Let a ∈ ℕ [setmn ] {0, 1} and let bn be a sequence of rational integers satisfying bn = O(η−2n) for every η ∈]0, 1[. We prove that the number S = [sum ]+∞n=0 1/(a2n + bn) is transcendental by using a special form of Mahler's transcendence method.

In this paper we consider group actions on generalized treelike structures (termed ‘pretrees’) defined simply in terms of betweenness relations. Using a result of Levitt, we show that if a countable group admits an archimedean action on a median pretree, then it admits an action by isometries on an ℝ-tree. Thus the theory of isometric actions on ℝ-trees may be extended to a more general setting where it merges naturally with the theory of right-orderable groups. This approach has application also to the study of convergence group actions on continua.

We study singularities of ruled surfaces in ℝ3. The main result asserts that only cross-caps appear as singularities for generic ruled surfaces.

Let F = (f1, …, fm): (Kn, 0) → (Km, 0), where K is either R or C, be an analytic mapping defined in a neighbourhood of the origin. Let Br ⊂ Kn be a closed ball of small radius r centred at the origin. For any regular value y ∈ Km close to the origin, the fibre Wy = F−1(y) ∩ Br is called the Milnor fibre of F. We assume that m [les ] n, because in the other case Wy is void.

Several authors investigated the topology of the Milnor fibres. Let us recall the most important results in the complex case. Let [Oscr ]C,0 denote the ring of germs of analytic functions f: (Cn, 0) → C.

Nash–Williams proved that the infinite trees are well-quasi-ordered (indeed, better-quasi-ordered) under the topological minor relation. We combine ideas of several authors into a more accessible and essentially self-contained short proof.

Let p be an odd prime. The results in this paper concern the units of the infinite extension of Qp generated by all p-power roots of unity. Let

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where μpn+1 denote the pn+1th roots of 1. Let [pscr ]n be the maximal ideal of the ring of integers of Φn and let Un be the units congruent to 1 modulo [pscr ]n.

Let ζn be a fixed primitive pn+1th root of unity such that ζpn = ζn − 1, ∀n [ges ] 1. Put πn = ζn − 1. Thus πn is a local parameter for Φn. Let

formula here

Kummer already exploited the obvious fact that every u0 ∈ U0 can be written in the form

formula here

where f0(T) is some power series in Zp[[T]]. Here of course, the power series f0(T) is not uniquely determined. Let

formula here

the inverse limit being with respect to norm maps. Coates and Wiles (see [3]) discovered that any unit u = (un) ∈ U∞ has a unique power series fu(T) in Zp[[T]] with fu(πn) = un. The uniqueness of such a power series is obvious by Weierstrass Preparation Theorem, but the existence is in no way obvious. They worked with the formal group of height one attached to an elliptic curve with complex multiplication at an ordinary prime, but their ideas apply to any Lubin–Tate group defined over Zp. Almost immediately, Coleman [4] gave a totally different proof of the existence of the fu(T), which holds for all Lubin–Tate groups. We refer to such a power series as a Coleman power series. In this paper we adopt the same approach as [3]. We first prove the following result which is stronger than the original one in [3].

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.

For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.

Here we show the following results:

(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;

(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;

(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).

In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

Let E be a modular elliptic curve over ℚ and let L(E, s) denote the associated L-function. The Birch and Swinnerton-Dyer conjecture then predicts that L(E, s) has a zero at s = 1 of order precisely equal to the rank of the Mordell–Weil group E(ℚ). According to Waldspurger's theorem [26] we know that there exists a real quadratic character χ such that the twisted L-function L(E, χ, s) does not vanish at s = 1. Recently Kolyvagin [16] has proved that E(ℚ) is finite provided L(E, 1) ≠ 0 and there exists a suitable real quadratic character χt such that L(E, χt, s) has a simple zero at s = 1. The latter condition was proved to be true for infinitely many t [4, 19]. Iwaniec [15] and Perelli and Pomykała [20] have proved quantitative results on this condition; Pomykała [21] has generalized it to the nth derivative of L(E, χ, s).

