Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

On the structure and volume growth of submanifolds in Riemannian manifolds

The aim of this note is two-fold. First, we investigate the relations between the volume growth of a submanifold and its second fundamental form. In the second part, we discuss the relations between the index of some Schrödinger operators and the structure of a submanifold, and prove some one-end theorems.     

Heavy tails for an alternative stochastic perpetuity model

In this paper we consider a stochastic model of perpetuity-type. In contrast to the classical affine perpetuity model of Kesten (1973) and Goldie (1991) all discount factors in the model are mutually independent. We prove that the tails of the distribution of this model are regularly varying both in the univariate and multivariate cases. Due to the additional randomness in the model the tails are not pure power laws as in the Kesten–Goldie setting but involve a logarithmic term.     

Endpoint estimates for multilinear fractional integrals with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rdwhich only satisfies the polynomial growth condition,the authors obtain the boundedness of the multilinear fractional integrals on Morrey spaces,weak-Morrey spaces and Lipschitz spaces associated with μ,which,in the case when μ is the d-dimensional Lebesgue measure,also improve the known results.     

Surface Tension of Ternary Liquid Mixtures

Abstract

Surface tension has been measured by the differential capillary rise method for three ternary mixtures containing alkanes (hexane + cyclohexane+benzene, pentane + hexane + benzene and cyclohexane + heptane + toluene at 298.15\pm 0.1°K). The sign and magnitude of the excess surface tension and excess volume depend ultimately upon the chain length of the component of the mixtures. The results of the surface tension were compared with theoretical values obtained from Flory theory, Sanchez method, Brock-Bird relation and volume fraction statistics. There is reasonable agreement between theory and experiment.     

Volumetric and Viscometric Studies on Some Inorganic Electrolytes in Water and in Water-SDS Solution Systems

The apparent molal volumes (φ _v ) of NaCl, NaNO₃, NH₄Cl, CuCl₂, CuSO₄, CoSO₄ and MgSO₄ in water and in water-SDS (Sodium dodecyl sulphate) solutions were determined from density measurements at 308.15, 313.15 and 323.15 K respectively. The limiting apparent molal volume at infinite dilution φ ^o _v which is practically equal to the partial molal volume V ^o ₂) of these electrolytes were found to be higher in water-SDS solution systems than those in water solutions. Viscosity coefficients (A and B) for these systems were also determined by Jones-Dole equation. All these electrolytes, except NH₄Cl exhibit structure making behaviour in water and in water-SDS solutions. Ammonium chloride showed structure breaking properties in water and in 0.01 molar water-SDS solutions. In 0.1 molar SDS solution, it showed structure making behaviour at the temperature range studied. The properties of these electrolytes in water and in water-SDS solution systems have been discussed in terms of the charge, size and hydrogen bonding effect.     

Spectrum of a Rudin–Shapiro-like sequence

We show that a recently proposed Rudin–Shapiro-like sequence, with balanced weights, has purely singular continuous diffraction spectrum, in contrast to the well-known Rudin–Shapiro sequence whose diffraction is absolutely continuous. This answers a question that had been raised about this new sequence.