93 results
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2. The Analytic Continuation of the Riemannliouville Integral in the Hyperbolic Case
- Author
-
Marcel Riesz
- Subjects
symbols.namesake ,General Mathematics ,Analytic continuation ,Riemann–Liouville integral ,Mathematical analysis ,Global analytic function ,Hyperbolic function ,symbols ,Inverse hyperbolic function ,Mathematics - Abstract
In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.
- Published
- 1961
3. Left Cauchy Integral Bases in Linear Topological Spaces
- Author
-
James A. Dyer
- Subjects
Pure mathematics ,General Mathematics ,Topological tensor product ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Infinite-dimensional holomorphy ,Topological space ,01 natural sciences ,Topological vector space ,Homeomorphism ,Continuous linear operator ,Locally convex topological vector space ,0101 mathematics ,Cauchy's integral theorem ,Mathematics - Abstract
The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.
- Published
- 1970
4. The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations
- Author
-
Meira Lavie
- Subjects
Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,01 natural sciences ,Domain (mathematical analysis) ,Linear differential equation ,0103 physical sciences ,Order (group theory) ,010307 mathematical physics ,Linear independence ,0101 mathematics ,Schwarzian derivative ,Complex plane ,Mathematics - Abstract
In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.
- Published
- 1969
5. A Geometrical Approach to the Second-Order Linear Differential Equation
- Author
-
J. E. Barry and C. M. Petty
- Subjects
Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,First-order partial differential equation ,Exact differential equation ,01 natural sciences ,Linear differential equation ,Homogeneous differential equation ,0103 physical sciences ,Riccati equation ,010307 mathematical physics ,0101 mathematics ,Universal differential equation ,Algebraic differential equation ,Mathematics - Abstract
In this paper various concepts intrinsically defined by the differential equation1.1are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).Throughout this paper,R(t)is a real-valued, continuous function ofton the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.
- Published
- 1962
6. A Generalized Integral II
- Author
-
R. D. James
- Subjects
Generalized inverse ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Line integral ,Singular integral ,01 natural sciences ,Integral equation ,Volume integral ,Dirichlet integral ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.
- Published
- 1950
7. Generalized Spectral Theory and Second Order Ordinary Differential Operators
- Author
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Héctor J. Sussmann
- Subjects
Oscillation theory ,Constant coefficients ,Spectral theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Microlocal analysis ,Spectral theorem ,Operator theory ,01 natural sciences ,Fourier integral operator ,0103 physical sciences ,Spectral theory of ordinary differential equations ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper continues the study, begun in [7], of the spectral theory of non-self-ad joint second order ordinary differential operators on a half-line. The case of a ‘Very small” potential was studied in [4; 5; 6]. The case considered in [7], and in the present paper, is that where the potential is not so small.
- Published
- 1973
8. Commutators of Operators on Hilbert Space
- Author
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Arlen Brown, Carl Pearcy, and P. R. Halmos
- Subjects
Pure mathematics ,Hilbert manifold ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Hilbert space ,Operator theory ,01 natural sciences ,Operator space ,Compact operator on Hilbert space ,Operator topologies ,symbols.namesake ,Bounded function ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Kuiper's theorem ,Mathematics - Abstract
The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.
- Published
- 1965
9. Arcs of Parabolic Order Four
- Author
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N. D. Lane
- Subjects
General Mathematics ,Mathematical analysis ,Order (group theory) ,Mathematics - Abstract
This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.
- Published
- 1964
10. Perturbation of the Continuous Spectrum of Systems of Ordinary Differential Operators
- Author
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John B. Butler
- Subjects
Constant coefficients ,General Mathematics ,010102 general mathematics ,Continuous spectrum ,Mathematical analysis ,Exact differential equation ,Perturbation (astronomy) ,Operator theory ,Differential operator ,01 natural sciences ,Fourier integral operator ,Poincaré–Lindstedt method ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Letbe an ordinary differential operator of order h whose coefficients are (η, η) matrices defined on the interval 0 ≤ x < ∞, hη = n = 2v. Let the operator L0 be formally self adjoint and let v boundary conditions be given at x = 0 such that the eigenvalue problem(1.1)has no non-trivial square integrable solution. This paper deals with the perturbed operator L∈ = L0 + ∈q where ∈ is a real parameter and q(x) is a bounded positive (η, η) matrix operator with piecewise continuous elements 0 ≤ x < ∞. Sufficient conditions involving L0, q are given such that L∈ determines a selfadjoint operator H∈ and such that the spectral measure E∈(Δ′) corresponding to H∈ is an analytic function of ∈, where Δ′ is a subset of a fixed bounded interval Δ = [α, β]. The results include and improve results obtained for scalar differential operators in an earlier paper (3).
