Cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function f∈ to be cyclic, i.e. for {pf: p is a polynomial} to be dense in . The proof is based on the notion of Bergman–Smirnov exceptional set introduced by Hedenmalm and Shields. Our methods yield the first known examples of such sets that are uncountable. One of the principal ingredients of the proof is a new converse to the strong-type inequality for capacity.     

Pluripolar hulls and fine analytic structure

We discuss the relation between pluripolar hulls and fine analytic structure. Our main result is the following. For each non polar subset S of the complex plane ? we prove that there exists a pluripolar set E ⊂ (S × ?) with the property that the pluripolar hull of E relative to ?² contains no fine analytic structure and its projection onto the first coordinate plane equals ?.     

Shcherbina’s theorem for finely holomorphic functions

We prove an analogue of Sadullaev’s theorem concerning the size of the set where a maximal totally real manifold M can meet a pluripolar set. M has to be of class C ¹ only. This readily leads to a version of Shcherbina’s theorem for C ¹ functions f that are defined in a neighborhood of certain compact sets ${K\subset\mathbb{C}}We prove an analogue of Sadullaev’s theorem concerning the size of the set where a maximal totally real manifold M can meet a pluripolar set. M has to be of class C ¹ only. This readily leads to a version of Shcherbina’s theorem for C ¹ functions f that are defined in a neighborhood of certain compact sets K ì \mathbbC{K\subset\mathbb{C}}. If the graph Γ _f (K) is pluripolar, then \frac?f?[`(z)]=0{\frac{\partial f}{\partial\bar z}=0} in the closure of the fine interior of K.     

On the duality between varieties and algebraic theories

It is proved that for any ultrametric space (X, d), the set L(X) of its closed balls is a lattice . It is complete, atomic, tree-like, and real graduated. For any such lattice , the set A(L) of its atoms can be naturally equipped with an ultrametric . These assignments are inverse of one another: where the first equality means an isometry while the second one is a lattice isomorphism. A similar correspondence established for morphisms, shows that there is an isomorphism of categories. The category ULTRAMETR of ultrametric spaces and non-expanding maps is isomorphic to the category LAT* of complete, atomic, tree-like, real graduated lattices and isotonic, semi-continuous, non-extensive maps. We describe properties of the isomorphism functor and its relations to the categorical operations and action of other functors. Basic properties of a space (such as completeness, spherical completeness, total boundedness, compactness, etc.) are translated into algebraic properties of the corresponding lattice L(X).     

An implication basis for linear forms

The main results of this paper are a generalization of the results of S. Fajtlowicz and J. Mycielski on convex linear forms. We show that if V_n is the variety generated by all possible algebras , where R denotes the real numbers and , for some , then any basis for the set of all identities satisfied by V_n is infinite. But on the other hand, the identities satisfied by V_n are a consequence of gL and μ_n, where μ_n is the n-ary medial law and the inference rule gL is an implication patterned after the classical rigidity lemma of algebraic geometry. We also prove that the identities satisfied by are a consequence of gL and μ_n iff {p₁, ... , p_n} is algebraically independent. We then prove analagous results for algebras of arbitrary type τ and in the final section of this paper, we show that analagous results hold for Abelian group hyperidentities. This paper is dedicated to Walter Taylor. Received July 16, 2005; accepted in final form January 12, 2006. The research of both authors was supported by an operating grant ODP0008215 from NSERC.     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006     

Auslander–Reiten sequences on schemes

Auslander–Reiten sequences are the central item of Auslander–Reiten theory, which is one of the most important techniques for the investigation of the structure of abelian categories. This note considers X, a smooth projective scheme of dimension at least 1 over the field k, and , an indecomposable coherent sheaf on X. It is proved that in the category of quasi-coherent sheaves on X, there is an Auslander–Reiten sequence ending in .     

