Canonizing structural Ramsey theorems

At the beginning of 1950s Erd?s and Rado suggested the investigation of the Ramsey-type results where the number of colors is not finite. This marked the birth of the so-called canonizing Ramsey theory. In 1985 Prömel and Voigt made the first step towards the structural canonizing Ramsey theory when they proved the canonical Ramsey property for the class of finite linearly ordered hypergraphs, and the subclasses thereof defined by forbidden substructures. Building on their results in this paper we provide several new structural canonical Ramsey results. We prove the canonical Ramsey theorem for the class of all finite linearly ordered tournaments, the class of all finite posets with linear extensions and the class of all finite linearly ordered metric spaces. We conclude the paper with the canonical version of the celebrated Ne?et?il–Rödl Theorem. In contrast to the “classical” Ramsey-theoretic approach, in this paper we advocate the use of category theory to manage the complexity of otherwise technically overwhelming proofs typical in canonical Ramsey theory.     

The intermediate disorder regime for a directed polymer model on a hierarchical lattice

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number $b\in \mathbb{N}$ and a segment number $s\in \mathbb{N}$. When $b\le s$ it is known that the model exhibits strong disorder for all positive values of the inverse temperature $\beta$, and thus weak disorder reigns only for $\beta =0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $\beta \equiv {\beta }_n$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b<s$ and $b=s$. In the case $b<s$ we prove that when the inverse temperature is taken to be of the form ${\beta }_n=\stackrel{?}{\beta }{(b/s)}^{n/2}$ for $\stackrel{?}{\beta }>0$, the normalized partition function of the system converges weakly as $n\to \infty$ to a distribution $L\left(\stackrel{?}{\beta }\right)$ and does so universally with respect to the initial weight distribution. We prove the convergence using renormalization group type ideas rather than the standard Wiener chaos analysis. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as ${\beta }_n=\stackrel{?}{\beta }/n$; for an explicitly computable critical value ${\kappa }_b>0$ the variance of the normalized partition function converges to zero with large $n$ when $\stackrel{?}{\beta }\le {\kappa }_b$ and grows without bound when $\stackrel{?}{\beta }>{\kappa }_b$. Finally, we prove a central limit theorem for the normalized partition function when $\stackrel{?}{\beta }\le {\kappa }_b$.     

Edelstein's fixed point theorem in topological spaces

An extension to topological spaces of a wellknown fixed point theorem of M. Edelstein for contractive mappings on metric spaces is presented. Results based on the generalized Edelstein's theorem are also established concerning the existence of fixed points of continuous selfmaps on a topological space. As a special case a compact starshaped subset of a linear topological space is considered. The results extend the fixed point theoremsfor nonexpansive mappings on a compact metric space of L.F.Guseman, Jr. and B.C. Peters, Jr.     

Lattice point theory in polyhedra

The exact order of the remainder term is determined in the formula for the number of lattice points in the region α₁∥u₁ + b₁∥ + α₂∥u₂ + b₂∥ + … + α_r∥u_r + b_r∥ ≤ x in dependence on the arithmetical properties of the coefficients α₁, α₂,…, α_r.     

Rapid solidification processing in molten salts chemistry: X-ray analysis of deeply undercooled cryolite-alumina melts

Rapid solidification processing (cooling rate from the interval 10⁵–10⁶ K s⁻¹) was used to prepare deeply undercooled cryolite-alumina melts. Such prepared samples were analyzed by the XRD method. Besides cryolite, XRD patterns belonging to ι-Al₂O₃ were recorded. The influence of annealing on the XRD patterns of deeply undercooled melts was also investigated.     

Mathematical modelling of selected characterisation procedures for oil fractions

Models of two important characterisation procedures for oil and oil fractions, ASTM D86 (ASTM = American Society for Testing of Materials) and EFV (Equilibrium Flash Vaporization), were presented. The purpose was to provide a basis for the construction of substitute mixtures of real components used in modelling and simulation of chemical engineering processes instead of original complex mixtures. It was shown how to deal with a possible problem of higher index of differential-algebraic model of the ASTM D86. A number of examples proved the adequacy of the presented models. Presented at the 34th International Conference of the Slovak Society of Chemical Engineering, Tatranské Matliare, 21–25 May 2007.     

Analysis and modification of nonhyperbolic kinetic models

A nonhyperbolic model of Rabin (Biochem J 1967, 102, 22C–23C) in which four conformational states are considered has been simplified to the three‐state one. Only multistate models (if the concentration of enzyme or receptor is assumed to be low) can yield nonhyperbolic kinetics. The simplified model has been shown to retain all the essential properties of the original four‐state model. The model yields a diversity of nonhyperbolic dose–response curves both with higher and lower steepness than that of Henri‐type ones. The three‐state model can be further reduced to the hyperbolic one by making identical the different (unliganded) states. The bistate model in which protein concentration is assumed to be high yields nonhyperbolic kinetics as well; the model is reduced to the hyperbolic one if the protein concentration is assumed to be low. © 2008 Wiley Periodicals, Inc. Int J Chem Kinet 40: 253–258, 2008     

Nanotubes Made from Deeply Undercooled Cryolite/Alumina Melts

The rapid‐solidification processing (by a cooling rate of 10⁵–10⁶ K/s) was used for the preparation of deeply undercooled cryolite/alumina (Na₃AlF₆/Al₂O₃) melts. We found a mass of nanotubes on the surface of these undercooled melts. The nanotubes were preferentially located on the defect places of the surface with the following approximate dimensions: base≈100×100 nm, length≈1000 nm. The solidified samples with the nanotubes on the surface were analyzed by scanning electron microscopy (SEM), energy‐dispersive X‐ray spectroscopy (EDX), X‐ray diffraction (XRD), and infrared spectroscopy (IR).     

Generalized localization and convergence tests for double trigonometric Fourier series of functions fromL p,p>1

. E , f(x)L _p (T ^N ),P1,f(x)=0 E (E— ^N =[-, ]^N) E , . , .

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In closing the author thanks V. A. Il'in and . A. Alimov for their constant attention paid to the present work.