bluuebunny/crossref_metadata_2025 huggingface dataset:
On Class Sums inp-Adic Group Rings
Authors: Sehgal Sudarshan K.
Published: December 2010

In this note we prove that an isomorphism ofp-adic group rings of finitep-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of thep-adic group ring of a finitep-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of thep-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.

Gdenotes a finitep-group.

Zdenotes the ring of (rational) integers.

ZPdenotes the ring ofp-adic integers.

Qpdenotes thep-adic number field.
Going Down in Polynomial Rings
Authors: McAdam Stephen
Published: December 2010

In this paper, R ⊂ T will be commutative domains having a common identity.

Definition. Suppose that R is a subdomain of T.

(i) If P is a prime ideal of R and Q is a prime ideal of T, we say that Q lies over P if Q ∩ R = P.

(ii) If every prime of R has a prime of T lying over it, we say that R ⊂ T has lying over.

(iii) If there is a unique prime of T lying over P in R, we say that P is unibranched in T.

(iv) If every prime of R is unibranched in T we say that R ⊂ T is unibranched.
Finite Linear Groups of Degree Six
Authors: Lindsey J. H.
Published: December 2010

In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).
Counting Coloured Graphs. III
Authors: Wright E. M.
Published: December 2010

In an earlier paper [4], we found an asymptotic expansion for Mn = Mn(k), the number of coloured graphs on n labelled nodes, when n is large. Such a graph is a set of n distinguishable objects called nodes, and a set of “edges”, that is, undirected pairs of nodes. The nodes are mapped onto k colours. Every pair of nodes of different colours may or may not be joined by an edge, but no edge can join a pair of nodes of the same colour.

We write mn for the number of these graphs which are connected, Fn for the number which use all k colours (i.e., at least one node in each graph is mapped onto each of the k colours), and fn for the number of connected graphs which use all k colours.
Properties of Quotient Rings
Authors: Page S.
Published: December 2010

In [1; 2 ; 7] Gabriel, Goldman, and Silver have introduced the notion of a localization of a ring which generalizes the usual notion of a localization of a commutative ring at a prime. These rings may not be local in the sense of having a unique maximal ideal. If we are to obtain information about a ring R from one of its localizations, Qτ (R) say, it seems reasonable that Qτ(R) be a tractable ring. This, of course, is what Goldie, Jans, and Vinsonhaler [4; 3; 8] did in the special case for Q(R) the classical ring of quotients.
Ergodic Theory and Averaging Iterations
Authors: Koliha J. J.
Published: December 2010

Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.
The Cohomological Dimension of a Directed Set
Authors: Mitchell Barry
Published: December 2010

Let R be a ring with identity, and let C be a small, nonempty category. We denote the category of right R-modules by AbR and the category of contravariant functors C → AbR by AbRC*. The limit functor

is left exact, and its kth right derived functor is denoted by colimk. The R-cohomological dimension of C is defined by

If there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.
Lie Solvable Group Rings
Authors: Passi I. B. S. , Passman D. S. , Sehgal S. K.
Published: December 2010

Let K[G] denote the group ring of G over the field K. One of the interesting problems which arises in the study of such rings is to find precisely when they satisfy polynomial identities. This has been solved for char K = 0 in [1] and for char K = p > 0 in [3]. The answer is as follows. If p > 0 we say that group A is p-abelian if A', the commutator subgroup of A, is a finite p-group. Moreover, for convenience, we say A is 0-abelian if and only if it is abelian.
On Real Zeros of Dedekind ζ-Functions
Authors: Heilbronn H.
Published: December 2010

Let K be a finite normal extension of an algebraic number field k; let k2 be the compositum of all quadratic extensions of k which are contained in K. Let ζk(s), ζK(s) and ζk2(s) denote the Dedekind ζ-functions of these fields.
Some Remarks on the Nijenhuis Tensor
Authors: Stone A. P.
Published: December 2010

