A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes

2021 ◽  
pp. 1-8
Author(s):  
Martin Lind
Keyword(s):  
Convergence Rate ◽  
Sharp Estimate ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Pseudorandom Numbers ◽  
Asymptotic Formulae ◽  
Congruential Method ◽  
Inversive Congruential Method ◽  
Star Discrepancy

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

scholarly journals Analysis of a Two-Level Algorithm for HDG Methods for Diffusion Problems

2016 ◽  
Vol 19 (5) ◽  
pp. 1435-1460 ◽  
Author(s):  
Binjie Li ◽  
Xiaoping Xie ◽  
Shiquan Zhang
Keyword(s):  
Convergence Rate ◽  
Numerical Experiments ◽  
Sharp Estimate ◽  
Model Problem ◽  
The Other ◽  
Extended Version ◽  
Weak Galerkin ◽  
Hybridizable Discontinuous Galerkin ◽  
Theoretical Results ◽  
Diffusion Problems

AbstractThis paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also are applied to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

An exploration-enhanced elephant herding optimization

2019 ◽  
Vol 36 (9) ◽  
pp. 3029-3046 ◽  
Author(s):  
Islam A. ElShaarawy ◽  
Essam H. Houssein ◽  
Fatma Helmy Ismail ◽  
Aboul Ella Hassanien
Keyword(s):  
Evolutionary Algorithms ◽  
Convergence Rate ◽  
Search Space ◽  
Exploration And Exploitation ◽  
Content Type ◽  
Discrepancy Measure ◽  
The One ◽  
Star Discrepancy ◽  
Exploration Phase

Purpose The purpose of this paper is to propose an enhanced elephant herding optimization (EEHO) algorithm by improving the exploration phase to overcome the fast-unjustified convergence toward the origin of the native EHO. The exploration and exploitation of the proposed EEHO are achieved by updating both clan and separation operators. Design/methodology/approach The original EHO shows fast unjustified convergence toward the origin specifically, a constant function is used as a benchmark for inspecting the biased convergence of evolutionary algorithms. Furthermore, the star discrepancy measure is adopted to quantify the quality of the exploration phase of evolutionary algorithms in general. Findings In experiments, EEHO has shown a better performance of convergence rate compared with the original EHO. Reasons behind this performance are: EEHO proposes a more exploitative search method than the one used in EHO and the balanced control of exploration and exploitation based on fixing clan updating operator and separating operator. Operator γ is added to EEHO assists to escape from local optima, which commonly exist in the search space. The proposed EEHO controls the convergence rate and the random walk independently. Eventually, the quantitative and qualitative results revealed that the proposed EEHO outperforms the original EHO. Research limitations/implications Therefore, the pros and cons are reported as follows: pros of EEHO compared to EHO – 1) unbiased exploration of the whole search space thanks to the proposed update operator that fixed the unjustified convergence of the EHO toward the origin and the proposed separating operator that fixed the tendency of EHO to introduce new elephants at the boundary of the search space; and 2) the ability to control exploration–exploitation trade-off by independently controverting the convergence rate and the random walk using different parameters – cons EEHO compared to EHO: 1) suitable values for three parameters (rather than two only) have to be found to use EEHO. Originality/value As the original EHO shows fast unjustified convergence toward the origin specifically, the search method adopted in EEHO is more exploitative than the one used in EHO because of the balanced control of exploration and exploitation based on fixing clan updating operator and separating operator. Further, the star discrepancy measure is adopted to quantify the quality of exploration phase of evolutionary algorithms in general. Operator γ that added EEHO allows the successive local and global searching (exploration and exploitation) and helps escaping from local minima that commonly exist in the search space.

scholarly journals ON THE DISTRIBUTION OF NONLINEAR CONGRUENTIAL PSEUDORANDOM NUMBERS IN RESIDUE RINGS

2006 ◽  
Vol 02 (01) ◽  
pp. 163-168 ◽  
Author(s):  
EDWIN D. EL-MAHASSNI ◽  
ARNE WINTERHOF
Keyword(s):  
Pseudorandom Number ◽  
Attractive Alternative ◽  
Pseudorandom Numbers ◽  
Ring Of Integers ◽  
Number Generation ◽  
New Type ◽  
Congruential Method ◽  
Pseudorandom Number Generation ◽  
Linear Congruential

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers over a ring of integers ℤM.

ALGORITHM FOR EXACT LONG TIME CHAOTIC SERIES AND ITS APPLICATION TO CRYPTOSYSTEMS

2004 ◽  
Vol 14 (10) ◽  
pp. 3607-3611 ◽  
Author(s):  
SHUNJI KAWAMOTO ◽  
TAKESHI HORIUCHI
Keyword(s):  
Time Series ◽  
Logistic Map ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Chaotic Time Series ◽  
High Dependency ◽  
Key Space ◽  
Long Time ◽  
Computer Environment ◽  
Numerical Generation

It is said that the numerical generation of exact chaotic time series by iterating, for example, the logistic map, will be impossible, because chaos has a high dependency on initial values. In this letter, an algorithm to generate them without the accumulation of inevitable round-off errors caused by the iteration is proposed, where rational numbers are introduced. Also, it is shown that the period of the chaotic time series depends on the rational numbers including large prime numbers, which are fundamentally related to the Mersenne and the Fermat prime ones. Since the time series are numerically regenerated by the proposed algorithm in an usual computer environment, it could be applied to cryptosystems which do not need the synchronization, and have a large key-space by using large prime numbers.

scholarly journals О размере множества произведения множеств рациональных чисел

2020 ◽  
Vol 21 (1) ◽  
pp. 357-363
Author(s):  
Юрий Николаевич Штейников
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Mathematical Expectation ◽  
Lower Estimate ◽  
Main Tool ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Random Subset ◽  
First Time

For the first time  in the article [1] was established non-trivial lower bounds on the size of the set of products of rational numbers, the numerators and denominators of which are limited to a certain quantity $Q$. Roughly speaking, it was shown that the size of the product deviates from the maximum by no less than $$\exp \Bigl\{(9 + o(1)) \frac{\log Q}{\sqrt{\log{\log Q}}}\Bigl\}$$ times. In the article [7], the index of $ \log{\log Q} $ was improved from $ 1/2 $ to $ 1 $, and the proof of the main result on the set of fractions was fundamentally different. This proof, its argument was based on the search for a special large subset of the original set of rational numbers, the set of numerators and denominators of which were pairwise mutually prime numbers. The main tool was the consideration of random subsets. A lower estimate was obtained for the mathematical expectation of the size of this random subset. There, it was possible to obtain an upper bound for the multiplicative energy of the considered set. The lower bound for the number of products and the upper bound for the multiplicative energy of the set are close to optimal results. In this article, we propose the following scheme. In general, we follow the scheme of the proof of the article [1], while modifying some steps and introducing some additional optimizations, we also improve the index from $1/2$ to $1-\varepsilon$ for an arbitrary positive $\varepsilon>0$.

scholarly journals SHORT INTERVALS ASYMPTOTIC FORMULAE FOR BINARY PROBLEMS WITH PRIME POWERS, II

2019 ◽  
Vol 109 (3) ◽  
pp. 351-370 ◽  
Author(s):  
ALESSANDRO LANGUASCO ◽  
ALESSANDRO ZACCAGNINI
Keyword(s):  
Asymptotic Formula ◽  
Prime Numbers ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Representations Of Integers ◽  
Binary Problems

AbstractWe improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

scholarly journals Divisibility Networks of the Rational Numbers in the Unit Interval

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1879
Author(s):  
Pedro A. Solares-Hernández ◽  
Miguel A. García-March ◽  
J. Alberto Conejero
Keyword(s):  
Real Life ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Unit Interval ◽  
Natural Numbers ◽  
Scale Free ◽  
Free Distribution ◽  
Human Interventions ◽  
Diagonal Argument ◽  
Infinite Sets

Divisibility networks of natural numbers present a scale-free distribution as many other process in real life due to human interventions. This was quite unexpected since it is hard to find patterns concerning anything related with prime numbers. However, it is by now unclear if this behavior can also be found in other networks of mathematical nature. Even more, it was yet unknown if such patterns are present in other divisibility networks. We study networks of rational numbers in the unit interval where the edges are defined via the divisibility relation. Since we are dealing with infinite sets, we need to define an increasing covering of subnetworks. This requires an order of the numbers different from the canonical one. Therefore, we propose the construction of four different orders of the rational numbers in the unit interval inspired in Cantor’s diagonal argument. We motivate why these orders are chosen and we compare the topologies of the corresponding divisibility networks showing that all of them have a free-scale distribution. We also discuss which of the four networks should be more suitable for these analyses.

scholarly journals An improved bound on the sum of prime numbers

2021 ◽  
Author(s):  
Monica Feliksiak
Keyword(s):  
Upper Bound ◽  
Prime Numbers ◽  
Asymptotic Formulae

We derive two asymptotic formulae, for the upper bound on the sum of the first n primes. Both the Supremum and the Estimate of the sum are superior to known bounds. The Estimate bound had been derived to promote the efficiency of estimation of the sum.

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

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