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 = ∞.     

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.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Codeterminants and <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We study codeterminants in the q-Schur algebra S _q (n,r) and prove that the standard ones form a basis of S _q (n,r), using a quantized version of the Désarménien matrix. We find elements of the form F _S 1_λ E _T in Lusztig’s modified enveloping algebra of gl(n), which, up to powers of q, map to the basis of standard codeterminants, where F _S ∈U ^– and E _T ∈U ⁺ are explicitly given products of root vectors, depending on Young tableaux S and T.     

Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Factorization of the r-matrix for the quantum algebra <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

We propose a method for construction of the general solution of the Yang–Baxter equation with the U _q (sℓ _n ) symmetry algebra. This method is based on the factorization property of the corresponding L-operator. We present a closed-form expression for the universal R-matrix in the form of a difference operator acting on the space of functions of n(n − 1) variables. Bibliography: 16 titles.     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">h</Emphasis>, 1, 1)-labeling of outerplanar graphs

An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, . . . , λ} to the nodes of the graph such that adjacent nodes are assigned integers of at least distance h ≥ 1 apart and all nodes of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λ _{h, 1, 1} and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n ³), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h, 1, 1)-labeling for outerplanar graphs that is within additive constants of the optimum values. This research is partially supported by the European Research Project Algorithmic Principles for Building Efficient Overlay Computers (AEOLUS) and was done during the visit of Richard B. Tan at the Department of Computer Science, University of Rome “Sapienza”, supported by a visiting fellowship from the University of Rome “Sapienza”.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

A Lower Bound of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Norms in the CLT for Strongly Mixing Random Variables

We derive a lower bound of L _p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X ₁,..., X _n with under the boundedness condition ℙ{|X _i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ _r⩾1 r ²α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.     

Generalized fuzzy <Emphasis Type="Italic">n</Emphasis>-ary subhypergroups of a commutative <Emphasis Type="Italic">n</Emphasis>-ary hypergroup

In this paper, by means of a new idea, the concept of (invertible) (∈,∈∨q)-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are investigated. A kind of n-ary quotient hypergroup by an (∈, ∈∨q)-fuzzy n-ary subhypergroup is provided and the relationships among (∈, ∈∨q)-fuzzy n-ary subhypergroups, n-ary quotient hypergroups and homomorphism are investigated. Several isomorphism theories of n-ary hypergroups are established.     

A lower bound for the <Emphasis Type="Italic">r</Emphasis>-order of a matrix modulo <Emphasis Type="Italic">N</Emphasis>

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity. Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy     

Lifting elementary embeddings <Emphasis Type="Italic">j</Emphasis>: <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript> → <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript>

We describe a fairly general procedure for preserving I₃ embeddings j: V _λ → V _λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I₃ embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency was established assuming an I₁ embedding.      

A generalized Kiepert formula for <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">ab</Emphasis></Subscript> curves

Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface which, outside from one point, has a smooth polynomial equation in the plane, in the sense that we characterize the points whose n-th multiple in the Jacobian belongs to the theta divisor.     

The Dirichlet series for the exterior square <Emphasis Type="Italic">L</Emphasis>-function on GL(<Emphasis Type="Italic">n</Emphasis>)

We present an elementary derivation of the Jacquet–Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients A(m ₁,…,m _n−1).     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Reduction for primitive flag-transitive (<Emphasis Type="Italic">v</Emphasis>, <Emphasis Type="Italic">k</Emphasis>, 4)-symmetric designs

It has been shown that if a (v, k, λ)-symmetric design with λ ≤ 3 admits a flag-transitive automorphism group G which acts primitively on points, then G must be of affine or almost simple type. Here we extend the result to λ = 4.