bluuebunny/crossref_metadata_2025 huggingface dataset:
A Combinatorial Decomposition Theory
Authors: Cunningham William H. , Edmonds Jack
Published: December 2010

Given a finite undirected graphGandA⊆E(G),G(A)denotes the subgraph ofGhaving edge-setAand having no isolated vertices. For a partition {E1, E2}ofE(G),W(G; E1)denotes the setV(G(E1))⋂V(G(E2)). We say thatGisnon-separableif it is connected and for every proper, non-empty subsetAofE(G), we have |W(G;A)| ≧ 2. Asplitof a non-separable graphGis a partition {E1, E2} ofE(G)such that

|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.

Where {E1, E2} is a split ofG, W(G; E2)= {u, v}, andeis an element not inE(G),we form graphsGii= 1 and 2, by addingetoG(Ei)as an edge joiningutov.
Euler Classes of Combinatorial Manifolds
Authors: Penna Michael A.
Published: December 2010

Every finite simplicial complex has a tangent bundle in the category of simplicial bundles (see [9]). The goal of this paper is to classify simplicial bundles, and, as an application of this result, to construct Euler classes for a large class of combinatorial manifolds. This construction is closely related to [3] and [4].
Biplanar Surfaces of Order Three II
Authors: Bisztriczky Tibor
Published: December 2010

A surface of order three F in the real projective three-space P3 is met by every line, not in F, in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ1 and τ2.

In [2], we examined the biplanar surfaces containing the line τ1 ⌒ τ2. In the present paper, we classify and describe the biplanar F with the property that τ1 ⌒ τ2⌒ F = {v}.

We denote the planes, lines and points of P3 by the letters α, β, …, L, M, … and p, q, … respectively. For a collection of flats α, L, p, …, 〈α, L, p, …〉 denotes the flat of P3 spanned by them. For a set M in P3, denotes the flat of P3 spanned by the points of .
High-Power Analogues of the Turán-Kubilius Inequality, and an Application to Number Theory
Authors: Elliott P. D. T. A.
Published: December 2010

An arithmetic function ƒ(n) is said to be additive if it satisfies ƒ(ab) = ƒ(a) + ƒ(b) whenever a and b are coprime integers. For such a function we define

A standard form of the Turán-Kubilius inequality states that

(1)

holds for some absolute constant c1, uniformly for all complex-valued additive arithmetic functions ƒ (n), and real x ≧ 2. An inequality of this type was first established by Turán [11], [12] subject to some side conditions upon the size of │ƒ(pm)│. For the general inequality we refer to [10].

This inequality, and more recently its dual, have been applied many times to the study of arithmetic functions. For an overview of some applications we refer to [2]; a complete catalogue of the applications of the inequality (1) would already be very large. For some applications of the dual of (1) see [3], [4], and [1].
A Note on Buchsbaum Rings and Localizations of Graded Domains
Authors: Daepp U. , Evans A.
Published: December 2010

Let R = ⊕i ≧0Ri be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R(p) = {r/t| r ∈ Ri, t ∈ Ri\p} be the geometric local ring at p and let Rp = {r/t| r ∈ R, t ∈ R\p} be the arithmetic local ring at p. Under the mild restriction that there exists an element r1 ∈ R1\p, W. E. Kuan [2], Theorem 2, showed that r1 is transcendental over R(p) and

where S is the multiplicative system R\p. It is also demonstrated in [2] that R(p) is normal (regular) if and only if Rp is normal (regular). By looking more closely at the relationship between R(p) and R(p), we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.
Group Rings over Z(p) with FC Unit Groups
Authors: Merklen H. , Polcino Milies C.
Published: December 2010

Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.

In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.

One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].
Free Subgroups in the Unit Groups of Integral Group Rings
Authors: Hartley B. , Pickel P. F.
Published: December 2010

Let G be a group, ZG the group ring of G over the ring Z of integers, and U(ZG) the group of units of ZG. One method of investigating U(ZG) is to choose some property of groups and try to determine the groups G such that U(ZG) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(ZG) to be nilpotent (see also [7]), and the same authors have investigated when U(ZG) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(ZG) contains a free subgroup of rank 2. We conjecture that if this does not happen, then every finite subgroup of G is normal, from which various other conclusions then follow (see Lemma 4).
Fixed Point Principles for Cones of a Linear Normed Space
Authors: Fournier Gilles
Published: December 2010

In [8] and [9], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [11], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to condensing maps and he applied this theorem to prove the existence of periodic solutions of nonlinear autonomous functional differential equations.

