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We classify the real and strongly real conjugacy classes in GL n( q) and SL n( q). In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This paper is the first of two that together classify the real and strongly real classes in GL n( q), SL n( q), PGL n( q), PSL n( q), and all quasi-simple covers of PSL n( q). [ABSTRACT FROM AUTHOR]

Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers. [ABSTRACT FROM AUTHOR]

We give a version in characteristic p > 0 of Mumford's theorem characterizing a smooth complex germ of surface ( X, x) by the triviality of the topological fundamental group of U = X\{ x}. [ABSTRACT FROM AUTHOR]

We show that the nilpotency of the metacyclic subgroups of a hypercyclic group implies the hypercentrality of the whole group. We also construct a center-free metabelian group all of whose polycyclic subgroups are abelian. [ABSTRACT FROM AUTHOR]

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if | G : A| ⩽ exp( A), the exponent of A, then A contains a non-trivial normal subgroup of G. This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by | G : A| < Exp( G), where Exp( G) denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G. We will use these results to derive some facts about transitive permutation groups. [ABSTRACT FROM AUTHOR]

Let ΦΦ( x) be the number of the integers n such that φφ( n) ≤≤ x, where φφ denotes the Euler function. In this paper we give an explicit form of Bateman's theorem for the remainder term ΦΦ( x) -- Ax where A == ζζ(2) ζζ(3)/ ζζ(6). [ABSTRACT FROM AUTHOR]

Recently, Cilleruelo proved that the product is a square only for n == 3. In this note, using similar techniques, we prove that for the positive integer m whose divisors are of the form of 4 q++1, the product is not a square for n sufficiently large. As a corollary, we prove that for is not a square for all n. [ABSTRACT FROM AUTHOR]

Qualified residue difference sets of power n are known to exist for n == 2, 4, 6, as do similar sets that include the zero element, while both classes of set are known to be nonexistent for n == 8 and n == 10. Both classes of set are proved nonexistent for n == 12. [ABSTRACT FROM AUTHOR]

A vector bundle E on a smooth irreducible algebraic variety X is called a Steiner bundle of type ( F0, F1) if it is defined by an exact sequence of the form where s, t ≥ 1 and ( F0, F1) is a strongly exceptional pair of vector bundles on X such that is generated by global sections. Let X be a smooth irreducible projective variety of dimension n with an n-block collection , of locally free sheaves on X which generate Db( X – mod). We give a cohomological characterisation of Steiner bundles of type on X, where 0 ≤ a < b ≤ n and 1 ≤ i0 ≤ αa, 1 ≤ j0 ≤ αb. [ABSTRACT FROM AUTHOR]

Let αα in (0, 1]] and ββ > 1 be algebraic numbers. We study the asymptotic behaviour of the function that counts the number of digit changes in the ββ-expansion of αα. [ABSTRACT FROM AUTHOR]

Let FG be the group algebra of a group G without 2-elements over a field F of characteristic p ≠ 2 endowed with the canonical involution induced from the map g ↦ g–1, g ∈ G. Let ( FG)– and ( FG)+ be the sets of skew and symmetric elements of FG, respectively, and let P denote the set of p-elements of G (with P = 1 if p = 0). In the present paper we prove that if either P is finite or G is non-torsion and ( FG)– or ( FG)+ is Lie solvable, then FG is Lie solvable. The remaining cases are also settled upon small restrictions. [ABSTRACT FROM AUTHOR]

We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups. [ABSTRACT FROM AUTHOR]

In reviewing the foundations of probability, to establish the underpinnings of the Monte Carlo method, I found some basic results for which I could not find a simple, clear, and concise derivation in the literature. This paper is an attempt to present a sufficient collection of such results. The presentation is pitched at the level of Monte Carlo practitioners, rather than mathematical probabilists. [ABSTRACT FROM AUTHOR]

In this article, we show a truncated Second Main Theorem of meromorphic mappings from n into N () for moving targets. The moving targets are only assumed to be nondegenerate. 2000 Mathematics Subject Classification: 32H30, 32A22; 30D35. [ABSTRACT FROM AUTHOR]

One of the famous open problems in Finite Geometry is the classification of ovoids in a projective 3-space PG(3, q) over the finite field with q elements, q even. For q odd, this classification was obtained in 1955 by A. Barlotti [ A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo. Boll. Un. Mat. Ital. (3) 10 (1955), 498–506. MR0075606 (17,776b) Zbl 0066.38901 ] and G. Panella [ G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito. Boll. Un. Mat. Ital. (3) 10 (1955), 507–513. MR0075607 (17,776c) Zbl 0066.38902 ]. A breakthrough result was a recent one of M. R. Brown, who showed in [ M. R. Brown, Ovoids of PG (3, q), q even, with a conic section. J. London Math. Soc. (2) 62 (2000), 569–582. MR1783645 (2001i:51012) Zbl 1038.51008 ] that when such an ovoid has a conic plane section, the ovoid must be an elliptic quadric. Even more recently, M. R. Brown and M. Lavrauw [ M. R. Brown, M. Lavrauw, Eggs in PG(4 n − 1, q), q even, containing a pseudo-conic. Bull. London Math. Soc. 36 (2004), 633–639. MR2070439 (2005k:51009) Zbl 1064.51005 ] generalized this theorem by obtaining a similar result for higher dimensions. Both results have equivalent statements in the theory of (translation) generalized quadrangles. In this note, we improve these results by showing that an elation generalized quadrangle of order ( q, q2), q even, with a subGQ containing the elation point arises from a non-singular elliptic quadric in PG(5, q). The theorem itself arises as a corollary of a more general observation which works for all characteristics. There is a wealth of consequences. [ABSTRACT FROM AUTHOR]