Universality and asymptotics of graph counting problems in non-orientable surfaces

Bender-Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants tg and pg for the orientable and the non-orientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series solution u(z) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of tg to all orders in 1/g for large g. The paper introduces a formal power series solution v(z) of a Riccati equation, gives a non-linear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence pg and v(z). Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2-dimensional projective plane. Our conjecture implies analyticity of the O(N)- and Sp(N)-types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann-Hilbert approach, and provide ample numerical evidence for our results.     

On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions

We provide universal algebraic characterizations (in the sense of not involving any “logical notion”) of some elementary classes of structures whose definitions involve universal d-Horn sentences and universally closed disjunctions of atomic formulas. These include, in particular, the classes of fields, of non-trivial rings, and of directed graphs without loops where every two elements are adjacent. The classical example of this kind of characterization result is the HSP theorem, but there are myriad other examples (e.g., the characterization of elementary classes using isomorphic images, ultraproducts and ultrapowers due to Keisler and Shelah).     

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), ${\Omega\subset{\mathbb R}^N}The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), W ì \mathbb R^N{\Omega\subset{\mathbb R}^N} , with p = N, that is, the case of Pohozhaev–Trudinger–Moser inequality. Similarly to the case p < N where the loss of compactness in W^1,p(\mathbb R^N){W^{1,p}({\mathbb R}^N)} occurs due to dilation operators u ?t^(N-p)/pu(tx){u {\mapsto}t^{(N-p)/p}u(tx)} , t > 0, and can be accounted for in decompositions of the type of Struwe’s “global compactness” and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W₀^1,N{W_0^{1,N}} over a ball in \mathbb R^N{{\mathbb R}^N} . We give a one-parameter scale of Hardy–Sobolev functionals, a “p = N”-counterpart of the H?lder interpolation scale, for p > N, between the Hardy functional ò\frac|u|^p|x|^p dx{\int \frac{|u|^p}{|x|^p}\,{\rm d}x} and the Sobolev functional ò|u|^pN/(N-mp)  dx{\int |u|^{pN/(N-mp)} \,{\rm d}x} . Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W ^1,N -norms under Hardy–Sobolev constraints.     

The multicriteria analysis for construction of benchmarkers to support the Clinical Engineering in the Healthcare Technology Management

In developing countries, approximately half of the healthcare equipments are not in full use and the main cause of this is the inadequate management of them. The Clinical Engineering might face this problem in the healthcare environment analyzing the equipment with a health technological process perspective to identify opportunities of improvements. Generally, the inadequate management is a result of lack of systemized and contextualized information about the health technological process. The effort to make an adequate management generates an increasing interest in the use of benchmarker. Currently, the benchmarkers used by the Clinical Engineering are not representative of a health technological process as a whole. Health technological process concept is stated in this article. The multicriteria analysis methodology MCDA (Multicriteria Decision Aid) is used to obtain benchmarkers and to identify opportunities of improvements, thus generating conditions for that the Clinical Engineering consolidates its relevant contribution for the healthcare.     

Thin film photochromic materials: Effect of the sol-gel ormosil matrix on the photochromic properties of naphthopyrans

Photochromic naphthopyran derivatives have been embedded in sol-gel prepared organically modified thin films. The introduction of organic functional groups into a silica matrix allows tailoring the surface of its pores and the polarity of the environment of the embedded host molecules. The photochromic properties of the naphthopyran molecules, such as the spectral properties of the coloured forms and the kinetics of the thermal bleaching, depend strongly on the polarity of the pores where the molecules are located and, hence, on the nature and loading of organic functional groups in the composition of the ormosil matrix. Important changes in the photochromic properties of the films have also been induced by modifications in the sol-gel preparation and processing parameters. The photostability of the photochromic molecules upon prolonged exposition to UV light is strongly related to the nature of the embedding ormosil matrix. The introduction of organic functional groups into the inner pore surface of the matrix, where the dye molecules will be located, affects the stability of the molecules, in terms of the effectiveness of the interaction between the photochromic molecules and the pore surface.