bluuebunny/crossref_metadata_2025_split huggingface dataset:
Mixed Problems for Hyperbolic Equations of General Order
Authors: Duff G. F. D. 
Published: December 2010

The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.
A Class of Solvable Groups
Authors: Gorenstein Daniel , Herstein I. N. 
Published: December 2010

Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB, where A and B are subgroups of G. The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B.More recently the structure of finite groups G which have a representation in the form ABA, where A and B are subgroups of G, has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba', where a, a’ are in A, and b ≠ 1 is in B.
On Some Properties of Functions Analytic in a Half-Plane
Authors: Rooney P. G. 
Published: December 2010

The spaces , w real, 1 ≤ p < ∞, consist of those functions f(s), analytic for Re s > w, and such that μp(f;x) is bounded for x > w, where1.1Doetsch (1) has shown that if e-wtϕ(t) ∈ Lp (0, ∞), 1 < p ≤ 2, and f is the Laplace transform of ϕ, that is,then f ∈ , where1.2and that conversely if f ∈ , 1 < p ≤ 2, then there is a function ϕ, with e-wtϕ(t) ∈ Lq (0, ∞), such that f is the Laplace transform of ϕ.
The Regular Maps on a Surface of Genus Three
Authors: Sherk F. A. 
Published: December 2010

A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and completed by Coxeter (6; 7, p. 141). In addition Coxeter enumerated the regular maps on the simplest non-orientable surfaces (7, pp. 116, 139), and constructed tables of some interesting families of regular maps (3; 7, p. 140).Most of the regular maps on a surface of genus 3 have appeared in these papers, but no systematic enumeration of them seems to have been attempted. The ultimate goal of this paper is a complete list of these regular maps. However, the families of maps {j.p,q}and {j.p,j.q} which are defined in § 4 and listed in Tables I and II are of considerable interest in themselves. Also of some importance is the complete list of regular maps of type {p,3} with six or fewer faces (§ 5 and Table III).
Convex Structures and Continuous Selections
Authors: Michael Ernest 
Published: December 2010

This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If X and Y are topological spaces, and ϕ a function (called a carrier) from X to the space 2Y of non-empty subsets of F, then a selection for ϕ is a continuous f: X → Y such that f(x) ∈ ϕ(x) for every x ∈ X. For reasons which are explained in (13), we restrict our attention to carriers which are lower semi-continuous (l.s.c), in the sense that, whenever U is open in Y, then is open in X. Our purpose in these papers is to find conditions for the existence and extendability of selections.The principal purpose of this paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).
A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc
Authors: Bing R. H. 
Published: December 2010

One of the unsolved problems of plane topology is the following:Question. What are the homogeneous bounded plane continua?A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.
A Linear Diophantine Problem
Authors: Johnson S. M. 
Published: December 2010

Let a1, a2, … , at be a set of groupwise relatively prime positive integers. Several authors, (2; 3; 5; 6), have determined bounds for the function F(a1, …, at) defined by the property that the equation1has a solution in positive integers X1, …, xt for n > F(a1, ..., at). If F(a1, …, at) is a function of this type, it is easy to see that2is the corresponding function for the solvability of (1) in non-negative x's.It is well known that a1a2 is the best bound for F(a1, a2) and a1a2 — a,1 — a2 for G(a1, a2). Otherwise only in very special cases have the best bounds been found, even for t = 3.In the present paper a symmetric expression is developed for the best bound for F(a1, a2, a3) which solves that problem and gives insight on the general problem for larger values of t. In addition, some relations are developed which may be of interest in themselves.
On Nilpotent Products of Cyclic Groups
Authors: Struik Ruth Rebekka 
Published: December 2010

In this paperG=F/Fnis studied forFa free product of a finite number of cyclic groups, andFnthe normal subgroup generated by commutators of weightn. The case ofn =4 is completely treated(F/F2is well known;F/F3is completely treated in (2)); special cases ofn >4 are studied; a partial conjecture is offered in regard to the unsolved cases. Forn= 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.
Intersection Irreducible Ideals of a Non-Commutative Principal Ideal Domain
Authors: Feller Edmund H. 
Published: December 2010

Let R always denote a fixed non-commutative principal ideal domain. A right (left) ideal aR (Ra) is termed right (left) ∩ irreducible provided it is not the intersection of two right (left) ideals that properly include it. In this case, the element a is called right (left) ∩ irreducible.Since R satisfies the A.C.C. for right ideals every right ideal aR can be written in the form aR = a,1R ∩ a2R ∩ … ∩ anR, where atR properly include aR and is right ∩ irreducible, i = 1,2, … ,n. We shall investigate properties (including primary properties) of right ∩ irreducible one-sided and two-sided ideals of R. These properties will depend on the results given in (1) and (2, chapter III).An element a is irreducible if it is not zero or a unit and has no proper factors. In this case aR (Ra) is a maximal right (left) ideal.
Continuous Translation of Hölder and Lipschitz Functions
Authors: Mirkil H. 
Published: December 2010