- Published
- 1962
11. Ergodic Theory and Averaging Iterations
- Author
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J. J. Koliha
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Applied mathematics ,Ergodic theory ,010307 mathematical physics ,0101 mathematics ,Stationary ergodic process ,01 natural sciences ,Mathematics - Abstract
Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.
- Published
- 1973
12. A Cosine Functional Equation with Restricted Argument
- Author
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L. B. Etigson
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,010102 general mathematics ,Characteristic equation ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Integro-differential equation ,Argument ,Functional equation ,Riccati equation ,Discrete Mathematics and Combinatorics ,Trigonometric functions ,0101 mathematics ,Mathematics - Abstract
We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.The purpose of this paper is to present a functional equation satisfied only by cosine functions.
- Published
- 1974
13. The Oscillatory Behavior of a First Order Non-Linear Differential Equation with Delay
- Author
-
Peter J. Ponzo and Forbes J. Burkowski
- Subjects
Liénard equation ,Non linear differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,0101 mathematics ,First order ,01 natural sciences ,Mathematics - Abstract
SynopsisThis paper establishes the existence of an infinite set of zeros for the solution of a certain functional differential equation. The primary condition assuring this oscillatory behavior is expressed in terms of the magnitude of the delay.
- Published
- 1974
14. Oscillation Criteria for Quasilinear Equations
- Author
-
W. Allegretto
- Subjects
Oscillation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Several authors have recently considered the problem of establishing sufficient criteria to guarantee the oscillation or non-oscillation of all solutions of a second order elliptic equation or system. We mention in particular the papers of C. A. Swanson, [15; 16], K. Kreith [9], Kreith and Travis [10], Noussair and Swanson [13], Allegretto and Swanson [3], Allegretto and Erbe [2] and the references therein.
- Published
- 1974
15. A Necessary and Sufficient Condition for the Oscillation of an Even Order Nonlinear Delay Differential Equation
- Author
-
Bhagat Singh
- Subjects
Oscillation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Delay differential equation ,01 natural sciences ,Nonlinear system ,Bounded function ,0103 physical sciences ,Order (group theory) ,010307 mathematical physics ,Half line ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation(1)where(i)denotes the order of differentiation with respect tot. The delay termsτiσiare assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constantMon the half line (t0, + ∞ ) for somet0≧ 0.
- Published
- 1973
16. Almost Convergence, Summability And Ergodicity
- Author
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J. Peter Duran
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Ergodicity ,Applied mathematics ,010307 mathematical physics ,Convergence (relationship) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The notion of almost convergence introduced by Lorentz [15] has been generalized in several directions (see, for example [1; 8; 11 ; 14; 17]). I t is the purpose of this paper to give a generalization based on the original definition in terms of invariant means. This is effected by replacing the shift transformation by an "ergodic" semigroupof positive regular matrices in the definition of invariant mean. The resulting "- invariant means" give rise to a summability method which we dub-almost convergence.
- Published
- 1974
17. Generalizations of Noshiro's Theorem and Their Applications
- Author
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Hidenobu Yoshida
- Subjects
Algebra ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
Meier [8, Hauptsatz] proved a remarkable theorem concerning the boundary behavior of functions meromorphic in the upper half plane; but his techniques are very complicated. So Noshiro [10, p. 72-73] proved an analogous (but somewhat weaker) result to Meier's by a simple method using the theorem of Gross and Iversen.In this paper, we sharpen and generalize Noshiro's theorem in some directions by making use of the notion “porosity”, and we state some applications.