A differential form α of degree r on an n-manifold is exact if there exists a form β of degree r — 1 such that α = dβ and is closed if dα = 0. Since d-d = 0 any exact form is closed. The Poincaré lemma asserts that a closed differential form of positive degree is locally exact. There is also a complex form, proved by Cartan-Grothendieck, of the Poincaré lemma in which the operator d has a decomposition into components and .
Polynomial Ideals in Group Rings
Authors: Parmenter M. M. , Passi I. B. S. , Sehgal S. K.
Published: December 2010

Letf(x1, x2, … , xn) be a polynomial in n non-commuting variables x1, x2, … , xn and their inverses with coefficients in the ring Z of integers, i.e. an element of the integral group ring of the free group on X1, x2, … , xn. Let R be a commutative ring with unity, G a multiplicative group and R(G) the group ring of G with coefficients in R.
A Characterization of Some Geometries of Chains
Authors: Chen Yi
Published: December 2010

The geometries considered here are the Möbius plane
M() (W. Benz [1]),
the Laguerre plane L() (W. Benz and H.
Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G.
Kaerlein [18]) over a field . All of them are
geometries of an algebra with identity over a field.

The characterization of the projective plane over a field by the proposition
of Pappus first gave a close relation between algebraic and geometric structures.
B. L. v. d. Waedern and L. J. Smid [28] presented a further example by
characterizing the Möbius and Laguerre plane with incidence axioms and the
"complete" proposition of Miquel.
Symmetrized Kronecker Products of Group Representations
Authors: Butler P. H. , King R. C.
Published: December 2010

Certain phases are associated with the Kronecker squares and cubes of
representations of the finite and of the compact semi-simple groups. These phases
are important in giving the symmetry properties of the 1 —
jm and 3 — jm symbols of
the groups [4; 9]. It is our primary purpose to evaluate these phases.

The Frobenius-Schur invariant [12, p. 142] for an irreducible representation
of group
G

(1.1)
On Odd Functions of Bounded Boundary Rotation
Authors: Leach Ronald J.
Published: December 2010

Let VK denote the class of functions

that are analytic in the unit disc U, satisfy f′(z) ≠ 0 in U, and map U onto a domain with boundary rotation at most Kπ (for a definition of this concept, see [9]). V. Paatero [9] showed that f(z) ∊ VK if and only if

1.1
Capacity Estimates for Planar Cantor-Like Sets
Authors: Minda Carl David
Published: December 2010

Upper and lower bounds for the capacity of planar Cantor-like sets are presented. Chebichev polynomials are the principal tool employed in the derivation of these estimates. A necessary and sufficient condition for certain planar Cantor-like sets to have positive capacity is obtained. Related one-sided capacitary estimates for more general Cantor-like sets can be found in [3, pp. 106-109]. Techniques analogous to those used in this paper yield similar results for linear Cantor-like sets which are well-known [2, pp. 150-161]. The use of Chebichev polynomials to obtain these results provides an alternate, possibly more elementary, approach to these linear problems.
Recursively Generated Periodic Sequences
Authors: Kurshan R. P. , Gopinath B.
Published: December 2010

A sequence (xn) (n = 1, 2, . . .) is periodic if xn+p = xn for some p and all n. Periodic sequences arise naturally in geometry and arithmetic in the study of mosaic patterns [4], continued fractions and frieze patterns [3; 5]. Some digital oscillators and tone generators also generate periodic sequences. In these cases one computes the period p of the sequence in question. On the other hand, in pseudo random sequences and cryptography [8] it is required to recursively generate sequences of large periods.
Weakly Isotopic Planar Ternary Rings
Authors: Stevenson Frederick W.
Published: December 2010

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.
The Explicit Fourier Decomposition of L2SO(n)/SO(n - m))
Authors: Strichartz Robert S.
Published: December 2010

The decomposition of L2SO(n)/SO(n - m)) into a direct sum of irreducible representations of SO(n) is given abstractly by the branching theorem and the Frobenius reciprocity theorem [1]. The goal of this paper is to obtain this decomposition explicitly, generalizing the theory of spherical harmonics (m = 1). The case m = 2 has been studied in Levine [6], and the case 2m ≦ n in Gelbart [3]. Our results shed more light on these cases as well as revealing new phenomena which only occur when 2m > n.
Extensive Subcategories of the Category of T1-spaces
Authors: Hong Sung Sa
Published: December 2010

