On Bayes estimates

In this paper the author tries to give general conditions for the existence of Bayes estimates and for the consistency of sequences of Bayes estimates.In Section 3 we prove existence theorems for Bayes estimates, which contain those of DeGroot and Rao [3], as a special case. The proof is based on a theorem of Landers [5].Section 4 gives a characterization of Bayes estimates with convex loss and linear decision space. This theorem is also a generalization of a similar theorem of DeGroot and Rao [3].In Section 5 we generalize the theory of minimum contrast estimates (the foundations of which were laid by Huber [4], cf. Pfanzagl [6]) in such a way that we can apply it to the theory of Bayes estimates.Section 6 tries to give a general theory of consistency for Bayes estimates using the martingale argument of Doob [1] and the theory of minimum contrast estimates. Confer in this connection the results of Schwartz [8].Section 7 contains some auxiliary results.     

Fractional derivatives of the H-function of several variables

In the present paper we derive a number of key formulas involving fractional derivatives for the H-function of several variables, which was introduced and studied in a series of papers by 11., 12., 13., 14., 15., 9., 261–277].We make use of the generalized Leibniz rule for fractional derivatives in order to obtain one of the aforementioned results, which involves a product of two multivariable H-functions. Each of these results is shown to apply to yield interesting new results for certain multivariable hypergeometric functions and, in addition, several known results due, for example, to J. L. Lavoie, T. J. Osler and R. Tremblay [SIAM Rev.18 (1976), 240–268], 4., 5., 371–382] and R. K. Raina and C. L. Koul [Jñānābha7 (1977), 97–105].     

On measures of fuzziness of solutions of fuzzy relation equations with generalized connectives

By using a general class of fuzzy connectives of Yager [Fuzzy Sets and Systems4 (1980), 235–242], Pedrycz [Fuzzy relational equations with generalized convectives and their applications, Fuzzy Sets and Systems10 (1983), 185–201] has shown that the classical fuzzy relation equations of Sanchez [in “Fuzzy Automata and Decision Processes” (M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds.), pp. 221–234, North-Holland, Amsterdam, 1977] can be considered as a particular case of a more extensive class of fuzzy equations. For such types of equations, in this paper the solutions having the greatest energy measure and the smallest possible entropy measure of fuzziness are characterized.     

The periodic and eigen solutions of the equation governing a nonrotating viscoelastic earth model under transient force

The problem of existence of the periodic solution of the equation governing a nonrotating viscoelastic earth model under transient force is examined. By first formulating the governing equations, using the methods of Coleman and Noll (Rev. Modern Physics33 (2) (1961), 239–249), Dahlen and Smith (Philos. Trans. Roy. Soc. London A279 (1975), 583–624), and Biot (“Mechanics of Incremental Deformations,” Wiley, New York, 1965), these equations are subjected to oscillatory displacement resulting in an eigenvalue problem whose solutions are the viscoelastic-gravitational displacement eigenfunctions U(x) with associated eigenfrequencies ω. A theorem is then proved to show the existence of a periodic solution.     

A note on the generalization of the summation formulas for Appell's and Kampé de Fériet's hypergeometric functions

In the present paper, a summation formula of a general triple hypergeometric series F⁽³⁾(x, y, z) introduced by Srivastava [10] is obtained. A particular case of this formula corresponds to a result of Shah [7] involving Kampé de Fériet's double hypergeometric function which can further be specialized to yield summation formulas of Srivastava [11] and Bhatt [2] for Appell's function F₂.     

Nonlinear differential equations with negative power nonlinearities

Differential equations involving the term y^?m, where m is a positive integer, are solved by the decomposition method 1., 3., 441–452).     

Asymptotic behaviour of certain second order integro-differential equations

The asymptotic behaviour of certain second order integro-differential equations which are more general than those equations studied in [R. P. Agarwal, J. Math. Anal. Appl.86 (1982), 471–475] and [S. R. Grace and B. S. Lalli, J. Math. Anal. Appl.76 (1980), 84–90] are discussed. It is pointed out that a defect appeared in the basic Assumption 1 made in both papers, and we avoid this defect in our discussion by using more natural conditions.     

Existence results without convexity conditions for general problems of optimal control with singular components

This note presents a new, quick approach to existence results without convexity conditions for optimal control problems with singular components in the sense of E. J. McShane (SIAM J. Control5 (1967), 438–485). Starting from the resolvent kernel representation of the solutions of a linear integral equation, a version of Fatou's lemma in several dimensions is shown to lead directly to a compactness result for the attainable set and an existence result for a Mayer problem. These results subsume those of L. W. Neustadt (J. Math. Anal. Appl.7 (1963), 110–117), C. Olech (J. Differential Equations2 (1966), 74–101), M. Q. Jacobs (“Mathematical Theory of Control,” pp. 46–53, Academic Press, 1967), L. Cesari (SIAM J. Control12 (1974), 319–331) and T. S. Angell (J. Optim. Theory Appl.19 (1976), 63–79).     

