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We consider the standard five-point finite difference method for solving the Poisson equation with the Dirichlet boundary condition. Its associated matrix is a typical ill-conditioned matrix whose size of the condition number is as big as . Among ILU, SGS, modified ILU (MILU) and other ILU-type preconditioners, Gustafson shows that only MILU achieves an enhancement of the condition number in different order as . His seminal work, however, is not for the MILU but for a perturbed version of MILU and he observes that without the perurbation, it seems to reach the same result in practice. In this work, we give a simple proof of Gustafsson's conjecture on the unnecessity of perturbation in case of Poisson equation on rectangular domains. Using the Cuthill-Mckee ordering, we simplify the recursive equation in two dimensional grid nodes into a recursive one in the level that is one-dimensional. Due to the simplification, our proof is easy to follow and very short.

Partial functions are abundant in mathematics and program specifications. Unfortunately, their importance has been mostly ignored in automated theorem proving. In this paper, we develop a theorem proving strategy for Partial Classical Logic (PCL). Proof search takes place in Kleene Logic. We show that PCL theories can be translated into equivalent sets of formulas in Kleene logic. For proof search we use a three-valued adaptation of geometric resolution. We prove that the calculus is sound and complete.

This technical note presents a useful methodology for studying how the variance of hydraulic and/or reactive attributes of an aquifer are linked to the multiscaled and hierarchical sedimentary architecture of the aquifer. A new recursive equation is derived which quantitatively describes how the variance is related to sedimentary facies defined at all scales across an entire stratal hierarchy. As compared to prior published equations that emphasize differences in means among facies populations within a hierarchical level, it emphasizes differences across levels. Because of the hierarchical relationships among the terms of the equation, we find it to be useful for conducting a holistic analysis of the relative contributions to the variance arising from all facies types defined across all scales. The methodology is demonstrated using appropriate field data, and is shown to be useful in defining parsimonious classification systems.

We study the problem of stationarity and ergodicity for autoregressive multinomial logistic time series models which possibly include a latent process and are defined by a GARCH-type recursive equation. We improve considerably upon the existing results related to stationarity and ergodicity conditions of such models. Proofs are based on theory developed for chains with complete connections. This approach is based on a useful coupling technique which is utilized for studying ergodicity of more general finite-state stochastic processes. Such processes generalize finite-state Markov chains by assuming infinite order models of past values. For finite order Markov chains, we also discuss ergodicity properties when some strongly exogenous covariates are considered in the dynamics of the process.

As the book’s title implies, recursive is a key word in hyperspectral processing, which means that the data processing involved in hyperspectral imagery can be implemented by recurrence relations. One way to realize a recurrence relation is to use a recursive equation that specifies how a recurrence relation takes place. One of most significant advantages that can be gained from recursive processing is that previously processed information is stored in recurrence relations so that there is no need to reprocess all the data that had already been processed. As a consequence of using recurrence relations, data processing need not revisit all the past data sample vectors; rather, it focuses on incoming data sample vectors. Accordingly, all of the data processing can be updated by including only the information, referred to as innovations information, that is contained in the incoming new data sample vectors but not in the already-processed past information. With this unique feature data processing can be implemented in real time via recurrence relations specified by recursive equations. This chapter is devoted to exploring the concept of recursive processing, particularly to the concept of discrete-time Kalman filtering, which has been widely applied in statistical signal processing to the design and development of hyperspectral processing algorithms, presented in subsequent chapters in Parts II-V.

We study the relationships between interest and inflation rates using a recursive equation approach that takes into account both Fisher and Wicksell effects. Extending previous work, a state space representation is used to estimate time-varying ex post Fisher and Wicksell equation effects. We subsequently recover ex ante interest and inflation rate series. Using these ex ante rate series, we estimate an ex ante Fisher equation, including both time-varying intercept estimates of the ex ante real interest rates and time-varying Fisher coefficients. Our results for the U.S. and three other countries support the Fisher propositions after taking into account Wicksell effects.

