Some new kinds of pseudoprimes

We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow -pseudoprimes and elementary Abelian -pseudoprimes. It turns out that every which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow -pseudoprime to two of these bases for an appropriate prime

We also give examples of strong pseudoprimes to many bases which are not Sylow -pseudoprimes to two bases only, where or

     

Background Independence in Quantum Gravity and Forcing Constructions

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ⁴. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory.     

Cohen-Macaulay Properties of Thom-Boardman Strata II: The Defining Ideals of {Sigma}i, j

We consider the Zariski closure of Thom–Boardman strata^{i, j} in the jet space for map-germs (Cⁿ, 0) (C^p, 0). We applythe geometric technique of calculating syzygies to the desingularizationof ^{i, j} constructed by Ronga. We investigate when the normalizationof the coordinate ring of ^{i, j} is rational in the cases i =n – p + 1 or i = j = 1. We also consider when these coordinaterings are normal with rational singularities. 2000 MathematicsSubject Classification 13D02, 14B05, 14M05, 14M12, 58C25 (primary),58K20 (secondary).     

Differences in Channelling of H0 and H+ in Si Single Crystal

The differences in the RBS angular yields measured for 100 keV hydrogen atoms H⁰ and protons H⁺ backscattered from Si crystal are reported and analysed. It was shown that H⁰ atom beam is better channelled in the pure crystal and is much more sensitive to the crystal surface coverage, particularly, Au layer than the H⁺ ion beam. The deep crystal regions seem to strengthen this differences.     

From noncommutative κ-Minkowski to Minkowski space–time

We show that free κ-Minkowski space field theory, discussed in the context of κ-Poincaré algebra and Doubly Special Relativity is equivalent to a relativistically invariant free field theory on Minkowski space–time. The field theory we obtain has in spectrum a relativistic mode of arbitrary mass m and a Planck mass tachyon. We show that while the energy–momentum for the relativistic mode is essentially the standard one, it diverges for the tachyon, so that there are no asymptotic tachyonic states in the theory. It also follows that the dispersion relation is not modified, so that, in particular, in this theory the speed of light is energy-independent.     

Inheriting independence and chi-squaredness under certain matrix orderings

Let x ～ N(μ, Z), and let S = (Σ:μ). It is shown that if x′A₁x is independent of x′Bx (x′A₁x is distributed as a chi-square variable), then this property is inherited by every x′A₂x for which S′A₂S precedes S′A₁S with respect to the range preordering (with respect to the rank subtractivity partial ordering).     

Exponential-like responses analysis method

An exponential-like responses analysis method is presented. The method enables one: 1) to evaluate the similarity of a curve to the ideal exponential characteristic; 2) to find the most appropriate final value of a curve; and 3) to find the value of Τ giving the best approximation between the analyzed curve and the ideal exponential one.