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.     

On the structure of periodic solutions of differential equations

This paper studies the “internal structure” of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay.Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x′ = f(x) and the fact that the origin belongs to the range of f. This situation is explored in [19].     

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].     

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.     

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).     

The higher reciprocity laws: An example

In Section 128 of Smith's “Report on the Theory of Numbers” (Chelsea, New York, 1965) one finds a certain theta function whose coefficients are multiplicative arithmetic functions. It is shown in this paper that this is an elementary example of the well-known connection between the higher reciprocity laws of number fields, Artin L-functions, and modular forms.     

Nonstationary value-iteration and adaptive control of discounted semi-Markov processes

We consider in this paper discounted-reward, denumerable state space, semi-Markov decision processes which depend on unknown parameters. The problems we are interested in are: Given that the true parameter value is unknown, (I) give an iterative scheme to determine the total maximal discounted reward, and (II) find an asymptotically discount optimal (adaptive) policy. Our solutions are inspired by the nonstationary value iteration (NVI) scheme of Federgruen and Schweitzer (J. Optim. Theory Appl.34 (1981), 207–241) combined with the ideas of Schäl (Preprint No. 428, Inst. Angew. Math. Univ. Bonn, 1981) concerning the “principle of estimation and control” for the adaptive control of semi-Markov processes.     

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.     

On systems of bilinear forms whose minimal division-free algorithms are all bilinear

In this paper we prove that for a certain class of systems of bilinear forms, all minimal division-free algorithms are essentially bilinear. This class includes systems for computing products in finite algebraic extension fields, and systems for computing the products of Toeplitz and Hankel matrices with vectors. Our results, together with the classification theorems of 10., 12., 169–180) completely describe all minimal division-free algorithms for computing these systems. We also prove, as an immediate consequence of our results, that the multiplicative complexity of the quaternion product over a real field is 8.     

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.     

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.     

Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups

The theory of symmetric local semigroups due to A. Klein and L. Landau (J. Funct. Anal.44 (1981), 121–136) is generalized to semigroups indexed by subsets of $\text{R}$ⁿ for n > 1. The result implies a similar result of A. E. Nussbaum (J. Funct. Anal.48 (1982), 213–223). It is further generalized to semigroups that are symmetric local in some directions and unitary in others. The results are used to give a simple proof of A. Devinatz's (Duke Math. J.22 (1955), 185–192) and N. I. Akhiezer's (“the Classical Moment Problem and Some Related Questions,” Hafner, New York, 1965) generalization of a theorem of Widder concerning the representation of functions as Laplace integrals. This result is extended to the representation as a Laplace integral of a function taking values in $\text{B}$($\text{R}$), the set of bounded linear operators on a Hilbert space $\text{R}$. Also, a theorem is proved encompassing both the result of Devinatz and Akhiezer, and Bochner's theorem on the representation of positive definite functions as Fourier integrals.     

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).     

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.     

Combinatorial maps

The classical approach to maps, as surveyed by Coxeter and Moser (“Generators and Relations for Discrete Groups,” Springer-Verlag, 1980), is by cell decomposition of a surface. A more recent approach, by way of graph embedding schemes, is taken by Edmonds (Notices Amer. Math. Soc.7 (1960), 646), Tutte (Canad. J. Math.31 (5) (1979), 986–1004), and others. Our intention is to formulate a purely combinatorial generalization of a map, called a combinatorial map. Besides maps on orientable and nonorientable surfaces, combinatorial maps include polytopes, tessellations, the hypermaps of Walsh, higher dimensional analogues of maps, and certain toroidal complexes of Coxeter and Shephard (J. Combin. Theory Ser. B.22 (1977), 131–138) and Grünbaum (Colloques internationaux C.N.R.S. No. 260, Problèmes Combinatoire et Théorie des Graphes, Orsay, 1976). The concept of a combinatorial map is formulated graph theoretically. The present paper treats the incidence structure, the diagram, reduciblity, order, geometric realizations, and group theoretic and topological properties of combinatorial maps. Another paper investigates highly symmetric combinatorial maps.     

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 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.     

Stability and forced oscillations

In this paper we derive a differential-difference equation for a circuit involving a lossless transmission line and we give conditions for global asymptotic stability of an equilibrium point, existence and stability of forced oscillations. Some of such problems have been investigated for an equation obtained by R. K. Brayton [Quart. J. Appl. Math.24 (1967), 289–301; O. Lopes, SIAM J. Appl. Math., to appear; M. Slemrod, J. Math. Anal. Appl.36 (1971), 22–40] but, for ours (which governs the same physical problem), better results can be proved. By using suitable Liapunov functionals, we reduce the problem of stability and uniform ultimate boundedness to a scalar ordinary differential inequality.     

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.