We introduce new variants of Kleene star and omega iteration for the case where the iterated operator is neither associative nor has a neutral element. The associated repetition algebras are used to give closed semantic expressions for the Until and While operators of the temporal logic \(\mathsf {CTL}^*\) and its sublogics \(\mathsf {CTL}\) and \(\mathsf {LTL}\). Moreover, the relation between the semantics of these logics can be expressed by homomorphisms between repetition algebras, which is a more systematic and compact approach than the ones taken in earlier papers.

In previous chapters, recursive hyperspectral band processing (RHBP) was developed for subpxiel detection, RHBP of constrained energy minimization in Chap. 13, RHBP of anomaly detection in Chap. 14, unsupervised target detection, RHBP of an automatic target generation process in Chap. 15, and mixed pixel detection/classification, RHBP of orthogonal subspace projection in Chap. 16. This chapter concludes the last piece of RHBP’s applications to the well-known technique linear spectral mixture analysis (LSMA). It develops a new approach, called RHBP of LSMA (RHBP-LSMA), which can process data unmixing according to the band-sequential (BSQ) format. This new concept is different from band selection (BS), which must select bands from a fully collected band set according to a band optimization criterion. There are several advantages of using RHBP-LSMA over BS. In particular, it allows users to perform LSMA using available bands without waiting for a complete collection of full bands. In doing so, an innovation information update recursive equation is further derived and can process LSMA progressively as well as recursively as its band processing taking place. The resulting LSMA becomes RHBP-LSMA. To be more specific, RHBP-LSMA can be carried out by updating LSMA results recursively band by band in the same way that a Kalman filter does in updating data information in a recursive fashion. Consequently, RHBP-LSMA can provide progressive band-varying profiles of LSMA results so that significant bands can also be detected and identified by inter-band changes in LSMA results without BS.

We develop a coalgebraic theory of Kleene algebra with tests (\(\mathsf {KAT}\)) along the lines of Rutten (1998) for Kleene algebra (\(\mathsf {KA}\)) and Chen and Pucella (Electron Notes Theor Comput Sci 82(1), 2003) for a limited version of \(\mathsf {KAT}\), resolving some technical issues raised by Chen and Pucella. Our treatment includes a simple definition of the Brzozowski derivative for \(\mathsf {KAT}\) expressions and an automata-theoretic interpretation involving automata on guarded strings. We also give a complexity analysis, showing that an efficient implementation of coinductive equivalence proofs in this setting is tantamount to a standard automata-theoretic construction. It follows that coinductive equivalence proofs can be generated automatically in PSPACE. This matches the bound of Worthington (2008) for the automatic generation of equational proofs in \(\mathsf {KAT}\).

Kleene algebra with tests (KAT) was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete actions guarded by Boolean tests.

The paper presents a dynamic model of modal split in a multimodal freight transport system, which supposes that the evolution over time of transport demand is accompanied by a corresponding evolution of transport modes, and that users react with delay to cost variations. Starting with these hypotheses, and following the paradigm of random utility, a recursive equation is obtained, whose iterated application furnishes the sequence of the demand fractions on the various transport modes in the successive epochs of the time period during which the evolution of the transport system is studied and enables forecasting the future modal split evolution.

In this paper we generalize the theorem that previously obtained by the authors for a wider class of linear orders using X-limitwise monotonic functions relative to the Kleene’s Ordinal Notation System.

We present examples from a reference implementation of variants of Kleene algebras and Tarski's relation algebras in the theorem proving environment Isabelle/HOL. For Kleene algebras we show how mo ...

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving transient heat conduction problems with heat sources. First, by expanding physical quantities at discrete time intervals, the recursive formulations of the governing equation are derived. Then, the recursive equation is solved by the RIBEM. Finally, a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time-step size. Numerical results show that the present approach can obtain satisfactory performance.

A new analytical approach based on dynamic programming method for the solution of cubic objective function with equality and inequality constraints is presented in this paper. A generalized recursive equation is derived to solve the problem. The proposed approach is non-iterative and directly gives the solution without any initial assumption. This method is implemented for economic power dispatch problem to minimize fuel cost and economic dispatch problem of chiller plant for saving energy. The superiority of this method is demonstrated by comparing the experimental results of the proposed method with lambda logic based algorithm and with gradient search method reported in the literature.

