Let B=(Bt)t[ges ]0 be standard Brownian motion started at zero. We prove

formula here

for all c>1 and all stopping times τ for B satisfying E(τr)<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

τ*=inf{t>0[mid ]St [ges ]u*, Xt =1∨αSt},

where u*=1+1/ec(c−1) and α=(c−1)/c for c>1, with Xt=[mid ]Bt[mid ] and St= max0[les ]r[les ]t[mid ]Br[mid ]. Likewise, we prove

formula here

for all c>1 and all stopping times τ for B satisfying E(τr<∞ for some r>1/2. This inequality is sharp, and equality is attained at the stopping time

σ*=inf{t>0[mid ]St [ges ]v*, Xt =αSt},

where v*=c/e(c−1) and α=(c−1)/c for c>1. These results contain and refine the results on the Llog L-inequality of Gilat [6] which are obtained by analytic methods. The method of proof used here is probabilistic and is based upon solving the optimal stopping problem with the payoff

formula here

where F(x) equals either xlog+x or xlog x. This optimal stopping problem has some new interesting features, but in essence is solved by applying the principle of smooth fit and the maximality principle. The results extend to the case when B starts at any given point (as well as to all non-negative submartingales).

Suppose that M is a finite module under the Galois group of a local or global field. Ever since Tate's papers [17, 18], we have had a simple and explicit formula for the Euler–Poincaré characteristic of the cohomology of M. In this note we are interested in a refinement of this formula when M also carries an action of some algebra [Ascr ], commuting with the Galois action (see Proposition 5.2 and Theorem 5.1 below). This refinement naturally takes the shape of an identity in a relative K-group attached to [Ascr ] (see Section 2). We shall deduce such an identity whenever we have a formula for the ordinary Euler characteristic, the key step in the proof being the representability of certain functors by perfect complexes (see Section 3). This representability may be of independent interest in other contexts.

Our formula for the equivariant Euler characteristic over [Ascr ] implies the ‘isogeny invariance’ of the equivariant conjectures on special values of the L-function put forward in [3], and this was our motivation to write this note. Incidentally, isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer) was also a motivation for Tate's original paper [18]. I am very grateful to J-P. Serre for illuminating discussions on the subject of this note, in particular for suggesting that I consider representability. I should also like to thank D. Burns for insisting on a most general version of the results in this paper.

Some years ago, Blatter [1] gave a result of the form

formula here

for any function f regular and univalent in D[ratio ][mid ]z[mid ]<1, where ρ is the hyperbolic distance between z1 and z2. Kim and Minda [5] pointed out that the multiplier on the right is incorrect. They say that Blatter's proof gives the correct multiplier, but Blatter's formula for ρ in terms of z1, z2 is wrong. Kim and Minda proved the generalized formula

formula here

where D1(f)=f′(z) (1−[mid ]z[mid ]2), valid for p[ges ]P with some P, 1<P[les ]3/2. In each case there was an appropriate equality statement. Kim and Minda made the important and easily verified remark that these problems are linearly invariant in the sense that if the result is proved for f, then it follows for f˜=UfT, where U is a linear transformation of the plane onto itself and T is a linear transformation of D onto itself. This means that we need to prove such results only in an appropriately normalized context.

We characterize composition operators on spaces of real analytic functions which at the same time have closed image and are open onto their images. Under some mild assumptions, we also characterize composition operators with closed range and composition operators open onto their images.

As a special case of a well-known conjecture of Artin, it is expected that a system of R additive forms of degree k, say

formula here

with integer coefficients aij, has a non-trivial solution in ℚp for all primes p whenever

formula here

Here we adopt the convention that a solution of (1) is non-trivial if not all the xi are 0. To date, this has been verified only when R=1, by Davenport and Lewis [4], and for odd k when R=2, by Davenport and Lewis [7]. For larger values of R, and in particular when k is even, more severe conditions on N are required to assure the existence of p-adic solutions of (1) for all primes p. In another important contribution, Davenport and Lewis [6] showed that the conditions

formula here

are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N[Gt ]R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints

formula here

to be sufficient for p-adic solubility of (1).

A noticeable feature of these results is that for even k, one always encounters a factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can reach the expected order of magnitude k2.