It is well known that epimorphisms in the category Top (Top1, respectively) of topological spaces (T1spaces, respectively) and continuous maps are precisely onto continuous maps. Since every mono-reflective subcategory of a category is also epi-reflective and every embedding in Top (Top1, respectively) is a monomorphism, there is no nontrivial reflective subcategory of Top (Top1 respectively) such that every reflection is an embedding. However, in the category Top0 of T0-spaces and continuous maps as well as in the category Haus of Hausdorff spaces and continuous maps, there are epimorphisms which are not onto. Moreover, every reflection of a reflective subcategory of Top0, which contains a non T1-space, is an embedding [16].
On the Ramsey Number r(F, Km) Where F is a Forest
Authors: Stahl Saul
Published: December 2010

The graphs considered here are finite and have no loops or multiple edges. In particular, Km denotes the complete graph on m vertices. For any graph G,V(G) and E(G) denote, respectively, the vertex and edge sets of G. A forest is a graph which has no cycles and a tree is a connected forest. The reader is referred to [1] or [4] for the meaning of terms not defined in this paper.
Unequivocal Rings
Authors: Divinsky N.
Published: December 2010

For any radical property Q, a nonzero simple ring (all rings in this paper are assumed to be associative) must make up its mind so to speak and must be either Q radical or Q semi-simple. Every Q thus divides the class of all nonzero simple rings into two disjoint classes. Conversely any partition of the nonzero simple rings into two disjoint classes leads to at least two radicals [1, p. 16].
Metrization of Symmetric Spaces
Authors: Harley P. W. , Faulkner G. D.
Published: December 2010

A distance function d on a set X is a function X × X → [0, ∞ ) satisfying (1) d(x, y) = 0 if and only if x = y, and (2) d(x, y) = d(y, x). Such a function determines a topology T on X by agreeing that U is an open set if it contains an ∈-sphere N(p; ∈)( = {x: d(p, x) < ∈﹜} about each of its points. Equivalently, F is closed if and only if d(x, F) > 0 for each x ∈ X — F. A topological space is symmetrizable via a distance function d if its topology is determined by d as above, and semi-metrizahle via d if x ∈ Ā is equivalent to d(x, A) = 0.
The Residual Finiteness of the Classical Knot Groups
Authors: Mayland E. J.
Published: December 2010

The purpose of this paper is to extend the class of knot groups whose commutator subgroups are known to be residually a finite pgroup (i.e., residually of order a power of the prime p). Such a knot group is known to be residually finite (see, e.g., [10]), and although this class is quite restricted we will show that it includes all the groups of knots in the classical knot table [15].
The Spectral Resolution of Some Non-Selfadjoint Partial Differential Operators
Authors: Huige Gustavus E.
Published: December 2010

Let ffl and be Hilbert spaces, the set of all densely defined linear operators from to , and its subset of bounded ones. Let and σ(T) denote the adjoint, range, domain, closure and spectrum of T respectively. Ri(z) will denote the resolvent (z — Ti)-1.
Dihedral Field Extensions of Order 2p Whose Class Numbers are Multiples of p
Authors: Callahan T.
Published: December 2010

If L is a cyclic extension of Q of prime degree p, then the class number of L is divisible by p if and only if more than one prime divides the discriminant D, of L. If p ≠ 2, then this condition is equivalent to the existence of more than one cyclic extension of Q of degree p with discriminant equal to D. In this paper we generalize these results to non-galois extensions of Q of degree p whose normal closures have degree 2p over Q.
Stable Extensions and Fields with the Global Density Property
Authors: Fried Michael , Jarden Moshe
Published: December 2010

