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 $(g, \nabla, \nabla^{*})$ of a Riemannian metric $g$ and two affine connections $\nabla$ and $\nabla^{*}$. 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 $\nabla$- and $\nabla^{*}$-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 $\mathbb{C}^n$, and $dV(z)$ normalized Lebesgue measure on $B$. For $\alpha > -1$, define $dV^\alpha (z)=(1-|z|^2)^\alpha dV(z)$. Let ${\cal H}(B)$ denote the space of holomorhic functions on $B$, and for $0 < p <\infty$, let ${\mathcal{A}}^p(dV_\alpha)$ denote $L^p(dV_\alpha)\cap {\cal H}(B)$. In this note we characterize ${\mathcal{A}}^p(dV_\alpha)$ as those functions in ${\cal H}(B)$ whose images under the action of a certain set of differential operators lie in $L^p(dV_\alpha)$. This is valid for $1 \le p <\infty$. We also show that the Ces\`aro operator is bounded on ${\mathcal{A}}^p(dV_\alpha)$ for $0<p<\infty$. Analogous results are given for the polydisc.

We show that for any field $k$, separable $\mathbb{A}^1$-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 [4], we obtain structure theorems for spaces of half-integral weight cusp forms $S_{k/2}{4N}{\chi}$ where $k$ and $N$ are odd nonnegative integers with $k \geq 3$, and $\chi$ 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 $\chi$, by giving conditions under which a newform has an equivalent cusp form in $S_{k/2}{4N}{\chi}$. 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 $\chi$. 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 $\{B_t^{(\mu)},t\geqq0\}$ be a one-dimensional Brownian motion with constant drift $\mu\in{\bf R}$ starting from $0$. In this paper we show that $$ Z_t^{(\mu)} = \exp(-B_t^{(\mu)}) \int_0^t \exp(2B_s^{(\mu)}) ds $$ 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 $\{Z_t^{(\mu)},t\geqq0\}$ 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 $\widehat{G}$ introduced by Arakelyan which is by definition the union of all sets ${1\over w }G$ with $w \in G^c$. The main result is the description of all isomorphisms between these multipliers algebras. As a consequence one obtains: If two multiplier algebras $M(H(G_1 ))$ and $M(H(G_2 ))$ are isomorphic then $\widehat{G_1 }$ is equal to $\widehat{G_2}$. Two algebras $H(G_1 )$ and $H(G_2)$ are isomorphic with respect to the Hadamard product if and only if $G_1$ is equal to $G_2$. Further the following uniqueness theorem is proved: If $G_1$ is a domain containing $0$ and if $M(H(G))$ is isomorphic to $H(G_1 )$ then $G_1$ is equal to $\widehat{G}$.

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