bluuebunny/crossref_metadata_2025 huggingface dataset:
Inequalities and Inverse Theorems in Restricted Rational Approximation Theory
Authors: Borwein Peter
Published: December 2010

The following lemma, part a) due to S. N. Bernstein and part b) due to A. A. Markov, is fundamental to the proofs of many inverse theorems in polynomial approximation theory.

LEMMA 1. [2, p. 62 and p. 67] Let ∏n denote the real polynomials of degree at most n. Let p ∈ ∏n, then

a)

b)

From the lemma one can deduce, for example:

THEOREM 1. Iƒ there is a sequence of polynomials pn ∈ ∏n and a δ > 0 so that ‖f – pn‖[a, b] ≧ A/nk + δ then f is k times continuously differentiate on (a, b).

We shall refer to inequalities, such as those of Lemma 1 that bound the derivative r′ of a rational function of degree n in terms of its supremum norm ‖r‖[a, b] and n, as Bernstein-type inequalities.
The Erdmann Condition and Hamiltonian Inclusions in Optimal Control and the Calculus of Variations
Authors: Clarke Frank H.
Published: December 2010

Consider the basic problem in the calculus of variations, that of minimizing

1.1

over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that

1.2

along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that

1.3

a relation which always evokes classical Hamiltonian mechanics and conservation laws.
Propriétés Presque Sûres et Quasi-Sûres Des Séries de Dirichletet et des Produits D'Euler
Authors: Queffélec Hervé
Published: December 2010

Il y a en mathématiques plusieurs théorèmes d'existence: ceux liés à la notion de point fixe (théorème des fonctions implicites, solution locale des équations différentielles y’= f(x, y)) ou ceux liés à la notion de cardinalité (théorèmes de Chevalley et Warning, de Sylow, existence de nombres réels transcendants) par exemple. Nous nous intéressons ici exclusivement à deux sortes de théorèmes d'existence: ceux liés au théorème de Baire (méthodes quasi-sûres) et ceux liés à la théorie des probabilités (méthodes presque sûres). Ces deux méthodes ont déjà donné lieu à de nombreux théorèmes: existence de fonctions continues partout sans dérivée, existence de fonctions C∞partout non analytiques, existence d'ensembles de Kronecker parfaits (Kaufman) par le quasi-sûr, existence de réels t tels que (tn) soit équirépartie modulo 1 (Koksma), existence de nombres normaux au sens de Borel, existence d'ensembles de nonsynthèse par le presque sûr, etc …
Relativized Weak Mixing of Uncountable Order
Authors: McMahon Douglas
Published: December 2010

We show that if Y is a metric minimal flow and θ: Y→Z in an open homomorphism that has a section (i.e., a RIM), and if S(θ)= R(θ),then °YΩ contains a dense set of transitive points, where Ω is the first uncountable ordinal

YΩ = П{Y:1 ≦ α < Ω and α not a limit ordinal}, and

°YΩ = {y ∈ YΩ:θ(yα)= θ(yβ)for 1 ≦ α,β < Ω and α, β not limit ordinals},

S(θ) is the relativized equicontinuous structure relation, and

R(θ)= {(y1,y2) ∈ Y X Y:θ(y1) = θ(y2)}.

We use this to generalize a result of Glasner that a metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI, [5].

I would like to thank Professor T. S. Wu for making helpful suggestions, and thank the referee for his time and effort.
Spaces of Orderings IV
Authors: Marshall Murray
Published: December 2010

A major goal of this paper is to give a proof of the following isotropy criterion: Let X = (X,G) be a space of orderings in the terminology of [9] or [10], and let f be a form defined over G.Then f is anisotropic over X if and only if f is anisotropic over some finite subspace of X.This is the content of Theorem 1.4, and generalizes [1, Corollary 3.4]. Moreover, in view of the known structure of finite spaces (see [9]), this has, essentially, the strength of [2, Satz 3.9] or [12, Theorem 8.12]. The technique used to prove this criterion is roughly patterned on that of [6], and yields some interesting by-products: An interesting invariant of a space of orderings called the chain length is introduced (Definition 1.1) and spaces of orderings with finite chain length are classified (Theorem 1.6).
Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation
Authors: Webb G. F.
Published: December 2010

In this paper we study the nonlinear initial boundary value problem

(1.1) ωtt— αΔ ωt— Δω= f(ω), t> 0

ω(x, 0) = ϕ(x), x∈ Ω

ωt(x, 0) = ψ (x), x∈ Ω

ω(x, t ) = 0, x ∈ ∂Ω, t ≥ 0.

