Mathematics Commons^™

A Theory Of Testability With Application To Fault Coverage Analysis, Sharad Seth, Vishwani Agrawal, Hassan Farhat

Mathematics Faculty Publications

When test vectors are applied to a circuit, the fault coverage increases. The rate of increase, however, could be circuit-dependent. In fact, the actual rise of fault coverage depends on the characteristics of vectors, as well as, on the circuit. The paper shows that the average fault coverage can be computed from circuit testability. A relationship between fault coverage and circuit testability is derived. The mathematical formulation allows computation of coverage for deterministic and random vectors. Applications of this analysis include: determination of circuit testability from fault simulation, coverage prediction from testability analysis, prediction of test length, and test generation …

Pixley-Roy Hyperspaces Of Ω-Graphs, Joe Mashburn

Mathematics Faculty Publications

The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two ω-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of Rⁿ are also shown to be homeomorphic.

Optimal Intervals For Third Order Lipschitz Equations, Paul W. Eloe, Johnny Henderson

Mathematics Faculty Publications

For the third order differential equation (see PDF), subintervals of (a,b) of maximal length are characterized, in terms of the Lipschitz coefficients (see PDF) on which certain boundary value problems possess unique solutions. The techniques for determining best interval length involve applications of the Pontryagin Maximum Principle along with uniqueness implies existence arguments.

Test Generation By Fault Sampling, Vishwani Agrawal, Hassan Farhat, Sharad C. Seth

Mathematics Faculty Publications

This paper presents a novel technique of generating tests from a random sample of faults. The entire fault population of the circuit is randomly divided into two groups. Only one group, usually the smaller one, is used for test generation by the test-generator and fault-simulator programs. This group is known as the sample and its coverage is deterministic. The coverage of faults in the remaining group is similar to that of random vectors and is estimated from the distribution of fault detection probabilities in the circuit. As the sample size increases, the fraction of unsampled faults reduces. At the same …

Periodic Solutions Of Volterra Integral Equations, Muhammad Islam

Mathematics Faculty Publications

Consider the system of equations

x(t)=f(t)+∫−∞tk(t,s)x(s)ds,           (1)

and

x(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds.        (2)

Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are obtained using the …

The Persistence Of Universal Formulae In Free Algebras, Anthony M. Gaglione, Dennis Spellman

Mathematics Faculty Publications

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements ...

Unitary Weighted Composition Operators, Valentin Matache

Mathematics Faculty Publications

A necessary and sufficient condition for a weighted composition operator to be unitary is given in terms of the weight-function and of the composition function.

Commutator Identities Obtained By The Magnus Algebra, Anthony M. Gaglione, Dennis Spellman

Mathematics Faculty Publications

In this paper, two related commutator identities are established through the use of the Magnus Algebra (the algebra of noncommutative formal power series with integral coefficients).

A Note On Irreducibility And Weak Covering Properties, Joe Mashburn

Mathematics Faculty Publications

A space X is irreducible if every open cover of X has a minimal open refinement. Interest in irreducibility began when Arens and Dugendji used this property to show that metacompact countably compact spaces are compact. It was natural, then, to find out what other types of spaces would be irreducible and therefore compact in the presence of countable compactness or Lindelof in the presence of N₁-compactness. …

It is shown in this paper that T₁ δθ -refinable spaces and T1 weakly δθ-refinable spaces are irreducible. Since examples of Lindelof spaces that are neither T₁ nor …

Homomorphisms For Equidistance Relations, Thomas Q. Sibley

Mathematics Faculty Publications

This paper presents necessary and sufficient conditions for the existence of homomorphisms for equidistance relations in terms of the closed subsystems (the Fundamental Theorem of Homomorphisms). Further, it shows that every closed subsystem of a 1-point homogenous equidistance system is a coset of a unique homomorphism. Affine spaces and other incidence geometries can be seen as examples of equidistance systems.

The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn

Mathematics Faculty Publications

A partially ordered set P is ω-chain complete if every countable chain (including the empty set) in P has a supremum. … Notice that an ω-chain continuous function must preserve order. P has the (least) fixed point property for ω-chain continuous functions if every ω-chain continuous function from P to itself has (least) fixed point.

It has been shown that a partially ordered set does not have to be ω-chain complete to have the least fixed point property for ω-chain continuous functions. This answers a question posed by G. Plotkin in 1978. I.I. Kolodner has shown that an ω-chain complete …

Equidistance Relations: A New Bridge Between Geometric And Algebraic Structures, Thomas Q. Sibley

Mathematics Faculty Publications

This paper investigates the transformations of certain geometric structures into algebraic ones and conversely. The algebraic notion of absolute value corresponds with the geometric one of equidistance. Further, latin squares with absolute values correspond to regular equidistance relations, "near groups" yield 1 point homogenous equidistance relations, and "near commutative groups" yield 2-point homogenous equidistance relations.

Ultrafilter Limits And Finitely Additive Probability, Thomas Q. Sibley

Mathematics Faculty Publications

Ultrafilter limits provide the natural convergence notion for finitely additive probability. The finitely additive infinitely divisible laws are closed under ultrafilter limits. The characteristic function of any convolution of finitely additive probability measures is the product of their characteristic functions.

Conjugate Type Boundary Value Problems For Functional-Differential Equations, Paul W. Eloe, Louis J. Grimm

Mathematics Faculty Publications

Two-point boundary value problems (BVPs) for delay differential equations have been studied extensively, beginning with the work of G. A. Kamenskiï, S. B. Norkin and others which was motivated by variational problems and problems in oscillation theory. L. J. Grimm and K. Schmitt and Ju. I. Kovac and L. I. Savcenko employed solutions of various differential inequalities for the study of two-point problems with retarded argument. In this paper, we show how a bilateral iteration procedure can be developed to yield existence and inclusion theorems for multipoint boundary value problems of conjugate type for nonlinear functional-differential equations.

Dissertation: The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn

Mathematics Faculty Publications

The basic definitions are given in the first section, including those for ω-chain continuity, ω-chain completeness, and the least fixed point property for ω-chain continuous functions. Some of the relations between completeness and fixed point properties in partially ordered sets are stated and it is briefly shown how the question basic to the dissertation arises.

In the second section, two examples are given showing that a partially ordered set need not be ω-chain complete to have the least fixed point property for ω-chain continuous functions.

Retracts are discussed in section 3, where it is seen that they are not sufficient …

Three Counterexamples Concerning Ω-Chain Continuous Functions And Fixed-Point Properties, Joe Mashburn

Mathematics Faculty Publications

A partially ordered set is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.

On The Derivative Of Bounded Functions, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.

On The Univalence Of A Certain Integral, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.

Extremal Problems For Functions Starlike In The Exterior Of The Unit Circle, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.

Some Applications Of The Weierstrass Mean Value Theorem, V. F. Cowling, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.

Coefficient Problems For Functions Regular In An Ellipse, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.

Convexity And Starlikeness Of Analytic Functions, Wimberly C. Royster

Mathematics Faculty Publications

No abstract provided.