Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).     

Dimensional reduction of the 5D Kaluza–Klein geodesic deviation equation

In the work of Kerner et al. (Phys Rev D 63:027502, 2001) the problem of the geodesic deviation in a 5D Kaluza–Klein background is faced. The 4D space–time projection of the resulting equation coincides with the usual geodesic deviation equation in the presence of the Lorenz force, provided that the fifth component of the deviation vector satisfies an extra constraint which takes into account the q/m conservation along the path. The analysis was performed setting as a constant the scalar field which appears in Kaluza–Klein model. Here we focus on the extension of such a work to the model where the presence of the scalar field is considered. Our result coincides with that of Kerner et al. when the minimal case f = 1{\phi=1} is considered, while it shows some departures in the general case. The novelty due to the presence of f{\phi} is that the variation of the q/m between the two geodesic lines is not conserved during the motion; an exact law for such a behaviour has been derived.     

Morphology Evolution During Polymer Crystallization Simultaneous Calorimetric and Optical Measurements

The crystallization of the isotactic poly(propylene) (I-PP) has been studied carrying out measurements by means of a special calorimeter connected to a microscope and a digital acquisition system of images. To authors' knowledge, this is the first time that simultaneous calorimetric and optical measurements are carried out on polymers. The analysis of Polarized Optical Microscopy images has allowed the appraisal of nucleation density and growth rate in isothermal and non isothermal conditions. The results obtained in isothermal conditions have been analyzed through the Kolmogoroff model and the crystallinity calculated from the model has been compared with that obtained from the calorimetric measurements.     

Effect of delivery system on the pharmacokinetic and phototherapeutic properties of bis(methyloxyethyleneoxy)silicon-phthalocyanine in tumor-bearing mice

A Si(IV)-phthalocyanine bearing two methoxyethyleneglycol axial ligands bound to the central metal ion (SiPc) has been prepared by chemical synthesis and analyzed for its phototherapeutic activity after administration in a Cremophor or liposome formulation to C57B1/6 mice bearing a subcutaneously transplanted Lewis lung carcinoma (LLC). The maximum drug accumulation in the tumor is found at 24 h after intraperitoneal injection, independent of the delivery system. However, the tumor concentration of SiPc in the Cremophor formulation is about two-fold higher, while the drug concentration in liver and skin shows similar trends with the two delivery systems. The drug accumulation and retention in the brain is much larger when using Cremophor emulsion. Photodynamic therapy (672 nm, 370 mW m⁻², 360 J cm⁻²) at 24 h after the injection of Cremophor emulsion- or DPPC liposome-formulated SiPc causes a very efficient and similar response for the LLC (8 versus 22 mm mean tumor diameter for the control groups at 21 days after phototreatment). These very promising effects, obtained both at higher and lower tumor drug concentrations, clearly demonstrate the potential phototherapeutical activity of the newly synthesized SiPc.     

Half-Range Expansions for an Astrophysical Problem

Problems of spectral analysis are studied for an indefinite singular boundary value problem coming from astrophysical theory of particle acceleration around shocks. This leads to a nonclassical initial-boundary value problem for a partial differential equation that can bereduced by separation of variables to an indefinite Sturm–Liouville problem for which we establish Riesz basis properties of the eigen- and associated functions and formulate completeness and expansion theorems.     

Cyclic k‐cycle systems of order 2kn + k: A solution of the last open cases

We exhibit cyclic (K_v, C_k)‐designs with v > k, v ≡ k (mod 2k), for k an odd prime power but not a prime, and for k = 15. Such values were the only ones not to be analyzed yet, under the hypothesis v ≡ k (mod 2k). Our construction avails of Rosa sequences and approximates the Hamiltonian case (v = k), which is known to admit no cyclic design with the same values of k. As a particular consequence, we settle the existence question for cyclic (K_v, C_k)‐designs with k a prime power. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 299–310, 2004.     

On the Posets ( \user1Wk2 , < ){\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)} and their Connections with Some Homogeneous Inequalities of Degree 2

A class of ranked posets {(D _h ^k , ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D ₂ ^k , which are related to systems of the form {x _a x _b * _abcd x _c x _d : 0 ≤ a, b, c, d ≤ k, * _abcd ∈ {<, >}, 0 < x ₀ < x ₁ < ...< x _k}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets . These structures, whose elements are all the D ₂ ^k 's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.     

Combinatorial Aspects of Total Weight Orders over Monomials of Fixed Degree

Among all the restrictions of weight orders to the subsets of monomials with a fixed degree, we consider those that yield a total order. Furthermore, we assume that each weight vector consists of an increasing tuple of weights. Every restriction, which is shown to be achieved by some monomial order, is interpreted as a suitable linearization of the poset arising by the intersection of all the weight orders. In the case of three variables, an enumeration is provided. For a higher number of variables, we show a necessary condition for obtaining such restrictions, using deducibility rules applied to homogeneous inequalities. The logarithmic version of this approach is deeply related to classical results of Farkas type, on systems of linear inequalities. Finally, we analyze the linearizations determined by sequences of prime numbers and provide some connections with topics in arithmetic.