bluuebunny/crossref_metadata_2025_split huggingface dataset:
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 formwhere 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 fieldwhere 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 ifconverges 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 equationswhere 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.
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 andso 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 asthen 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 thatS 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)whereIt 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 formwith k = 0, 1,2, … and n = 0, ±l, ±2 , …, where Hk(α, n)(θiθ) is defined by the generating functionSince Hk(0,0)(eiθ) = Pk(cos θ), where Pk(x) is the Legendre polynomial of degree k, and these functions satisfy the orthogonality relationGreiner 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 functionis 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 functionsis 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 asand 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 isLet Ā be the integral closure of A. If P1, P2, …, Ps are the minimal primes of A thenwhere 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, 15and 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 form1a1 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.
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.
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:0.1Dans [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
Toeplitz Operators on Bergman Spaces
Authors: Axler Sheldon , Conway John B. , McDonald Gerard 
Published: December 2010

Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L2(G) is given by the norm ‖h‖2 of a function h in L2(G) is given by ‖h‖2 = (∫G|h|2)1/2.The Bergman space of G, denoted La2(G), is the set of functions in L2(G) that are analytic on G. The Bergman space La2(G) is actually a closed subspace of L2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.