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Full-Text Articles in Mathematics
Differential Inequalities And Flow-Invariance Via Linear Functionals, V. Lakshmikantham, Jerome Eisenfeld
Differential Inequalities And Flow-Invariance Via Linear Functionals, V. Lakshmikantham, Jerome Eisenfeld
Mathematics Technical Papers - Archive
This paper deals with flow-invariance properties with respect to [see pdf for notation] and resulting comparison inequalities. Both the order relation He and the regularity conditions on x are defined, in a general manner, in terms of a specified set of linear functionals, S. This permits greater flexibility for applications e.g., a proper choice of S gives the PDE operators of the form [see pdf for notation] where [see pdf for notation], an appropriate quasimonotonicity property.
System Identification Problems And The Method Of Moments, Jerome Eisenfeld, S. W. Cheng, Stephen R. Bernfeld
System Identification Problems And The Method Of Moments, Jerome Eisenfeld, S. W. Cheng, Stephen R. Bernfeld
Mathematics Technical Papers - Archive
Let X(t) and W(t) be vectors of dimension N > 0. We are concerned with the problem of computing an N x N matrix A such that [see pdf for notation](1.1) where X'(t) is the rate of change of X(t) with respect to time t. Such problems frequently arise in the biosciences although it is not always immediately evident that they may be posed in the form of Eq. (1.1). In applications the data for X(t) and W(t) is obtained from experiments and the collection of such data is not always performed over equal time intervals nor is it always the …
Existence And Uniqueness Of Solutions Of Delay Differential Equations On A Closed Subset Of A Banach Space, V. Moauro, S. Leela, V. Lakshmikantham
Existence And Uniqueness Of Solutions Of Delay Differential Equations On A Closed Subset Of A Banach Space, V. Moauro, S. Leela, V. Lakshmikantham
Mathematics Technical Papers - Archive
In an earlier work [5], sufficient conditions for the existence of solutions in a closed subset F of a Banach space E for the Cauchy problem (1.1) [see pdf for notation] where [see pdf for notation], are obtained by requiring f to satisfy (i) a compactness-type condition in terms of the Kuratowski measure of noncompactness and (ii) a boundary condition, namely, (1.2) [see pdf for notation] for every [see pdf for notation]. In this paper, we wish to explore the other direction followed in the study of the Cauchy problem for differential equations in a Banach space [4,6,7], that is, …
Existence Of Solutions Of Boundary Value Problems For Nonlinear Second Order Systems In A Banach Space, A. Richard Mitchell, V. Chandra
Existence Of Solutions Of Boundary Value Problems For Nonlinear Second Order Systems In A Banach Space, A. Richard Mitchell, V. Chandra
Mathematics Technical Papers - Archive
This paper is concerned with the existence of solutions of boundary value problems (BVP, for short) for nonlinear second order ordinary differential equations of the type (1.1) [see pdf for notation] (1.2) [see pdf for notation] where [see pdf for notation] is a real Banach space. In case [see pdf for notation], existence was proved by first obtaining a priori bounds for [see pdf for notation] of a solution of (1.1) and (1.2) and then employing a theorem of Scorza-Dragoni [3,7,16]. The methods involve assuming inequalities in terms of the second derivative of Lyapunov-like functions relative to H, using comparison …
On The Existence Of Weak Solutions Of Differential Equations In Nonreflexive Banach Spaces, V. Lakshmikantham, Evin Bronson, A. Richard Mitchell
On The Existence Of Weak Solutions Of Differential Equations In Nonreflexive Banach Spaces, V. Lakshmikantham, Evin Bronson, A. Richard Mitchell
Mathematics Technical Papers - Archive
The study of the Cauchy problem for differential equations in a Banach space relative to the strong topology has attracted much attention in recent years [2,4,5,7]. This study has taken two different directions. One direction is to impose compactness type conditions that guarantee only existence and the corresponding results are extensions of the classical Peano's Theorem. The other approach is to utilize dissipative type conditions that assure existence and uniqueness of solutions, and the corresponding results are extensions of the classical Picard's Theorem. However, a similar study of the Cauchy problem in a Banach space relative to the weak topology …
Existence Of Solutions In A Closed Set For Delay Differential Equations In Banach Spaces, V. Moauro, S. Leela
Existence Of Solutions In A Closed Set For Delay Differential Equations In Banach Spaces, V. Moauro, S. Leela
Mathematics Technical Papers - Archive
The study of the Cauchy problem for ordinary differential equations in a Banach space has been extensive [1,3-7,9-12]. The two main directions that are followed in such a study are (i) finding monotonicity type conditions which guarantee the existence as well as uniqueness of solutions and (ii) finding compactness type conditions which assure only the existence of solutions [3,4]. It is also known [10,12] that in order to prove the existence of solutions in a closed subset F of the Banach space, a boundary condition of the type [see pdf for notation] is required.
Stability Of Differential Systems With Impulsive Perturbations In Terms Of Two Measures, S. Leela
Stability Of Differential Systems With Impulsive Perturbations In Terms Of Two Measures, S. Leela
Mathematics Technical Papers - Archive
The study of differential systems of the form (1.1) [see pdf for notation] where [see pdf for notation] denotes the distributional derivative of [see pdf for notation], a function of bounded variation (that is, differential systems with impulsive perturbations, also called measure differential equations), is both interesting and important because most models for biological neural nets,pulse frequency modulation systems, automatic control problems with impulsive inputs and many physical processes are best described by such equations [1-3,8,10,12,13]. Since the solutions of (1.1) are discontinuous (that is, functions of bounded variation), the investigation of the stability properties of (1.1) by the usual …
Mathematical Modeling In Medicine, Jerome Eisenfeld
Mathematical Modeling In Medicine, Jerome Eisenfeld
Mathematics Technical Papers - Archive
Although the involvement of mathematics in medicine is still relatively recent, the discipline has become attractive to the mathematics community, and in fact, medically oriented articles presently appear in several mathematics journals. We call the reader's attention to the following journals: Bulletin of Mathematical Biology (continues The Bulletin of Mathematical Biophysics), Biomathematics (also called Revue de Bio-Mathematique), SIAM Journal in Applied Mathematics (a new series devoted to biomathematics will appear shortly), Math Biosciences, Biomedical Engineering, IEEE Transactions in Biomedical Engineering, Journal of Biomechanics, Biological Cybernetics, Computers in Biology and Medicine, Journal of Theoretical Biology, and Biometrics. Each year there are …
Cone-Valued Lyapunov Functions, S. Leela, V. Lakshmikantham
Cone-Valued Lyapunov Functions, S. Leela, V. Lakshmikantham
Mathematics Technical Papers - Archive
It is very well known that employing a single Lyapunov function and the theory of scalar differential inequality offers a useful mechanism to study a variety of qualitative problems of differential equations in a unified way [10]. Nevertheless, when using this powerful technique for concrete problems, the main difficulty we face is the lack of general method of constructing a Lyapunov function. This naturally beads to the development of the method of vector Lyapunov functions which utilizes several Lyapunov-like functions and the theory of vector differential inequalities in a fruitful manner [5,8-12]. This method offers a more flexible mechanism to …
A Monotone Method For Infinite System Of Nonlinear Boundary Value Problems, V. Lakshmikantham, Jagdish Chandra, S. Leela
A Monotone Method For Infinite System Of Nonlinear Boundary Value Problems, V. Lakshmikantham, Jagdish Chandra, S. Leela
Mathematics Technical Papers - Archive
Monotone iterative methods have been successfully used to generate improvable two-sided point-wise bounds on solutions of nonlinear boundary value problems for both ordinary and partial differential equations. While such procedures take a simple form when the nonlinearities are independent of gradient terms [6,9], the extension of such techniques to fully nonlinear problems has been quite formidable. In the case of scalar ordinary differential equations of the type (1.1) [see PDF for equation] such results have been obtained making use of either a linear maximum principle (3,1] or a nonlinear maximum principle [4]. In either case an essential use is made …
Minimal And Maximal Solutions Of Nonlinear Boundary Value Problems, Jagdish Chandra, Stephen R. Bernfeld
Minimal And Maximal Solutions Of Nonlinear Boundary Value Problems, Jagdish Chandra, Stephen R. Bernfeld
Mathematics Technical Papers - Archive
This paper is concerned with the construction of the minimal and the maximal solutions of the nonlinear boundary value problem [see PDF for equation] under tether mild assumptions of f. in particular, no assumption of monotonicity is made on [see PDF for equation] either in u or [see PDF for equation].
Generalized Stability Of Motion And Vector Lyapunov Functions, Deborah A. Pace, Roger W. Mitchell
Generalized Stability Of Motion And Vector Lyapunov Functions, Deborah A. Pace, Roger W. Mitchell
Mathematics Technical Papers - Archive
The direct theory for stability of motion in terms of vector Lyapunov functions and the general comparison method is well-developed [4, 5, 6, 7] and effectively applied for large scale dynamical systems [1, 3]. However, the problem of constructing vector Lyapunov functions and developing appropriate perturbation theory has seen very little progress. Perhaps one of the reasons for this situation is that the stability definitions by means of the standard norm are not flexible enough for such a venture.