Nussbaum's proof makes an essential use of the difficult Zabreiko and Krasnosel'skiĭ, and Steinlein (mod p)-theorem for the fixed point index [13 -16]. In [6], Fournier and Peitgen proved two different versions of this theorem for completely continuous maps each one being sufficient for Nussbaum's applications. The proofs of these two theorems are much less involved and, although they are different, they make use of the same easier generalized Lefschetz number calculations (see [12] for (mod p) and [5] for compact attractor).
Correction to on the Brauer Group of Algebras Having a Grading and an Action
Authors: Orzech Morris
Published: December 2010

Let R be a commutative ring, G a finite abelian group. Let A be an R-algebra which is graded by G (i.e. A = Σ⊕σ∈GAσ, where AσAτ ⊂ Aστ for σ, τ in G) and for which A1 is an R-module of finite type. In Remark 4.1 (a) of [1] we asserted that under these hypotheses if u is in A and u + pA is homogeneous in A/pA for each maximal ideal p of R then u is homogeneous in A. We used this assertion for u a unit in A such that a → uau–1 is a grading-preserving homomorphism. K. Ulbrich has kindly pointed out a counterexample to the assertion: R = Z/4Z, G = {1, σ};, u = 2σ + 1, p = 2R. Proposition 4.2 of [1] uses the erroneous result and is in turn invoked later in the paper.
On Finitely Generated Lattices of Finite Width
Authors: Poguntke W. , Sands B.
Published: December 2010

The width of a lattice L is the maximum number of pairwise noncomparable elements in L.

It has been known for some time ([5] ; see also [4]) that there is just one subdirectly irreducible lattice of width twro, namely the five-element nonmodular lattice N5. It follows that every lattice of width two is in the variety of N5, and that every finitely generated lattice of width two is finite.

Beginning a study of lattices of width three, W. Poguntke [6] showed that there are infinitely many finite simple lattices of width three. Further studies on width three lattices were made in [3], where it was asked whether every finitely generated simple lattice of width three is finite. In this paper we will show that, in fact, more is true:

THEOREM 1.1. Every finitely generated subdirectly irreducible lattice of width three is finite.
Knots with Free Period
Authors: Hartley Richard
Published: December 2010

At the Georgia conference in 1961 Fox presented a paper, “Knots and periodic transformations”, in which he asked which knots may be fixed by a periodic transformation of the 3-sphere. He distinguished eight cases according to the type of fixed point set of T and its relationship to the knot. Except for case a), all these cases have since received some attention and conditions have been given for knots to fall into each of these classes. In fact, the problem of deciding which knots fall into category d) (that is “periodic knots“; I will refer to them as cyclically periodic knots) has been the subject of at least six papers [2], [3], [13], [15], [17], [22], but measured by their effectiveness at determining the periods of the knots to nine crossings, the theorems contained in these papers are not entirely satisfactory.
Algebras of Acyclic Type
Authors: Hanlon Phil
Published: December 2010

In this paper we consider the problem of determining when an algebra of formal power series over a commutative ring R is the homomorphic image of a reduced incidence algebra P(R, ∽). The question of when two such algebras are isomorphic is answered in Section 8 of [1]. A slight generalization of their notion of full binomial type is introduced here.

Section 1 contains background material together writh a summary of the results of [1]. In Section 2 we present the desired characterization, and to conclude an application appears in Section 3. In Section 3 the tools of Section 2 are used to derive an equation of R. W. Robinson and R. P. Stanley which counts labelled, acyclic digraphs.
Some Properties of θ-Closure
Authors: Espelie M. Solveig , Joseph James E.
Published: December 2010

The concept of θ-closure was introduced by Velicko to study H-closed spaces and to generalize Taimanov's extension theorem [11], [12]. More recently, this notion has been used by Dickman and Porter [1] to characterize those Hausdorff spaces in which the Fomin H-closed extension operator commutes with the projective cover (absolute) operator and [2] to study extentions of functions. If X is a topological space and A ⊂ X, we let Σ(A) and Γ(A) represent, respectively, the family of open subsets which contain A and closed subsets which contain some element of Σ(A). The θ-closure of A ⊂ X, denoted by clθ(A) (clθ(v) if A = {v}), is {x ∈ X: each V ∈ Γ(x) satisfies V ∩ A ≠ ∅} and A is called θ-closed in case clθ(A) = A.
Topologically Versal Deformations of Matrices; Codimension at most Two
Authors: Bass D. W.
Published: December 2010