Alexis Fontaine's route to the calculus of several variables

In the years around 1732 Alexis Fontaine des Bertins (1704–1771) member of the French Royal Academy of Sciences from 1733 on, worked out a dualoperator differential calculus to solve problems in “families of curves.” Within six years a calculus of several variables had emerged out of this work. These developments, discussed at greater length elsewhere [Greenberg, 1981, Greenberg, 1981, Greenberg, 1981], are summarized, with additional general reflections and the correction of one error. Together the discussions complement the excellent, recently published account of “families of curves” and the origins of partial differentiation in the works of Leibniz, the various Bernoullis, and Euler [Engelsman 1982]. Common concerns motivate the works of all of these mathematicians. At the same time, certain differences in conception in Fontaine's work highlight the creativity of one of the lesser known eighteenth-century mathematicians.     

Richardson extrapolation and Romberg integration

A review, supplementing previous accounts, is given of the historical background of some related techniques of numerical analysis chiefly associated with L. F. Richardson and W. Romberg. The application of extrapolation processes in connection with the classical problem of the quadrature of the circle is discussed in some detail, as is the development of some of the most frequently employed rules for approximate integration. It is also shown that many of the principal features of the foregoing numerical analysis techniques developed in modern times were anticipated in the books of Colin Maclaurin [1742] and Saigey, 1859, Vincent and Saigey, 1856.     

A class of infinite dimensional linear programming problems

We study the infinite dimensional linear programming problem. The previous work done on this subject defined the dual problem in a small space and derived duality results for such pairs of problems. But because of that and of the strong requirements on the functions involved, those theorems do not actually hold in many applications. With our formulation, we define the dual problem in a larger space and obtain new duality results under, generally, mild assumptions. Furthermore, the solutions turn out to be extreme points of the unbounded, but w¹-locally compact, feasibility set. For this purpose, we did not try a constructive proof of our duality results, but instead we examine the problem from a more abstract point of view and derive results using general ideas from the theory of convex analysis in normed spaces [R. T. Rockafellar, “Conjugate Duality and Optimization,” SIAM, Philadelphia, Penn., 1973, and R. Holmes, “Geometric Functional Analysis,” Springer-Verlag, New York, 1975]. Our work extends previous results in this area, which appeared in [N. Levinson, J. Math. Anal. Appl.16 (1965) 73–83, and W. Tundall, SIAM J. Appl. Math.13 (1965), 644–666].     

On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

Decomposition into a direct sum of irreducible representations of the representation of the full collineation group of a finite Desarguesian plane, as a group of matrices permuting the flags of the plane and the simple components of the corresponding commutant algebra, have been worked out here for the projective plane PG(2, 2) and the affine plane EG(2, 3). The dimension and the components of the covariance matrix of the observations from a design derived from such a plane, which commutes with such a permutation representation of the full collineation group of the plane, are thus determined. This paper is in the spirit of earlier works by, James (1957), Mann (1960), 6., 7., McLaren (1963), and Sysoev and Shaikan (1976). A. T. James, Ann. Math. Statist.28 (1957), 993–1002, H. B. Mann, Ann. Math. Statist.31 (1960), 1–15, E. J. Hannan, Research Report (Part. (I)), Summer Research Institute, Australian Math. Soc. and Methuen's Monographs on Applied Probability and Statistics, Supplementary Review Series in Applied Probability, Vol. 3, A. D. McLaren, Proc. Cambridge Philos. Soc.59 (1963), 431–450, and L. P. Sysoev and M. E. Shaikin, Avtomat. i Telemekh.5 (1976), 64–73.     

Asymptotic normal distribution of multidimensional statistics of dependent random variables

A central limit theorem for multidimensional processes in the sense of [9], [10] is proved. In particular the asymptotic normal distribution of a sum of dependent random functions of m variables defined on the positive part of the integral lattice is established by the method of moments. The results obtained can be used, for example, in proving the asymptotic normality of different statistics of n₀-dependent random variables as well as to determine the asymptotic behaviour of the resultant of reflected waves of telluric type.     

