Generalized Padé-Type Approximation and Integral Representations

In several complex variables, the multivariate Padé-type approximation theory is based on the polynomial interpolation of the multidimensional Cauchy kernel and leads to complicated computations. In this paper, we replace the multidimensional Cauchy kernel by the Bergman kernel function K (z,x) into an open bounded subset of C ⁿ and, by using interpolating generalized polynomials for K (z,x), we define generalized Padé-type approximants to any f in the space OL ²() of all analytic functions on which are of class L ². The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for OL ²() matches the Fourier series expansion of f as far as possible. After studying the error formula and the convergence problem, we show that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator – the so-called generalized Padé-type operator – which maps every f OL ²() to a generalized Padé-type approximant to f. By the continuity of this operator, we obtain some convergence results about series of analytic functions of class L ². Our study concludes with the extension of these ideas into every functional Hilbert space H and also with the definition and properties of the generalized Padé-type approximants to a linear operator of H into itself. As an application we prove a Painlevé-type theorem in C ⁿ and we give two examples making use of generalized Padé-type approximants.     

Measurement of the X‐ray mass attenuation coefficients of silver in the 5–20 keV range

The X‐ray mass attenuation coefficients of silver were measured in the energy range 5–20 keV with an accuracy of 0.01–0.2% on a relative scale down to 5.3 keV, and of 0.09–1.22% on an absolute scale to 5.0 keV. This analysis confirms that with careful choice of foil thickness and careful correction for systematics, especially including harmonic contents at lower energies, the X‐ray attenuation of high‐Z elements can be measured with high accuracy even at low X‐ray energies (<6 keV). This is the first high‐accuracy measurement of X‐ray mass attenuation coefficients of silver in the low energy range, indicating the possibility of obtaining high‐accuracy X‐ray absorption fine structure down to the L₁ edge (3.8 keV) of silver. Comparison of results reported here with an earlier data set optimized for higher energies confirms accuracy to within one standard error of each data set collected and analysed using the principles of the X‐ray extended‐range technique (XERT). Comparison with theory shows a slow divergence towards lower energies in this region away from absorption edges. The methodology developed can be used for the XAFS analysis of compounds and solutions to investigate structural features, bonding and coordination chemistry.     

Numerical analysis of a quadratic matrix equation

The quadratic matrix equation AX²+ BX + C = 0in n x nmatricesarises in applications and is of intrinsic interest as oneof the simplest nonlinear matrix equations. We give a completecharacterization of solutions in terms of the generalized Schurdecomposition and describe and compare various numerical solutiontechniques. In particular, we give a thorough treatment offunctional iteration methods based on Bernoulli’s method.Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions.We show that functional iteration applied to the quadraticmatrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem (²A + B + C)x = 0.     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

Deflationist Truth is Substantial

Deflationism is usually thought to differ from the correspondence theory over whether truth is a substantial property. However, I argue that this notion of a ‘substantial property’ is tendentious. I further argue that the Equivalence Schema alone is sufficient to lead to idealism when combined with a pragmatist theory of truth. Deflationism thus has more powerful metaphysical implications than is generally thought and itself amounts to a kind of correspondence theory.     

The cycle structure of two rows in a random Latin square

Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r,s be arbitrary distinct rows of L. We study the distribution of σ_r,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1 ‐ o(1) as n → ∞: (i) σ_r,s has more than (log n)^1?c cycles, (ii) σ_r,s has fewer than 9 cycles, (iii) L has fewer than n^5/2 intercalates (latin subsquares of order 2). We also show that the probability that σ_r,s is an even permutation lies in an interval and the probability that it has a single cycle lies in [2n^‐2,2n^‐2/3]. Indeed, we show that almost all derangements have similar probability (within a factor of n^3/2) of occurring as σ_r,s as they do if chosen uniformly at random from among all derangements of {1,2,…,n}. We conjecture that σ_r,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure of latin squares of orders n ≤ 11. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2‐connected (simple) graphs on n vertices and m edges, provided that m ‐ n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2‐edge‐connectedness is treated similarly. We also obtain formulae for the number of 2‐connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

In silico design of peptidic inhibitors targeting estrogen receptor alpha dimer interface

Human estrogen receptor alpha (ERα), which acts as a biomarker and as a therapeutic target for breast cancers, is activated by agonist ligands and co-activator proteins. Selective estrogen receptor modulators (SERM) act as antagonists in specific tissues and tamoxifen, a SERM, has served as a drug for decades for ERα-positive breast cancers. However, the ligand-selective and tissue-specific response of ERα biological activity and the resistance to tamoxifen treatment in advanced stages of ERα-positive breast cancers underscores the need to find a ligand-independent inhibitor for ERα. Here we present a ligand-independent approach of inhibiting ERα transactivation targeting its dimerization-a key process of ERα biological activity. Using in silico techniques, we first elucidated the hydrogen bond interactions involved in dimerization and identified three interfacial sequence motifs, where sequence I (DKITD) and sequence II (QQQHQRLAQ) of one monomer form hydrogen bonding with sequence II and sequence I of the second monomer, respectively, and sequence III (LSHIRHMSNK) hydrogen bonds with the same from the second monomer. Studying the structural stability and the binding affinity of the peptides derived from these sequence motifs, we found that an extended and ARG mutated version (LQQQHQQLAQ) of sequence II can act as a suitable template for designing peptidic inhibitors. It provides additional structural stability and interacts more strongly with ERα dimer interface groove formed by helices 9 and 10/11 and prevent ERα dimerization. Our result provides a novel therapeutic designing pipeline for ligand-independent inhibition of ERα.     

Constraining Cyclic Peptides To Mimic Protein Structure Motifs

Many proteins exert their biological activities through small exposed surface regions called epitopes that are folded peptides of well‐defined three‐dimensional structures. Short synthetic peptide sequences corresponding to these bioactive protein surfaces do not form thermodynamically stable protein‐like structures in water. However, short peptides can be induced to fold into protein‐like bioactive conformations (strands, helices, turns) by cyclization, in conjunction with the use of other molecular constraints, that helps to fine‐tune three‐dimensional structure. Such constrained cyclic peptides can have protein‐like biological activities and potencies, enabling their uses as biological probes and leads to therapeutics, diagnostics and vaccines. This Review highlights examples of cyclic peptides that mimic three‐dimensional structures of strand, turn or helical segments of peptides and proteins, and identifies some additional restraints incorporated into natural product cyclic peptides and synthetic macrocyclic peptidomimetics that refine peptide structure and confer biological properties.     

A genetic algorithm for the structural optimization of Morse clusters

This article describes the application of a genetic algorithm for the structural optimization of 19–50-atom clusters bound by medium-range and short-range Morse pair potentials. The GA is found to be efficient and reliable for finding the geometries corresponding to the previously published global minima [Doye JPK, Wales DJ (1997) J Chem Soc Faraday Trans 93: 4233]. Using the genetic algorithm, only a relatively small number of energy evaluations and minimizations are required to find the global minima. By contrast, a simple random search algorithm often cannot find the global minima of the larger clusters, even after many thousands of searches. Received: 27 October 1999 / Accepted: 7 December 1999 / Published online: 19 April 2000