1. Introduction. The reduction of a matrix to its Jordan normal form is an unstable operation in that both the normal form itself and the reducing mapping depend discontinuously on the elements of the original matrix. For example, the matrix trivially reduces to itself in Jordan form, but there are arbitrarily small perturbations of this matrix that reduce to the form which is certainly not a small perturbation of the original matrix, and moreover the reducing mapping is not a small perturbation of the identity. In [1], Arnol'd derives the simplest possible normal forms to which families of matrices may be linearly reduced in a ‘stable’ manner. In this paper, we consider a ‘topological' version of the problem, using the classification of matrices up to topological conjugacy given in [8] and the classification of linear dynamical systems up to orbital equivalence given in [9].
Gleason Parts of Real Function Algebras
Authors: Kulkarni S. H. , Limaye B. V.
Published: December 2010

Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].

In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.
On the Structure of Finitely Presented Lattices
Authors: Gratzer G. , Huhn A. P. , Lakser H.
Published: December 2010

A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)

It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.

In this paper we prove a structure theorem that could be said to verify this conjecture.

THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.
On Explicit Bounds in Landau's Theorem. II
Authors: Jenkins James A.
Published: December 2010

Quite some years ago a number of mathematicians were interested in obtaining explicit expressions for the bounds in Schottky's and Landau's theorems, specifically numerically évaluable bounds of a particular form. The best bounds of this type in Schottky's theorem were given by the author [3]. For Landau's theorem the chosen form is as follows. Let F(Z) be regular in |Z| < 1, omit the values 0 and 1 and have Taylor expansion about Z = 0

Then

Using the same method employed for Schottky's theorem the author showed that one can take K = 5.94. By a slight modification of the author's method Lai [6] gave the further value K = 4.76.
Developability and Some New Regularity Axioms
Authors: Heldermann N. C.
Published: December 2010

In a recent publication H. Brandenburg [5] introduced D-completely regular topological spaces as a natural extension of completely regular (not necessarily T1) spaces: Whereas every closed subset A of a completely regular space X and every x ∈ X\A can be separated by a continuous function into a pseudometrizable space (namely into the unit interval), D-completely regular spaces admit such a separation into developable spaces. In analogy to the work of O. Frink [16], J. M. Aarts and J. de Groot [19] and others ([38], [46]), Brandenburg derived a base characterization of D-completely regular spaces, which gives rise in a natural way to two new regularity conditions, D-regularity and weak regularity.
On Uniqueness Sets for Expansions in Sequences of Functions Arising from Singular Generating Functions
Authors: Wimp Jet
Published: December 2010

Let {pn(z)}; be a sequence of functions analytic in a region D. A problem in analysis which has received much attention is the following: describe the sets Z ⊂ D for which

(1)

implies hn is 0 for all n, (To make the problem interesting, only those situations are studied where finite subsets of the pn(z) are linearly independent in D.) Another way of phrasing this is: Characterize the uniqueness sets of pn(z), a uniqueness set Z being a set in D such that the restriction of {pn(z)}; to Z is linearly independent. If Z is not a uniqueness set then for some {hn}; not all 0, we have

(2)

This formula is called a non-trivial representation of 0 (on Z).
Criteria for Total Projectivity
Authors: Hill Paul
Published: December 2010

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.
Planar Maps are Well Labeled Trees
Authors: Cori Robert , Vauquelin Bernard
Published: December 2010

In the theory of enumeration, the part devoted to the counting of planar maps gives rather surprising results. Of special interest to the combinatorialists is the conspicuous feature of counting numbers associated with families of maps as discussed in the papers of Tutte, Brown and Mullin. These formulas are by no means easy to prove; this is also a part of their charm. We are convinced therefore that these maps possess deep combinatorial properties. We discuss such properties in this paper.

