bluuebunny/crossref_metadata_2025_split huggingface dataset:
Application of a method of Szemeredi
Authors: Halberstam H. 
Published: May 2009

Let ℬ = {bi:b1 <b2<…} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so thatAssume also thatLet A=Aℬ denote the sequence of ℬ-free numbers, that is, of positive integers divisible by no element of ℬ. This concept, generalizing square-free and k-free numbers, derives from Erdös [2] who proved in 1966 that there exists a constant c, 0<c<l, independent of ℬ, such that the interval (x, x+xc) contains elements of A provided only that x is large enough. This result of Erdös was shown by Szemeredi [7] in 1973 to hold with c=½+ε, if x≥xo(ε, ℬ), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε.
The order of magnitude of the mth coefficients of cyclotomic polynomials
Authors: Montgomery H. L. , Vaughan R. C. 
Published: May 2009

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler's function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m
Sur la lacunarité des puissances de η
Authors: Serre Jean-Pierre 
Published: May 2009

La fonction η de Dedekind est définie paroù , Im(z)>0. C'est une forme modulaire parabolique de poids 1/2. Si r est un entier, la puissance r–ième de η s'écrit;où les coefficients pr(n) sone définis par l'identité.
On Sands' questions concerning strong and hereditary radicals
Authors: Puczyłowski E. R. 
Published: May 2009

[7] Sands raised the following questions:(1) Must a hereditary radical which is right strong be left strong?(2) Must a right hereditary radical be left hereditary?(3) (Example 6) Does there exist a right strong radical containing the prime radical β which is not left strong or hereditary?Negative answers to questions (1) and (2) were given by Beidar [1].In this paper we present different examples to answer (1) and (2), and we answer (3). We prove that the strongly prime radical defined in [4, 5] is right but not left strong. In the proof we use an example given by Parmenter, Passman and Stewart [6]. The same example and the strongly prime radical are used to answer (2) and (3).
Some characterizations of hereditarily artinian rings
Authors: van Huynh Dinh 
Published: May 2009

Throughout this note, rings will mean associative rings with identity and all modules are unital. A ringRis calledright artinianifRsatisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ringRis right artinian thenRis calledhereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.
A note on continuation problems
Authors: Nugari Rita 
Published: May 2009

Recently M. Martelli [6] and M. Furi and M. P. Pera [1] proved some interesting results about the existence and the global topological structure of connected sets of solutions to problems of the form:Lx = N(λ, x)with L:E → F a bounded linear Fredholm operator of index zero (where E, F are real Banach spaces), and N:ℝ × E → F a nonlinear map satisfying suitable conditions.While the existence of solution sets for this kind of problem follows from the Leray–Schauder continuation principle, it is our aim to show in this note that their global topological structure can be obtained as a consequence of the theory developed by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli in [3, 4] about parameter dependent compact vector fields in Banach spaces.
A Simple proof of the Goldberg–Straus theorem on numerical radii
Authors: Tam Bit-Shun 
Published: May 2009

Let Mn(ℂ) be the algebra of n × n complex matrices, and let be its unitary group. Given A, B ε Mn(ℂ), the A-numerical radius of B is the nonnegative quantityIn particular, for A = diag(1, 0, …, 0) it reduces to the classical numerical radius r(B) = max||x*Bx|:x*x = 1}. In [1] Goldberg and Straus proved that rA is a generalized matrix norm (i.e. a positive definite seminorm) on Mn(ℂ) if and only if A is nonscalar and tr A ≠ 0. This result agrees with the well-known fact that the classical numerical radius r is a generalized matrix norm. The nontrivial part of the proof is to show that if A is nonscalar and tr A ≠ 0 then rA is positive definite; that is, for any B ε Mn(ℂ), tr(AU*BU) = 0 for all U ε implies B = 0. The proof given in [1] is computational and involves the use of differentiation on matrices. Later Marcus and Sandy [2] gave three elementary proofs of the result. Their proofs are still computational in nature and two of them need knowledge of multilinear algebra.
Inequalities for the Schatten p-norm II
Authors: Kittaneh Fuad 
Published: May 2009

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.
Coefficient conditions for starlike functions
Authors: Ruscheweyh S. 
Published: May 2009

