Finite Chain Research Articles

The lifetime distribution of finite fission chain group around the prompt criticality

In order to describe the behavior of finite fission chains in a fission system initiated by a δ neutron source, we present the conception of the finite fission chains group (FFCG). The group is the collectivity of the fission chains initiated by plenty of neutrons at the same time. Based on the result of the solving generating function, we deduced the lifetime distribution formula of FFCG based on the point kinetics model. Finally, the formula is validated by the experiment with a single pulse neutron source in CFBR-Ⅱ.

Additively generated discrete quasi-overlap functions

In order to meet the needs of solving qualitative information aggregation problems, the theoretical research on usual aggregation operators on finite chains has never been interrupted. As part of the previous research category, this study focuses on additive generators of discrete quasi-overlap functions recently proposed by the author. The main results are: (1) it introduces the concepts of additive generators, 12-min-additive generators and concave additive generators of discrete quasi-overlap functions; (2) it provides several basic properties of additively generated, 12-min-additively generated and concave additively generated discrete quasi-overlap functions; (3) it investigates migrativity property of additively generated quasi-overlap functions.

Scaling limits of loop-erased Markov chains on resistance spaces via a partial loop-erasing procedure

We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-erasure of the paths of the diffusion process. Finally, we show that the scaling limit of the loop-erased random walks on the Sierpiński carpet graphs exists, and is equivalent to the loop-erased random paths on the Sierpińksi carpet.

Edge state behavior in a Su–Schrieffer–Heeger like model with periodically modulated hopping

Su–Schrieffer–Heeger (SSH) model is one of the simplest models to show topological end/edge states and the existence of Majorana fermions. Here we consider a SSH like model both in one and two dimensions where a nearest neighbor hopping features spatially periodic modulations. In the 1D chain, we witness appearance of new in-gap end states apart from a pair of Majorana zero modes (MZMs) when the hopping periodicity go beyond two lattice spacings. The pair of MZMs, that appear in the topological regime, characterize the end modes each existing in either end of the chain. These, however, crossover to both-end end modes for small hopping modulation strength in a finite chain. Contrarily in a 2D SSH model with symmetric hopping that we consider, both non-zero and zero energy topological states appear in a finite square lattice even with a simple staggered hopping, though the zero energy modes disappear in a ribbon configuration. Apart from edge modes, the 2D system also features corner modes as well as modes with satellite peaks distributed non-randomly within the lattice. In both the dimensions, an increase in the periodicity of hopping modulation causes the zero energy Majorana modes to become available for either sign of the modulation. But interestingly with different periodicity for hopping modulations in the two directions, the zero energy modes in a 2D model become rarer and does not appear for all strength and sign of the modulation.

Reliability analysis of multi-state balanced systems with standby components switching mechanism

An increasing attention has been paid to the reliability analysis of balanced system. In previous studies, there were three rebalancing strategies for balanced systems: forcing down the unbalanced pair of subsystems, restarting components that have been shut down, and dynamically adjusting the state of the working components. In this paper, a new rebalancing strategy is proposed by switching the standby components for a multi-state balanced system. The system consists of m subsystems, each with one working component and n-1 standby components. The system is balanced when the working components do not fail and the system's balanced degree doesn't exceed a threshold. Two cases of system balance conditions are considered: one is based on component states and the other is based on symmetric component states. If the system is imbalanced or fails, the system needs to adjust the state of the working component by switching switches to let the system operate normally. The switching rules for each case are given separately. In this paper, the finite Markov chain imbedding approach (FMCIA) is used to derive the system reliability. Finally, numerical examples and sensitivity analysis are given.

Evolution of discrete symmetries.

Symmetries are known to dictate important physical properties and can be used as a design principle in particular in wave physics, including wave structures and the resulting propagation dynamics. Local symmetries, in the sense of a symmetry that holds only in a finite domain of space, can be either the result of a self-organization process or a structural ingredient into a synthetically prepared physical system. Applying local symmetry operations to extend a given finite chain we show that the resulting one-dimensional lattice consists of a transient followed by a subsequent periodic behavior. Due to the fact that, by construction, the implanted local symmetries strongly overlap the resulting lattice possesses a dense skeleton of such symmetries. We proof this behavior on the basis of a class of local symmetry operations allowing us to conclude upon the "asymptotic" properties such as the final period, decomposition of the unit cell and the length and appearance of the transient. As an example case, we explore the corresponding tight-binding Hamiltonians. Their energy eigenvalue spectra and eigenstates are analyzed in some detail, showing in particular the strong variability of the localization properties of the eigenstates due to the presence of a plethora of local symmetries.

