bluuebunny/crossref_metadata_2025 huggingface dataset:
Range inclusion for normal derivations
Authors: Fong C. K.
Published: May 2009

For a (bounded, linear) operator A in a (complex, infinite-dimensional, separable) Hilbert space ℋ, the inner derivation DA as an operator on ℬ(ℋ), is defined by DAX = AX – XA. Johnson and Williams [4] showed that, when A is a normal operator, range inclusion DBℬ(ℋ)⊆DA(ℋ)⊆ is equivalent to the condition that B = f(A), where f is a Lipschitz function on σ(A) such that t(z, w)(f(z)–f(w))/(z–w) is a trace class kernel on L2(μ) whenever t(z, w) is such a kernel. (Here μ is the dominating scalar valued spectral measure of A constructed in multiplicity theory). This result is deep and its proof is difficult. In the present paper, we establish the following analogous result which is easier to prove: for a normal operator A, range inclusion DB℘2(ℋ) holds if and only if B = f(A) for some Lipschitz function f on σ(A). Here ℘(ℋ) stands for the Hilbert-Schmidt class of operators on ℋ. As by-products of our argument, we generalize some results in [4], [8], [9] concerning the non-existence of a one-sided ideal contained in certain derivation ranges; for example, we show that if A is hyponormal and if the point spectrum σP(A*) of A* is empty, then DAℬ(ℋ) does not contain any nonzero right ideal.
On the joint spectra of doubly commuting n-tuples of semi-normal operators
Authors: Chō Muneo , Dash A. T.
Published: May 2009

Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that

.

A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such that

A point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ j ≤ n.
A decomposition theorem for submeasures
Authors: Khan A. R. , Rowlands K.
Published: May 2009

In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been proved for group-valued measures. For example, Traynor [4], [6] has established Lebesgue decomposition theorems for exhaustive groupvalued measures on a ring using (1) algebraic and (2) topological notions of continuity and singularity, and generalizations of the Hewitt-Yosida theorem have been given by Drewnowski [2], Traynor [5] and Khurana [3]. In this paper we consider group-valued submeasures and in particular we have established a decomposition theorem from which analogues of the Lebesgue and Hewitt-Yosida decomposition theorems for submeasures may be derived. Our methods are based on those used by Drewnowski in [2] and the main theorem established generalizes Theorem 4.1 of [2].
Integral Harnack inequality
Authors: Rao Murali
Published: May 2009

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f then

where ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.
Representations of finite abelian groups Cnm,p
Authors: Saeed-ul-Islam M.
Published: May 2009

Let C= CmXCmX…XCm be the finite abelian group of order mn generated by n elements w1…,wn of order m. Let C be the field of complex numbers and P a projective representation of G with factor set α over C (see Morris [2]). Further let

and
Inequalities for the Schatten P-norm
Authors: Kittaneh Fuad
Published: May 2009

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1<p<∞, si(A)p <∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).
On the asymptotic behaviour of nonlinear systems of ordinary differential equations
Authors: Athanassov Zhivko S.
Published: May 2009

In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:

where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).
Algebraic points on quartic curves over function fields
Authors: Bremner Andrew
Published: May 2009

The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?

It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q.
When does an affine curve have an algebraic integer point?
Authors: Birch B. J.
Published: May 2009

The purpose of this note is to draw attention to the question in the title. If C⊆Kn is an (absolutely) irreducible affine curve, defined by equations over a number field K, an algebraic integer point of C is a point P = (x1, …, xn) with all of x1, …, xn integers of some finite extension L of K. For such an algebraic integer point P to exist, there are obviously necessary local conditions: for every prime p of K there must exist a prime B above p and a corresponding finite extension LB of the completion Kp such that C has a B-adic integer point. We would like to know whether these obviously necessary local conditions are also sufficient.
On unramified Am-extensions of quadratic number fields
Authors: Elstrodt J. , Grunewald F. , Mennicke J.
Published: May 2009

Number fields such as described in the title play a rôle in the study of Artin L-functions and automorphic forms for the groups SL2 over rings of integers in quadratic extensions of ℚ. They are also of some interest on their own. We have not found many examples in the literature. Lang [4] mentions an unramified A5-extension of a real quadratic number field which is due to E. Artin.
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 that

Assume also that

Let 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 equation

and we write

where ϕ is Euler's function.

