An optimistic decision-making in fuzzy environment

Bellman and Zadeh have originated three systems of multistage decision processes in a fuzzy environment: deterministic, stochastic and fuzzy systems. In this article, we consider an optimization problem with an optimistic criterion on a fuzzy system. By making use of minimization–maximization expectation in a fuzzy environment, we derive a recursive equation for the fuzzy decision process through invariant imbedding approach. By illustrating a three-state, two-decision and two-stage model, we give an optimal solution through dynamic programming. The optimal solution is also verified by the method of multistage fuzzy decision tree-table.     

The unified approach in a semi-infinite atmosphere

The unified approach of invariant imbedding is extended to the case of a semi-infinite atmosphere. The integral equation satisfied by the reflection function from a semi-infinite atmosphere is derived by virtue of the approach. For stationary and non-stationary radiation fields, the equations obtained in this article coincide with those already obtained.     

On some properties of the functions from Sobolev-Morrey type spaces

In this paper the spaces of type Sobolev-Morrey-W _p,a,г,τ ^l (Q,G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G∋R ⁿ satisfies the flexible λ-horn condition.     

Cauchy system for Fredholm resolvents with composite displacement kernels

In recent years, the invariant imbedding approach to initial-value solutions of Fredholm integral equations with degenerate or semidegenerate kernels has been discussed with emphasis. In the present paper, with the aid of invariant imbedding, the Cauchy problem for Fredholm resolvents with composite displacement kernels is reduced to that for generalized Chandrasekhar'sX-functions andY-functions. In other words, it is shown how to convert the initial-value solution of the resolvent into a system of simultaneous nonlinear integrodifferential equations ofX-functions andY-functions of a single argument. From the computational aspect, the result seems to be more tractable than the original ones.Dedicated to R. BellmanThis work was partially supported by the Ministry of Education of Japan, Grant No. 503540. The author is indebted to Professor R. E. Kalaba, University of Southern California, and Dr. H. H. Kagiwada, Hughes Aircraft Company, for their helpful comments and kind interest in the present paper.     

Sobolev Type Spaces on the Dual of the Laguerre Hypergroup

Sobolev type spaces E ^s,p (0, sR, p[1,+]) are defined on R×N by using the Fourier transform and its inverse on the Laguerre hypergroup. An analogue of H ^s (R ⁿ ), denoted by H ^s is investigated in this paper. Some properties including completeness and imbedding results for these spaces are given, Reillich-type theorem and Poincaré's inequality are proved. Also, global regularity results for certain differential operators are obtained.     

A note on the imbedding theorem of Browder and Ton

The imbedding theorem of Browder and Ton states that for any real separable Banach space there exist a real separable Hilbert space and a compact linear injection such that is dense in We shall give a short and elementary new proof to this result. We also briefly discuss the corresponding result without the completeness assumption.

     

On the formulation of invariant imbedding problems

The formulation of an invariant imbedding problem from a given linear, two-point boundary-value problem is not unique. In this paper, we illustrate how the formulation of the problem by partitioning the original vectory(z) into [u(z),v(z)], can affect the numerical accuracy. In fact, the partitioning, the choice of theR, O system orS, T system of equations in Scott's method, the location and number of switch points, and the switching procedure, all influence the numerical results and the ease of obtaining solutions. A new method of switching and the appropriate formulas are described, namely, the repeated switching from theR, Q system to theR, Q system of equations or from theS, T system to theS, T system of equations.     

Enumerating the orientable 2‐cell imbeddings of complete bipartite graphs

A formula is developed for the number of congruence classes of 2‐cell imbeddings of complete bipartite graphs in closed orientable surfaces. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 77–90, 1999     

Preduals of Campanato Spaces and Sobolev-Campanato Spaces: A General Construction

In this paper we describe two limiting processes for families of Banach spaces closely related to the standard definition of projective and inductive limits. These processes lead again to Banach spaces. Information about linear operators and duality between basic families of spaces is carried over to the corresponding limit spaces. The abstract results are shown to be applicable to Campanato spaces and Sobolev-Campanato spaces. In particular, we obtain the existence and a characterization of predual spaces. Some imbedding relations are investigated in more detail.