In (1.1) Ω is a smooth bounded domain in Rn, n = 1, 2, 3, α > 0, and f ∈ C1(R;R) with f‘(x) ≦ co for all x ∈ R (where c0 is a nonnegative constant), lim sup|x|→+∞f(x)/x ≦0, and f(0) = 0. Our objective will be to establish the existence of unique strong global solutions to (1.1) and investigate their behavior as t→ +∞.

Our approach takes advantage of the semilinear character of (1.1) and reformulates the problem as an abstract ordinary differential equation in a Banach space.
On Definability of Nonmeasurable Sets
Authors: Friedman Harvey
Published: December 2010

In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.

In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.
(D-2)-Extreme Points and a Helly-Type Theorem for Starshaped Sets
Authors: Breen Marilyn
Published: December 2010

We begin with some preliminary definitions. Let S be a subset of Rd. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by Sx = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{Sx: x in S}.

Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧ 2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in Rd, d ≧ 2, then ker S = ⋂{Se: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected.
Modular Representations of Finite Groups
with Unsaturated Split (B,N)-Pairs
Authors: Tinberg N. B.
Published: December 2010

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if

(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”

(ii) H= ⋂n∈N n-1Bn

(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.

A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)
Conformal Geometry and the Cyclides of Dupin
Authors: Cecil Thomas E. , Ryan Patrick J.
Published: December 2010

A Riemannian manifold (M, g) is said to be conformally flat if every point has a neighborhood conformai to an open set in Euclidean space. Over the past thirty years, many papers have appeared attacking, with varying degrees of success, the problem of classifying the conformally flat spaces which occur as hypersurfaces in Euclidean space. Most of these start from the following pointwise result of Schouten.
Locally Flat Vector Lattices
Authors: Anderson Marlow
Published: December 2010

Let G be a lattice-ordered group (l-group). If X ⊆ G, then let

Then X’ is a convex l-subgroup of G called a polar. The set P(G) of all polars of G is a complete Boolean algebra with ‘ as complementation and set-theoretic intersection as meet. An l-subgroup H of G is large in G (G is an essential extension of H) if each non-zero convex l-subgroup of G has non-trivial intersection with H. If these l-groups are archimedean, it is enough to require that each non-zero polar of G meets H. This implies that the Boolean algebras of polars of G and H are isomorphic. If K is a cardinal summand of G, then K is a polar, and we write G = K⊞K'.
Longest Cycles in 3-Connected 3-Regular Graphs
Authors: Bondy J. A. , Simonovits M.
Published: December 2010

In this paper, we study the following question: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to Tait [6], who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such that the length of a longest cycle in each member of the family is at most nc, where c = 1 – 2–17 and n is the number of vertices. The exponent c was subsequently reduced by Walther [8, 9] and by Grùnbaum and Walther [4].
Fredholm Toeplitz Operators and Slow Oscillation
Authors: Power S. C.
Published: December 2010

The purpose of this paper is to show how Fred hoi m criteria for Toeplitz operators, whose symbols lie in an algebra,A, may often be generalized to cover a larger symbol algebra generated by A and SO, the slowly oscillating functions. Mere A and SO are algebras of continuous functions on the real line, so that we are concerned principally with the effect of a single discontinuity in the symbol function.

We shall treat the cases when A is the almost periodic functions, the semi-almost periodic functions and the multiplicatively periodic functions. Sufficient criteria for Fredholmness are obtained in Section 5. The more difficult task of establishing necessary and sufficient criteria is only achieved here for the slowly oscillating almost periodic functions and this is done in Section 6.
On James' Quasi-Reflexive Banach Space as a Banach Algebra
Authors: Andrew Alfred D. , Green William L.
Published: December 2010

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.

In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.
The Spectra for the Conjugate Invariant Subgroups of n2 × 4 Orthogonal Arrays
Authors: Lindner C. C. , Mullin R. C. , Hoffman D. G.
Published: December 2010

An n2 × k orthogonal array is a pair (P, B) where P = {1, 2, …, n} and B is a collection of k-tuples of elements from P (called rows) such that if i < j ∈ {1, 2, …, k} and x and y are any two elements of P (not necessarily distinct) there is exactly one row in B whose ith coordinate is x and whose jth coordinate is y. We will refer to the ith coordinate of a row r as the ith column of r. The number n is called the order (or size) of the array and k is called the strength.
A Characterization of Identities Implying Congruence Modularity I
Authors: Day Alan , Freese Ralph
Published: December 2010

In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jónsson showed in [10] that from this “congruence modularity” of a variety of algebras one can even deduce the (stronger) Arguesian identity.