The FE-closure operator on set P N of functions of countable-valued logic is considered. It is proved that any general recursive operator (defined in the Herbrand-Godel formalism) can be realized in the language of FE-closure. It is established that the class of relations definable in the language of FE-closure coincides with class Σ 1 1 in the Kleene analytic hierarchy. It is shown that the membership in an FE-closed class of any nontrivial homogeneous function implies that the entire set of homogeneous functions is contained in this class.

We consider a class of discrete-time two person zero-sum Markov games with Borel state and action spaces, and possibly unbounded payoffs. The game evolves according to the recursive equation x n + 1 = F ( x n , a n , b n , ξ n ) , n = 0 , 1 , … , where the disturbance process { ξ n } is formed by independent and identically distributed R k -valued random vectors, which are observable but their common density ρ ∗ is unknown for both players. Combining suitable methods of statistical estimation of ρ ∗ with optimization procedures, we construct a pair of average optimal strategies.

Summary Reliable survival parameter estimation is an essential part of building predictive models for microbial survival. It has been demonstrated that these parameters can be accurately identified using a one-step regression approach that fits a survival model to multiple dynamic data sets at once. However, the existing methods are not quite user-friendly because their application requires relatively high computer skills. In this study, a recursive equation for the Weibull model was used to construct microbial survival curves under dynamic conditions. Based on this, a procedure was developed to estimate survival parameters by fitting the equation to dynamic survival data sets using the built-in functions and Solver of Microsoft Excel. The results showed that the method provided an easy and accurate way for estimating microbial survival parameters.

In reference to our earlier work [1], Pierre McKenzie and Sambuddha Roy pointed out that the proofs of statements (b) and (c) in Theorem 7.3 are buggy. The main flaw is that the identity e of the group F may not be the identity of the monoid, and so the claim that w ∈ (AF,r )∗ ⇐⇒w ∈ Test does not work. Herewith, we show: • With a slight change to Definition 7.1, the statement of Theorem 7.3 holds unchanged. In our opinion, this is the most interesting way to correct the error in the original paper. We present a complete proof below. For completeness, we also mention another way to correct the error: • Leaving Definition 7.1 unchanged, a weaker version of Theorem 7.3 holds (with only minor adjustments to the proof given in the paper).

This paper is a sequel to Kageyama et al. [1], in which a Markov-type hybrid process has been constructed and the corresponding discounted total reward has been characterized by the recursive equation. The objective of this paper is to formulate a hybrid decision process and to give the existence and characterization of optimal policies.

Kleene algebra with tests (KAT) is an equational system that combines Kleene and Boolean algebras. One can model basic programming constructs and assertions in KAT, which allowed for its application in compiler optimization, program transformation and dataflow analysis. To provide semantics for KAT expressions, Kozen first introduced emph{automata on guarded strings}, showing that the regular sets of guarded strings plays the same role in KAT as regular languages play in Kleene algebra. Recently, Kozen described an elegant algorithm, based on ``derivatives'', to construct a deterministic automaton that accepts the guarded strings denoted by a KAT expression. This algorithm generalizes Brzozowski's algorithm for regular expressions and inherits its inefficiency arising from the explicit computation of derivatives. In the context of classical regular expressions, many efficient algorithms to compile expressions to automata have been proposed. One of those algorithms was devised by Berry and Sethi in the 80's (we shall refer to it as Berry-Sethi construction/algorithm, but in the literature it is also referred to as position or Glushkov automata algorithm). In this paper, we show how the Berry-Sethi algorithm can be used to compile a $KAT$ expression to an automaton on guarded strings. Moreover, we propose a new automata model for KAT expressions and adapt the construction of Berry and Sethi to this new model.