A mean-value theorem, an inverse mapping theorem and an implicit mapping theorem are established here in a class of complex locally convex spaces, including the spaces of test functions in distribution theory. Our main tool is the integral formula and the invariance of the domain, derived from topological degrees, rather than from fixed points of contractions in Banach spaces.

Let Γ be a Fuchsian group. We denote by ax(Γ) the set of axes of hyperbolic elements of Γ. Define Fuchsian groups Γ1 and Γ2 to be isoaxial if ax(Γ1)=ax(Γ2). The main result in this note is to show (see Section 2 for definitions) the following.

Theorem 1.1. Let Γ1and Γ2be isoaxial arithmetic Fuchsian groups. Then Γ1and Γ2are commensurable.

This result was motivated by the results in [6], where it is shown that if Γ1 and Γ2 are finitely generated non-elementary Fuchsian groups having the same non-empty set of simple axes, then Γ1 and Γ2 are commensurable. The general question of isoaxial was left open. Theorem 1.1 is therefore a partial answer. Arithmetic Fuchsian groups are a very special subclass of Fuchsian groups (for example, there are at most finitely many conjugacy classes of such groups of a fixed signature), and the general case at present seems much harder to resolve.

The technical result of this paper which implies Theorem 1.1 is Theorem 2.4 (below), proved in Section 4. Denoting the commensurability subgroup by Comm(Γ) and the subgroup of PGL(2, R) which preserves the set ax(Γ) by Σ(Γ) (careful definitions are given below), we have the following theorem which describes the commensurability subgroup of an arithmetic Fuchsian group geometrically.

Theorem 2.4. Let Γ be an arithmetic Fuchsian group. Then Comm(Γ)=Σ(Γ).

The problem of ‘isoaxial implies commensurable’ is like a dual problem to ‘isospectral implies commensurable’. It was shown in [8] that this latter statement holds for arithmetic Fuchsian groups. As in the case considered here, the general statement about isospectrality implying commensurability is still open.

Let $A_n=K\langle x_1,\ldots,x_n \rangle$ be a free associative algebra over a field $K$. In this paper, examples are given of elements $u \in A_n$, $n \ge 3$, such that the factor algebra of $A_n$ over the ideal generated by $u$ is isomorphic to $A_{n-1}$, and yet $u$ is not a primitive element of $A_n$ (that is, it cannot be taken to $x_1$ by an automorphism of $A_n$). If the characteristic of the ground field $K$ is $0$, this yields a negative answer to a question of G. Bergman. On the other hand, by a result of Drensky and Yu, there is no such example for $n=2$. It should be noted that a similar question for polynomial algebras, known as the embedding conjecture of Abhyankar and Sathaye, is a major open problem in affine algebraic geometry.

This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as $l^1(q)$ , and this definition is extended to all height $m$ polynomials as \[ l^m(q):= \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^1 + \cdots + \epsilon_n q^n,\, \epsilon_i \in {\bb Z},\, \vert\epsilon_i\vert \leqslant m,\, y \ne 0). \] A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of $l^m(q)$ when $q$ is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows.

THEOREM. Let $q$ be a quadratic Pisot number that satisfies a polynomial of the form $p(x) = x^2 - ax \pm 1$ , with conjugate $r$ . Let $q$ have convergents $\{C_k/D_k\}$ and let $k$ be the maximal integer such that \[ \vert D_k r - C_k\vert \leqslant m \frac{1}{1-\vert r\vert}; \] then \[ l^m (q) = \vert D_k q - C_k \vert. \]

A value related to $l(q)$ is $a(q)$ , the minimal non-zero value when all ${\pm}1$ polynomials are evaluated at $q$ . Formally, this is \[ a(q) := \inf(\vert y\vert: y = \epsilon_0 + \epsilon_1 q^2 + \cdots + \epsilon_n q^n,\, \epsilon_i = {\pm} 1,\, y \ne 0). \] An open question concerning how often $a(q) = l(q)$ is also answered in this paper.

