On three conjectures by K. E. Shuler

Some fifteen years ago, Shuler formulated three conjectures relating to the large-time asymptotic properties of a nearest-neighbor random walk on ² that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition. We also prove a strong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple random walk on . We briefly discuss extensions to higher dimension and to other types of random walk.Dedicated to Prof. K. E. Shuler on the occasion of his 70th birthday, celebrated at a Symposium in his honor on July 13, 1992, at the University of California at San Diego, La Jolla, California.     

Operator Semigroup Compactifications

A weakly continuous, equicontinuous representation of a semitopological semigroup on a locally convex topological vector space gives rise to a family of operator semigroup compactifications of , one for each invariant subspace of . We consider those invariant subspaces which are maximal with respect to the associated compactification possessing a given property of semigroup compactifications and show that under suitable hypotheses this maximality is preserved under the formation of projective limits, strict inductive limits and tensor products.

     

A strong limit theorem for generalized Cantor-like random sequences

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The O(N) vector model in the large-N limit revisited: multicritical points and double scaling limit

The multicritical points of the O(N)-invariant N vector model in the large-N limit are re-examined. Of particular interest are the subtleties involved in the stability of the phase structure at critical dimensions. In the limit N → ∞ while the coupling g → g_c in a correlated manner (the double scaling limit) a massless bound state O(N) singlet is formed and powers of 1/N are compensated by IR singularities. The persistence of the N → ∞ results beyond the leading order is then studied with particular interest in the possible existence of a phase with propagating small mass vector fields and a massless singlet bound state. We point out that under certain conditions the double scaled theory of the singlet field is non-interacting in critical dimensions.     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

Preparation and Characterisation of Meloxicam Hydroxy Propyl <Emphasis Type="Italic">β</Emphasis>-Cyclodextrin Inclusion Complex

Meloxicam is a non steroidal anti inflammatory drug, used in the treatment of rheumatoid and osteoarthiritis. It is practically insoluble in water and its prolonged use is associated with the incidence of side effects like gastro intestinal perforations, ulcerations and bleeding. Therefore, an attempt has been made to improve the aqueous solubility of the drug by making an inclusion complex using hydroxy propyl β cyclodextrin(HPβ-CD). The complexes were prepared by physical mixture and freeze drying method. The different methods employed for evaluation such as DSC, XRD, SEM and FT-IR studies indicated complete formation of the complex by freeze drying method in a molar ratio of 1:2. The prepared complexes showed improved in-vitro dissolution profile as compared to the pure drug.     

Spherical models with a Gates-Penrose-type phase transition

Gates and Penrose have given criteria under which classical gases with weak long-range interactions fail to be described by the van der Waals equation with Maxwell's rule. Unfortunately, examples of equations of state for such systems have not yet been produced. This paper examines the Gates-Penrose class of interactions-i.e.,U (r)=q(r)+(r), in the limit0, where the Fourier transform (p) has a minimum at a nonzero value ofp-for the spherical model on a one-dimensional lattice. Free energy and magnetization isotherms are computed; it is seen that there is a phase transition, but that the zero-field spontaneous magnetization is always zero (a parahelicoidal phase). However, the pair-correlation function may exhibit either long-range order or the appearance of oscillation.     

New indices for analysing partial ranking diagrams

Interest is growing in decision making strategies and several techniques are now available. The assessment of priorities is a typical premise before final decisions are taken. Total and partial order ranking (POR) strategies, which from a mathematical point of view are based on elementary methods of discrete mathematics, appear as an attractive and simple tool to asses priorities. Despite the well-known total ranking strategies, which are scalar methods combining the different criteria values into a global index which always ranks elements in an ordered sequence, the partial order ranking is a vectorial approach which recognises that not all the elements can be directly compared with all the others. In fact when many criteria are considered, contradictions in the ranking are bound to exist and the higher the number of criteria, the higher the probability that contradictions in the ranking occur. The Hasse diagram technique (HDT) is a very useful tool to perform partial order ranking. The results of the partial order ranking are visualised in a diagram, called Hasse diagram. Incomparable elements are located at the same geometrical height and as high as possible in the diagram, thus incomparable elements are arranged in levels. The quality of a ranking procedure has to be evaluated by a deep analysis and by several indices, i.e. scalar functions that describe features of an ordered set and allow comparison among different rankings. For this purpose, new indices for ranking analysis are proposed here, compared with the ones found in literature and tested on theoretical examples and on real data.     

Effect of temperature on the solubility of carnallite type double salts

A method based on Pitzer's model has been used for calculation of the solubilities of carnallite type double salts crystallizing in the systems MeX–MgX₂–H₂O (Me=Li, NH₄, K, Rb, Cs; X=Cl, Br). The solubility of congruently soluble double salts was also determined experimentally at 25–75°C. The results from studies of the solubility diagrams of ternary carnallite type water-salt systems over a wide temperature range are summarized. It is found that the width of the crystallization region for each of the salts can be explained by the relative differences in the solubilities of the ternary solution components (MeX, MgX₂·6H₂O and MeX·MgX₂·6H₂O) and by a change of temperature, and by the effect of temperature on the solubility. A method is proposed for calculating Pitzer's ternary parameters of interionic interaction (_MN and _MNX) on the basis of experimental data for the solubility in water of the double salts crystallizing in the corresponding ternary water-salt systems.     

Peculiarities of changes of some physicomechanical characteristics of monoisotopes B, B and natural β-boron

Real structure and some physicomechanical characteristics of the samples of natural beta-rhombohedral boron B as well as of its ¹⁰B and ¹¹B monoisotopes have been studied. It was shown that the influence of ¹⁰B and ¹¹B isotopes on physicomechanical properties of boron had a different character. In particular, the samples enriched with ¹¹B had high values of microhardness, shear modulus (SM) and elastic limit if compared to those of boron, while the samples enriched with ¹⁰B monoisotopes were characterized with high values of thermal expansion coefficient (TEC) and thermal conductivity; lattice parameters a and c increased by the sequence: ¹¹B, B, ¹⁰B. It was established that TEC, thermal conductivity, microhardness, SM and shear elastic limit increased in all samples at annealing for 5 h at 1500 °C regardless of isotope content.