The main result of this article is the construction of a bisection between planar maps and a special family of trees: their vertices are labeled with numbers that differ by at most 1 on adjacent vertices. These trees are said to be well labeled. Combinatorialists usually claim that theorems specifying the “geometry” of objects lead to enumerating formulas. Our result obeys this general rule: we can indeed obtain a new proof of the formula counting the number of rooted planar maps with m edges. Four different proofs of this formula are known to date.
Limits on Pairwise Amicable Orthogonal Designs
Authors: Wolfe Warren
Published: December 2010

Anorthogonal design in order n of type(u1, …,ut) on the commuting variablesx1, …,xtis ann×nmatrixXwith entries 0, ±x1, …, ±xtsuch that

In [5] Geramita and Wallis show that ifn= 24a+b·n0, wheren0is odd and 0 ≦b> 4, thent≧ρ(n)= 8a+ 2b. The result is essentially Radon's limit on the number of anti-commuting, real, anti-symmetric, orthogonal matrices in ordern. Garamita and Pullman show that this limit is sharp for orthogonal designs: i.e., givenn, there exists an orthogonal design in ordernwithρ(n)variables [6].

Two orthogonal designs,XandF, are calledamicableifXYt=YXt.

Such pairs of orthogonal designs are especially useful in generating new orthogonal designs [5] or [6].
A Classification of Homogeneous Surfaces
Authors: Huckleberry A. T. , Livorni E. L.
Published: December 2010

Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.
On the Monodromy Groups of Riemann Surfaces of Genus ≧1
Authors: Kuiken Kathryn
Published: December 2010

It is well-known [5, 19] that every finite group can appear as a group of automorphisms of an algebraic Riemann surface. Hurwitz [9, 10] showed that the order of such a group can never exceed 84 (g – 1) provided that the genus g is ≧2. In fact, he showed that this bound is the best possible since groups of automorphisms of order 84 (g – 1) are obtainable for some surfaces of genus g. The problems considered by Hurwitz and others can be considered as particular cases of a more general question: Given a finite group G, what is the minimum genus of the surface for which it is a group of automorphisms? This question has been completely answered for cyclic groups by Harvey [7]. Wiman's bound 2(2g + 1), the best possible, materializes as a consequence. A further step was taken by Maclachlan who answered this question for non-cyclic Abelian groups.
Generators of Un(V) Over a Quasi Semilocal Semihereditary Ring
Authors: Ishibashi Hiroyuki
Published: December 2010

Let o be a quasi semilocal semihereditary ring, i.e., o is a commutative ring with 1 which has finitely many maximal ideals {Ai|i ∊ I} and the localization oAi at any maximal ideal Ai is a valuation ring. We assume 2 is a unit in o. Furthermore * denotes an involution on o with the property that there exists a unit θ in o such that θ* = –θ. V is an n-ary free module over o with f : V × V → o a λ-Hermitian form. Thus λ is a fixed element of o with λλ* = 1 and f is a sesquilinear form satisfying f(x, y)* = λf(y, x) for all x, y in V. Assume the form is nonsingular; that is, the mapping M → Hom (M, A) given by x → f( , x) is an isomorphism. In this paper we shall write f(x, y) = xy for x, y in V.
Une Remarque Sur Une Classe D'Algebres De Dirichlet Faibles* De Fonctions Analytiques
Authors: Savoyant Michel
Published: December 2010

Soit K un compact du plan complexe C, d'intérieur non vide et connexe. A(K) désigne l'algèbre uniforme des fonctions continues sur et analytiques dans . Pour chaque , on note λz la mesure harmonique sur la frontière de du point z, et on pose λ = λz0, z0 étant un point fixé de . On dit que A(K) est une algèbre de Dirichlet faible* sur (— désigne le conjugué) est faiblement* dense dans L∞(λ) (la topologie faible* sur L∞(λ) est la topologie faible pour la dualité entre L1(λ) et L∞(X)); vu les inégalités de Harnack pour les fonctions harmoniques cette propriété est indépendante de la mesure harmonique du point z0 fixé dans .
Direct Product of Derived Steiner Systems Using Inversive Planes
Authors: Phelps K. T.
Published: December 2010

A Steiner system S(t, k, v) is a pair (P, B) where P is a v-set and B is a collection of k-subsets of P (usually called blocks) such that every t-subset of P is contained in exactly one block of B. As is well known, associated with each point x ∈ P is a S(t � 1, k � 1, v � 1) defined on the set Px = P\{x} with blocks

B(x) = {b\{x}|x ∈ b and b ∈ B}.

The Steiner system (Px, B(x)) is said to be derived from (P, B) and is called (obviously) a derived Steiner (t – 1, k – 1)-system. Very little is known about derived Steiner systems despite much effort (cf. [11]). It is not even known whether every Steiner triple system is derived.

