The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple (g,∇, ∇*) of a Riemannian metric g and two affine connections ∇ and ∇*. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning ∇- and ∇*-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.

Let B denote the unit ball in ℂn, and dV(z) normalized Lebesgue measure on B. For α > -1, define dVα(z) = (1 - \z\2)αdV(z). Let (B) denote the space of holomorhic functions on B, and for 0 < p < ∞, let p(dVα) denote Lp(dVα) ∩ (B). In this note we characterize p(dVα) as those functions in (B) whose images under the action of a certain set of differential operators lie in Lp(dVα). This is valid for 1 < p < oo. We also show that the Cesàro operator is bounded on p(dVα) for 0 < p < oo. Analogous results are given for the polydisc.

We show that for any field k, separable A1-forms over commutative k-algebras are trivial.

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2(4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

We construct a measure over the string bundle associated to the loop space of a Riemannian manifold. We deduce a representation of a finite energy Kac-Moody group analoguous to the energy representation.

Let be a one-dimensional Brownian motion with constant drift µ ∈ R starting from 0. In this paper we show that

gives rise to a diffusion process and we explain how this result may be considered as an extension of the celebrated Pitman’s 2M - X theorem. We also derive the infinitesimal generator and some properties of the diffusion process and, in particular, its relation to the generalized Bessel processes.

Let G be a domain in the complex plane containing zero and H(G) be the set of all holomorphic functions on G. In this paper the algebra M(H(G)) of all coefficient multipliers with respect to the Hadamard product is studied. Central for the investigation is the domain introduced by Arakelyan which is by definition the union of all sets with w ∈ Gc. The main result is the description of all isomorphisms between these multipliers algebras. As a consequence one obtains: If two multiplier algebras M(H(G1)) and M(H(G2)) are isomorphic then is equal to Two algebras H(G1) and H(G2) are isomorphic with respect to the Hadamard product if and only if G1 is equal to G2. Further the following uniqueness theorem is proved: If G1 is a domain containing 0 and if M(H(G)) is isomorphic to H(G1) then G1 is equal to .

We give a purely algebraic proof that the rational functions Pn(t), Qn(t) inductively defined by the recurrence relation (1), (2) respectively, are polynomials. The proof reveals the Hirota bilinear relations satisfied by the τ-functions.