These and similar results [3; 5; 9; 12; 18; 21] induced Jónsson in [17; 18] to introduce the following notions. For a variety of algebras , is the (congruence) variety of lattices generated by the class () of all congruence lattices θ(A), . Secondly if is a lattice identity, and Σ is a set of such, holds if for any variety implies .
Structure of Some Noetherian Injective Modules
Authors: Sarath B.
Published: December 2010

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. IfQis an injective indecomposable noetherian module, thenQcontains a non-zero submoduleQ0such that the endomorphism rings ofQ0and all its submodules are skewfields. Over a commutative ring, such aQ0is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.
On Extending the Trace as a Linear Functional, II
Authors: Elliott George A.
Published: December 2010

A positive bounded selfadjoint operator is in the trace class of von Neumann and Schatten ([4]) if the sum of its diagonal matrix elements with respect to some orthonormal basis is finite, and the trace is then defined to be this sum, which is independent of the basis. A bounded selfadjoint but not necessarily positive operator x is in the trace class if in the decomposition x = x+ – x−, with x+ and x− positive and x+x− = 0, both x+ and x−are in the trace class; the trace of x is then defined to be the difference of the finite traces of x+ and x−. The trace defined in this way is a linear functional on the trace class, and is unitarily invariant; if u is a unitary operator, the trace of uxu−1 is the same as the trace of x.
Projective Geometries that are Disjoint Unions of Caps
Authors: Kestenband Barbu C.
Published: December 2010

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.

A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).

Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:

All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.
Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions
Authors: Bshouty D. , Hengartner W. , Schober G.
Published: December 2010

Let S be the set of all normalized univalent analytic functions ƒ(z) = z + a2z2 + … in the open unit disk U. Then ƒ(U) contains the disk . Here is the best possible constant and is referred to as the Koebe constant for S. On the other extreme, ƒ(U) cannot contain the disk {|w| < 1}; unless ƒ is the identity mapping.

In order to interpolate between the class S and the identity mapping, one may introduce the families , of functions ƒ ∈ S such that ƒ(U) contains the disk {|w| < d};. Then S(d1) ⊃ S(d2) for d1 < d2, and S(1) contains only the identity mapping. It is obvious that d is the “Koebe constant” for S(d). The relation between d and the second coefficient a2 has been studied by E. Netanyahu [5, 6].
On Nonreflexive Banach Spaces Which Contain No c0 or lp
Authors: Casazza P. G. , Lin Bor-Luh , Lohman R. H.
Published: December 2010

The first infinite-dimensional reflexive Banach space X such that no subspace of X is isomorphic to c0 or lp, 1 ≦ p < ∞, was constructed by Tsirelson [8]. In fact, he showed that there exists a Banach space with an unconditional basis which contains no subsymmetric basic sequence and which contains no superreflexive subspace. Subsequently, Figiel and Johnson [4] gave an analytical description of the conjugate space T of Tsirelson's example and showed that there exists a reflexive Banach space with a symmetric basis which contains no superreflexive subspace; a uniformly convex space with a symmetric basis which contains no isomorphic copy of lp, 1 < p < ∞; and a uniformly convex space which contains no subsymmetric basic sequence and hence contains no isomorphic copy of lp, 1 < p < ∞. Recently, Altshuler [2] showed that there is a reflexive Banach space with a symmetric basis which has a unique symmetric basic sequence up to equivalence and which contains no isomorphic copy of lp, 1 < p < ∞.
Maximal Left Ideals and Idealizers in Matrix Rings
Authors: Stone David R.
Published: December 2010

Let R be a ring with identity, Mn(R) the ring of n × n matrices over R. The lattice of two-sided ideals of R is carried via A → Mn(A) to form the lattice of two-sided ideals of Mn(R). We wish to study the more complex left ideal structure of Mn(R). For example, if K is a commutative field, then Mn(K) has non-trivial left ideals. In particular Mn(K) has the maximal left ideal consisting of all matrices with some designated column zero. Or for any ring with maximal left ideal M, Mn(R) has the maximal left ideal consisting of all matrices with some column's entries from M. In Theorem 1.2 we characterize the maximal left ideals of Mn(R) in terms of those of R. We briefly study some contraction properties of maximal left ideals in matrix rings. For R commutative we “count” the maximal left ideals of Mn(R) and describe the idealizer of any such ideal; in the case where K is a field we see that the collection of maximal left ideals of Mn(K) can be naturally identified with Pn–1(K) (projective space).
Minimal Operators for Schrodinger-Type Differential Expressions with Discontinuous Principal Coefficients
Authors: Faierman M. , Knowles I.
Published: December 2010

