Strengthening Quality of Chaotic Bit Sequences

We discuss chaos and its quality as measured through the 0-1 test for chaos. When the 0-1 test indicates deteriorating quality of chaos, because of the finite precision representations of real numbers in digital implementations, then the process may eventually lead to a periodic sequence. A simple method for improving the quality of a chaotic signal is to mix the signal with another signal by using the XOR operation. In this paper, such mixing of weak chaotic signals is considered, yielding new signals with improved quality (with K values from the 0-1 test close to 1). In some sense, such a mixing of signals could be considered as a two-layer prevention strategy to maintain chaos. That fact may be important in those applications when the hardware resources are limited. The 0-1 test is used to show the improved chaotic behavior in the case when a continuous signal (for example, from the Chua, Rössler or Lorenz system) intermingles with a discrete signal (for example, from the logistic, Tinkerbell or Henon map). The analysis is presented for chaotic bit sequences. Our approach can further lead to hardware applications, and possibly, to improvements in the design of chaotic bit generators. Several illustrative examples are included.

Linear-Time Recognition of Double-Threshold Graphs

A graph $$G = (V,E)$$ G = ( V , E ) is a double-threshold graph if there exist a vertex-weight function $$w :V \rightarrow \mathbb {R}$$ w : V → R and two real numbers $$\mathtt {lb}, \mathtt {ub}\in \mathbb {R}$$ lb , ub ∈ R such that $$uv \in E$$ u v ∈ E if and only if $$\mathtt {lb}\le \mathtt {w}(u) + \mathtt {w}(v) \le \mathtt {ub}$$ lb ≤ w ( u ) + w ( v ) ≤ ub . In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in $$O(n^{3} m)$$ O ( n 3 m ) time, where n and m are the numbers of vertices and edges, respectively.

New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

In this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.

Theorems on analogous of Ramanujan’s remarkable product of theta-function and their explicit evaluations

In this article, we define Em,n for any positive real numbers m and n involving Ramanujan’s product of theta-functions ψ(−q) and f(q), which is analogous to Ramanujan’s remarkable product of theta-functions and establish its several properties by Ramanujan. We establish general theorems for the explicit evaluations of Em,n and its explicit values.

Static absorber modelling

Introduction/purpose: A static absorber is capable of neutralizing any signal either in the physical or virtual domain and its analysis will be presented in this paper. Methods: The approach used here includes purely scientific thoughts as well as a model with its explanations evaluated step by step applying highly sophisticated computer tools for design and simulation. No empirical results will be attached, only claims with their evidence. Results: The crucial outcome of this research is a completely new approach to binary systems that are now observed as a set of real numbers. Conclusion: The purpose of this research is to introduce something brand-new that can be used in cyber industry while a dynamic variation of the absorber is still under development.

Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime

<abstract><p>We study the existence of solutions to the cubic Schrödinger system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u_i = \sum\limits_{j = 1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i = 0\ \hbox{on}\ \partial\Omega,\ i = 1,\dots,m, $\end{document} </tex-math></disp-formula></p> <p>when $ \Omega $ is a bounded domain in $ \mathbb R^4, $ $ \lambda_i $ are positive small numbers, $ \beta_{ij} $ are real numbers so that $ \beta_{ii} &gt; 0 $ and $ \beta_{ij} = \beta_{ji} $, $ i\neq j $. We assemble the components $ u_i $ in groups so that all the interaction forces $ \beta_{ij} $ among components of the same group are attractive, i.e., $ \beta_{ij} &gt; 0 $, while forces among components of different groups are repulsive or weakly attractive, i.e., $ \beta_{ij} &lt; \overline\beta $ for some $ \overline\beta $ small. We find solutions such that each component within a given group blows-up around the same point and the different groups blow-up around different points, as all the parameters $ \lambda_i $'s approach zero.</p></abstract>

Topological models for stable motivic invariants of regular number rings

For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

Cartesian Products of Some Regular Graphs Admitting Antimagic Labeling for Arbitrary Sets of Real Numbers

An edge labeling of graph G with labels in A is an injection from E G to A , where E G is the edge set of G , and A is a subset of ℝ . A graph G is called ℝ -antimagic if for each subset A of ℝ with A = E G , there is an edge labeling with labels in A such that the sums of the labels assigned to edges incident to distinct vertices are different. The main result of this paper is that the Cartesian products of complete graphs (except K 1 ) and cycles are ℝ -antimagic.

Predicting Logarithm of Real Numbers to any Real Number Base

This paper aims to propose an algorithm that can predict the values of logarithmic function for any domain and for any bases with minimum possible error using basic mathematical operations. It talks about a univariate quadratic function that overlaps with the graph of the common logarithmic function to some extent, and using this resemblance to fullest advantage. It proposes a formula and certain algorithms based on the same formula.