Synthesis and structural characterization of novel,fluorinated poly(phthalazinone ether)s containing perfluorophenylene moieties

The synthesis and structural characterization of a series of novel, fluorinated poly(phthalazinone ether)s containing perfluorophenylene moieties are described. The monomers, 4‐(4′‐hydroxyaryl)phthalazin‐1(2H)‐ones ( 2a – 2d ), were conveniently and efficiently synthesized from phenols and phthalic anhydride in two steps via 2‐(4′‐hydroxybenzoyl)benzoic acids, which were first obtained by the Friedel–Crafts reaction in good yields and with high stereoselectivity and were then converted into 2a – 2d by fusion with hydrazine. All the polymers were prepared by nucleophilic aromatic substitution (S_NAr) polycondensation between the compounds perfluorobiphenyl and 4‐(4′‐hydroxyaryl)phthalazin‐1(2H)‐ones ( 2a ‐ 2d ). The resulting fluorinated polymers were readily soluble in common organic solvents (e.g., CHCl₃, tetrahydrofuran, dimethylformamide, dimethyl sulfoxide, N‐methylpyrrolidone, etc.) at room temperature. Their weight‐average molecular weights and the polydispersities ranged from (7.96–18.25) × 10³ to 1.31–2.71, respectively. Their glass‐transition temperatures varied from 213 to 263 °C. They were all stable up to 390 °C both in air and in argon. The 5% weight‐loss temperatures of these polymers in air and argon ranged from 393–487 to 437–509 °C, respectively. Wide‐angle X‐ray diffraction studies indicated they were all amorphous and could be attributed to the presence of kink nonplanar moiety, phenyl phthalazinone along the polymer backbone. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 925–932, 2004     

On division rings with algebraic commutators of bounded degree

We shall answer several questions concerning additive or multiplicative commutators in a division ring which are algebraic of bounded degree over its center.     

Seizure warning algorithm based on optimization and nonlinear dynamics

There is growing evidence that temporal lobe seizures are preceded by a preictal transition, characterized by a gradual dynamical change from asymptomatic interictal state to seizure. We herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing EEG signals. Such, the algorithm constitutes a seizure warning system. The algorithm estimates STL_max, a measure of the order or disorder of the signal, of EEG signals recorded from individual electrode sites. The optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures. Specifically, a quadratically constrained quadratic 0-1 programming problem is formulated to identify critical electrode sites. The automated seizure warning algorithm was tested in continuous, long-term EEG recordings obtained from 5 patients with temporal lobe epilepsy. For individual patient, we use the first half of seizures to train the parameter settings, which is evaluated by ROC (Receiver Operating Characteristic) curve analysis. With the best parameter setting, the algorithm applied to all cases predicted an average of 91.7% of seizures with an average false prediction rate of 0.196 per hour. These results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes.Mathematics Subject Classification (1991):20E28, 20G40, 20C20     

On Einstein four-manifolds with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-actions

We study closed Einstein 4-manifolds which admit S¹ actions of a certain type, i.e., warped products. In particular, we classify them up to isometry when the fixed point of the S¹ action satisfies certain natural geometric conditions. The proof uses the Bochner-Weitzenböck formula for 1-forms and the theory of minimal surfaces in 3-manifolds.in final form: 22 January 2003     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal     

Bounded Positive Solutions of Schrödinger Equations in Two-Dimensional Exterior Domains

We prove under quite general assumptions the existence of a bounded positive solution to the semilinear Schrödinger equation in a two-dimensional exterior domain.     

On a Definitizable Analog of the Trigonometric Moment Problem Generating an Indefinite Toeplitz Form

We prove the existence of an integro-polynomial representation for a sequence of numbers such that there exists a difference operator mapping this sequence to a sequence that generates the solvable trigonometric moment problem. A similar result related to the power moment problem was given in [12].     

Pseudorandom properties of prime factors

In this paper we study some pseudo-random properties related to the prime factors of integers. We prove some results concerning the well distribution of the greatest prime factor modulo 4, and give some computational insight on the correlations, which seem to be beyond reach of proof.     

Minimal expansions in redundant number systems: Fibonacci bases and Greedy algorithms

We study digit expansions with arbitrary integer digits in base q (q integer) and the Fibonacci base such that the sum of the absolute values of the digits is minimal. For the Fibonacci case, we describe a unique minimal expansion and give a greedy algorithm to compute it. Additionally, transducers to calculate minimal expansions from other expansions are given. For the case of even integer bases q, similar results are given which complement those given in [6].