The objective of this paper is to extend the recent results [7, 8, 9] concerning the self-adjointness of Schrödinger-type operators with singular potentials to a more general setting. We shall be concerned here with formally symmetric elliptic differential expressions of the form

1.1

where x = (x1, …, xm) ∈ Rm (and m ≧ 1), i = (–1)1/2, ∂j = ∂/∂xj, and the coefficients ajk, bj and q are real-valued and measurable on Rm.

The basic problem that we consider is that of deciding whether or not the formal operator defined by (1.1) determines a unique self-adjoint operator in the space L2(Rm) of (equivalence classes of) square integrable complex-valued functions on Rm.
Operators which Factor through Convex Banach Lattices
Authors: Reisner Shlomo
Published: December 2010

We investigate here classes of operators T between Banach spaces E and F, which have factorization of the form

where L is a Banach lattice, V is a p-convex operator, U is a q-concave operator (definitions below) and jF is the cannonical embedding of F in F”. We show that for fixed p, q this class forms a perfect normed ideal of operators Mp, q, generalizing the ideal Ip,q of [5]. We prove (Proposition 5) that Mp, q may be characterized by factorization through p-convex and q-concave Banach lattices. We use this fact together with a variant of the complex interpolation method introduced in [1], to show that an operator which belongs to Mp, q may be factored through a Banach lattice with modulus of uniform convexity (uniform smoothness) of power type arbitrarily close to q (to p). This last result yields similar geometric properties in subspaces of spaces having G.L. – l.u.st.
On Characterizing Injective Sheaves: Corrigendum
Authors: Dobbs David E.
Published: December 2010

B. Banaschewski [1] has produced a counterexample to [2, Theorem 6]. As noted in [1, Remark 1], our error occurs in the final paragraph of the purported proof of [2, Theorem 6], for P need not be a subpresheaf of I. Accordingly, it remains an open problem to find an analogue of [2, Proposition 1] in the context of Boolean spaces. We hope that attacks on this problem will be facilitated by the (valid) initial three paragraphs of the argument given for [2, Theorem 6].

The following alterations to [2] are in order. Example 5, being a corollary of Theorem 6, remains doubtful, although the special case noted on pp. 1034-1035 is not affected. In Corollary 7, the assertion that j preserves infectives remains doubtful, although the proof for divisibility of j(M)(G) is valid.
On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field
Authors: Takeuchi Hiroshi
Published: December 2010

Let p be an integer and let H(p) be the class-number of the field

where ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).

In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).

We designate by h(p) the class-number of the real quadratic field
Strong Convergence of Pramarts in Banach Spaces
Authors: Egghe Leo
Published: December 2010

Let E be a Banach space and be an adapted sequence on the probability space We denote by T the set of all bounded stopping times with respect to . is called a pramart if

converges to zero in probability, uniformly in τ ≧ σ. The notion of pramart was introduced in [6]. A good property is the optional sampling property (see Theorem 2.4 in [6]). Furthermore the class of pramarts intersects the class of amarts, and every amart is a pramart if and only if dim E < ∞ ([2], see also [4]). Pramarts behave indeed quite differently than amarts. Although the class of pramarts is large, they have good convergence properties as is seen in the next two results of Millet-Sucheston, [6], [7].

THEOREM 1.1. Let be a real-valued pramart of class (d), i.e.,
On the Number and Distribution of Simultaneous Solutions to Diagonal Congruences
Authors: Spackman Kenneth W.
Published: December 2010

Two aspects, and their connections, of the problem of enumerating solutions to certain systems of congruences are explored in this paper. Although slightly more general cases are mentioned, the basic object of study is a system of diagonal equations

where d1, d2, …, dt are positive integers and the coefficient matrix [aij] has entries from Fp = GF(p), and for which solutions x = (x1, x2 …, xt) ∈ Fpt are sought. Speaking loosely, such a system usually has approximately pt–n solutions in the sense that the difference between pt–n and the correct value becomes small in comparison with pt–n as p becomes large. A parameter is introduced which measures the extent to which the matrix [aij] is non-singular over Fp. The effect of this parameter on the size of the error term (the difference between the number of solutions to (1) and pt–n) is the first aspect of the enumeration problem to be treated.
Uniqueness of Certain Spherical Codes
Authors: Bannai Eiichi , Sloane N. J. A.
Published: December 2010

