Nilpotent fusion categories

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.     

Analysis of thermal phases in calcareous Iron Age pottery using FT-IR and Raman spectroscopy

The work deals with the FT-IR and micro-Raman spectroscopy study of the pseudo-amorphous and crystalline thermal phases in the composition of calcareous Iron Age pottery from the Galilee. The application of second-derivative and curve-fitting techniques improves the identification of the thermal phases in the composition of the pottery and makes it possible to analyze the pseudo-amorphous phases which are formed during the firing of the clayey raw material to pottery. This technique makes it possible to distinguish between meta-smectite and meta-kaolinite and to estimate the firing temperature of the pottery. The Micro-Raman spectroscopy is sensitive to the structural degree of ordering of the thermal phases and enables point analysis of peculiar components in the composition of the pottery. Based on the spectroscopic study, it is concluded that the calcareous pottery contained large amounts of microcrystalline-recarbonated calcite mixed with the meta-clay. The large amount of recarbonated calcite in the pottery material and the relatively low firing temperature indicates that instead of sintering the clay, lime technology was used for the cementation of the calcareous vessels. This process took place after the firing by recarbonation of the decomposed calcite which leads to cementation of the vessels with microcrystalline calcite.     

Response times in gated M/G/1 queues: The processor-sharing case

A class of single server queues with Poisson arrivals and a gated server is considered. Whenever the server becomes idle the gate separating it from the waiting line opens, admitting all the waiting customers into service, and then closes again. The batch admitted into service may be served according to some arbitrary scheme. The equilibrium waiting time distribution is provided for the subclass of conservative schemes with arbitrary service times and the processor-sharing case is treated in some detail to produce the equilibrium time-in-service and response time distributions, conditional on the length of required service. The LIFO and random order of service schemes and the case of compound Poisson arrivals are treated briefly as examples of the effectiveness of the proposed method of analysis. All distributions are provided in terms of their Laplace transforms except for the case of exponential service times where the L.T. of the waiting time distribution is inverted. The first two moments of the equilibrium waiting and response times are provided for most treated cases and in the exponential service times case the batch size distribution is also presented.     

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.     

Rectified approximations for the solution of nonlinear equations

In this paper we introduce the ‘rectification method’ for the construction of algorithms of pre-designed order r for the solution of nonlinear equations f(x) = 0. Our method is based upon the derivation of a rectified approximation g(x) to f(x), via Padé formulas, such that the application of the Newton-Raphson iterations to g generates the desired rth order algorithm. Various properties of g are explored as are recursive relations among rectified approximations associated with successive orders of convergence. It is demonstrated that the use of g in favor of f can relax standard sufficient conditions assuring convergence of the iterations.     

Anisotropic percolation and the d-dimensional surface roughening problem

We review recent numerical simulations of several models of interface growth in d-dimensional media with quenched disorder. These models belong to the universality class of anisotropic diode-resistor percolation networks. The values of the roughness exponent δ=0.63±0.01 (d=1+1) and δ=0.48±0.02 (d=2+1) are in good agreement with our recent experiments. The values of δ in higher dimensions (δ=0.38±0.03 in d=4 and δ=0.27±0.05 in d=5) do not support a recent theoretical conjecture.     

Activation of tertiary paraffins by elemental fluorine

Elemental fluroine was found to substitute, in a regio and stereospecific way, tertiary unactivated hydrogens in alkanes - the most unreactive family of organic compounds.     

The complexity of identifying redundant and essential elements

In many optimization problems a solution is a subset of optimum number of elements satisfying some desired property. An element is redundant if it does not belong to any solution of the problem. An element is essential if it belongs to every solution of the problem. We consider the complexity of indentifying redundant and essential elements in a sample of NP-Hard optimization problems. It is shown that these identification problems are also NP-Hard. The proofs are based on an analysis of the original reductions of Cook [The complexity of theorem proving procedures, in “Proceedings, Third Annual Assoc. Comput. Mach. Symposium on Theory of Computing”, pp. 151–158, Assoc. Comput. Mach., New York 1971] and Karp [Reducibility among combinational problems, in “Complexity of Computer Computations” (R. E. Miller and J. W. Thatcher, Eds.), pp. 85–104, Plenum, New York 1972].