bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Valence of Sums and Products
Authors: Goodman A. W. 
Published: December 2010

A function ƒ(z) is said to be p-valent in a region if it is regular in if the equation1has p distinct roots in for some particular w0, and if for each complex w0, equation (1) does not have more than p roots in . The function ƒ(z) is also said to have valence p in . In the case when p = 1, the function is said to be univalent in .
Almost Convergence and Well-Distributed Sequences
Authors: Zame Alan 
Published: December 2010

A sequence (xn) of real numbers is said to be well-distributed modulo 1 (abbreviated w.d.) if for each subinterval I = [a, b] of [0, 1] we have thatwhere χI is the characteristic function of I modulo 1. A sequence (rn) of positive numbers is lacunary if
The Measure Algebra as an Operator Algebra
Authors: Ramirez Donald E. 
Published: December 2010

In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G).Let △ denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of △ in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative B*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space.
A Presentation of the Groups PSL(2, p)
Authors: Behr H. , Mennicke J. 
Published: December 2010

In the present paper, we shall prove the following result.Theorem A. The groups PSL(2, p) can be presented by the following system of generators and relations:1.1This theorem considerably improves earlier results of Bussey, Frasch, and Todd (cf. 2, pp. 93-96). The presentation of Frasch reads as follows:1.2
Two-Dimensional Linear Groups Over Local Rings
Authors: Lacroix Norbert H. J. 
Published: December 2010

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .
Normal Completions of Small Categories
Authors: Kennison J. F. 
Published: December 2010

In (3), Isbell proposed a stronger definition for the term “complete category” and obtained many nice theorems for the resulting notion of a completion. In particular, he showed (3, Theorem 3.20) that completions of small categories satisfy a strong normality condition.In this paper we shall always use the term “complete” in the weaker sense of Freyd (1). (In (3), Isbell used the term “small-complete” for this weaker notion.) We shall prove that the completions, in the sense of Freyd, of small categories also enjoy the same normality condition, provided they admit at least one bicategory structure. (The complete categories in the sense of Isbell always admit bicategory structures; see the remark following Proposition 2.4.)In what follows, we let mean that is a full subcategory of . Moreover, if , then means that each object of is equivalent to an object in .
Decomposition of Metric Spaces with a 0-Dimensional Set of Non-Degenerate Elements
Authors: Lamoreaux Jack W. 
Published: December 2010

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.
Sur les Ensembles de Sommets Indépendants Dans les Graphes Chromatiques Minimaux
Authors: Brown W. G. , Moon J. W. 
Published: December 2010

f(n, k) sera l'entier maximum t tel qu'il existe un graphe G ayant les propriétés suivantes:(a) G possède n sommets;(b) le nombre chromatique de G est égal à k;(c) Gest minimal par rapport à la propriété (b); c'est-à-dire, la suppression ede n'importe quelle arête rend G(k – 1)-colorable;(d) il existe t sommets indépendants de G, c'est-à-dire dont nulle paire ne se joigne par une arête.Un graphe sera k-minimal s'il possède les propriétés (b) et (c). Puisque les graphes 3-minimaux sont tous des polygones impairs, il s'ensuit que f(n, 3) = [n/2] (n = 3, 5, 7, …), Il y a quelque temps T. Gallai a posé la conjecture:1.1M. Simonovits a réfuté l'inégalité stricte dans (1.1) en prouvant que1.2
Two Results Concerning the Zeros of Functions with Finite Dirichlet Integral
Authors: Caughran James G. 
Published: December 2010

A function f, analytic in the unit disk, is said to have finite Dirichlet integral if1Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by . The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn, then if and only if Σn|an|2 < ∞. Thus, is contained in the Hardy class H2.In particular, every such function has boundary valuesalmost everywhere and log |f(eiθ)| ∊ L1(dθ).The zeros zn of a function must satisfy the Blaschke conditionand f(s) = B(z)F(z), where F(z) has no zeros andis the Blaschke product with zeros zn; see (5).
Finite Groups the Centralizers of whose involutions have Normal 2-Complements
Authors: Gorenstein Daniel 
Published: December 2010

In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity,
we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition.As is well-known, the automorphism group G = PΓL(2, q) of H= PSL(2, q), q= pn, is of the form G = LF, where L = PGL(2, q), L ⨞ G, F is cyclic of order n, L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q) acts; cf. (3, Lemma 2.1) or (7, Lemma 3.3). It follows at once (4, Lemma 2.1; 8, Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.
On the Bernstein-Szegö Theorem for Complex Polynomials
Authors: Brannan D. A. , Pommerenke Ch. 
Published: December 2010

