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This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media which are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. The formulation allows a generalization of the principles of virtual work, and virtual stresses, which, in turn, allow us to extend the extremum and variational theorems of elasticity and plasticity, as well as handle flows in fractal porous media. In all the cases, the derived relations depend explicitly on D, d and R, and, upon setting D=3 and d=2, reduce to conventional forms of governing equations for continuous media with Euclidean geometries. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure making it capable of grasping local material anisotropy. The product measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress, as noted in condensed matter physics. On this basis, a framework of micropolar mechanics of fractal media is developed.

Let (Ω, F, P) be a probability space and let X : Ω → ℝ2 be a vector valued function, i.e., the values are vectors in ℝ2. For each w ∈ Ω, let X1(w) = (X1(w), X2(w)), where X 1 (w)) and X2(w) are the components of X(w) with respect to the canonical basis of ℝ2 consisting of el = (1, 0) and e2 = (0, 1).

9.1 Let µ be a measure on a semiring S whose underlying set is X. Let Y be a µ-measurable subset of X. Let T be a semiring whose underlying set is Y. Assume that µ, Y, and T are endowed with adequate properties. We define the measure µ/ T induced by µ on T and see how to integrate with respect to this induced measure (Theorem 9.1.1).

The following discussion will form the background for an application of certain of the results of statistical information theory. The results in this section for the particular case of discrete spaces are essentially those for contingency tables in Kullback (1959, Chapter 8). For a similar discussion see Dobrushin (1959), Pinsker (1964). We use the notation in Halmos (1950).

It will be convenient to extend the domain of an integral to include certain ℝ* valued functions, and to extend the integral to an ℝ* valued functional on the larger domain. We make this extension and subsequently phrase the Beppo Levi theorem and Fatou’s lemma in this context. A more serious use of the new construct is then made in the study of product integrals and product measures.

A Boolean measure space is a Boolean σ-space X together with a normalized measure on the σ-field of Borel sets in X, such that non-empty open sets have positive measure and nowhere dense Borel sets have measure zero. The last condition is a very strange one. At first glance it might seem that since a nowhere dense set is topologically small and a set of measure zero is measure-theoretically small, it is fitting and proper that the one should imply the other. A little measure-theoretic experience (with Lebesgue measure in Euclidean spaces, for instance) shows, however, that the implication is not at all likely to hold. The results of this section will show that Boolean measure spaces, in which the implication is assumed to hold, have rather pathological and almost paradoxical properties. The reason for considering them anyway is that measure algebras are important, and, as it turns out, Boolean measure spaces are exactly the duals of measure algebras.

The exclusion process differs from the spin systems which are the subject of the previous five chapters in that two (rather than one) coordinates of η1 change at a time. In order to describe this process, let p(x, y) be the transition probabilities for a discrete time Markov chain on the countable set 5: $$ p(x,y) \geq 0,{\text{ }}and{\text{ }}\sum\limits_y {p(x,y) = 1.} $$

For one dimensional data, the notion of a record is natural and, universally agreed upon but this is not the case when the data is in two or more dimensions. We outline several plausible definitions of a record and settle on one induced by a partial ordering: For observations X1,…,Xn we say Xn is a record if each component of Xn is bigger than the corresponding components of previous observations. Some properties of such records of iid observations are reviewed and used to study the shape of the convex hull of the first n observations, as n→∞.

For many families of p-measures occurring in the literature the tangent cones are, in fact, linear spaces. This chapter contains some examples of this kind. Several families do, however, not share all of the additional properties discussed in Chapter 1, such as loc. as. symmetry or loc. as. convexity.

For i ∈ {1,. . .,m} let (xi, Ai) be measurable spaces. In the following, sums Σ and products ×,Π over i always run from 1 to m. Let β be a family of p-measures on ×Ai, and κ: β → IR a functional. Our problem is to estimate κ(P) under various conditions on β.