Sequences of words characterizing finite solvable groups

There are two sequences in two variables which characterize the solvability of finite groups. Namely, the sequence of Bandman, Greuel, Grunewald, Kunyavskii, Pfister and Plotkin which is defined by u ₁ = x ⁻² y ⁻¹ x and and the sequence of Bray, Wilson, and Wilson defined by s ₁ = x and . We define new sequences and proof that six of them characterize the solvability of finite groups.      

Carrier recombination and high-barrier Schottky diodes on silicon

Small-area high-barrier Schottky diodes have a very high dynamic resistance. Consequently, special care is needed when measuring the current-voltage characteristic of such diodes. The reported observation of carrier recombination in the depletion layer of high-barrier IrSi/Si Schottky diodes at room temperature is shown to be due to instrumental loading of the diodes. Careful measurements show that carrier recombination is observed only below 200 K and is dependent on the dimension of the diode.     

Shattering News

 We explore the concepts of shattered and order-shattered sets. In particular, for every family ℱ of subsets of {1,2,…,m} there are exactly |ℱ| subsets of {1,2,…,m} order-shattered by ℱ. This provides proofs and strengthenings of the result of Sauer, Perles and Shelah, Vapnik and Chervonenkis (sometimes known as Sauer's lemma) and a new approach to the reverse Sauer Inequality of Bollobás and Radcliffe. We characterize those sets which can be order-shattered by a uniform family and those sets which can be order-shattered by an antichain. We also give an algebraic interpretation of order shattering using Gr?bner bases. This results in sharpening of a theorem of Frankl and Pach. It also points out an interesting and promising connection between combinatorial and algebraic objects. Received: May 9, 2000 Final version received: July 3, 2001     

Grobner-Shirshov Basis for Degenerate Ringel-Hall Algebras of Type F4

In this paper, by using the Frobenius morphism and the multiplication formulas of the generic extension monoid algebra, the authors first give a presentation of the degenerate Ringel-Hall algebra, and then construct the Gr\"{o}bner-Shirshov basis for degenerate Ringel-Hall algebras of type $F_4.$     

Ideal and subalgebra coefficients

For an ideal or -subalgebra of , consider subfields , where is generated - as ideal or -subalgebra - by polynomials in . It is a standard result for ideals that there is a smallest such . We give an algorithm to find it. We also prove that there is a smallest such for -subalgebras. The ideal results use reduced Gröbner bases. For the subalgebra results we develop and then use subduced SAGBI (bases), the analog to reduced Gröbner bases.

     

Theory of fluidity of liquids, glass transition, and melting

This is a presentation of a rigorous theory of fluidity of liquids, glass transition and melting of solids in the frame of an asymmetric double well potential model. Potential wells are doubled time to time by the local density fluctuations caused by the thermal longitudinal waves. The average frequency of doubling of potential wells is equal to the frequency of the most energetic waves which obey a law similar to Wein’s displacement law in black body radiation. Based on the equilibrium thermodynamic theory of fluctuations and the displacement law, a law of linear pre-diffusion mean-square displacement of particles in a solid is derived: the mean-square displacement of molecules within their potential wells increases linearly with temperature. It is shown that when this is broken-down (where the mean-square displacement at a certain temperature rapidly changes its slope as a function of temperature) glass devitrifies and crystal melts, and all possible solid-liquid transitions of a substance occur at the same critical mean-square displacement: any solid (not only crystals) transforms into liquid when the mean-square displacement, as a fraction of the average intermolecular distance, acquires a certain universal critical value - the same for different substances. It is proved that molecules in a liquid perform specific Brownian motion. The average jump distance is a function of temperature and it is much smaller than the nearest intermolecular distances. At a certain temperature, shown to be the Kauzmann temperature, the average jump distance of Brownian motion becomes equal to zero: the super-cooled liquid undergoes glass transition. The transition was proven to be a phase transition of the fourth order: the free energy of the system and its first, second and third derivatives are all continuous functions, but its fourth derivative with respect to temperature is discontinuous. Molecular mobility, diffusion and viscosity are obtained as functions of temperature.     

Synthesis and characterization of <Emphasis Type="Italic">S,N</Emphasis>-heteroacenes by Bischler-Napieralski reaction

Bischler-Napieralski cyclization has been successfully employed to prepare planar S,N-heteroacenes (SNHs) with the analogue structures to 1,10-phenanthroline. The opto-/electrical properties and the packing modes of these S,Nheteroacenes were examined, which were dependent on the structure of the fused groups. SNH-based copolymers with high molecular weight and good solubility in common organic solvents or water have been synthesized by Suzuki-Miyaura- Schlüter polycondensation. The photophysical property studies revealed that the copolymers may be a promising chemosensor material.     

Effect of interface states on negative capacitance characteristics in GaAs homojunction far-infrared detectors

Bias-, frequency- and temperature-dependent capacitance characteristics of p-GaAs homojunction interfacial work-function internal photoemission (HIWIP) far-infrared detectors are reported. A strong negative capacitance phenomenon has been observed. The origin of this effect is believed to be due to the carrier capture and emission at interface states, and has been confirmed by a comparison study of capacitance characteristics on p-GaAs HIWIP detectors with different interface state densities. A fitting data based on charging–discharging current and the inertial conducting current model show good agreement with the experimental observations. Received: 15 November 1999 / Accepted: 20 April 2000 / Published online: 5 October 2000     

On Decomposition of Algebraic PDE Systems into Simple Subsystems

In this paper we present an algorithmization of the Thomas method for splitting a system of partial differential equations and (possibly) inequalities into triangular subsystems whose Thomas called simple. The splitting algorithm is applicable to systems whose elements are differential polynomials in unknown functions and polynomials in independent variables. Simplicity properties of the subsystems make easier their completion to involution. Our algorithmization uses algebraic Gröbner bases to avoid some unnecessary splittings.     

Gr?bner bases in difference-differential modules with coefficients in a commutative ring

Insa and Pauer presented a basic theory of Grbner bases for differential operators with coefficients in a commutative ring and an improved version of this result was given by Ma et al.In this paper,we present an algorithmic approach for computing Grbner bases in difference-differential modules with coefficients in a commutative ring.We combine the generalized term order method of Zhou and Winkler with SPoly method of Insa and Pauer to deal with the problem.Our result is a generalization of theories of Insa and Pauer,Ma et al.,Zhou and Winkler and includes them as special cases.