Let p(z) be a complex polynomial of degree less than or equal to n. Generalizing the well-known Bernstein theorem, Szegö (3) has shown thatWe shall give a partial generalization of this result.THEOREM. Let p(z) be a polynomial of degree at most n. Let R be the radius of the largest disc contained in G = {p(z): |z| < 1}. ThenSince R ≦ max|2|=1 |Re p(z)|, we obtain Szegö's result, but with a worse constant. It would be interesting to see whether it is possible to replace the constant e by 1. If so, Rzn would be an extremal for all n, and another extremal for even n.The proof is based on the following result of Ahlfors (1) (cf., e.g., 2, p. 321). This result corresponds to the estimate f or the Landau constant.
On Order Paracompact Spaces
Authors: McCandless Byron H. 
Published: December 2010

In (2), Fitzpatrick and Ford defined a class of spaces which they called order totally paracompact. This class of spaces is significant since the order totally paracompact metric spaces constitute the largest known class of spaces for which the small and large inductive dimensions coincide. In this paper we shall consider a class of spaces containing the class of order totally paracompact spaces; we shall call these spaces order paracompact. We shall establish the relationship of this class to the more familiar classes of spaces and obtain an invariance theorem for order paracompact spaces analogous to theorems obtained for various other classes.In what follows, no separation axioms are assumed unless specifically mentioned. Thus, for example, a Lindelöf space is not taken to be Hausdorff without making the further assumption. Therefore, a Lindelöf space is not necessarily paracompact.
Isometric Immersions of almost Hermitian Manifolds
Authors: Gray Alfred 
Published: December 2010

The Lefschetz theorem on hyperplane sections, as proved by Andreotti and Frankel (1), depends upon the following result.THEOREM. If M is a non-singular affine algebraic variety of real dimension 2k of complex n-space, thenThis theorem, which is interesting in itself, has been strengthened by Milnor (7), who showed that M has the homotopy type of a k-dimensional CW-complex.In this paper we generalize the above theorem in two directions. First, we replace complex n-space by some other complete simply connected Riemannian manifold which either has non-positive curvature or is a compact symmetric space. Secondly, we allow M and to be quasi-Kâhlerian (see below) instead of Kählerian.We first introduce some notation. Let M and be C∞ Riemannian manifolds with M isometrically immersed in . Denote by 〈, 〉 the metric tensor of either M or .
Galois Connections and Pair Algebras
Authors: Derderian J. C. 
Published: December 2010

Unless further restricted, P, Q, and R denote arbitrary partially ordered sets whose order relations are all written “≦” .An isotone mapping ϕ: P → Q is said to be residuated if there is an isotone mapping ψ: Q → P such that(RM 1) xϕψ ≧ x for all x i n P;(RM 2) yψϕ ≦ for all y in Q.Let Q* denote the partially ordered set with order relation dual to that of Q.(A) The following conditions are equivalent:(i) ϕ: P → Q* is a Galois connection;(ii) ϕ: P → Q is a residuated mapping;(iii) Max{z ∈ P: zy ≦ y} exists for all y in Q and is equal to yψ.Since ψ is uniquely determined by ϕ, it will be denoted by ϕ+.
The Exponents of Strongly Connected Graphs
Authors: Bender Edward A. , Tucker Thomas W. 
Published: December 2010

A directed graphG is a set of vertices V and a subset of V × V called the edges of G. A path in G of length k,is such that (vi, vi+1) is an edge of G for 1 ≦ i ≦ k. A directed graph G is strongly connected if there is a path from every vertex of G to every other vertex. A circuit is a path whose two end vertices are equal. An elementary circuit has no other equal vertices. See (1) for a fuller discussion.Let G be a finite, strongly connected, directed graph (fscdg). The kth power Gk of G is the directed graph with the same vertices as G and edges of the form (i, j) where G has a path of length k from i to j.
A Class of Compact Rigid 0-Dimensional Spaces
Authors: Lozier F. W. 
Published: December 2010

A topological space is called “rigid” if its autohomeomorphism group is trivial. In (1), de Groot and McDowell showed that there are rigid, 0- dimensional spaces of arbitrarily high cardinality but left open the question of whether or not there are compact,rigid, 0-dimensional spaces of arbitrarily high cardinality, pointing out that an affirmative answer implies the existence of arbitrarily large Boolean rings with trivial automorphism groups. In this paper we construct a class of rigid, 0-dimensional spaces X αof arbitrary infinite cardinality and show that their Stone-Cech compactifications βX αare also rigid, thus answering the above question affirmatively.I would like to thank S. W. Willard, J. R. Isbell, and the referee for their careful readings of preliminary versions of this paper.
A Note on Strongly Regular Function Algebras
Authors: Wilken Donald R. 
Published: December 2010

