<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Classification of Material <Emphasis Type="Italic">G</Emphasis>-Structures

A geometrical interpretation of the G-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

On eigenvalues of <Emphasis Type="Italic">p</Emphasis>-adic curvature

We study the eigenvalues of the p-adic curvature transformationson buildings. In particular, we determine the maximal eigenvalues ofthese transformations.     

Rate of convergence to stationarity of the system <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">N</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">R</Emphasis>

We consider the M/M/N/N+R service system, characterized by N servers, R waiting positions, Poisson arrivals and exponential service times. We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and study its behaviour as a function of R, N and the arrival rate λ, allowing λ to be a function of N.     

Commutativity of Almost <Emphasis Type="Italic">F</Emphasis>-algebras

Let A be an Archimedean vector lattice, let be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ ₀:A × A→ B is a positive orthosymmetric bimorphism, then there exists a positive bimorphism extension Ψ of Ψ ₀ to × in B which is orthosymmetric. This leads to a new and short proof of the commutativity of the almost f-algebras multiplications.     

Presenting affine <Emphasis Type="Italic">q</Emphasis>-Schur algebras

We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine . Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto.     

Lie superalgebras graded by the root system B(<Emphasis Type="Italic">m,n</Emphasis>)

We determine the Lie superalgebras that are graded by the root system B(m,n) of the orthosymplectic Lie superalgebra osp(2m + 1,2n). Mathematics Subject Classification (2000) Primary 17B70, Secondary 17A70     

Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an l_p-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on l_p that is quasinilpotent at a non-zero positive vector x₀ has a non-trivial invariant closed subspace. In this paper, it is proved that if is a collection of continuous operators with moduli on l_p that is finitely modulus-quasinilpotent at a non-zero positive vector x ₀ then and its right modulus sub-commutant have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on l _p whose moduli commute with a non-zero positive operator I on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace, so that all positive operators on l _p which commute with a non-zero positive operator S on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace. This research was supported by the Natural Science Foundation of Hunan Province of P. R. China (04JJ6004), the Foundation of Education Department of Hunan Province of P. R. China (04C002) and the Natural Science Foundation of P. R. China (10671147). Received: 4 December 2005 Revised: 19 June 2006     

On the cogeneration of <Emphasis Type="Italic">t</Emphasis>-structures

Let C be a set of objects in a triangulated compactly generated category We denote by the smallest suspended subcategory closed under coproducts which contains C (the smallest cosuspended subcategory closed under products which contains C). We prove that if C is a set of compacts objects then is a t-structure in where T_C I is the dual of C with respect to an injective cogenerator I in the category Mod C. Moreover, we show that: C is a tilting set if and only if And, this is equivalent to T_CI is a cotilting object in Received: 28 March 2003     

Different prime <Emphasis Type="Italic">N</Emphasis>-ideals and IFP <Emphasis Type="Italic">N</Emphasis>-Ideals

We provide some characterizations of completely prime (completely semiprime) and 3-prime (3-semiprime) N-groups. The relationship between a 3-prime (completely prime) N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N is investigated. Moreover, the notion of IFP N-ideal is defined. We prove that the concept of IFP N-ideal occurs naturally where N is a left permutable (left self distributive, subcommutative) near-ring and Γ a monogenic N-group. Also, we obtain some relationships between an IFP N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N.     

Zinbiel algebras under <Emphasis Type="Italic">q</Emphasis>-commutators

An algebra with the identity t ₁(t ₂ t ₃) = (t ₁ t ₂+t ₂ t ₁)t ₃ is called Zinbiel. For example, ℂ[x] under the multiplication is Zinbiel. Let a ○ _q b = a ○ b + q b ○ a be a q-commutator, where q ∈ ℂ. We prove that for any Zinbiel algebra A the corresponding algebra under the commutator A ⁽⁻¹⁾ = (A, ○₋₁) satisfies the identities t ₁ t ₂ = −t ₂ t ₁ and (t ₁ t ₂)(t ₃ t ₄) + (t ₁ t ₄)(t ₃ t ₂) = jac(t ₁, t ₂, t ₃)t ₄ + jac(t ₁, t ₄, t ₃)t ₂, where jac(t ₁, t ₂, t ₃) = (t ₁ t ₂)t ₃ + (t ₂ t ₃)t ₁ + (t ₃ t ₁)t ₂. We find basic identities for q-Zinbiel algebras and prove that they form varieties equivalent to the variety of Zinbiel algebras if q ² ≠ 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 57–78, 2005.     

Packet loss characteristics for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis> queueing systems

In this contribution we investigate higher-order loss characteristics for M/G/1/N queueing systems. We focus on the lengths of the loss and non-loss periods as well as on the number of arrivals during these periods. For the analysis, we extend the Markovian state of the queueing system with the time and number of admitted arrivals since the instant where the last loss occurred. By combining transform and matrix techniques, expressions for the various moments of these loss characteristics are found. The approach also yields expressions for the loss probability and the conditional loss probability. Some numerical examples then illustrate our results.     

Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis>

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle where is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone coincides with the cone of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space of the irreducible representation space V of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of at all smooth points xG/P. Equivalently, an integral variety of is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.     

Groups with less than <Emphasis Type="Italic">n</Emphasis> subgroups of index <Emphasis Type="Italic">n</Emphasis>

We characterise (residually-finite) groups which possess less than n subgroups of index n for almost all n ∈ ℕ.     

Positive Operator Majorization and <Emphasis Type="Italic">p</Emphasis>-hyponormality

In this note we examine the relationships between p-hyponormal operators and the operator inequality . This leads to a method for generating examples of p-hyponormal operators which are not q-hyponormal for any . Our methods are also shown to have implications for the class of Furuta type inequalities.     

Charakterisierung der <Emphasis Type="Italic">q</Emphasis>-Orthogonalpolynome in <Emphasis Type="Italic">x</Emphasis>

Characterization of q-Orthogonal Polynomials. Im Anschluß an die Arbeit Orthogonalpolynome in x und q^–x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung (Monatsh. Math. 132, 123–140 (2001); im folgenden als [4] zitiert) werden alle Möglichkeiten für q-Orthogonalpolynome in x als Lösungen von q-Operatorgleichungen zweiter Ordnung angegeben (Orthogonalität im positiv definiten Sinne). Dabei erfolgt die Numerierung der Abschnitte und die Angabe der Formel-nummern unter Einbeziehung von [4].     

An Estimate of Growth Bound of Positive <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-Semigroup on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Space and its Applications

Let be a positive C ₀-semigroup on L ^p (Ω), with infinitesimal generator A. In this paper, it is proved that if there exists a such that and , where A ^* is the adjoint of A, then the growth bound of T(t) is upper bounded by b when p = 1, and by when 1 lt; p lt; α and c D(A), where . This is an operator version of a classical stability result on Z-matrix. As application examples, some new results on the asymptotic behaviours of population system and neutron transport system are obtained. Submitted: March 1, 2001?Revised: August 28, 2002     

The <Emphasis Type="Italic">N</Emphasis>-wave equations with <Emphasis Type="Italic">PT</Emphasis> symmetry

We study extensions of N-wave systems with PT symmetry and describe the types of (nonlocal) reductions leading to integrable equations invariant under the P (spatial reflection) and T (time reversal) symmetries. We derive the corresponding constraints on the fundamental analytic solutions and the scattering data. Based on examples of three-wave and four-wave systems (related to the respective algebras sl(3,C) and so(5,C)), we discuss the properties of different types of one- and two-soliton solutions. We show that the PT-symmetric three-wave equations can have regular multisoliton solutions for some specific choices of their parameters.