Using ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].

In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

Extreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.

The spectrum of the Laplacian or more generally of a Schrödinger operator on an open manifold may have possibly a complicated aspect. For example, a Cantor set in the real axis may appear as the spectrum even for an innocent looking potential on a standard Riemannian manifold (see J. Moser [10]). The fundamental result of the spectral theory of periodic Schrödinger operators, however, says that the picture of the spectrum of a Schrödinger operator on ℝn with a periodic potential is simple; indeed the spectrum consists of a series of closed intervals of the real axis without accumulation, separated in general by gaps outside the spectrum (see M. Reed and B. Simon [13] or M. M. Skriganov [15] for instance).

Let 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

Let Sbe a topological semigroup, K a compact convex subset of a separated convex space Eand T: S x K → K an affine action (denoted by (s, x) → Ts(x),s ∈ S, x ∈ K) of S as continuous affine maps on K. It is shown in A. Lau and J. Wong [22] that the weakly left uniformly measurable functions WLUM(S) on S has a left invariant mean iff Shas the fixed point property for weakly measurable affine actions, i.e. affine actions such that the scalar function s → x*Ts(x) is measurable for each x ∈ K and x*∈ E* (the dual of E) with respect to the Borel sets in S. It is natural to ask for a “strongly” measurable analogue of this result. There are a number of ways to define such actions and the corresponding functions on S. In this paper, we obtained a neat analogue of this fixed point theorem by a suitable choice of strong measurability which naturally leads to another new fixed point theorem for separable actions. Also, we shall unify these and many known fixed point theorems and extend and generalise them to anti-actions of S as bounded linear operators on Banach spaces

In this note we give a symplectic approach to Yang Mills theory for non commutative n-tori, inspired by the classical theory of Atiyah and Bott.

Let = ℝ or ℂ. Let U be an open subset of n. Let X be a closed analytic subset of U and let Z be a proper closed analytic subset of X. Let M(X;Z) denote the ring of meromorphic functions on Xwhose poles lie in Z. Let M be the families of formal power series generated by a finite sequence ƒi,… ,ƒq ∈ M(X; Z) ⟦ y⟧p (For details, see § 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.

We present a series of conjectures for immanants, together with the supporting evidence we possess for them. The conjectures are loosely organized into three families. The first concerns inequalities involving the immanants of totally positive matrices (Le.,real matrices with nonnegative minors). This includes, for example, the conjecture that immanants of totally positive matrices are nonnegative. The second family involves the immanants of Jacobi-Trudi matrices. These conjectures were suggested by a previous conjecture of Goulden and Jackson (recently proved by C. Greene) that the immanants of Jacobi-Trudi matrices are polynomials with nonnegative coefficients. The third family involves geometric and combinatorial structures associated with total positivity and paths in acyclic digraphs.

By exploiting the known quasi-heredity of Schur algebras, the structure of basic algebras of the Schur algebras Sk(n, p) for n ≥ p over an algebraically closed field k is completely determined.

Let G be a profinite group. We define an extension (E, J) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of K satisfying: for some open normal subgroup Sof G, and the map is continuous (A being discrete).

In this paper a q-analogue of Gegenbauer's addition formula for Bessel functions is obtained by using the orthogonality relation for the q-Ultraspherical polynomials of Rogers'. Also some product formulas and an integral representation for the Hahn-Exton q-Bessel functions are obtained.

Our main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

In [Con2] Connes introduced cyclic cohomology HC*(A) for an associative algebra A. When A is a complex algebra he constructed a Chern character for p-summable Fredholm modules over A taking values in HC*(A). As a very special case, when X is a closed C∞-manifold and A = C∞ (X), this construction recovers the usual Chern character, which is a rational isomorphism from the K-homology K0(X) to , the even dimensional deRham homology of X.

Let π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

We first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

In [6] and [7], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [9], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to k-setcontraction maps, k < 1. Other versions for completely continuous maps were given by Fournier-Peitgen [2] and G. Fournier [1].

The purpose of this paper is to generalise some of these results to upper semi continuous multivalued maps which are K-set contractions, k < 1, and differentiable at the origin and infinity.

In this paper the question of weak and strong local optimality of a Lipschitz (as opposed to C1 ) extremal is addressed. We show that the classical Jacobi sufficient conditions can be extended to the case of Lipschitz candidates. The key idea for this achievement lies in proving that the “generalized” strengthened Weierstrass condition is equivalent to the existence of a “feedback control” function at which the maximum in the “true” Hamiltonian is attained. Then the Hamilton-Jacobi approach is pursued in order to conclude the result.

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).

Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.