bluuebunny/crossref_metadata_2025 huggingface dataset:
Variety Invariants for Modular Lattices
Authors: Wille Rudolf
Published: December 2010

A variety (primitive class) is a class of abstract algebras which is closed under the formation of subalgebras, homomorphic images, and products. For a given variety we shall call a function μ*, which assigns to each algebra a natural number or ∞, denoted by μ*(A), a variety invariant if for every natural number n the class of all with μ*(A) ≦ n is again a variety. In this paper, a general method of finding variety invariants for the variety of all modular lattices will be developed. This method will be based on the concept of a quotient tree of a modular lattice. As examples of variety invariants we shall define, using the general result, the primitive length and the primitive width of modular lattices.
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 if

1

Geometrically, 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 values

almost everywhere and log |f(eiθ)| ∊ L1(dθ).

The zeros zn of a function must satisfy the Blaschke condition

and f(s) = B(z)F(z), where F(z) has no zeros and

is 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 that

We 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}. Then

Since 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.
Finite Direct Sums of Complete Matrix Rings over Perfect Completely Primary Rings
Authors: Courter Richard
Published: December 2010

The main theorem of this paper names seven ring classifications which coincide with the class of rings named in the title. Three of them are (1) rings over which every module is rationally complete, (2) left and right perfect rings over which every module is corationally complete, and (3) right perfect rings over which every right module is rationally complete. Corational completeness (introduced in this paper) and rational completeness are generalizations of projectivity and injectivity, respectively. One recalls that a proper subclass of the rings investigated in this paper, the artinian rings with zero radical, is known to have one-sided characterizations in terms of the injectivity (projectivity) of modules. An example proves, however, that the class of rings over which every right module is rationally complete properly contains the rings of the title.
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, then

This 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+.
Recurrent Transformation Groups
Authors: Christiansen R. A.
Published: December 2010

Let (X, T, π) denote a flow, whereXis a compact topological space metrizable byd, andTis a closed non-trivial subgroup of the reals under addition.Tisrecurrentif and only if for eachands> 0, there existst>ssuch thatx∈Ximplies. IfTis almost-periodic, thenTis both recurrent and distal. In§§4 and 5, it is shown that, under more stringent hypotheses, the recurrence ofTis neither a necessary nor a sufficient condition forTto be distal. LetSbe a closed non-trivial subgroup ofT. It is shown in§3 thatTis recurrent if and only ifSis recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).
Some Semigroups Having Quasi-Frobenius Algebras. II
Authors: Wenger R.
Published: December 2010

The investigation of finite semigroups S with quasi-Frobenius (q.-F.) algebras F(S) over a field F was begun in (7; 8). The problem for commutative semigroups was reduced (7, Theorem 3) to the study of semigroups of the form S = G ∪ S1, where G is a group and S1 is either the null set or is a nilpotent ideal in S (i.e., S1n = {0} for some positive integer n). Such semigroups were called “of type C”. The question is “When does a semigroup of type C have a q.-F. algebra over a field?” (7, Theorem 4) shows that no distinction need be made between the properties q.-F. and Frobenius for commutative algebras.
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.
Rings of Quotients of Group Rings
Authors: Burgess W. D.
Published: December 2010

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.
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
Zur Geometrie der Körpererweiterungen
Authors: Benz Walter
Published: December 2010

Seiein (nicht notwendig kommutativer) Körper, seiein (ebenfalls nicht notwendig kommutativer) echter Unterkörper von. Mitbezeichnen wir die projektive Gerade über, d.h. die Menge, wo ∞ ein neues (in die Betrachtung hereinkommendes) Element, genannt “unendlich”, darstellt. Ist entsprechend, so ist also(dasZeichen ⊂ bezeichne echtes Enthaltensein). Die Elemente vonnennen wirPunkte, die Punktmengendie projektive Gruppe von(s. § 1) darstellt, nennen wirKetten. Eine eineindeutige Abbildung der Menge der Punkte auf sich, die in beiden Richtungen Ketten in Ketten überführt, heiße eineKettenverwandtschaft.Diese Forderung, daß nämlich Bilder und Urbilder von Ketten wieder Ketten sind, kann (s. § 4) (wennim Zentrum vonliegt) reduziert werden so, daß nur Ketten in Ketten übergehen sollen. Dann liegt nämlich bereits eine Kettenverwandtschaft vor.
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 conditions

for all i, j ≦ n. A skew H-matrix is an H-matrix of the form

where 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)).
On Morita Duality
Authors: Bruno J. Mùller
Published: December 2010

