bluuebunny/crossref_metadata_2025_split huggingface dataset:
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 → ∞.
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”.
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 thatIf 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].
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.
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]).
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.
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 idealof 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.
Répartition Des Nombres Hautement Composés de
Ramanujan
Authors: Nicolas Jean-Louis 
Published: December 2010

On dit qu'un nombre entier A est hautement composé si tout
nombre M plus petit que A a moins de
diviseurs que A. Si l'on définit d(n) =
nombre de diviseurs de n, on sait que, si la décomposition
en facteurs premiers de n est n = Πi
piai, on a:La définition devient: A est hautement composé si et
seulement si:1Ramanujan [8] a défini et étudié les nombres hautement
composés, démontrant les propriétés suivantes:Si A = 2a2 3a3 … pkapk est un nombre hautement composé, on a:et à l'exception de A = 4 et A = 36, on a:
apk = 1 [8, §8].
Periodic Solutions of Non-Linear Evolution Equations in Banach
Spaces
Authors: Ton Bui An 
Published: December 2010

In this paper the theory of Browder [2] and of Lions
[3] on periodic solutions of non-linear evolution equations
in Banach spaces is put in a more general framework so as to include the
Navier-Stokes equations and their variants.An abstract existence theorem is proved in § 1. Applications are given in §
2. The existence of periodic solutions of the Navier-Stokes equations
without any restriction on the dimension of the space domain is established.
Application of the abstract theorem to the following problem is given:1. Let H be a Hilbert space and (., .)H the inner product in H. Let V and
W be two reflexive separable Banach spaces with
W ⊂ V ⊂ H. W is dense
in V and V is dense in H.
Ramification Groups of Abelian Local Field Extensions
Authors: Marshall Murray A. 
Published: December 2010

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.
On the Exponent of an Osculatory Packing
Authors: Boyd David W. 
Published: December 2010

Suppose that U is an open set in Euclidean N-space which has a finite volume |U|. A complete packing of U is a sequence of disjoint N-spheres C = {Sn} which are contained in U and whose total volume equals that of U. In an osculatory packing, the spheres are chosen recursively so that for all n larger than a certain value m, Sn has the largest radius of all spheres contained in U\(S1– ∪ … ∪ Sn-1–) (S– is the closure of S). An osculatory packing is simple if m = 1. If rn denotes the radius of Sn, the exponent of the packing is defined by:This quantity is of considerable interest since it measures the effectiveness of the packing of U by C.
Partial Spreads and Replaceable Nets
Authors: Bruen A. 
Published: December 2010

A blocking set S in a projective plane Π is a subset of the points of Π such that every line of Π contains at least one point of S and at least one point not in S. In previous papers [5; 6], we have shown that if Π is finite of order n, then n + √n + 1 ≦ |S| ≦ n2 – √n (see [6, Theorem 3.9]), where |S| stands for the number of points of S. This work is concerned with some applications of the above result to nets and partial spreads, and with some examples of partial spreads which give rise to unimbeddable nets of small deficiency.In the next section we re-prove a well known result of Bruck which states that if N is a replaceable net of order n and degree k then k ≧ √n + 1, and show how this bound can be improved if n = 7, 8, or 11.
Biconnected Multifunctions of Trees which have an end Point as Fixed Point or Coincidence
Authors: Schirmer Helga 
Published: December 2010

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].
Quasi-Splitting Exact Sequence
Authors: Hou Hsiang-Dah 
Published: December 2010

Let R be a ring with 1 ≠ 0 and α, β, γ R-endomorphisms of R-modules A, B, and C respectively. We shall denote by M(R) the category of R-modules, and by End(R) the category of R-endomorphisms. For objects α and β of End(R) a morphism λ: α → β is an R-homomorphism such that λα = β λ. We shall denote by Idm(R) the full subcategory of End(R) whose objects are idempotents. Idm(R) is an abelian category, ker, coker and im are constructed in the naive way and henceis exact in M(R) if and only ifis exact in Idm(R), where the domains of α,β, and γ are A, B, and C respectively. One sees that End (R) as well as Idm(R) is abelian.
On the Number of Prolongations of a Finite Rank Valuation
Authors: Wright Michael J. 
Published: December 2010

A (non-archimedean) valuation v on a field K is said to be henselian if it has a unique prolongation to a valuation on Ka, the algebraic closure of K. A henselization (Kh, vh) of a valuated field (K, v) is a smallest separable extension of K containing a henselian prolongation vh of v. (Kh, vh) is unique up to K-isomorphism, and (Kh, vh) = (K, v) if and only if v is henselian. In this paper we confine ourselves to valuations of finite rank.If v is a non-henselian rank one valuation on K, and if [Ks:K] = ∞, Ks being the separable closure of K, then it is known that v has infinitely many prolongations to Ka [1, (27.11)]. We shall see that this is no longer true if the rank of v is greater than one.
First Homology of Irreducible 3-Manifolds
Authors: Evans Benny 
Published: December 2010

In [2], J. Gross provides an infinite collection of topologically distinct irreducible homology 3-spheres. In this paper, we construct for any finitely generated abelian group A, an infinite collection {Mi} of topologically distinct irreducible closed 3-manifolds such that H1(Mi) = A for each i.The proof consists of first constructing a closed irreducible 3-manifold MA with H(MA) = A, and then providing a method for producing more such manifolds with the same first homology group.All maps and spaces in this paper are assumed to be in the piecewise linear category, and all subspaces are assumed to be piecewise linear subspaces.A 3-manifold M is irreducible if each 2-sphere in M bounds a 3-cell in M. A compact 2-manifold (or surface) F in a compact 3-manifold M is properly embedded in M if F ∩ bdM = bdF.