- Published
- 1973
18. A Ratio Limit Theorem for Approximate Martingales
- Author
-
Charles W. Lamb
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,010307 mathematical physics ,Limit (mathematics) ,0101 mathematics ,01 natural sciences ,Squeeze theorem ,Mathematics - Abstract
It has been proved [3, p. 630] that the martingale convergence theorem obtained by Andersen and Jessen [1, p. 5] follows from the classical theory developed by Doob. By using some results of Yosida and Hewitt [9] on finitely additive set functions, Johansen and Karush [7] proved that the identification of the limit function as a derivative in the approach of Andersen and Jessen can be obtained in the general case. In this paper we sharpen the methods of Andersen and Jessen to obtain a ratio limit theorem for “approximate martingales”.
- Published
- 1973
19. An Intermediate Value Property for Operators with Applications to Integral and Differential Equations
- Author
-
J. S. Muldowney and D. Willett
- Subjects
Constant coefficients ,Parametrix ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Microlocal analysis ,Operator theory ,01 natural sciences ,Integral equation ,Fourier integral operator ,Stochastic partial differential equation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Numerical partial differential equations ,Mathematics - Abstract
It is well known that a real valued continuous function f on a closed interval S assumes every value between its maximum and minimum on S, i.e. if ξ is such that f(α) ≦ ξ ≦ f(β) then there exists γ between α and β such that f(γ) = ξ. The purpose of this paper is to develop the existence theory associated with differential and integral inequalities in the context of an intermediate value property for operators on partially ordered spaces. This has the advantage of allowing rather simple proofs of known results while in most cases giving slight improvements, and in some cases substantial improvements, in these results. Classical and recent results from different areas are unified under one principle.
- Published
- 1974
20. Asymptotic Solution Of Differential Equations In a Domain Containing a Regular Singular Point
- Author
-
N. D. Kazarinoff and R. McKelvey
- Subjects
Equilibrium point ,Singular perturbation ,Asymptotic analysis ,Regular singular point ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Singular point of a curve ,01 natural sciences ,Method of matched asymptotic expansions ,Singular solution ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Numerical partial differential equations - Abstract
1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation.Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.
- Published
- 1956
21. On a Geometrical Theorem in Exterior Algebra
- Author
-
Daniel Pedoe
- Subjects
Filtered algebra ,Symmetric algebra ,Lie coalgebra ,Multivector ,Pure mathematics ,Differential form ,Kelvin–Stokes theorem ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Division algebra ,Exterior algebra ,Mathematics - Abstract
In this paper we shall give necessary and sufficient conditions for three lines, passing respectively through the vertices of a proper triangle PQR in the real Euclidean plane, to be concurrent. Of course, the theorem of Ceva deals with this problem, but it is useful to have a criterion which involves only vectors localized at a point O of the plane, and the exterior products of these vectors. Applications are made to theorems which are not easily proved by other methods.
- Published
- 1967
22. A Simple Bounding Formula for Integrals
- Author
-
J. L. Synge
- Subjects
Order of integration (calculus) ,Bounding overwatch ,Simple (abstract algebra) ,General Mathematics ,Mathematical analysis ,Applied mathematics ,Function (mathematics) ,Mathematics - Abstract
In this paper I establish the following bounding formula for the integral of a function of n variables:(1.1)
- Published
- 1953
23. Derivatives and Integrals with Respect to a Base Function of Generalized Bounded Variation
- Author
-
R. L. Jeffery and H. W. Ellis
- Subjects
Integro-differential equation ,General Mathematics ,Bounded function ,010102 general mathematics ,Mathematical analysis ,Bounded variation ,Base function ,Bounded deformation ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we consider measures determined by arbitrary functions G(x) for which finite right and left limits exist everywhere and indicate how some of these measures permit the definition of generalized integrals of constructive or Denjoy type. These definitions are related to corresponding descriptive definitions based on the Perron approach as given by Ward (6) and Henstock (2). An exposition of the introductory theory is given in (1).
- Published
- 1967
24. Singular Perturbations of Non-Linear Elliptic and Parabolic Variational Boundary-Value Problems
- Author
-
Bui An Ton
- Subjects
Nonlinear system ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,010307 mathematical physics ,Boundary value problem ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Singular perturbations of linear elliptic and parabolic boundary-value problems have been studied extensively by Visik and Lyusternik (7), Huet (5), and others. It is the purpose of this paper to extend the results of (5) to the non-linear elliptic and parabolic variational boundary-value problems considered during the last few years by Browder (2, 4).In §1, we give the notations and state the main assumptions on the nonlinearity of the elliptic operators. In §2 we study the singular perturbations of non-linear elliptic variational boundary problems. In §3, we consider the case of non-linear parabolic variational boundary problems with a small parameter.
- Published
- 1966
25. Oscillation Criteria for Second Order Nonlinear Delay Equations
- Author
-
Lynn Erbe
- Subjects
Nonlinear system ,Oscillation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,Order (group theory) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Nonlinear differential equations ,Mathematics - Abstract
It is the purpose of this paper to establish oscillation criteria for second order nonlinear differential equations with retarded argument. Specifically, we consider the equation1.1where f ∊ C[0, + ∞) x R2, g ∊ C[0, + ∞), and1.2We shall restrict attention to solutions of (1.1) which exist on some ray [T, + ∞). A solution of (1.1) is called oscillatory if it has no largest zero.
- Published
- 1973
26. On Some New Generalizations of the Functional Equation of Cauchy
- Author
-
Gy. Muszély and P. Fischer
- Subjects
Cauchy problem ,General Mathematics ,010102 general mathematics ,Functional equation ,Mathematical analysis ,Applied mathematics ,Cauchy distribution ,Cauchy principal value ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Examining certain problems in physics M. Hosszu [l] obtained the functional equation(1)where x, y, f are real.In another paper M. Hosszu [2] proved that the equation (1) is equivalent to the functional equation of Cauchy; i. e., to the equation(1)under the assumption that x is real and f is real and continuous.
- Published
- 1967
27. A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces
- Author
-
L. Sario and Young Koan Kwon
- Subjects
Harmonic coordinates ,symbols.namesake ,Maximum principle ,Subharmonic function ,Harmonic function ,General Mathematics ,Mathematical analysis ,symbols ,Maximum modulus principle ,Harmonic measure ,Dirichlet distribution ,Potential theory ,Mathematics - Abstract
Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).
- Published
- 1970
28. Dimension of Ideals in Polynomial Rings
- Author
-
Alex Rosenberg and Maurice Auslander
- Subjects
Ring theory ,General Mathematics ,Polynomial ring ,010102 general mathematics ,Mathematical analysis ,Semiprime ring ,Artinian ring ,01 natural sciences ,Global dimension ,Combinatorics ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Maximal ideal ,010307 mathematical physics ,0101 mathematics ,Commutative algebra ,Dimension theory (algebra) ,Mathematics - Abstract
A well-known theorem asserts that if K is a field, a prime ideal in the polynomial ring S = K[X1, … Xn] and d the transcendence degree of S / over K n = rank + d. In the first half of this paper we extend this result to the case of arbitrary commutative noetherian K, as well as giving a purely homological proof of the classical theorem. In the second half we use our first result to compute the analogue of the dimension of the product and intersection of two affine varieties when K is a Dedekind ring.
- Published
- 1958
29. On the Location of Singularities of a Class of Elliptic Partial Differential Equations in Four Variables
- Author
-
R. P. Gilbert
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,First-order partial differential equation ,01 natural sciences ,Parabolic partial differential equation ,Stochastic partial differential equation ,Semi-elliptic operator ,Elliptic partial differential equation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Symbol of a differential operator ,Separable partial differential equation ,Mathematics ,Numerical partial differential equations - Abstract
In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation(1.1)where A (r2), C(r2) are entire functions of the complex variable
- Published
- 1965
30. Riemannian Structures Subordinate to Certain Almost Tangent Structures
- Author
-
M. P. Closs
- Subjects
Mathematics::Probability ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Tangent ,Mathematics::Differential Geometry ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Almost tangent structures have been studied by Eliopoulos [1] and certain Riemannian structures subordinate to almost tangent structures have been studied by Closs [2]. In this paper we investigate those subordinate Riemannian structures for which the underlying almost tangent structure is without torsion and those for which the fundamental form is closed.Similar studies have been carried out with respect to Riemannian structures subordinate to almost complex structures by Lichnerowicz [4] and with respect to Riemannian structures subordinate to almost product structures by Legrand [5].
