Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat

Semantics of a Typed Algebraic Lambda-Calculus

Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations

Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Quantum computations without definite causal structure

We show that quantum theory allows for transformations of black boxes that cannot be realized by inserting the input black boxes within a circuit in a pre-defined causal order. The simplest example of such a transformation is the classical switch of black boxes, where two input black boxes are arranged in two different orders conditionally on the value of a classical bit. The quantum version of this transformation-the quantum switch-produces an output circuit where the order of the connections is controlled by a quantum bit, which becomes entangled with the circuit structure. Simulating these transformations in a circuit with fixed causal structure requires either postselection, or an extra query to the input black boxes.Comment: Updated version with expanded presentatio

Molecular excitation in the Interstellar Medium: recent advances in collisional, radiative and chemical processes

We review the different excitation processes in the interstellar mediumComment: Accepted in Chem. Re

Cap-Gly Proteins at Microtubule Plus Ends: Is EB1 Detyrosination Involved?

Localization of CAP-Gly proteins such as CLIP170 at microtubule+ends results from their dual interaction with α-tubulin and EB1 through their C-terminal amino acids −EEY. Detyrosination (cleavage of the terminal tyrosine) of α-tubulin by tubulin-carboxypeptidase abolishes CLIP170 binding. Can detyrosination affect EB1 and thus regulate the presence of CLIP170 at microtubule+ends as well? We developed specific antibodies to discriminate tyrosinated vs detyrosinated forms of EB1 and detected only tyrosinated EB1 in fibroblasts, astrocytes, and total brain tissue. Over-expressed EB1 was not detyrosinated in cells and chimeric EB1 with the eight C-terminal amino acids of α-tubulin was only barely detyrosinated. Our results indicate that detyrosination regulates CLIPs interaction with α-tubulin, but not with EB1. They highlight the specificity of carboxypeptidase toward tubulin

High latitude gas in the Beta Pictoris system. A possible origin related to Falling Evaporating Bodies

The presence of off-plane Ca II ions in the Beta Pictoris disk, and the non-detection of off-plane Na I atoms, can be explained as a consequence of the evaporation process of Falling Evaporating Bodies (FEBs). In the star-grazing regime, the FEBs are subject to inclination oscillations up to 30 - 40 degrees that causes most metallic species released by sublimation to move off plane The ions are be stopped at about 100 AU from the star. We show that collisions with a neutral medium can stop the ions. The required H I column density is reduced to 10^17 cm^-2, one order of magnitude below present detection limits. We also investigate the possibility that the ions are slowed down magnetically. While the sole action of a magnetic field of the order of 1 microGauss is not effective, the combined effect of magnetic and collisional deceleration processes lead to an additional lowering of the required H I column density.Comment: 17 page

Non-real zeros of derivatives of meromorphic functions

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane