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The paper discusses long time behavior and error bounds for discretized Volterra equations. A key property is positivity of the quadrature which is combined with monotonicity properties of the nonlinearities in the equations. It is shown how the positivity of discretization quadrature is linked with \(A\)-stability property of linear multistep methods for ordinary differential equations. Some of the results are new when applied to differential equations with monotone nonlinearities and \(A\)-stable discretizations.

The paper discusses long time behavior and error bounds for discretized Volterra equations. A key property is positivity of the quadrature which is combined with monotonicity properties of the nonlinearities in the equations. It is shown how the positivity of discretization quadrature is linked with \(A\)-stability property of linear multistep methods for ordinary differential equations. Some of the results are new when applied to differential equations with monotone nonlinearities and \(A\)-stable discretizations.

Let G denote the dyadic group, which has as its dual group the Walsh(-Paley) functions. In this paper we formulate a condition for functions in L1(G) which implies that their Walsh-Fourier series converges in L1(G)-norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L1(G) convergence and we prove that the assumption on the L1(G) modulus of continuity in this theorem cannot be weakened. Similar results also hold for functions on the circle group T and their (trigonometric) Fourier series.

Several representations for a complete residue system in the Euclidean domain Z(ω) are presented in this paper.

In [1] E. L. Reiss obtained a multivariable expansion of the problem L[y]≡y″+2ϵy′+y=0, t>0;y(0;ϵ)=0, y′(0,ϵ)=1 by means of a technique which does not employ the traditional method of using a secularity condition in obtaining higher order approximations. In this paper Reiss' technique is formalized and new results are established.

In this paper the author shows that a well known sufficient condition for strict cycliclty of a weighted shift on ℓp is not a necessary condition for any p with 1

This paper deals with the characterizations of the complete residue system mod. G, where G is any n×n matrix, in the ring of n×n matrices.

Some inequalities for the Wallis functions are proved. The results of this paper are consequences of some characterization of convex functions. A generalization of a result of Boyd (1) and an extentlon of an inequality of Gantschi (3) are obtained.

In this paper the classical theorem a conservative holonomic dynamic system is invariantly connected with a certain differential form is generalized to group variables. This general theorem is then used to reduce the order of a Hamiltonian system by the use of the integral of energy. Equations of motion of the reduced system so obtained are derived which are the so-called generalized Whittaker's equations. Finally an example is given as an application of the theory.

In this paper we obtain the general solution of scalar, first-order differential equations. The method is variation of parameters with asymptotic series and the theory of partial differential equations.

In this paper, we continue the study of quasi-complemented algebras and complemented algebras. The former are generalizations of the latter and were introduced in [4] and studied in [4] and [11]. Some results are proved.

In this paper, the results of the first detailed and systematic study of the family of fifth order implicit linear multistep methods requiring function evaluations at four backpoints are given. Also described is a practical and efficient nonlinear optimization procedure which made it possible to locate in a precise manner the methods of this type which possess optimal ranges of relative stability in the sense-that the corresponding relative stability regions contain maximal disks centered at the origin.

A theory is presented of the generation and propagation of the two and the three dimensional tsunamis in a shallow running ocean due to the action of an arbitrary ocean floor or ocean surface disturbance. Integral solutions for both two and three dimensional problems are obtained by using the generalized Fourier and Laplace transforms. An asymptotic analysis is carried out for the investigation of the principal features of the free surface elevation. It is found that the propagation of the tsunamis depends on the relative magnitude of the given speed of the running ocean and the wave speed of the shallow ocean. When the speed of the running ocean is less than the speed of the shallow ocean wave, both the two and the three dimensional free surface elevation represent the generation and propagation of surface waves which decay asymptotically as t−12 for the two dimensional case and as t−1 for the three dimensional tsunamis. Several important features of the solution are discussed in some detail. As an application of the general theory, some physically realistic ocean floor disturbances are included in this paper.

In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.