A contra variant category-equivalence between categories of right R-modules and left S-modules (all rings have units, all modules are unitary) that contain RR, SS and are closed under submodules and factor modules, is naturally equivalent to a functor Horn (–, U) with a bimodule SUR such that SU, UR are injective cogenerators with S = End UR and R = End SU, and all modules in are [U-reflexive. Conversely, for any SSUR, Hom(–, U) is a contravariant category equivalence between the categories of [U-reflexive modules, and if U has the properties just stated, then these categories are closed under submodules, factor modules, and finite direct sums and contain RR, UR,SS, and SU. Such a functor will be called a (Morita) duality between R and S induced by U (see (5)).
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.
The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra
Authors: Hedrlín Z. , Mendelsohn E.
Published: December 2010

By a graph we mean a pair (X, R) where X is a non-void set and R ⊂ X × X. A mapping f: X → Y is called a compatible map (or morphism) from (X, R) into (Y, S) if 2f(R) ⊂ S, where 2f: X2 → Y2 is defined by 2f((x1, x2)) = (f(x1),f(x2)). The set of all compatible maps from (X, R) into itself forms a monoid (semigroup with a unit element) under composition, which is denoted by M(X, R). A graph (X1, R1) is said to be a full subgraph of (X, R) if X1 ⊂ X and R1 = R ∩ (X1 × X1). A graph (X, R) is said to be without loops if (x, x) ∉ R for all x ∈ X.
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.
Compactness in Hom(G, H)
Authors: Corson H. H. , Glicksberg I.
Published: December 2010

Let G be a locally compact abelian group with Bohr compactification Ga. Then [3, Theorem 1.2] any subset F of G compact in Ga is necessarily compact in G; alternatively, any closed non-compact subset F of G has its closure F– in Ga ≠ F; hence F –\F ≠ ø. One of our aims in the present note is to give a result (Corollary 6) which asserts that F –\F has no points which are Gδs, so that F–\F is a perfect set. Another aim is to give an extension of a cited result of [3] in which commutativity and local compactness are essentially irrelevant, and to unify the proofs.
On Absolute Summability by Riesz and Generalized Cesàro Means. I
Authors: Körle H.-H.
Published: December 2010

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.

We shall consider series

(1)

with complex terms an. Throughout, we will assume that

(2)

and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means

(of the partial sums of (1)) tend to s as x → ∞.
Singular Integrals are Perron Integrals of a Certain Type
Authors: Pfeffer W. F.
Published: December 2010

In [7] a Perron-like integral was denned in an arbitrary topological space and many of its basic properties were established. In this paper we shall show (the theorem in § 2) that in a suitable setting the integral from [7] includes a class of so-called singular integrals, i.e., generalized forms of the Cauchy principal value of an integral. Thus, the powerful machinery of Perron integration, e.g., the monotone and dominant convergence theorems, can be automatically applied to these singular integrals.
M-Primary Elements of a Local Noether Lattice
Authors: Johnson E. W. , Johnson J. A.
Published: December 2010

In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.
Characterizations of Spherical Neighbourhoods
Authors: Petty C. M. , Crotty J. M.
Published: December 2010

If Σ is a specified class of metric spaces and M ∈ Σ, then the characterization problem is to find necessary and sufficient conditions which distinguish the spherical neighbourhoods (open spheres) of M among a specified class of subsets of M.

