It is shown that the compact matrix quantum groups SUq(2) are non-isomorphic to each other for q∈[−1, 1][setmn ]{0}, and that the compact matrix quantum groups SUq(n) are non-isomorphic to each other for q∈(0, 1]. Some invariants for compact quantum groups are also discussed.

Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, $h_1(p)$ , is further examined. Whereas Kummer conjectured that $h_1(p) \sim G(p) := 2p(p/4\pi^2)^{(p-1)/4}$ it is shown, under certain plausible assumptions, that there exist constants $a_\alpha, b_\alpha$ such that $h_1(p) \sim \alpha G(p)$ for $\sim a_\alpha x/\log^{b_\alpha} x$ primes $p \le x$ whenever $\log \alpha$ is rational. On the other hand, there are $\ll_A x/\log^A x$ such primes when $\log \alpha$ is irrational. Under a weak assumption it is shown that there are roughly the conjectured number of prime pairs $p, mp\pm 1$ if and only if there are $\gg_m x/\log^2 x$ primes $p \le x$ for which $h_1(p) \sim e^{\pm 1/2m} G(p)$ .

Given a directed graph, there exist a universal operator algebra and universal $C^*$-algebra associated to the directed graph. In this paper we give intrinsic constructions for these objects. We also provide an explicit construction for the maximal $C^*$-algebra of an operator algebra. We discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isomorphism invariant for the universal operator algebra of a directed graph. We show that the underlying undirected graph is a Banach algebra isomorphism invariant for the universal $C^*$-algebra of a directed graph.

it is conjectured that a rational map whose coefficients are algebraic over $\mathbb q_p$ has no wandering components of the fatou set. benedetto has shown that any counterexample to this conjecture must have a wild recurrent critical point. we provide the first examples of rational maps whose coefficients are algebraic over $\mathbb q_p$ and that have a (wild) recurrent critical point. in fact, it is shown that there is such a rational map in every one-parameter family of rational maps that is defined over a finite extension of $\mathbb q_p$ and that has a misiurewicz bifurcation.

Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm [mid ]·[mid ]:

formula here

It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V[rarrhk ]H[rarrhk ]V*, t [ges ] 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·)[ratio ]H×R+→R1 which satisfies the following conditions:

formula here

formula here

where [Lscr ] is the infinitesimal generator of the Markov process Xt and ci, ki, μi, i = 1, 2, 3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, μi = 0) are considered.

Characterizations of subnormality due to Trent are extended to finite systems of operators, thus generalizing results of Gãvruţã and Suciu (subnormal pairs) and Binzar and Pãunescu (subnormal triplets). Vector and operator multidimensional moment problems with jointly subnormal solutions are characterized entirely in terms of the initial data. Bounded as well as unbounded solutions are taken into consideration.

Our main aim in this paper is to show that there is a partition of the reals into finitely many classes with ‘many’ forbidden distances, in the following sense: for each positive real x, there is a natural number n such that no two points in the same class are at distance $x/n$. In fact, more generally, given any infinite set $\{ c_n:n<\omega\}$ of positive rationals, there is a partition of the reals into three classes such that for each positive real x, there is some n such that no two points in the same class are at distance $c_nx$. This result is motivated by some questions in partition regularity.

This paper deals with some character values of the symmetric group Sn as well as its double cover S˜n.

Let χλ(ρ) be the irreducible character of Sn, indexed by the partition λ and evaluated at the conjugacy class ρ. Comparing the character tables of S2 and S4, one observes that

for ρ = (2), 2ρ = (4) and ρ = (12), 2ρ = (22). A number of such observations lead to what we call Littlewood's multiple formula (Theorem 1.1). This formula appears in Littlewood's book [2]. We include a proof that is based on an ‘inflation’ of the variables in a Schur function. This is different from one given in [2], and we claim that it is more complete than the one given there.

Our main objective is to obtain the spin character version of Littlewood's multiple formula (Theorem 2.3).

It is proved that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X [oplus ] Y, then there exists a completely bounded automorphism τ: X [oplus ] Y → X [oplus ] Y with completely bounded inverse such that τZ = X0 [oplus ] Y0, where X0 and Y0 are completely complemented subspaces of X and Y, respectively. If X and Y are homogeneous, the existence is proved of such a τ under a weaker assumption that any operator from X to Y is strictly singular. An upper estimate is obtained for ∥τ∥cb∥τ−1∥cb if X and Y are separable homogeneous Hilbertian operator spaces. Also proved is the uniqueness of a ‘completely unconditional’ basis in X [oplus ] Y if X and Y satisfy certain conditions.

