Ramanujan's theories of elliptic functions to alternative bases
Bruce C.
Berndt;
S.
Bhargava;
Frank G.
Garvan
4163-4244
Abstract: In his famous paper on modular equations and approximations to $ \pi$, Ramanujan offers several series representations for $1/\pi$, which he claims are derived from "corresponding theories" in which the classical base $q$ is replaced by one of three other bases. The formulas for $1/\pi$ were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.
On the classification of $(n-k+1)$-connected embeddings of $n$-manifolds into $(n+k)$-manifolds in the metastable range
Rong
Liu
4245-4258
Abstract: For an $(n - k + 1)$-connected map $f$ from a connected smooth $ n$-manifold $M$ to a connected smooth $ (n + k)$-manifold $ V$, where $M$ is closed, we work out the isotopy group ${[M \subset V]_f}$ in the metastable range $n \leqslant 2k - 4$. To prove our results, we develop the Hurewicz-type theorems which provide us with the efficient methods of computing the homology groups with local coefficients from the homotopy groups.
The order bidual of almost $f$-algebras and $d$-algebras
S. J.
Bernau;
C. B.
Huijsmans
4259-4275
Abstract: It is shown in this paper that the second order dual $A''$ of an Archimedean (almost) $ f$-algebra $A$, equipped with the Arens multiplication, is again an (almost) $f$-algebra. Also, the order continuous bidual $ (A')_n'$ of an Archimedean $ d$-algebra $A$ is a $d$-algebra. Moreover, if the $d$-algebra $A$ is commutative or has positive squares, then $ A''$ is again a $ d$-algebra.
Formes diff\'erentielles non commutatives et cohomologie \`a coefficients arbitraires
Max
Karoubi
4277-4299
Abstract: The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between $ K$-theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes différentielles non commutatives et opérations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm. Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836.
Characterizations of Bergman spaces and Bloch space in the unit ball of ${\bf C}\sp n$
Cai Heng
Ouyang;
Wei Sheng
Yang;
Ru Han
Zhao
4301-4313
Abstract: In this paper we prove that, in the unit ball $B$ of $ {{\mathbf{C}}^n}$, a holomorphic function $f$ is in the Bergman space $L_a^p(B),\;0 < p < \infty$, if and only if $\displaystyle \int_B {\vert\tilde \nabla } f(z){\vert^2}\vert f(z){\vert^{p - 2}}{(1 - \vert z{\vert^2})^{n + 1}}d\lambda (z) < \infty ,$ where $\tilde \nabla$ and $\lambda$ denote the invariant gradient and invariant measure on $B$, respectively. Further, we give some characterizations of Bloch functions in the unit ball $ B$, including an exponential decay characterization of Bloch functions. We also give the analogous results for $\operatorname{BMOA} (\partial B)$ functions in the unit ball.
The alternative torus and the structure of elliptic quasi-simple Lie algebras of type $A\sb 2$
Stephen
Berman;
Yun
Gao;
Yaroslav
Krylyuk;
Erhard
Neher
4315-4363
Abstract: We present the complete classification of the tame irreducible elliptic quasi-simple Lie algebras of type ${A_2}$, and in particular, specialize on the case where the coordinates are not associative. Here the coordinates are Cayley-Dickson algebras over Laurent polynomial rings in $\nu \geqslant 3$ variables, which we call alternative tori. In giving our classification we need to present much information on these alternative tori and the Lie algebras coordinatized by them.
The complex zeros of random polynomials
Larry A.
Shepp;
Robert J.
Vanderbei
4365-4384
Abstract: Mark Kac gave an explicit formula for the expectation of the number, ${\nu _n}(\Omega )$, of zeros of a random polynomial, $\displaystyle {P_n}(z) = \sum\limits_{j = 0}^{n - 1} {{\eta _j}{z^j}} ,$ in any measurable subset $\Omega$ of the reals. Here, $ {\eta _0}, \ldots ,{\eta _{n - 1}}$ are independent standard normal random variables. In fact, for each $n > 1$, he obtained an explicit intensity function $ {g_n}$ for which $\displaystyle {\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{g_n}(x)\,dx.}$ Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset $\Omega$ of the complex plane $\mathbb{C}$. Namely, we show that $\displaystyle {\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{h_n}(x,y)\,dxdy + \int_{\Omega \cap \mathbb{R}} {{g_n}(x)\,dx,} }$ where ${h_n}$ is an explicit intensity function. We also study the asymptotics of ${h_n}$ showing that for large $n$ its mass lies close to, and is uniformly distributed around, the unit circle.
On invariants for $\omega\sb 1$-separable groups
Paul C.
Eklof;
Matthew
Foreman;
Saharon
Shelah
4385-4402
Abstract: We study the classification of $ {\omega _1}$-separable groups by using Ehrenfeucht-Fraïssé games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming $\diamondsuit $.
Wiman-Valiron theory in two variables
P. C.
Fenton
4403-4412
Abstract: Inequalities are obtained for the coefficients of the Taylor series of an entire function of two complex variables and used to obtain an inequality for the maximum modulus of the function in terms of the maximum term of the series.
