis of average-based distributed algorithms

We analyze average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value, which means that they all know a property of the global system. Using a well-chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds on the convergence time. We apply these general results to both the proportion problem and the system size problem.

Models of true arithmetic are integer parts of models of real exponentation

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

On new arithmetic function relative to a fixed positive integer. Part 1

The main purpose of this note is to define a new arithmetic function relative to a fixed positive integer and to study some of its properties.

Retraction Note: Proofs to one inequality conjecture for the non-integer part of a nonlinear differential form

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13660-021-02555-5

The integer part of nonlinear forms with prime variables

<abstract><p>In this paper, we discuss problems that integer part of nonlinear forms with prime variables represent primes infinitely. We prove that under suitable conditions there exist infinitely many primes $ p_j, p $ such that $ [\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^k] = p $ and $ [\lambda_1p_1^3+\cdots+\lambda_4p_4^3+\lambda_5p_5^k] = p $ with $ k\geq 2 $ and $ k\geq 3 $ respectively, which improve the author's earlier results.</p></abstract>

On a sum involving the number of distinct prime factors function related to the integer part function

In this paper, we obtain asymptotic formula on the sum \sum\limits_{n\leq x}\omega \left( \left\lfloor \frac{x}{n}\right\rfloor \right) , where \omega \left( n\right) denote the number of distinct prime divisors of n and \left\lfloor t\right\rfloor denotes the integer part of t.

Parametric Decimal Division using Hardware Description Language

In this work we describe a fast and high-precision algorithm written in VHDL Hardware Description Language to perform the division between two_nite decimal numbers, i.e. numbers composed of an integer part and a decimal one, under the scheme of a fixed point representation. The algorithm proposed is not an approximation one as it is usually considered. To do so, the size of the bits of the operands can be tunned by means of a couple of parameters N and M, according to which the latency of the calculation will depend. The project is _nally sinthesized in a _eld programmable gate array or FPGA of the type SPARTAN 3E from XILINX.

An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications

Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) \lambda )_{n \in \mathbf{Z}}$ is well-distributed $\bmod \, 1$ for all but countably many $\lambda \in{\mathbf R}$ if and only if $\lim\nolimits _{\substack{|n| \rightarrow \infty \\ n \notin J}} |q(n)| = \infty $ for some (possibly empty) set $J$ having zero natural density in $\mathbf{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.