The purpose of this paper is to derive a nonlinear partial differential equation for which λ given by (1.3), is one value of the solution. In Section 2, we derive this equation using a straightforward dynamic programming approach. In Section 3, we discuss some computational aspects of derermining the solution of this equation. In Section 4, we show that the same method may be applied to the nonlinear characteristic value problem. In Section 5, we discuss how the method may by applied to find the higher characteristic values. In Section 5, we discuss how the same method may be applied to some matrix problems. Finally, in Section 7, we discuss selective computation.

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extension RG(R may be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extension RQ over a ring R, where Q are the usual quaternions i,j,k and multiplication and addition are defined as quaternion algebras over a field. We shall show that RG has a unique separable idempotent if and only if G is abelian, that there are more than one separable idempotents for a separable quaternion ring RQ, and that RQ is separable if and only if 2 is invertible in R.

The fundamental homomorphism theorem for rings is not generally applicable in hemiring theory. In this paper, we show that for the class of N-homomorphism of hemirings the fundamental theorem is valid. In addition, the concept of N-homomorphism is used to prove that every hereditarily semisubtractive hemiring is of type (K).

In the last few years various infinite dimensional extensions to Krasnoselski's Theorem on starshaped sets [14] have been made. These began with a paper by Edelstein and Keener [8] and have culminated in the papers by Borwein, Edelstein and O'Brien [3] [4] by Edelstein, Keener and O'Brien [9] and finally by O'Brien [16].

Some characteristic approximation properties of quadratic irrationals are studied in this paper. It is shown that the limit points of the sequence δn form a subset C(x), and D(x) can be generated from C(x) in a relatively simple way. Another proof of Lekkerkerker's theorem is given using relations between δn−1, δn, δn+1 which are independent of x and n.

In this paper we give a further result which states sufficient conditions for the theory of convergence of minimizing sequences to be applicable, develop the theory further, and give an application.

A necessary and sufficient condition for a bounded operator to be a composition operator is investigated in this paper. Normal, quasi-hyponormal, paranormal composition operators are characterised.

Let G and A be abelian torsion groups. In[5], R. S. Pierce develops a complete set of invariants for Hom(G, A). To compute these invariants he introduces, and uses extensively, the group of small homomorphisms of G into A. Also, using some of Pierce's methods, Fuchs characterizes this group in [1]. Our purpose in this paper is to characterize Hom(G, A) in what seems to be a more natural manner than either of the treatments just mentioned.

Let πb,c(x) denote the number of primes ≤x and ≡c(modb), and for positive integers x let Δb(x,c,l)=πb,c(x)−πb,l(x). Negative values of Δ4(x,3,1) less than 1012 occur in six widely spaced regions. The first three regions, investigated by Leech [6], Shanks [9] and Lehmer [6 ], contain only a few thousand negative values of Δ4(x,3,1). However, the authors [1] have recently discovered 3 new regions, the sixth occurring before 20 billion and containing more than half a billion negative values of Δ4(x,3,1). In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus 8 are presented. Previously, no negative values have been found for Δ8(x,c,1), c=3,5, or 7 and our search to 1012 reveals no such values for Δ8(x,3,1) or Δ8(x,7,1). For Δ8(x,5,1) we have discovered the first two regions of negative values. The first of these regions, beginning at x=588067889, contains 422,500 negative values of Δ8(x,5,1); the second occurs in the vicinity of 35 billion and contains more than a billion negative values of Δ8(x,5,1).

In literature, it is generally asserted that the principle of objectivity is valid for constitutive equations but not valid for general field equations. In this paper, the principle of objectivity is formulated clearly to emphasize the correct significance, and necessity of the principle and to discuss its validity in physical theories. A discussion is made to show how successfully the principle of objectivity is also applicable to non-relativistic classical physics.

Suppose ∑n=0∞anzn has radius of convergence R and σN(z)=|∑n=N∞anzn|. Suppose |z1|σN(z) for zϵT}. Two questions are asked: (a) can S be cofinite? (b) can S be infinite? This paper provides some answers to these questions. The answer to (a) is no, even if T=z2. The answer to (b) is no, for T=z2 if liman=a≠0. Examples show (b) is possible if T=z2 and for T a neighborhood of z2.

