Given a computably enumerable set W, there is a Turing degree which is the least jump of any set in which W is computably enumerable, namely 0′. Remarkably, this is not a phenomenon of computably enumerable sets. It is shown that for every subset A of ℕ, there is a Turing degree, c′μ(A), which is the least degree of the jumps of all sets X for which A is [sum ]01(X). In addition this result provides an isomorphism invariant method for assigning Turing degrees to certain torsion-free abelian groups.

A method is developed for obtaining Ramanujan's mock theta functions from ordinary theta functions by performing certain operations on their q-series expansions. The method is then used to construct several new mock theta functions, including the first ones of eighth order. Summation and transformation formulae for basic hypergeometric series are used to prove that the new functions actually have the mock theta property. The modular transformation formulae for these functions are obtained.

(Almost strongly minimal) generalized n-gons are constructed for all n [ges ] 3 for which the automorphism group acts transitively on the set of ordered ordinary (n + 1)-gons contained in it, a new class of BN-pairs thus being obtained. Through the construction being modified slightly, 2ℵ0 many non-isomorphic almost strongly minimal generalized n-gons are obtained for all n [ges ] 3, none of which interprets an infinite group. Furthermore, a characterization is given of all graphs whose simple cycles all have length 2n for some n [ges ] 3.

In this paper we develop a structure theory for transitive permutation groups definable in o-minimal structures. We fix an o-minimal structure [Mscr ], a group G definable in [Mscr ], and a set Ω and a faithful transitive action of G on Ω definable in [Mscr ], and talk of the permutation group (G, Ω). Often, we are concerned with definably primitive permutation groups (G, Ω); this means that there is no proper non-trivial definable G-invariant equivalence relation on Ω, so definable primitivity is equivalent to a point stabiliser Gα being a maximal definable subgroup of G. Of course, since any group definable in an o-minimal structure has the descending chain condition on definable subgroups [23] we expect many questions on definable transitive permutation groups to reduce to questions on definably primitive ones.

Recall that a group G definable in an o-minimal structure is said to be connected if there is no proper definable subgroup of finite index. In some places, if G is a group definable in [Mscr ] we must distinguish between definability in the full ambient structure [Mscr ] and G-definability, which means definability in the pure group G:= (G, .); for example, G is G-definably connected means that G does not contain proper subgroups of finite index which are definable in the group structure. By definable, we always mean definability in [Mscr ]. In some situations, when there is a field R definable in [Mscr ], we say a set is R-semialgebraic, meaning that it is definable in (R, +, .). We call a permutation group (G, Ω) R-semialgebraic if G, Ω and the action of G on Ω can all be defined in the pure field structure of a real closed field R. If R is clear from the context, we also just write ‘semialgebraic’.

Commutative rings for which every ideal can be written as a finite product of invertible and prime ideals are characterized.

In [6] S. Shelah showed that in the endomorphism semi-group of an infinitely generated algebra which is free in a variety one can interpret some set theory. It follows from his results that, for an algebra Fℵ which is free of infinite rank ℵ in a variety of algebras in a language L, if ℵ > |L|, then the first-order theory of the endomorphism semi-group of Fℵ, Th(End(Fℵ)), syntactically interprets Th(ℵ,L2), the second-order theory of the cardinal ℵ. This means that for any second-order sentence χ of empty language there exists χ*, a first-order sentence of semi-group language, such that for any infinite cardinal ℵ > |L|,

formula here

In his paper Shelah notes that it is natural to study a similar problem for automorphism groups instead of endomorphism semi-groups; a priori the expressive power of the first-order logic for automorphism groups is less than the one for endomorphism semi-groups. For instance, according to Shelah's results on permutation groups [4, 5], one cannot interpret set theory by means of first-order logic in the permutation group of an infinite set, the automorphism group of an algebra in empty language. On the other hand, one can do this in the endomorphism semi-group of such an algebra.

In [7, 8] the author found a solution for the case of the variety of vector spaces over a fixed field. If V is a vector space of an infinite dimension ℵ over a division ring D, then the theory Th(ℵ, L2) is interpretable in the first-order theory of GL(V), the automorphism group of V. When a field D is countable and definable up to isomorphism by a second-order sentence, then the theories Th(GL(V)) and Th(ℵ, L2) are mutually syntactically interpretable. In the general case, the formulation is a bit more complicated.

The main result of this paper states that a similar result holds for the variety of all groups.

The existing results are surveyed and several new results are proved for the cardinality of the restricted doubling 2ˆA = {a′ + a″:a′,a″ ∈ A,a′ ≠ a″}, where A ⊆ G is a subset of the set of elements of an (additively written) group G. In particular, known estimates for G = ℤ and G = ℤ/pℤ are improved and a first-of-a-kind general estimate valid for arbitrary G is given.

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)&→HH(R), from the algebraic K-theory of a ring to its Hochschild homology, which can be used to obtain information about K(R) from HH(R). In [1] Bökstedt described a factorization of this trace map. The intermediate functor THH(HR) is called the topological Hochschild homology of the Eilenberg–MacLane spectrum HR associated with R, because it is constructed similarly to Hochschild homology with the tensor product replaced by the smash product of spectra.