Representation Formulae for Solutions to Some Classes of Higher Order Systems and Related Liouville Theorems

Let m ≥ 1 be an integer and N > 2m. Let μ be a positive Radon measure on . We study necessary and sufficient conditions on possible distributional solutions of , that guarantee the validity of the representation formula a.e. on , where and c(2m) is a positive constant depending on m and N. Several consequences are derived. In particular we prove Liouville theorems for systems of higher order elliptic inequalities and weighted form of Hardy-Littlewood-Sobolev systems of integral equations. Received: March 2008     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Identification of a time-dependent UV-photon source in an interstellar cloud

Abstract Consider an interstellar cloud that occupies the region , bounded by the known surface , and assume that the scattering cross section σ_s and the total cross section σ are also known. Then, we prove that it is possible to identify the source q=q(x,t) that produces UV-photons inside the cloud, provided that the UV-photon distribution function arriving at a location , far from the cloud, is measured at times , , ..., . Keywords: Photon transport, Semigroups and linear evolution equations, Inverse problems Mathematics Subject Classification (2000): 82A25, 82C70, 34K29, 65M32     

A new method to exploit the Entropy Principle and galilean invariance in the macroscopic approach of Extended Thermodynamics

Abstract In extended thermodynamic the entropy principle and the Galilean invariance dictate respectively constraints for the constitutive equations and the velocity dependence. The entropy principle in particular requires the existence of a privileged field, the main field u^′, such that the original system becomes symmetric hyperbolic and is generated by four potentials. It is not easy to solve the restrictions of both principles, if we use as field the non convective main field and the velocity v. This is due to the fact that are not independent. Rather its components satisfy three scalar constraints. The aim of this paper is to solve the full problem using as new strategy to consider as independent variables and requiring an appropriate differential constraint. This new procedure is very efficient and we are able to solve the problem of 13 moments in the full non linear case (far from equilibrium). It turns out that the knowledge of only the equilibrium state function is sufficient to close the system. Keywords: Extended Thermodynamics, Entropy Principle, Galilean invariance, Rarefied Gas, Hyperbolic systems Mathematics Subject Classification (2000): 74A20, 76P05, 35l60     

On stationary thermo-rheological viscous flows

Abstract We study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor S has the form where u is the vector velocity, P is the pressure, θ is the temperature and μ ,p and τ are the given coefficients depending on the temperature. D and I are respectively the rate of strain tensor and the unit tensor. We prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions. Keywords: Non-Newtonian fluids, Nonlinear thermal diffusion equations, Heat and mass transfer Mathematics Subject Classification (2000): 76A05, 76D07, 76E30, 35G15     

The parabolic analytic functions and the derivative of real functions

Among the bidimensional hypercomplex-number systems defined as the parabolic (dual) numbers are introduced with the rule α = 0. As well as the functions of a complex variable, the analytic functions of a parabolic variable can be introduced as analytic continuation of the real functions of a real variable. These functions hold the property that the “imaginary” part is linked to the derivative of the “real” part. In this paper we will show how this property allows one to demonstrate in an algebraic way some rules of the differential calculus for the real functions of a real variable.     

Codimension-<Emphasis Type="Italic">p</Emphasis> Paley–Wiener theorems

We obtain the generalized codimension-p Cauchy–Kovalevsky extension of the exponential function in R ^m =R ^p ⊕R ^q , where p>1, , and prove the corresponding codimension-p Paley–Wiener theorems.     

An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem:

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let be the Euclidean Dirac operator in the n-dimensional flat space the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials. Received: 27 October 2005     

Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

We introduce the set of bicomplex numbers which is a commutative ring with zero divisors defined by where We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr?dinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr?dinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symmetries. We obtain the standard Born’s formula for the class of bicomplex wave functions having a null hyperbolic angle.     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

Projections in the space<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces

LetM be a non-compact connected Riemann surface of a finite type, andR⋐M be a relatively compact domain such thatH ₁(M,Z)=H ₁(R,Z). Let be a covering. We study the algebraH ^∞(U) of bounded holomorphic functions defined in certain subdomains . Our main result is a Forelli type theorem on projections inH ^∞(D). Research supported in part by NSERC.