Algebras of Symmetric Holomorphic Functions on lp

The authors study the algebra of uniformly continuous holomorphicsymmetric functions on the ball of l_p, investigating in particularthe spectrum of such algebras. To do so, they examine the algebraof symmetric polynomials on l_p-spaces, as well as finitely generatedsymmetric algebras of holomorphic functions. Such symmetricpolynomials determine the points in l_p up to a permutation.     

The Determination of Consistent Orderings for the SOR Iterative Method

A p-cyclic matrix A can always be transformed into a consistentlyordered form PAP^T, where P is a permutation matrix. In thispaper simple and systematic techniques are described for establishingthe cyclicity of a matrix and for determining P. The techniquesdepend on the use of ordering vectors, and formulae for determiningthese vectors are derived.     

The Minimal Base Size of Primitive Solvable Permutation Groups

A base of a permutation group G is a sequence B of points fromthe permutation domain such that only the identity of G fixesB pointwise. Answering a question of Pyber, we prove that allprimitive solvable permutation groups have a base of size atmost four.     

Uncountable Cofinalities of Permutation Groups

A sufficient criterion is found for certain permutation groupsG to have uncountable strong cofinality, that is, they cannotbe expressed as the union of a countable, ascending chain (H_i)_i     

On Principal Blocks of p-Constrained Groups

A theorem of K. W. Roggenkamp and L. L. Scott shows that fora finite group G with a normal p-subgroup containing its owncentralizer, any two group bases of the integral group ringZG are conjugate in the units of Z_pG. Though the theorem presentsitself in the work of others and appears to be needed, thereis no published account. There seems to be a flaw in the proof,because a ‘theorem’ appearing in the survey [K.W. Roggenkamp, ‘The isomorphism problem for integral grouprings of finite groups’, Progress in Mathematics 95 (Birkhäuser,Basel, 1991), pp. 193--220], where the main ingredients of aproof are given, is false. In this paper, it is shown how toclose this gap, at least if one is only interested in the conclusionmentioned above. Therefore, some consequences of the resultsof A. Weiss on permutation modules are stated. The basic stepsof which any proof should consist are discussed in some detail.In doing so, a complete, yet short, proof of the theorem isgiven in the case that G has a normal Sylow p-subgroup. 2000Mathematical Subject Classification: primary 16U60; secondary20C05.     

A Characterisation of p-Soluble Groups

If p is a prime, then a finite group is p-soluble if each ofits composition factors is either a p-group or has order coprimeto p. For example, soluble groups are p-soluble. However, thereare many insoluble groups that are p-soluble. We shall provethe following result. 1991 Mathematics Subject Classification20D10.     

Some Remarks on p-Adic Analytic Groups

A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and p^e exactly divides |F|, then a Z_pF-latticeM is determined up to isomorphism by its finite quotient M/p^e+1M.If M is a free Z_p-module of rank d, this is equivalent to sayingthat representations of F in GL_d(Z_p) are determined up to equivalenceby their images modulo p^e+1. 1991 Mathematics Subject Classification20E18, 22E20.     

Shift-Generated Random Permutations and The M/D/1 Queue

We consider a simple shift procedure which transforms any mappingf:{0, ..., n}{0, ..., n} into a permutation of 0, ..., n. Westudy this shift under the assumption that the values f(0),..., f(n) are uniformly and independently chosen random numbersin {0, ..., n}. The exact and asymptotic distributions of therequired shift sizes are derived, and a connection to queueingtheory is exhibited. 1991 Mathematics Subject Classification60C05, 60K25.     

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p₁,...,p_m, where each p_j isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors p_j are not unique butthe number m of them is, as is the set of the degrees of thep_j. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced.     

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.     

A refined bound for sum-free sets in groups of prime order

Improving upon earlier results of Freiman and the present authors,we show that if p is a sufficiently large prime and A is a sum-freesubset of the group of order p, such that n: = |A| > 0.318p,then A is contained in a dilation of the interval [n, p –n](mod p).     

Combining Perfect Systems of Difference Sets

We use sharply 2-transitive permutation groups to constructan additive sequence of permutations from a system of differencesets, each component of which has size one less than a primepower. This allows us to combine perfect systems of differencesets to form other perfect systems. In particular, if thereexists a perfect (m, n + 1, 1)-system and a perfect (q, n +1, 1)-system then there exists a perfect (mqn(n + 1) + m + q,n + 1, 1)-system.     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

A Note on Osserman Lorentzian Manifolds

Let p be a point of a Lorentzian manifold M. We show that ifM is spacelike Osserman at p, then M has constant sectionalcurvature at p; similarly, if M is timelike Osserman at p, thenM has constant sectional curvature at p. The reverse implicationsare immediate. The timelike case and 4-dimensional spacelikecase were first studied in [3]; we use a different approachto this case. 1991 Mathematics Subject Classification 53B30,53C50.     

On Sylow Subgraphs of Vertex-Transitive Self-Complementary Graphs

One of the basic facts of group theory is that each finite groupcontains a Sylow p-subgroup for each prime p which divides theorder of the group. In this note we show that each vertex-transitiveself- complementary graph has an analogous property. As a consequenceof this fact, we obtain that each prime divisor p of the orderof a vertex-transitive self-complementary graph satisfies thecongruence p^m 1(mod 4), where p^m is the highest power of pwhich divides the order of the graph. 1991 Mathematics SubjectClassification 05C25, 20B25.     

Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication

Let E/Q be an elliptic curve. For a prime p of good reduction,let E(F_p) be the set of rational points defined over the finitefield F_p. Denote by (#E(F_p)) the number of distinct prime divisorsof #E(F_p). For an elliptic curve with complex multiplication,the normal order of (#E(F_p)) is shown to be log log p. The normalorder of the number of distinct prime factors of the exponentof E(F_p) is also studied. 2000 Mathematics Subject Classification11N37, 11G20.     

Sharkovskii Type of Cycles

The Sharkovskii type of a map of an interval is the Sharkovskii-greatestinteger t such that it has a periodic point of period t. TheSharkovskii type of a cycle (that is, a cyclic permutation)is the Sharkovskii type of the 'connect the dots' map determinedby it. For n 2, let c(n) denote the finite set of integerswhich are Sharkovskil types of n-cycles. We give an internalcharacterization of c(n) and an n⁴-time algorithm for determiningthe Sharkovskii type of an n-cycle.     

A Congruence for Factorials

The methods of p-adic analysis are used to prove a congruencefor (pn)!(pⁿn!) modulo a power of a prime p.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.