Let {ak} be a sequence of non-negative real numbers satisfying a1 = l and(1)Brannan [1] proved that the function(2)is close-to-convex univalent in the unit disc D. The exampleshows that the conclusion in Brannan's theorem is sharp in that sense that “close-to-convex” cannot be replaced by the stronger one: “starlike”. It is therefore of interest to see which additional condition can guarantee this stronger conclusion.
Classification of p-groups of automorphisms of Riemann surfaces and their lower central series
Authors: Zomorrodian Reza 
Published: May 2009

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.
On normal derivations of Hilbert–Schmidt type
Authors: Kittaneh Fuad 
Published: May 2009

Let H denote a separable, infinite dimensional Hilbert space. Let B(H), C2 and C1 denote the algebra of all bounded linear operators acting on H, the Hilbert–Schmidt class and the trace class in B(H) respectively. It is well known that C2 and C1 each form a two-sided-ideal in B(H) and C2 is itself a Hilbert space with the inner productwhere {ei} is any orthonormal basis of H and tr(.) is the natural trace on C1. The Hilbert–Schmidt norm of X ∈ C2 is given by ⅡXⅡ2=(X, X)½.
On formulae of Macbeath and Hussein
Authors: Stothers W. W. 
Published: May 2009

In his thesis [1], Hussein considered regular permutations of order 2 and 3 in Sn whose product is an n-cycle. For such a pair, we must havefor some g ≥ 1. Such a permutation pair corresponds to a free cycloidal subgroup of the classical modular group (see, e.g., [3]). Previously the free subgroups and the cycloidal subgroups of fixed genus had been enumerated ([4], [5]).
A Counterexample in the theory of derivations
Authors: Wenying Feng , Guoxing Ji 
Published: May 2009

Let B(H) be the algebra of all bounded linear operators on a separable, infinite dimensional complex Hilbert space H. Let C2 and C1 denote respectively, the Hilbert–Schmidt class and the trace class operators in B(H). It is known that C2 and C1 are two-sided*-ideals in B(H) and C2 is a Hilbert space with respect to the inner product(where tr denotes the trace). For any Hilbert–Schmidt operator X let ∥X∥2=(X, X)½ be the Hilbert-Schmidt norm of X.For fixed A ∈ B(H) let δA be the operator on B(H) defined byOperators of the form (1) are called inner derivations and they (as well as their restrictions have been extensively studied (for example [1–3]). In [1], Fuad Kittaneh proved the following result.
Quotient rings, chain conditions and injective ring endomorphisms
Authors: Wilkinson J. C. 
Published: May 2009

In this paper, the situation we shall be concerned with is that of a ring R, with a ring monomorphism α: R → R, which will not be assumed to be surjective.Much work has been done on the skew polynomial ring R[x, α] and the skew Laurent polynomial ring R[x, x-1, α], where α is an automorphism—see [3] for example. However, the fact that α is not surjective renders the study of these objects much more difficult.
Error estimates for the spectral Galerkin approximations of the solutions of Navier–Stokes type equations
Authors: Salvi R. 
Published: May 2009

In [8], [9] R. Rautmann has given a systematic development of error estimates for the spectral Galerkin approximations of the solution of the Navier–Stokes equations (spectral in the sense that one chooses as basis functions the eigenfunctions of the Stokes operator).Error estimates are presented locally in time, valid on a finite interval determined by certain norms of the data. If one assumes the solution to be approximated is uniformly regular fort ≥ 0, the method gives an error estimate which grows exponentially with time. Without further assumptions this is the best estimate that one can expect. However, as pointed out in [2] by J. G. Heywood, if one assumes, additionally, that the solution to be approximated is stable, then one obtains an error estimate which is uniform in time.
A Proof of some identities of Ramanujan using modular forms
Authors: Biagioli Anthony J. F. 
Published: May 2009

In 1974 B. J. Birch [1] published a description of some manuscripts of Ramanujan which contained, among other things, a list of forty identities involving the Rogers-Ramanujan functionsAt that time nine of these had been proven, and since then twenty-two more of them have been proven, fifteen of them by David Bressoud in his thesis [2]. Bressoud gives a synopsis of the extant proofs, where he attributes proofs to H. B. C. Darling [3], L. J. Rogers [4], L. J. Mordell [5], and G. N. Watson [6].
A noncommutative theory of Bade functionals
Authors: Hadwin Don , Orhon Mehmet 
Published: May 2009

Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have(i) α (ax) ≥0,(ii) if α (ax) = 0, then ax = 0.
Cyclic surgery on satellite knots
Authors: Zhang Xingru 
Published: May 2009

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).
Homomorphisms of (0, 1)-lattices with a given sublattice and quotient
Authors: Koubek Václav 
Published: May 2009

Let us recall the notions of full embedding and universality of categories we will be using throughout.A full embedding is a functor F taking the objects of a source category A injectively to objects of a target category B and the hom-sets HomA(a, b) bijectively to the hom-sets HomR(F(a), F(b)). If A is a subcategory of B and the corresponding inclusion functor is a full embedding then A is said to be a full subcategory of B. In this case we have HomA(a, b) = HomB(a, b) for any a, b in A; that is to say, a full subcategory is completely determined, within a given category, by specifying the class of its objects. A category U is termed universal if an arbitrary category of algebras can be fully embedded in U.
On the construction of matrix representations and their extensions II
Authors: Kuisch Eric B. 
Published: May 2009

In [5] we exhibited the construction of faithful irreducible matrix representations of p-groups E and constructed their extensions to a semidirect product E. H, in case E and H satisfied suitable conditions. One of the major conditions was that the prime p had to be odd.In this paper we assume the same conditions as in [5], but now with p = 2, in order to see if similar results can be obtained. Henceforth we will work with the following hypothesis.
On the Cauchy problem for the differential equation f(t, x, x′, …, x(k)) = 0
Authors: Ricceri Biagio 
Published: May 2009

In the sequel, given k, n ∈ ℕ, p ∈ [1, ∞] and a compact real interval I, we denote by Wk, p(I, ℝn) (simply by Wk,p(I if n = 1) the space of all functions u ∈ Ck−1(I, ℝn) such that uk−1 is absolutely continuous in I and u(k) ∈ Lp (I, ℝn).Very recently, in [11], J. R. L. Webb and S. C. Welsh obtained the following existence result.
The Mazur property and completeness in the space of Bochner-integrable functions L1(μ, X)
Authors: Schlüchtermann G. 
Published: May 2009

A locally convex space (E, ) has the Mazur Property if and only if every linear -sequential continuous functional is -continuous (see [11]).In the Banach space setting, a Banach space X is a Mazur space if and only if the dual space X* endowed with the w*-topology has the Mazur property. The Mazur property was introduced by S. Mazur, and, for Banach spaces, it is investigated in detail in [4], where relations with other properties and applications to measure theory are listed. T. Kappeler obtained (see [8]) certain results for the injective tensor product and showed that L1(μ, X), the space of Bochner-integrable functions over a finite and positive measure space (S, σ, μ), is a Mazur space provided X is also, and ℓ1 does not embed in X.
Homological characterizations of the approximation property for Banach spaces
Authors: Selivanov Yu. V. 
Published: May 2009

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).
A functorial approach to weak amenability for commutative Banach algebras
Authors: Runde Volker 
Published: May 2009

Let A be a commutative algebra, and let M be a bimodule over A. A derivation from A into M is a linear mapping D: A→M that satisfiesIf M is only a left A-module, by a derivation from A into M we mean a linear mapping D: A→M such thatEach A-bimodule M is trivially a left module. However, unless it is commutative, i.e.the two classes of linear operators from A into M characterized by (1) and (2), respectively, need not coincide.
On right duo p.p. rings
Authors: Chatters A. W. , Xue Weimin 
Published: May 2009

Throughout the paper, rings are associative rings with identity. A ring is called right duo if every right ideal is two-sided, and it is called right p.p. if every principal right ideal is projective. A left duo (p.p.) ring is denned similarly, and a duo (p.p.) ring will mean a ring which is both right and left duo (p.p.). There is a right p.p. ring that is not left p.p. (see Chase [2[). Small [9] proved that right p.p. implies left p.p. if there are no infinite sets of orthogonal idempotents, and Endo [5, Proposition 2] has shown the same implication in the case where each idempotent in the ring is central. Since Courter [3, Theorem 1.3] noted that every idempotent in a right duo ring is central, we can simply speak of right duo p.p. rings. A typical example of a right duo ring which is not left duo is the following. Let F be a field and F(x) the field of rational functions over F. Let R = F(x)× F(x) as an additive group and define the multiplication as follows:Then R is a local artinian ring with c(RR) = 2 and c(RR)= 3. Thus R is right duo but not left due.
Matrix invariants and complete intersections
Authors: le Bruyn Lieven , Teranishi Yasuo 
Published: May 2009

