Measure-theoretic pressure for subadditive potentials

By using the definition of topological pressure for subadditive potentials, we introduce a new definition of measure-theoretic pressure for ergodic measures of subadditive potentials on a compact metric space. Under some assumption, we show that this definition has a similar formalism to the definition of measure-theoretic pressure for ergodic measures of additive potentials on a compact metric space. As an application, assume that the subadditive potential is Hölder continuous, then for an expansive homeomorphism with the specification property, we study the relationship between the measure-theoretic pressure and the periodic points.     

Upper escape rate of Markov chains on weighted graphs

We obtain an upper escape rate function for a continuous time minimal symmetric Markov chain defined on a locally finite weighted graph. This upper rate function, which has the same form as the manifold setting, is given in terms of the volume growth with respect to an adapted path metric. Our approach also gives a weak form of Folz’s theorem on the conservativeness as a consequence.     

Entropy in the cusp and phase transitions for geodesic flows

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one hand on symbolic methods and Markov partitions, and on the other on geometric techniques and approximation properties at the level of groups.     

A Convenient Way to Characterize Equivalent Martingale Measures in Incomplete Markets

It is an empirical fact that the (empirically) relevant models for asset prices often describe markets that are incomplete in terms of their underlying assets, yielding many possible equivalent martingale measures under the no-arbitrage assumption. By using actual derivative prices, i.e., prices as observed in the market, additional information about the empirically relevant equivalent martingale measures might be obtained. In order to be able to process such information easily one needs a convenient way to represent all possible equivalent martingale measures in relation to derivative prices. In this paper we present such a convenient characterization. Conceptually, our characterization is not different from existing characterizations using, for example, Radon–Nikodym derivatives of martingale measures with respect to objective probabilities, but our characterization offers some advantages. The main advantage is that pricing derivatives is split up into two steps. The first step is solving a related complete markets pricing problem. This is a well-studied problem, so that it can easily be solved generally. In the second step a weighted average of the first step complete markets price must be calculated. Pricing under different equivalent martingale measures in the original market only differs with respect to the second step. The empirically relevant weighting can be determined by confronting the theoretical with the actually observed prices. As a byproduct we obtain a new and natural definition of idiosyncratic risk, which we show to be in line with the use of this term in the literature.To illustrate the ideas we discuss several examples. Among others we obtain the Hull–White formula for options on assets with stochastic volatility under close to minimal conditions that (for example) do not rely on a specification of the processes in terms of Itô diffusion.we relax the assumption of no-correlation between asset prices and volatilities in the Hull–White framework; we consider the case where the stochastic volatility does bear a risk-premium; we discuss pricing under stochastic interest rates; and we consider square-root type processes. All these pricing problems, and many more, can conveniently be handled using the approach based on our characterization of the equivalent martingale measures in continuous time markets that are incomplete in the underlying assets.     

On power-law fluids with the power-law index proportional to the pressure

In this short note we study special unsteady flows of a fluid whose viscosity depends on both the pressure and the shear rate. Here we consider an interesting dependence of the viscosity on the pressure and the shear rate; a power-law of the shear rate wherein the exponent depends on the pressure. The problem is important from the perspective of fluid dynamics in that we obtain solutions to a technologically relevant problem, and also from the point of view of mathematics as the analysis of the problem rests on the theory of spaces with variable exponents. We use the theory to prove the existence of solutions to generalizations of Stokes’ first and second problem.     

Escape Rate of Markov Chains on Infinite Graphs

We study the escape rate of continuous time symmetric Markov chains associated with weighted graphs. The upper rate functions are given in terms of volume growth of the weighted graphs. For a class of symmetric birth and death processes, we obtain sharp upper rate functions.     

Continuous dependence on parameters of certain self-affine measures,and their singularity

In this paper, we first prove that the self-affine sets depend continuously on the expanding matrix and the digit set, and the corresponding self-affine measures with respect to the probability weight behave in much the same way. Moreover, we obtain some sufficient conditions for certain self-affine measures to be singular.     

Growth and Approximation of Generalized Bi-Axially Symmetric Potentials

The paper deals with growth estimates and approximation (not necessarily entire) of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials (GBASP's). To obtain more refined measure of growth, we have defined $q$-proximate order and obtained the characterization of generalized $q$-type and generalized lower $q$-type with respect to $q$-proximate order of a GBASP in terms of approximation errors and ratio of these errors in sup norm.     

Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.     