- Published
- 1972
31. Space of Solutions of Homogeneous Elliptic Equations
- Author
-
T. Husain and Ed Dubinsky
- Subjects
Quarter period ,Simultaneous equations ,Homogeneous ,General Mathematics ,Mathematical analysis ,Space (mathematics) ,Homogeneous distribution ,Mathematics - Abstract
This is the continuation of our paper [1] and includes the results promised there. As in [1], we consider a homogeneous elliptic equation in two variables. In [1] we showed that all solutions of such equations can be written in a specific form, viz. in the form of an infinite series in certain specific polynomials. Here we first establish that a common solution of any two positive powers of any two linearly independent, linear elliptic polynomials can be expressed as a polynomial (Lemma 2).
- Published
- 1971
32. A Generalization of an Inversion Formula for the Gauss Transformation
- Author
-
P. G. Rooney
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Gauss ,Mathematical analysis ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,Quadratic Gauss sum ,01 natural sciences ,Inversion (discrete mathematics) ,Mathematics - Abstract
In an earlier paper [3] we considered an inversion formula for the Gauss transformation G defined by1.1We noted there that formally G is inverted by,1.2and we showed that if e-D2 is interpreted via the power series for the exponential function, that is if1.3then under certain conditions on φ,1.4
- Published
- 1963
33. An Integral Representation for the Product of Spectral Measures
- Author
-
N. A. Derzko
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Hilbert space ,Disjoint sets ,01 natural sciences ,Semiring ,symbols.namesake ,Operator (computer programming) ,Set function ,Product (mathematics) ,Bounded function ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Integration by parts ,010307 mathematical physics ,0101 mathematics ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
Let be a Hilbert space with inner product (•, •) and let E(•) and E0(•) be spectral measures in corresponding to self-adjoint operators and . In this paper we consider the set function ƒ(I × J) = E(I)E0(J) defined on the semiring of bounded rectangles, and obtain an integral representation for this set function for disjoint I, J under the hypotheses that H — H0 is a type of Carleman operator.
- Published
- 1968
34. Tests for the Scale Parameter of the Truncated Normal
- Author
-
Irwin Guttman
- Subjects
Truncated normal distribution ,General Mathematics ,Mathematical analysis ,Scale parameter ,Mathematics - Abstract
This paper continues the work begun in [l] and examines the loss of power when using tests based on the assumption that the variable being sampled has a “complete” normal distribution, when in fact, sampling is from a symmetrically truncated distribution. The hypothesis considered here is the one - sided test for the variance of a normal distribution. Some tables have been computed and they show that appreciable losses in size occur. Some loss occurs in the power too, but this decreases with the alternative value of the variance and the degree of truncation.
- Published
- 1960
35. Minimum and Conjugate Points in Symmetric Spaces
- Author
-
Richard J. Crittenden
- Subjects
Tangent bundle ,Pure mathematics ,Complex conjugate ,Triple system ,General Mathematics ,010102 general mathematics ,Conjugate points ,Mathematical analysis ,01 natural sciences ,Hermitian matrix ,Symmetric space ,0103 physical sciences ,Simply connected space ,Tangent space ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.
- Published
- 1962
36. Series Expansions for Dual Laguerre Temperatures
- Author
-
Deborah Tepper Haimo
- Subjects
Differential heat ,Series (mathematics) ,General Mathematics ,Mathematical analysis ,Laguerre polynomials ,Heat equation ,Characterization (mathematics) ,Series expansion ,Mathematics ,Dual (category theory) - Abstract
In a recent paper [2], the author, with F. M. Cholewinski, derived criteria for the series expansions of solutions u(x, t) of the Laguerre differential heat equation xuxx + (α + 1 - x)ux = ut in terms of the Laguerre heat polynomials and of their temperature transforms. Our present goal is the characterization of those solutions which are representable in a Maclaurin double series in xe-t and in 1 — e-t Some of the results are analogous to those derived by D. V. Widder in [4] for the classical heat equation and by the author in [1] for the generalized heat equation.