A finite-horizon, periodic-review inventory model with inflow forecasting updates following the Martingale Model of Forecast Evolution (MMFE) in multiresevoirs is considered. This model introduces a new method of determining an operating policy in which the policy is based on the Dynamic Programming (DP) model with a physical equation and a recursive equation. Some important constraints are described for the multireservoirs operation. We propose two methods to adjust the decision variable during the optimization. The adjustment policy is essential to improve the profit and decrease the shortage in multireservoirs management. Finally, to assess the effectiveness of the policies, the model is compared with other models and is applied to the Chinese South-North Water Diversion project.

This paper studies endomorphism monoids of Kleene algebras. The main result is that these algebras form an almost universal variety k, from which it follows that for a given monoid M there is a proper class of non-isomorphic Kleene algebras with endomorphism monoid M+ (where M+ denotes the extension of M by a single element that is a right zero in M+). Kleene algebras form a subvariety of de Morgan algebras containing Boolean algebras. Previously it has been shown the latter are uniquely determined by their endomorphisms, while the former constitute a universal variety, containing, in particular, arbitrarily large finite rigid algebras. Non-trivial algebras in K always have non-trivial endomorphisms (so that universality of K is ruled out) and unlike the situation for de Morgan algebras the size of End(L) for a finite Kleene algebra L necessarily increases as |L| does. The paper concludes with results on endomorphism monoids of algebras in subvarieties of the variety of MS-algebras.

This paper proposes a framework to recommend personal sightseeing route as well as to objectively obtain the tourist's utility to sightseeing plans and sightseeing spots from subjective comparison. The subjective comparison is qualitatively performed using a scale of measurement such as Likert scale, and the mathematical programming problem including this comparison as constraints is introduced. In the case of route planning, traveling and sightseeing times are randomly changed dependent on current traffic and congestion conditions, and hence, Time-Expanded Network (TEN) to represent these traffic conditions in the underlying static network with each discrete time step is introduced. In addition, the network optimization problem is introduced to obtain the personal appropriate sightseeing route. This problem is formulated as a nonlinear and discrete optimization problem, and it is hard to solve it directly and efficiently. Therefore, an efficient algorithm is also developed based on dynamic programming and transformation of the main problem into the recursive equation.

Two fundamental results of classical automata theory are the Kleene theorem and the Buchi-Elgot-Trakhtenbrot theorem. Kleene's theorem states that a language of finite words is definable by a regular expression iff it is accepted by a finite state automaton. Buchi-Elgot-Trakhtenbrot's theorem states that a language of finite words is accepted by a finite-state automaton iff it is definable in the weak monadic second-order logic. Hence, the weak monadic logic and regular expressions are expressively equivalent over finite words. We generalize this to words over arbitrary linear orders.

Cauchy unital tree valuation monoids are introduced as weight structures for weighted tree automata. Rational tree series over this kind of monoids are defined and Kleene’s classical theorem is proved for this setting: a tree series over a Cauchy unital tree valuation monoid is recognizable if and only if it is rational.

International audience; Step-indexing has proven to be a powerful technique for defining logical relations for languages with advanced type systems and models of expressive program logics. In both cases, the model is stratified using natural numbers to solve a recursive equation that has no naive solutions. As a result of this stratification, current models require that each unfolding of the recursive equation – each logical step – must coincide with a concrete reduction step. This tight coupling is problematic for applications where the number of logical steps cannot be statically bounded. In this paper we demonstrate that this tight coupling between logical and concrete steps is artificial and show how to loosen it using transfinite step-indexing. We present a logical relation that supports an arbitrary but finite number of logical steps for each concrete step.

We analyze here different types of fractional differential equations, under the assumption that their fractional order $\nu \in (0,1] $ is random\ with probability density $n(\nu).$ We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process $N(t),t>0.$\ We prove that, for a particular (discrete) choice of $n(\nu)$, it leads to a process with random time, defined as $N(% \widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}(t)),t>0.$ The distribution of the random time argument $\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}(t)$ can be expressed, for any fixed $t$, in terms of convolutions of stable-laws. The new process $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}})$ is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}})$, as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see \cite{Vib}%). In view of the previous results it is natural to consider diffusion-type fractional equations of distributed order. We present here an approach to their solutions in terms of composition of the Brownian motion $B(t),t>0$ with the random time $\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}$. We thus provide an alternative to the constructions presented in Mainardi and Pagnini \cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order case., Comment: 26 pages

