A Simple Recurrence for Covers of the Sphere With Branch Points of Arbitrary Ramification

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800’s. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-Mélou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called m-Eulerian trees. In this paper, we give a simple partial differential equation for Bousquet-Mélou and Schaeffer’s generating series, and for Goulden and Jackson’s generating series, as well as a new proof of the result by Bousquet-Mélou and Schaeffer. We apply algebraic methods based on Lagrange’s theorem, and combinatorial methods based on a new use of Bousquet-Mélou and Schaeffer’s m-Eulerian trees. Supported by a Discovery Grant from NSERC. Research supported by a Postgraduate Scholarship from NSERC. Received October 8, 2005     

A New and Self-Contained Proof of Borwein’s Norm Duality Theorem

Borwein’s norm duality theorem establishes the equality between the outer (inner) norm of a sublinear mapping and the inner (outer) norm of its adjoint mappings. In this note we provide an extended version of this theorem with a new and self-contained proof relying only on the Hahn-Banach theorem. We also give examples showing that the assumptions of the theorem cannot be relaxed. This author is supported by Grant BES-2003-0188 from FPI Program of MEC (Spain).     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

A simple proof of some ergodic theorems

Some ideas of T. Kamae’s proof using nonstandard analysis are employed to give a simple proof of Birkhoff’s theorem in a classical setting as well as Kingman’s subadditive ergodic theorem.     

Families of vector measures of uniformly bounded variation

We characterize families of vector measures of uniformly bounded variation and semivariation in terms of additivity properties. A classical theorem of Rickart and a simplified proof of Nikodym’s boundedness theorem follow. Received: 3 January 2006; Revised: 31 March 2006 The second named author was supported by Estonian Science Foundation Grant 5704.     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

Two applications of Boolean models

Semantical arguments, based on the completeness theorem for first-order logic, give elegant proofs of purely syntactical results. For instance, for proving a conservativity theorem between two theories, one shows instead that any model of one theory can be extended to a model of the other theory. This method of proof, because of its use of the completeness theorem, is a priori not valid constructively. We show here how to give similar arguments, valid constructively, by using Boolean models. These models are a slight variation of ordinary first-order models, where truth values are now regular ideals of a given Boolean algebra. Two examples are presented: a simple conservativity result and Herbrand's theorem. Received December 5, 1995     

Quantization of lie Bialgebras via the Formality of the operad of Little Disks

We give a proof of the Etingof–Kazhdan theorem on quantization of Lie bialgebras based on the formality of the chain operad of little disks and show that the Grothendieck–Teichmüller group acts non-trivially on the corresponding quantization functors. Partially supported by an NSF Grant and A. Sloan Research Fellowship Received: November 2004 Revision: April 2006 Accepted: May 2006     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

An Inductive Proof of the Berry-Esseen Theorem for Character Ratios

Bolthausen used a variation of Stein’s method to give an inductive proof of the Berry-Esseen theorem for sums of independent, identically distributed random variables. We modify this technique to prove a Berry-Esseen theorem for character ratios of a random representation of the symmetric group on transpositions. An analogous result is proved for Jack measure on partitions. Received March 11, 2005     

The translation representation theorem

Short of a new theorem on semigroups of operators, a new proof of an old theorem on this subject is a suitable offering to Einar Hille on his 85th birthday.The work of both authors was supported in part by the National Science Foundation, the first author under Grant No. MCS-76-07039 and the second author under Grant No. MCS-77-04908 A 01.     

Caractérisation spectrale de la forme de Jordan

Soit , où désigne l'ensemble des matrices n×n à coefficients complexes. Nous montrons qu'on peut complètement caractériser la forme de Jordan de A en examinant le polynôme caractéristique de tA+X pour tous les tC et tous les . Ceci nous permet de donner une démonstration plus élémentaire d'un théorème de Baribeau et Ransford sur les transformations holomorphes de qui préservent le spectre.

Denote by the set of complex n×n matrices, and let . We give a variational, purely spectral characterization of the Jordan form of A by examining the characteristic polynomial of the perturbed matrices tA+X for tC and . This allows us to give a more elementary proof of a theorem of Baribeau and Ransford on spectrum-preserving holomorphic maps on .     

Packing Non-Returning <Emphasis Type="Italic">A</Emphasis>-Paths*

Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint nonzero A-paths in group-labeled graphs [1], which is a generalization of Mader's theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside , which is given by an oracle for non-bipartite matching. * Research is supported by OTKA grants T 037547 and TS 049788, by European MCRTN Adonet, Contract Grant No. 504438 and by the Egerváry Research Group of the Hungarian Academy of Sciences. † The author is a member of the Egerváry Research Group (EGRES).     

Independence numbers of graphs and generators of ideals

This article investigates the generators of certain homogeneous ideals which are associated with graphs with bounded independence numbers. These ideals first appeared in the theory oft-designs. The main theorem suggests a new approach to the Clique Problem which isNP-complete. This theorem has a more general form in commutative algebra dealing with ideals associated with unions of linear varieties. This general theorem is stated in the article; a corollary to it generalizes Turán’s theorem on the maximum graphs with a prescribed clique number. Research supported in part by NSF Grant MCS77-03533.     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

Subspace theorems for homogeneous polynomial forms

We prove a subspace theorem for homogeneous polynomial forms which generalizes Schmidt’s subspace theorem for linear forms. Further, we formalize the subspace theorem into a form which is just the counterpart of a second main theorem in Nevanlinna’s theorem, and also suggest a problem. The work of the first author was partially supported by NSFC of China: Project. No. 10371064. The second author was partially supported by a UGC Grant of Hong Kong: Project No. 604103.     

Hyperfinite construction of G-expectation

The hyperfinite G-expectation is a nonstandard discrete analogue of G-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time G-expectation operator is defined as a hyperfinite G-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time G-expectation. We develop the basic theory for hyperfinite G-expectations and prove an existence theorem for liftings of (continuous-time) G-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the G-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of G-expectations, Stoch. Process. Appl. 122(2) (2012), pp. 664–675]).     

Jensen measures and analytic multifunctions

This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions. The author was partially supported by an NSF Grant.     

A short proof of Halin’s grid theorem

We give a short proof of Halin’s theorem that every thick end of a graph contains the infinite grid.