A horadam-based pseudo-random number generator

Uniformly distributed pseudo-random number generators are commonly used in certain numerical algorithms and simulations. In this article a random number generation algorithm based on the geometric properties of complex Horadam sequences was investigated. For certain parameters, the sequence exhibited uniformity in the distribution of arguments. This feature was exploited to design a pseudo-random number generator which was evaluated using Monte Carlo π estimations, and found to perform comparatively with commonly used generators like Multiplicative Lagged Fibonacci and the 'twister' Mersenne

On the masked periodicity of horadam sequences: A generator-based approach

The Horadam sequence is a general second order linear recurrence sequence, dependent on a family of four (possibly complex) parameters|two recurrence coe cients and two initial conditions. In this article we examine a phenomenon identi ed previously and referred to as `masked' periodicity, which links the period of a self-repeating Horadam sequence to its initial conditions. This is presented in the context of cyclicity theory, and then extended to periodic sequences arising from recursion equations of degree three or more.The Horadam sequence is a general second order linear recurrence sequence, dependent on a family of four (possibly complex) parameters|two recurrence coe cients and two initial conditions. In this article we examine a phenomenon identi ed previously and referred to as `masked' periodicity, which links the period of a self-repeating Horadam sequence to its initial conditions. This is presented in the context of cyclicity theory, and then extended to periodic sequences arising from recursion equations of degree three or more

On the structure of periodic complex horadam orbits

Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multisymmetric patterns can be recovered for selected parameter values. Some applications are also suggested.Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multisymmetric patterns can be recovered for selected parameter values. Some applications are also suggested

On the Dynamic Geometry of Kasner Quadrilaterals with Complex Parameter

We explore the dynamics of the sequence of Kasner quadrilaterals (AnBnCnDn)n≥0 defined via a complex parameter α. We extend the results concerning Kasner triangles with a fixed complex parameter obtained in earlier works and determine the values of α for which the generated dynamics are convergent, divergent, periodic, or dense

Local maximizers of generalized convex vector-valued functions

Any local maximizer of an explicitly quasiconvex real-valued function is actually a global minimizer, if it belongs to the intrinsic core of the function's domain. In this paper we show that similar properties hold for componentwise explicitly quasiconvex vector-valued functions, with respect to the concepts of ideal, strong and weak optimality. We illustrate these results in the particular framework of linear fractional multicriteria optimization problems.Any local maximizer of an explicitly quasiconvex real-valued function is actually a global minimizer, if it belongs to the intrinsic core of the function's domain. In this paper we show that similar properties hold for componentwise explicitly quasiconvex vector-valued functions, with respect to the concepts of ideal, strong and weak optimality. We illustrate these results in the particular framework of linear fractional multicriteria optimization problems

An equivalent property of a Hilbert-type integral inequality and its applications

Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions

On Horadam Sequences with Dense Orbits and Pseudo-Random Number Generators

Horadam sequence is a general recurrence of second order in the complex plane, depending on four complex parameters (two initial values and two recurrence coefficients). These sequences have been investigated over more than 60 years, but new properties and applications are still being discovered. Small parameter variations may dramatically impact the sequence orbits, generating numerous patterns: periodic, convergent, divergent, or dense within one dimensional curves. Here we explore Horadam sequences whose orbit is dense within a 2D region of the complex plane, while the complex argument is uniformly distributed in an interval. This enables the design of a pseudo-random number generator (PRNG) for the uniform distribution, for which we test periodicity, correlation, Monte Carlo estimation of π, and the NIST battery of tests. We then calculate the probability distribution of the radii of the sequence terms of Horadam sequences. Finally, we propose extensions of these results for generalized Horadam sequences of third order

Inertial Krasnosel’skiĭ-Mann iterative algorithm with step-size parameters involving nonexpansive mappings with applications to solve image restoration problems

In this work, we propose and study an inertial Krasnosel’ski˘ ı-Mann iterative algorithm with step-size parameters involving nonexpansive mapping to find a solution of a fixed point problem of a nonexpansive mapping in the frame work of Hilbert spaces. Strong convergence of the new proposed algorithm is proved under some useful properties of a nonexpansive mapping and inequalities on real Hilbert spaces together with the appropriate conditions of scalar controls without relying on the concept of viscosity methods. For the applications, we employ the obtained results to find a zero point of some monotone inclusion problems and then apply to solve image restoration problems. For representing the advantage of the new algorithm, the signal to noise ratio (SNR) with various types of blurring operators and some numerical experiments are presented to compare and illustrate the behavior of the new algorithm with numerical results of some previous existing algorithms

Disrupting resilient criminal networks through data analysis: The case of sicilian mafia

Compared to other types of social networks, criminal networks present particularly hard challenges, due to their strong resilience to disruption, which poses severe hurdles to LawEnforcement Agencies (LEAs). Herein, we borrow methods and tools from Social Network Analysis (SNA) to (i) unveil the structure and organization of Sicilian Mafia gangs, based on two real-world datasets, and (ii) gain insights as to how to efficiently reduce the Largest Connected Component (LCC) of two networks derived from them. Mafia networks have peculiar features in terms of the links distribution and strength, which makes them very different from other social networks, and extremely robust to exogenous perturbations. Analysts also face difficulties in collecting reliable datasets that accurately describe the gangs' internal structure and their relationships with the external world, which is why earlier studies are largely qualitative, elusive and incomplete. An added value of our work is the generation of two realworld datasets, based on raw data extracted from juridical acts, relating to a Mafia organization that operated in Sicily during the first decade of 2000s. We created two different networks, capturing phone calls and physical meetings, respectively. Our analysis simulated different intervention procedures: (i) arresting one criminal at a time (sequential node removal); and (ii) police raids (node block removal). In both the sequential, and the node block removal intervention procedures, the Betweenness centrality was the most effective strategy in prioritizing the nodes to be removed. For instance, when targeting the top 5% nodes with the largest Betweenness centrality, our simulations suggest a reduction of up to 70% in the size of the LCC. We also identified that, due the peculiar type of interactions in criminal networks (namely, the distribution of the interactions' frequency), no significant differences exist between weighted and unweighted network analysis. Our work has significant practical applications for perturbing the operations of criminal and terrorist networks

Criminal networks analysis in missing data scenarios through graph distances

Data collected in criminal investigations may suffer from issues like: (i) incompleteness, due to the covert nature of criminal organizations; (ii) incorrectness, caused by either unintentional data collection errors or intentional deception by criminals; (iii) inconsistency, when the same information is collected into law enforcement databases multiple times, or in different formats. In this paper we analyze nine real criminal networks of different nature (i.e., Mafia networks, criminal street gangs and terrorist organizations) in order to quantify the impact of incomplete data, and to determine which network type is most affected by it. The networks are firstly pruned using two specific methods: (i) random edge removal, simulating the scenario in which the Law Enforcement Agencies fail to intercept some calls, or to spot sporadic meetings among suspects; (ii) node removal, modeling the situation in which some suspects cannot be intercepted or investigated. Finally we compute spectral distances (i.e., Adjacency, Laplacian and normalized Laplacian Spectral Distances) and matrix distances (i.e., Root Euclidean Distance) between the complete and pruned networks, which we compare using statistical analysis. Our investigation identifies two main features: first, the overall understanding of the criminal networks remains high even with incomplete data on criminal interactions (i.e., when 10% of edges are removed); second, removing even a small fraction of suspects not investigated (i.e., 2% of nodes are removed) may lead to significant misinterpretation of the overall network. Copyright