Erdös and Vaughan [3] have shown that

uniformly in n as m-→∞, where

and that for every large m
The classical modular group as a subgroup of GL(2, ℤ)
Authors: Newman Morris
Published: May 2009

The title is somewhat misleading, since the classical modular group Г= PSL(2, ℤ) is certainly not a subgroup of GL(2, ℤ). What is meant of course are the faithful representations of F as a subgroup of GL(2, ℤ), where Г is to be thought of as the free product of a cyclic group of order 2 and a cyclic group of order 3. No such representation is possible as a subgroup of SL(2, ℤ); it is necessary to have matrices of determinant −1 as well.
The distribution of certain special values of the cubic Legendre symbol
Authors: Patterson S. J.
Published: May 2009

Let ω be a primitive cube root of unity. We define the cubic residue symbol (Legendre symbol) on ℤ[ω] as follows. Let πεℤ[ω] be a prime, (3, π)=1. For α ε ℤ[ω] such that (α, π)=1 we let be that third root of unity so that
Sur la lacunarité des puissances de η
Authors: Serre Jean-Pierre
Published: May 2009

La fonction η de Dedekind est définie par

où , 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

.
Represéntations Galoisiennes paires
Authors: Vignéras M.-F.
Published: May 2009

On présente des exemples de représentations dede dimension 2, de déterminant pair, qui sont de type diédral (I) ou de conducteur premier et de type quelconque (II), en imitant la construction de Tate (Serre [11]) de représentations de déterminant impair.
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).
Bounded completeness and Schauder's basis for C[0, 1]
Authors: Holub J. R.
Published: May 2009

A basis , for a Banach space X is said to be boundedly complete [4, p. 284] if whenever is a sequence of scalars for which converges. It is well-known [2, p. 70] that if is a boundedly complete basis for X then X is isometric to a conjugate space; in fact, X = [fi]*, where is the sequence of coefficient functionals associated with the basis It follows that no basis for C[0,1] can be boundedly complete since no separable conjugate space contains C0[l], yet C[0,1] is a separable space which contains c0.
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.
Symbols for trace class Hankel operators with good estimates for norms
Authors: Bonsall F. F. , Walsh D.
Published: May 2009

Peller [4, 5] has proved that a Hankel operator S on the Hardy space H2 is in the trace class if and only if with h analytic on the open unit disc Dand with its second derivative belonging to the Bergman space L1a. This theorem does not include an estimate for the trace class norm ∥S∥1, of the operator in terms of the symbol function. In fact it is clear that cannot give an estimate for since the first two terms in the coefficient sequence of the Hankel operator have been removed by differentiation.
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.
Finitely spectral operators
Authors: Nagy B.
Published: May 2009

In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.
The compact range property and C0
Authors: Gretsky Neil E. , Ostroy Joseph M.
Published: May 2009

The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.
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 quantity

In 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.
Irreducible representations of the generalized symmetric group Bmn
Authors: Saeed-ul-Islam M.
Published: May 2009

This paper is devoted to the determining of the irreducible linear representations of the generalized symmetric group (elsewhere written as , Cm ≀ Sn or G(m, 1, n)) by considering the conjugacy classes of and then constructing the same number of inequivalent irreducible linear representations of . These have previously been determined by Kerber [2, Section 5] using Clifford's theory applied to wreath products.
Almost nilpotent Lie algebras
Authors: Towers David A.
Published: May 2009

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.
On approximation in weighted spaces of continuous vector-valued functions
Authors: Khan Liaqat Ali
Published: May 2009

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.
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 example

shows 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 form

The 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 product

where {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.
Some finiteness conditions for automorphism groups
Authors: Franciosi Silvana , de Giovanni Francesco
Published: May 2009

Many authors have investigated the behaviour of the elements of finite order of a group G when finiteness conditions are imposed on the automorphism group Aut G of G. The first result was obtained in 1955 by Baer [1], who proved thata torsion group with finitely many automorphisms is finite. This theorem was generalized by Nagrebeckii in [6], where he proved that if the automorphism group Aut G is finite then the set of elements of finite order of G is a finite subgroup.
A note on Dunford-Pettis operators
Authors: Holub J. R.
Published: May 2009

Talagrand has shown [4, p. 76] that there exists a continuous linear operator from L1[0, 1] to c0 which is not a Dunford-Pettis operator. In contrast to this result, Gretsky and Ostroy [2] have recently proved that every positive operator from L[0, 1] to c0 is a Dunford-Pettis operator, hence that every regular operator between these spaces (i.e. a difference of positive operators) is Dunford-Pettis.
Normal radicals and normal classes of modules
Authors: Nicholson W. K. , Watters J. F.
Published: May 2009

The study of special radicals was begun by Andrunakievič [1]. A class of prime rings is called special if it is hereditary and closed under prime extensions. The upper radicals determined by special classes are called special. In later works Andrunakievič and Rjabuhin [2] and [3] defined the concept of a special class of modules.
On the tensor product of quaternion algebras of characteristic two
Authors: Mammone P.
Published: May 2009