In NURBS interpolation, real-time parameter update is an indispensable step which affects not only feedrate fluctuation but also contour error. Using Taylor approximation interpolation method to find the next interpolation point causes a large feedrate fluctuation due to the accumulation and truncation errors. This paper presents a new, simple, and precise NURBS interpolator for CNC systems. The proposed interpolation algorithm does not use Taylor’s expansion, but the recursive equation of the NURBS formula. A simulation study is conducted to demonstrate the advantages of this proposed interpolator compared with those using Taylor’s equation. It is readily seen that this interpolator using the new concept of interpolation for modern CNC systems is simple and precise. The proposed method can be used for interpolating a continuous NURBS curve.

This paper presents a performance evaluation of a NURBS interpolator that uses not Taylor’s expansion method but the recursive characteristics of NURBS. Taylor’s expansion has mostly been used for NURBS interpolation. However, it is very complicated and gives an indispensable truncation error. A fast and precision NURBS interpolator replacing Taylor’s expansion is presented for CNC systems in robots and CNC machine tools. The presented interpolation algorithm uses the recursive equation of the NURBS formula rather than Taylor’s expansion. A simulation study is conducted to demonstrate the advantages of this proposed interpolator compared with those using Taylor’s equation. The feedrate error and calculation time are simulated in the performance evaluation. The recursive method of NURBS interpolation is faster and more accurate than Taylor’s expansion.

The class of regular Kleene lattices is the least strict quasi-variety of De Morgan lattices. In this paper we give necessary and sufficient conditions for two finite regular Kleene lattices to share the same monoid of endomorphisms.

In this paper, we study about the ordered structure of rough sets determined by a quasi order. A characterization theorem for rough sets of an approximation space (U, R) based on a quasi order R is given in Nagarajan and Umadevi (2010). Then using the characterization of rough sets determined by a quasi order, its rough sets system is represented by a new construction. This construction is generalized and abstracted into a new method of constructing Kleene based algebraic structures from dually isomorphic distributive lattices. Then by using different varieties of distributive lattices, we obtain various Kleene based algebraic structures. By this construction, we give various algebraic structures to the rough sets system determined by a quasi order R.

This paper considers equations of the form Conditions on f and which guarantee global asymptotic stability of positive solutions are provided. The results generalize recent work in the literature for equations of the form . Asymptotic periodicity for solutions is also considered.

Yang et al. [J.P. Yang, S.H. Cheng, Q. Wu, Recursive equations for compound distribution with the severity distribution of the mixed type, Science in China Series A 48 (2005) 594–609] investigated a recursive procedure for a kind of compound distributions with the number of claims belonging to ( a , b ) -family and the severity distribution of the mixed type. In this paper, we extend their results by assuming that the claim number belongs to a larger class. As applications, the excess-of-loss reinsurance treaty is discussed and concrete examples are considered in some detail.

In this note we consider several kind of partition functions of one-dimensional models with nearest — neighbor interactions In, n ∈ Z and spin values ±1. We derive systems of recursive equations for each kind of such functions. These systems depend on parameters In, n ∈ Z. Under some conditions on the parameters we describe solutions of the systems of recursive equations.

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form f(bn)=R(f(n),f(n),…,f(b−1)n)+g(n), where b is a positive integer ≥2; R a rational function in b−1 variables and g a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.

The work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus. The resulting algebraic structure is called synchronous Kleene algebra. Models are given in terms of sets of synchronous strings and finite automata accepting synchronous strings. The extension of synchronous Kleene algebra with Boolean tests is presented together with models on sets of guarded synchronous strings and the associated automata on guarded synchronous strings. Completeness w.r.t. the standard interpretations is given for each of the two new formalisms. Decidability follows from completeness. Kleene algebra with synchrony should be included in the class of true concurrency models. In this direction, a comparison with Mazurkiewicz traces is made which yields their incomparability with synchronous Kleene algebras (one cannot simulate the other). On the other hand, we isolate a class of pomsets which captures exactly synchronous Kleene algebras. We present an application to Hoare-like reasoning about parallel programs in the style of synchrony.

Lindley's recursion is an explicit recursive equation that describes the recursive relationship between consecutive waiting times in a single-stage single-server queue. In this paper, we develop explicit recursive representations for multiserver tandem queues with blocking. We demonstrate the application of these recursive representations with simulations of large-scale tandem queueing networks. We compare the computational efficiency of these representations with two other popular discrete-event simulation approaches, namely, event scheduling and process interaction. Experimental results show that these representations are seven (or more) times faster than their counterparts based on the event-scheduling and process-interaction approaches.

The development of world construction projects at this time that established by widespread methods of physical implementation which more excellent and sophisticated. The contractors are competing to obtain optimum results, and reduce losses more lower. The results obtained should not give benefit one person only, the other one should not to lose and suffer. Islam guides human how they work give benefit to their own and others. Many methods, equipment, and expertise deployed by the contractor for their work more optimal, one method for optimizing the physical implementation of the project is Dynamic Programming. Dynamic Programming to solve the problem optimization is not only with one step, but the events that took place linked to each other. Case study in this paper is the application of dynamic programming of implementation method of concrete casting on floor slab of Green House building, it’s belongs to Department of Biology, UIN Maliki Malang with backward induction phase. In modern century as this era has many fast-growing method of implementation that are categorized as advanced, but the problem that the methods actually optimal in terms of cost and time, due to cost and time parameters are very important in the progress of construction projects. The methods is very sophisticated so make project to be done very fast but it is very expensive financing, or the method is still conventional and traditional so that project costs will be reduced as cheaply, but the project need long time to be finished. Where is the decision that one will be expected to demand that we use what method. Mathematical formulas of dynamic programming as recursive equation with change big problem into smaller sub-problems, so the completion of smaller problems will be found solving of the original problem. Optimization process in this case that slab casting which solved in two methods, manually calculations with a recursive formula and use software DS Win 2.0. Optimization result of the two method are compared, and the two method produce similar results. Optimal or not parameter of the two method is optimum time and cost.

Among different ways to obtain the thermodynamic properties of a fluid in the liquid phase, there is the possibility to solve, numerically, a system of two differential equations where density and specific heat capacity are the unknown variables expressed as functions of temperature and pressure. By means of general methods, like Predictor–Corrector or Runge–Kutta, this system can be solved integrating along isothermal paths, once the initial conditions are known along an isobar and the speed of sound values are measured over the p – T region of interest. In this work a new perturbative method, called recursive equation method (REM), based on the analytic recursive determination of the coefficients describing density and specific heat capacity, is proposed. The main advantage of REM is the possibility to also get the value of the uncertainty associated with the solutions, obtained by means of standard methods of error analysis. Another improvement introduced by this algorithm is the possibility to solve the system of equations following arbitrary integration paths. In particular, a comparison between the results of density and heat capacity, obtained by integrating along isentropic and isothermal paths, is presented, both for water and for acetone.

We generalise the designs of the Unifying Theories of Programming (UTP) by defining them as matrices over semirings with ideals. This clarifies the algebraic structure of designs and considerably simplifies reasoning about them, for example, since they form a Kleene and omega algebra and a test semiring. We apply our framework to investigate symmetric linear recursion and its relation to tail-recursion. This substantially involves Kleene and omega algebra as well as additional algebraic formulations of determinacy, invariants, domain, pre-image, convergence and Noetherity. Due to the uncovered algebraic structure of UTP designs, all our general results also directly apply to UTP.

In this paper, we discuss a method of constructing a credibilistic process which is a family of fuzzy variables on credibility space. Applying the extension theorem in credibility theory, the finite or infinite horizon credibilistic process is made up from a family of credibilistic kernels. Also, for the Markov case, convergence theorems are given and credibilistic risk models for reward processes are considered, whose risk is completed by the recursive equation.

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 k − 1) or 6(2 k − 1) for some k ≥ 1. The existence of solutions with such periods is also considered.