Optimum Structural Probability Bounds For The Right-Tail Of The Paretian Law, Danny D. Dyer
Optimum Structural Probability Bounds For The Right-Tail Of The Paretian Law, Danny D. Dyer
Mathematics Technical Papers - Archive
The two-parameter Pareto density function [see PDF for equation] is widely used in applied statistics. For example, in reliability theory (1.1) represents the failure-time distribution of a "work-hardened" component whose.hazard function is inversely proportional to time. Because of its long right-tail, (1.1) is often a probability model for certain socioeconomic phenomena (e.g., distribution of city population sizes, distribution of personal incomes which exceed tax-exempt levels). One of the more recent characterizations of the Pareto distribution is due to Revankar, et. at. [15]. They show that within the population of individuals that under-report their income for income tax purposes, the average …
A Nonlinear Variation Of Constants Method For Integro-Differential And Integral Equations, M. E. Lord, Stephen R. Bernfeld
A Nonlinear Variation Of Constants Method For Integro-Differential And Integral Equations, M. E. Lord, Stephen R. Bernfeld
Mathematics Technical Papers - Archive
It is well known that a very important technique in obtaining the asymptotic behavior of solutions of linear and nonlinear ordinary differential equations under perturbations is through the use of the variation of constants formula (see [1] and [14]). Miller [16] used a well known representation formula for perturbed linear Volterra integral equations in terms of the resolvent which has been successfully used in recent years to analyze stability behavior of solutions (see, for example, [12] and [17] and references therein). In addition, Bownds and Cushing [4] obtained another variation of constants formula for linear integral equations utilizing a fundamental …
On The Construction Of A Norm Associated With The Measure Of Noncompactness, Stephen R. Bernfeld, V. Lakshmikantham, Jerome Eisenfeld
On The Construction Of A Norm Associated With The Measure Of Noncompactness, Stephen R. Bernfeld, V. Lakshmikantham, Jerome Eisenfeld
Mathematics Technical Papers - Archive
It is shown that the bounded, nonempty subsets of a reflexive Banach space g can be imbedded in another Banach space B(E) in such a manner so that the measure of noncompactness corresponds to the norm in B(E). The results are applied to ordinary differential equations theory.
The Extendability And Uniqueness Of Solutions Of Ordinary Differential Equations, Stephen R. Bernfeld
The Extendability And Uniqueness Of Solutions Of Ordinary Differential Equations, Stephen R. Bernfeld
Mathematics Technical Papers - Archive
In a recent paper [1] the author obtained results on the extendability of solutions of perturbed differential equations. The question of the extendability of solutions of differential equations is a fundamental and important property since questions of stability and boundedness require extendability. In this paper we continue our study of extendability of perturbed scalar differential equations. Our somewhat surprising results also extend to the question of uniqueness of the zero solution of perturbed equations satisfying an Osgood condition [4] (See also [2] for recent results on the uniqueness of perturbed systems.) Examples are provided to demonstrate the strength of our …
Structural Inference On Reliability In A Lognormal Model, Danny D. Dyer
Structural Inference On Reliability In A Lognormal Model, Danny D. Dyer
Mathematics Technical Papers - Archive
The theory of structural inference, as developed by Fraser (1968), is based on a group-theoretic approach using invariant Haar measures to Fisher's fiducial theory. Structural inference theory constructs a unique distribution, conditional on the given sample information only, for the parameters of a measurement model. Based on the structural density for the two-parameter lognormal distribution, the structural density and distribution function for the reliability function are derived. Consequently, expressions for structural point and interval estimates of the reliability function are developed. Approximations for large sample sizes and/or moderately reliable components are also discussed. An example based on lognormal data is …
On The Zeros Of Monotone Operators Of Retarded Type In A Banach Space, Patrick Sutherland, V. Lakshmikantham
On The Zeros Of Monotone Operators Of Retarded Type In A Banach Space, Patrick Sutherland, V. Lakshmikantham
Mathematics Technical Papers - Archive