The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a finite field has kq points where k is a small number and q is a prime.

Let (A, [mfr ]) be a local ring. For convenience we will assume throughout this paper that the residue field of A is infinite.

Let I be an ideal of A. An ideal J ⊆I is called a reduction of I if JIn = In+1 for some integer n. The least number n with this property is denoted by rJ (I). A reduction of I is said to be minimal if it does not contain any other reduction of I. The reduction number r(I) of I is the minimum of rJ(I) for all minimal reductions J of I. A minimal reduction of I usually has better properties than I. It can be viewed as an approximation of I and the reduction number is a measure for how it is different from I. The minimal number of generators of every minimal reduction of I is equal to the dimension of the fibre ring [oplus ]n[ges ]0In/[mfr ]In. This invariant is called the analytic spread of I and denoted by [lscr ](I). All these notions have played an important role in ideal theory since their introduction by Northcott and Rees [16].

The moduli problem for (algebraic completely) integrable systems is introduced. This problem consists in constructing a moduli space of affine algebraic varieties and explicitly describing a map which associates to a generic affine variety, which appears as a level set of the first integrals of the system (or, equivalently, a generic affine variety which is preserved by the flows of the integrable vector fields), a point in this moduli space. As an illustration, the example of an integrable geodesic flow on SO(4) is worked out. In this case, the generic invariant variety is an affine part of the Jacobian of a Riemann surface of genus 2. The construction relies heavily on the fact that these affine parts have the additional property of being 4:1 unramified covers of Abelian surfaces of type (1,4).

The closure of the periodic points of rational maps over a non-archimedean field is studied. An analogue of Montel's theorem over non-archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non-archimedean field is contained in the closure of the periodic points.

It is proved that if an algebra R over a field can be endowed with a pointed and finite-dimensional ℕn-filtration such that the associated ℕn-graded algebra T is semi-commutative, then R is left and right finitely partitive. In order to do this, a multi-variable Poincaré series for every finitely generated graded T-module is considered and it is shown that this Poincaré series is a rational function. The methods apply to some iterated Ore extensions such as quantum matrices and quantum Weyl algebras as well as to the quantized enveloping algebra of [sfr ][lfr ](ν+1).

Let L1(x), L2(x), …, LN(x) be N real linear forms in N variables defined by

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such that the associated N × N matrix of coefficients C = (cmn) is unimodular. In the classical theory of geometry of numbers, Minkowski's well-known linear forms theorem asserts that for any positive numbers ε1, ε2, …, εN satisfying ε1, ε2, …, εN [ges ] 1, there exists a lattice point p ∈ ℤN, p ≠ 0, such that

formula here

In 1937, Mordell [7] posed the following question which may be viewed, in some sense, as a converse to Minkowski's result. Does there exist a constant ΔN such that for each N × N unimodular real matrix C, there exist positive real numbers ε1, ε2, …, εN satisfying

formula here

such that the only integer solution to the inequalities of (1.1) is p = 0? Moreover, if ΔN does exist, what is the best possible value for ΔN, that is, what is the supremum of all such admissible values of ΔN? Clearly, by Minkowski's theorem, if ΔN exists, then ΔN < 1.

If C is a reduced curve which is invariant by a one-dimensional foliation [Fscr ] of degree d[Fscr ] on the projective space then it is shown that d[Fscr ]−1+a is a bound for the quotient of the two coefficients of the Hilbert–Samuel polynomial for C, where a is an integer obtained from a concrete problem of imposing singularities to projective hypersurfaces, and so a bound is obtained for the degree of C when it is a complete intersection. Concrete values of a can be derived for several interesting applications. The results are presented in the form of intersection-theoretical inequalities for one-dimensional foliations on arbitrary smooth algebraic varieties.