Consider the vector space of m-tuples of n by n matrices.The linear group GLn(C) acts on Xm, n by simultaneous conjugation. The corresponding ring of polynomial invariantswill be denoted by C(n, m) and is called the ring of matrix invariants of m-tuples of n by n matrices. C. Procesi has shown in [8] that C(n, m) is generated by traces of products of the corresponding generic matrices and, as such, coincides with the center of the trace ring of m generic n by n matrices R (n, m) which is also the ring of equivariant maps from Xm, n to Mn(ℂ).
The λ-property in schreier's space S and the Lorentz space d(a, 1)
Authors: Shura Thaddeus J. , Trautman David 
Published: May 2009

AbstractWe add Schreier's space S and the Lorentz space d(a, 1) to the list of classical Banach spaces which enjoy the λ-property, investigate the extreme point structure of S, and show that d(a, 1) has a λ-function which is continuous on Sd(a, 1), though not even uniformly so.
A note on GV-modules with Krull dimension
Authors: Huynh van Dinh , Smith Patrick F. , Wisbauer Robert 
Published: May 2009

AbstractExtending a result of Boyle and Goodearl in [1] on V-rings it was shown in Yousif [11] that a generalized V-module (GV-module) has Krull dimension if and only if it is noetherian. Our note is based on the observation that every GV-module has a maximal submodule (Lemma 1). Applying a theorem of Shock [6] we immediately obtain that a GV-module has acc on essential submodules if and only if for every essential submodule K ⊂ M the factor module M/K has finitely generated socle. Yousif's result is obtained as a corollary.Let R be an associative ring with unity and R-Mod the category of unital left R-modules. Soc M denotes the socle of an R-module M. If K ⊂ M is an essential submodule we write K⊴M.An R-module M is called co-semisimple or a V-module, if every simple module is M-injective ([2], [7], [9], [10]). According to Hirano [3] M is a generalized V-module or GV-module, if every singular simple R-module is M-injective. This extends the notion of a left GV-ring in Ramamurthi-Rangaswamy [5].It is easy to see that submodules, factor modules and direct sums of co-semisimple modules (GV-modules) are again co-semisimple (GV-modules) (e.g. [10, § 23]).
Components in vector lattices and extreme extensions of quasi-measures and measures
Authors: Lipecki Z. 
Published: May 2009

We develop some ideas contained in the author's paper [8] which was, in turn, inspired by Bierlein and Stich [5]. The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.
An infinitesimal proof of the implicit function theorem
Authors: Cutland Nigel J. , Hanqiao Feng 
Published: May 2009

We give a short and constructive proof of the general (multi-dimensional) Implicit Function Theorem (IFT), using infinitesimal (i.e. nonstandard) methods to implement our basic intuition about the result. Here is the statement of the IFT, quoted from [4];Theorem. Let A ⊂ ℝn × ℝmbe an open set and let F:A → ℝ be a function of class Cp (p≥1). Suppose that (xO, yO) ε A with F(xO, yO) = 0 (xO ε ℝn, yO ε ℝm) and that the Jacobian determinantis not zero at (xO, yO). Then there is an open neighbourhood U of xO and a unique function f:U→ ℝmwithF(x, f(x)) = 0for all x ε U. Moreover, f is of class Cp.
Finite coverings by subgroups with a given property
Authors: Brodie Marc A. , Kappe Luise-Charlotte 
Published: May 2009

Letbe a group-theoretic property. We say a group has a finite covering by-subgroups if it is the set-theoretic union of finitely many-subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups,n-abelian subgroups or 2-central subgroups.R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC(G)] finite implies the existence of a finite covering by subgroups of nilpotency classc, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, butZ2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.
Banach spaces with property (w)
Authors: Leung Denny H. 
Published: May 2009

A Banach spaceEis said to haveProperty(w) if every operator fromEintoE' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach spaceEtoC(K, E).