bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Geometrical Approach to the Second-Order Linear Differential Equation
Authors: Petty C. M. , Barry J. E. 
Published: December 2010

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.
Approximate Continuity and Differentiation
Authors: Sion Maurice 
Published: December 2010

The relation between the notions of measurability and continuity of a function has received a great deal of attention. The best-known result in this connection is the Vitali-Lusin theorem. Various versions of it can be found in (2; 3; 5; 8; 9). We prove one in this paper (Theorem 3.5) under very weak assumptions and state the classical one in Corollary 3.6. Functions satisfying the property stated in the theorem are frequently called quasicontinuous.Here we are interested in the notion of approximate continuity, which we call μ-continuity for short and which is closely connected to differentiation. It was first introduced by Lebesgue (4) and it has so far required knowledge of density theorems in order to prove its relation to measurability. In this paper, making use of a certain type of Vitali property, we prove first the relation between μ-measurable functions and those μ-continuous almost everywhere (Theorems 3.8 and 3.9). Then, in §4, with Theorem 3.8 as the main tool, we derive several theorems about density and differentiation which extend results found in (1; 3; 7; 8).
Connectivity and Reducibility of Graphs
Authors: Johnson Diane M. , Dulmage A. L. , Mendelsohn N. S. 
Published: December 2010

Corresponding to every graph, bipartite graph, or directed bipartite graph there exists a directed graph which is connected if and only if the original graph is connected.In this paper, it is shown that for every directed graph there exists a certain bipartite graph such that the directed graph is connected if and only if the bipartite graph is irreducible. Other connections between reducibility and connectivity are established.
Trace Forms on Lie Algebras
Authors: Block Richard 
Published: December 2010

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient
trace form if and only if there exists a Lie algebra L with a representation Δ such that
The General Static Spherically Symmetric Solution of the "Weak" Unified Field Equations
Authors: Vanstone J. R. 
Published: December 2010

In 1947 Einstein and Strauss (2) proposed a unified field theory based on a four-dimensional manifold characterized by a nonsymmetric tensor gij and a non-symmetric connection , where(1)Using a variational principle in which gij and are independently varied, the above authors obtain the equivalent of the following field equations:(2a)(2b)In these equations a comma denotes partial differentiation with respect to the co-ordinates of the manifold, Wij is the Ricci tensor formed from and the notationfor the symmetric and skew-symmetric parts of geometric objects Q is employed.
Some New Difference Sets
Authors: Gordon Basil , Mills W. H. , Welch L. R. 
Published: December 2010

A difference set is a set D = {d1, d2, … , dk] of k distinct residues modulo v such that each non-zero residue occurs the same number of times among the k(k — 1) differences di — dj, i ≠ j. If λ is the number of times each difference occurs, then(1)When we wish to emphasize the particular values of v, k, and λ involved we will call such a set a (v, k, λ) difference set. Another (v, k, λ) difference set E = {e1, e2, … ek} is said to be equivalent to the original one if there exist a and t such that (t, v) = 1 and E = {a + td1, … , a + tdk}. If t = 1 we will call the set E a slide of the set D. If D = E, then t is called a multiplier of D.
Asymptotic Behaviour of Solutions of Parabolic Differential Inequalities
Authors: Lees Milton 
Published: December 2010

Let there be given a parabolic differential operatorwhere A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality(1.1)Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.
Osculating Spaces
Authors: Scherk Peter 
Published: December 2010

In this paper an attempt is made to prove some of the basic theorems on the osculating spaces of a curve under minimum assumptions. The natural approach seems to be the projective one. A duality yields the corresponding results for the characteristic spaces of a family of hyperplanes. A duality theorem for such a family and its characteristic curve also is proved. Finally the results are applied to osculating hyperspheres of curves in a conformai space.The analytical tools are collected in the first three sections. Some of them may be of independent interest.
Integral Bases for Quadratic Forms
Authors: Chalk J. H. H. 
Published: December 2010

Letbe an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :
On the Sum of the Reciprocals of the Fermat Numbers and Related Irrationalities
Authors: Golomb Solomon W. 
Published: December 2010

In (1), P. Erdös showed that the functiontakes on irrational values whenever z = 1/t, t = 2 , 3 , 4 , 5 , . . . . The method of proof uses Lambert's identity,where d(n) is the number of divisors of n; and it is shown that
The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes
Authors: Foulser David A. 
Published: December 2010

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).
Connections Satisfying a Generalized Ricci Lemma
Authors: Vanstone J. R. 
Published: December 2010

In this paper we shall consider a generalization of a very old problem in differential geometry; namely, given a second-order covariant tensor field aij(x) on an n-dimensional manifold, when does there exist a connection such that the covariant derivative, defined byvanishes?The earliest question of this type arose in the case when is symmetric and positive definite. A solution connection of the problem is then given by the Christoffel symbols
On the Enumeration of Rooted Non-Separable Planar Maps
Authors: Brown W. G. , Tutte W. T. 
Published: December 2010

It has been shown elsewhere (1, 4) that the number of rooted non-separable planar maps with n edges isIn the present paper we improve upon this result by finding the number fi,j of rooted non-separable planar maps with i + 1 vertices and j + 1 faces. We use the definitions of (1).Among the non-separable planar maps only the loop-map and the link-map have i = 0 or j = 0. We therefore confine our attention to the case in which i and j are both positive.
Summability- (Z, p) and Sequences of Periodic Type
Authors: Hill J. D. , Sledd W. T. 
Published: December 2010

We shall say that the sequence x = {sk} is summable- (Z, p) to the value s ifwhere p is a positive integer andIgnoring the values of n, 1 — p ≤ n < 0, which are clearly irrelevant, the transformation (Z, p) coincides with the regular Nörlund transformation defined by the sequence ( 1 , 1 , . . . , 1, 0, 0 , . . .) containing p initial 1's. This class of methods was first studied systematically by Silverman and Szász (8).
A Property of a Triangle Inscribed in a Convex Curve
Authors: Zirakzadeh A. 
Published: December 2010

The purpose of this paper is to prove the following theorem:Theorem. Given a convex curve C, of perimeter length I and three points M, N, and P which divide the perimeter of C into three parts of equal length, the perimeter length of the triangle MNP is never less than ½l. Equality holds only in the case where C is an equilateral triangle and M, N, and P are the mid-points of the three sides.
Sur Une Famille de Groupes de Permutations Doublement Transitifs
Authors: Ree Rimhak 
Published: December 2010

Dans (6), l'auteur a défini, pour chaque entier n ≥ 0, un groupe fini Gn d'ordre q3(q —1)(q3 + 1), où q = 32n+1. Les groupes G1, G2, . . . sont simples, tandis que est simple.En réalisant G = Gn comme un groupe de permutations, opérant à droite, de l'ensemble de q3 + 1 classes à droite modulo le normalisateur N(P) d'un 3-sous-groupe de Sylow de G, on voit aisément que G satisfait aux conditions (0.1)-(0.4) suivantes :(0.1)G est un groupe de permutations doublement transitif d'un ensemble E de m + 1 lettres, où m ≥ 3;(0.2)si a et b sont deux lettres distinctes quelconques de E, le sous-groupe formé de la totalité des permutations dans G qui laissent invariantes a et b contient une, et une seule, permutation ≠ 1 qui laisse au moins 3 lettres invariantes;(0.3)toute involution dans G laisse au moins 3 lettres invariantes (une involution dans un group est par définition un élément d'ordre 2);(0.4)m est impair.
A Property of Two Chords which Divide a Convex Curve into Four Arcs of Equal Length
Authors: Eggleston H. G. , Zirakzadeh A. 
Published: December 2010

It will be shown in this paper that if two chords of a closed plane convex curve θ divide θ into four arcs of equal length and intersect inside the domain bounded by θ, then the sum of the lengths of the two chords is at least equal to (√5 — 2)½ times the length of θ. We shall show firstly that we need only consider the case when θ is a convex (possibly degenerate) quadrilateral and then prove the result in this case.This result is related to a conjecture of P. Ungar in which the chords are assumed to be perpendicular and the factor (√5 — 2)½ is replaced by §. But Ungar's conjecture is neither proved nor disproved by this result.
On the Total Area of the Faces Of a Four-Dimensional Polytope
Authors: Fejes Tóth L. 
Published: December 2010

Let L be the total length of the edges of a convex polyhedron containing a unit sphere. If the number of edges is small, the edges must be, on the average, comparatively long. If, on the other hand, the edges are short, their number must be great. So the problem arises to find a polyhedron with a possibly small value of L.By a simple argument the author (5; 6) proved that L > 20 and announced the conjecture that L ≥ 24 with equality only for the cube (of in-radius 1). The same argument shows that for trigonal-faced polyhedra L > 28. This supports the conjecture that for such polyhedra L ≥ 12√6 = 29.4 . . . , with equality only for the tetrahedron and octahedron.
An Extension of the Class of Alternative Rings
Authors: Outcalt D. L. 
Published: December 2010

Define R to be a ring of characteristic not 2 or 3 satisfying the identity(1) (x,y,z) = (y,z,x)for all x, y, z ∊ R, where by characteristic not p is meant x —> px is a one-toone mapping of R upon R. The associator (a, b, c) of R is defined by (a, b, c) = ab·c — a·bc. lf R also satisfies the identity(2) (x, x, x) = 0
Ideal Decompositions in Noetherian Rings
Authors: Barnes Wilfred E. , Cunnea William M. 
Published: December 2010

An interesting identity is obtained for ideals A and B in a Noetherian ring :A = (A + Bn) ∩ (A : Bn)for sufficiently large n. This identity is applied to obtain Fuchs' quasi-primary decomposition of A in an improved form, and to obtain Krull's theorem on the intersection of the powers of A, both developments making no use of the Noetherian primary decomposition of A. Finally, the identity is used to obtain the primary decomposition without reference to irreducible ideals, in a largely constructive manner which yields the decomposition in an illuminating, automatically normal form, and which, subject to certain simple conditions, is unique.
The Genus of the n-Cube
Authors: Beineke Lowell W. , Harary Frank 
Published: December 2010

The definition of the genus γ(G) of a graph G is very well known (König
2): it is the minimum genus among all orientable surfaces in which G can be
drawn without intersections of its edges. But there are very few graphs whose
genus is known. The purpose of this note is to answer this question for one
family of graphs by determining the genus of the n-cube.The graph Qn called the n-cube has 2n vertices each of which is a binary
sequence a1a2. . . an of length n, where ai = 0 or 1.
On Measures of Symmetry of Convex Bodies
Authors: Chakerian G. D. , Stein S. K. 
Published: December 2010

Let K be a convex body (compact, convex set with interior points) in n-dimensional Euclidean space En, and let V(K) denote the volume of K. Let K′ be a centrally symmetric body of maximum volume contained in K (in fact, K′ is unique; see 2 or 9), and definec(K) = V(K′)/V(K)Letc(n) = inf{c(K) : K ⊂ En}.
Hereditary Radicals in Associative and Alternative Rings
Authors: Anderson T. , Divinsky N. , Suliński A. 
Published: December 2010

In the first part of this paper we shall consider associative rings, pointing out where associativity is required. In the second part we shall consider not necessarily associative rings and in particular alternative rings.A property S of rings is said to be a radical property, in the sense of Kurosh (4), if it satisfies the following three conditions:(a) Every homomorphic image of an S-ring (i.e. a ring with property S) is again an S-ring.
On Tchebycheff Quadrature
Authors: Erdös Paul , Sharma A. 
Published: December 2010

Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula(1)is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).
Colouring of Trivalent Polyhedra
Authors: Kotzig Anton 
Published: December 2010

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?
The Construction of Fields with Infinite Cyclic Automorphism Group
Authors: Kuyk Willem 
Published: December 2010

This paper deals with a problem raised in a paper by J. de Groot (1): Do there exist fields Ω whose full automorphism group is isomorphic to the additive group of integers Z?The answer to this question is yes. In this paper we construct, given any subfield k of the complex numbers, extension fields Ω of k such that the automorphism group G(Ω/k) of Ω with respect to k is infinite cyclic. Fields having the infinite cyclic group as a full group of automorphisms are obtained by choosing the base field k in such a way that it does not contain any subfield k0 so that k possesses non-trivial automorphisms leaving k0 pointwise fixed.
Regular Rank Rings
Authors: Halperin Israel 
Published: December 2010

1.1. Throughout this note, will denote an associative ring but we shall not require to possess a unit.If A and B are subsets of , then A + B will denote the set {x + y| x ∊ A, y ∊ B}. Aτ will denote the set {u ∊ | au = 0 for all a ∊ A} .Elements a and b will be said to be orthogonal if ab = ba = 0.
A Note on Well-Distributed Sequences
Authors: Murdoch B. H. 
Published: December 2010

A sequence is said to be well distributed (mod 1) (3, 4, 5) if the limit(1.1)exists, uniformly in p ≥ 0, for all intervals I in [0, 1], with length |I|, characteristic function where (x) is the fractional part of x. If (1.1) is true for p = 0 and all I in [0, 1] we say that is uniformly distributed (mod 1).
Affine Subplanes of Finite Projective Planes
Authors: Rigby J. F. 
Published: December 2010

Letπbe a finite projective plane of orderncontaining a finite projective subplane π* of orderu<n. Bruck has shown (1, p. 398) that ifπcontains a point that does not lie on any line of π*, thenn≥u2+u, while if every point ofπlies on a line of π* thenn=u2.Letπbe a finite projective plane of orderncontaining a finiteaffinesubplane π0of orderm<n.
Two Theorems on Mosaics
Authors: Gordon B. , Robertson M. M. 
Published: December 2010

The concept of a mosaic was recently introduced by A. A. Mullin (1). By the fundamental theorem of arithmetic, every integer n > 1 can be uniquely expressed in the formwhere the pi are primes satisfying p1 < p2 < . . . < pr. We then express any exponents aj which are greater than unity in the same manner, and continue in this way until the process terminates. The resulting planar configuration of primes is called the mosaic of n.
A Normal form for Riemann Matrices
Authors: Albert A. A. 
Published: December 2010

A matrix ω having p rows and 2p columns of complex number elements is called a Riemann matrix of genus p if there exists a rational 2p-rowed skew matrix C such that(1)is positive definite Hermitian. The matrix C is then called a principal matrix of ω. If ω and ω0 are two Riemann matrices of the same genus, we say that ω and ω0 are isomorphic if there exists a non-singular p-rowed complex matrix α, and a non-singular 2p-rowed rational matrix A, such that(2)
On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, II: Rectangular Matrices
Authors: Majindar Kulendra N. 
Published: December 2010

In this paper we establish a connection between rectangular integer matrices and incidence matrices of resolvable balanced incomplete block designs. The definition of these terms has been given in paper I of this series.Our theorem can be stated as follows:THEOREM 2. Let A be a v X b matrix with integer elements such that2.1
On Ingham's Summation Method
Authors: Segal S. L. 
Published: December 2010

Ingham (2) has defined the following summation method. A series ∑ an will be called summable (I) to s ifwhere as usual [x] is the greatest integer ⩽ x. (An equivalent method was described somewhat earlier by Wintner (7), who called it “an Eratosthenian method“; however, the notation (I) and the name “Ingham summability” introduced by Hardy (1) seem to have become usual.)
Convex Polyhedra with Regular Faces
Authors: Johnson Norman W. 
Published: December 2010

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.
Isotopy of 2-Dimensional Cones
Authors: Copeland Arthur H. 
Published: December 2010

Knowing the isotopy of cones is a crucial first step in knowing the isotopy of finitely triangulable spaces, for the cones are exactly the stars of vertices. Furthermore, they are the simplest examples of contractible spaces, and the non-triviality of the contractible spaces is one of the distinguishing characteristics of isotopy theory as contrasted with homotopy theory.The present paper is concerned with the cones over 1-dimensional finitely triangulable spaces. It is clear that homeomorphic spaces have homeomorphic cones, hence cones of the same isotopy type. The surprising result of §2 is that there are very few exceptions to the converse statement. The exceptional isotopy classes of cones all contain cones over spaces that are themselves cones.
On the Continuity and Self-Injectivity of a Complete Regular Ring
Authors: Utumi Yuzo 
Published: December 2010

Let S be a ring, and let (ei) be an orthogonal system of a finite number of idempotents. Then e = Σei has the following properties:(i) Se Σ Sei and eS = Σ ei S.(ii) The mappings v: Se → Π Sei and w: eS → Π ei S defined by v(x) = [xei] and w(x) = [ei x] respectively are isomorphisms.Next assume that (ei)i∈I is a set of idempotents indexed by a totally ordered set I such that ei ej = 0 for every i < j. If I is finite, it is evident thathas the above two properties.
Pairs of Bilinear Equations in a Finite Field
Authors: Duane Porter A. 
Published: December 2010

Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations1.1where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.
Inequalities for the Permanental Minors of Non-Negative Matrices
Authors: Brualdi R. A. , Newman M. 
Published: December 2010

Let A be an n X n non-negative matrix, that is, a matrix whose entries are non-negative numbers. The permanent of A is the scalarvalued function of A defined bywhere the summation extends over all permutations i1 … , in of the integers 1, … , n. The purpose of this paper is to prove several inequalities involving the permanent of A and the permanent of submatrices of A when suitable restrictions are placed on the row sums.
A Note on Convergence Fields
Authors: Berg I. D. 
Published: December 2010

The purpose of this note is to show that the (bounded) convergence field of a conservative matrix is closed under a certain diagonalization procedure.As an application of the above result we establish a conjecture of Hill and Sledd in (1) and obtain a result of Lorentz originally proved in (2).First we introduce some notation and definitions, most of which are standard. Let l∞ denote the Banach space of bounded sequences with the supremum norm and let c denote the closed subspace of l∞ consisting of convergent sequences.