Jack B. Brown

Order independence and factor convergence in iterative scaling

Abstract We prove two results concerning iterative scaling which have been claimed, but not proved, by other authors in the field. Iterative scaling is a procedure which begins with a probability measure Q on a finite set X and a finite set of partition constraints on X. It produces a sequence of iterates P0 = Q, P1,…,Pt,… in which each Pt (t #62; 0) is an adjustment (scaling) of Pt−1 to fit a single set of prescribed partition constraints. Under fairly general hypotheses, the iterates Pt converge to the unique probability P∗ which simultaneously satisfies the given constraints and is exponentially equivalent to the starting measure Q (or equivalently, which minimizes the I-divergence D(P‖Q) over all probabilities P on X which satisfy the constraints). The first main result (Theorem 3.1) states that the convergence of iterative scaling to the unique limit is independent of the order in which the adjustments for the prescribed partition constraints are made, provided each adjustment is made infinitely many times. The second main result (Theorem 4.2) is that, under certain commonly satisfied conditions, each factor of the exponential form being built up within the algorithm (one factor for each partition constraint) converges separately. A third group of results (Sections 5,6) provides an account of the relationship between the probability measures which satisfy all of the constraints and those which are exponentially equivalent to Q.

Measurable Darboux functions

We investigate how certain Darboux-like properties of real functions (including connectivity, almost continuity, and peripheral continuity) are related to each other within certain measurability classes (including the classes of Lebesgue measureable, Borel, and Baire-1 functions).

Differentiable restrictions of real functions

Some new theorems about differentiable, continuously differentiable, or highly differentiable restrictions of continuous and measurable real functions are presented.

Negligible sets for real connectivity functions

Introduction. Only functions from the intervalfI= [0, 1] into I will be considered in this paper, and no distinction will be made between a function and its graph. For real functions f from I into I, the property of connectivity is intermediate to that of continuity and the Darboux property, and is equivalent to the property of f being a connected set in the plane. For some connectivity functions g from I into I, it can be observed that there is a subset M of I such that every function from I into I wlhich agrees with g on I-M is a connectivity function (in this case, M will be said to be g-negligible). For example, if M is a Cantor subset of I and g is a function from I into I such that if x belongs to some component (a, b) of I-M, g(x) = I sin {cot [r(x-a)/(b-a] } f, then M is g-negligible. The following theorems will be proved:

Smooth Interpolation, Holder Continuity, and the Takagi-van der Waerden Function

If r is odd, set nk = 1J pi 2k+1. Then nk 3(4) (since 32 = 1(4)). But no pi divides nk for 1 Pr. If j : k, say j > k, the assumption that qk also divides nj leads to the same contradiction as earlier: since nk nj = 2j+1 2k+1 = 2k+1(2j-k 1), we have qk I 2j-k 1 and hence qk < Pr. Thus, there are at least [log2(4r)j + 1 distinct primes of the form 4? + 3 in (Pr, fj Pi). If r is even, the same argument applied to r 1 shows that there

A MEASURE THEORETIC VARIANT OF BLUMBERG'S THEOREM

It is the purpose of this note to present a measure theoretic variant of Blumbergs theorem about continuous restrictions of arbitrary real functions.

Intersections of continuous, Lipschitz, Hölder class, and smooth functions

We present some improvements of known theorems and examples concerning intersections of continuous or Lipschitz functions with smooth functions or intersections of smooth functions or Holder class functions with smoother functions. We are particularly concerned with our ability to force the projection of the intersection to be uncountable within a given set M which is either large in measure or in category (or both).

Geologic waste disposal and a model for the surface movement of radionuclides

In the following, a model for the surface movement of radionuclides is presented. This model, which is referred to as the Pathways Model, was constructed at Sandia Laboratories in a project funded by the Nuclear Regulatory Commission to develop a methodology to assess the risk associated with the geologic disposal of high level radioactive waste. The methodology development involves work in two major areas: a) models for physical processes and b) statistical techniques for the use and assessment of these models. The following presentation of the Pathways Model will involve topics from both areas. Additional information on the entire project can be obtained from its interim report (1). The work introduced in this paper is described in greater detail in various referenced technical reports; these reports should be consulted for more detailed information than can be provided here.