Applied Mathematics Commons^™

Mathematics At Chartres Cathedral, Richard Stout

ACMS Journal 2004

Chartres Cathedral is highly regarded as a magnificent Gothic structure including beautiful stained glass windows and striking sculptures. Mathematics, especially geometry, played a central role in the design of this cathedral. This paper explores that mathematics and how it contributes to the overall effectiveness of the design.

Mathematics And Values: Can Philosophy Guide Projects?, Michael H. Veatch

ACMS Journal 2004

The philosophy of mathematics has provided insight on questions of foundations and mathematical truth; however, it has not been very fruitful in guiding the practice of mathematics. This paper attempts to find points of contact between a Christian worldview and the choice of mathematical projects and methods. Three areas are considered: (i) dubitability in current research, (ii) the intrinsic value of contemporary mathematics to contemporary society, and (iii) the affirmation of human value in the use of mathematics. Finally, a framework for valuing mathematics is proposed as an encouragement to think more deeply about how a Christian might choose a …

Travelling Fronts In Cylinders And Their Stability, Jerrold W. Bebernes, Comgming Li, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

The Nordstrom–Robinson Code Is Algebraic-Geometric, Judy L. Walker

Department of Mathematics: Faculty Publications

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite field since the early 1980’s. Recently, there has been an increased interest in the study of linear codes over finite rings. In a previous paper [10], we combined these two approaches to coding theory by introducing and studying algebraic-geometric codes over rings. In this correspondence, we show that the Nordstrom–Robinson code is the image under the Gray mapping of an algebraic geometric code over Z = 4Z.

Hoffman’S Error Bounds And Uniform Lipschitz Continuity Of Best L(P) -Approximations, H. Berens, M. Finzel, W. Li, Y. Xu

Mathematics & Statistics Faculty Publications

In a central paper on smoothness of best approximation in 1968 R. Holmes and B. Kripke proved among others that on ℝⁿ, endowed with the l_ρ-norm, 1< p < ∞, the metric projection onto a given linear subspace is Lipschitz continuous where the Lipschitz constant depended on the parameter p. Using Hoffman’s Error Bounds as a principal tool we prove uniform Lipschitz continuity of best l_ρ -ap- proximations. As a consequence, we reprove and prove, respectively, Lipschitz. continuity of the strict best approximation (sba, p = ∞ and of the natural best approximation (nba, p = 1.

Superconvergence Of The Iterated Collocation Methods For Hammerstein Equations, Hideaki Kaneko, Richard D. Noren, Peter A. Padilla

Mathematics & Statistics Faculty Publications

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].

A Dual Approach To Constrained Interpolation From A Convex Subset Of Hilbert Space, Frank Deutsch, Wu Li, Joseph D. Ward

Mathematics & Statistics Faculty Publications

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyx∈Xfrom the setK≔C∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb∈Y. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y …

Positive Solution Curves Of Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the positive solutions to the semilinear equation:

-Δu(x) = λf(u(x)) for x ∈ Ω

u(x) = 0 for x ∈ ∂Ω

where Ω denotes a smooth bounded region in R^N (N > 1) and λ > 0. Here f :[0, ∞)→R is assumed to be monotonically increasing, concave and such that f(0) < 0 (semipositone). Assuming that f'(∞) ≡ lim t→∞ f'(t) > 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)'. When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how …

On The Level Crossing Of Multi-Dimensional Delayed Renewal Processes, Jewgeni H. Dshalalow

Mathematics and System Engineering Faculty Publications

The paper studies the behavior of an (l+3)th-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the crossing level and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier …

The Effect Of Three-Dimensional Freestream Disturbances On The Supersonic Flow Past A Wedge, Peter W. Duck, D. Glenn Lasseigne, M. Y. Hussaini

Mathematics & Statistics Faculty Publications

The interaction between a shock wave (attached to a wedge) and small amplitude, three-dimensional disturbances of a uniform, supersonic, freestream flow are investigated. The paper extends the two-dimensional study of Duck et al. [P W. Duck, D. G. Lasseigne, and M. Y. Hussaini, ''On the interaction between the shock wave attached to a wedge and freestream disturbances,'' Theor. Comput. Fluid Dyn. 7, 119 (1995) (also ICASE Report No. 93-61)] through the use of vector potentials, which render the problem tractable by the same techniques as in the two-dimensional case, in particular by expansion of the solution by means of …

The Hadamard Matroid And An Anomaly In Its Single Element Extensions, C. H. Cooke

Mathematics & Statistics Faculty Publications

A nonstandard vector space is formulated, whose bases afford a representation of what is called a Hadamard matroid, M_p. For prime p, existence of M_p is equivalent to the existence of both a classical Hadamard matrix H(p,p) and a certain affine resolvable, balanced incomplete block design AR(p). An anomaly in the representable single element extension of a Hadamard matroid is discussed.

Collected Papers, Vol. 2, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

An Invariance Property Of Common Statistical Tests, N. Rao Chaganty, A. K. Vaish

Mathematics & Statistics Faculty Publications

Let A be a symmetric matrix and B be a nonnegative definite (nnd) matrix. We obtain a characterization of the class of nnd solutions Σ for the matrix equation AΣA = B. We then use the characterization to obtain all possible covariance structures under which the distributions of many common test statistics remain invariant, that is, the distributions remain the same except for a scale factor. Applications include a complete characterization of covariance structures such that the chisquaredness and independence of quadratic forms in ANOVA problems is preserved. The basic matrix theoretic theorem itself is useful in other characterizing …

The Duals Of Warfield Groups, Peter Loth

Mathematics Faculty Publications

A Warfield group is a direct summand of a simply presented abelian group. In this paper, we describe the Pontrjagin dual groups of Warfield groups, both for the p-local and the general case. A variety of characterizations of these dual groups is obtained. In addition, numerical invariants are given that distinguish between two such groups which are not topologically isomorphic.

Eigenfunction And Harmonic Function Estimates In Domains With Horns And Cusps, Michael Cranston, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

Monotone Iterations For Differential Equations With A Parameter, Tadeusz Jankowski, V. Lakshmikantham

Mathematics and System Engineering Faculty Publications

Consider the problem {y′(t)=f(t,y(t),λ),t∈J=[0,b],y(0)=k0,G(y,λ)=0. Employing the method of upper and lower solutions and the monotone iterative technique, existence of extremal solutions for the above equation are proved.