Vector valued logarithmic residues and the extraction of elementary factors

An analysis is presented of the circumstances under which, by the extraction of elementary factors, an analytic Banach algebra valued function can be transformed into one taking invertible values only. Elementary factors are generalizations of the simple scalar expressions λ – α, the building blocks of scalar polynomials. In the Banach algebra situation they have the form e – p + (λ – α)p with p an idempotent. The analysis elucidates old results (such as on Fredholm operator valued functions) and yields new insights which are brought to bear on the study of vector-valued logarithmic residues. These are contour integrals of logarithmic derivatives of analytic Banach algebra valued functions. Examples illustrate the subject matter and show that new ground is covered. Also a long standing open problem is discussed from a fresh angle.analytic vector-valued function;annihilating family of idempotents;elementary factor;generalizations of analytic functions;idempotent;integer combination of idempotents;logarithmic residue;plain function;resolving family of traces;topological algebras

Quench dynamics across quantum critical points

We study the quantum dynamics of a number of model systems as their coupling constants are changed rapidly across a quantum critical point. The primary motivation is provided by the recent experiments of Greiner et al. (Nature 415, 39 (2002)) who studied the response of a Mott insulator of ultracold atoms in an optical lattice to a strong potential gradient. In a previous work (cond-mat/0205169), it had been argued that the resonant response observed at a critical potential gradient could be understood by proximity to an Ising quantum critical point describing the onset of density wave order. Here we obtain numerical results on the evolution of the density wave order as the potential gradient is scanned across the quantum critical point. This is supplemented by studies of the integrable quantum Ising spin chain in a transverse field, where we obtain exact results for the evolution of the Ising order correlations under a time-dependent transverse field. We also study the evolution of transverse superfluid order in the three dimensional case. In all cases, the order parameter is best enhanced in the vicinity of the quantum critical point.Comment: 10 pages, 6 figure

The Receptor for Advanced Glycation End Products (RAGE) and the Lung

The receptor for advanced glycation end products (RAGE) is a member of the immunoglobulin superfamily of cell surface molecules. As a pattern-recognition receptor capable of binding a diverse range of ligands, it is typically expressed at low levels under normal physiological conditions in the majority of tissues. In contrast, the lung exhibits high basal level expression of RAGE localised primarily in alveolar type I (ATI) cells, suggesting a potentially important role for the receptor in maintaining lung homeostasis. Indeed, disruption of RAGE levels has been implicated in the pathogenesis of a variety of pulmonary disorders including cancer and fibrosis. Furthermore, its soluble isoforms, sRAGE, which act as decoy receptors, have been shown to be a useful marker of ATI cell injury. Whilst RAGE undoubtedly plays an important role in the biology of the lung, it remains unclear as to the exact nature of this contribution under both physiological and pathological conditions

Asymptotics of block Toeplitz determinants and the classical dimer model

We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice ($t=0$) and the triangular lattice ($t=1$), and we obtain the asymptotics for $0<t\le 1$. For $0<t<1$ we apply the Szeg\"o Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form

Logarithmic residues, Rouché’s theorem, and spectral regularity: The C∗-algebra case

AbstractUsing families of irreducible Hilbert space representations as a tool, the theory of analytic Fredholm operator valued function is extended to a C∗-algebra setting. This includes a C∗-algebra version of Rouché’s Theorem known from complex function theory. Also, criteria for spectral regularity of C∗-algebras are developed. One of those, involving the (generalized) Calkin algebra, is applied to C∗-algebras generated by a non-unitary isometry

Logarithmic residues of analytic Banach algebra valued functions possessing a simply meromorphic inverse

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter.Logarithmic residues;Cauchy domains;analytic Banach algebra valued function;meromorphic inverse

Logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encoutered in previous investigations. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. The set of sums of idempotens and the set of logarithmic residues have an intriguing topological structure.Banach algebra;Logarithmic residues;sums of idempotents