In this paper we will show that partially ordered monads contain sufficient structure for modelling monadic topologies, rough sets and Kleene algebras. Convergence represented by extension structures over partially ordered monads includes notions of regularity and compactness. A compactification theory can be developed. Rough sets [Z. Pawlak, Rough sets, Int. J. Computer and Information Sciences 5 (1982) 341356] are modelled in a generalized setting with set functors. Further, we show how partially ordered monads can be used in order to obtain monad based examples of Kleene algebras building upon a wide range of set functors far beyond just strings [S. C. Kleene, Representation of events in nerve nets and finite automata, In: Automata Studies (Eds. C. E. Shannon, J. McCarthy), Princeton University Press, 1956, 3-41] and relations [A. Tarski, On the calculus of relations, J. Symbolic Logic 6 (1941), 65-106].

We derive Schützenberger’s characterisation of the set of recognizable formal power series as a formal corollary from Kleene’s characterisation of the set of regular languages.

A short recursive procedure for calculating the effectiveness factor for enzymes immobilized in porous spherical particles is presented. The method is mathematically simple and very precise; it is valid for reversible Michaelis–Menten kinetics including, as particular situations, simple Michaelis–Menten and product competitive inhibition kinetics. The procedure is a modification of the two-parameter model previously published by the authors. The definition of an auxiliary dimensionless concentration leads to a recursive equation which simplifies the resolution of the model by means of elementary numerical methods. The solution algorithm has been transformed into a computer program, whose source code is appended. The exact values of the effectiveness factors for zero and first order kinetics, as a function of the Thiele module, were compared with those obtained with the numerical procedure proposed in this work. The good agreement between both results demonstrates the validity of the method.

Consider a discrete time risk model U n = ( U n - 1 + X n ) ( 1 + I n ) - Y n , n = 1 , 2 , … , where U 0 ≔ M > 0 is the initial reserve of an insurance company, X n the total amount of premiums, Y n the total amount of claims, I n the interest rate and U n the reserve at time n. The time of ruin is denoted by τ M ≔ inf { n ⩾ 1 ; U n 0 } . In this paper, the recursive equations for finite time ruin probabilities and bounds for ultimate ruin probabilities are provided. When { Y n } are heavy tailed, we also give reasons for the asymptotic estimate P ( τ M ∞ ) ≈ M - λ , where λ is a specific positive parameter. A more general risk model from Nyrhinen [1999. On the ruin probabilities in a general economic environment, Stachastic Process. Appl. 83, 319–330] is also discussed, and similar asymptotic estimate for ultimate ruin probabilities is given.

The proofs of major results of Computability Theory like Rice, Rice-Shapiro or Kleene's fixed point theorem hidemore information of what is usually expressed in theirrespective statements. We make this information explicit, allowing to state stronger, complexity theoretic-versions of all these theorems. In particular, we replace the notion of extensional set of indices of programs, by a set of indices of programs having not only the same extensional behavior but also similar complexity (Complexity Clique). We prove, under very weak complexity assumptions, that any recursive Complexity Clique is trivial, and any r.e. Complexity Clique is an extensional set (and thus satisfies Rice-Shapiro conditions). This allows, for instance, to use Rice's argument to prove that the property of having polynomial complexity is not decidable, and to use Rice-Shapiro to conclude that it is not even semi-decidable. We conclude the paper with a discussion of "complexity-theoretic" versions of Kleene's Fixed Point Theorem.

The paper deals with active fault detection and dual control in the multiple model framework. A monitored and controlled system is described by a discrete-time linear stochastic model at each step of a finite time horizon. The model belongs to an a priori given set of models, and known transient probabilities describe switching between the models. The goal is to design an active detector and controller that processes all available information and generates decisions and inputs. The decisions inform whether a fault has occurred in the system, and the inputs should simultaneously control and excite the system. As the control is in conflict with the excitation, the dual control problem arises. It is shown that both active fault detection and dual control can be solved using Bellman's principle of optimality, and a corresponding backward recursive equation is derived. The approximative solution of the backward recursive equation is discussed, and an algorithm based on an application of rolling horizon and nonlinear filtering techniques is presented. The presented approach is illustrated in a simple numerical example.