A mixed singular/switching control problem with terminal cost for modulated diffusion processes

In this paper, we study the regularity of the value function associated with a stochastic control problem involving two controls that act simultaneously on a modulated multidimensional diffusion process. The first is a switching control modelling a random clock. Whenever the random clock rings, the generator matrix is replaced by another, leading to a different dynamic for the finite state Markov chain of the modulated diffusion process. The second is a singular stochastic control that is executed on the process within each regime.

Joint signatures of two or more semi-coherent systems with shared components

In this paper, for two or more semi-coherent systems with shared components, three signature measures–joint signature, joint survival signature and joint cumulative signature–are presented and their relationship to joint reliability and joint unreliability are also discussed. This relationship provides a joint (survival/cumulative) signature computational method which transforms related computation to the computation of joint reliability/unreliability of the systems. Based on that, further discussions on computation of the joint reliability are presented for two different types of consecutive-k systems, namely, linear (n, f, k): F systems and linear <n, f, k>: F systems, by using finite Markov chain imbedding approach (FMCIA) in a new and novel way. Besides, for modular systems that can be regarded as a series/parallel connection of two groups of subsystems with shared components, their joint (survival/cumulative) signature is derived based on the joint survival/cumulative signatures of the subsystems, and this provides another computational method for joint signatures. Related results are applied to calculate joint signatures of a series/parallel connection of (n, f, k): F subsystems and <n, f, k>: F subsystems by applying the two computational methods together. Finally, some potential applications and generalizations are mentioned.

Difference operators on lattices

A differential operator of weight [Formula: see text] is the algebraic abstraction of the difference quotient [Formula: see text], including both the derivation as [Formula: see text] approaches to [Formula: see text] and the difference operator when [Formula: see text]. Correspondingly, differential algebra of weight [Formula: see text] extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight [Formula: see text] on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains.

Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain

Let [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in the second and third sections we give a necessary and sufficient condition for an element in CPn to be regular and characterize all the Green’s relations in the semigroup CPn, respectively. In the subsequent fourth and fifth sections we obtain analogous results for the subsemigroups of order preserving and/or order reversing contractions, and that of full contractions, respectively. Finally, in the last section, we characterize all the maximal subgroups of these semigroups.

On Hamming distance distributions of repeated-root constacyclic codes of length 3ps over [formula omitted

Let p≠3 be any prime. The algebraic structures of all λ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm are provided by classifying all ideals of the local ring R[x]〈x3ps−λ〉. Using these structures, in this paper, the exact values of Hamming distances of all such λ-constacyclic codes of length 3ps are established. As an application, we identify all maximal distance separable λ-constacyclic codes of length 3ps.

Reliability analyses of linear two-dimensional consecutive k-type systems

AbstractIn this paper, several linear two-dimensional consecutive k-type systems are studied, which include the linear connected-(k, r)-out-of- $(m,n)\colon\! F$ system and the linear l-connected-(k, r)-out-of- $(m,n)\colon\! F$ system without/with overlapping. Reliabilities of these systems are studied via the finite Markov chain imbedding approach (FMCIA) in a novel way. Some numerical examples are provided to illustrate the theoretical results established here and also to demonstrate the efficiency of the developed method. Finally, some possible applications and generalizations of the developed results are pointed out.