In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R8 (resp. R24) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in R8 (resp. R24) so that they all touch two further, touching spheres. The following tight spherical t-designs are unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23, and the 11-design in Ω24. It was shown in [20] that the maximum number of nonoverlapping unit spheres in R8 (resp. R24) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E8 and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.
On a Conjecture of Erdös, Faber, and Lovász about n-Colorings
Authors: Hindman Neil
Published: December 2010

Let be a finite family of finite sets with the property that |A ∩ B| ≦ 1 whenever A and B are distinct members of . It is a conjecture of Erdös, Faber, and Lovász ([1] a 50 pound problem and [2] a 100 dollar problem) that there is an n-coloring of (i.e., a function such that A ∩ B = ∅ whenever A and B are distinct members of with f(A) = f(B). They actually state the conjecture in a different form. They actually state the conjecture in a different form. Namely, if n is a positive integer and is a family of n sets satisfying (1) |Bp| = n for each p and (2) |Bp ∩ Bq| ≦ 1 when p ≠ q, then there is an n-coloring of the elements of so that each set Bp gets all n colors. The equivalence of the two forms is easily seen by interchanging the roles of elements and sets.
Locally Bounded Topologies on the Ring of Integers of a Global Field
Authors: Cohen Jo-Ann D.
Published: December 2010

A subset A of a topological ring R is bounded if given any neighborhood U of zero, there exists a neighborhood V of 0 such that AV ⊆ U and VA ⊆ U. The topology on R is locally bounded if there exists a bounded neighborhood of 0.

We recall that a seminorm ‖··‖ on a ring R is a function from R into the non-negative real numbers satisfying ‖x‖ = 0 if x = 0, ‖–x‖ = ‖x‖, ‖x + y‖ ≦ ‖x‖ + ‖y‖ and ‖xy‖ ≦ ‖x‖ ‖y‖ for all x, y in R. A seminorm ‖··‖ on R is a norm on R if ‖x‖ = 0 implies x = 0. We note that a seminorm ‖··‖ on R defines a locally bounded topology on R, and a norm on R defines a Hausdorff, locally bounded topology on R. Two norms on R are said to be equivalent if they define the same topology on R.
Some Properties of Generalized Euler Numbers
Authors: Leeming D. J. , Macleod R. A.
Published: December 2010

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by

(1.1)

where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is

(1.2)

Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).

Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).
On a Problem of Turán about Polynomials II
Authors: Pierre R. , Rahman Q. I.
Published: December 2010

1. It was proved by A. A. Markov [3] that if is a polynomial of degree at most n and |pn(x)| ≦ 1 in the interval –1 ≦ x ≦ 1, then in the same interval

(1)

The problem was proposed by the chemist Mendeleev who knew the answer for polynomials of degree 2. For a historical background of the problem see [1].

A. A. Markov's younger brother W. A. Markov considered the problem of determining exact bounds for the j–th derivative of pn(x) at a given point X0 in [ – 1, 1]. His results appeared in a Russian journal in the year 1892; a German version of his remarkable paper was later published in [4]. Amongst other things he proved the following two theorems.
On Linear Functionals and Summability Factors for Strong Summability II
Authors: Balser W. , Jurkat W. B. , Peyerimhoff A.
Published: December 2010

The first part of this paper, which will be referred to by I, appeared in Volume 30 of this journal. The present paper will use the same bibliography as I.

Theorem 1 in I shows that the knowledge of all continuous linear functionals in o[A]p is essential in determining convergence and summability factors for strong summability. So far, Theorem 7 in I was for a general A the only tool in deciding whether a given sequence ∈ generates such a functional. We mentioned in a remark following Theorem 7 the difficulties in verifying the conditions of this theorem (two parameters are involved). In the present paper, we study continuous linear functionals in o[A]1 in more detail, and we obtain in a corollary to Theorem 22 a condition which appears to be a more satisfactory answer to the question, whether a given sequence ∈ generates a continuous linear functional in o[A]1.
On the Algebra of Multipliers
Authors: Khalil Roshdi
Published: December 2010

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.

1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).
Projective Elements in Categories with Perfect θ-Continuous Maps
Authors: Vermeer Hans , Wattel Evert
Published: December 2010

In 1958 Gleason [6] proved the following :

THEOREM. In the category of compact Hausdorff spaces and continuous maps, the projective elements are precisely the extremally disconnected spaces.

