Finite Chain Research Articles

Modeling and analysis of gradient metamaterials for broad fusion bandgaps

A gradient metamaterial with varying-stiffness local resonators is proposed to open the multiple bandgaps and further form a broad fusion bandgap. First, three local resonators with linearly increasing stiffness are periodically attached to the spring-mass chain to construct the gradient metamaterial. The dispersion relation is then derived based on Bloch’s theorem to reveal the fusion bandgap theoretically. The dynamic characteristic of the finite spring-mass chain is investigated to validate the fusion of multiple bandgaps. Finally, the effects of the design parameters on multiple bandgaps are discussed. The results show that the metamaterial with a non-uniform stiffness gradient pattern is capable of opening a broad fusion bandgap and effectively attenuating the longitudinal waves within a broad frequency region.

ROLE OF INTERVENTION STRATEGIES AND PSYCHOLOGICAL EFFECT ON THE CONTROL OF INFECTIOUS DISEASES IN THE RANDOM ENVIRONMENT

Infectious disease outbreaks have caused significant losses to populations worldwide in terms of morbidity, social and economic costs. This has prompted governments to seek suitable intervention policies to suppress the outbreaks. This paper aims to explore the intrinsic impact of intervention strategies and the psychological behavior of people on the spread of infectious diseases. Employing the finite state Markov chain, we formulate a new stochastic SIS epidemic model under regime switching by using the Beddington–DeAngelis transmission function, in the presence of some governmental control measures. The study of the model is centered on three different scenarios in terms of varying two control parameters. We analyze the deterministic system for the global dynamics in terms of basic reproductive number. Some new threshold parameters are defined to discuss the extinction and the persistence of the disease in the stochastic setting. Numerical assessments were undertaken to validate theoretical findings for the deterministic and stochastic systems. Our findings strongly suggest that heightened awareness among susceptible individuals contributes significantly to the eradication of infection and fosters a stable epidemiological situation. As levels of public awareness increase, particularly within the susceptible population, a discernible trend toward the eradication of the infection emerges, ultimately ensuring sustained stability within the epidemiological landscape.

Spectral properties and non-Hermitian skin effect in the Hatano–Nelson model

At present, non-Hermitian topological systems continue to be actively studed. In a rigorous approach, we study one of the key non-Hermitian systems — the Hatano–Nelson model $H$. We find the Green function for this Hamiltonian. Using the Green function, we analytically obtain the eigenvalues and eigenfunctions of $H$ for finite and semi-infinite chains, as well as for an infinite chain with a local potential. We discuss the non-Hermitian skin effect for the models mentioned above. We also describe the boundary between localized and resonant eigenfunctions (for the zero spectral parameter, this is the boundary between non-Hermitian topological phases).

A survey on the enumeration of classes of logical connectives and aggregation functions defined on a finite chain, with new results

The enumeration of logical connectives and aggregation functions defined on a finite chain has been a hot topic in the literature for the last decades. Multiple advantages can be derived from knowing a general formula about their cardinality, for instance, the ability to anticipate the computational cost required for generating operators with different properties. This is of paramount importance in image processing and decision making scenarios, where the identification of the most optimal operator is essential. Furthermore, it facilitates the examination of how constraining a certain property is in relation to its parent class. As a consequence, this paper aims to compile the main existing formulas and the methodologies with which they have been derived. Additionally, we introduce some novel formulas for the number of smooth discrete aggregation functions with neutral element or absorbing element, idempotent conjunctions, and commutative and idempotent conjunctions.

Intrinsically localized modes of bilinear FPU chains: Analytical study

Present study concerns the analysis of stationary intrinsically localized modes emerging in bilinear and symmetric Fermi–Pasta–Ulam chains. Each element of the chain is coupled to its nearest neighbors through identical and symmetric (tension/compression) bilinear springs. The intrinsically localized modes are the time-periodic, localized vibration states which are manifested by the extreme energy localization on one and two bonds of the chain for bond - centered and site - centered symmetries, respectively. Approximate and exact analytical solutions of intrinsic localized modes are derived for the infinite as well as the finite chains, correspondingly. The derived analytical approximations allow to describe explicitly the amplitude wave profiles of these special localized states as well as to establish their zones of existence in the space of system parameters. In the second part of the study, we derive the exact analytical solutions for the same type of site- and bond- centered intrinsically localized modes supported by the finite bilinear Fermi–Pasta–Ulam chains and analyze their stability properties using Fillipov method.

Introduction of Local Resonators to a Nonlinear Metamaterial With Topological Features

Abstract Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasi-periodic metamaterial with periodic local resonator attachments. We model a one-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasi-periodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasi-periodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.