The purpose of this note is to generalize to fields of characteristic two the results obtained in [4]. We obtain necessary and sufficient conditions involving quadratic forms for certain tensor products of quaternion algebras to be division algebras.
Embedding any countable semigroup without idempotents in a 2-generated simple semigroup without idempotents
Authors: Byleen Karl
Published: May 2009

Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollák [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.
Representations of spaces as function spaces
Authors: Stannett M. P.
Published: May 2009

Given a topological space X, we can consider the group G(X) of all autohomeomorphisms of X. Much is known about the relationship between X and G(X) for certain restricted classes of the space X; Whittaker [7] has shown that the existence of an isomorphism between any two sufficiently large subgroups of G(X) and G(Y) implies that X and Y are actually homeomorphic, whenever these are both compact, locally Euclidean manifolds, with or without boundary; Fine and Schweigert [1] give a detailed analysis of G(ℝ); recently, Neumann [4], Mekler [3] and Truss [6] have considered in depth the group G(ℚ).
Strongly analytic spaces in spectral decomposition
Authors: Lange Ridgley , Wang Shengwang
Published: May 2009

It is now well-known that decomposable operators have a rich structure theory; in particular, an operator is decomposable iff its adjoint is [3]. There are many other criteria for decomposability [8], [9]. In Theorem 2.2 of this paper (see below) we give several new ones. Some of these (e.g. (ii), (iii)) are “relaxations” of conditions given in [7] and [8]. Assertion (vi) is a version of a result in [10]. Characterizations (iv)and (v) are novel in two respects. For instance, (v) states that an operator Tcan be “patched” together into a decomposable operator if it has an invariant subspace Y such that T | Y and the coinduced operator T | Y are both decomposable. Secondly, in this way the strongly analytic subspace appears in the theory of spectral decomposition.
Trivial action on the tensor product of finite groups
Authors: Higgs R. J.
Published: May 2009

Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (G ⊗ H); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).
A remark on the Shimura correspondence
Authors: Kohnen Winfried
Published: May 2009

In [4] an identity is given which relates the product of two Fourier coefficients of a Hecke eigenform g of half-integral weight and level 4N with N odd and squarefree to the integral of a Hecke eigenform f of even integral weight associated to g under the Shimura correspondence along a geodesic period on the modular curve X0(N) This formula contains as a special case a refinement of a result of Waldspurger [6] about special values of L-series attached to f at the central point.
On asymptotic values of certain sets of attached prime ideals
Authors: Taherizadeh A.-J.
Published: May 2009

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.
An extreme positive operator on a polyhedral cone
Authors: Robertson A. Guyan
Published: May 2009

In [2], R. Loewy and H. Schneider studied positive linear operators on circular cones. They characterised the extremal positive operators on these cones and noticed that such operators preserve the set of extreme rays of the cone in this case. They then conjectured that this property of extremal positive operators is true in general.
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 have

for 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]).
Root closure in integral domains, II
Authors: Anderson David F.
Published: May 2009

In this note, we give an elementary procedure for constructing n-root closed integral domains. We then use this construction to give two interesting examples. First, we give an example of a root closed integral domain which is not quasinormal. Secondly, we show that for any subset 5 of odd positive primes there is a one-dimensional affine domain which is p-root closed for a prime p if and only if p ∈ S.
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 by

Operators 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.
Symplectic bilinear forms on affine real algebraic surfaces
Authors: Kucharz W.
Published: May 2009

Given a commutative ring A with identity, let W–1(A) denote the Witt group of skew-symmetric bilinear forms over A (cf. [1] or [7] for the definition of W–1 (A)).
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 contribution to the solution of the compact correction problem for operators on a Banach space
Authors: Searcóid Mícheál Ó
Published: May 2009

We consider the hypothesis that an operator T on a given Banach space can always be perturbed by a compact operator K in such a way that, whenever a complex number A is in the semi-Fredholm region of T + K, then T + K – λ is either bounded below or surjective. The hypothesis has its origin in the work of West [11], who proved it for Riesz operators on Hilbert space. In this paper, we reduce the general Banach space problem to one of considering only operators of a special type, operators which are, in a spectral sense, natural generalizations of the Riesz operators studied by West.
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 functions

At 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].
Elliptically embedded subgroups and nilpotent products
Authors: Smith Howard
Published: May 2009

A recent paper of Rhemtulla and Wilson [4] is concerned with elliptically embedded subgroups of groups. A subgroup H of a group G is elliptically embedded in G if, for each subgroup K of G, there is some integer m such that (H, K) = (HK)m. Some sufficient conditions for elliptic embedding are given in Section 2 of [4], and some consequences of the presence of this property are to be found in Theorems 1 and 2 of the same paper and in the main theorem of [5]. It is evident from all of these results that the property of being elliptically embedded is closely related to the nilpotency and subnormality of certain subgroups. One of the questions considered here i s the following.
Commutators in groups of order-preserving permutations
Authors: Droste Manfred , Shortt R. M.
Published: May 2009