The projective elements in many topological categories with perfect continuous functions as morphisms have been found since that time. For example: In the following categories the projective elements are precisely the extremally disconnected spaces:

(i) The category of Tychonov spaces and perfect continuous functions. [4] [11].

(ii) The category of regular spaces and perfect continuous functions. [4] [12].

(iii) The category of Hausdorff spaces and perfect continuous functions. [10] [1].

(iv) In the category of Hausdorff spaces and continuous k-maps the projective members are precisely the extremally disconnected k-spaces. [14].

In 1963 Iliadis [7] constructed for every Hausdorff space X the so called Iliadis absolute E[X], which is a maximal pre-image of X under irreducible θ-continuous maps.
A Specialised Net of Quadrics Having Selfpolar Polyhedra, with Details of the Fivedimensional Example
Authors: Edge W. L.
Published: December 2010

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)

osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]

Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number and

so that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics

(1.1)

in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.
Irreducible Modules for Polycyclic Group Algebras
Authors: Musson I. M.
Published: December 2010

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.

We introduce some terminology.
On an Irreducibility Theorem of A. Cohn
Authors: Brillhart John , Filaseta Michael , Odlyzko Andrew
Published: December 2010

In [1, b.2, VIII, 128] Pólya and Szegö give the following interesting result of A. Cohn:

THEOREM 1. If a prime p is expressed in the decimal system as

then the polynomial irreducible inZ[x].

The proof of this result rests on the following theorem of Pólya and Szegö [1, b.2, VIII, 127] which essentially states that a polynomial f(x) is irreducible if it takes on a prime value at an integer which is sufficiently far from the zeros of f(x).

THEOREM 2. Let f(x) ∊ Z[x] be a polynomial with the zeros α1, α2, …, αn. If there is an integer b for which f(b) is a prime, f(b – 1) ≠ 0, and for 1 ≦ i ≦ n, then f(x) is irreducible inZ[x].
Distinguished Subfields of Intermediate Fields
Authors: Deveney James K. , Mordeson John N.
Published: December 2010

Let L be a finitely generated extension of a field K of characteristic p ≠ 0. If L/K is algebraic, then there is a unique intermediate field S such that

S is just the maximal separable extension of K in L. If L/K is not algebraic, then Dieudonne [4] showed there exist maximal separable extensions D of K in L such that L ⊆ Kp–∞⊗KD. In general, not every maximal separable extension of K in L has the property. Those which do have the property are called distinguished. Kraft [7] established that a maximal separable extension D of K in L is distinguished if and only if [L:D] is as small as possible. If the minimum of the [L:D] is pr, r is called the order of inseparability of L/K, denoted inor (L/K).

Let L1 be an intermediate field of L/K. If L/K is algebraic, then the maximal separable extension S1 of K in L1 is contained in the maximal separable extension S of K in L, and moreover S is separable over S1.
On BMOA for Riemann Surfaces
Authors: Metzger Thomas A.
Published: December 2010

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if

(1)

where

It is known (see [3, p. 96]) that (1) is equivalent to

(2)
Orthogonality of Certain Functions with Respect to Complex Valued Weights
Authors: Gasper George
Published: December 2010

In his work on the Dirichlet problem for the Heisenberg group Greiner [5] showed that each Lα-spherical harmonic is a unique linear combination of functions of the form

with k = 0, 1,2, … and n = 0, ±l, ±2 , …, where Hk(α, n)(θiθ) is defined by the generating function

Since Hk(0,0)(eiθ) = Pk(cos θ), where Pk(x) is the Legendre polynomial of degree k, and these functions satisfy the orthogonality relation

Greiner raised the question of whether the functions Hk(0,0)(eiθ) are orthogonal or biorthogonal with respect to some complex valued weight function.
Are Non-Commutative Lp Spaces Really Non-Commutative?
Authors: Katavolos A.
Published: December 2010

1. The central objects in integration theory can be considered to be an abelian Von Neumann algebra, L∞, of the measure space, together with a (not necessarily finite-valued) positive linear functional on it, the integral (see [10]). It is natural, therefore, to attempt to construct a “non-commutative” integration theory starting with a non-abelian Von Neumann algebra. Segal [9] and Dixmier [2] have developed such a theory, and constructed the Non-Commutative Lp spaces associated with a Von Neumann algebra M and a normal, faithful, semifinite trace (i.e. a unitarily invariant weight) t on M. They show that there exists a unique ultra-weakly dense *-ideal J of M such that t (extends to) a positive linear form on J . A generalisation of the Hölder inequality then shows that, for 1 ≦ p < ∞, the function

is a norm on J, denoted by || • ||p.
Some Results on the Center of an Algebra of Operators on VN(G) for the Heisenberg group
Authors: Cecchini C. , Zappa A.
Published: December 2010

Let G be an amenable locally compact group. We will use the terminology of [3] and denote by VN(G) the Von Neumann algebra of the regular representation and by A(G) its predual, which is the algebra of the coefficients of the regular representation. The Von Neumann algebra VN(G) is, in a natural fashion, a module with respect to A(G) [3].