- Published
- 1972
37. The Cauchy Problem for a Hyperbolic Second Order Equation with Data on the Parabolic Line
- Author
-
M. H. Protter
- Subjects
Cauchy problem ,Cauchy's convergence test ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Ultraparallel theorem ,01 natural sciences ,Elliptic partial differential equation ,Parabolic cylindrical coordinates ,0103 physical sciences ,Initial value problem ,Cauchy boundary condition ,010307 mathematical physics ,0101 mathematics ,Hyperbolic partial differential equation ,Mathematics ,Mathematical physics - Abstract
In this paper we consider the Cauchy problem for the equation(1) h(x, y) K(y) vxx − vyy + a(x, y) vx + b(x, y) vy + c(x, y) v + f(x, y) = 0with initial values prescribed on a segment of the x-axis. The coefficients in (1) are assumed to possess two continuous derivatives with respect to x and one continuous derivative with respect to y in the closure of the domain under consideration.
- Published
- 1954
38. Non-Desarguesian Projective Plane Geometries Which Satisfy The Harmonic Point Axiom
- Author
-
N. S. Mendelsohn
- Subjects
Projective harmonic conjugate ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fano plane ,01 natural sciences ,Blocking set ,Real projective plane ,Duality (projective geometry) ,0103 physical sciences ,Projective space ,010307 mathematical physics ,Projective plane ,0101 mathematics ,Non-Desarguesian plane ,Mathematics - Abstract
1. Introduction and summary. In her papers (12) and (13) R. Moufang discusses projective plane geometries which satisfy the axiom of the uniqueness of the fourth harmonic point. Her main result is that in such geometries non-homogeneous co-ordinates may be assigned to the points of the plane (except for the “line at infinity”) in such a way that straight lines have equations of the forms aαx + y + β = 0, or x + γ − 0.
- Published
- 1956
39. On the Relation Between Boundedness and Oscillation of Differential Equations of Second Order
- Author
-
A. G. Kartsatos
- Subjects
Relation (database) ,Differential equation ,Oscillation ,General Mathematics ,Mathematical analysis ,Order (group theory) ,Mathematics - Abstract
In this paper we are dealing with differential equations of the forms:Eiwhere the functions pi are positive.By a solution of an equation of the above forms, we mean a function x(t) ∈ C2 [c, + ∞) where c is a non-negative constant, which satisfies the corresponding equation on the whole interval [c, +∞). By an oscillatory solution of (Ei), we mean a solution with arbitrarily large zeros.
- Published
- 1967
40. The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution
- Author
-
Paul M. Gauthier
- Subjects
Distribution (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Maximum modulus principle ,Modulus ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Value (mathematics) ,Mathematics ,Meromorphic function - Abstract
Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulusand the minimum modulusWhen no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then(1)This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.
- Published
- 1970
41. On The Potential Theory Of Coclosed Harmonic Forms
- Author
-
G. F. D. Duff
- Subjects
Harmonic fields ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Harmonic (mathematics) ,Poisson distribution ,01 natural sciences ,Potential theory ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Boundary value problem ,0101 mathematics ,Variety (universal algebra) ,Mathematics - Abstract
1. Introduction. The potential theory of real harmonic tensors, which was first studied by Hodge (5), offers a variety of problems by no means all of which have yet been examined. In the present paper there are formulated the solutions of some boundary value problems for the Poisson equations associated with coclosed harmonic forms. These problems include as special cases a number of previous results on coclosed harmonic forms and harmonic fields.
- Published
- 1955
42. Constant Holomorphic Curvature
- Author
-
N. S. Hawley
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Holomorphic function ,Hermitian manifold ,Landau's constants ,Curvature form ,Sectional curvature ,Curvature ,Manifold ,Scalar curvature ,Mathematics - Abstract
We shall present in this paper a certain theorem concerning complex manifolds provided with an Hermitian metric satisfying the Kaehler restriction. The variables z1, z2, …, zn denote local complex coordinates in the manifold and their conjugates. The subscripts a, b, c, … run from 1 to n and by .
- Published
- 1953
43. The Hilbert-Schmidt Property for Embedding Maps between Sobolev Spaces
- Author
-
Colin Clark
- Subjects
Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Sobolev inequality ,Sobolev space ,0103 physical sciences ,Interpolation space ,Embedding ,010307 mathematical physics ,Birnbaum–Orlicz space ,0101 mathematics ,Mathematics - Abstract
Let H0m(Ω) denote the so-called Sobolev space consisting of functions denned on a region Ω in n-dimensional Euclidean space, which together with their generalized derivatives of all orders ⩽m belong to , and which vanish in a certain sense on the boundary ∂Ω. (Precise definitions are given in the next section.) For each pair m, k of non-negative integers the inclusion H0m+k(Ω) ⊂ H0m(Ω) defines a natural “embedding” map. For the case of a bounded region Ω it is well known that these maps are completely continuous, and even, for sufficiently large k, of Hilbert-Schmidt type. We have discussed complete continuity in the case of unbounded regions in an earlier paper; here we consider conditions on Ω which imply the Hilbert-Schmidt property for embeddings.
