Klass has established decoupling inequalities for discrete time processes with independent increments. We extend his result to continuous time processes with independent increments. This extension enables us to obtain BDG inequalities for exponential functions instead of moderate functions.

The aim of this paper is to study properties of sequences that are recursively defined by a linear equation and their applications to the truncated moment problem in connection with the problem of subnormal completion of the truncated weighted shifts. Special cases are considered and some classical results due to Stampfli, Curto and Fialkow are recovered using elementary techniques.

Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. We prove the conjecture for every arithmetic progression whose modulus is a power of 2.

It is well known that the second homology group of any Kac–Moody Lie algebra and the Virasoro algebra is trivial. This is equivalent to saying that any Kac–Moody Lie algebra (or the Virasoro algebra) is its own universal covering algebra; see, for example, [1] and [4]. Recently, Loday and Pirashvili in [12] proved that the second Leibniz homology group of the Virasoro algebra is also trivial. The Leibniz homology is a noncommutative analogue of Lie algebra homology developed by J-L. Loday (see [11] and [12]). It can be extended for a larger class of algebras called Leibniz algebras.

In this paper, we study the second Leibniz homology group HL2(g(A)) of a Kac–Moody Lie algebra g(A). We shall show that HL2(g(A)) = 0 for any non-affine Kac–Moody algebra g(A), but HL2(g(A)) ≠ 0 for any affine Kac–Moody algebra g(A). In the latter case, we use Wilson's idea [15] to carry out HL2(g(A)) explicitly in terms of certain subspaces of Kähler differentials over the Laurent polynomial ring. More precisely, we have the following result.

This paper is concerned with the chromatic polynomials of ‘bracelets’: specifically, graphs constructed by taking n copies of a complete graph and linking them together in a ring. Using a sieve method, explicit formulae for the dominant and subdominant terms of the chromatic polynomial are obtained. Finally, a simple description of the effect of twisting the links is obtained.

It is shown that Voiculescu's topological entropy for the canonical endomorphism of a simple Cuntz–Krieger algebra [Oscr ]A equals the logarithm of the spectral radius of A.

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.

Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side. Let μ(m, n) be the expected number of survivors. Clearly, we have boundary conditions:

μ(m, 0)=m, μ(0, n)=n. (1.1)

We also have the equation

formula here

This is because the probability that the first successful shot is made by the side with m gunmen is m/(m+n). On using the recurrence relation (1.2) together with the boundary condition (1.1), the computer produces Table 1 below, in which

m=8192+k, n=8192−k, d(m, n) =√(m2−n2 =128√(2k).

The L1 means of various exponential sums with arithmetically interesting coefficients have been investigated in many recent papers. For example, Balog and Perelli proved in [1] that

formula here

for a suitable positive number c. The method of proving the lower bound in [1] is rather flexible and can work well with many multiplicative functions in place of μ(n), the Möbius function, whose Dirichlet series have a suitable expression by the Riemann ζ-function.

In this short note we improve on the above lower bound.

A proof is given to show that the operator Hilbert space $\operatorname{OH}$ does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge's recent result, which admits such an embedding in the non-semi-finite case.

the cohomology of a right-angled artin group with group ring coefficients is explicitly presented in terms of the cohomology of its defining flag complex.

An explanation of a duplication identity in law involving the variances of the Brownian bridge and Brownian motion is given, with the help of an elementary transformation in time and space of a two-dimensional Brownian motion.

It is proved that two random elements of the Nottingham group generate a free group with probability 1. This answers a question of A. Shalev.

functional identities for the classical rogers dilogarithm associated to simply laced dynkin diagrams are proved uniformly in the setting of $y$-systems related to clusters.

As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp forms of half-integral weight, it is shown that any sufficiently large number $n\equiv 3$ (mod 24) with $5 \nmid n$ is expressible as a sum of three squares of integers having at most 521 prime factors.

Suppose that K is a linear space of functions analytic in some domain D in the complex plane. A sequence Λ = (λk) of distinct points from D is said to be a set of uniqueness for K if ƒ∈K and ƒ(λk) = 0 for all k imply ƒ≡0. Depending on the dispersion and the density of Λ on the one hand, and the growth of the functions in K on the other, one may often require only |ƒ(λk)| [les ]ak for some sequence of positive numbers ak, and still conclude that ƒ≡0 for ƒ∈K. Of particular interest are sharp conditions on the decay of ak, which reflect the interplay between growth and decay of analytic functions.

The purpose of this paper is to show how the methods of motivic integration of Kontsevich, Denef–Loeser (Invent. Math. 135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290) and Looijenga (Astérisque 276 (2002) 267–297) can be adapted to prove the McKay–Ruan correspondence, a generalization of the McKay–Reid correspondence to orbifolds that are not necessarily global quotients.

A dinilpotent group is a group that can be written as the product of two nilpotent subgroups. There is an extensive literature dealing with such groups (see, for example, the recent book of Amberg, Franciosi and de Giovanni [1]). In 1955, Itô proved in [6] that the product of two abelian groups is always metabelian, and in the following year, Hall and Higman proved, as a special case of [3, Theorem 1.2.4], that if G=AB is a finite soluble group with A and B nilpotent of coprime orders and classes c and d respectively, then G has derived length at most c+d. Wielandt [9] showed that if the finite group G=AB has A and B nilpotent and of coprime orders, then G is necessarily soluble. Kegel [7] then showed that the condition that the orders be coprime was unnecessary. A natural next question to ask is if the derived length of a finite dinilpotent group is bounded by some function of the classes of the factors, and, in the light of the Hall–Higman result and the result of Itô, the following conjecture seems natural. (The first explicit reference to this conjecture that we know of is in Kegel [8]; see also, for example, Problem 5 on page 36 of [1].)

We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius.

In this paper, some new results concerning the existence and uniqueness of nontrivial solutions to nonlinear Volterra integral equations with convolution kernels are presented.