Further arguments for slice convergence in nonreflexive spaces

It is shown that no notion of set convergence at least as strong as Wijsman convergence but not as strong as slice convergence can be preserved in superspaces. We also show that such intermediate notions of convergence do not always admit representations analogous to those given by Attouch and Beer for slice convergence, and provide a valid reformulation. Some connections between bornologies and the relationships between certain gap convergences for nonconvex sets are also observed.Research supported in part by an NSERC research grant and by the Shrum endowment.NSERC postdoctoral fellow.     

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.     

The rectangle intersection problem revisited

We take another look at the problem of intersecting rectangles with parallel sides. For this we derive a one-pass, time optimal algorithm which is easy to program, generalizes tod dimensions well, and illustrates a basic duality in its data structures and approach.The work of the first author was supported by the DAAD (Deutscher Akademischer Austauschdienst) and carried out while visiting McMaster University as a postdoctoral fellow. The work of the second author was supported by a Natural Sciences and Engineering Research Council of Canada Grant No. A-7700.     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

A note on a Monte-Carlo estimator of darling and robbins

Darling and Robbins, in discussing certain sequential tests that do not necessarily terminate with probability one, give a family of Monte-Carlo procedures for estimating the probability of termination. The choice of estimator is left open, although one presumably would like to have a small variance (the estimators are unbiased). As a contribution to this problem, we show that these estimators do not have moments higher than the first. Work done while a postdoctoral fellow of the National Science Foundation.     

A Hermitian Cauchy formula on a domain with fractal boundary

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called Dirac operator, which factorizes the Laplacian; monogenic functions may thus also be seen as a generalization of holomorphic functions in the complex plane. Hermitian Clifford analysis offers yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitian (or h-) monogenic functions, of two Hermitian Dirac operators which are invariant under the action of the unitary group. In Brackx et al. (2009) [8] a Clifford-Cauchy integral representation formula for h-monogenic functions has been established in the case of domains with smooth boundary, however the approach followed cannot be extended to the case where the boundary of the considered domain is fractal. At present, we investigate an alternative approach which will enable us to define in this case a Hermitian Cauchy integral over a fractal closed surface, leading to several types of integral representation formulae, including the Cauchy and Borel-Pompeiu representations.     

Algebraic analysis of Hermitian monogenic functions

In this Note we present an algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary group, both in the case of one and several variables. To cite this article: A. Damiano et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On Eigenvector Bounds

We show under very general assumptions that error bounds for an individual eigenvector of a matrix can be computed if and only if the geometric multiplicity of the corresponding eigenvalue is one. Basically, this is true if not computing exactly like in computer algebra methods. We first show, under general assumptions, that nontrivial error bounds are not possible in case of geometric multiplicity greater than one. This result is also extended to symmetric, Hermitian and, more general, to normal matrices. Then we present an algorithm for the computation of error bounds for the (up to normalization) unique eigenvector in case of geometric multiplicity one. The effectiveness is demonstrated by numerical examples.This revised version was published online in October 2005 with corrections to the Cover Date.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

Almost hermitian manifolds and Osserman condition

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively(δ,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied. The first author is partially supported by SFS, Project #04M03.     

Hamiltonian systems of PDEs with selfdual boundary conditions

Selfdual variational calculus is developed further and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, and Hamiltonian systems of PDEs. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as “periodic orbits up to an isometry” for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, without destroying the selfdual character of the system. N. Ghoussoub was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. A. Moameni’s research was supported by a postdoctoral fellowship at the University of British Columbia.     

Some examples of Hermitian surfaces

The Riemannian version of the Goldberg-Sachs theorem says that a compact Einstein Hermitian surface is locally conformal Kähler. In contrast to the compact case, we show that there exists an Einstein Hermitian surface which is not locally conformal Kähler. On the other hand, it is known that on a compact Hermitian surface M ⁴, the zero scalar curvature defect implies that M ⁴ is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.     

Block projection operators in normed solid spaces of measurable operators

We prove a Hermitian analog of the well-known operator triangle inequality for vonNeumann algebras. In the semifinite case we show that a block projection operator is a linear positive contraction on a wide class of solid spaces of Segal measurable operators. We describe some applications of the results.     

Spectral geometry for almost isospectral hermitian manifolds

The following question was posed by M. Berger: Is it possible to determine from the spectrum of the real Laplacian whether or not a manifold is Kähler? The Kähler condition for Hermitian manifolds is found out from the invariants of the spectrum of some differential operators acting on forms of type (p, q). P. Gilkey and H. Donnelly proved the Berger conjecture for the complex Laplacian and the reduced complex Laplacian respectively. In this paper we consider the Berger conjecture of almost isospectral Hermitian manifolds about the complex Laplacian acting on forms of type (p, q). Then we can show that a closed complexm(≥ 3)-dimensional Hermitian manifold which is strongly (?2/m)-isospectral to the complex projective space CP ^m with the Fubini-Study metric is holomorphically isometric to CP ^m .     

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).     

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded almost surely. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type.     

On vanishing theorems for Higgs bundles

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the base manifold was Kähler, also holds when the manifold is compact complex. From this fact and some basic properties of Hermitian Higgs bundles, we conclude several results. In particular we show that, in analogy to the classical case, there are vanishing theorems for invariant sections of tensor products of Higgs bundles. Then, we prove that a Higgs bundle admits no nonzero invariant sections if there is a condition of negativity on the greatest eigenvalue of the Hitchin–Simpson mean curvature. Finally, we prove that the invariant sections of certain tensor products of a weak Hermitian–Yang–Mills Higgs bundle are all parallel in the classical sense.     

On products of two Hermitian operators

In this paper, we study the problem of characterizing the bounded linear operators on a Hilbert space that admit a factorization as a product of two Hermitian operators. It is shown that a normal operator can be decomposed as a product of two Hermitian operators if and only if it is similar to its adjoint. Some partial results about hyponormal operators are obtained.     

Some aspects of Hermitian Jacobi forms

We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show its commutation with certain Hecke operators and use it to construct a map from elliptic cusp forms to Hermitian Jacobi cusp forms. We construct Hermitian Jacobi forms as the image of the tensor product of two copies of Jacobi forms and also from the differentiation of the variables. We determine the number of Fourier coefficients that determine a Hermitian Jacobi form and use the differential operator to embed a certain subspace of Hermitian Jacobi forms into a direct sum of modular forms for the full modular group.     

A low-cost-memory CUDA implementation of the conjugate gradient method applied to globally supported radial basis functions implicits

Hermitian radial basis functions implicits is a method capable of reconstructing implicit surfaces from first-order Hermitian data. When globally supported radial functions are used, a dense symmetric linear system must be solved. In this work, we aim at exploring and computing a matrix-free implementation of the Conjugate Gradients Method on the GPU in order to solve such linear system. The proposed method parallelly rebuilds the matrix on demand for each iteration. As a result, it is able to compute the Hermitian-based interpolant for datasets that otherwise could not be handled due to the high memory demanded by their linear systems.