Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and .     

On analytic functions related with generalized Robertson functions

Analytic functions f are called Robertson functions for which zf^′ is α-spiral-like. This concept is generalized by several authors and a class real, of analytic functions is introduced and studied. It is noted that the functions in are of bounded boundary rotation and consists of Robertson functions.In this paper, we use the class to define a new class of analytic functions which unifies a number of classes previously studied such as the class of close-to-convex functions of higher order. Some interesting properties of this class, including coefficient problems, inclusion results and a sufficient condition for univalency are studied.     

On a universal differential equation for the analytic terms of C-superpositions on the real line

It is proved that every continuous function on the real line can be approximated uniformly (in the sense of a specific norm) by superpositions of analytic functions, which are solutions of a single universal differential equation. Every superposition is some function belonging to . This improves a former result of the author, from which the superpositions are known to be continuous.     

Asymptotic expansion of the period function

Let P be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane. Suppose that the singularities which appear after desingularization of the vertices of P are formally linearizable. Consider the function T defined by the return time near P. It is shown that the function T and its derivative T′ have asymptotic expansions in and . It is also shown that under some other conditions imposed on the polycycle vertices, the asymptotic expansions of T and T′ converge absolutely and uniformly to these functions, respectively. These results are applied to the polycycles of the analytic vector fields which have a Darboux first integral. In particular, it is obtained that if P is a polycycle of a Hamiltonian vector field with an analytic (polynomial if P is unbounded) Hamiltonian function, T is a nonoscillating function. Another application concerns the nilpotent centers or focus, since the singularities which appear after desingularization of such a singularity have analytic first integrals.     

The symmetric Meixner-Pollaczek polynomials with real parameter

The Symmetric Meixner-Pollaczek polynomials for λ>0 are well-studied polynomials. These are polynomials orthogonal on the real line with respect to a continuous, positive real measure. For λ?0, are also polynomials, however they are not orthogonal on the real line with respect to any real measure. This paper defines a non-standard inner product with respect to which the polynomials for λ?0, become orthogonal polynomials. It examines the major properties of the polynomials, for λ>0 which are also shared by the polynomials, for λ?0.     

Semicrossed products generated by two commuting automorphisms

In this paper, we study the semicrossed product of a finite dimensional C∗-algebra for two types of -actions, and identify them with matrix algebras of analytic functions in two variables. We look at the connections with semicrossed by -actions.     

A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space

This paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous Navier-Stokes equations in the half-space in the case of small data with critical regularity. In dimension n?3, we state that if the initial density ρ₀ is close to a positive constant in and the initial velocity u₀ is small with respect to the viscosity in the homogeneous Besov space then the equations have a unique global solution. The proof strongly relies on new maximal regularity estimates for the Stokes system in the half-space in , interesting for their own sake.     

Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

Let , where is a random symmetric matrix, a random symmetric matrix, and with being independent real random variables. Suppose that , and are independent. It is proved that the empirical spectral distribution of the eigenvalues of random symmetric matrices converges almost surely to a non-random distribution.     

Weak solutions to the Cauchy problem for the diffusive discrete coagulation-fragmentation system

The initial value problem for the discrete coagulation-fragmentation system with diffusion is studied. This is an infinite countable system of reaction-diffusion equations describing coagulation and fragmentation of discrete clusters moving by spatial diffusion in all space . The model considered in this work is a generalization of Smoluchowski's discrete coagulation equations. Existence of global-in-time weak solutions to the Cauchy problem is proved under natural assumptions on initial data for unbounded coagulation and fragmentation coefficients. This work extends existence theory for this system from the case of clusters distribution on bounded domain subject to no-flux boundary condition to the case of all      

Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality . We relate this to the number of integral solutions to the Diophantine inequality in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.     

Reflection about k-jets and holomorphic extension of CR mappings

Let be a CR mapping between real analytic generic submanifolds M, M₁ of and , respectively. According to Webster's theory (Proc. Amer. Math. Soc. 86 (1982) 236-240) and its further developments, f has holomorphic extension to a full neighborhood of M in when the following requirements are fulfilled: f extends to a wedge W continuous up to M; f is of class C^k; (where denotes the complex tangent bundle); M₁ is “k-nondegenerate.” We deal here with the case where is strictly smaller than but is still real analytic in suitable sense. We show that a suitably refined condition of k-nondegeneracy still entails holomorphic extension of f.     

On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceH^N(S¹; ), (N?1), Hamiltonian (q):=∫_S1((∂_xq(x))²+q(x)³) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN?1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH¹(S¹; ), the KdV-Hamiltonian (and similarly any Hamiltonian in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vectorfield of KdV.     

Toeplitz operators on the space of analytic functions with logarithmic growth

Continuous and compact Toeplitz operators for positive symbols are characterized on the space of analytic functions with logarithmic growth on the open unit disc of the complex plane. The characterizations are in terms of the behaviour of the Berezin transform of the symbol. The space was introduced and studied by Taskinen. The Bergman projection is continuous on this space in a natural way, which permits to define Toeplitz operators. Sufficient conditions for general symbols are also presented.     

Exact solutions and branching of singularities for some hyperbolic equations in two variables

Exact global propagators are constructed for the singular hyperbolic operators in two variables , λ a real parameter, and for the degenerate hyperbolic operators . Qualitative phenomena such as uniqueness in the Cauchy problem and branching of singularities vary with λ, as shown earlier by Treves and by Taniguchi and Tozaki.     

Universality limits in the bulk for varying measures

Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form in the region where universality is desired. Wn does not need to be analytic, nor possess more than one derivative—and then only in the region where universality is desired. We deduce universality in the bulk for a large class of weights of the form , for example, when W=e−Q where Q is convex and Q^′ satisfies a Lipschitz condition of some positive order. We also deduce universality for a class of fixed exponential weights on a real interval.     

Inheritance of surjectivity for partial differential operators on spaces of real analytic functions

For an open set let A(Ω) be the space of real analytic functions on Ω. Improving our previous results, we prove a new quantitative characterization of the linear partial differential operators P(D) which are surjective on A(Ω). This implies that P(D) is surjective on if P(D) is surjective on A(Ω) for some Ω≠∅. Further inheritance properties for the surjectivity of P(D) on A(Ω) are also obtained.