All functions will be complex, periodic, integrable (on [0,2π]) functions of a real variablex. Moreover, we shall require that every function have mean zero on [0,2π], so that in particular non-zero constants are excluded.1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f ofbounded variationon [0,2π]. These f form a Banach spaceFwhen each f is normed by its total variation on [0,2π]. And translations define a natural one-parameter group of isometries onF.
On Lie Semi-Groups
Authors: Langlands R. P. 
Published: December 2010

Suppose we have a semi-group structure defined ona subset of real Euclidean n-space, En, by (p, q) → F (p, q) = poq. In this note we shall be concerned with a representation T(.) of π as a semi-group of bounded linear operators on a Banach space 𝒳. More particularly, we suppose that postulates P1, P2, P3, P5 and P6 of chapter 25 of (2) are satisfied so that, by Theorem 25.3.1 of that book, there is a continuous function, f(.), defined on π such that f((ρ + σ)a) = f(ρa)o f(σa) for a ∈ π, ρ,σ ≥ 0; that the representation is strongly continuous in a neighbourhood of the origin and that T(0) = I.
The Expression of Trigonometrical Series in Fourier Form
Authors: Cross George 
Published: December 2010

In a paper published in 1936 Burkill (2) proved that, if the trigonometrical series1.1is bounded except on a countable set and if the series obtained by integrating series (1.1) once converges everywhere, then the coefficients can be written in Fourier form using the C1P-integral. In §3 of this paper an analogous result is shown to be true when (1.1) is bounded (C, k), k < 0. The proof of this depends on generalizations of theorems by Verblunsky and Zygmund and both of these generalizations are obtained in §2.
The Analytic Continuation of the Riemannliouville Integral in the Hyperbolic Case
Authors: Riesz Marcel 
Published: December 2010

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.
Integration of Subspaces Derived from a Linear Transformation Field
Authors: Kobayashi Edward T. 
Published: December 2010

The problem we study is a generalization of a problem first solved by Tonolo (6), then generalized successively by Schouten (5), Nijenhuis (4), Haantjes (3), and Nijenhuis-Frölicher (2). The Tonolo- Schouten approach is distinct from that of Nijenhuis-Haantjes-Frölicher in the sense that the former consider the problem on a Riemannian space, while the latter consider it on a manifold without any further structure.The object of investigation is the integrability of the distribution θ of vector subspaces θP of the tangent space Tp to a manifold M, when θP is intrinsically related to a given field h on M, of linear transformations hp on Tv. The research has so far been restricted to certain types of h.
On the Structure of Semi-Prime Rings and their Rings of Quotients
Authors: Lambek Joachim 
Published: December 2010

We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.
Sur La Representation Integrale Des Certaines Operations Lineaires. IV Operations Lineaires Sur L'espace Lap
Authors: Dinculeanu N. , Foias C. 
Published: December 2010

La première partie de cet article a un caractère introductif; on reprend l'étude des mesures vectorielles sur un espace localement compact, présenté dans (8) pour les espaces compacts.Dans la seconde partie on donne un théorème de représentation intégrale d'une mesure vectorielle par rapport à sa variation (théorème 2), duquel on déduit une généralisation du théorème de Lebesgue-Nikodym (théorème 3) et une généralisation du théorème de Lebesgue sur la décomposition d'une mesure (théorème 5). Ces théorèmes ont été déjà démontrés par les auteurs sous des conditions un peu plus restrictives, (10) et (13), mais ici les démonstrations sont nouvelles, et l'ordre dont on les déduit l'un de l'autre, est inversée.
Study of Certain Similitudes
Authors: Wonenburger Maria J. 
Published: December 2010

One important point in the determination of the automorphisms of the classical groups is the study of group-theoretic properties of the elements of order 2, that is, the involutions. The group of similitudes and the projective group of similitudes of a non-degenerate quadratic form Q are extensions of the orthogonal group and the projective orthogonal group, respectively. These extended groups may contain involutions which do not belong to the orthogonal group or the projective orthogonal group. To study the automorphisms of such groups, group-theoretic properties of some of these new involutions should be established.The aim of this paper is to give some properties of a similitude T of ratio ρ whose square T2 is equal to the scalar multiplication by — p (it is always assumed that we are dealing with a field of characteristic ≠2). If ρ = — 1, T is an involution in the group of similitudes and for any ρ the coset of T in the projective group of similitudes is an involution.
The Permanent Function
Authors: Marcus Marvin , May F. C. 
Published: December 2010

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by1.1where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.
On the Measure of Sets of Parallel Linear Subspaces in Affine Space
Authors: Santaló L. A. 
Published: December 2010

LetEnbe then-dimensional euclidean real space andthe group of unimodular affine transformations which operates on it. It is known that the sets of linearh-spacesLh(0 <h < n) have no invariant measure with respect tou(5). We wish now to consider sets of elements1.1composed byqparallel subspaces of dimensionsh1, h2, … ,hqwhich transform transitively by.
An Integral Over Function Space
Authors: Eberlein W. F. 
Published: December 2010