Variants and generalizations are possible. For instance, Friedberg and Hoffstein [10] established a nonvanishing theorem for quadratic twists of the L-series of an arbitrary cuspidal automorphic form on GL(2) over any number field. In some cases such a result can be applied to the construction of l-adic representations associated to modular forms over imaginary quadratic fields [25]. Further examples as well as some perspective concerning the higher rank case is discussed in an excellent survey article [5].

We should also mention that there are examples of irreducible cuspidal automorphic representations of GL(2) over a number field such that the corresponding twisted L-function vanishes at the centre of the critical strip for all quadratic characters [22].

Here is our contribution to the above picture.

Let M be a pure motive over ℚ, and let L(M, s) denote the corresponding L-function. In this paper we prove, under certain assumptions, (a quantitative version of) a nonvanishing theorem for n-th derivatives of quadratic twists of L(M, s) at the centre of the critical strip (Theorem 3). Also, assuming the Birch and Swinnerton-Dyer conjecture for abelian varieties over ℚ, we obtain a bound for the order of the (twisted) Shafarevich–Tate group (Theorems 4 and 5).

All the results of this paper except Section 5 could be stated (and proved) without using any motivic language. Almost all we need to know about motives concerns the conjectural description of their L-functions (see Section 1).

Let [Fscr ] be the category of functors from the category of finite-dimensional [ ]2-vector spaces to [ ]2-vector spaces. The concept of &∇tilde;-nilpotence in the category [Fscr ] is used to define a ‘dimension’ for the category of analytic functors which has good properties. In particular, the paper shows that the tensor product F [otimes ] G of analytic functors which are respectively &∇tilde;s and &∇tilde;t nilpotent is &∇tilde;s+t − 1-nilpotent.

The notion of &∇tilde;-nilpotence is extended to define a dimension in the category of unstable modules over the mod 2 Steenrod algebra, which is shown to coincide with the transcendence degree of an unstable Noetherian algebra over the Steenrod algebra.

Let A ⊂ B be a homogeneous inclusion of standard graded algebras with A0 = B0. To relate properties of A and B we intermediate with another algebra, the associated graded ring G = grA1B(B). We give criteria as to when the extension A ⊂ B is integral or birational in terms of the codimension of certain modules associated to G. We also introduce a series of multiplicities associated to the extension A ⊂ B. There are applications to the extension of two Rees algebras of modules and to estimating the (ordinary) multiplicity of A in terms of that of B and of related rings. Many earlier results by several authors are recovered quickly.

The main focus in this paper is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-lv higher Chow groups to mod-lv étale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p [ges ] 0 with l ≠ p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-lv motivic cohomology and mod-lv algebraic K-theory of projective nonsingular linear varieties to be free ℤ/lv-modules.

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.

Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:

([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional co-commutative Hopf algebra B. Our method is based on that of Benson–Carlson–Rickard [BCR1], who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

The geometry of a desingularization Ym of an arbitrary subvariety of a generic hypersurface Xn in an ambient variety W (e.g. W = ℙn+1) has received much attention over the past decade or so. Clemens [5] has proved that for m = 1, n = 2, W = ℙ3 and X of degree d, Y has genus g [ges ] 1 + d(d − 5)/2; Xu [13, 14] improved this to g [ges ] d(d −3)/2 − 2 for d [ges ] 5 and showed that if equality holds for d [ges ] 6 then Y is planar; he also gave some lower bounds on the geometric genus pg(Y) in case m = n − 1. Voisin [10, 11] proved that, for X of degree d in ℙn+1, n [ges ] 3, m [les ] n − 2 then then pg(Y) > 0 if d [ges ] 2n + 1 − m and KY separates generic points if d [ges ] 2n + 2 − m (see also [1, 4]). For Xn a generic complete intersection of type (d1, …, dk) in any smooth polarized (n + k)-fold M, Ein [6, 7] proved that pg(Y) > 0 if d1 + … + dk [ges ] 2n + k − m + 1 and Y is of general type if d1 + … + dk [ges ] 2n + k − m + 2.

