On the diameter and radius of randon subgraphs of the cube

The n-dimensional cube Qⁿ is the graph whose vertices are the subsets of {1,…n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Qⁿ has diameter and radious n. Write M = n2^n-1 = e(Qⁿ) for the size of Qⁿ. Let Q = (Q_t)_o^M be a random Qⁿ-process. Thus Q^t is a spanning subgraph of Qⁿ of size t, and Q_t is obtained from Q_t–1 by the random addition of an edge of Qⁿ not in Q_t–1, Let t^(k) = τ(Q;δ?k) be the hitting time of the property of having minimal degree at least k. We show that the diameter d_t = diam (Q_t) of Q_t in almost every Q? behaves as follows: d_t starts infinite and is first finite at time t⁽¹⁾, it equals n + 1 for t⁽¹⁾ ? t⁽²⁾ and d_t, = n for t ? t⁽²⁾. We also show that the radius of Q_t, is first finite for t = t⁽¹⁾, when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Q_t, for t = Ω(M). © 1994 John Wiley & Sons, Inc.     

Talbot self-imaging in fractional Fourier planes of real and complex orders

A study of Talbot self-imaging associated with fractional Fourier transforms (FRFTs) of real and complex orders, as realised by bulk lenses, is presented. Self-images of a periodic object, corresponding to different orders, can be observed in different FRFT planes, herein called Talbot-FRFT planes. The spatial scaling factor of the self-images is shown to be a harmonic function of the FRFT order.     

Risk-neutral valuation with infinitely many trading dates

The first Fundamental Theorem of Asset Pricing establishes the equivalence between the absence of arbitrage in financial markets and the existence of Equivalent Martingale Measures, if appropriate conditions hold. Since the theorem may fail when dealing with infinitely many trading dates, this paper draws on the A.A. Lyapunov Theorem in order to retrieve the equivalence for complete markets such that the Sharpe Ratio is adequately bounded.     

Synthesis and polymerization of o-hexylaniline: Characterization of the corresponding polyaniline

In the field of research on soluble conducting polymers, the poly(o-alkylanilines) are very interesting because we can expect them to give more soluble polymers and new properties. Like poly(o-propylaniline) (POP), which is more soluble than polyaniline (PANi), poly(o-hexylaniline) (POH) appears to be more soluble in organic solvents than POP because of the longer alkyl groups in the 2-position. The higher solubility confers better processability on this new polymer, and because of this solubility, an NMR study in solution became possible.The nitration of hexylbenzene and the reduction of the resulting product to o-hexylaniline were performed according to the literature. The chemical polymerization was easy and it is possible to produce this polymer in large quantities.The polymerization carried out in anhydrous NH₄F, 2.35 HF medium and in 5 M perchloric acid gave a polymer with almost quantitative yield. The electrochemical behaviour of POH displayed faster electron transfers than PANi in organic solvents, depending on the acido-basicity level of the aqueous solutions. Unlike PANi, fractal growth was not observed.     

Magnetic calorimeter for registration of small energy release

A scheme of magnetic calorimeter for registration of rare events characterized by small energy release (cosmic rays, WIMPs, solitary X-ray quanta) is proposed. The calorimeter is brought to operation by adiabatic demagnetization, and its magnetic response is measured by a quantum interferometer (SQUID, A. Barone and G. Paterno, Physics and applications of Josephson Effect). Special consideration is given to the specific features of calorimeter operation in the ferromagnetic transition region. The trigger registration of ultrasmall energy release by a ferromagnetic system in the metastable state is described.     

Path properties of l_p-valued Gaussian random fields

General limit theorems are established for l~p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l~p-valued Gaussian random fields under(?)explicit conditions.     

Micellar effect in hydroformylation of high olefin catalysed by water-soluble rhodium complexes associated with sulfonated diphosphines

High linear alkenes (1-octene and 1-decene) have been hydroformylated using water-soluble rhodium complexes associated with sulfonated diphosphines in the presence of ionic surfactants or methanol. In all cases, the hydroformylation activities were higher than in experiments without additives. The selectivity in aldehydes was higher when we used cetyltrimethylammonium hydrogensulfate (CTAHSO₄) as the surfactant or methanol as the co-solvent.     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.