For a field M we denote by Ms and respectively the separable closure and the algebraic closure of M. If F is a variety which is defined over M, then we denote by V(M) the set of all if-rational points of V. M is said to be pseudo-algebraically closed (PAC) field, if V(M) ≠ θ for every non-void abstract variety V defined over M. It can be shown that then is dense in V(M) in the Zariski M -topology.
Local Minimal Overrings
Authors: Papick Ira J.
Published: December 2010

Let R be a (commutative integral) domain having quotient field K. A domain S satisfying , is called an overring of R. We say R has a minimal overring T, in case and there are no domains properly between R and T. The purpose of this paper is the study of certain classes of coherent domains having local minimal overrings; that is, having minimal overrings with unique maximal ideals.
Nonlinear Oscillation of Fourth Order Differential Equations
Authors: Kusano Takaŝi , Naito Manabu
Published: December 2010

In this paper we are concerned with the fourth order nonlinear differential equation

where the following conditions are always assumed to hold:

(a) r(t) is continuous and positive for t ≠ 0, and
Cyclic Element Theory in Connected and Locally Connected Hausdorff Spaces
Authors: Lehman B.
Published: December 2010

G. T. Whyburn, in 1926, began the development of cyclic element theory for Peano continua. This theory proved fruitful in the study of Peano spaces and a comprehensive development of the theory for metric spaces was presented in [6]. An excellent history of the theory is to be found in [4]. In [7] and [5] the generalization of cyclic element theory to more general spaces was begun. However, in each of these papers only basic definitions were set forth and fundamental results obtained. In this paper, we concern ourselves primarily with connected and locally connected Hausdorff spaces, developing the cyclic element theory initiated in [7] and demonstrating that the theory has many of the applications to connected and locally connected Hausdorff spaces that the classical theory has to Peano spaces.
Saturation and Inverse Theorems for Combinations of a Class of Exponential-Type Operators
Authors: May C. P.
Published: December 2010

Rates of convergence, saturation theorems and the socalled “inverse problems” for Bernstein polynomials have been intensively studied (see, e.g., [1 ; 4; 8; 14; 17]). The same problems for some other positive operators have also been investigated by many authors. In this paper, we shall use a uniform approach to study the saturation and inverse problems for a class of linear combinations of operators including Bernstein polynomials, and Szâsz, Post-Widder, Gauss-Weierstrass and Baskakov operators.
Representations by Hermitian Forms in a Finite Field of Characteristic Two
Authors: Fulton John D.
Published: December 2010

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.
The 3-Irreducible Partially Ordered Sets
Authors: Kelly David
Published: December 2010

The dimension [4] of a partially ordered set (poset) is the minimum number of linear orders whose intersection is the partial ordering of the poset. For a positive integer m, a poset is m-irreducible[10] if it has dimension m and removal of any element lowers its dimension. By the compactness property of finite dimension, every m-irreducible poset is finite and every poset of dimension ≧ m contains an m-irreducible subposet.
Symmetrizable, -, and Weakly First Countable Spaces
Authors: Stephenson R. M.
Published: December 2010

A number of results are given concerning the character and cardinality of symmetrizable and related spaces. An example is given of a symmetrizable Hausdorff space containing a point that is not a regular Gδ , and a proof is given that if a point p of a symmetrizable Hausdorff space has a neighborhood base of cardinality , then p is a Gδ . It is shown that for each cardinal number m there exists a locally compact, pseudocompact, Hausdorff -space X with |X| ≧ m. Several questions of A. V. Arhangel'skii and E. Michael are partially answered.
Symmetric Conference Matrices of Order pq 2 + 1
Authors: Mathon Rudolf
Published: December 2010

A conference matrix of order
n is a square matrix C
with zeros on the diagonal and ±1 elsewhere, which satisfies the orthogonality
condition CCT = (n — 1)I. If in
addition C is symmetric, C =
CT, then its order
n is congruent to 2 modulo 4 (see
[5]). Symmetric conference
matrices (C) are related to several
important combinatorial configurations such as regular two-graphs, equiangular
lines, Hadamard matrices and balanced incomplete block designs
[1; 5; and 7, pp. 293-400]. We
shall require several definitions.
Generic Free Resolutions II
Authors: Buchsbaum David A.
Published: December 2010