A Markov chain approximation of switched Fokker–Planck equations for a model of on–off intermittency in the postural control during quiet standing

The intermittent on–off switching of feedback control is considered as a major mechanism of postural stabilization during human quiet standing, which can be modeled by switched-type hybrid stochastic delay differential equations with unstable subsystems. Dynamics of the model can also be described by the corresponding switched-type Fokker–Planck (FP) equations. Here, we develop a comprehensive numerical recipe to simulate switched-type FP equations in the case that the probability current is conserved at the switching boundary, as is the case for noise-free models exhibiting C0-continuity for solutions at the boundary, but in a way extendable to cases with discontinuous jumps. Specifically, the FP equations are approximated by a finite state Markov chain model using the finite element method. Then, dynamics of the Markov chain model, including time evolution of probability density function (PDF), stationary PDF, and power spectrum of postural sway are analyzed. We further investigate how the stationary PDF alters as values of important parameters of the model change. Dynamics of the Markov chain model are compared with Monte Carlo-based dynamics of the model, by which the developed numerical recipe is validated. The obtained Markov chain model forms a basis of our future investigations of the intermittent postural control as a Markov decision process.

British Put Option On Stocks Under Regime-Switching Model

In a plain vanilla option, its holder is given the right, but not the obligation, to buy or sell the underlying stock at a specified price (strike price) at a predetermined date. If the exercise date is at maturity, the option is called a European; if the option is exercised anytime prior to maturity, it is called an American. In a British option, the holder can enjoy the early exercise feature of American option whereupon his payoff is the ‘best prediction’ of the European payoff given all the information up to exercise date under the hypothesis that the true drift of the stock equals a specified contract drift. In this paper, in contrast to the constant interest rate and constant volatility assumptions, we consider the British option by assuming that the economic state of the world is described by a finite state continuous-time Markov chain. Also, we provide a solution to a free boundary problem by using PDE arguments. However, closed form expression for the arbitrage-free price are not available in our setting.

On Automorphisms of Chain Rings

Suppose R is a finite chain ring with invaraints p,n,r,k,k′,m. Suppose G is also the subset of all φ∈ Aut(R), the automorphism group of R, such that φ(πk′)=πk′, where π is a generator of the maximal ideal of R. It was found that G is a group that is, in some sense, the set of all symmetries of {πk′}. The main purpose of this article is to describe the structure of G. The subgroup G helps us understand the structure of Aut(R) in the general case which in turn provides immediate results in classifying chain rings.

Expanders on Matrices over a Finite Chain Ring, II

Expanders on Matrices over a Finite Chain Ring, II

A Comparison of Sets of Recognizable Weighted Tree Languages Over Specific Sets of Bounded Lattices

We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.

Beyond neurons and spikes: cognon, the hierarchical dynamical unit of thought

AbstractFrom the dynamical point of view, most cognitive phenomena are hierarchical, transient and sequential. Such cognitive spatio-temporal processes can be represented by a set of sequential metastable dynamical states together with their associated transitions: The state is quasi-stationary close to one metastable state before a rapid transition to another state. Hence, we postulate that metastable states are the central players in cognitive information processing. Based on the analogy of quasiparticles as elementary units in physics, we introduce here the quantum of cognitive information dynamics, which we term “cognon”. A cognon, or dynamical unit of thought, is represented by a robust finite chain of metastable neural states. Cognons can be organized at multiple hierarchical levels and coordinate complex cognitive information representations. Since a cognon is an abstract conceptualization, we link this abstraction to brain sequential dynamics that can be measured using common modalities and argue that cognons and brain rhythms form binding spatiotemporal complexes to keep simultaneous dynamical information which relate the ‘what’, ‘where’ and ‘when’.

Endomorphisms of semigroups of monotone transformations

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

A Sequential Test of Traffic Intensity for the M/M/1 Queueing System

This paper deals with testing traffic intensity, which is the most important parameter of the queue system. The Wald-type sequential probability ratio test (SPRT) is performed to determine traffic intensity by taking advantage of the fact that the total number of customers arriving up to the kth service period, including the kth service period, has a negative binomial distribution. Acceptance and rejection limits obtained from the SPRT are arranged as a multiple sampling plan. This plan is applied with a finite Markov chain. Then, the Operating Characteristic (OC) function and the Average Sample Number (ASN) are obtained precisely. Obtaining the standard errors of the sample number compared to the studies on SPRT applications is the novelty brought by this paper. The sequential analysis method based on the finite Markov chain is also applied to a numerical sample.