The second law of thermodynamics revisited I. The absolute zero of potential and the general expression for the efficiency of universal reversible cycle

The fundamental equation of thermodynamics expresses the internal energy of a system as a function of all the extensive parameters of the system. The differential form of this equation is referred to as the Gibbs equation. We have integrated this equation and have used it to derive an expression characterising the efficiency of a system for any kind of cyclical process by which work is produced. This system has access to two reservoirs at low and high thermodynamics potential respectively. It is claimed that all thermodynamic potentials (temprature, chemical potential, hydrostatic pressure, electric potential, etc) can have an absolute value of zero which we defined as the value of the potential in the lower reservoir when the efficiency of the cycle is 1. It is also shown that the classical Carnot machine, in which heat is converted into mechanical work, is an example of the general expression.     

Independent sets in tensor graph powers

The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gⁿ)/|V(Gⁿ)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 ( 5 ), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory     

On the number of rectangulations of a planar point set

We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n+1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schröder number, and the total number of rectangulations is O(n²⁰/n⁴).     

Extension of Arrow's theorem to symmetric sets of tournaments

Arrow's impossibility theorem [K.J. Arrow, Social Choice and Individual Values, Wiley, New York, NY, 1951] shows that the set of acyclic tournaments is not closed to non-dictatorial Boolean aggregation. In this paper we extend the notion of aggregation to general tournaments and we show that for tournaments with four vertices or more any proper symmetric (closed to vertex permutations) subset cannot be closed to non-dictatorial monotone aggregation and to non-neutral aggregation. We also demonstrate a proper subset of tournaments that is closed to parity aggregation for an arbitrarily large number of vertices. This proves a conjecture of Kalai [Social choice without rationality, Reviewed NAJ Economics 3(4)] for the non-neutral and the non-dictatorial and monotone cases and gives a counter example for the general case.     

Hidden convexity in some nonconvex quadratically constrained quadratic programming

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs. This author's work was partially supported by GIF, the German-Israeli Foundation for Scientific Research and Development and by the Binational Science Foundation. This author's work was partially supported by National Science Foundation Grants DMS-9201297 and DMS-9401871.     

Growing ZnO crystals on magnetite nanoparticles

We report herein on the oriented growth of ZnO crystals on magnetite nanoparticles. The ZnO crystals were grown by hydrolyzing a supersaturated aqueous solution of zinc nitrate. The seeds for the growth were magnetite nanoparticles with a diameter of 5.7 nm and a narrow size distribution. Hollowed ZnO hexagons of 0.15 microm width and 0.5 microm length filled with Fe(3)O(4) particles were obtained. HR-TEM (high-resolution transmission electron microscopy) and selected-area EDS (energy-dispersive spectroscopy) show that the nanoparticles are homogenously spread in the ZnO tubes. Zeta potential measurements were employed to understand the relationship between the nanoparticles and the oriented growth of the ZnO crystals. The results show that the surfactants induced the directional growth of the ZnO crystals.     

Reactions under autogenic pressure at elevated temperature (RAPET) of various alkoxides: formation of metals/metal oxides-carbon core-shell structures

The RAPET reaction at 700 degrees C, of different alkoxides (Ti, V, and Si) led to three different nanocomposites. Carbon is the element common to all three structures. Carbon was found as the core in the decomposition of TEOS, as the shell in the decomposition of VO(OEt)3 and embedded in, or mixed completely with, TiO2 in the decomposition of Ti(OiPr)4. This novel method using only metallic alkoxide precursor, in the absence of catalyst, leads in a one-step process to core-shell structures.     

Multilinear mappings and estimates of multiplicity

A method for estimating multiplicity of operators in terms of certain multilinear mappings is introduced. Several applications are given which include the following result: IfT is a completely non-unitary contraction on a complex Hilbert space, and for some vectorx ₀, then there exists a constant >0, such that for every positive integern, the multiplicity of the operatorT ⁿ is not less thann. This extends a result of B. Sz. Nagy and Foia [10].     

Microwave-assisted selective preparation and characterization of Li21Si5 and Li17Sn4

Two types of intermetallic lithium alloys, Li₂₁Si₅ and Li₁₇Sn₄ (previously Li₂₂Si₅ and Li₂₂Sn₅), were prepared for the first time using microwave-assisted solid-state reaction. The optimum oven power for their preparation is 80-60%, and the irradiation times are 5 min for Li₂₁Si₅ or 10 min for Li₁₇Sn₄. A cheap alumina crucible was found to be the most suitable container in quick (less than 10 min) microwave reactions for Li-containing alloys. The synthesized compounds were characterized by PXRD. Mössbauer spectroscopy was used to characterize Li₁₇Sn₄ under different conditions. The hyperfine interaction parameters of ¹¹⁹Sn in Li₁₇Sn₄ show a typical Li-Sn alloy Sn isomer shift (1.88 mm/s). The oxidization processes of the two intermetallic lithium alloys in air were investigated. The microwave method was found to be simple, fast and efficient, with high selectivity for the preparation of these compounds.     

Magnetic circular dichroism of I2 in the cordes band energy region

The absorption and magnetic circular dichroism (MCD) of iodine in the gas phase was recorded over the wavelength range 1950-1780 Å. The strong MCD si