In [2] the first two authors applied the classical method of focal loci of families as in Segre [8] and Ciliberto–Sernesi [3] (really normal bundle considerations) to give a new proof of one of the main results of Xu in [13, 14] and to extend the lower bounds for the genus in the cases of general surfaces in a component of the Noether–Lefschetz locus in ℙ3 and of general projectively Cohen–Macaulay surfaces in ℙ4.

In this paper we introduce some notions of ‘filling’ subvarieties and use them to prove two new results which inter alia give another perspective on the above results, especially the genus bounds. The general philosophy is as usual that as Y moves with X, sections of KY (or some twist) can be produced through differentiation; but here this is implemented by exploiting and extending an elementary but perhaps surprising technique in the spirit of classical projective geometry which goes back to [2] and which gives a useful lower bound, depending on the dimension of the projective span of Y, on the number of independent sections of KY produced by the differentiation process.

As to our results, in Theorem 1 below we extend and refine in the above sense most of the above-quoted results (not including Voisin's), by giving a lower bound on the number of sections of a certain twist of KY involving KW (for a recent extension, including the proof of a conjecture of Clemens, see [15]). Our second result (Theorem 2), based on the notion of ‘r-filling’ subvariety, indicates an apparently new and unexpected direction as it deals with some higher-order tensors on Y, manifested in the form of an effective divisor on a Cartesian product Yr having certain vanishing order on a diagonal locus as well as on a ‘double point’ locus associated to the map Y → X. As one application, we conclude a lower bound on the number of quadrics (and higher-degree hypersurfaces) containing certain ‘adjoint-type’ projective images of Y (and even on the dimension of the kernel of certain ‘symmetric Gaussian’ maps) (Corollary 2·1 below). Our feeling, however, is that the latter is only the tip of an iceberg and we hope to explore further in this direction in the future.

This article is concerned with homotopy properties of H-spaces X that are reflected in the module of indecomposables QH*(X; [ ]p). It is shown that mod p H-spaces X of finite type with finite transcendence degree mod p cohomology and locally finite QH*(X; [ ]p) are Bℤ/p-null spaces, Eilenberg–MacLane spaces K(ℤˆp, 2), K(ℤ/pr, 1), and extensions of those. If we restrict attention to H-spaces with noetherian mod p cohomology algebra, then we are left with finite mod p H-spaces and Eilenberg–MacLane spaces.

Let S (respectively, S′) be a finite subset of a compact connected Riemann surface X (respectively, X′) of genus at least two. Let [Mscr ] (respectively, [Mscr ]′) denote a moduli space of parabolic stable bundles of rank 2 over X (respectively, X′) with fixed determinant of degree 1, having a nontrivial quasi-parabolic structure over each point of S (respectively, S′) and of parabolic degree less than 2. It is proved that [Mscr ] is isomorphic to [Mscr ]′ if and only if there is an isomorphism of X with X′ taking S to S′.

We show that, for any braided compact closed bicategory [Bscr ], the bicategory Ladj ([Bscr ]) of left adjoints in [Bscr ] also admits a braided compact closed structure.

Let N ⊆ M be a pair of closed smooth manifolds and L an algebraic model for the submanifold N. In this paper, we will give an obstruction to finding an algebraic model X of M so that the submanifold N corresponds in X to an algebraic subvariety isomorphic to L.

Let ℝd* = ℝd ∪ {[midast ]} be the one-point compactification of Euclidean space ℝd and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]f(ℝd*) denote the set of all sets C ⊆ ℝd* for which there is a series [sum ]an in ℝd with zero sum such that C is the cluster set in ℝd* of the sequence of partial sums of [sum ]af(n). Every C ∈ [Cscr ]f(ℝd*) is non-empty, connected and closed in ℝd*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝd* belong to [Cscr ]f(ℝd*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]f(ℝd*) which contain [midast ].