Let {Xk} be independent random variables with EXk=0 for all k and let {ank:n≥1, k≥1} be an array of real numbers. In this paper the almost sure convergence of Sn=∑k=1nankXk, n=1,2,…, to a constant is studied under various conditions on the weights {ank} and on the random variables {Xk} using martingale theory. In addition, the results are extended to weighted sums of random elements in Banach spaces which have Schauder bases. This extension provides a convergence theorem that applies to stochastic processes which may be considered as random elements in function spaces.

This study is concerned with the theory of parametric coupling of waves in a hot inhomogeneous magnetized plasma in which the temperature gradient has been taken into account. The general dispersion relation and the polarization of the ordinary and the extra-ordinary wave modes are discussed. The eigen-mode solutions of the coupled differential equations for the wave amplitudes are obtained in the terms of the so called three wave interaction matrix elements. The theory of nonlinear wave-wave interactions, which has been extended to the case of an inhomogeneous magnetized plasma, is used to determine the threshold value of the electric field and the frequency shift. The results of this paper are also compared with the other known results. It is shown that the findings of this study are in excellent agreement with the results of earlier investigators.

We show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem Δu+λu=0 in D⊂Rn and u=0 on ∂D, is not bounded generally if one imposes the norm ∫Du2p(x)dx=1, p=(1),2,3,…. The same holds with the norm ∫D|gradu|2pdx=1 when n>4p−1. On the other hand, if D⊂R2, resp. R3 the norm ∫D|gradu|2dx=1 implies max|uk|→k→∞0, resp. max|uk|=0(1).

This paper summarizes most of the results to date on convergence space compactifications, and establishes necessary and sufficient conditions for the existence of largest and smallest compactifications subject to various conditions imposed upon the compactifications.

This paper contains the solution of a functional equation which is a generalization of a functional equation arising in a characterization of directed divergence and inaccuracy.

Let S be a subset of a metric space (X,d) and T:S→X be a mapping. In this paper, we define the notion of lower directional increment QT(x,y] of T at x∈S in the direction of y∈X and give sufficient conditions for T to have a fixed point when QT(x,Tx]

In this paper we investigate some structure properties of the tail σ-field and the invariant σ-field of both homogeneous and nonhomogeneous Markov chains as representations for asymptotic events, descriptions of completely nonatomic and atomic sets and global characterizations of asymptotic σ-fields. It is shown that the Martin boundary theory can provide a unified approach to the asymptotic σ-fields theory.

This paper deals with the problems of representing an arbitrary double differential of the second kind, defined on a surface which is the topological product of two curves, in terms of the products of simple differentials of the second kind on the two curves. The curves are assumed to be non-singular and irreducible in a complex projective 2-space.

A well-known theorem of N. Jacobson states that any periodic associative ring is commutative. Several authors (most notably Kaplansky and Herstein) generalized the periodic polynomial condition and were still able to conclude that the rings under consideration were commutative. (See [3]) In this paper we develop a structure theory for a class of rings which properly contains the periodic rings. In particular, an associative ring R is said to be a quasi-anti-integral (QAI) ring if for every a≠0 in R there exist a positive integer k and integers n1,n2,…,nk (all depending on a), so that 0≠n1a=n2a2+…+nkak. In the main theorems of this paper, we show that any QAl-ring is a subdirect sum of prime QAl-rings, which in turn are shown to be left and right orders in division algebras which are algebraic over their prime fields.

In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] where B=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that when B is itself a real separable Hilbert space is also illustrated.

In this paper a general canonical form for elements in a ring Euclidean with respect to a real valuation is established. It is also shown that this form is unique and minimal thus gives the arithmetical weight of an element with respect to a radix.

This paper is concerned with the continuous Legendre transform, derived from the classical discrete Legendre transform by replacing the Legendre polynomial Pk(x) by the function Pλ(x) with λ real. Another approach to T.M. MacRobert's inversion formula is found; for this purpose an inverse Legendre transform, mapping L1(ℝ+) into L2(−1,1), is defined. Its inversion in turn is naturally achieved by the continuous Legendre transform. One application is devoted to the Shannon sampling theorem in the Legendre frame together with a new type of error estimate. The other deals with a new representation of Legendre functions giving information about their behaviour near the point x=−1.