In a metric space M the notation pqr means p ≠ q ≠ r and pq + qr = pr.M is said to be uniformly locally externally convex if there exists δ > 0 such that if p, q ∈ M, p ≠ q, and pq < δ, then there exists r ∈ M such that the relation pqr subsists. We will prove the following result.
Characterization of Non-Linear Transformations Possessing Kernels
Authors: Mizel Victor J.
Published: December 2010

Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form

(1.1)

with a continuous “kernel” φ: R → R,or

(1.2)

with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see [8]) and to the theory of fading memory in continuum mechanics [3]. However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form

(1.3)

where the kernel φ: R × T → R satisfies “Carathéodory conditions”.
Distributive Projective Lattices
Authors: Baker Kirby A. , Hales Alfred W.
Published: December 2010

Two basic unsolved problems of lattice theory are (1) the characterization of sublattices of free lattices and (2) the characterization of projective lattices. A solution to an important case of the first problem has been provided by Galvin and Jónsson [3], who characterize distributive sublattices of free lattices. In this paper, we solve the same case of the second problem by characterizing distributive projective lattices (Theorem 4.1). An interesting corollary is the verification for distributive lattices of the conjecture that a, finite lattice is projective if and only if it is a sublattice of a free lattice.
On the ARF Invariant of an Involution
Authors: Orlik Peter
Published: December 2010

Let Σ4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S4k+1k ≧ 1, and T: Σ4k+1 → Σ4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ4k+1 and the properties of the free involution T.

To be more specific, recall [5] that the h-cobordism classes of smooth manifolds homeomorphic (or, what is the same, homotopy equivalent) to S4k+1, k ≧ 1, form a finite abelian group θ4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP4k+2 is either zero or cyclic of order 2. In the latter case the two distinct homotopy spheres are distinguished by the Arf invariant of the parallelizable manifolds they bound.
Linearization of the Product of Jacobi Polynomials. II
Authors: George Gasper
Published: December 2010

Let [3, p. 170, (16)]

(1.1)

denote the Jacobi polynomial of order (α, β), α, β > – 1, and let g(k, m, n; α, β) be denned by

(1.2)

where Rn(α, β)(x) = Pn(α, β)(x)/Pn(α, β)(1). It is well known [1; 2; 4; 5; 6] that the harmonic analysis of Jacobi polynomials depends, at crucial points, on the answers to the following two questions.

Question 1. For which (α, β) do we have

(1.3)

Question 2. For which (α, β) do we have

(1.4)

where G depends only on (α, β)?
Strongly Regular Graphs Derived from Combinatorial Designs
Authors: Goethals J. M. , Seidel J. J.
Published: December 2010

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.

In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.
Localization Problem of the Absolute Riesz and Absolute Nörlund Summabilities of Fourier Series
Authors: Izumi Masako , Izumi Shin-Ichi
Published: December 2010

1.1. Let Σ an be an infinite series and sn its nth partial sum. Let (pn) be a sequence of positive numbers such that

If the sequence

(1)

is of bounded variation, that is, Σ |tn – tn –1| < ∞, then the series Σ an is said to be absolutely (R, pn, 1) summable or |R, pn, 1| summable.

Let ƒ be an integrable function with period 2π and let its Fourier series be

(2)

Dikshit [4] (cf. Bhatt [1] and Matsumoto [7]) has proved the following theorems.

THEOREM I. Suppose that (i) the sequence (pn/Pn) is monotone decreasing, (ii) mn > 0, (iii) the sequence (mnpn/Pn) decreases monotonically to zero, and (iv) the series Σ mnPn/Pn) is divergent.
Brownian Motion and Dimension of Perfect Sets
Authors: Kaufman Robert
Published: December 2010

Let X(t) denote real-valued Brownian motion on the interval 0 ≦ t ≦ 1, so normalized that E(X2(t)) = t. We prove some theorems about transforms X(F) of closed sets F: in general, F is not known in advance but depends on X. The main point of comparison among sets is taken to be their Hausdorff dimension, and in this respect the linear process is quite different from the planar. We state and discuss briefly two theorems.

