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.     

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].     

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).     

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.     

Ergodic theorems for tensor products

An ergodic theorem is proved for tensor products of Banach spaces. As a special case, an ergodic theorem is proved for vector-valued L^p-spaces. This theorem generalizes results of Aribaud, J. Funct. Anal.5 (1970), 395–411, and Dinculeanu, J. Funct. Anal.12 (1973), 229–235.     

Gδ-embeddings in Hilbert space

It is shown that a separable Banach space X has the point of weak to norm continuity property (resp. the Radon-Nikodym property) if and only if there exists a compact G_δ-embedding (resp. an H^δ-embedding) from X into l₂. This solves several questions of J. Bourgain and H. P. Rosenthal (J. Funct. Anal.52 (1983)). It is also shown that every non-relatively compact sequence in a Banach space with property (PC) has a difference subsequence which is a boundedly complete basic sequence. This solves a question of Pelczynski and extends some results of W. B. Johnson and H. P. Rosenthal (Studia Math.43 (1972), 77–92). Various related questions asked in the above Bourgain-Rosenthal reference and by G. A. Edgar and R. F. Wheeler (Pac. J. Math.115 (1984)) and N. Ghoussoub and H. P. Rosenthal (Math. Ann.264 (1983), 321–332) are also settled.     

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.     

Les espaces Lp d'une algèbre de von Neumann définies par la derivée spatiale

This work answers a question raised by A. Connes (on the spatial theory of von Neumann algebras, preprint, Inst. Hautes Études Sci., France) and generalizes for a general von Neumann algebra the theory of non-commutative integration of J. Dixmier (Bull. Soc. Math. France81 (1953)) and I. Segal (Ann. of Math.57 (1973)).     

The free fermion field as a Markov field

A definition of a Markov field is given which allows for noncommuting fields. In the commutative case, we recover Nelson's definition (E. Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis12 (1973), 97–112). Conditional expectations are shown to exist in a regular probability gage space, and, using an independence property of these in the free fermion gage space, it is shown that the free fermion field over $\text{H}$^?1($\text{R}$^d) is a Markov field.     

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].     

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.     

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.     

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.     

On factorial states of operator algebras

Results for the factorial state space of a C¹-algebra A which are analogous to results of 11., 12., 572–612),Tomiyama and Takesaki (Tohôku Math. J. (2) 13 (1961), 498–523) for the pure state space. It is shown that A is prime if and only if the (type I) factorial states are dense in the state space. It follows that every factorial state is a w¹-limit of type I factorial states. The factorial state space of a von Neumann algebra is determined, and it is shown that if A is unital and acts non-degenerately on a Hilbert space then the factorial state space of the generated von Neumann algebra restricts precisely to the factorial state space of A. It is shown that the set of factorial states is w¹-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.     

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.     

Upper semicontinuous fuzzy sets and applications

Let (X, τ) be a generalized topological space of type $\text{V}$_α (see A. Appert and K. Fan, “Espaces topologiques intermédiares,” Herman, Paris, 1951) and (L, ?) be a complete Brouwerian lattice such that the dual lattice of (L, ?) is also Brouwerian. We prove that every upper semicontinuous L-fuzzy subset of X can be represented by a τ-closed random set. As an important application we obtain a fuzzification of measurable spaces as well as of topological spaces. In particular a concept of measurable (open) L-fuzzy sets is developed.     

A note on the oscillatory property for nonlinear difference equations and differential equations

Criteria are given to determine the oscillatory property of solutions of the nonlinear difference equation: Δ^du_n + ∑_{i = 1}^mp_inf_i(u_n, Δu_n,…,Δ^{d ? 1}u_n) = 0, n = 0, 1, 2,…, where d is an arbitrary integer, generalizing results that have been obtained by B. Szmanda (J. Math. Anal. Appl.79 (1981), 90–95) for d = 2. Analogous results are given for the differential equation: u^(d) + ∑_{i = 1}^mp_i(t)f_i(u, u′,…, u^{(d ? 1)}) = 0, t ? t₀, which coincide with the criteria given by 2., 3., 599–602) and 4., 5., 6., 715–719) for the case m = 1.     

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 class of nonlinear elliptic problems

We obtain a strict coercivity estimate, (generalizing that of T. I. Seidman [J. Differential Equations 19 (1975), 242–257] in considering spatial variation) for second order elliptic operators A: u ? ?▽ · γ(·, ▽u) with γ “radial in the gradient” ?γ(·, ξ) = a(·, |ξ|)ξ for ξ ? $\text{R}$^m. The estimate is then applied to obtain existence of solutions of boundary value problems: $\text{?▽ ·}\text{a}\text{?}\text{(·, u, |▽u|) ▽u = f(·, u, ▽u)}$ with Dirichlet conditions.