This paper discusses the existence and decay of solutions u(t) of the variational inequality of parabolic type: ≧0for ∀v∈Lp([0,∞);V)(p≧2) with v(t)∈K a.e. in [0,∞), where K is a closed convex set of a separable uniformly convex Banach space V, A is a nonlinear monotone operator from V to V* and B is a nonlinear operator from Banach space W to W*. V and W are related as V⊂W⊂H for a Hilbert space H. No monotonicity assumption is made on B.

In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.

Let B be a commutative ring with 1, and G(={σ}) an automorphism group of B of order 2. The generalized quaternion ring extension B[j] over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free B-module with a basis {1,j}, and (2) j2=−1 and jb=σ(b)j for each b in B. The purpose of this paper is to study the separability of B[j]. The separable extension of B[j] over B is characterized in terms of the trace (=1+σ) of B over the subring of fixed elements under σ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved.

In this paper, a new class of normalized univalent functions is introduced. The properties of this class and its relationship with some other subclasses of univalent functions are studied. The functions in this class are close-to-convex.

This paper involves characterizations of a class of continuous flows. The flows considered are those in which the positive prolongation of each point coincides with the trajectory through the point. The characterizations are based on the theory of prolongation.

The objectives of the paper are to study the behavior of Wronskians of solutions of the fourth order differential equations and to relate this behavior with the oscillations of these equations, as well as to the structure of the subspaces of the solution spaces of the equations.

Let GF(pz) denote the finite field of pz elements. Let A1 be s×m of rank r1 and A2 be s×n of rank r2 with elements from GF(pz). In this paper, formulas are given for finding the number of X1,X2 over GF(pz) which satisfy the matric equation A1X1=A2X2, where X1 is m×t of rank k1, and X2 is n×t of rank k2. These results are then used to find the number of solutions X1,…,Xn, Y1,…,Ym, m,n>1, of the matric equation A1X1…Xn=A2Y1…Ym.

Recently Chatterjea (1) has proved a theorem to deduce a bilateral generating function for the Ultraspherical polynomials. In the present paper an attempt has been made to give a general version of Chatterjea's theorem. Finally, the theorem has been specialized to obtain a bilateral generating function for a class of polynomials {Pn(x;α,β)} introduced by Bhattacharjya (2).

Let S be a subset of a metric space X and let B(X) be the class of all nonempty bounded subsets of X with the Hausdorff pseudometric H. A mapping F:S→B(X) is a directional contraction iff there exists a real α∈[0,1) such that for each x∈S and y∈F(x), H(F(x),F(z))≤αd(x,z) for each z∈[x,y]∩S, where [x,y]={z∈X:d(x,z)+d(z,y)=d(x,y)}. In this paper, sufficient conditions are given under which such mappings have a fixed point.

Epstein and Horn ([6]) proved that a Post algebra is always a P-algebra and in a P-algebra, prime ideals lie in disjoint maximal chains. In this paper it is shown that a P-algebra L is a Post algebra of order n≥2, if the prime ideals of L lie in disjoint maximal chains each with n−1 elements. The main tool used in this paper is that every bounded distributive lattice is isomorphic with the lattice of all global sections of a sheaf of bounded distributive lattices over a Boolean space. Also some properties of P-algebras are characterized in terms of the stalks.

The rate of growth of Hankel determinant for close-to-convex functions is determined. The results in this paper are best possible.

In this paper the following Cauchy problem, in a Hilbert space H, is considered: (I+λA)u″+A2u+[α+M(|A12u|2)]Au=fu(0)=u0u′(0)=u1

This paper presents a method of the determination of the steady state response for a class of nonlinear systems. The response of a nonlinear system to a given input is first obtained in the form of a series solution in the multidimensional frequency domain. Conditions are then determined for which this series solution will converge. The conversion from multidimensions to a single dimension is then made by the method of association of variables, and thus an equivalent linear model of the nonlinear system is obtained. The steady state response is then found by any technique employed with linear system.