The algebra of bounded linear operators on VN(G), which commute
with the action of A(G), has been studied in [6] and in [1]. If UCB(Ĝ) is the space of the elements of VN(G) of the form vT, for some v in A(G) and some T in VN(G) (see for instance [4]), in [6] and in [1] it is proved that, for any amenable locally compact group there exists an isometric bijection between and UCB(Ĝ)*.
On Spreads Admitting Projective Linear Groups
Authors: Jha Vikram , Kallaher Michael J.
Published: December 2010

In [8] Jha raised the following problem.

(*) Let Γ be a spread whose components are subspaces of V2n(GF(q)). Suppose G ≦ Aut Γ leaves a set of q + 1 components invariant while acting transitively on Γ\Δ.

Find the possibilities for Γ or, more generally, the possibilities for (G, Γ, n, q).

Many special cases of (*) have been settled. For instance, Cohen et al [1] have shown that if G fixes two non-zero points of V, that do not both lie in the same component of Γ, then Γ is the spread associated with either a Hall plane or the Lorimer-Rahilly plane of order 16 (LR-16) [14], [18].

Another such result is given in [8]; there it is shown that if q is a prime number and G is a one-dimensional projective unimodular group then Γ is the spread associated with one of the following translation planes:

(1) the Desarguesian planes of order 4, 8, or 9;

(2) the nearfield plane of order 9;

(3) LR-16;

(4) the translation plane JW-16, obtained by transposing the slope maps of LR-16 [19].
Holomorphic Functions with Positive Real Part
Authors: Sawyer Eric
Published: December 2010

The main purpose of this note is to prove a special case of the following conjecture.

Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).

Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functions

is in Hp(B2) for 0 < p < 3/2.
Commutativity Conditions on Rings with Involution
Authors: Misso Paola
Published: December 2010

Let R be a ring with involution *. We denote by S, K and Z = Z(R) the symmetric, the skew and the central elements of R respectively.

In [4] Herstein defined the hypercenter T(R) of a ring R as

and he proved that in case R is without non-zero nil ideals then T(R) = Z(R).

In this paper we offer a partial extension of this result to rings with involution.

We focus our attention on the following subring of R:

(We shall write H(R) as H whenever there is no confusion as to the ring in question.)

Clearly H contains the central elements of R. Our aim is to show that in a semiprime ring R with involution which is 2 and 3-torsion free, the symmetric elements of H are central.
Rational Equivalence of Fibrations with Fibre G/K
Authors: Halperin Stephen , Thomas Jean Claude
Published: December 2010

Let be two Serre fibrations with same base and fibre in which all the spaces have the homotopy type of simple CW complexes of finite type. We say they are rationally homotopically equivalent if there is a homotopy equivalence between the localizations at Q which covers the identity map of BQ.

Such an equivalence implies, of course, an isomorphism of cohomology algebras (over Q) and of rational homotopy groups; on the other hand isomorphisms of these classical algebraic invariants are usually (by far) insufficient to establish the existence of a rational homotopy equivalence.

Nonetheless, as we shall show in this note, for certain fibrations rational homotopy equivalence is in fact implied by the existence of an isomorphism of cohomology algebras. While these fibrations are rare inside the class of all fibrations, they do include principal bundles with structure groups a connected Lie group G as well as many associated bundles with fibre G/K.
Ordinary Singularities with Decreasing Hilbert Function
Authors: Roberts Leslie G.
Published: December 2010

Let A be the co-ordinate ring of a reduced curve over a field k. This means that A is an algebra of finite type over k, A has no nilpotent elements, and that if P is a minimal prime ideal of A, then A/P is an integral domain of Krull dimension one. Let M be a maximal ideal of A. Then G(A) (the graded ring of A relative to M) is defined to be . We get the same graded ring if we first localize at M, and then form the graded ring of AM relative to the maximal ideal MAM. That is

Let Ā be the integral closure of A. If P1, P2, …, Ps are the minimal primes of A then

where A/Pi is a domain and is the integral closure of A/Pi in its quotient field.
On an Identity Relating to Partitions and Repetitions of Parts
Authors: Kirdar M. S. , Skyrme T. H. R.
Published: December 2010

This note is concerned with a simple but rather surprising identity which emerged unexpectedly from the work of one of the authors on the characterisation of characters. Consider, for example, the seven partitions of 5. These are

(1) 5, 4 1, 3 2, 3 12, 22 1, 2 13, 15

and with each of these we can associate a product of factorials of the numbers of repetitions, respectively

(2) 1!, (1!)(1!), (1!)(1!), (1!)(2!), (2!)(1!), (1!)(3!), 5!

