(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 [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|,

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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.

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].

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 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).

It is shown that D. Cohen's inequality bounding the isoperimetric function of a group by the double exponential of its isodiametric function is valid in the more general context of locally finite simply connected complexes. It is shown that in this context this bound is ‘best possible’. Also studied are second-dimensional isoperimetric functions for groups and complexes. It is shown that the second-dimensional isoperimetric function of a group is bounded by a recursive function. By a similar argument it is shown that the area distortion of a finitely presented subgroup of a finitely presented group is recursive. Cohen's inequality is extended to second-dimensional isoperimetric and isodiametric functions of 2-connected simplicial complexes.

It is shown that if a locally compact group acts isometrically on a Banach space X leaving a closed subspace M invariant, and if the induced actions on M and X/M are strongly continuous, then the action on X is strongly continuous. Since this may be of interest for one-parameter semigroups, similar results are proved for actions of suitable topological semigroups. Other generalizations are given for (suitable) non-isometric actions, non-locally compact groups, and non-Banach spaces; corollaries concerning 1-cocycles and uniformly continuous actions are given.

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.

The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F' = f almost everywhere and there exists a differentiable function H such that F – H is absolutely continuous. They strengthen this definition by proving that F – H can have arbitrary small variation. Nevertheless, their definition still needs a choice of a differentiable function.

Ritt has shown that any complex polynomial p can be written as the composition of polynomials p1,…,pm, where each pj is prime in the sense that it cannot be written as a non-trivial composition of polynomials. The factors pj are not unique but the number m of them is, as is the set of the degrees of the pj. The paper extends Ritt's theory and, in particular, a third invariant of the decomposition is introduced.

Let [Gscr ] be a family of functions analytic in a domain D in the complex plane. It is proved that [Gscr ] is a normal family, provided that for each f∈[Gscr ], there exists k = k(f) > 1 such that the kth iterate fk has no repulsive fixpoint in D. A new proof of a result of Bergweiler and Terglane concerning the dynamics of entire functions is also given.

The definiteness of the Peano kernel is proved for a functional associated with the mean-value property of Picone and Bramble and Payne for polyharmonic functions in the ball. An important corollary of this is that if a function f satisfying (−1)pΔpf>0 vanishes on p concentric spheres centered at 0, then f(0)>0. This generalizes a well-known property of subharmonic functions (which arise in the special case p = 1).

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved.

In particular, let ly = (1/r)(−(py′)′+qy), l˜y = (1/r˜)(−(p˜y′)′+q˜y),

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where det Δ = δ > 0, c ≠ 0, det [sum ] > 0, t ≠ 0 and (cs + dr − au − tb)2 < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define (l˜; α; Δ) as above but with l replaced by l˜. Let wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(1)+cpw′n(1). Define w˜n and γ˜n similarly.

As sample results, it is proved that if (l; α; Δ) and (l˜; α; Δ) have the same spectrum, and (l; α; Σ) and (l˜; α; Σ) have the same spectrum or ∫10[mid ]wn[mid ]2rdt+([mid ]γn[mid ]2/δ) = ∫10[mid ]w˜n[mid ]2r˜dt+([mid ]γ˜n[mid ]2/δ) for all n, then q/r = q˜/r˜.

It is shown that every irreducible admissible representation of Sp4(F), where F is a p-adic field, has at most one Fourier Jacobi model for given Fourier Jacobi data.

In this paper we study the abstract quasi-linear evolution equation of second order

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in a general Banach space Z. It is well-known that the abstract quasi-linear theory due to Kato [10, 11] is widely applicable to quasi-linear partial differential equations of second order and that his theory is based on the theory of semigroups of class (C0). (For example, see the work of Hughes et al. [9] and Heard [8].) However, even in the special case where A(t, w, v) = A is independent of (t, w, v), it is found in [2] and [14] that there exist linear partial differential equations of second order for which Cauchy problems are not solvable by the theory of semigroups of class (C0) but fit into the mould of well-posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of A. (See also work by Krein and Khazan [13] and Fattorini [6, Chapter 8].) This kind of Cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. The theory of integrated semigroups was studied intensively by Arendt [1] and that of regularized semigroups was initiated by Da Prato [3] and renewed by Davies and Pang [4]. For the theory of regularized semigroups we refer the reader to [5] and [16].

In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem

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where Δ = [sum ]Ni=1δ2/δx2i is the Laplace operator in ℝN, 2 < q < ∞ for N = 1, 2, 2 < q < 2N/(N−2) for N[ges ]3, and Q(x) is a bounded positive continuous function on ℝN satisfying the following conditions.

(Q1) Q has a strict local minimum at some point x0∈ℝN, that is, for some δ > 0

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for all 0 < [mid ]x−x0[mid ] < δ.

(Q2) There are constants C, θ > 0 such that

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for all [mid ]x−x0[mid ] [les ] δ, [mid ]y−y0[mid ] [les ] δ.

Our aim here is to show that corresponding to each strict local minimum point x0 of Q(x) in ℝN, and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided ε is sufficiently small, that is, a solution with k-maximum points converging to x0, while vanishing as ε → 0 everywhere else in ℝN.

Conditions for pointwise Fourier inversion using spherical Cesàro means of a given degree are established on rank 1 noncompact symmetric spaces.

A linear operator between two Banach spaces X and Y is strictly-singular (or Kato) if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion for Banach lattices introduced in [8] is the following one: an operator T from a Banach lattice X to a Banach space Y is said to be disjointly strictly-singular if there is no disjoint sequence of non-null vectors (xn)n∈ℕ in X such that the restriction of T to the subspace [(xn)∞n=1] spanned by the vectors (xn)n∈ℕ is an isomorphism. Clearly every strictly-singular operator is disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion Lq[0, 1][rarrhk ]Lp[0, 1] for 1[les ]p<q<∞). In the special case of considering Banach lattices X with a Schauder basis of disjoint vectors both concepts coincide. The notion of disjointly strictly-singular has turned out to be a useful tool in the study of lattice structure of function spaces (cf. [7–9]). In general the class of all disjointly strictly-singular operators is not an operator ideal since it fails to be stable with respect to the composition on the right.

The aim of this paper is to study when the inclusion operators between arbitrary rearrangement invariant function spaces E[0, 1] ≡ E on the probability space [0, 1] are disjointly strictly-singular operators.

Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A−∞, but it need not be hypercyclic on radial weighted algebras of entire functions.