(A) It is almost sure that, for every closed set F in [0, 1],

(B) For each closed set F in (—00,00) and number a ,

Plainly, statements (A) and (B) are nearly best possible. For the planar process dimX(F) = 2 dim F (with the same quantification as in (A)) [6].
A Mapping Problem and Jp-Index. II
Authors: Wakae Masami , Hamara Oma
Published: December 2010

In [9], indices for equivariant mappings have been denned in the case that the transformation groups are cyclic. Thus a question will naturally arise as to the generalization of [4, Theorem 2] or [8, § IV, Theorem 2.8]. In this paper we will generalize the above result when the transformation groups are of order paqb, p, q are odd prime numbers. The method used here can be used directly for more general cyclic groups, say, of order n = p1α1 … pmαm. However, the results are too complicated to be of interest.
On a Class of Perfect Rings
Authors: Dlab Vlastimil
Published: December 2010

In [3], the perfect rings of Bass [1] were characterized in terms of torsions in the following way:

A ring R is right perfect if and only if every (hereditary) torsion in the categoryMod Rof all left R-modules is fundamental (i.e. generated by some minimal torsions) and closed under taking direct products; as a consequence, the number of all torsions inMod Ris finite and equal to 2n for a natural n.

Here, we present a simple description of those rings R which allow only two (trivial) torsions, viz. 0 and Mod R (and thus, are right perfect by [3]). Finite direct sums of these rings represent a natural generalization of completely reducible (i.e. artinian semisimple) rings (cf. Theorem 2) and we shall call them for that matter π-reducible rings. They can also be characterized in terms of their idempotent two-sided ideals.
An Abstract Concept of the Sum of a Numerical Series
Authors: Ruckle William H.
Published: December 2010

Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems.

1.1 The Köthe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l1 (absolutely summable sequences) for each s∈T. It is known that if cs or bs is used in place of l1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a rôle analogous to l1? This problem is treated in [6].

1.2. Let {xn, fn} be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.
Elations of Designs
Authors: Kantor William M.
Published: December 2010

An elation of a design is an automorphism γ of fixing some block X pointwise and some point x on X blockwise. Luneburg [4] and I [2] have proved results which state that a design admitting many dations and having additional properties must be the design of points and hyperplanes of a finite desarguesian projective space. In this note, additional results of this type will be proved and applied to yield a generalization of a previous result on Jordan groups [3]. The proofs were suggested by a result of Hering on dations of finite projective planes [1, pp. 122, 190].
Extensions of a Brandt Semigroup by Another
Authors: Lallement Gérard , Petrich Mario
Published: December 2010

One possible first step in considering the structure of a class of semigroups is to study the ideal extensions (here simply called “extensions“) of simple or 0-simple semigroups in by another if the latter are of known structure. Extensions of a semigroup by another were first studied by Clifford (see [1, 4.4 and 4.5]). In his constructions, an extension of a semigroup S by a semigroup T with zero is given by a function (satisfying certain conditions) from T* = T\0 into the translational hull of S.

We use certain results (refining those of Clifford) established in [2] and a description of the translational hull of a Brandt semigroup given in [9] (see also [8]), to construct all extensions V of a Brandt semigroup S having a finite number of idempotents by any Brandt semigroup T (cf. [10]).
More on Extending Continuous Pseudometrics
Authors: Shapiro H. L.
Published: December 2010

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.
Maximality In Function Algebras
Authors: Blumenthal Robert G.
Published: December 2010

In this paper we prove that the proper Dirichlet subalgebras of the disc algebra discovered by Browder and Wermer [1] are maximal subalgebras of the disc algebra (Theorem 2). We also give an extension to general function algebras of a theorem of Rudin [4] on the existence of maximal subalgebras of C(X). Theorem 1 implies that every function algebra defined on an uncountable metric space has a maximal subalgebra.

