Abstract: 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.

Abstract: Let 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

Abstract: In this paper, we study two generalizations of the Stirling numbers of the first and second kinds, inspired from their combinatorial interpretation in terms of 0-1 tableaux. They are the 𝔄-Stirling numbers and the partial Stirling numbers. In particular, we give a q and a p, q-analogue of convolution formulae for Stirling numbers of the second kind, due to Chen and Verde-Star, and we extend these formulae to Stirling numbers of the first kind. Included in this study are the a, d-progressive Stirling numbers, corresponding to 0-1 tableaux with column lengths from an arithmetic progression ﹛a + id﹜i≥0.

Abstract: Coercive closed forms on L 2 -spaces are studied whose associated L 2 -semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

Abstract: In his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujan's values are established; four are proved for the first time, while the remaining four had been previously proved by Ramanathan by entirely different methods. Our proofs employ some of Ramanujan's beautiful eta-function identities, which have not been heretofore used for evaluating continued fractions.

Abstract: We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.

Abstract: An important question in the study of moment problems is to determine when a fixed point in ℝn lies in the moment cone of vectors , with μ a nonnegative measure. In associated optimization problems it is also important to be able to distinguish between the interior and boundary of the moment cone. Recent work of Dachuna-Castelle, Gamboa and Gassiat derived elegant computational characterizations for these problems, and for related questions with an upper bound on μ. Their technique involves a probabilistic interpretation and large deviations theory. In this paper a purely convex analytic approach is used, giving a more direct understanding of the underlying duality, and allowing the relaxation of their assumptions.

Abstract: This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

Abstract: For every m-tuple of operators acting on a Hilbert space, it is shown that there exists a common dilation of these operators to mcommuting normal operators on some larger Hilbert space. We then introduce a norm on the m-fold cartesian product of ℬ(ℋ) that is defined to be, for a given w-tuple, the infimum of the joint spectral radii of all joint normal dilations of the m operators. This norm has several good features, one of which is that it is invariant under the passage to adjoints.

Abstract: Resume We consider surfaces in ℝ3 with generic wave front singularities, called A-mersions (A for Arnold who classified these singularities), we classify A-mersions up to generic homotopy. For the sphere they are classified by the degree of the Gauss map, and for higher genus surfaces the degree of the Gauss map and the number of zig-zags classify the A-mersion.

Abstract: Let m, n be nonnegative integers and B (m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = P m,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B (m+n). If m = mv , n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B (m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n (0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n (z)| when z ∈ K. These results generalize properties of the classical Pade approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.

Abstract: An additive subgroup P of a skew field F is called a prime of F if P does not contain the identity, but if the product xy of two elements x and y in F is contained in P, then x or y is in P. A prime segment of F is given by two neighbouring primes P 1 ⊃ P 2; such a segment is invariant, simple, or exceptional depending on whether A(P 1) = {a ∈ P 1 | P 1 aP 1 ⊂ P 1} equals P 1, P 2 or lies properly between P 1 and P 2. The set T(F) of all primes of F together with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fields F of fractions of group rings of certain right ordered groups. In particular, groups G of affine transformations on ordered vector spaces V are considered, and the relationship between properties of Dedekind cuts of V, certain right orders on G, and chains of prime segments of F is investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.

Abstract: In this paper, we study the boundedness of є-families of operators on Triebel-Lizorkin with wide range of parameters. We also prove that є -families of operators are bounded from Triebel-Lizorkin spaces into (generalized) tent spaces, and obtain a characterization of certain Triebel-Lizorkin spaces in terms of tent spaces. In particular, the boundedness of fractional operators in Triebel-Lizorkin, and a sharp version of T\theorem for generalized Calderon-Zygmund operators on Triebel-Lizorkin spaces can be considered as applications of (proofs of) these results.

Abstract: The intriguing relation between the theory of quadratic forms and Galois theory has been of interest for a long time. (See for example [Wi:1936], [Wr:1979], [Wr:1983], [Wr:1985], [JWr:1989], [AEJ:1984], among others.) However, recently the connection between the Witt ring structure for a field F (of characteristic not 2) and its Galois groups was made quite precise. If L is a Galois field extension of F , with Gal(L/F ) ∼= G, we call L a G-extension of F . Given a field F , with charF 6= 2, one can consider the field extension F /F which is the compositum over F of all Z/2Z, Z/4Z-, and D4-extensions of F . (Here D4 denotes the dihedral group of order 8.) One can then show that the Galois group GF of F (3) over F , hereafter referred to as the W-group of F , is determined by W (F ), and that GF determines W (F ) except in the case when the level s(F ) of F is ≤ 2 and the form 〈1, 1〉 is universal. (In this paper basic knowledge of quadratic form theory and profinite groups will be assumed. See [La:1973] or [Sc:1985] for the former and [N:1971] and [Se:1965] for the latter. Throughout we will assume all fields to be of characteristic not 2.) In other words, knowledge of GF is essentially equivalent to knowledge of W (F ). (See [MiSm:1993], [:1990], [MiSp:1995], [Sm:1988], [Sp:1987].) This relationship between a specific Galois group of F and W (F ) opened a new way of attacking some questions in quadratic from theory and posed other new questions. In particular, it allows one to use the techniques of inverse Galois theory to study some classical problems in the theory of Witt rings. One of the most outstanding problems is the characterization of Witt rings in the category of all rings. In spite of many efforts, very little is known. Indeed, we know only

Abstract: The topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

Abstract: We extend the min-max methods used in the critical point theory of differentiable functionals on smooth manifolds to the case of continuous functionals on a complete metric space. We study the topological properties of the min-max generated critical points in this new setting by adopting the methodology developed by Ghoussoub in the smooth case. Many old and new results are extended and unified and some applications are given.

Abstract: From Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big q-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomi- als, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polyno- mial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.

Abstract: Let T = (ju)uçjd be a semigroup of measure preserving transformations on a measure space (Q, f, ji). The main result of the paper is the proof of a.e. convergence for the moving averages #/„ where {Fjn} is a superadditive process and {/„} is a sequence of cubes in Z+ satisfying the \"cone-condition\". The identification of the limit is given. A moving local theorem is also proved.

Abstract: The operator lm is defined as ra-fold indefinite integration with zero constants of integration. The zero distribution of Im(p) for polynomials p is studied in general, and for two special classes of polynomials in detail. The main results are: (i) The zeros of In(Pn), where Pn(z) is the n-th Legendre polynomial, converge to a certain algebraic curve; (ii) the zeros of Y%Ln+\\{nz) /k\\ (c > 2 an integer) converge to pieces of a circle and of two \"Szegô curves\".

Abstract: The well-known Tschirnhausen transformation, , eliminates the second term of the polynomial xn + axn-l + …. By a mere repeated application of this transformation, one can decide whether a given element of k[[x,y]] is prime (irreducible) or not. Here K is an algebraically closed field of characteristic 0. A generalised version of Hensel's Lemma is developed for the proofs. The entire paper can be understood by undergraduate students.