On the lattice of fuzzy rough sets

By the means of lower and upper fuzzy approximations we define quasiorders. Their properties are used to prove our main results. First, we characterize the pairs of fuzzy sets that form fuzzy rough sets w.r.t. a t-similarity relation θ on U, for certain t-norms and implicators. If U is finite or the range of θ and of the fuzzy sets is a fixed finite chain, we establish conditions under which fuzzy rough sets form lattices. We show that this is the case for the min t-norm and any S-implicator defined by an involutive negator and the max co-norm.

The Influence of Lobbies: Analyzing Group Consensus from a Physics Approach

In this paper, we study the influence of a small group of agents (i.e., a lobby) that is trying to spread a rumor in a population by using the known model proposed by Serge Galam. In particular, lobbies are modeled as subgroups of individuals who strategically choose their seating in the social space in order to protect their opinions and influence others. We consider different social gatherings and simulate, using finite Markovian chains, opinion dynamics by comparing situations with a lobby to those without a lobby. Our results show how the lobby can influence opinion dynamics in terms of the prevailing opinion and the mean time to reach unanimity. The approach that we take overcomes some of the problems that behavioral economics and psychology have recently struggled with in terms of replicability. This approach is related to the methodological revolution that is slowly changing the dominant perspective in psychology.

Solving systems of algebraic equations over finite commutative rings and applications

AbstractSeveral problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

Wave propagation in finite discrete chains unravelled by virtual measurement of dispersion properties

AbstractTravelling waves in circuit chains are studied to measure continuous dispersion. A lock‐in frequency meter (LIF) is suitable for precisely determining k for each set of waves in finite alternate LC chains, where LIF has been proven to be more accurate than the fast Fourier transform. In addition to the –k measurement, the wave impedance spectrum of the travelling wave can be measured simultaneously, for investigating the dispersion and splitting of pulse propagation. The measured dispersion is validated to be consistent with the derived theoretical equations. The result provides an independent way to precisely obtain dynamical system properties for chains composed of non‐ideal components, such as resistors for researching non‐Hermitian behaviour under dissipation. Systematical mapping of relative deviation dependence of wave dispersion measurement with LIF on different chain length and component variation is studied, indicating boundaries of 1%, 0.1%, and 0.01% precision for guidance of experiments.

Synthesis and Crystallographic Characterization of Heteroleptic Ir(III) Complexes Containing the N-oxide Functional Group and Crystallographic Characterization of Ir(III) N-oxide Precursors

The N-oxide functional group has been exploited for synthetic strategies and drug design, and it has been utilized in imaging agents. Herein, we present rare examples of neutral heteroleptic cyclometallated Ir(III) compounds that contain an uncoordinated N-oxide functional group. These species, along with others described within, were verified by NMR, EA, HRMS, and single-crystal X-ray analysis. N-oxide-containing Ir(III) species were prepared selectively in high yields > 66% from chloro-bridged Ir(III) dimers with Acipimox, a picolinate-type ligand containing the N-oxide functional group. Non-N-oxide analogs were synthesized in a similar fashion (yields > 77%). Electrochemical comparison (cyclic voltammetry) indicates that the presence of an N-oxide functional group anodically shifts the reduction potential, suggesting that the N-oxide is acting as an electron-withdrawing group in these species. Crystallographic studies were pursued to examine the coordination behavior of these N-oxides compared to their non-oxidized congeners. The Ir(III) complexes with Acipimox indeed leave the N-oxide uncoordinated and exposed on the complexes. The uncoordinated N-oxide group is influential in directing the packing structures of these complexes directly through C-H···O and O···π interactions at the N-oxide. The crystallographic characterization of cationic Ir(III) compounds with uncoordinated nitrogen atoms is also presented. The C-H···N interactions between these complexes form a variety of dimers, finite chains, and continuous chains. Future work will focus on functionalizing the cationic Ir(III) species into their corresponding N-oxide derivatives and rigorously characterizing how the N-oxide functional group impacts the optical properties of transition metal compounds in both cationic and neutral complexes.

Abelian and consta-Abelian polyadic codes over affine algebras with a finite commutative chain coefficient ring

This paper studies Abelian and consta-Abelian polyadic codes over rings defined as affine algebras over chain rings. For this purpose, we use the classical construction via splittings and multipliers of the underlying Abelian group. We also derive some results on the structure of the associated polyadic codes and the number of codes under these conditions.