Let A be a commutative ring. A graded A-algebra U = [oplus ]n[ges ]0 is a standard A-algebra if U0 = A and U = A[U1] is generated as an A-algebra by the elements of U1. A graded U-module F = [oplus ]n[ges ]0Fn is a standard U-module if F is generated as a U-module by the elements of F0, that is, Fn = UnF0 for all n [ges ] 0. In particular, Fn = U1Fn−1 for all n [ges ] 1. Given I, J, two ideals of A, we consider the following standard algebras: the Rees algebra of I, [Rscr ](I) = [oplus ]n[ges ]0Intn = A[It] ⊂ A[t], and the multi-Rees algebra of I and J, [Rscr ](I, J) = [oplus ]n[ges ]0([oplus ]p+q=nIpJqupvq) = A[Iu, Jv] ⊂ A[u, v]. Consider the associated graded ring of I, [Gscr ](I) = [Rscr ](I) [otimes ] A/I = [oplus ]n[ges ]0In/In+1, and the multi-associated graded ring of I and J, [Gscr ](I, J) = [Rscr ](I, J) [otimes ] A/(I+J) = [oplus ]n[ges ]0([oplus ]p+q= nIpJq/(I+J)IpJq). We can always consider the tensor product of two standard A-algebras U = [oplus ]p[ges ]0Up and V = [oplus ]q[ges ]0Vq as a standard A-algebra with the natural grading U [otimes ] V = [oplus ]n[ges ]0([oplus ]p+q=nUp [otimes ] Vq). If M is an A-module, we have the standard modules: the Rees module of I with respect to M, [Rscr ](I; M) = [oplus ]n[ges ]0InMtn = M[It] ⊂ M[t] (a standard [Rscr ](I)-module), and the multi-Rees module of I and J with respect to M, [Rscr ](I, J; M) = [oplus ]n[ges ]0([oplus ]p+q=nIpJqMupvq) = M[Iu, Jv] ⊂ M[u, v] (a standard [Rscr ](I, J)-module). Consider the associated graded module of M with respect to I, [Gscr ](I; M) = [Rscr ](I; M) [otimes ] A/I = [oplus ]n[ges ]0InM/In+1M (a standard [Gscr ](I)-module), and the multi-associated graded module of M with respect to I and J, [Gscr ](I, J; M) = [Rscr ](I, J; M) [otimes ] A/(I+J) = [oplus ]n[ges ]0([oplus ]p+q= nIpJqM/ (I+J)IpJqM) (a standard [Gscr ](I, J)-module). If U, V are two standard A-algebras, F is a standard U-module and G is a standard V-module, then F [otimes ] G = [oplus ]n[ges ]0([oplus ]p+q= nFp [otimes ] Gq) is a standard U [otimes ] V-module.

Denote by π[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I, J; M) and σ[ratio ][Rscr ](I, J; M) → [Rscr ](I+J; M) the natural surjective graded morphisms of standard [Rscr ](I) [otimes ] [Rscr ](J)-modules. Let ϕ[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I+J; M) be σ∘π. Denote by &πmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I, J; M) and &σmacr;[ratio ][Gscr ](I, J; M) → [Gscr ](I+J; M) the tensor product of π and σ by A/(I+J); these are two natural surjective graded morphisms of standard [Gscr ](I) [otimes ] [Gscr ](J)-modules. Let &ϕmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I+J; M) be &σmacr;∘&πmacr;. The first purpose of this paper is to prove the following theorem.

A classical result of M. Gerstenhaber [5] states that finite-dimensional semisimple complex Lie algebras are rigid. One interpretation of this fact is as follows. Let [Lfr ] be a semisimple complex Lie algebra of dimension d and let [Bfr ] be a [Copf ]-basis of [Lfr ]. Let L(d) ⊂ [Copf ]d·(d2) denote the affine variety of all structure constants of [Copf ]-Lie algebras of dimension d and let c ∈ L(d) denote the point corresponding to the structure constants of [Lfr ] with respect to [Bfr ]. Then there exists an open neighbourhood in the metric topology U ⊂ L(d) of c ∈ L(d) such that [Lfr ]c* is isomorphic to [Lfr ] for all c* ∈ U, where [Lfr ]c* denotes the Lie algebra defined by the structure constants c* ∈ L(d). Our aim is to generalize this result to semisimple Lie algebras over discrete valuation domains and to apply these results to powerful pro-p-groups.

Let A be an abelian group, F be a field of characteristic 0, and α, β be linearly independent additive maps from A to F, and let δ∈ker(α)\{0}. Then there is a Lie algebra L = L(A, α, β, δ) = [oplus ]x∈AFex under the product

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If, further, β(δ) = 1, and β(A) = Z, there is a subalgebra L+:=L(A+, α, β, δ) = [oplus ]x∈A+Fex, where A+ = {x∈A|β(x)[ges ]0}. The necessary and sufficient conditions are given for L' = [L, L] and L+ to be simple, and all semi-simple elements in L' and L+ are determined. It is shown that L' and L+ cannot be isomorphic to any other known Lie algebras and L' is not isomorphic to any L+ , and all isomorphisms between two L' and all isomorphisms between two L+ are explicitly described.

Since the remarkable discovery of the relevance of derived equivalences in the theory of p-blocks of finite groups, where p is a prime, by J. Rickard in [15, 16], various attempts have been made to understand this phenomenon. In particular, J. Rickard defines in [18] a certain class of derived equivalences between the derived module categories of p-blocks of finite groups that he calls splendid equivalences (and that we are going to call splendid derived equivalences in this paper) which take into account the local structure, that is, which under suitable hypotheses induce a family of derived equivalences at all ‘local levels’ of the considered p-blocks (see [18] for a more detailed motivation). The main condition for a derived equivalence to be splendid is that it is given by a two-sided tilting complex consisting of p-permutation bimodules (see Definitions 1.3 and 1.4 below for the precise terminology).

Some subnormal embeddings of groups into groups with additional properties are established, and in particular, embeddings of countable groups into two-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, H. Neumann and G. Higman on embeddings of that type. Through the consideration of subnormal embeddings of finite groups into finite groups with additional properties, a result of H. Heineken and J. Lennox is illustrated.