bluuebunny/crossref_metadata_2025_split huggingface dataset:
Solvable Subgroups and their Lie Algebras in Characteristic p
Authors: Winter David J. 
Published: December 2010

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .
Multiplicative Groups Under Field Extension
Authors: May Warren 
Published: December 2010

Let K be a field and L an extension field. L. Fuchs [2, Problem 98] has suggested studying the change in multiplicative groups in going from K* to L*. We wish to indicate difficulties that arise in trying to relate the group theoretic structure of L* to that of K*, even when K* has particularly simple structure and the extension is quadratic.First let us note a trivial fact. If [L : K] = n < ∞ and K* has a free direct factor A, then L* has a free direct factor isomorphic to A. To see this, let ϕ be the composite L* → K* → A of the norm map followed by the projection map. Then L* has a free direct factor isomorphic to ϕ(L*). But the image of the norm map contains (K*)n, hence ϕ(L*) ≅ A.
Graded π-rings
Authors: Anderson D. D. , Matijevic J. 
Published: December 2010

All rings considered will be commutative with identity. By a graded ring we will mean a ring graded by the non-negative integers.A ring R is called a π-ring if every principal ideal of R is a product of prime ideals. A π-ring without divisors of zero is called a π-domain. A graded ring (domain) is called a graded π-ring (-domain) if every homogeneous principal ideal is a product of homogenous prime ideals. A ring R is called a general ZPl-ring if every ideal is a product of primes. A graded ring is called a graded general ZPl-ring if every homogenous ideal is a product of homogeneous prime ideals.
Neighbor Relation and Neighbor Homomorphism of Hjelmslev Groups
Authors: Knüppel Frieder , Kunze Michael 
Published: December 2010

The geometry of Hjelmslev groups is a comprehensive plane metric geometry. It supplies, for example, an approach to euclidean, hyperbolic, elliptic, Minkowskian and Galilean geometry.The subject of this geometry are Hjelmslev groups and their group planes. (The definition of Hjelmslev groups and some basic concepts and propositions of this theory can be found in the second edition of Bachmann's book [1] (pages 318-328). A similar report in English is the lecture [3]. A comprehensive introduction is developed in the key work [2]. The first part of this work was translated by Garner [4]. An abstract is given in Math. Rev. 52, 9066 (1976).) The group plane arises from giving geometric names to some group theoretical facts. The group plane of a Hjelmslev group is an incidence structure with orthogonality, and the Hjelmslev group acts on this plane as a group of motions.
L-Indistinguishability For SL (2)
Authors: Labesse J-P. , Langlands R. P. 
Published: December 2010

The notion of L-indistinguishability, like many others current in the study of L-functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL(2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [12].The phenomena which the notion is intended to express have been met–and exploited–by others (Hecke [5] § 13, Shimura [17]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and the algebraic closure of F then two elements of G(F) may be conjugate in without being conjugate in G(F).
A Remark on Disintegrations With Almost All Components Non -Additive
Authors: Dang-Ngoc Nghiem 
Published: December 2010

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:(i) ∀x ∊ X , σx(.) is a finitely additive measure .(ii) ∀A ∊ , σ. (A) is constant on each -atom.(iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and (iv)σx(B) = 1 if x ∊ B ∊ .
The Group of Units in K-Theory Modulo an Odd Prime
Authors: Steiner R. J. 
Published: December 2010

1. Introduction. There are several multiplicative cohomology theories for which the group of units in the zeroth term is the zeroth term of another cohomology theory. Examples, due to Segal, May and others, are given by ordinary cohomology with rather general graded coefficients, real and complex K-theory with integral coefficients, and various bordism theories, also with integral coefficients [8, 7, 2, 5, IV]. The object of this paper is to show that complex K-theory modulo an odd prime p provides a counter-example.To state the theorem precisely we recall the result of Araki and Toda that there is a unique anticommutative associative admissible multiplication in K*( ;Z/p) for p an odd prime [3, 3, 7, 10]; admissible is defined in [3] and means essentially that the reduction homomorphism K*( ) → K*( ; Z/p) preserves products.
Quadratic forms over Quadratic Extensions of Fields with Two Quaternion Algebras
Authors: Cordes Craig M. , Ramsey John R. 
Published: December 2010

In this paper, we analyze what happens with respect to quadratic forms when a square root is adjoined to a field F which has exactly two quaternion algebras. There are many such fields—the real numbers and finite extensions of the p-adic numbers being two familiar examples. For general quadratic extensions, there are many unanswered questions concerning the quadratic form structure, but for these special fields we can clear up most of them.It is assumed char F ≠ 2 and K = F (√a) where a ∊ Ḟ – Ḟ2. Ḟ denotes the non-zero elements of F. Generally the letters a, b, c, … and α, β, … refer to elements from Ḟ and x, y, z, … come from .
Subgroups of the Power Semigroup of a Finite Semigroup
Authors: Putcha Mohan S. 
Published: December 2010

Throughout this paper, S will denote a finite semigroup and Z+ the set of positive integers. E = E(S) denotes the set of idempotents of S. Let . If , then let AB = {ab| a ∈ A, b ∈ B}. has been studied by many authors, including [2, 3, 5, 6, 7]. If X is a set, then |X| denotes the cardinality of X. For undefined terms in this paper, see [1,4].THEOREM 1. Let I be an ideal of S, a subgroup of . Then has a normal subgroups such that is isomorphic to a subgroup of and is isomorphic to a subgroup of .
A Decomposition for Sets Having a Segment Convexity Property
Authors: Breen Marilyn 
Published: December 2010

Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments.
Non-Amphicheiral Codimension 2 Knots
Authors: González-Acuña F. , Montesinos José M. 
Published: December 2010

An n-knot (Sn+2, Sn) is amphicheiral if there is an orientation reversing autohomeomorphism of Sn+2 leaving Sn invariant as a set. It is invertible if there is an orientation preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation reversing autohomeomorphism of Sn onto itself. In 1961 Fox [8, Problem 35] asked if there exist non-amphicheiral locally flat 2-knots. We will prove the following THEOREM 1. For any integer n there are smooth n-knots which are neither amphicheiral nor invertible.A knot (Sn+2, Sn) is + amphicheiral (resp. —amphicheiral) if there is an orientation reversing autohomeomorphism f of Sn+2 leaving Sn invariant such that f| Snpreserves (resp. reverses) orientation.
Generalized Directional Derivatives and Subgradients of Nonconvex Functions
Authors: Rockafellar R. T. 
Published: December 2010

Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ: x →∂ƒ(x) is the sub differential of ƒ.Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ1 + ƒ2,ƒ = g o A, etc.), comprise the sub differential calculus.
The Duality of Distributive Continuous Lattices
Authors: Banaschewski B. 
Published: December 2010

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.
Some Properties of the q-Hermite Polynomials
Authors: Allaway WM. R. 
Published: December 2010

Heine [7, p. 93] gave the following representation for the Legendre Polynomial {Pn(x)}∞n=owhere fo,n = 1 andSzegö [7, p. 96] generalized this result to the Ultraspherical Polynomial set ﹛Cnλ(x)﹜∞n=o and obtainedwhere
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.
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 defineA 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) andwhere 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].
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).
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.
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.
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 = 0ThenUsing 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.
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).
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 thatIn [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].
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 blocksB(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.