On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds

This paper continues the discussion, begun in J. Schwartz and M. Sharir [Comm. Pure Appl. Math., in press], of the following problem, which arises in robotics: Given a collection of bodies B, which may be hinged, i.e., may allow internal motion around various joints, and given a region bounded by a collection of polyhedral or other simple walls, decide whether or not there exists a continuous motion connecting two given positions and orientations of the whole collection of bodies. We show that this problem can be handled by appropriate refinements of methods introduced by A. Tarski [“A Decision Method for Elementary Algebra and Geometry,” 2nd ed., Univ. of Calif. Press, Berkeley, 1951] and G. Collins [in “Second GI Conference on Automata Theory and Formal Languages,” Lecture Notes in Computer Science, Vol. 33, pp. 134–183, Springer-Verlag, Berlin, 1975], which lead to algorithms for this problem which are polynomial in the geometric complexity of the problem for each fixed number of degrees of freedom (but exponential in the number of degrees of freedom). Our method, which is also related to a technique outlined by J. Reif [in “Proceedings, 20th Symposium on the Foundations of Computer Science,” pp. 421–427, 1979], also gives a general (but not polynomial time) procedure for calculating all of the homology groups of an arbitrary real algebraic variety. Various algorithmic issues concerning computations with algebraic numbers, which are required in the algorithms presented in this paper, are also reviewed.     

Éclosions combinatoires appliquées à l'inversion multidimensionnelle des séries formelles

Using general methods from the theory of combinatorial species, in the sense of A. Joyal (Adv. in Math.42 (1981), 1–82), symmetric powers of suitably chosen differential operators are interpreted combinatorially in terms of “éclosions” (bloomings) of certain kinds of points, called “bourgeons” (buds), into certain kinds of structures, called “gerbes” (bundles). This gives rise to a combinatorial setting and simple proof of a general multidimensional power series reversion formula of the Lie-Gröbner type (14., 15.). Some related functional equations are also treated and an adaptation of the results to the reversion of cycle index (indicatrix) series, in the sense of Pólya-Joyal (Joyal, loc. cit.), is given.     

Cubical sperner lemmas as applications of generalized complementary pivoting

Two cubical versions of Sperner's lemma, due to Kuhn and Fan, are proved constructively without resorting to a simplicial decomposition of the cube, presenting examples of generalized complementary pivoting discussed by Todd (Math. Programming6 (1974)). The first version is essentially equivalent to Sperner's lemma in that it implies Brouwer's fixed point theorem, thereby answering a question raised by Kuhn (IBM J. Res. Develop.4 (1960)). The second has the property that although the structure is that of generalized complementarity, there is a uniquely defined path or algorithm associated with it. The basic structure used is a cubical decomposition of the cube, a special case of a cubical pseudomanifold, presented by Fan (Arch. Math.11 (1960)). Given the existence of a constructive algorithm for Sperner's lemma (see Cohen, J. Combinatorial Theory2 (1967)) and its generalization by Fan J. Combinatorial Theory2 (1967)) allied to the large amount of recent progress in complementary pivot theory, resulting in particular from the works of Lemke (Manage. Sci.11 (1965)) and Scarf (“The Computation of Economic Equilibria”) the computational attractions of a simplicial decomposition have become apparent. However, a cubical decomposition leads to certain advantages when a search for more than one “completely labeled” region is required, and no simplicial construction for the Fan lemma is known.     

Fuzzy measures and measures of fuzziness

In this paper, the fuzzy integral defined by Z.-X. Wang (Fuzzy Math. Wuhan, China, in press), which is different from that defined by M. Sugeno (“Theory of Fuzzy Integrals and Its Applications,” Ph. D., Tokyo Inst. of Technology, 1974), is further considered, and it is shown that the fuzzy measures of ordinary sets and fuzzy sets can be determined by each other. Summing up the results on the measure of fuzziness by A. DeLuca and S. A. Termini (Inform. and Control20 (1972), 301–312), Z.-X. Wang (op. cit.) and R. R. Yager (Internat. J. Gen. Systems5 (1979), 221–229; Inform. and Control44 (1980), 236–260), the axioms for measures of fuzziness are given. Furthermore, as an application of the furry integrals, a measure of fuzziness is defined. Inversely, it is proven that a measure of fuzziness satisfying some conditions can surely be expressed as a fuzzy integral with respect to some fuzzy measure.     

A priori bounds for approximations of Markov programs

This note determines a priori bounds for B. L. Fox's [J. Math. Anal. Appl., 34 (1971), 665–670] scheme of approximating discounted Markov programs, thus refining bounds recently obtained by D. J. White (Notes in Descision Theory No. 43, University of Manchester, 1977). The approximation scheme focuses careful attention on only a subset of the state space and uses a fixed function to characterize future returns outside the designated subset. The a priori bounds are useful to design the specific approximation, that is, to select the appropriate subset on which the approximation is based.     

Geometry in Grassmannians and a generalization of the dilogarithm

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811). In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.     

Fuzzy measures as extensions of stochastic measures

Links between fuzzy measures (cf. Höhle, Z. Wahrsch. Verw. Gebiete36 (1976), 179–188) and stochastic measures (cf. Morando, Lecture Notes in Mathematics No. 88, pp. 190–216, Springer-Verlag, Berlin/New York, 1969) are specified. In particular, a class of stochastic measures, from which fuzzy measures can be derived quite naturally, is exhibited.