Let ƒ:H → (—∞,∞] be lower semicontinuous, where H is a real Hilbert space. An approach based upon nonsmooth analysis and optimization is used in order to characterize monotonicity of ƒ with respect to a cone, as well as Lipschitz behavior and constancy. The results, which involve hypotheses on the proximal subgradient ∂ πƒ, specialize on the real line to yield classical characterizations of these properties in terms of the Dini derivate. They also give new extensions of these results to the multidimensional case. A new proof of a known characterization of convexity in terms of proximal subgradient monotonicity is also given.

Recently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.

1. Introduction. A space of orderings will refer to what was called a “set of quasi-orderings” in [5]. That is, a space of orderings is a pair (X, G) where G is an elementary 2-group (i.e. x2 = 1 for all x ∈ G) with a distinguished element – 1 ∈ G, and X is a subset of the character group x(G) = Horn (G, {1, –1};) satisfying the following properties:

01: X is a closed subset of χ(G).

02: σ(−l) = −1 holds for all σ ∊ X.

03: X⊥ = {a ∊ G|χa = 1 for all a ∊ X} = 1.

04: If f and g are forms over G and if x ∊ Df⊗g, then there exist y ∊ Df and z ∊ Dg such that x ∊ D(y, z).

Soient E un espace compact (par exemple l' intervalle et μ une mesure de Radon sur E. Une fonction numérique f intégrable pour μ définit une mesure sur l'ensemble des parties mesurables (pour μ) de E. On peut aussi considérer la forme linéaire qu'elle définit sur l'espace C des fonctions numériques continues dans E, et cette forme linéaire est continue pour la topologie définie par la norme . Si on appelle encore “mesure” une forme linéaire continue sur C, le théoréme de Lebesgue-Nikodym classique caractérise celles de ces “mesures” forme qui sont de la . par une condition de “continuité absolue” par rapport à μ.

The q-shifted factorial (a)n or (a; q)n is

and an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials

1.1

where

1.2

and

1.3

We shall refer to these polynomials as the Askey-Wilson polynomials or the orthogonal 4ϕ3 polynomials. They generalize the 6 — j symbols and are the most general classical orthogonal polynomials, [2].

We study a class of generalized hypergeometric functions in several variables introduced by A. Korânyi. It is shown that the generalized Gaussian hypergeometric function is the unique solution of a system partial differential equations. Analogues of some classical results such as Kummer relations and Euler integral representations are established. Asymptotic behavior of generalized hypergeometric functions is obtained which includes some known estimates.

The following result is well known (as usual, [x]denotes the integral part of x):

(A) Let α and β be positive irrational numbers satisfying

1

Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.

The theorem has a converse, and the following holds:

(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

The work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

In (8) M. V. Menon investigates the diagonal equivalence of a non-negative matrix A to one with prescribed row and column sums and shows that this equivalence holds provided there exists at least one non-negative matrix with these row and column sums and with zeros in exactly the same positions A has zeros. However, he leaves open the question of when such a matrix exists. W. B. Jurkat and H.J. Ryser in (7) study the convex set of all m × n non-negative matrices having given row and column sums.

Let

1

2

Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

Let A be an artin algebra. Let ( be a component of the stable Auslander- Reiten quiver of A. If is periodic, then the structure of G is known. Here, we are going to consider the case when is non-periodic: we will show that is isomorphic to Z with a valued quiver. In particular, there is no cyclic path in .

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm

(f ∊ ).

It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).

With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F. V. Atkinson.

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.

The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.

The object of this note is to prove the following theorem.

THEOREM. Let A be a division ring with centre Z, and suppose that for every x in A, some power (depending on x) is in . Then A is commutative.

We study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

In this paper, we study the boundary version of the classical Cartan theorem. We show that for some weakly pseudoconvex domains, when a holomorphic self-mapping has a sufficiently high order of contact (which depends only on the geometric properties of the domains) with the identical map at some boundary point, then it must coincide with the identity.