Book Ramsey numbers. I

A book B_p is a graph consisting of p triangles sharing a common edge. In this paper we prove that if p ≤ q/6 ?o(q) and q is large, then the Ramsey number r (B_p,B_q) is given by r (B_p,B_q) = 2q+3, and the constant 1/6 is essentially best possible. Our proof is based on Szemerédi's uniformity lemma and a stability result for books. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

We consider finite groups which have connected transversals to subgroups whose order is a product of two primespandq. We investigate those values ofpandqfor which the group is soluble. We can show that the solubility of the group follows ifq = 2 andp ≤ 61,q = 3 andp ≤ 31,q = 5 andp ≤ 11. We then apply our results on loop theory and we show that if the inner mapping group of a finite loop has orderpqwherepandqare as above then the loop is soluble.     

The Graded Algebra Generated by Two Eulerian Derivatives

We study the algebra R _p,q generated by the Eulerian derivatives for two parameters p and q. Subject to certain conditions on the parameters, we show that R _p,q is a finitely presented N-graded algebra of Gelfand–Kirillov dimension 3. We establish a criterion for the cyclic module R _p,q /R _p,q f to be Noetherian, where f is homogeneous of degree 1. For some choices of the parameters, this criterion always holds and we know of no situation where it fails. It is not known whether R _p,q is Noetherian. We classify the point modules for R _p,q and determine the normal elements and graded automorphisms for R _p,q .     

Some permutation groups of degreep=6q+1

Transitive permutation groups of degrees 43, 67, 79, 103 and 139 are classified.In this note we consider insoluble transitive permutation groups of degreeq = 6q+1 wherep andq are primes and summarise the computations whereby these groups have been classified for some small values ofq. The result which allows progress on this problem is due to McDonough [1]; he showed that if such a group has a Sylowp-normaliser of order 3p then it is isomorphic either toPSL(3, 3) orPAL(3, 5) (of degrees 13, 31 respectively). Using this theorem machine computations along the lines of those done by Parker, Nikolai and Appel [3, 2] for degreesp=2q+1 andp=4q+1 give the following     

On <Emphasis Type="Italic">p</Emphasis>-adic quasi Gibbs measures for <Emphasis Type="Italic">q</Emphasis> + 1-state Potts model on the Cayley tree

In the present paper we introduce a new kind of p-adic measures, associated with q + 1-state Potts model, called p-adic quasi Gibbs measure, which is totally different from the p-adic Gibbs measure. We establish the existence of p-adic quasi Gibbs measures for the model on a Cayley tree. If q is divisible by p, then we prove the occurrence of a strong phase transition. If q and p are relatively prime, then there is a quasi phase transition. These results are totally different from the results of [F. M. Mukhamedov and U. A. Rozikov, Indag. Math. N. S. 15, 85–100 (2005)], since when q is divisible by p, which means that q + 1 is not divided by p, so according to a main result of the mentioned paper, there is a unique and bounded p-adic Gibbs measure (different from p-adic quasi Gibbs measure)     

Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

Archimedean classes in integral commutative residuated chains

This paper investigates a quasi‐variety of representable integral commutative residuated lattices axiomatized by the quasi‐identity resulting from the well‐known Wajsberg identity (p → q) → q ≤ (q → p) → p if it is written as a quasi‐identity, i. e., (p → q) → q ≈ 1 ? (q → p) → p ≈ 1 . We prove that this quasi‐identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi‐variety is in fact a variety and provide an axiomatization. The obtained results shed some light on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL‐algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL‐algebras is generated by its Archimedean members (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 = ∞.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

An upper bound for the minimum weight of dual codes of Figueroa planes

In this note we give an explicit construction for words of weight 2q³ - q² - q in the dual p-ary code of the Figueroa plane of order q³, where q > 2 is any power of the prime p. When p is odd this then allows us, for the Figueroa planes, to improve on the previously known upper bound of 2q³ for the minimum weight of the dual p-ary code of any plane of order q³. The construction is the same as one that applies to desarguesian planes of order q³ as described in [3].     

The H?lder-Poincaré duality for L q,p -cohomology

We prove the following version of Poincaré duality for reduced L _q,p -cohomology: For any 1 < q, p < ∞, the L _q,p -cohomology of a Riemannian manifold is in duality with the interior L _p',q'-cohomology for 1/p + 1/p′ = 1/q + 1/q′ = 1.     

Estimates of Disjoint Sequences in Banach Lattices and R.I. Function Spaces

We introduce UDS _p -property (resp. UDT _q -property) in Banach lattices as the property that every normalized disjoint sequence has a subsequence with an upper p-estimate (resp. lower q-estimate). In the case of rearrangement invariant spaces, the relationships with Boyd indices of the space are studied. Some applications of these properties are given to the high order smoothness of Banach lattices, in the sense of the existence of differentiable bump functions     

Conditional pressure and coding

We use pressure to obtain invariants for bounded-to-one block homomorphisms between Markov shifts. These invariants enable us to show that if there is a bounded-to-one block homomorphism between Bernoulli shifts given by probability vectorsp andq thenq may be obtained fromp by a permutation. The invariants may be viewed as conditional pressures; a convergence theorem for eigenmeasures of Ruelle operators motivates the definition of conditional pressure and helps establish our invariants for regular isomorphism of Markov shifts. It follows that Bernoulli shifts given by probability vectorsp andq are regularly isomorphic iffq is a permutation ofp. We employ our invariants also in the context of a finite equivalence. Finally we indicate that ratio variational principles yield further invariants.     

On r*-sequenceability and symmetric harmoniousness of groups. II

In this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*-sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*-sequenceable is R*-sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley & Sons, Inc.     

Relative Position,Sequences and Operators in Banach Spaces

A (p + q) × (p + q) matrix-valued inner function S in the unit disc ?? is called (p, q)-type Arov-inner if in the block partition . the p × p diagonal block S₁₁ and the q × q diagonal block S₂₂ are outer matrix-valued functions. A holomorphic p × q matrix-valued function f in ?? is called Arov-completable if there is a (p, q)-type Arov-inner function S such that S₁₂ = f Arov-completability of a given p × q Schur function f is characterized in terms of a (p + q)-variate stationary sequence (X_{n) ? Z}) in Hilbert space which is naturally associated with f. The necessary and sufficient condition for Arov-completability is an orthogonality condition for certain backward and forward innovation vectors generated by (X_{n) ? Z}.     

On an inclusion between operator ideals

Let 1 ⩽ q < p < ∞ and 1/r:= 1/p max(q/2, 1). We prove that L _r,p ^(c), the ideal of operators of Gel’fand type l _r,p , is contained in the ideal Π _p,q of (p, q)-absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.     

Uniform convexity of unitary ideals

ItE is a symmetric Banach sequence which isq-concave with the constant equal to 1 (where 2≦q<∞), thenS _E isq-PL-convex. IfE isq-concave andp-convex with the constants equal to 1 (where 1<p≦2≦q<∞), thenS _E is uniformly convex with modulus of convexity of power typeq and uniformly smooth with modulus of smoothness of power typep.