Real functionsmay be identified with elements x = (x0, x1, x2, … ) of the sequence space l1 Since the unit sphere S∞ of l1 is compact under the weak* topology = topology of co-ordinatewise convergence, a countably additive measure on S∞ is induced by a positive linear functional E (integral) on C(S∞), the weak* continuous real-valued functions on S∞. There exists a natural integral over S∞ reducing towhen f is a function of x0 alone. The partial sums Sn = Sn(x) of the power series for x(t) then form a martingale and zero-or-one phenomena appear. In particular, if R(x) is the radius of convergence of the series and e is the base of the natural logarithms, it turns out that R(x) = e for almost all x in S∞. Applications of the integral to the theory of numerical integration, the original motivation, will appear in a later paper.
On a Class of Finitely Presented Groups
Authors: Macdonald I. D. 
Published: December 2010

The groups in question are generated by elements A and B subject to the relationsin which α and β are fixed integers. We prove:Theorem. Each group of the class just presented is finite when neither α nor β equals 1, and is nilpotent. Its order is a factor of 27 (α — 1) (β — l)∈8where ∈ is the greatest common divisor of α — 1 and β — 1, and its nilpotency class is at most 8.We denote the commutator X-1Y-1XY by [X, Y], and Y-1XY by XY. If X1, X2, … , Xr are elements of some group then {X1, X2, … , Xr} will mean the subgroup which they generate.
Fractional Integration and Dual Integral Equations
Authors: Erdélyi A. , Sneddon I. N. 
Published: December 2010

In the analysis of mixed boundary value problems by the use of Hankel transforms we often encounter pairs of dual integral equations which can be written in the symmetrical form(1.1)Equations of this type seem to have been formulated first by Weber in his paper (1) in which he derives (by inspection) the solution for the case in which α — β = ½, v = 0, F ≡ 1, G ≡ 0.The first direct solution of a pair of equations of this type was given by Beltrami (2) for the same values of α— β and v with G(p) ≡ 0 but with F(ρ) arbitrary.
A Census of Slicings
Authors: Tutte W. T. 
Published: December 2010

A band is a closed connected set in the 2-sphere, bounded by one or more disjoint simple closed curves.Consider a band B with bounding curves J1, J2, … , Jk. On each curve Ji let there be chosen mi ≥ 0 points to be called vertices, with the restriction that the sum of the k integers mi is to be even. Write(1)Next consider a set of n disjoint open arcs in the interior of B which join the 2n vertices in pairs and partition the remainder of the interior of B into simply connected domains. We call the resulting dissection of B a slicing with respect to the given set of vertices. The arcs are the internal edges of the slicing and the simply connected domains are its internal faces, or slices.
The Distribution of Sequences and Summability
Authors: Dowidar A. F. , Petersen G. M. 
Published: December 2010

We suppose that 0 <sn≤ 1 for everyn, and denote byn(α, β) the number ofS0, S1,S2, . . . ,Snwhich fall in the interval 0 ≤ α <x≤ β ≤ 1. If there exists a functiong(t), 0 ≤t≤ 1, such thatfor every interval (α, β] with 0 ≤ β — α ≤ 1, the sequence (Sn) is said to have a distribution functiong(t), 0 ≤t≤ 1, in the interval [0, 1], (see 9, p. 87).
Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators
Authors: Mitchell Josephine 
Published: December 2010

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by
Enumeration of Bicolourable Graphs
Authors: Harary Frank , Prins Geert 
Published: December 2010

In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.
Motions of Matrix Rings
Authors: Keller Roy F. 
Published: December 2010

Metric spaces in which the distances are not real numbers have been studied by several people (2, 3, 4, 7, 9). Any ringRtogether with a mapping,X—>ϕ(X), ofRinto a lattice A with 0 and 1 satisfyingis called a "lattice-valued ring," where the operations union, ∪, and intersection, ∩, are the usual lattice operations. The mapping ϕ is called a "valuation" andAis a 'Valuation lattice." IfRis a lattice-valued ring and a mappingdis defined by
Convolutions as Bilinear and Linear Operators
Authors: Edwards R. E. 
Published: December 2010

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].
Linear Relations Between Higher Matrix Commutators
Authors: Khan Nisar A. 
Published: December 2010

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, letdefine the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation
Arcs of Parabolic Order Four
Authors: Lane N. D. 
Published: December 2010

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.
On Two Problems Concerning the Generalized Lototsky Transforms
Authors: Meir Amram 
Published: December 2010

The generalized Lototsky transform (or the [F, dn] transform) of a sequence {sn} into a sequence {tn) was defined in (2) in the following way: Let {dn) (dn ≠ —1) be a fixed complex sequence and letthen the sequence {tn} is defined byIn a recent paper by V. F. Cowling and C. L. Miracle (1) on these transformations, the following two problems were left open.