Recent interest in the Cauchy problem for differential equations in a Banach space [9] has stimulated a similar interest for differential equations of a retarded type in a Banach space. The difficulty in imposing assumptions due to a different range and domain space has been overcome in [4] where existence of solutions is established using a monotoni- city type condition in terms of norm and weaker forms of differential inequalities. The theory of existence of solutions of differential equations has been used in [1,2,5,6,7] to obtain existence of zeros and fixed points for nonlinear operators from E into E. In …
A Survey Of Techniques For Determining Lower Confidence Bounds On Series System Reliability Based On Subsystem Test Data, Danny D. Dyer
A Survey Of Techniques For Determining Lower Confidence Bounds On Series System Reliability Based On Subsystem Test Data, Danny D. Dyer
Mathematics Technical Papers - Archive
A problem of considerable interest and for which a great deal of research has been expended is that of determining lower confidence bounds on series system reliability based on subsystem failure data. With regard to the types of failure data taken on the subsystems, primary consideration is given to binomial (pass-fail) data, exponentially distributed time-to-fail data for both Type I censoring (fixed test times) and Type II censoring (fixed number of failures), or any mixture thereof. In this survey we shall examine optimum solutions whenever they exist, approximate optimum solutions, and nonoptimum solutions. Numerical examples are given to illustrate the …
On The Convergence Of Successive Approximations For Quasi-Nonexpansive Mappings Through Abstract Cones, B. B. Williams, J. C. Bolen
On The Convergence Of Successive Approximations For Quasi-Nonexpansive Mappings Through Abstract Cones, B. B. Williams, J. C. Bolen
Mathematics Technical Papers - Archive
In a recent paper, Petryshyn and Williamson [4] investigated the convergence of successive approximations of quasi- nonexpansive mappings in a Banach space. This paper contains an outline, in chronological order, of the main results concerning the convergence of iteration method and consequently includes a number of references. Perov and Kibenko [3] employed generalized Banach spaces to extend contraction mapping principal and to show the flexibility of such an approach in applications. See also Bernfeld and Lakshmikantham [1]. More recently, Eisenfeld and Lakshmikantham [5,6] proved some fixed point theorems in abstract cones which extend and generalize many known results. In this …
Remarks On Nonlinear Contraction And Comparison Principle In Abstract Cones, V. Lakshmikantham, Jerome Eisenfeld
Remarks On Nonlinear Contraction And Comparison Principle In Abstract Cones, V. Lakshmikantham, Jerome Eisenfeld
Mathematics Technical Papers - Archive
The contraction mapping principle and the Schauder principle can both be viewed as a comparison of maps. For the former one has a condition of the type [see pdf for notation] and for the latter one has a condition of the type [see pdf for notation] where p is the metric and y is the Kuratowski measure of noncompactness. If p is a linear map [see pdf for notation] from the nonnegative reals [see pdf for notation] into itself then the map T satisfying (1.1) is said to be k-contractive and the map satisfying (1.2) is said to be k-set …
Maximal And Minimal Solutions And Comparison Principle For Differential Equations In Abstract Cones, Roger W. Mitchell, A. Richard Mitchell, V. Lakshmikantham
Maximal And Minimal Solutions And Comparison Principle For Differential Equations In Abstract Cones, Roger W. Mitchell, A. Richard Mitchell, V. Lakshmikantham
Mathematics Technical Papers - Archive
Existence of maximal and minimal solutions for differential equations in abstract cones is established without requiring uniform continuity. Utilizing such a result an abstract comparison principle is developed. The results of the paper significantly improve earlier results of the authors and also simplify the proofs of other known results.
On A Measure Of Nonconvexity And Applications, V. Lakshmikantham, Jerome Eisenfeld
On A Measure Of Nonconvexity And Applications, V. Lakshmikantham, Jerome Eisenfeld
Mathematics Technical Papers - Archive
The measure of noncompactness which was introduced by Kuratowski [8] (in 1930) has now become an important tool in nonlinear analysis (although its value in that regard was not appreciated until much later). Following Kuratowski we introduce a measure of nonconvexity which has many properties in common with the measure of noncompactness and therefore we may now have "convex" where previously we had "compact" in the statements of some theorems.