Steiner systems are closely connected to equational classes of algebras (see [7]) for certain values of k.
Transitivity and Ortho-Bases
Authors: Kofner Jacob
Published: December 2010

Throughout this paper “space” means “T1 topological space.“

1. The concept of an ortho-base was introduced by W. F. Lindgren and P. J. Nyikos.

Definition 1. A base of a space X is called an ortho-base provided that for each subcollection either is open or is a local base of a point x ∊ X [17].

Ortho-bases are related to interior-preserving collections which have been known for some time.

Definition 2. A collection of open sets of a space X is called interior-preserving provided that the intersection of any subcollection is open. A space X is called orthocompact provided that each open cover has an open interior-preserving refinement.

It was proved in [17], in particular, that each space with an ortho-base is orthocompact, and each orthocompact developable space (which is the same as a non-archimedean quasi-metrizable developable space [4]) has an ortho-base.
Criteria for σ-Smoothness, τ-Smoothness, and Tightness of Lattice Regular Measures, with Applications
Authors: Bachman George , Stratigos Panagiotis D.
Published: December 2010

Consider an arbitrary set X and an arbitrary disjunctive lattice of subsets of X, ℒ. The algebra of subsets of X generated by ℒ is denoted by , the set of all ℒ-regular measures on , by MR(ℒ), and the associated Wallman space, a compact T1 space, by IR(ℒ); assume X is embedded in IR(ℒ) (otherwise, consider the image of X in IR(ℒ)).

In part of an earlier paper [4] the work of Knowles [15] and Gould and Mahowald [11] was generalized from the explicit topological setting of X, a Tychonoff space, with ℒ the lattice of zero sets of X, to the above setting, with the added assumption that ℒ was also δ and normal. This was done so that the important Alexandroff Representation Theorem [1] could be utilized in order to induce two associated measures , and defined on and respectively, where W(ℒ) is the Wallman lattice in IR(ℒ).
Isomorphic Group Rings of Free Abelian Groups
Authors: Krempa Jan
Published: December 2010

S. K. Sehgal ([9], Problem 26) proposed the following question : Let A, B be rings and X an infinite cyclic group. Does AX ⋍ BX imply A ⋍ B? M. M. Parmenter and S. K. Sehgal (cf. [9], Chapter 4) proved that, under some strong assumptions concerning rings A, B, the answer is affirmative. In this paper, we show that the assumptions concerning the ring B may be omitted in the above mentioned results. Moreover, it is proven that if (AX)X ⋍ BX then AX ⋍ B for all rings A, B. If A is commutative and noetherian then a partial answer to Problem 27, [9] follows from our results.

Recently, L. Griinenfelder and M. M. Parmenter constructed nonisomorphic rings A, B for which the group rings AX, BX are isomorphic, [2], We give a new class of rings of this type. Our examples are noncommutative domains and algebras over finite fields. That also gives a negative answer to Problem 29, [9].
Factor-Correspondences in Regular Rings
Authors: Berberian S. K.
Published: December 2010

Factor-correspondences are nothing more than a way of describing isomorphisms between principal ideals in a regular ring. However, due to a remarkable decomposition theorem of M. J. Wonenburger [7, Lemma 1], they have proved to be a highly effective tool in the study of completeness properties in matrix rings over regular rings [7, Theorem 1]. Factor-correspondences also figure in the proof of D. Handelman's theorem that an ℵ0-continuous regular ring is unitregular [4, Theorem 3.2].

The aim of the present article is to sharpen the main result in [7] and to re-examine its applications to matrix rings. The basic properties of factor-correspondences are reviewed briefly for the reader's convenience.

Throughout, R denotes a regular ring (with unity).

Definition 1 (cf. [5, p. 209ff], [7, p. 212]). A right factor-correspondence in R is a right R-isomorphism φ : J → K, where J and K are principal right ideals of R (left factor-correspondences are defined dually).
On Asymptotic Behavior of Induced Representations
Authors: Baggett Larry , Taylor Keith F.
Published: December 2010

This paper is devoted to the proof of the following theorem.