Spectral representation of semistable processes,and semistable laws on Banach spaces

We introduce the notion of semistable processes and semistable random measures; and give a characterization of semistable laws on Banach spaces. Using this charcterization, we discuss the existence of semistable random measures, define the stochastic integrals with respect to these measures, and obtain the spectral representations of arbitrary (not necessairly symmetric) semistable and stable processes. In addition, we give a criterion of independence for stochastic integrals.     

Escape rate and Hausdorff measure for entire functions

The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.     

K-拟可加模糊数值积分及其收敛性

在K-拟可加模糊测度空间上,针对一类(?)-可积模糊数值函数,建立了所谓的K-拟可加模糊数值积分,并通过引入诱导算子K,获得这种积分的转换定理,进而研究这种K-拟可加模糊数值积分的一些重要性质,同时给出了它的一系列收敛定理,从而丰富了模糊数学的积分理论。     

Classical statistics of one-component systems with model potentials

We obtain a functional integral representation on the configuration integral for one-component classical systems with two-particle potentials admitting the Fourier expansion. In this representation, the integrand is factored with respect to the atomic coordinates. The “monoatomic” factors are universal (i.e., independent of the explicit form of the interatomic potential). We obtain a sufficient condition for spontaneous symmetry breaking in continuous classical statistics models. We investigate the case of model potentials with a nonnegative Fourier transform. The functional integral for this class of potentials is calculated using the saddle-point method. We prove the existence of phase transitions for some model potentials. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 167–176, April, 1999.     

Integral representations of generalized axially symmetric potentials in a simply connected domain

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a complex variable that take real values on the real axis and generalized axially symmetric potentials of certain classes.     

Convex Potentials and Optimal Shift Generated Oblique Duals in Shift Invariant Spaces

We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in \(L^2(\mathbb {R}^k)\). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.     

Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. II

We continue the investigation of the problem of energy minimum for condensers began in the first part of the present work. Condensers are treated in a certain generalized sense. The main attention is given to the case of classes of measures noncompact in the vague topology. In the case of a positive-definite kernel, we develop an approach to this minimum problem based on the use of both strong and vague topologies in the corresponding semimetric spaces of signed Radon measures. We obtain necessary and (or) sufficient conditions for the existence of minimal measures. We describe potentials for properly determined extremal measures.     

From discrete to continuum: A Young measure approach

The passage from atomistic to continuum models is usually done via G\Gamma-convergence with respect to the weak topology of some Sobolev space; the obtained continuum energy, in a one dimensional model, is then convex. These kind of results are not optimal for problems related to materials which may undergo to phase transitions. We present here a new simple way for dealing with these problems. Our method consists in rewriting the discrete energy in terms of particular measures and taking the G\Gamma-limit with respect to the weak * convergence of measures. The continuum energy arising from a linear chain of discrete mass points interacting with only the nearest neighbours turns out to be written in terms of Young measures. While, if the discrete mass points interact not only with the nearest neighbours but also with the second nearest neighbours we obtain a continuum problem in which appears a ``multiple Young measure" representing multiple levels of interaction. In this way we obtain a novel continuum problem which is able to capture the ``microstructure" at two different levels.     

Nonlinear Transformations of Evolution Families Generated by Diffusion Processes

We derive closed equations for some nonlinear transformations of solutions to forward and backward Kolmogorov equations (which are parabolic equations with respect to measures and functions). In particular, we obtain closed equations for the logarithmic derivatives of smooth diffusion measures and for the Radon–Nikodym derivatives of a pair of absolutely continuous diffusion measures. Similar results are obtained for the backward Kolmogorov equation. Bibliography: 14 titles.     

Active classification using belief functions and information gain maximization

Obtaining reliable estimates of the parameters of a probabilistic classification model is often a challenging problem because the amount of available training data is limited. In this paper, we present a classification approach based on belief functions that makes the uncertainty resulting from limited amounts of training data explicit and thereby improves classification performance. In addition, we model classification as an active information acquisition problem where features are sequentially selected by maximizing the expected information gain with respect to the current belief distribution, thus reducing uncertainty as quickly as possible. For this, we consider different measures of uncertainty for belief functions and provide efficient algorithms for computing them. As a result, only a small subset of features need to be extracted without negatively impacting the recognition rate. We evaluate our approach on an object recognition task where we compare different evidential and Bayesian methods for obtaining likelihoods from training data and we investigate the influence of different uncertainty measures on the feature selection process.     

Representations of Positive Projections II

In this paper we obtain a number of Maharam-type slice integral representations, with respect to scalar measures, for positive projections in Dedekind complete vector lattices and f-algebras. AMS Classification: 47B65, 46A40, 06F25