Statistical Analysis Of Fourier Coefficients In A Restricted Harmonic Dial, Danny D. Dyer
Statistical Analysis Of Fourier Coefficients In A Restricted Harmonic Dial, Danny D. Dyer
Mathematics Technical Papers - Archive
Due to the physical nature of certain periodic data, the harmonic dial points (the Fourier coefficients obtained from harmonic analysis of the data) are sometimes restricted to circular regions in the dialplane. It is proposed that a circular normal distribution (CND) truncated outside a circular region be used to describe the probabilistic behavior of the random phenomena. Recurrence relations for the population moments of a CND truncated outside a circular region are derived. These recurrence relations are used to obtain consistent asymptotically (jointly) normal estimators of the unknown parameters of the distribution. A numerical example based on the harmonic dial …
Vector Lyapunov Functions And Perturbations Of Nonlinear Systems, Marion E. Moore, Roger W. Mitchell
Vector Lyapunov Functions And Perturbations Of Nonlinear Systems, Marion E. Moore, Roger W. Mitchell
Mathematics Technical Papers - Archive
Two recent papers [2,3] have combined the techniques of the Lyapunov method and the nonlinear variation of parameters to study the effects of perturbations of nonlinear differential systems. In this paper, we generalize the results obtained in [2,3] by the use of several Lyapunov functions and the concept of a generalized norm. To convince ourselves that we do have more flexibility in working with the generalized norm and vector Lyapunov functions see [1,4].
Block Diagonalization And Eigenvalues, Jerome Eisenfeld
Block Diagonalization And Eigenvalues, Jerome Eisenfeld
Mathematics Technical Papers - Archive
Let A denote an Algebra with an identity element. Consider an [see pdf for notation] matrix [see pdf for notation] with a partitioning [see pdf for notation] where E and H have respective orders [see pdf for notation] and [see pdf for notation]. We seek to obtain conditions under which A is similar to a matrix D of the form [see pdf for notation] where [see pdf for notation] denotes the zero [see pdf for notation] matrix over A. Some advantage is gained in working in a general algebra. The algebra A may be taken as a Banach algebra of …
Unitary Perfect Polynomials Over Gf(Q), Jacob T. B. Beard
Unitary Perfect Polynomials Over Gf(Q), Jacob T. B. Beard
Mathematics Technical Papers - Archive
For monic polynomials A(x), B(x) e GF[q,x], call B(x) a unitary divisor of A(x) provided (B(x),A(x)/B(x)) = 1 . The polynomial A(x) is called unitary perfect over GF(q) if and only if the sum [see pdf for notation] of the distinct unitary divisors of A(x) equals A(x). Principal results characterize all unitary perfect polynomials over GF(p) which split in GF[p,x].
On The Existence Of Zeros Of Lyapunov-Monotone Operators, S. Leela, V. Lakshmikantham
On The Existence Of Zeros Of Lyapunov-Monotone Operators, S. Leela, V. Lakshmikantham
Mathematics Technical Papers - Archive
Consider a nonlinear operator T from a Banach space into itself. The study of the existence of zeros of T plays an important role in yielding fixed points of nonlinear operators. The operator T has a zero if and only if the initial value problem [see pdf for notation],has a constant solution. If T is a monotone operator then (1.1) has a unique solution [see pdf for notation] defined on [see pdf for notation] and the solution operator [see pdf for notation] is nonexpansive for all [see pdf for notation]. Imposing further assumptions one can show that U(t) must have …
On The Existence Of Solutions Of Differential Equations And Zeros Of Operators In K-Banach Spaces, Evin Bronson, R. L. Tennison, A. Richard Mitchell
On The Existence Of Solutions Of Differential Equations And Zeros Of Operators In K-Banach Spaces, Evin Bronson, R. L. Tennison, A. Richard Mitchell
Mathematics Technical Papers - Archive
The theory of existence of solutions of differential equations in a Banach space employing norm as a measure is sufficiently well known [5, 6, 8, 9]. Also utilizing this theory one can prove the existence of zeros of operators [2, 7, 8, 9, 11]. The advantage of using a generalized norm as a candidate in discussing the qualitative theory of differential equations is also known [1]. These thoughts naturally lead to the use of cone-valued norms as a measure since this approach unifies the existing theories as well as offers a more flexible mechanism for applications. In this paper, we …
Fixed Point Theorms Of Operators With Ppf Dependence In Banach Spaces, Stephen R. Bernfeld
Fixed Point Theorms Of Operators With Ppf Dependence In Banach Spaces, Stephen R. Bernfeld
Mathematics Technical Papers - Archive
In this paper we develop a theory of fixed points of a nonlinear operator, T, whose domain is the Banach space of continuous functions defined on an interval [a,b] with range in a Banach space E denoted by [see pdf for notation] and the range of the nonlinear operator T is in E. As we shall see delay differential equations form an important example of such a nonlinear operator. We shall obtain analogues of the contraction mapping principle, Krasnoselskii's fixed point theorem as well as a result on the convergence of iterations of quasi-nonexpansive mappings.