A function algebra A on X is a uniformly closed, point-separating subalgebra of C(X), containing the constants, where X is a compact Hausdorff space. If A and B are function algebras on X, A ⊂ B, A ≠ B, we say A is a maximal subalgebra of B if whenever C is a function algebra on X with A ⊂ C ⊂ B, either C = A or C = B.
Functions of Bounded mean Square, and Generalized Fourier-Stieltjes Transforms
Authors: Henniger J.
Published: December 2010

A complex function on the real line is said to be bounded in mean square if it is locally in L2 (i.e. on each finite interval) and satisfies

(1.1)

The set of all such functions clearly forms a linear space over the complex numbers and is a Banach space B under the norm ‖·‖B defined by (1.1). This space, among others, has been discussed by Beurling in [1], where it was shown to be the dual, in the Banach space sense, of a certain Banach (convolution) algebra of functions. We have used Beurling's characterization of B and others of his results throughout this paper, and indeed the essence of one or two of the proofs has been derived from his theorems.
Indefinite Finsler Spaces and Timelike Spaces
Authors: Beem John K.
Published: December 2010

In this paper we investigate indefinite Finsler spaces in which the metric tensor has signature n — 2. These spaces are a generalization of Lorentz manifolds. Locally a partial ordering may be defined such that the reverse triangle inequality holds for this partial ordering. Consequently, the spaces we study may be made into what Busemann [3] terms locally timelike spaces. Furthermore, sufficient conditions are obtained for an indefinite Finsler space to be a doubly timelike surface (see [2; 4]). In particular, all two-dimensional pseudo-Riemannian spaces are shown to be doubly timelike surfaces.
Algèbres Commutatives Engendrées Par Leurs Éléments Idempotents
Authors: Keimel Klaus
Published: December 2010

Dans ce travail, R désignera toujours un anneau commuta tif ayant un élément unité 1. Les R-algèbres considérées seront supposées associatives. Si A est une R-algèbre, nous supposerons toujours 1 · a = a quel que soit a ∊ A. Si B ⊂ A, nous désignerons par Ann(B) l'ensemble des r ∊ R tels que rB = {0}.

Soit A une R-algèbre commutative (avec ou sans élément unité). Nous désignerons par EA l'ensemble des éléments idempotents de A. Si l'on définit pour e, f ∊ EA,

alors EA devient un treillis distributif relativement complémenté dont 0 est le plus petit élément [2].

Nous nous intéresserons aux R-algèbre commutatives engendrées par leurs éléments idempotents.
A Construction in General Radical Theory
Authors: Ortiz Augusto H.
Published: December 2010

Given an arbitrary associative ring R we consider the ring
R[x] of polynomials over
R in the commutative indeterminate x.
For each radical property S we define the function
S* which assigns to each ring R the ideal

of R. It is shown that the property SA (that a ring R be equal to S*(R)) is a radical property. If
S is semiprime, then SA is semiprime also. If S is a special radical,
then SA is a special radical. SA is always contained in S. A necessary and
sufficient condition that S and SA coincide is given.

The results are generalized in the last section to include extensions of
R other than R[x], One
such extension is the semigroup ring R[A],
where A is a semigroup with an identity adjoined.
Remarks on Op and Towber Rings
Authors: Lissner David , Geramita Anthony
Published: December 2010

In this paper all rings considered have identity and are commutative, and
all modules are finitely generated. We shall make liberal use of the
definitions and notation established in [6; 7].

Towber observed in [9] that a local Outer Product ring
(OP-ring) must have v-dimension ≦ 2, and so a local OP-ring is either
regular of global dimension ≦ 2 or it has infinite global dimension. Since
the global dimension of a noetherian ring is the supremum of the global
dimensions of its localizations, we immediately obtain the following
result.

