Hemisystems on the Hermitian Surface

The natural geometric setting of quadrics commuting with a Hermitiansurface of PG(3,q²), q odd, is adopted and a hemisystem on theHermitian surface H(3,q²) admitting the group P^–(4,q)is constructed, yielding a partial quadrangle PQ((q–1)/2,q²,(q–1)²/2) and a strongly regular graph srg((q³+1)(q+1)/2,(q²+1)(q–1)/2,(q–3)/2,(q–1)²/2).For q>3, no partial quadrangle or strongly regular graphwith these parameters was previously known, whereas when q=3,this is the Gewirtz graph. Thas conjectured that there are nohemisystems on H(3,q²) for q>3, so these are counterexamplesto his conjecture. Furthermore, a hemisystem on H(3,25) admitting3.A₇.2 is constructed. Finally, special sets (after Shult) andovoids on H(3,q²) are investigated.     

The 2-Modular and 3-Modular Decomposition Numbers for the Sporadic Simple O'Nan Group and its Triple Cover

The 2-modular and 3-modular decomposition numbers for the sporadicsimple O'Nan group O'N and its triple cover are determined.The results are obtained with the help of the computer algebrapackages MeatAxe and GAP. Permutation modules and tensor productsof modules have been condensed. The result relies on the constructionof a 153-dimensional representation for 3·O'N over GF(4)and a 154-dimensional representation for O'N over GF(3).     

On the Girth of Graphs Critical with Respect to Edge-Colourings

A graph G with maximum valency r is called critical if r + 1colours are needed for an edgecolouring, but every proper subgraphrequires at most r. In this note we consider the minimum orderf(r, g) of a critical graph of maximum valency r and girth g.We show that f(r, 3) = r+1 or r+2 according as r is even orodd, f(r, 4) = 2r+1,f(3, 5) = 9 and f(3, 6) = 15.     

Two New Constructions of the O'Nan Group

A construction is described of a 153-dimensional representationof the triple cover of O'Nan's sporadic simple group, over thefield of order 4. This is then extended to a 306-dimensionalrepresentation of 3·O'N:2 over GF(2). A similar constructionin characteristic 3 gives an unrelated representation of theO'Nan group in 154 dimensions over GF(3), which extends to O'N:2provided that the field is extended to GF(9).     

On the Geometry of the Automorphism Group of a Free Group

The groups Aut(F₃) and Out(F₃) satisfy strictly exponentialisoperimetric inequalities; in particular, they are not automatic.For n 3, Aut (F_n) and Out (F_n) do not admit bounded bicombingsof sub-exponential length, hence they cannot act properly andcocompactly by isometries on any simply-connected space of non-positivecurvature, and they are not biautomatic.     

An algebraic loop theorem and the decomposition of PD3-pairs

Let (Y, X) denote a three-dimensional Poincaré pair (PD³-pair).By the work of Eckmann, Müller and Linnell we may suppose,up to a homotopy equivalence, that the boundary X is a closed2-manifold. We show that if a component of X fails to be ₁-injectivein Y, then there is an essential simple loop in X which is nullhomotopicin Y. It follows that there is a finite process of attaching2-disks along essential simple loops on X, and filling sphericalcomponents of X, which transforms (Y, X) into a PD³-pair (Y',X') with aspherical incompressible boundary X' and such that₁(Y) = ₁(Y'). The PD³-pair (Y', X') then admits a canonicaldecomposition as a connected sum of a finite number of asphericalPD³-pairs with incompressible boundary, together with a PD³-pairhaving virtually free (possibly finite) fundamental group andboundary a (possibly empty) disjoint union of projective planes.     

On the Small Squaring and Commutativity

Let G be a group and let k > 2 be an integer, such that (k²– 3)(k – 1) < |G|/15 if G is finite. Supposethat the condition |A²| k(k + 1)/2 + (k – 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets.     

Characterization of the Mod 3 Cohomology of the Compact, Connected, Simple, Exceptional Lie Groups of Rank 6

It is shown that the mod 3 cohomology of a 1-connected, homotopyassociative mod 3 H-space that is rationally equivalent to theLie group E₆ is isomorphic to that of E₆ as an algebra. Moreover,it is shown that the mod 3 cohomology of a nilpotent, homotopy-associativemod 3 H-space that is rationally equivalent to E₆, and whosefundamental group localized at 3 is non-trivial, is isomorphicto that of the Lie group Ad E₆ as a Hopf algebra over the mod3 Steenrod algebra. It is also shown that the mod 3 cohomologyof the universal cover of such an H-space is isomorphic to thatof E₆ as a Hopf algebra over the mod 3 Steenrod algebra. 2000Mathematics Subject Classification 57T05, 57T10, 57T25.     

On the Magnitude of the Sum of Bases of the Natural Numbers

We construct two bases of the natural numbers B₁ and B₂, eachof order two, such that (B₁ + B₂ (n) <n^{+c/(log n)}. For alower estimate, it is proved that if B₂ and are two bases, eachof order two, then (B₁+B₂)(n) > n. Generalisations to sumsof bases of order h > 2 are also given.     