On minimal sets of scalar parabolic equations with skew-product structures
Wen Xian
Shen;
Yingfei
Yi
4413-4431
Abstract: Skew-product semi-flow ${\Pi _t}:X \times Y \to X \times Y$ which is generated by $\displaystyle \left\{ \begin{gathered}{u_t} = {u_{xx}} + f(y \cdot \,t,x,u,{u_x... ...D\;{\text{or }}N\;{\text{boundary conditions}} \end{gathered} \right.$ is considered, where $X$ is an appropriate subspace of $ {H^2}(0,1),\;(Y,\,\mathbb{R})$ is a minimal flow with compact phase space. It is shown that a minimal set $E \subset X \times Y$ of ${\Pi _t}$ is an almost $1{\text{ - }}1$ extension of $Y$, that is, set $ {Y_0} = \{ y \in Y\vert\operatorname{card} (E \subset {P^{ - 1}}(y)) = 1\}$ is a residual subset of $Y$, where $ P:X \times Y \to Y$ is the natural projection. Consequently, if $(Y,\mathbb{R})$ is almost periodic minimal, then any minimal set $ E \subset X \times Y$ of $ {\Pi _t}$ is an almost automorphic minimal set. It is also proved that dynamics of $ {\Pi _t}$ is closed in the category of almost automorphy, that is, a minimal set $ E \subset X \times Y$ of $ {\Pi _t}$ is almost automorphic minimal if and only if $(Y,\mathbb{R})$ is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.
Uniformisations partielles et crit\`eres \`a la Hurewicz dans le plan
Dominique
Lecomte
4433-4460
Abstract: Résumé: On donne des caractérisations des boréliens potentiellement d'une classe de Wadge donnée, parmi les boréliens à coupes verticales dénombrables d'un produit de deux espaces polonais. Pour ce faire, on utilise des résultats d'uniformisation partielle.
On some subalgebras of $B(c\sb 0)$ and $B(l\sb 1)$
F. P.
Cass;
J. X.
Gao
4461-4470
Abstract: For a non-reflexive Banach space $X$ and $w \in {X^{{\ast}{\ast}}}$, two families of subalgebras of $ B(X),\;{\Gamma _w} = \{ T \in B(X)\vert{T^{{\ast}{\ast}}}w = kw\;{\text{for some}}\;k \in \mathbb{C}{\text{\} }}$, and $ {\Omega _w} = \{ T \in B(X)\vert{T^{{\ast}{\ast}}}w \in w \oplus \hat X\}$ for $w \in {X^{{\ast}{\ast}}}\backslash \hat X$ with $ {\Omega _w} = B(X)$ for $w \in \hat X$, were defined originally by Wilansky. We consider $X = {c_0}$ and $X = {l_1}$ and investigate relationships between the subalgebras for different $w \in {X^{{\ast}{\ast}}}$. We prove in the case of ${c_0}$ that, for $w \in {X^{{\ast}{\ast}}}\backslash \hat X$, all $ {\Gamma _w}$'s are isomorphic and all $ {\Omega _w}$ 's are isomorphic. For $X = {l_1}$, where it is known that not all ${\Gamma _w}$'s are isomorphic and not all ${\Omega _w}$ 's are isomorphic, we show, surprisingly, that subalgebras associated with a Dirac measure on $\beta \mathbb{N}\backslash \mathbb{N}$, regarded as a functional on $ l_1^{\ast}$, are isomorphic to those associated with some Banach limit (i.e., a translation invariant extended limit). We also obtain a representation for the operators in the subalgebras $\{ \cap {\Gamma _f}\vert f\;{\text{is a Banach limit}}\}$ and $\{ \cap {\Omega _f}\vert f\;{\text{is a Banach limit}}\}$ of $B({l_1})$.
The Kechris-Woodin rank is finer than the Zalcwasser rank
Haseo
Ki
4471-4484
Abstract: For each differentiable function $f$ on the unit circle, the Kechris-Woodin rank measures the failure of continuity of the derivative function $f'$ while the Zalcwasser rank measures how close the Fourier series of $f$ is to being a uniformly convergent series. We show that the Kechris-Woodin rank is finer than the Zalcwasser rank. Roughly speaking, small ranks mean the function is well behaved and big ranks imply bad behavior. For each countable ordinal, we explicitly construct a continuous function with everywhere convergent Fourier series such that the Zalcwasser rank of the function is bigger than the ordinal.
Minimization problems for noncoercive functionals subject to constraints
Khoi Le
Vy;
Klaus
Schmitt
4485-4513
Abstract: We consider noncoercive functionals on a reflexive Banach space and establish minimization theorems for such functionals on smooth constraint manifolds. These results in turn yield critical point theorems for certain classes of homogeneous functionals. Several applications to the study of boundary value problems for quasilinear elliptic equations are included.
${\bf R}$-trees, small cancellation, and convergence
Andrew
Chermak
4515-4531
Abstract: The "metric small cancellation hypotheses" of combinatorial group theory imply, when satisfied, that a given presentation has a solvable Word Problem via Dehn's Algorithm. The present work both unifies and generalizes the small cancellation hypotheses, and analyzes them by means of group actions on trees, leading to the strengthening of some classical results.
Triangles of groups
Andrew
Chermak
4533-4558
Abstract: Given a certain commutative diagram of groups and monomorphisms, does there necessarily exist a group in which the given diagram is essentially a diagram of subgroups and inclusions? In general, the answer is negative, but J. Corson, and Gersten and Stallings have shown that in the case of a "non-spherical triangle" of groups the answer is positive. This paper improves on these results by weakening the non-sphericality requirement.