Recursive functions of context free languages (Ⅰ)——The definitions of CFPRF and CFRF

It is intended to establish the recursive function theory on context free languages (CFLs). In this paper, the function class CFRF and its proper subclass CFPRF were defined on CFLs; it is quite straightforward to use them for describing non-numerical algorithms. In fact, they are respectively the partial recursive functions and primitive recursive functions of context free languages. The structure induction method for proving CFPRF function properties was presented. A method for CFL sentence enumeration was given, the minimization operator was defined. Based on CFL sentence enumeration, the minimization operator evaluation method was given. Finally, the design and implementation principles of executable specification languages with the CFRF as theoretical basis were discussed.

Recursive functions of context free languages (Ⅱ)——Validity of CFPRF and CFRF definitions

In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains . It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.

PROPAGATION CHARACTERISTICS OF A CIRCULAR MULTILAYER CHIRAL DIELECTRIC WAVEGUIDE

An efficient recursive eigenfunction solution is described for the propagation characteristics of a circular multilayer chiral dielectric waveguide. A characteristic equation is derived for the propagation constant of an N-layer chiral dielectric waveguide. Numerical results are presented for several examples.

Decision Making as Theorem Proving

We present a method for using type theory to solve decision making problem. Our method is based on the view that decision making is a special kind of theorem proving activity. An isomorphism between problems and types, and solutions and programs has been established to support this view which is much similar to the Curry-Howard isomorphism between propositions and types, and proofs and programs. To support our method, a proof development system called PowerEpsilon has been developed, and the synthesis of a decision procedure for validity of first-order propositional logic is discussed to show the power of the system.

Introducing a Function with Plural Derivatives

It is a solid truth in mathematics that the derivative of a function is unique. We want to show that there exist particular functions all of which have the same form but their derivatives are different. Even though this may seem quite novel, such function could be crucial for the purpose of describing the world such as related to mental phenomena where the logic of the current mathematics is not adequate.

Reduced order model for unsteady aerodynamic performance of compressor cascade based on recursive RBF

Based on Recursive Radial Basis Function(RRBF)neural network,the Reduced Order Model(ROM)of compressor cascade was established to meet the urgent demand of highly efficient prediction of unsteady aerodynamics performance of turbomachinery.One novel ROM called ASA-RRBF model based on Adaptive Simulated Annealing(ASA)algorithm was developed to enhance the generalization ability of the unsteady ROM.The ROM was verified by predicting the unsteady aerodynamics performance of a highly-loaded compressor cascade.The results show that the RRBF model has higher accuracy in identification of the dimensionless total pressure and dimensionless static pressure of compressor cascade under nonlinear and unsteady conditions,and the model behaves higher stability and computational efficiency.However,for the strong nonlinear characteristics of aerodynamic parameters,the RRBF model presents lower accuracy.Additionally,the RRBF model predicts with a large error in the identification of aerodynamic parameters under linear and unsteady conditions.For ASA-RRBF,by introducing a small-amplitude and highfrequency sinusoidal signal as validation sample,the width of the basis function of the RRBF model is optimized to improve the generalization ability of the ROM under linear unsteady conditions.Besides,this model improves the predicting accuracy of dimensionless static pressure which has strong nonlinear characteristics.The ASA-RRBF model has higher prediction accuracy than RRBF model without significantly increasing the total time consumption.This novel model can predict the linear hysteresis of dimensionless static pressure happened in the harmonic condition,but it cannot accurately predict the beat frequency of dimensionless total pressure.

Accurate prediction of different forecast horizons wind speed using a recursive radial basis function neural network

Environmental considerations have prompted the use of renewable energy resources worldwide for reduction of greenhouse gas emissions.An accurate prediction of wind speed plays a major role in environmental planning,energy system balancing,wind farm operation and control,power system planning,scheduling,storage capacity optimization,and enhancing system reliability.This paper proposes an accurate prediction of wind speed based ona Recursive Radial Basis Function Neural Network(RRBFNN)possessing the three inputs of wind direction,temperature and wind speed to improve modern power system protection,control and management.Simulation results confirm that the proposed model improves the wind speed prediction accuracy with least error when compared with other existing prediction models.

Pattern Matching Compilation of Functions Definedin Context-Free Languages

LFC is a functional language based on recursive functions defined in context-free languages. In this paper, a new pattern matching algorithm for LFC is presented, which can represent a sequence of patterns as an integer by an encoding method. It is a rather simple method and produces efficient case-expressions for pattern matching definitions of LFC. The algorithm can also be used for other functional languages, but for nested patterns it may become complicated and further studies are needed.

Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem

We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward-backward stochastic differential equations with jump processes,in which the coefficients depend on the marginal law of the state process through its expected value.The control variable is allowed to enter both diffusion and jump coefficients.Moreover,the cost functional is also of mean-field type.Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques.As an application,time-inconsistent mean-variance portfolio selectionmixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.

Function Definition Language FDL andIts Implementation

A Function Definition Language (FDL) is presented. Though de-signed for describing specifications, FDL is also a general-purpose functional pro-gramming language. It uses context-free language as data type, supports patternmatching definition of functions, offers several function definition forms, and is exe-cutable. It is shown that FDL has strong expressiveness, is easy to use and describesalgorithms concisely and naturally. An interpreter of FDL is introduced. Experi-ments and discussion are included.

The M-computations induced by accessibility relations in nonstandard models M of Hoare logic

Hoare logic is a logic used as a way of specifying semantics of programming languages, which has been extended to be a separation logic to reason about mutable heap structure. In a model M of Hoare logic, each program α induces an M-computable function fα M on the universe of M; and the M-recursive functions are defined on M. It will be proved that the class of all the M-computable functions fα M induced by programs is equal to the class of all the M- recursive functions. Moreover, each M-recursive function is ∑ 1 NM -definable in M, where the universal quantifier is a num- ber quantifier ranging over the standard part of a nonstandard model M.