Let A be a uniformly closed subalgebra of C(X), the algebra of all complex-valued continuous functions on a compact Hausdorff space X. If A separates the points of X and contains the constant functions, A is called a function algebra. The algebra A is said to be strongly regular on X if it has the following property.Property. For each f in A, each point x in X, and every , there is a neighbourhood U of x and a function g in A with g(y) = f(x) for all y in U and for all y in X.That is, each function in A is uniformly approximate on X by functions in A which are constant near any point of X. Stated in terms of ideals, strong regularity means that, for each x, the ideal of functions vanishing in a neighbourhood of x is uniformly dense in the maximal ideal at x.
Extremal Point and Edge Sets in n-Graphs
Authors: Sauer N. 
Published: December 2010

A set of points (edges) of a graph is independent if no two distinct members of the set are adjacent. Gallai (1) observed that, if A0 (B0) is the minimum number of points (edges) of a finite graph covering all the edges (points) and A1 (B1) is the maximum number of independent points (edges), then:holds, where m is the number of points of the graph.The concepts of independence and covering are generalized in various ways for n-graphs. In this paper we establish certain connections between the corresponding extreme numbers analogous to the above result of Gallai.Ray-Chaudhuri considered (2) independence and covering problems in n-graphs and determined algorithms for finding the minimal cover and some associated numbers.
The Coarseness of the Complete Bipartite Graph
Authors: Beineke Lowell W. , Guy Richard K. 
Published: December 2010

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar graphs whose union is G. The coarseness of the complete graph has been investigated elsewhere (1; 2). We consider the coarseness of the complete bipartite, or 2-coloured, graph, Km,n, consisting of sets of mand nvertices, each member of one set being joined by an edge to each member of the other. No members of one set are joined to each other.Our results are summarized in the following theorem, where square brackets denote “integer part”.THEOREM. If m= 3p + d, 0 ≦ d≦ 2, and n = 3q + e, 0 ≦ e ≦ 2, then for d = 0 or 1 and e = 0 or 1,1
Construction of Transverse Fields
Authors: Putz H. 
Published: December 2010

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold,Mn, in Euclidean space, which are sufficient to prove thatMnhas a transverse field (see 1.1 and 1.2, definitions).In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifoldMn.Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).After the research for this paper was completed, a paper of Davis (2) came to our attention.
Commutators of Certain Finitely Generated Soluble Groups
Authors: Rhemtulla A. H. 
Published: December 2010

Groups in which the commutator subgroup coincides with the set of commutators have been studied to a certain extent by several authors. It was shown in (2; 4; 6; 7) that various types of known simple groups have this property. In (3), Macdonald has considered certain soluble groups with this property, and Hall has shown that any group can be embedded as a subgroup of a simple group of this type. Here we shall be concerned with the class C of groups defined as follows.For any positive integer n, denote by Cn the class of all groups in which every element of the commutator subgroup can be expressed as a product of at most n commutators. It is not difficult to show that Cn is a proper subclass of Cn+1 for all n. Let so that a group G ∊ C if and only if G ∊ Cn for some n.
Groups with Metacyclic Sylow 2-Subgroups
Authors: Camina A. R. , Gagen T. M. 
Published: December 2010

A group S is said to be metacyclic if it contains a normal cyclic subgroup N such that S/N is cyclic. In this note the following theorem is proved.THEOREM. Let G be a group, S a metacyclic Sylow 2-subgroup of G. If S has a cyclic normal subgroup N such that S/N is cyclic of order greater than 2, then G is soluble.Remark. We show that such a group G contains a 2-nilpotent normal subgroup of index a divisor of 6. The solubility of these groups requires the solubility of groups of odd order unavoidably.Notation. All groups considered will be finite. Let G be a group, S a subset of G, A and B subgroups of G, N a normal subgroup of G.〈S〉: the subgroup of G generated by S.NG(S): the normalizer of S in G.CG(S): the centralizer of S in G.
Algebras of Holomorphic Functions in Ringed Spaces, I
Authors: Shauck Maxwell E. 
Published: December 2010

A pair () is a ringed space if it is a subsheaf of rings with 1 of the sheaf of germs of continuous functions on X. If U is an open subset of X, we denote the set of sections over U relative to by . If , then implies that there exists some open neighbourhood V of u, V ⊂ U, and some g continuous on V such that the germ of g at u, ug is ϕ(u). Now we define ϕ(u) (u) to be g(u) and in this way we obtain, in a unique fashion, a continuous complex-valued function on U. The collection of all such functions for a given set is denoted by and is called the -holomorphic functions on U.THEOREM. Let X be a locally connected Hausdorff space and () a ringed space.
Non-Isomorphic Non-Hyperfinite Factors
Authors: Ching Wai-Mee 
Published: December 2010

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.
A Skew Hadamard Matrix of Order 52
Authors: Blatt D. , Szekeres G. 
Published: December 2010

1. A Hadamard (H-) matrix H = (hij) of order n is an n × n square matrix satisfying the conditionsfor all i, j ≦ n. A skew H-matrix is an H-matrix of the formwhere I is the identity matrix and S’ the transpose of 5. In particular,Skew H-matrices have applications in the theory of finite projective planes (2) and tournaments (4), also in the construction of H-matrices of certain orders. For example, if there is a skew H-matrix of order n, then there is an H-matrix of order n(n – 1) (Williamson, see (1, p. 213)).
A Classification of Semi-Translation Planes
Authors: Johnson Norman Lloyd 
Published: December 2010

The classification of certain types of projective planes has recently been of considerable interest to both geometers and group theorists. Due in part to the current general interest in finite mathematics and the developments connecting group theory and finite geometry, the Lenz-Barlotti classification of finite projective planes (2; 10), in particular, has generated a tremendous amount of research. A great deal of this research has been related to the construction of non-Desarguesian planes.Fryxell (6), Hughes (7), Luneburg (11), and Ostrom (13; 15; 18) have found examples of projective planes, all of which are of a general type that we call semi-translation planes. Many of these planes are of the same Lenz-Barlotti class I-1. (The Lùneburg planes are translation planes. However, the planes dual to the Luneburg planes are semi-translation planes as well as dual translation planes.)
Unique Addition Rings
Authors: Stephenson W. 
Published: December 2010

A semigroup (R, ⋅) is said to be a unique addition ring (UA-ring) if there exists a unique binary operation + making (R, ⋅, + ) into a ring. All our results can be presented in this semigroup theoretic setting. However, we prefer the following equivalent ring theoretic formulation: a ring R is a UA-ring if and only if any semigroup isomorphism α: (R, ⋅) ≅ (S, ⋅) with another ring S is always a ring isomorphism.UA-rings have been studied in (8; 4) and are also touched on in (1; 2; 6; 7). In this note we generalize Rickart's methods to much wider classes of rings. In particular, we show that, for a ring R with a 1 and n ≧ 2, the (n × n) matrix ring over R and its subring of lower triangular matrices are UA-rings.
Partial Solution to Mackey's Problem about Modular Pairs and Completeness
Authors: Holland Samuel S. 
Published: December 2010

Two elements A, B of a lattice are said to form a modular pair when (X ∨ A) Λ B = X ∨ (A Λ B) holds for all X ≦ B, and are said to form a dual-modular pair when (X Λ A) ∨ B = X Λ (A ∨ B) holds for all X ≧ B.We are concerned here with a particular lattice, the lattice of closed subspaces of a normed linear space, and with a question posed by Mackey in 1945 (6, p. 206, problem 2), namely:“Are there any incomplete normed linear spaces in whose lattices of closed subspaces modularity and d-modularity are equivalent?”.The principal result of this paper is the following.
On Purely Inseparable Extensions of Unbounded Exponent
Authors: Haddix G. F. , Mordeson J. N. , Vinograde B. 
Published: December 2010

Let L/K be a field extension of characteristic p ≠ 0. If L/K is purely inseparable and of bounded exponent, then the property that L has a subbasis over K (11, p. 436) is of significance in the theory of higher derivations (11) and in the theory of Hopf algebras (9; 10). In this case, where L/K is of bounded exponent, it has been shown independently in (1; 9; 5) that L/K having a sub-basis is equivalent to the property that L/K is modular (9, p. 401). Our aim in this paper is to extend and apply these properties for L/K purely inseparable and of unbounded exponent.In Theorem 1, we give several conditions on a p-basis of L which are equivalent to the property that L/K has a sub-basis. In Theorem 2, we give a sequence of implications starting with L/K has a sub-basis.
Reconstruction of Trees
Authors: Manvel Bennet 
Published: December 2010

Every tree T determines a set of distinct maximal proper subtrees Ti = T — vi, which are obtained by the deletion of an endpoint of T. In this paper we prove that a tree is almost always uniquely determined by this set of its subtrees, and point out two interesting consequences of this result.In [5], Ulam proposed the following conjecture, which we state in a slightly stronger form due to Harary [1].ULAM'S CONJECTURE. A graph G with at least three points is uniquely determined up to isomorphism by the subgraphs Gi = G — vi.Kelly [4] proved the conjecture for trees and Harary and Palmer [3] showed that not all of the Gi are needed in that case by proving Corollary 1 below. If we remove from the list of subgraphs Gi of a graph G all but one graph of each isomorphism type, we obtain a set of Gi which are distinct up to isomorphism.