- Published
- 1966
44. Indices of Function Spaces and their Relationship to Interpolation
- Author
-
David W. Boyd
- Subjects
Pure mathematics ,General theorem ,Function space ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,Space (mathematics) ,01 natural sciences ,Linear map ,0103 physical sciences ,Interpolation space ,010307 mathematical physics ,0101 mathematics ,Special case ,Mathematics ,Interpolation - Abstract
A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q,q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper (5), as part of a more general theorem, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the functions in X.One of Calderόn's results implies that if X is a function space in the sense of Luxemburg (9), then X must be a rearrangement-invariant space.
- Published
- 1969
45. Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity
- Author
-
C. Nasim
- Subjects
General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Function (mathematics) ,Infinity ,Inversion (discrete mathematics) ,symbols.namesake ,Fourier transform ,Bounded variation ,symbols ,Sine ,Mathematics ,Sine and cosine transforms ,media_common - Abstract
It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results iswhere f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.
- Published
- 1969
46. Boundary Value Problems Associated With the Tensor Laplace Equation
- Author
-
G. F. D. Duff
- Subjects
Laplace's equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mixed boundary condition ,01 natural sciences ,Green's function for the three-variable Laplace equation ,Exact solutions in general relativity ,Laplace transform applied to differential equations ,0103 physical sciences ,Symmetric tensor ,010307 mathematical physics ,Boundary value problem ,0101 mathematics ,Tensor density ,Mathematics - Abstract
The boundary value problems considered in this paper relate to harmonic p-tensors on Riemannian manifolds with boundary. We study the equation of Beltrami-Laplace
- Published
- 1953
47. A Convergence Theorem for Double L 2 Fourier Series
- Author
-
Richard P. Gosselin
- Subjects
General Mathematics ,Projection-slice theorem ,Mathematical analysis ,Convergence (routing) ,Applied mathematics ,Fourier series ,Mathematics ,Parseval's theorem - Abstract
Our aim in this paper is to extend a known theorem about the convergence of subsequences of the partial sums of the Fourier series in one variable of class L 2 to Fourier series in two variables of the same class, (1, p. 396). The theorem asserts that for each function ƒ in L 2, there is a sequence {m V } of positive integers of upper density one such that Smv(X;ƒ) converges to ƒ almost everywhere where sm(x;f) denotes the mth partial sum of the Fourier series of ƒ.
- Published
- 1958
48. On The Extension Of Measure By The Method Of Borel
- Author
-
L. LeBlanc and G. E. Fox
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Measure (mathematics) ,Lebesgue–Stieltjes integration ,Borel equivalence relation ,Random measure ,Transverse measure ,Borel hierarchy ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Borel set ,Borel measure ,Mathematics - Abstract
Introduction. This paper concerns the problem of extending a given measure defined on a Boolean ring to a measure on the generated σ-ring. Two general methods are familiar to the literature, that of Lebesgue (outer measure) and a method proposed by Borel using transfinite induction (4, 49-134; 2, 228-238).
- Published
- 1956
49. Disconjugacy Conditions for the Third Order Linear Differential Equation
- Author
-
Lynn Erbe
- Subjects
Third order ,Linear differential equation ,Homogeneous differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
An nth order homogeneous linear differential equation is said to be disconjugate on the interval I of the real line in case no non-trivial solution of the equation has more than n - 1 zeros (counting multiplicity) on I. It is the purpose of this paper to establish several necessary and sufficient conditions for disconjugacy of the third order linear differential equation(1.1)where pi(t) is continuous on the compact interval [a, b], i = 0, 1, 2.
- Published
- 1969
50. Approximation by Unimodular Functions
- Author
-
Stephen Fisher
- Subjects
Pure mathematics ,Unimodular matrix ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.
- Published
- 1971
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