Laplace's rule of succession in information geometry

Laplace's "add-one" rule of succession modifies the observed frequencies in a sequence of heads and tails by adding one to the observed counts. This improves prediction by avoiding zero probabilities and corresponds to a uniform Bayesian prior on the parameter. The canonical Jeffreys prior corresponds to the "add-one-half" rule. We prove that, for exponential families of distributions, such Bayesian predictors can be approximated by taking the average of the maximum likelihood predictor and the \emph{sequential normalized maximum likelihood} predictor from information theory. Thus in this case it is possible to approximate Bayesian predictors without the cost of integrating or sampling in parameter space

Distributary Channel Networks as Moving Boundaries: Causes and Morphodynamic Effects

We propose an exploratory model to describe the morphodynamics of distributary channel network growth on river deltas. The interface between deep channels and the shallow, unchannelized delta front deposits is modeled as a moving boundary. Steady flow over the unchannelized delta front is friction dominated and modeled by Laplace\u27s equation. Shear stress along the network boundary produces nonlinear erosion rates at the interface, causing the boundary to move and network elements (channels and branches) to form. The model was run for boundary conditions resembling the Wax Lake Delta in coastal Louisiana, 20 parameterizations of sediment transport, and 3 parameterizations of discharge. In each case, the model produced a complex channel network with channel number, width, bifurcation angle, and channel shape depending on the sediment transport formula. For reasonable sediment transport parameters and gradually increasing water discharge, the model produced network characteristics and progradation rates similar to the Wax Lake Delta. This suggests that the model contains the processes responsible for network growth, despite its abstract formulation

The hour of traces: an analysis of the presuppositions, axiomatic formalization, and ontological implications of Reichenbach's frequency interpretation of probability

Thesis (M.A.)--Boston Universit

The Theory of Determinants: Some Special Forms and Applications

The theory of determinants is a relatively new branch of mathematics; it was not firmly established until a brief 200 years ago. It was not until the middle of the 18th century that Cramer, one of the independent discoverers of the fundamental idea, was fortunate enough to attract attention to the theory of determinants. Much of the literature is written on the assumption that its reader has a good background in the subject. Then other authors just give the fundamental properties of determinants without proof of their statements. These latter authors are not interested in the mathematical theory of determinants, but are interested only in using them as a tool. In this thesis no attempt is made to cover the entire field of the theory of determinants, but an attempt is made to summarize the material in certain phases of the theory of determinants. This monograph is divided into the following parts: (1) A brief history of the early development s in the theory of determinants. (2) The statement and proof of the general properties of determinants, and the definition of the common terms. (3) A brief treatment of the most common special forms of determinants. (4) A few of the many applications of determinants and an indication of others in various phases of mathematics and other sciences

Fourier Series and Some Applications.

The object of the author in writing this thesis was to make a compilation of material necessary to work certain problems in mathematical physics with one use of Fourier series. Although whole books have been written on different aspects of this thesis, all had some intervening material which was too advanced for the average student. Any students interested in mathematics or physics cannot fail to see the importance of Fourier series. To mention a few applications, the series is used in finding the solution to various problems in heat distribution, electrical transmission, and aircraft construction

Manifest: A computer program for 2-D flow modeling in Stirling machines

A computer program named Manifest is discussed. Manifest is a program one might want to use to model the fluid dynamics in the manifolds commonly found between the heat exchangers and regenerators of Stirling machines; but not just in the manifolds - in the regenerators as well. And in all sorts of other places too, such as: in heaters or coolers, or perhaps even in cylinder spaces. There are probably nonStirling uses for Manifest also. In broad strokes, Manifest will: (1) model oscillating internal compressible laminar fluid flow in a wide range of two-dimensional regions, either filled with porous materials or empty; (2) present a graphics-based user-friendly interface, allowing easy selection and modification of region shape and boundary condition specification; (3) run on a personal computer, or optionally (in the case of its number-crunching module) on a supercomputer; and (4) allow interactive examination of the solution output so the user can view vector plots of flow velocity, contour plots of pressure and temperature at various locations and tabulate energy-related integrals of interest

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks

Numerical simulation of low gravity draining

A boundary value problem was solved numerically for a liquid that is assumed to be inviscid and incompressible, having a motion that is irrotational and axisymmetric, and having a constant (5 degrees) solid-liquid contact angle. The avoidance of excessive mesh distortion, encountered with strictly Lagrangian or Eulerian kinematics, was achieved by introducing an auxiliary kinematic velocity field along the free surface in order to vary the trajectories used in integrating the ordinary differential equations simulating the moving boundary. The computation of the velocity potential was based upon a nonuniform triangular mesh which was automatically revised to varying depths to accommodate the motion of the free surface. These methods permitted calculation of draining induced axisymmetric slosh through the many (or fractional) finite amplitude oscillations that can occur depending upon the balance of draining, gravitational, and surface tension forces. Velocity fields, evolution of the free surface with time, and liquid residual volumes were computed for three and one half decades of Weber number and for two Bond numbers, tank fill levels, and drain radii. Comparisons with experimental data are very satisfactory