THEOREM 1.1. Let H be a closed subgroup of a connected Lie group G, let N denote the largest (closed) subgroup of H which is normal in all of G, and suppose that π is a unitary representation of H whose restriction to N is a multiple of a character χ of N. Then every matrix coefficient of the induced representation Uπ vanishes at infinity modulo the kernel of Uπ providing that the following two conditions hold:

i) N is almost-connected (finite modulo its connected component).

ii) The subgroup Hk is “regularly related” to the diagonal subgroup D in Gk for at least one integer k ≧ k0 where k0 is determined by G and H.
Localizations of Linked Quaternionic Mappings
Authors: Yucas Joseph
Published: December 2010

Let G and B be abelian groups with G having exponent 2 and a distinguished element –1. In [7] we defined a linked quaternionic mapping to be a map q : G × G → B satisfying the following properties:

(A) q is symmetric and bilinear

(B) q(a, a) = q(a, – 1) for every a ∈ G, and

(L) q(a, b) = q(c, d) implies there exists x ∈ G such that q(a, b) = q(a, x) and q(c, d) = q(c, x).

A form (of dimension n over q) is a symbol φ = 〈a1, …, an〉 with a1, …, an ∈ G. The determinant and Hasse invariant of such a form φ are
On Automorphism Groups of Divisible Designs
Authors: Jungnickel Dieter
Published: December 2010

A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ2) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [50] obtained the first result in this direction for the finite projective spaces).

As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [11] and [20].
Necessary and Sufficient Conditions for the Equality of L(f) and l1
Authors: Deeb Waleed
Published: December 2010

Introduction. Let f be a modulus, ei = (δij) and E = {ei, i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp, 0 < p ≦ 1 is an L(f) space with f(x) = xp, and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l1, for all f modulus (see [2]), and l1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.
Two Undecidability Results using Modified Boolean Powers
Authors: Burris Stanley , Lawrence John
Published: December 2010

In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction.
A Class of Loops Which are Isomorphic to all Loop Isotopes
Authors: Goodaire Edgar G. , Robinson D. A.
Published: December 2010

It is convenient and not without precedent (see [2], [1], and also [6]) to call a loop which is isomorphic to all of its loop isotopes a G-loop. Since all groups are readily seen to be G-loops, the only interest in such loops, from a loop-theoretic standpoint, resides with those which are not associative. Examples and ad hoc constructions of such loops have appeared sporadically in the literature (see, for instance, [1], [2], [4], [6], [8], [9], and [13]).

Any finite loop of order n < 5 is a group and, hence, must also be a G-loop. R. L. Wilson [11, 12, 13] proved that a finite G-loop of prime order is necessarily a group; he also exhibited for each even integer n > 5 a G-loop of order n which is not associative and then raised questions concerning the existence of finite G-loops which are not groups for every possible composite order n > 5.
Spectral Inclusion and C.N.E.
Authors: Lubin A. R.
Published: December 2010

1. An n-tuple S = (S1, …, Sn) of commuting bounded linear operators on a Hilbert space H is said to have commuting normal extension if and only if there exists an n-tuple N = (N1, …, Nn) of commuting normal operators on some larger Hilbert space K ⊃ H with the restrictions Ni|H = Si, i = 1, …, n. If we take

the minimal reducing subspace of N containing H, then N is unique up to unitary equivalence and is called the c.n.e. of S. (Here J denotes the multi-index (j1, …, jn) of nonnegative integers and N*J = N1*jl … Nn*jn and we emphasize that c.n.e. denotes minimal commuting normal extension.) If n = 1, then S1 = S is called subnormal and N1 = N its minimal normal extension (m.n.e.).
On a Problem of Turán about Polynomials HI
Authors: Pierre R. , Rahman Q. I.
Published: December 2010

Let

where x: = cos θ, denote the nth degree Chebyshev polynomials of the first and second kind, respectively. Further, let

Given non-negative integers λ and μ we define

and
On Polynomials with Curved Majorants
Authors: Newman D. J. , Rivlin T. J.
Published: December 2010

A well-known result of Chebyshev is that if pn ∊ Pn, (Pn is the set of polynomials of degree at most n) and

(1)

then an(pn), the leading coefficient of pn, satisfies

(2)

with equality holding only for pn = ±Tn, where Tn is the Chebyshev polynomial of degree n. (See [6, p. 57].) This is an example of an extremal problem in which the norm of a given linear operator on Pn is sought. Another example is A. A. Markov's result that (1) implies that

(3)

There are also results for the linear functionals pn(k)(x0), x0 real, k = 1, … n – 1 ([8]).

Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized to

as suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?
The Effective Version of Brooks' Theorem
Authors: Schmerl James H.
Published: December 2010

One of the fundamental results on graph coloring is the following classical theorem of Brooks.

BROOKS’ THEOREM. Suppose that k ≧ 3 and that G is a k-regular graph which does not induce a (k + 1)-clique. Then G is k-colorable.

Brooks proved his theorem in [1]; several more recent proofs have appeared in [3], [4] and [5]. All the proofs of this theorem have the common feature of applying only to finite graphs; the transition to infinite graphs can be accomplished by a very standard implementation of the Compactness Theorem (or some other equally noneffective device such as the theorem of deBruijn and Erdös [2] asserting that a graph is k-colorable if and only if each of its finite subgraphs is). Thus, it is not immediately apparent that an effective version of Brooks’ Theorem exists. It is our purpose to show, however, that the effective analogue of Brooks' Theorem is indeed true.
Dense Subalgebras of Left Hilbert Algebras
Authors: Daele A. van
Published: December 2010

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).
Latin Squares with Prescribed Diagonals
Authors: Hilton A. J. W. , Rodger C. A.
Published: December 2010

1. Introduction. An incomplete latin rectangle on t symbols σ1 …, σt of size r × s is an r × s matrix in which each cell is occupied by exactly one of the symbols σ1 …, σt in such a way that no symbol occurs more than once in any row or more than once in any column. If r = t or s = t then it is a latin rectangle; if r = s < t it is an incomplete latin square; if r = s = t it is a latin square. The diagonal of a latin square consists of the cells (i, i) (1 ≦ i ≦ t) together with the symbols occupying those cells. Let an allowed sequence of length t be a sequence of length t in which no symbol occurs exactly t – 1 times. Let an allowed diagonal of length t be a diagonal occupied by an allowed sequence.
Decomposition of Witt Rings
Authors: Carson Andrew B. , Marshall Murray A.
Published: December 2010

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].
The Pointwise Ergodic Theorem for Transformations whose Orbits contain or are contained in the Orbits of a Measure-Preserving Transformation
Authors: Kieffer John C. , Rahe Maurice
Published: December 2010

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,

where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.
A Compactification with θ-Continuous Lifting Property
Authors: Kent D. C. , Richardson G. D.
Published: December 2010

1. Let X be a topological space, and let X′ be the set of all non-convergent ultrafilters on X. If A ⊆ X, let , and A* = A ∪ A′. If is a filter on X such that for all , then let. be the filter on X* generated by ; let be the filter on X* generated by . If exists then ; otherwise, .

A convergence is defined on X* as follows: If x ∈ X, then a filter A → x in X* if and only if , where Vx(x) is the X neighborhood filter at x; , then in X* if and only if .
Some Separable Spaces and Remote Points
Authors: Dow Alan
Published: December 2010

0. Introduction. A point p ∈ βX\X is called a remote point of X if P ∉ clβXA for each nowhere dense subset A of X. If X is a topological sum Σ{Xn : n ∈ ω} we call nice if {n : F ∩ Xn = ∅} is finite for each . We call remote if for each nowhere dense subset A of X there is an with F ∩ A = ∅ and n-linked if each intersection of at most n elements of is non-empty.
Produit Tensoriel Topologique de Corps Values
Authors: Fresnel Jean , Matignon Michel
Published: December 2010

Soient K un corps local, sa clôture algébrique, R, L, M trois corps tels K ⊂ R ⊂ L ⊂ et K ⊂ R ⊂ M ⊂ . Supposons que L et M soient linéairement disjoints sur R. Quel est le complété du corps LM (compositum de L et M) pour une norme de R-algèbre qui coïncide avec la valeur absolue sur L ∪ M? On sait que LM est isomorphe en tant que R-algèbre au produit tensoriel L ⨂RM et il est facile de montrer que la valeur absolue est la plus petite norme du type considéré ci-dessus et que la norme tensorielle en est la plus grande. C'est pourquoi nous étudions d'abord le complété de L ⊗RM pour la norme tensorielle. Citons dans ce cas les résultats essentiels lorsque la caractéristique de K est nulle.
On a Realization of Prime Tangles and Knots
Authors: Van Quach Thi Cam
Published: December 2010

The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.

The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles.
On Integration in Partially Ordered Groups
Authors: Pavlakos Panaiotis K.
Published: December 2010

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.

The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.

Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.
Groupoid Enriched Categories and Homotopy Theory
Authors: Fantham P. H. H. , Moore E. J.
Published: December 2010

We are concerned in this paper with category-theoretic aspects of homotopy theory. Originally, category theory developed as a simplifying language in the context of algebraic topology and yet one primary example: the category Π of spaces and homotopy classes of maps admits only limited use of the language owing to the very sparse occurrence of limits. Of course, full use has been made of them nevertheless: limits and colimits exist in the case of products and coproducts, and in almost no other case; yet, from this we obtain the theory of Samelson products, Whitehead products, and Hopf invariants which can all be expressed in Π see [8]. In addition, there are hosts of adjoint functors and yet the outcome is disappointing because the language applies only to special cases rather than to the situation as a whole.
Homomorphism-Compact Spaces
Authors: Babiker A. G. A. G. , Graf S.
Published: December 2010

In 1979 Edgar asked for a characterization of those completely regular Hausdorff topological spaces X which have the property that any Boolean σ-homomorphism from the Baire σ-field of X into the measure algebra of an arbitrary complete probability space can be realized by a measurable point-mapping. Those spaces X will be called homomorphism-compact or, for short, H-compact hereafter. It is wellknown that compact spaces are H-compact (cf. [4], p. 637, Proposition 3.4). We will show that the same is true for strongly measure compact spaces. On the other hand H-compact spaces are easily seen to be real-compact. Since the notions of measure-compactness and liftingcompactness (cf. [3]) also lie between strong measure-compactness and real-compactness it is natural to investigate the relations among these notions. Here the results are mainly negative (cf. Sections 4 and 6). Concerning the structural properties of H-compactness not very much can be said so far (cf. Section 7): it is, for instance, unknown whether the product of two H-compact spaces is again H-compact.
Algebraic Inner Derivations on Operator Algebras
Authors: Miers C. Robert , Phillips John
Published: December 2010

Let A be a C*-algebra, let p be a polynomial over C, and let a in M(A) (the multiplier algebra of A) be such that p(ad a) = 0. In this paper we study the following problem: when does there exist λ in Z(M(A)) (the centre of M(A)) such that p(a – λ) = 0? The first result of this type known to us is due to I. N. Herstein [7], who showed that for a simple ring with identity, such a λ always exists when p is of the form p(x) = xk for some positive integer k. Later, in [8], C. R. Miers showed that the result is true for any primitive unital C*-algebra and any polynomial whatever. It was also shown in [8] that if A is a unital C*-algebra acting on H and p is any polynomial, then such a λ exists in the larger algebra Z(A″). In particular, the strict result holds for any von Neumann algebra, A.
Scalar Actions
Authors: Lebow A. , Schreiber M.
Published: December 2010

The subject of this paper arises from the familiar process whereby an automorphism of a field generates new representations from old. One may think of that process spatially, as a change of vector space structure in the representation space by means of the automorphism. The operators of the representation acting in the “new“ space then constitute the new representation. This point of view makes visible an algebraic structure we call a scalar action. A scalar action f of a ring R (with unity) in an abelian group Kis a ring homomorphism f:R → End(V) taking the unity element of R to the identity operator in End(V). If f is a scalar action of a field F and ϕ is an automorphism of F then f ∘ ϕ is another scalar action of F, and it is this construction which is used to define the “new” representation space mentioned above. But the variety of scalar actions goes rather beyond that construction.
Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2
Authors: Herz Carl
Published: December 2010

It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ(W) of “maximal” forms. If ψ ∈ Ψ(W) then the subgroup of AUTC(W) consisting of the invertible complex-linear transformations S such that ψ(S•, S•, S•) = ψ(•, •, •) is denoted G(ψ), and, in Proposition 3.6. we prove

where G1(ψ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking.
On a Generalized Fundamental Equation of Information
Authors: Kannappan Pl. , Ng C. T.
Published: December 2010

The object of this paper is to determine the general solution of the functional equation

FE

where α is multiplicative. It turns out that non-trivial embeddings of the reals in the complex generate some interesting solutions.

In many applications, various special cases of (FE) have occurred ([1,3, 4, 6, 10, 11, 14]). The special case where f = g = h = k and α = the identity map is known as the fundamental equation of information, and has been extensively investigated by many authors ([5]). The case where f = g = h = k and α is multiplicative was treated in [13, 14]. The general solution of (FE) when α(1 – x) = (1 – x)β has been obtained in [9], except when β = 2.