In [1], a number of “multilinear” functors
Lp
q, denned for finitely generated free modules, were
introduced. They arose as cycles in a generic Koszul complex, and in turn gave
rise to a large family of other generic complexes. One of the things we will do in
this paper is study some of these new complexes in order to obtain new multilinear
functors on free modules which appear as their cycles.
Sur La Convergence Presque Partout des Suites de Fonctions Mesurables
Authors: Bucchioni D. , Goldman A.
Published: December 2010

L'objet de cet article est de donner quelques résultats concernant la
structure des suites de fonctions mesurables sur un espace mesuré abstrait
(X, ∑, μ), le théorème principal étant le
suivant:

Theoreme (A). Soit (ƒn) une suite de fonctions
mesurables sur un espace mesuré (X, ∑, μ) dont aucune
sous-suite ne converge presque partout. Il existe alors un élément Y ∊ ∑, μ
(Y) > 0, deux nombres r ∊ R, δ > 0 et
une partie infinie M de N tels
que, pour tout A ∊ ∑, ⊂ C
F, μ(A) > 0 et pour toute
partie infinie L CP.S. M (c'est-à-dire L\M est fini), on puisse trouver
x et y ∊
A vérifiant ƒn(x) > r +
δ et ƒn(y) < r pour une
infinité d'indices n ∊ L.
Suites D'Interpolation Pour les Classes de Bergman de la Boule et du Polydisque de Cn
Authors: Amar Eric
Published: December 2010

Soit Dn = ﹛z = (z1, . . . , zn) ∈ Cn, |zi| <> 1﹜ le polydisque de Cn et ƛn la mesure de Lebesgue de Cn normalisée sur Dn. Pour b > 0, .on définit les espaces de Bergman Ap(ƛn) de la manière suivante:

Ap(ƛn) est l'espace des fonctions analytiques dans Dn telles que:
Uniform Approximation by Meromorphic Functions on Closed Sets with Continuous extension into the Boundary
Authors: Roth Alice
Published: December 2010

Notation. For S a subset of the extended plane C* = C ∪ (∞ ) let S0 be its interior, its closure in C* and its boundary. H(S) and M(S) will denote the sets of all restrictions to S of functions which are, on a neighborhood of S, holomorphic or meromorphic, respectively. A (S) will stand for the set of functions from S to C* which are continuous on S and whose restriction to S° are holomorphic.
Analytic Sets, Baire Sets and the Standard Part Map
Authors: Henson C. Ward
Published: December 2010

The problems considered here arose in connection with the interesting use by Loeb [8] and Anderson [1], [2] of Loeb's measure construction [7] to define measures on certain topological spaces. The original problem, from which the results given here developed, was to identify precisely the family of sets on which these measures are defined.

To be precise, let be a set theoretical structure and * a nonstandard extension of , as in the usual framework for nonstandard analysis (see [10]). Let X be a Hausdorff space in and stx the standard part map for X, defined on the set of nearstandard points in *X. Suppose, for example, µ is an internal, finitely additive probability measure defined on the internal subsets of *X.
Stable Conjugacy: Definitions and Lemmas
Authors: Langlands R. P.
Published: December 2010

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.

The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.
Tensor Products of Holomorphic Discrete Series Representations
Authors: Repka Joe
Published: December 2010

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.

For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.