Morita Equivalent 3-Blocks of the 3-Dimensional Projective Special Linear Groups

If G is a projective special linear group PSL(3,q) with q 4or 7 (mod 9), then a Sylow 3-subgroup of G is elementary abelianof order 9. We show that the principal 3-blocks of any two suchgroups are Morita equivalent. This result and Okuyama's theoremfor PSL(3,4) prove the Broué conjecture for these blocks.1991 Mathematics Subject Classification: 20C05, 20C20.     

On the Location of Zeros of Certain Classes of Polynomials with Applications to Numerical Analysis

A polynomial is said to be of type (p₁, p₂, p₃) relative tothe unit circle if it has p₁ zeros interior to, p₂ on, and p₃exterior to the unit circle. Stability criteria frequently arisewhere a polynomial or a family of polynomials must be shownto be of type (p₁, p₂, 0) or of type (p₁, 0, 0). Here we reconsiderthe practical problem of showing that a polynomial is of oneor other of these types, and we show that the testing of a polynomialof degree n may always be reduced to the testing of one of degreen–1. The simplicity of the method is illustrated by itsapplication to several well known difference schemes for partialdifferential equations.     

Infinite Simple (2, 3, n)-Groups and Congruence Hulls in the Modular Group

We prove that if n > 66 and (n, 30) = 1, then there existuncountably many infinite simple (2, 3, n)- groups, that is,groups generated by a pair of elements x, y, say, where theorders of x, y and xy are 2, 3 and n, respectively. This extendsprevious results of Schupp and the authors. These results are used to prove the existence of subgroups ofthe modular group with special arithmetic properties. 1991 MathematicsSubject Classification 20F06.     

Abel Equations: Composition Conjectures and the Model Problem

We consider the Abel equation where p and q are polynomials, and a is a fixed constant. Wedenote the solution of (1) by y(x, c), where y(a, c) = c. Standardexistence theorems ensure that y(x, c) is well-defined and analyticin both its arguments, for c sufficiently small. If y(b, c)= c, then we call y(x, c) a periodic solution. Likewise, ify(b, c) c for all c close to 0, then we say that the systemhas a centre between a and b. The numbers a and b are not important;by a simple transformation, we can always choose a = 0 and b= 1, and we shall usually do so from now on. Abel equations arise in several circumstances, but perhaps themain reason for their recent study is connected to the familyof systems where M and N are homogeneous polynomials of the same degreen. A transformation due to Cerkas allows us to bring these systemsto the form (1), where p and q are now trigonometric polynomials.It is not hard to show that, setting a = 0 and b = 2, the definitionsof periodic solution and centre for (1) coincide with theirusual definitions in the planar system (2). There are also transformationsto Abel-type equations for more general systems; see [8, 10]. This trigonometric Abel equation has been used in a large numberof works in order to estimate the number of limit cycles orobtain centre conditions, as well as in more general investigationsrelating the derivatives of the return map with iterated integrals.However, studying system (2) in whatever form is by no meanseasy, and a natural question is to ask whether we can stillcapture the essence of this problem if we take p and q to bepolynomials, in the hope that the calculations will become easier.The recent series of investigations by Briskin, Françoiseand Yomdin [3, 4, 5] seems to indicate that this could be thecase. Our interest here is to see what conditions the existence ofa centre in (1) imposes on the defining equations. For easeof reference, we shall always denote the antiderivative of thepolynomials p and q as P and Q; that is, 2000 Mathematics SubjectClassification 34C25, 34C99.     

Extended Generalized Hexagons and the Suzuki Chain

Four extended generalized hexagons related to the simple groupsG₂(2)', PSU₄(3), HJ and Suz are characterized by the conditionthat any triple of points {x, y, z} is a clique of the pointgraph not in a circle of the extended hexagon if and only ifthe distance of y and z in the residue at x is 3.     

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S³ is singled out. Using the Yasuda–Shimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S³. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by Yasuda–Shimada. It isexplained why this family of Finslerian space forms is not projectivelyflat.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

The Dimension of the Space of Harmonic 2-Spheres in the 6-Sphere

Using the twistorial approach and some previous results, weprove the conjecture that the dimension of the moduli spaceof harmonic maps of area 4d from the 2-sphere to the 2n-sphereis 2d + n² for the particular case n = 3. 2000 Mathematics SubjectClassification 53A10 (primary), 53C42 (secondary).     

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

Some Results on the Oberwolfach Problem

The well-known Oberwolfach problem is to show that it is possibleto 2-factorize K_n (n odd) or K_n less a 1-factor (n even) intopredetermined 2-factors, all isomorphic to each other; a fewexceptional cases where it is not possible are known. A completelynew technique is introduced that enables it to be shown thatthere is a solution when each 2-factor consists of k r-cyclesand one (n–kr)-cycle for all n 6kr–1. Solutionsare also given (with three exceptions) for all possible valuesof n when there is one r-cycle, 3 r 9, and one (n–r)-cycle,or when there are two r-cycles, 3 r 4, and one (n–2r)-cycle.     

The Laplace Transform of the Macdonald Function of Argument x3/2 and the Airy Function Ai (x)

The Laplace transform of the Airy function of the first kindAi(x) is obtained conveniently as a special case of the Laplacetransform of the Macdonald function of argument 2x^3/2.