Sensitivity analysis of the differential matrix Riccati equation based on the associated linear differential system

Perturbation bounds are given for the solution of the nth order differential matrix Riccati equation using the associated linear 2nth order differential system. The new bounds are alternative to those existing in the literature and are sharper in some cases. This revised version was published online in June 2006 with corrections to the Cover Date.     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows. Received August 10, 2001; in revised form June 25, 2002     

Decay Law of Moving Unstable Particle

Quantum relativistic decay law of moving unstable particle is analytically calculated in the model case of the Breit–Wigner mass distribution. It turns out that Einstein time dilation of the moving particle decay holds approximately at times when the decay is exponential. The related correction is calculated analytically. Being very small at these times it is practically unobservable. It is shown that Einstein dilation fails for large times t when decay is not exponential. An unstable system of the kind of K ₀-meson (which is the superposition of K _s and K _I) is also considered. In this case, the violation of Einstein dilation is shown to be appreciable at all times under some condition     

Ion-driven ATP pump by self-organized hybrid membrane materials

We report new hybrid organic-inorganic materials, based on macrocyclic receptors 1-3 self-organized in tubular superstructures prepared by sol-gel process. Fourier transform infrared (FTIR) and NMR spectroscopic analyses demonstrate that the self-organization by hydrogen bonding of organogel superstructures of 2 and 3 were preserved in the hybrid materials throughout the sol-gel process. The molecular arrangement of heteroditopic receptors defines a particularly attractive functional transport device for both cation (tubular macrocycles) and anion (sandwich-urea) directional-diffusion transport mechanism in the hybrid membrane material. This system has been employed successfully to design a solid dense membrane, functioning as an ion-powered adenosine triphosphate (ATP(2)(-)) pump, and illustrates how a self-organized hybrid material performs interesting and potentially useful functions.     

A singular control model with application to the goodwill problem

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system’s controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system’s state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product’s image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second-order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems.     

Thermal and Emission Properties of a Series of Lanthanides Complexes with N-Biphenyl-Alkylated-4-Pyridone Ligands: Crystal Structure of a Terbium Complex with N-Benzyl-4-Pyridone

This work focuses on the investigation of the liquid crystalline behavior and luminescence properties of the lanthanide complexes of Eu(III), Sm(III) and Tb(III) with N-biphenyl-alkylated-4-pyridone ligands. The organic ligands having a biphenyl group attached via a long flexible spacer with either 9 or 10 carbon atoms were synthesized by the reaction between 4-hydroxypyridine and the corresponding bromide compounds. The chemical structures of the organic and lanthanide complexes were assigned based on elemental analysis, single-crystal X-ray diffraction, ¹H, ¹³C NMR and IR spectroscopies, and thermogravimetric analysis (TGA). The X-ray diffraction analysis of a parent compound shows that the lanthanide ions are surrounded by three monodentate pyridone ligands and three bidentate nitrate ions, giving a 9-coordinate environment. The mesogenic behavior and the type of liquid crystalline phases exhibited by the new complexes were analyzed by differential scanning calorimetry (DSC) and polarizing optical microscopy (POM), and powder X-ray diffraction (XRD) studies. Only the lanthanide complexes with longer spacer (10) display a monotropic SmA phase, typically on a short thermal range (less than 10 °C). The complexes with shorter flexible chains (9) show no liquid crystalline properties with melting temperatures lower than their analogs with longer spacers. The emission spectra recorded in solid state at room temperatures show typical emission bands for each lanthanide ion employed (Eu(III), Tb(III) and Sm(III)).     

On a question of A. Kameyama concerning self-similar metrics

A. Kameyama introduced the concept of self-similar topological system and asked the following fundamental question: given a topological self-similar system (K,_{(fi)i∈{1,2,...,N}})$(K,{(f_i)}_{i\in \{1,2,...,N\}})$, does there exist a metric on K   comparable to the topology such that all the functions _fi$f_i$ are contractions? We modify Kameyama's question (which has a negative answer for an arbitrary topological self-similar system) by weakening the requirement that the functions in the topological self-similar system be contractions to requiring that they be φ  -contractions. More precisely we give an affirmative answer to the following question: given a topological self-similar system (K,_{(fi)i∈{1,2,...,N}})$(K,{(f_i)}_{i\in \{1,2,...,N\}})$ does there exist a metric δ on K   which is compatible with the original topology and a comparison function φ:[0,∞)→[0,∞)$\phi :[0,\infty )\to [0,\infty )$ such that all the functions _fi:(K,δ)→(K,δ)$f_i:(K,\delta )\to (K,\delta )$ are φ  -contractions? Consequently the iterated function system ((K,d),_{(fi)i∈{1,2,...,N}})$((K,d),{(f_i)}_{i\in \{1,2,...,N\}})$, where d is a metric on K compatible with the original topology on K, is φ-hyperbolic.