Digital Commons Network^™

Computer Assistance In Discovering Formulas And Theorems In System Engineering Ii, J. W. Helton, Mark Stankus, Kurt Schneider

Mathematics

[HSWcdc94] focused on procedures for simplifying complicated expressions automatically. [HScdc95] turned to the adventurous pursuit of developing a highly computer assisted method for “discovering” certain types of formulas and theorems.

It is often the case that some variables in the formulation of a problem are not the natural “coordinates” for solution of the problem. Gröbner Basis Algorithms, which lie at the core of our method, are very good at eliminating unknowns, but have no way of finding good changes of variables. This paper gives a way of incorporating changes of variables into our method.

As an example, we “discover” the …

M-Isometric Transformations Of Hilbert Space, Iii, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

A Note On Distributions Of True Versus Fabricated Data, Theodore P. Hill

Research Scholars in Residence

New empirical evidence and statistical derivations of Benford’s Law have led to successful goodness-of fit tests to detect fraud in accounting data. Several recent case studies support the hypothesis that fabricated data does not conform to expected true digital frequencies.

Strongly-Consistent, Distribution-Free Confidence Intervals For Quantiles, David Gilat, Theodore P. Hill

Research Scholars in Residence

Strongly-consistent, distribution-free confidence intervals are derived to estimate the fixed quantiles of an arbitrary unknown distribution, based on order statistics of an iid sequence from that distribution. This new method, unlike classical estimates, works for totally arbitrary (including discontinuous) distributions, and is based on recent one-sided strong laws of large numbers.

Strong Laws For L- And U-Statistics, J. Aaronson, R. Burton, H. Dehling, D. Gilat, Theodore P. Hill, B. Weiss

Research Scholars in Residence

Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.

An Existence Result For Linear Partial Differential Equations With C∞ Coefficients In An Algebra Of Generalized Functions, Todor D. Todorov

Mathematics

We prove the existence of solutions for essentially all linear partial differential equations with C^∞-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy’s equation has solutions whenever its right-hand side is a classical C^∞-function.

The Closed Geodesic Problem For Compact Riemannian 2-Orbifolds, Joseph E. Borzellino, Benjamin G. Lorica

Mathematics

In this paper it is shown that any compact Riemannian 2-orbifold whose underlying space is a (compact) manifold without boundary has at least one closed geodesic.