Let (S, ≤) be a poset (partially ordered set),A(S) = Aut(S, ≤) its automorphism group andG⊆A(S) a subgroup. In the literature, various authors have studied sufficient conditions onGand the structure of (S, ≤) which imply thatGis simple or perfect. Let us call (S, ≤)doubly homogeneousif each isomorphism between two 2-subsets of 5 extends to an isomorphism of (S, ≤). Higman [8] proved that if (S, ≤) is a doubly homogeneous chain thenB(S), the group of all automorphisms of (S, ≤) with bounded support, is simple, and each element ofB(S) is a commutator inB(S). Droste, Holland and Macpherson [5] showed that if (S, ≤) is a doubly homogeneous tree then its automorphism group again contains a unique simple normal subgroup in which each element is a commutator. Dlab [3] established similar results for various groups of locally linear automorphisms of the reals. Further results in this direction are contained in Glass [7]. It is the aim of this note to establish a common generalization and sharpening of the previously mentioned results.
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.
Asymmetry in the lattice of kernel functors
Authors: de Viola-Prioli Ana M. , Viola-Prioli Jorge E.
Published: May 2009

Much of the research done by different authors on the lattice of kernel functors (equivalently, linear topologies) has been summarized by Golan in [2]. More recently, the rings whose lattices of kernel functors are linearly ordered were introduced in [3] as a categorical generalization of valuation rings in the non-commutative case. Results (and examples) in [3] show that there is an abundance of non-commutative rings R whose lattices (R), both in Mod-R and R-Mod, are simultaneously linearly ordered; however, the question of the symmetry of this condition remained open. Here we will prove that, for every natural number n≥3, there exists a ring Rn such that (Mod-Rn) is a linearly ordered lattice of n elements, whereas (Rn-Mod) is not linearly ordered.
Knots, groups, and spinning
Authors: Kearton Cherry
Published: May 2009

In this note we observe how results of Gordon, Luecke, Feustel, and Whitten establish the precise extent to which knowledge of the knot group determines the knot, in both the oriented and the unoriented case. This enables us to say, in terms of the knot group, exactly when two spun knots are the same.
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.
Unique factorization in Cayley arithmetics and cryptology
Authors: Lamont P. J. C.
Published: May 2009

Let be the classical Cayley algebra defined over the reals with basis where gives a quaternion algebra ℋ4 with i0 = 1, i1i2i3 = −1, i1i4 = i5, i2i4 = i6 and i3i4 = i7. The multiplication table of the imaginary basic units follows:
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.
Free products of inverse semigroups II
Authors: Jones Peter R. , Margolis Stuart W. , Meakin John , Stephen Joseph B.
Published: May 2009

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.
On the geometry of the unit spheres of the Lorentz spaces Lw,1
Authors: Carothers N. L. , Dilworth S. J. , Trautman D. A.
Published: May 2009

We identify the extreme points of the unit sphere of the Lorentz space Lw,1 This yields a characterization of the surjective isometries of Lw,1(0,1). Our main result is that every element in the unit sphere of Lw,1 is the barycenter of a unique Borel probability measure supported on the extreme points of the unit sphere of Lw,1.
The semigroup of continuous selfmaps ofIhas infinitely many ideals
Authors: Baker J. W. , Pym J. S.
Published: May 2009

LetIdenote the interval [0,1] in its usual topology and letS=S(I)be the semigroup of continuous mappings ofIinto itself with function composition as the semigroup operation. In a survey talk given at Oberwolfach in 1989 (see [5]), K. D. Magill Jr. pointed out that some elementary algebraic properties ofSare still unknown. We shall answer one of the questions he raised (which appears as Problem 4.6 in [5] and which he had asked earlier, as long ago as 1975 [3]) by showing thatShas infinitely many distinct two-sided ideals. In fact, we shall produce an infinite descending sequence of distinct ideals. As Magill points out, this also solves his Problem 4.5:Shas infinitely many distinct congruences. We believe thatSmust have c distinct ideals, but we have been unable to prove this.
Integral closures of ideals relative to Artinian modules, and exact sequences
Authors: Sharp R. Y. , Tiraş Y.
Published: May 2009

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].
The regular radical of semigroup rings of commutative semigroups
Authors: Kelarev A. V.
Published: May 2009

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.
On the Hirsch-Plotkin radical of a factorized group
Authors: Franciosi Silvana , de Giovanni Francesco
Published: May 2009

Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.
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 satisfies

If M is only a left A-module, by a derivation from A into M we mean a linear mapping D: A→M such that

Each 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.
A note on p-groups with power automorphisms
Authors: Bryce R. A. , Cossey John , Ormerod E. A.
Published: May 2009

Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).