The Hall–Paige conjecture deals with conditions under which a finite group $G$ will possess a complete mapping, or equivalently a Latin square based on the Cayley table of $G$ will possess a transversal. Two necessary conditions are known to be: (i) that the Sylow 2-subgroups of $G$ are trivial or non-cyclic, and (ii) that there is some ordering of the elements of $G$ which yields a trivial product. These two conditions are known to be equivalent, but the first direct, elementary proof that (i) implies (ii) is given here.

It is also shown that the Hall–Paige conjecture implies the existence of a duplex in every group table, thereby proving a special case of Rodney's conjecture that every Latin square contains a duplex. A duplex is a ‘double transversal’, that is, a set of $2n$ entries in a Latin square of order $n$ such that each row, column and symbol is represented exactly twice.

Wirtinger and Northcott introduced an inequality bounding the norm of a 2$\Omega$ periodic function by the norm of its derivatives. It is shown here that this type of result occurs for other operators, defined on various domains and spaces.

Let A be an order integral over a valuation ring V in a central simple F-algebra, where F is the fraction field of V. We show that (a) if (Vh, Fh) is the Henselization of (V, F), then A is a semihereditary maximal order if and only if A[otimes ]VVh is a semihereditary maximal order, generalizing the result by Haile, Morandi and Wadsworth, and (b) if J(V) is a principal ideal of V, then a semihereditary V-order is an intersection of finitely many conjugate semihereditary maximal orders; if not, then there is only one maximal order containing the V-order.

The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. An analogous problem concerning spreading models is also addressed.

a conjecture is proposed, bounding the number of cycles with label $w^n$ in a labeled directed graph. some partial results towards this conjecture are established. as a consequence, it is proved that $\langle a_1, a_2, \ldots\,{\mid}\,w^n\rangle$ is coherent for $n\,{\geq}\,4$. furthermore, it is coherent for $n\,{\geq}\,2$, provided that the strengthened hanna neumann conjecture holds.

For any x ∈ (0, 1], let the series [sum ]∞n=1 1/dn(x) be the Sylvester expansion of x. Galambos has shown that the Lebesgue measure of the set

[formula here]

is 1 when α = e, the base of the natural logarithm. This paper provides a proof that for any α [ges ] 1, A(α) has Hausdorff dimension 1 when α ≠ e.

It is shown that a Carter subgroup of a non-soluble unitary group can only be the normalizer of a Sylow 2-subgroup. This result, combined with previous results, implies that no finite simple group can be a minimal counterexample to the conjugacy conjecture of Carter subgroups in a finite group.The third author was partially supported by the University of Padova, the RFBR, grants 02-01-00495 and 02-01-06226, Ministry of Education RF, E00-1.0-77, ‘Universities of Russia’ UR.04.01.031.

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S, at least two distinct words of length n on these letters are equal in S. In particular, S is collapsing whenever it satisfies a law. Let [Uscr ](A) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: [Uscr ](A) is collapsing; [Uscr ](A) satisfies some semigroup law; [Uscr ](A) satisfies the Engel condition; [Uscr ](A) is nilpotent; A is nilpotent when considered as a Lie algebra.

In this note it is shown that for $k$ a field, and for the four-dimensional algebra $\Lambda=k\langle x,y\rangle /\langle x^2,y^2,xy+qyx\rangle$ when $q^n\neq 1,0$ for all $n$, there exist a two-dimensional module $M$ and a family of two-dimensional modules $M_i$, $i=1,2,\ldots$, such that $\dim_k\Ext^i_\Lambda(M,M_j)=1$ for $i$ equal to 0, $j$ and $j+1$, and $\dim_k\Ext^i_\Lambda(M,M_j)=0$ otherwise. This is probably the most straightforward example giving a negative answer to a question raised by Maurice Auslander.

We prove that a large and natural class of subhomogeneous C*-algebras with one-dimensional spectrum have the fragility property, that is, every morphism out of such a C*-algebra and into a C*-algebra of real rank zero can be approximated by a morphism with finite-dimensional range, provided that a K-theoretical obstruction vanishes.

In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations $Sq^k$ on the cohomology ring of any cocommutative Hopf algebra over $Z/2$. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that $Sq^0$ is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra $\cal {B}$ that is generated by elements $Sq^k$ and subject to Adem relations; it shows that the Cartan formula gives a well-defined coproduct on $\cal {B}$. Also, it is shown that $\cal {B}$ with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper.