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The structure of nondeterministic computations is extremely complicated. C-graphs are abstract representations of the branching structure of nondeterministic computations. The paper investigates the structure of finite state nondeterministic computations by showing that the complexity of the structure increases non-elementarily while the number of computation steps increases. This is achieved by establishing a recursive equation relating the number of C-graphs of a certain height to the number of C-graphs of smaller height.

A monolithic process is a single recursive equation with data parameters, which only uses non-determinism, action prefixing, and recursion. We present a technique that decomposes such a monolithic process into multiple processes where each process defines behaviour for a subset of the parameters of the monolithic process. For this decomposition we can show that a composition of these processes is strongly bisimilar to the monolithic process under a suitable synchronisation context. Minimising the resulting processes before determining their composition can be used to derive a state space that is smaller than the one obtained by a monolithic exploration. We apply the decomposition technique to several specifications to show that this works in practice. Finally, we prove that state invariants can be used to further improve the effectiveness of this decomposition technique., Comment: In Proceedings ICE 2021, arXiv:2109.14908. This is a peer reviewed version of arXiv:2012.06468 [cs.LO]. This revision contains more examples but omits proofs

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.

We obtain solutions to some conjectures about the onlinear difference equation $${x_{n + 1}} = \alpha + \beta {x_{n - 1}}{e^{ - x}}^{_n},n = 0,1, \ldots,\,\alpha,\beta >0.$$ More precisely, we get not only a condition under which the equilibrium point of the above equation is globally asymptotically stable but also a condition under which the above equation has a unique positive cycle of prime period two. We also prove some further results.

Most qualitative behaviours of difference equations are rapidly investigated nowadays. This can be attributed to the fact that it is often sophisticated to construct the exact solutions of most difference equations. This article is written to analyse the local stability, global attractor and the boundedness of the solution of the seventh order difference equation given by where the coefficients are supposed to be positive real numbers and the initial conditions are arbitrary non-zero real numbers. Under some suitable conditions, the stability, boundedness and a special case equation from the considered equation are presented in 2 D figures.

We consider the one-armed bandit problem, i.e., the two-armed bandit problem with known probability distribution of incomes corresponding to applying the first action. On the known control horizon one has to determine the most profitable action and to ensure its preferential use. We consider the problem in a batch processing application and, hence, distributions of incomes are Gaussian. In Bayesian setting, we obtain a recursive integro-difference equation for computing Bayesian strategy and Bayesian risk with respect to an arbitrary prior distribution. Then we obtain a recursive equation in invariant form with control horizon one and a second order partial differential equation in the limiting case. In minimax setting of the problem, we determine minimax strategy and minimax risk as Bayesian ones corresponding to the worst-case prior distribution. The redundancy that occurs in the problem manifests itself by the fact that instead of sequential one-by-one data processing one can use batch processing with virtually no increase in minimax risk. For example, numerical results show that processing data in 50 batches increases minimax risk by only 3%. If data can be processed in parallel then the total processing time is determined by the number of batches rather than by the total number of data.

The assessing resources dynamics problem, in the context of an economic system with Gaussian consumption and deterministic productivity, is considered in this paper. Basically it is presented a discrete time recursive equation that supports the recourse to the Ornstein-Uhlenbeck diffusion process. Some assumptions on the regeneration of the process are made, in order to observe the system equilibrium in what concerns the resources depreciation or accumulation. The objective of this work is to present a result on the sign of a ratio that can be used to evaluate harvest procedures in this context., Comment: 6 pages and no figures

This brief proposes a novel discrete-time realization of the spatial repetitive controller topology, an internal model principle-based regulator, which deals with the exogenous position-dependent periodic signals. A simple procedure is proposed in order to estimate the time-varying delay required to implement this spatial controller, resulting in a single recursive equation. A robust synthesis methodology based on a convex optimization problem constrained by linear matrix inequalities is also presented in order to guarantee the closed-loop system stability and performance in the presence of uncertain plant dynamics. The proposed design and implementation methods are validated on a direct current motor subject to angle-dependent disturbances.

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.

An algebra for a functor H is called completely iterative (cia, for short) if every flat recursive equation in it has a unique solution. Every cia is corecursive, i.e., it admits a unique coalgebra-to-algebra morphism from every coalgebra. If the converse also holds, H is called a cia functor. We prove that whenever the base category is hyper-extensive (i.e. countable coproducts are ‘well-behaved’) and H preserves countable coproducts, then H is a cia functor. Surprisingly few cia functors exist among standard finitary set functors: in fact, the only ones are those preserving coproducts; they are given by X ↦ W × ( − ) + Y for some sets W and Y.

A posteriori ratemaking is widely applied in the premium calculation of property and casualty products, particularly in third-party automobile insurance, which usually uses a bonus–malus system for premium adjustment. The present paper suggests an alternative to common frameworks that are designed as random walks on graphs of mostly finite states representing premium levels. The proposed premium calculation model is governed by the policyholder’s claim history through a recursive equation. This new autoregressive scheme is structurally different from the ones in use. Relevant metrics that measure the system’s optimality are evaluated, partially in analytical form. Through a comparison with existing models and parameterisation from real-life data, the new model is put into context and its practical relevance is investigated.

The probability hypothesis density filter with linear Gaussian jump Markov system multi-target models is an attractive approach to tracking multiple maneuvering targets in the presence of data association uncertainty, clutter, noise, and detection uncertainty. However, these models are not precise enough to describe moving targets on road networks in ground target tracking scenario. In this paper, the road map information is integrated into the jump Markov system Gaussian mixture probability hypothesis density (JMS-GM-PHD) filter, and a road-constraint JMS-GM-PHD filter for ground target tracking is proposed. In addition, we then derive the recursive equation of the proposed filter. Simulation results show that the proposed road-constrained JMS-GM-PHD filter is effective in tracking ground moving targets.

In this study we are analyzing the fuzzy dynamic programming based multi-stage transportation problem. The purpose of this paper is to evolve a method relating to fuzzy dynamic programming for solving the fuzzy least cost route problem. In this work the costs of transportation are represented as a generalized trapezoidal fuzzy number. Backward recursive equation characterized by fuzziness is used to determine the optimal solution of the given problem. The analyzed approach is explained by a numerical example. The special characteristic of the used method is the uncertainty in the least cost route models, and it is effectively eliminated by fuzzy dynamic programming.

Under the condition that the step size is less than one, a statistical tracking behavior analysis for the affine projection algorithm based on direction error is discussed. When the unknown true weight vector is modeled by the stochastic walk model, the mean weight error is derived under the four assumptions based on the deterministic recursive equation. Furthermore, the statistical tracking behavior of the steady state is analyzed for the affine projection algorithm based on direction error. Simulation analysis is shown to suppniort the mathematical results.

We got acquainted with the DP decomposition principle and Bellman’s recursive equation in Chap. 1. Now we turn to practical matters in terms of actual implementation and computational issues. As we have stressed, dynamic programming is a principle, rather than a specific algorithm that can be purchased off-the-shelves. Thus, careful modeling and analysis are needed on the one hand, and customized implementation is needed on the other one. Indeed, one of the main aims of this tutorial booklet is to encourage newcomers to experiment with DP, which is only possible by implementing it using any programming language or computing environment they like. Our choice is MATLAB, which is widely available and quite convenient as it yields short and intuitive code, not obscuring the underlying ideas.

The present paper investigates the quality of numberings measured in three different ways: (a) the complexity of finding witnesses of Kleene's Recursion Theorem in the numbering; (b) for which learning notions from inductive inference the numbering is an optimal hypothesis space; (c) the complexity needed to translate the indices of other numberings to those of the given one. In all three cases, one assumes that the corresponding witnesses or correct hypotheses are found in the limit and one measures the complexity with respect to the best criterion of convergence which can be achieved. The convergence criteria considered are those of finite, explanatory, vacillatory and behaviourally correct convergence. The main finding is that the complexity of finding witnesses for Kleene's Recursion Theorem and the optimality for learning are independent of each other. Furthermore, if the numbering is optimal for explanatory learning and also allows to solve Kleene's Recursion Theorem with respect to explanatory convergence, then it also allows to translate indices of other numberings with respect to explanatory convergence.

In the life-cycle analysis, the total cost of damage caused by earthquakes is a significant but highly uncertain component. In the current literature, the seismic risk analysis is largely limited to the evaluation of the average cost of damage, which is not informative about the full extent of variability in the cost. The paper presents a systematic development of the stochastic modeling of seismic risk analysis problem and reformulates the damage cost analysis as a superposition of compound Poisson processes. An explicit analytical solution for the distribution of damage cost is derived in form of a recursive equation. The proposed approach extends the capability of the existing framework of seismic risk analysis, which can be used to optimize initial design and retrofitting of structures.

In this paper, the two-dimensional (2D) crack problem under dynamic loading condition is studied. Firstly, by the precise time domain expanding algorithm, the recursive formulation of the governing equation is derived, and the recursive equation can be solved by FEM. Then, a new Symplectic Analytical Singular Element (SASE) defined by the analytical symplectic eigen expanding terms is further developed for the investigated problem. The area near the crack tip is represented by using the SASE, while the other area is meshed by conventional elements. Taking advantage of the SASE, the crack problem can be solved with higher accuracy. Simultaneously, the dynamic stress intensity factors (DSIFs) can be obtained directly without any post processing through the relationship between the DSIFs and the eigen expanding coefficients. The numerical accuracy and stability is investigated carefully through the conducted numerical examples. It has been demonstrated that the proposed SASE for dynamic crack problem is effective and efficient.

Recent works aimed to understand how to identify “milestone” scientific papers of great significance from large-scale citation networks. To this end, previous results found that global ranking metrics that take into account the whole network structure (such as Google’s PageRank) outperform local metrics such as the citation count. Here, we show that by leveraging the recursive equation that defines the PageRank algorithm, we can propose a family of local impact metrics. Our results reveal that the obtained PageRank-based local metrics outperform the citation count and other local metrics in identifying the seminal papers. Compared with global metrics, these local metrics can reach similar performance in the identification of seminal papers in shorter computational time, without requiring the whole network topology. Our findings could help to better understand the nature of groundbreaking research from citation network analysis and find practical applications in large-scale data.

This article proposes a two recursive equation model to correct for endogeneity in latent class Probit models. Concretely, it is assumed that there exists an endogenous and continuous variable defined as a predictor, while unobserved heterogeneity is conceptualized as a vector of parameters that varies across individuals following a discrete distribution. A Maximum Likelihood Estimator is provided to estimate the model parameters based on normally distributed random terms and a free code in R software is provided to carry out the estimation procedure. A small Monte Carlo experiment is carried out to analyze the properties of the estimator. Finally, the estimator is applied to analyze the heterogeneous effects of weight on mental well-being.

According to the characteristics of blood inventory control problem under the condition of blood shortage, the dynamic decision-making problem of blood supply chain is investigated in this paper. Firstly, based on the recursive equation, the state transition equations of two categories of blood demand under two inventory issue strategies (FIFO and LIFO) are given. The mathematical expressions of key indexes such as blood outdating and blood shortage are obtained. A blood collection decision-making method based on EWA (Estimated Withdrawal & Aging) strategy is proposed. Then, an optimal model of blood transshipment problem is established with the goal of the shortest transshipment time and the maximum freshness of the transported blood. In addition, an allocation planning model with multiple priority requirements and fairness concerns is established to achieve the best fairness and the minimum shortage. Besides, a discrete event system simulation (DESS) framework is designed according to the characteristics of the model. Finally, the effectiveness of the decision-making method and EWA inventory strategy are verified by numerical simulation. The results show that safety stock, target stock level and fluctuation range of demand have significant impacts on the control effect of blood inventory.

Gabbay and Ciancia (2011) presented a nominal extension of Kleene algebra as a framework for trace semantics with statically scoped allocation of resources, along with a semantics consisting of nominal languages. They also provided an axiomatization that captures the behavior of the scoping operator and its interaction with the Kleene algebra operators and proved soundness over nominal languages. In this paper, we show that the axioms proposed by Gabbay and Ciancia are not complete over the semantic interpretation they propose. We then identify a slightly wider class of language models over which they are sound and complete.

As a popular optimization tool for multi-stage sequential decision problems, dynamic programming (DP) has been widely used to handle with hydropower system operation problems. However, the DP computational burden shows an exponential growth with the increasing number of hydroplants, which results in “the curse of dimensionality” and limits its application to resolve large and complex hydropower operation problem. Thus, this paper presents a novel modified DP algorithm called uniform dynamic programming (UDP) to alleviate the dimensionality problem of dynamic programming. In UDP, the uniform design is first used to construct the state variables set of each period by selecting some small but representative discrete state combinations, and then the DP recursive equation is used to find an improved solution for the next computation cycle. The UDP method is tested in the Wu River cascaded hydropower system of southwest China. The results indicate that the proposed UDP algorithm has competitive performance in computational efficiency and convergence speed, which is an effective tool for 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.

Energy problems are important in the formal analysis of embedded or autonomous systems. With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce energy automata. These are finite automata whose edges are labeled with energy functions that define how energy levels evolve during transitions. Motivated by this application and in order to compute with energy functions, we introduce a new algebraic structure of *-continuous Kleene ω-algebras. These involve a *-continuous Kleene algebra with a *-continuous action on a semimodule and an infinite product operation that is also *-continuous. We define both a finitary and a non-finitary version of *-continuous Kleene ω-algebras. We then establish some of their properties, including a characterization of the free finitary *-continuous Kleene ω-algebras. We also show that every *-continuous Kleene ω-algebra gives rise to an iteration semiring-semimodule pair.

In the present paper, a system with components subject to soft and hard failures is considered. It is assumed that hard failures are revealed and fixed immediately and present an additional opportunity for inspection (opportunistic inspection), but soft failures are hidden and only corrected at periodic inspections. The objective is to find the optimal maintenance policy for all components and the optimal periodic inspection for the entire system. Two models are considered in this context. In the first model, hard-type and soft-type components are subject to minimal repair or corrective replacement, and soft-type components undergo opportunistic inspections. In the second model, in addition to the assumptions of the first model, hard-type components may be preventively replaced at periodic inspections. In our models, we base the maintenance decision for the soft-type components on the optimal number of minimal repairs until replacement, and for the hard-type components – on the optimal age before replacement. A recursive equation is provided for deriving the required expected values. Hidden failures preclude us from expressing the terms of the objective function in closed form. For this reason, the optimal periodic inspection interval for the system minimizing its total expected life cycle cost is found for both models using simulation.

We consider exponential two-armed bandit problem in which incomes are described by exponential distribution densities. We develop Bayesian approach and present recursive equation for determination of Bayesian strategy and Bayesian risk. In the limiting case as the control horizon goes to infinity, we obtain the second order partial differential equation in the domain of "close distributions". Results are compared with Gaussian two-armed bandit. It turned out that exponential and Gaussian two-armed bandits have the same description in the limiting case. Since Gaussian two-armed bandit describes the batch processing, this means that in case of exponential two-armed bandit batch processing does not enlarge Bayesian risk in comparison with one-by-one optimal processing as the total number of processed data items goes to infinity.

We prove that the automaton presented by Maslov [Soviet Math. Doklady 11 (1970) 1373–1375] meets the upper bound 3/4·2n on the state complexity of Kleene closure. Our main result shows that the upper bounds 2n − 1 + 2n − 1 − k on the state complexity of Kleene closure of a language accepted by an n -state DFA with k final states are tight for every k with 1 ≤ k ≤ n in the binary case. We also study Kleene Closure on prefix-free languages. In the case of prefix-free languages, the Kleene closure may attain just three possible complexities n − 2,n − 1, and n . We give some survey of our computations.

We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Moller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings , or gsp-strings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gsp-strings in parallel. To ensure that the model satisfies the weak exchange law ( x ‖ y ) ( z ‖ w ) ≤ ( x z ) ‖ ( y w ) of CKA, we make use of the subsumption order of Gischer on the gsp-strings. We also define deterministic branching automata and investigate their relation to (nondeterministic) branching automata. To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas and relate them to branching automata and to concurrent Kleene algebras with tests.

It is shown that antidomain semirings are more expressive than test semirings and that Kleene algebras with domain are more expressive than Kleene algebras with tests. It is also shown that Kleene algebras with domain are expressive for propositional Hoare logic whereas Kleene algebras with tests are not., Comment: Typos have been corrected

The subject matter of this paper involves the examination of the process of liquid drop formation at the outlet of a nozzle. A theoretical model of the formation of liquid droplet has been developed. The model is based on a recursive equation. Solution of this equation is obtained by the solution set, which is similar to the process of liquid droplets formation. The objective is to demonstrate that a recursive theoretical model can reproduce the chaotic behavior of physical phenomena, which is the formation of liquid droplets. In this paper, it is the assessment of the non-stationary characteristics in the process of drop formation. The potential for describing it in terms of principles familiar from deterministic chaos is presented. The paper contains the results of experiments which indicate that chaotic phenomena occur during the formation of liquid drops. The results of the research are elaborated and presented in the form of attractors and power spectrum diagrams. The ranges of the parameters are determined for which the time intervals between the successive drops tend to be non-stationary. Their presentation in the phase space has revealed the occurrence of chaotic phenomena. It was concluded that numerical calculations and the results gained from experiment are comparable. Consequently, it was indicated that relatively simple mathematical model can be used for simulation of the states of actual physical processes.

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.

Purpose: Double sampling   chart is a useful tool for detecting a relatively small shift in process mean when the process is controlled by multiple variables. This paper finds the optimal design of the double sampling   chart in both economical and statistical sense under Weibull failure model. Methods: The expected cost function is mathematically derived using recursive equation approach. The optimal designs are found using a genetic algorithm for numerical examples and compared to those of single sampling   chart. Sensitivity analysis is performed to see the parameter effects. Results: The proposed design outperforms the optimal design of the single sampling   chart in terms of the expected cost per unit time and Type-I error rate for all the numerical examples considered. Conclusion: Double sampling   chart can be designed to satisfy both economic and statistical requirements under Weibull failure model and the resulting design is better than the single sampling counterpart.

The dual reciprocity boundary element method (DRBEM) and the precise time-domain expanding algorithm are combined for solving transient heat conduction problems with heat sources. Firstly, the recursive formulation of the governing equation is derived by expanding time-dependent physical quantities at any discrete time interval. Then, the recursive equation is solved by the DRBEM, while a self-adaptive check scheme is used for estimating recursive times in a time step. Finally, the single and multi-connected domain problems are analyzed respectively with different kinds of boundary conditions. The results show that the proposed method can obtain very stable and accurate results with different time steps.

For the case that the prediction and update steps of sequence Monte Carlo Generalized Labeled Multi-Bernoulli filter (SMC-GLMB) require pruning respectively which causes large amount of calculation and low operation efficiency, a fast GLMB algorithm based on box particle filter for target tracking is proposed. First, a new recursive equation is derived based on the combination of prediction and update step, then the box particle filter is used to approximate the probability density of single target state, finally we use the new recursive equation to update the probability density of target state. Simulation results show that our proposed algorithm can effectively estimate the state of target, and the computational efficiency is significantly improved compared with the traditional SMC-GLMB filter.

We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean-time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations., Comment: 12 pages, 7 figures, to appear in Phys. Rev. E (2019)

For real-valued additive process ( X t ) t ≥ 0 a recursive equation is derived for the entire positive moments of functionals I s , t = ∫ s t exp ( − X u ) d u , 0 ≤ s t ≤ ∞ , in case the Laplace exponent of X t exists for positive values of the parameter. From the equation emerges an easy-to-apply sufficient condition for the finiteness of the moments. As an application we study first hit processes of diffusions.

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.

Although many optimization models and methods are applied to the optimization of reservoir operation at present, the optimal operation decision that is made through these models and methods is just a retrospective review. Due to the limitation of hydrological prediction accuracy, it is practical and feasible to obtain the suboptimal or satisfactory solution by the established operation rules in the actual reservoir operation, especially for the mid- and long-term operation. In order to obtain the optimized sample data with global optimality; and make the extracted operation rules more reasonable and reliable, this paper presents the multi-dimensional dynamic programming model of the optimal joint operation of cascade reservoirs and provides the corresponding recursive equation and the specific solving steps. Taking Li Xianjiang cascade reservoirs as a case study, seven uncertain problems in the whole operation period of the cascade reservoirs are summarized after a detailed analysis to the obtained optimal sample data, and two sub-models are put forward to solve these uncertain problems. Finally, by dividing the whole operation period into four characteristic sections, this paper extracts the operation rules of each reservoir for each section respectively. When compared the simulation results of the extracted operation rules with the conventional joint operation method; the result indicates that the power generation of the obtained rules has a certain degree of improvement both in inspection years and typical years (i.e., wet year; normal year and dry year). So, the rationality and effectiveness of the extracted operation rules are verified by the comparative analysis.

Orthogonal subspace projection (OSP) has found many applications in hyperspectral data exploitation. Its effectiveness and usefulness result from implementation of two stage processes, i.e., annihilation of undesired signal sources by an OSP via inverting a matrix in the first stage followed by a matched filter to extract the desired signal source in the second stage. This paper presents a theory of recursive OSP (ROSP) for hyperspectral imaging, which performs OSP recursively without inverting undesired signature matrices. This ROSP opens up many new dimensions in extending OSP. First of all, ROSP allows OSP to implement varying signatures via a recursive equation without reinverting undesired signature matrices. Second, ROSP can be further used to derive an unsupervised ROSP (UROSP) OSP, which allows OSP to find a growing number of unknown signal sources recursively while simultaneously determining a desired number of signal sources. As a result, the commonly used automatic target generation process (ATGP) can be extended to a recursive ATGP, which can be considered as a special case of UROSP. Third, for practical applications, UROSP can be also extended in two differ ent fashions to causal process and progressive process, which give rise to causal UROSP and progressive UROSP, respectively, both of which can be easily realized in hardware implementation. Finally, UROSP provides a feasible stopping rule via a recently developed UROSP-specified virtual dimensionality.

This paper presents a mechanically verified implementation of an algorithm for deciding the equivalence of Kleene algebra terms within the Coq proof assistant. The algorithm decides equivalence of two given regular expressions through an iterated process of testing the equivalence of their partial derivatives and does not require the construction of the corresponding automata. Recent theoretical and experimental research provides evidence that this method is, on average, more efficient than the classical methods based on automata. We present some performance tests, comparisons with similar approaches, and also introduce a generalization of the algorithm to decide the equivalence of terms of Kleene algebra with tests. The motivation for the work presented in this paper is that of using the libraries developed as trusted frameworks for carrying out certified program verification.

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Godel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of th...

This paper discusses the design and practical implementation of solving an external consensus problem for inhomogeneous networked multiagent systems with constant network delay in the output feedback. Based on the recursive equation, a novel prediction strategy is proposed using the transfer function form to overcome the effects of the network delay. By considering a single-input single-output linear system for each agent, the proposed strategy is simulated, demonstrated, and validated through a test-rig application. Concurrently, the stability of the control scheme and the criterion to determine the appropriate coupling gains value for the proposed consensus protocol are also studied.

Stock allocation is a system used to ensure that goods and services reach the ultimate users through efficient stocking in warehouses close to the consumers. The dire need for optimum distribution of goods to both retailers and consumers has caused a reasonable drift from ordinary allocation to developing a mathematical model that ensures efficient allocation of goods and services. Allocation of stock to warehouses is a complex problem that is broken down into simpler sub problems. Dynamic programming problem is a linear optimization method that obtains optimum solution of a multivariable problem by decomposing it into sub problems. A recursive equation links the different stages of the problem such that the optimum feasible solution of each stage is guaranteed to be the optimum feasible solution for the entire problem. This work will use the dynamic programming technique to develop a stock allocation model that would ensure optimum allocation of goods and services for maximum returns. Relevant related literature are presented and reviewed with the aim of using this research to improve stock allocation processes. A manufacturing company that has at least six distribution Original Research Article Emenonye and Chikwendu; JSRR, 6(5): 383-393, 2015; Article no.JSRR.2015.163 384 outlets is used as a case study. The model is applied to data collected from the firm to obtain an enhanced stock allocation.