Algebraic Properties of the Semigroup of Partial Isometries of a Finite Chain

This work analyses the algebraic properties of the sub-semi group of partial Isometries (〖DP〗_n) and of order preserving partial isometries (〖ODP〗_n) of a finite chain X_n={1,2,…n}, with a symmetric inverse semigroup I_n defined on it. It also investigates the subsemigroups 〖DP〗_n and 〖ODP〗_n for cycle structure and shows that 〖ODP〗_n is a 0 -E- unitary inverse semigroup.

Reliability analysis for a generalized sparse connection multi-state consecutive-k-out-of-n linear system

The utilization of binary states is a common practice in reliability analysis. However, in complex systems such as aviation, spaceflight, and watercraft, the prevalence of multi-state systems and units is more pronounced. Consequently, extensive research has been undertaken to explore reliability analysis with a focus on multi-state consecutive-k models across various disciplines. Notably, the concept of sparse connection has received considerable attention due to its relevance in practical applications like wireless communication, cloud computing networks, and oil pipeline systems. This paper aims to propose a generalized multi-state consecutive-k: G system model that incorporates sparse connection. Both the system and the units are allowed to be in one of the (M+1) possible states, wherein the order numbers 0,1,2,…,M indicate the performance measurement of the system or the units, i.e., 0 denotes the worst performance state while M denotes the best performance state. We classify the model into increasing, decreasing and non-monotonic consecutive-k systems corresponding to different mission requirements in practical applications. This paper utilizes the finite Markov chain imbedding (FMCI) method to derive explicit system state distributions for three types of multi-state consecutive-k-out-of-n: G systems. Subsequently, numerical examples are provided to demonstrate the proposed models and the corresponding results.

Dynamical quantum phase transitions from random matrix theory

We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transition persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.

Transient traveling breather response of strongly anharmonic array of self-sustained oscillators: Analytical study.

In the present study, we analyze the transient response of a locally excited chain of strongly anharmonic self-sustained oscillators. This discrete system under consideration models the dynamics of genuinely nonlinear, aeroelastic metamaterial. We particularly focus on the transient evolution of the traveling dissipative breathers, forming in locally excited, finite chains of self-sustained oscillators. The genuinely anharmonic nature of the system under consideration turns the asymptotic analysis of the transient regimes arising in this type of model into a highly challenging task. In the present study, we formulate a special analytical approach which allows for a simple, explicit, and fairly accurate analytical description of the amplitude evolution of the breather core towards the steady state as well as its instantaneous position.

Determinants of tridiagonal matrices over some commutative finite chain rings

Abstract Diagonal matrices and their generalization in terms of tridiagonal matrices have been of interest due to their nice algebraic properties and wide applications. In this article, the determinants of tridiagonal matrices over a finite field F q {{\mathbb{F}}}_{q} and a commutative finite chain ring R R are studied. The main focus is the enumeration of tridiagonal matrices with prescribed determinant. The number of tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and the number of non-singular tridiagonal matrices with prescribed determinant over R R are completely determined. For singular tridiagonal matrices with prescribed determinant over R R , bounds on the number of such matrices with prescribed determinant are given. Subsequently, the number of some special tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and R R is presented.

Reinforcement learning strategies using Monte-Carlo to solve the blackjack problem

Blackjack is a classic casino game in which the player attempts to outsmart the dealer by drawing a combination of cards with face values that add up to just under or equal to 21 but are more incredible than the hand of the dealer he manages to come up with. This study considers a simplified variation of blackjack, which has a dealer and plays no active role after the first two draws. A different game regime will be modeled for everyone to ten multiples of the conventional 52-card deck. Irrespective of the number of standard decks utilized, the game is played as a randomized discrete-time process. For determining the optimum course of action in terms of policy, we teach an agent-a decision maker-to optimize across the decision space of the game, considering the procedure as a finite Markov decision chain. To choose the most effective course of action, we mainly research Monte Carlo-based reinforcement learning approaches and compare them with q-learning, dynamic programming, and temporal difference. The performance of the distinct model-free policy iteration techniques is presented in this study, framing the game as a reinforcement learning problem.

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

Arrowhead matrices have attracted attention due to their rich algebraic structures and numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The number of n × n arrowhead matrices over Fq of a fixed determinant a is determined for all positiveintegers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are given. Finally, some open problems are presented.

MDS and MHDR Cyclic Codes over Finite Chain Rings

This work establishes a unique set of generators for a cyclic code over a finite chain ring. Towards this, we first determine the minimal spanning set and rank of the code. Furthermore, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code are obtained. Finally, to support our results, some examples of optimal cyclic codes are presented.