The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.
Examples and Questions in the Theory of Fixed Point Sets
Authors: Martin John R. , Nadler Sam B.
Published: December 2010

All spaces considered in this paper will be metric spaces. A subset A of a space X is called a fixed point set of X if there is a map (i.e., continuous function) ƒ: X → X such that ƒ(x) = x if and only if x ∈ A. In [22] L. E. Ward, Jr. defines a space X to have the complete invariance property (CIP) provided that each of the nonempty closed subsets of X is a fixed point set of X. The problem of determining fixed point sets of spaces has been investigated in [14] through [20] and [22]. Some spaces known to have CIP are n-cells[15], dendrites [20], convex subsets of Banach spaces [22], compact manifolds without boundary [16], and a class of polyhedra which includes all compact triangulable manifolds with or without boundary [18].
A Fundamental Solution for a Nonelliptic Partial Differential Operator
Authors: Greiner Peter C.
Published: December 2010

Let

(1)

and set

(2)

Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”

(3)

In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].
Polynomial Invariants of Finite Linear Groups of Degree Two
Authors: Huffman W. Cary
Published: December 2010

Recently invariant theory of linear groups has been used to determine the structure of several weight enumerators of codes. Under certain conditions on the code, the weight enumerator is invariant under a finite group of matrices. Once all the polynomial invariants of this group are known, the form of the weight enumerator is restricted and often useful results about the existence and structure of codes can be found. (See [5], [8], [14], and [15].) Many of the groups in these applications are of degree 2; in this paper all the invariants of finite 2 X 2 matrix groups over C are determined.
Intersections of Real Closed Fields
Authors: Craven Thomas C.
Published: December 2010

In this paper we wish to study fields which can be written as intersections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hereditarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythagorean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal , the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3].
On the Nonstandard Duality Theory of Locally Convex Spaces
Authors: Grainger Arthur D.
Published: December 2010

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).

The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).
Completely Regular Mappings and Homogeneous, Aposyndetic Continua
Authors: Rogers James T.
Published: December 2010

The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua.

The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these mappings were invented for quite another purpose [1]. The author believes that these maps are likely to provide even more new information about homogeneous continua.

A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h: M → M such that h(p) = q.
Infinitely Determined Mapgerms
Authors: Wilson Leslie C.
Published: December 2010

In differential analysis, it is very useful to have the local behavior of a differentiable map be determined by the derivatives of the map at a point. Hence we have the theories of finite and infinitely determined germs. Letmnpbe the space of germs ofC∞mapsf: (Rn, 0) → (Rp, 0) andGa group operating onmnp. A germfis calledfinitely G-determinedif there exists an integerksuch that every germ having the samek-jet asfisG-equivalentto (i.e., in the sameG-orbit as)f. A germfis called ∞-G-determinedif every germ having the same formal power series asfisG-equivalent tof.
A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials
Authors: Rahman Mizan
Published: December 2010

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:

with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:

(1.1)

Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion

(1.2)
On the Size of a Maximum Transversal in a Steiner Triple System
Authors: Brouwer A. E.
Published: December 2010

Let (X, ) be a Steiner triple system on v = |X| points, and suppose that is a partial parallel class (transversal, clear set, set of pairwise disjoint blocks) of maximum size . We want to derive a bound on . (I conjecture that in fact r is bounded, e.g., r ≦ 4 – 4 is attained for the Fano plane, but all that has been proved so far (cf. [1], [2]) are bounds r < C.v for some C. Here we shall prove r < 5v2/3.)

Define a sequence of positive real numbers by q0 = Q · r2/v, , where l is determined by ql ≧ 6, , i.e.,

(The constant Q will be chosen later.) Define inductively sets Ai, Ki and collections as follows. Let
Closed Lie Ideals in Operator Algebras
Authors: Miers C. Robert
Published: December 2010

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.
Recursive Colorings of Highly Recursive Graphs
Authors: Kierstead Henry A.
Published: December 2010

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.

A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].
Real Flexible Division Algebras
Authors: Benkart Georgia M. , Britten Daniel J. , Osborn J. Marshall
Published: December 2010

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].

All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.
The Associated Ultraspherical Polynomials and their q-Analogues
Authors: Bustoz Joaquin , Ismail Mourad E. H.
Published: December 2010

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that

1.1

A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence

1.2

with

1.3

Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by

1.4

with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.
Elements of Order Coxeter Number +1 in Chevalley Groups
Authors: Chang Bomshik
Published: December 2010

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.

Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrix

has order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.
Class Numbers and Biquadratic Reciprocity
Authors: Williams Kenneth S. , Currie James D.
Published: December 2010

0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,

(0.1)

with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by

(0.2)

Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by

(0.3)

These choices are assumed throughout.

The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].
Eisenstein Series for Reductive Groups Over Global Function Fields II: The General Case
Authors: Morris L. E.
Published: December 2010

This paper is a continuation of [5]. As stated there, the problem is to explicitly decompose the space L2 = L2(G(F)\G(A)) into simpler invariant subspaces, and to deal with the associated continuous spectrum in case G is a connected reductive algebraic group defined over a global function field. In that paper the solution was begun by studying Eisenstein series associated to cusp forms on Levi components of parabolic subgroups; these Eisenstein series and the associated intertwining operators were shown to be rational functions satisfying functional equations. To go further it is necessary to consider more general Eisenstein series and intertwining operators, and to show that they have similar properties. Such Eisenstein series arise from the cuspidal ones by a residue taking process, which is detailed in a disguised form suitable for induction in the first part of this paper: the main result is a preliminary form of the spectral decomposition.
Trivial Set-Stabilizers in Finite Permutation Groups
Authors: Gluck David
Published: December 2010

For which permutation groups does there exist a subset of the permuted set whose stabilizer in the group is trivial?

The permuted set has so many subsets that one might expect that subsets with trivial stabilizer usually exist. The symmetric and alternating groups are obvious exceptions to this expectation. Another, more interesting, infinite family of exceptions are the 2-Sylow subgroups of the symmetric groups on 2n symbols, in their natural representations on 2n points.

One of our main results, Corollary 1, sheds some light on this last family of groups. We show that when the permutation group has odd order, there is indeed a subset of the permuted set whose stabilizer in the group is trivial. Corollary 1 follows easily from Theorem 1, which completely classifies the primitive solvable permutation groups in which every subset of the permuted set has non-trivial stabilizer.
C-Nodal Surfaces of Order Three
Authors: Bisztriczky Tibor
Published: December 2010

The problem of describing a surface of order three can be said to originate in the mid-nineteenth century when A. Cayley discovered that a non-ruled cubic (algebraic surface of order three) may contain up to twenty-seven lines. Besides a classification of cubics, not much progress was made on the problem until A. Marchaud introduced his theory of synthetic surfaces of order three in [9]. While his theory resulted in a partial classification of a now larger class of surfaces, it was too general to permit a global description. In [1], we added a differentiability condition to Marchaud's definition. This resulted in a partial classification and description of surfaces of order three with exactly one singular point in [2]-[5]. In the present paper, we examine C-nodal surfaces and thus complete this survey.
The Enumeration of Non-Isomorphic 2-Connected Planar Maps
Authors: Liskovets V. A. , Walsh T. R. S.
Published: December 2010

Since Tutte initiated the systematic enumeration of planar maps [11], most of the literature on the subject has dealt with rooted maps (i.e., with maps whose automophism group has been trivialized by distinguishing a doubly-oriented edge). In particular, Tutte proved [11] that the number B′(n) of rooted planar 2-connected (i.e., non-separable) maps with n ≧ 1 edges is expressed by the formula

Recently one of the authors developed a general technique for enumerating unrooted planar maps considered up to orientationpreserving isomorphisms (see [6] and [8]). This technique, which is based on combinatorial map theory, Burnside’s lemma [3, p. 181] and the concept of a quotient map (see Section 1.4), was used to find, with little algebraic manipulation, simple counting formulae for the numbers of non-isomorphic planar maps of several types [7].
Merotopic Spaces and Extensions of Closure Spaces
Authors: Chattopadhyay K. C. , Njåstad Olav , Thron W. J.
Published: December 2010

Proximity spaces and contiguity spaces, and more recently nearness spaces, have been studied not just because they provide various approaches to uniform structure. Possibly of greater importance is that they can be used as a means of introducing compactifications and more general extensions of the topological spaces on which they are defined. Riesz [20] was probably the first to recognize this connection. Since then the idea was used by Freudenthal [9], Alexandroff [1], Smirnov [21], Leader [17] and Ivanov and Ivanova [13, 14, 15] among others.