It is then seen that the product of all the numbers occurring in (1) coincides with that of all the numbers in (2).

Generally, for any particular natural number n, the partitions can be written in the form

1a1 2a2 3a3…,

in which ak is the frequency of repetition of the part k, and are enumerated by the distinct sets {α} = {a1, a2, …,} with ak ≧ 0 and Σkak = n.
Divisibility Properties of Graded Domains
Authors: Anderson D. D. , Anderson David F.
Published: December 2010

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.
Approximation Harmonique, Approximation Holomorphe et Topologie
Authors: Gauthier P. M. , Hengartner W. , Labréche M.
Published: December 2010

Le but de cet article est de montrer qu'il n'y a pas de relation étroite entre ces trois concepts.

Soit K une partie compacte du plan complexe fini, et soit A(K) (resp. a(K)) l'ensemble des fonctions continues sur K et holomorphes (resp. harmoniques) à l'intérieur K0 de K. Nous dirons que K est un ensemble d'approximation holomorphe (resp. harmonique) si toute fonction de A(K) (resp. a(K)) peut être approchée uniformément sur K par des fonctions holomorphes (resp. harmoniques) sur K, i.e., dans un voisinage de K.

Pour un voisinage fixé Ω de K, le théorème de Mergelian nous donne des conditions nécessaires et suffisantes, purement topologiques, pour que toute fonction de A(K) puisse être approchée uniformément sur K par des fonctions holomorphes dans Ω. D'autre part, on sait (par exemple [4]) qu'il n'est pas possible de caractériser topologiquement les ensembles d'approximation holomorphe.
Rich Proximities and Compactifications
Authors: Carlson Stephan C.
Published: December 2010

Each Hausdorff compactification of a given Tychonoff space is the Smirnov compactification associated with a compatible proximity on the space. Also each realcompactification of a given Tychonoff space is the underlying topological space of the completion of a compatible uniformity on the space. But if T is a realcompactification of a Tychonoff space X which is contained in a particular compactification Z of X, then it is not always possible to find a compatible uniformity on X such that T is the underlying topological space of the completion of (X, ) and induces the proximity on X associated with Z. We shall call a Hausdorff compactification Z of a Tychonoff space X a rich compactification of X (and the associated proximity on X a rich proximity) if every realcompactification of X contained in Z can be obtained as the underlying topological space of the completion of a compatible uniformity on X which induces the proximity on X associated with Z.
Exponential Families and Game Dynamics
Authors: Akin Ethan
Published: December 2010

A symmetric game consists of a set of pure strategies indexed by {0, …, n} and a real payoff matrix (aij). When two players choose strategies i and j the payoffs are aij and aji to the i-player and j-player respectively. In classical game theory of Von Neumann and Morgenstern [16] the payoffs are measured in units of utility, i.e., desirability, or in units of some desirable good, e.g. money. The problem of game theory is that of a rational player who seeks to choose a strategy or mixture of strategies which will maximize his return. In evolutionary game theory of Maynard Smith and Price [13] we look at large populations of game players. Each player's opponents are selected randomly from the population, and no information about the opponent is available to the player. For each one the choice of strategy is a fixed inherited characteristic.
Une Generalisation d'Une Formule de Meixner-Tricomi
Authors: Batola Francois
Published: December 2010

Dans [11] (page 32 formule (5)), Tricomi donne la formule suivante:

0.1

Dans [8] (page 704, formule 36) Meixner donne une formule analogue.

La formule (0.1) peut donc être considérée comme la formule de Meixner-Tricomi, la formule de Meixner citée étant antérieure, semblet-il, à celle de Tricomi.

L'objet de ce travail est de généraliser le résultat (0.1) qui sera considéré comme la formule de Meixner-Tricomi.

Pour cela, on va considérer l'équation différentielle ordinaire suivante:

0.2
Pseudocomplemented and Implicative Semilattices
Authors: Hoo C. S.
Published: December 2010

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.