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This paper axiomatizes static and dynamic quantile preferences. Static quantile preferences specify that a prospect should be preferred if it has a higher $$\tau $$ -quantile, for some $$\tau \in (0,1)$$ , while its dynamic counterpart extends this to take into account a sequence of decisions and information disclosure. An important motivation for the axiomatization that leads to this preference is the separation of tastes and beliefs. We first axiomatize quantile preferences for the static case with finite state space and then extend the axioms to the dynamic context. The dynamic preferences induce an additively separable quantile model with standard discounting, that is, the recursive equation is characterized by the sum of the current period utility function and the discounted value of the certainty equivalent, which is a quantile function. These preferences are time consistent and have a simple quantile recursive representation, which gives the model the analytical tractability needed in several fields in financial and economic applications. Finally, we study the notion of risk attitude in both the static and recursive quantile models. In quantile models, the risk attitude is completely captured by the quantile $$\tau $$ , a single-dimensional parameter. This is simpler than in expected utility models, where in general the risk attitude is determined by a function.

The radiative–conductive transfer equation in the $$S_N$$ approximation for spherical geometry is solved using a modified decomposition method. The focus of this work is to show how to distribute the source terms in the recursive equation system in order to guarantee arithmetic stability and thus numerical convergence of the obtained solution, guided by a condition number criterion. Some examples are compared with results from literature and parameter combinations are analyzed, for which the condition number analysis indicates convergence of the solutions obtained by the recursive scheme.

The dimensionality problem is posing an enormous challenge for cascade hydropower reservoirs operation because the memory usage and execution time grow exponentially with the expansion of system scale. To effectively address this problem, this paper develops a novel Latin dynamic programming algorithm for dimensionality reduction in hydropower reservoir operation problem, where the Latin hypercube sampling method is firstly adopted to produce a subset of discrete state variables at each stage, and then the standard dynamic programming recursive equation is used to search for a modified trajectory around the newly-generated solutions, while the iterative search strategy is used to gradually enhance the solution quality. The results in a real-world hydropower system of China demonstrate that compared with the standard dynamic programming method, the execution efficiency of the presented method is significantly improved while the power generation is well maintained in different scenarios. Hence, the novelty of the paper is to provide an effective dimensionality reduction tool for solving the complex hydropower operation problem.

In this paper, we present sound and complete natural deduction systems for Fitting’s four-valued generalizations of Kleene’s three-valued regular logics.

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).

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 investigates bivariate recursive equations on excess–of–loss reinsurance. For an insurance portfolio, under the assumptions that the individual claim severity distribution has bounded continuous density and the number of claims belongs to R1(a, b) family, bivariate recursive equations for the joint distribution of the cedent’s aggregate claims and the reinsurer’s aggregate claims are obtained.

This paper considers the problem of estimating the moving average (MA) parameters of a two-dimensional autoregressive moving average (2-D ARMA) model. To solve this problem, a new algorithm that is based on a recursion relating the ARMA parameters and cepstral coefficients of a 2-D ARMA process is proposed. On the basis of this recursion, a recursive equation is derived to estimate the MA parameters from the cepstral coefficients and the autoregressive (AR) parameters of a 2-D ARMA process. The cepstral coefficients are computed benefiting from the 2-D FFT technique. Estimation of the AR parameters is performed by the 2-D modified Yule---Walker (MYW) equation approach. The development presented here includes the formulation for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. The proposed algorithm is computationally efficient especially for the higher-order 2-D ARMA models, and has the advantage that it does not require any matrix inversion for the calculation of the MA parameters. The performance of the new algorithm is illustrated by some numerical examples, and is compared with another existing 2-D MA parameter estimation procedure, according to three performance criteria. As a result of these comparisons, it is observed that the MA parameters and the 2-D ARMA power spectra estimated by using the proposed algorithm are converged to the original ones

We formulate a stochastic control problem with a general information structure, and show that an optimal law exists and is characterized as the unique solution of a recursive stochastic equation. For a special information structure of the “signal-plus-noise” type and with quadratic cost-functions, this recursive equation is solved for the value function of the control problem. This value function is then shown to satisfy the Mortensen equation of Dynamic Programming in function-space.

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.

In this paper we show how two fundamental operations used in formal language theory provide useful tools for the investigation of arithmetic complexity classes. More precisely, we use Kleene closure of languages and inversion of formal power series to investigate subclasses of the complexity class GapL. GapL is the complexity class that characterizes the complexity of computing the determinant; it corresponds to counting the number of accepting and rejecting paths of nondeterministic logspace-bounded Turing machines.) We define a counting version of Kleene closure and show that it is intimately related to inversion within the complexity classes GapL and GapNC^1. In particular, we prove that Kleene closure and inversion are both hard operations in the following sense There is a set in AC^0 for which Kleene closure is NL-complete and inversion is GapL-complete. There is a finite set for which Kleene closure is NC^1-complete and inversion is GapNC^1-complete. Furthermore, we classify the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid.

This paper shows how to define Petri nets through recursive equations. It specifically addresses this problem within the context of the box algebra, a model of concurrent computation which combines Petri nets and standard process algebras. The paper presents a detailed investigation of the solvability of recursive equations on nets in what is arguably the most general setting. For it allows an infinite number of possibly unguarded equations, each equation possibly involving infinitely many recursion variables. The main result is that by using a suitable partially ordered domain of nets, it is always possible to solve a system of equations by constructing the limit of a chain of successive approximations.

This paper develops general tools for the analysis of stochastic divide and conquer algorithms. We concentrate on the average performance and the distribution of the running time of the algorithm. As a special example we analyse the average performance and the running time distribution of the (2k + 1)-median version of Quicksort.

A review of investigations on the development of algebraic foundations of algorithmics is given. Results connected with the solution of the problem of functional completeness of a Kleene algebra generalization are considered. These results reflect recent advances in the investigation of algorithmic metaalgebras associated with well-known methods of design of algorithms and programs.

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.

That there are only three different kinds of α-tautologies in revised Kleene systems is proved first; then in finite case the diversity theorem of α-tautologies with α's larger than the mid-value is established. Lastly, also in finite case, a representation theorem of α-tautologies by means of tautologies is given.

An algebra is called affine complete if all its compatible (i.e. congruence-preserving) functions are polynomial functions. In this paper we characterize affine complete members in the variety of Kleene algebras. We also characterize local polynomial functions of Kleene algebras and use this result to describe locally affine complete Kleene algebras.

A new efficient algorithm for calculating the weighting coefficients of maximally linear, FIR digital differentiators is presented. Simple closed-form explicit and recursive formulas are derived in a very straightforward manner. Moreover, a simple recursive equation is established, relating coefficients of two digital differentiators of adjacent ranks.

Consider a Markov-modulated G/G/1 queueing system in which the arrival and the service mechanisms are controlled by an underlying Markov chain. The classical approaches to the waiting time of this type of queueing system have severe computational difficulties. In this paper, we develop a numerical algorithm to calculate the moments of the waiting time based on Gong and Hu's idea. Our numerical results show that the algorithm is powerful. A matrix recursive equation for the moments of the waiting time is also given under certain conditions.

In one of his papers, Dragalin proposed a very general approach to the construction of models for nonstandard logics of uniform-algebra type and, in particular, for intuitionistic logic (see [1]). The presentation (clear enough, but without the obvious details) is accompanied by a number of arithmetical examples (see also [1]). In the last of these examples, Kleene’s slash realizability is considered (see [2, column C]), and the author gives a “ . . . model corresponding to Kleene’s slash realizability . . . ” [1, p. 194]. However, the connection between this model and Kleene’s slash realizability is as follows: “ . . . ‖φ‖ = T ⇔ ((|φ)∧HA φ)”; (see also [1, p. 195]; cf. [3]). Of course, by means of the above-mentioned model (which corresponds exactly to the formula realizability from [3]) one can prove the properties of disjunctiveness and existentiality for arithmetic (this is just the result the author seeks to obtain, using a suitable uniform algebra). However, Kleene’s slash realizability does not coincide with deducibility in the intuitionistic HA arithmetic. As mentioned above, the paper [1] contains a number of examples of functional pseudoboolean algebras corresponding to various HA models (including, in particular, realizability models). Exact formulation of functional pseudoboolean algebra is given in [1], and here we shall only briefly mention facts needed in what follows. If F is the set of forms of a functional pseudoboolean algebra, then for any f and g from F (where f and g are forms of the same arity), there exist the forms f ∧ g , f ∨ g , f ⊃ g in F of equal arity. After that, a functional algebraic model for logic-mathematical language is defined, and for a given functional algebraic model, we define the value of any formula of our language in it. The main distinction from usual algebraic models is in the fact that the value ‖φ‖ of the formula φ is a certain form fφ from a functional pseudoboolean algebra. Now we can fix the language of arithmetic, the object area, and the function Ĉnst . After that, each model is defined by specifying the following set: B (a pseudoboolean algebra), F , and Pr-valuation of predicates. Let a given functional pseudoboolean algebra be a model for Kleene’s slash realizability, i.e., suppose formulas of the language can be mapped to the set of forms so that any formula φ is |realizable if and only if fφ ∈ 1 (the unit of algebra B). Consider two distinct statements, φ and ψ , undecidable in HA, i.e., HA φ, HA ψ, HA ¬φ, HA ¬ψ. Since φ and ψ cannot be deduced in HA, the formulas ¬φ and ¬ψ are nondeducible |-realizable formulas of the language of arithmetic. If the forms corresponding to them in the functional algebraic model are F¬φ and F¬ψ , respectively, then these forms belong to algebra’s 1 , and then the form F¬φ ∨F¬ψ = F¬φ∨¬ψ (which corresponds in the functional algebraic model to the formula ¬φ∨¬ψ , see [1, p. 187]), also belongs to our algebra’s unit, and hence, the formula ¬φ∨¬ψ is |realizable. But this implies that HA φ or HA ψ , which is impossible by the choice of φ and ψ . Thus, we have proved the following theorem

It is given an explicit description of coproducts in the category of Kleene algebras in terms of the dual topological spaces. As an application, a description of dual spaces of free Kleene algebras is given. It is also shown that the coproduct of a family of three-valued Łukasiewicz algebras in the category of Kleene algebras is the same as the coproduct in the subcategory of three-valued Łukasiewicz algebras.

An efficient algorithm for finding the Kleene closure of regular expressions matrices or fuzzy regular expression matrices is presented and illustrated by examples. Properties of Kleene closure are also investigated. The results may have useful applications in automata theory, pattern recognition, and pictorial information systems.

We give algorithms for computing the extent of similarity between two or three sequences of letters. The similarity measures we consider include a penalty for inserting gaps within the sequence in order to enhance similarity. The magnitude of the penalty for gaps is assumed to be independent of their size in order to accommodate certain biological applications. Our algorithm for three sequence comparisons, which is based on solving a system of recursive equations, improves upon the efficiency of existing methods. Although the system of recursive equations utilized by the algorithm is quite complicated as it stands, it has none the less been simplified by appeal to combinatorial considerations.

The problem of estimating trigonometric sums for sequences of elements in a finite field which satisfy a linear recursive equation with periodic coefficients is considered.