Recently Reed [19] using work of Bentley [2, 4] and Herrlich [11, 12] studied the 1 – 1 correspondence between the class of all cluster generated nearness spaces and the class all principal T1-extensions of a given T1-space. She succeeded in showing that the mapping induces a 1 – 1 correspondence between the contingual nearness spaces in and the compactifications in .
On the Regularity of the Kowalsky Completion
Authors: Lowen-Colebunders Eva
Published: December 2010

Cauchy spaces were introduced by Kowalsky in 1954 [9]. In that paper a first completion method for these spaces was given. In 1968 Keller [5] has shown that the Cauchy space axioms characterize the collections of Cauchy filters of uniform convergence spaces in the sense of [1]. Moreover in the completion theory of uniform convergence spaces the associated Cauchy structures play an essential role [12]. This fact explains why in the past ten years in the theory of Cauchy spaces, much attention has been given to the study of completions.
Associated Prime Ideals in Non-Noetherian Rings
Authors: Iroz Juana , Rush David E.
Published: December 2010

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].
A Generalization of an Addition Theorem for Solvable Groups
Authors: Yuster Thomas , Peterson Bruce
Published: December 2010

The “sets” in this paper are actually multi-sets. That is, we allow an element to occur several times in a set and distinguish between the number of elements in a set and the number of distinct elements in the set. On the few occasions when we need to avoid repetition we will use the term “ordinary set.“

Definition. Let G be a group and let S a set of elements of G. An r-sum in S is an ordered subset of S of cardinality r; the result of that r-sum is the product of its elements in the designated order.

Definition. If S is a set, r(x, S) denotes the number of times x appears in S and [x, S] is a set consisting of r(x, S) copies of x. An n-set or n-subset is a set consisting of n elements. Hence [x, S] is an r(x, S)-subset of S.
Injectivity in the Topos of Complete Heyting Algebra Valued Sets
Authors: Higgs Denis
Published: December 2010

Let be a complete Heyting algebra (CHA). An -valued set is a pair (X, δ) where X is a set and δ is a function from X × X to such that

for all x, y z in X. -valued sets form a category as follows: a morphism from (X, δ) to (Y, δ) is a function f from X × Y to such that

(i)

f(x) ∧ δ(x, x′) ≦ f(x′, y), f(x, y) ∧ δ(y, y′) ≧ f(x, y′),

(ii)

f(x, y) ∧ f(x, y′) ≦ δ(y, y′), and

(iii)
On the Difference Property of the Class of Pointwise Discontinuous Functions and of Some Related Classes
Authors: Laczkovich M.
Published: December 2010

Let R denote the set of real numbers. For f:R → R and h ∈ R, the difference function Δhf is defined by

The function H:R → R is called additive if it satisfies Cauchy's equation

Let ℱ be a class of real valued functions defined on R. ℱ is said to have the difference property if, for every function f:R → R satisfying Δhf ∈ ℱ for every h ∈ R, there exists an additive function H such that f — H ∈ ℱ

It was conjectured by P. Erdos that the class of continuous functions has the difference property. This conjecture was proved by N. G. de Bruijn in [1], where the difference property of several other classes was verified as well. (For other references, see [6].)
Modular and Admissible Semilattices
Authors: Hoo C. S. , Murty P. V. Ramana
Published: December 2010

We correct some errors in [1] and extend some of the results there. Generally, we shall follow the terminology and notation of [1]. There is an error in the proof of Lemma 3.13 there, and consequently the subsequent results which depend on it are incorrect as stated. However, they are correct if we replace the condition “a-admissible” by “strongly a-admissible” (see [3] where this notion was introduced). We also show that the results in [1] are correct if the semilattices are assumed to be modular.

We shall change the terminology in [3] slightly.

Definition 1 (see [3]). Let A be a Boolean algebra and let D be a meet semilattice with 1. An admissible map f.A X D → D is called strongly admissible if

where a’ is the complement of a in A.