THEOREM 1.1. The global dimension of a noetherian
OP-ring is either∞ or ≦ 2.
Growth and Decay Estimates near Non-Elementary Stationary
Points
Authors: Coleman Courtney
Published: December 2010

Local growth and decay estimates near the stationary point at the origin are
derived in § 3 for solutions of the vector system,

(1)

where A(x) and B(y) are homogeneous of
degree m > 1 in the components of x and
y, respectively, and f* and
g* are of order greater than m in
‖(x, y)‖ near the origin. It is assumed that
x = 0 is asymptotically stable and y =
0 is asymptotically unstable for the homogeneous systems of first
approximation,

(2)

In order to derive the estimates in § 3, various results are needed
concerning solutions of a homogeneous system such as (2) (a). These are
derived in § 2 and are based on work of Hahn [4; 5], Lefschetz
[8], and Zubov [12].
Ideal Extensions of Topological Semigroups
Authors: Christoph Francis T.
Published: December 2010

In the study of compact semigroups the constructive method rather than the
representational method is usually the better plan of attack. As it was
pointed out by Hofmann and Mostert in the introduction to their book
[10] this method is more productive than searching for a
representation theory. Hofmann and Mostert described a constructive method
called the Hormos and showed that any irreducible compact semigroup is
obtained by the Hormos construction. Many of the important examples of
irreducible semigroups which motivated their work were obtained by Hunter
[11; 12; 13; 14].

In this paper, we apply the constructive method of ideal extensions
[5] in algebraic semigroups to topological semigroups which
are not necessarily compact. Many of Hunter's examples and examples of the
Hormos technique can also be obtained by our method of topological ideal
extensions. The topological ideal extension method, however, is, in general,
a different type of construction technique.
Ideals in Rings of Analytic Functions with Smooth Boundary Values
Authors: Taylor B. A. , Williams D. L.
Published: December 2010

LetAdenote the Banach algebra of functions analytic in the open unit discDand continuous in. Iffand its firstmderivatives belong toA,then the boundary functionf(eiθ)belongs toCm(∂D). The spaceAmof all such functions is a Banach algebra with the topology induced byCm(∂D).If all the derivatives of/ belong toA,then the boundary function belongs toC∞(∂D), and the spaceA∞all such functions is a topological algebra with the topology induced byC∞(∂D). In this paper we determine the structure of the closed ideals ofA∞(Theorem 5.3).

Beurling and Rudin (see e.g. [7, pp. 82-89;10]) have characterized the closed ideals ofA, and their solution suggests a possible structure for the closed ideals ofA∞.
Invertibility and Class Number of Orders
Authors: Gorman Howard
Published: December 2010

This paper continues the work begun in [5] and concerns the
invertibility of modules in a finite-dimensional, symmetric algebra
L with 1 over a field. In particular, we continue the
work done in [5] which dealt with the connection between
invertibility in these algebras and a condition called the Brandt Condition,
which is a reformulation by Kaplansky [6] of some ideas of
Brandt.

We begin by proving some preliminary results on invertibility and some
equivalent conditions for the dual of an order to be principal.

Then, we define the class number of an order and reformulate the concept of
invertibility in terms of class number. In this terminology, we find some
equivalent conditions which ensure that an order in certain algebras
L (including commutative, symmetric algebras, and
algebras with a strong involution) has class number equal to 1 (i.e., all
modules principal), and we characterize a class of Brandt algebras over the
quotient fields of valuation rings as those in which all orders have class
number less than or equal to 2.
On Branched Coverings of S3
Authors: Burde Gerhard
Published: December 2010

In [3] Fox studied a certain class of irregular coverings of
S3 branched along some knot or link which turned out to be homotopy
spheres. By a simple geometric construction, it is shown in this paper that
these homotopy spheres are just 3-spheres, provided that the group of the
knot or link k in question cannot be generated by a number
of Wirtinger generators smaller than the minimal number of bridges of this
knot or link. The knots and links with two bridges provide examples for such
coverings. In the covering sphere there is a link covering k. With the help of braid
automorphisms, can be
determined. Figure 3 shows a link in a
5-sheeted covering over k = 41. Links over
31 and 61 in 3-sheeted coverings were determined by
Kinoshita [5] by a different method. The method used here is
applicable to these cases and confirms his results.