For every Hausdorff function h, the paper gives a construction of a compact metric space of infinite diameter-based h-packing measure which has no subsets of positive finite measure. It then indicates how such a construction can be modified to provide the same result for a radius-based packing measure in the case of certain Hausdorff functions which do not satisfy a doubling condition. This is in surprising contrast to another result which shows that for a particular radius-based definition of packing measure no such example can be found.

Let F1, …, Ft be diagonal forms of degree k with real coefficients in s variables, and let τ be a positive real number. The solubility of the system of inequalities

formula here

in integers x1, …, xs has been considered by a number of authors over the last quarter-century, starting with the work of Cook [9] and Pitman [13] on the case t = 2. More recently, Brüdern and Cook [8] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F1, …, Ft satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F1, …, Ft to have different degrees. However, with the recent work of Wooley [18,20] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [18] and [20]. Let λ1, …, λs and μ1, …, μs be real numbers such that for each i either λi or μi is nonzero. We define the forms

formula here

and consider the solubility of the system of inequalities

formula here

in rational integers x1, …, xs. Although the methods developed by Wooley [19] hold some promise for studying more general systems, we do not pursue this in the present paper. We devote most of our effort to proving the following theorem.

The paper considers new components of a key space of a moduli space of minimal surfaces in flat 4-tori and calculates their dimensions. An example of minimal surfaces in 4-tori is constructed, and an element of the moduli is obtained. In the process of construction, an example is given of minimal surfaces with a good property in a 3-torus that is distinct from classical examples.

Multidirectional mean value inequalities provide estimates of the difference of the extremal value of a function on a given bounded set and its value at a given point in terms of its (sub)-gradient at some intermediate point. A generalization of such multidirectional mean value inequalities is derived by using new infinitesimal conditions for a weak r-growth of the lower semicontinuous function along approximate trajectories of differential inclusions. This new form of the multidirectional mean value inequality does not rely on the linear structure of the underlying space and removes a traditional assumption of lower boundedness on the function.

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.

Those $(2m-1)$-edge-colourings of a spanning subgraph of $K_{2m}$, consisting of $K_{r}$ and independent edges, that can be embedded in a $(2m-1)$-edge-colouring of $K_{2m}$ are characterised.

Let [ ] denote the integer part. Among other results in [3] we gave a complete solution to the following problem.

PROBLEM. Given an increasing sequence an ∈ ℝ+, n = 1, 2, …, where an → ∞ as n → ∞, are there infinitely many primes in the sequence [αan] for almost all α?

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].

Hammocks have been considered by Brenner [1], who gave a numerical criterion for a finite translation quiver to be the Auslander–Reiten quiver of some representation-finite algebra. Ringel and Vossieck [11] gave a combinatorial definition of left hammocks which generalised the concept of hammocks in the sense of Brenner, as a translation quiver H and an additive function h on H (called the hammock function) satisfying some conditions. They showed that a thin left hammock with finitely many projective vertices is just the preprojective component of the Auslander–Reiten quiver of the category of [Sscr ]-spaces, where [Sscr ] is a finite partially ordered set (abbreviated as ‘poset’). An important role in the representation theory of posets is played by two differentiation algorithms. One of the algorithms was developed by Nazarova and Roiter [8], and it reduces a poset [Sscr ] with a maximal element a to a new poset [Sscr ]′=a∂[Sscr ]. The second algorithm was developed by Zavadskij [13], and it reduces a poset [Sscr ] with a suitable pair (a, b) of elements a, b to a new poset [Sscr ]′=∂(a,b)[Sscr ]. The main purpose of this paper is to construct new left hammocks from a given one, and to show the relationship between these new left hammocks and the Nazarova–Roiter algorithm. In a later paper [5], we discuss the relationship between hammocks and the Zavadskij algorithm.

Part of the Choquet theory of integral representations on a universal cap is presented here in a slightly improved form. This makes possible a rapid proof of a generalised Bochner theorem for non-separable locally compact groups and of two extensions to such groups of the classical Lévy–Cramér continuity theorem concerning the convergence of sequences of probabilities.

It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety [Bfr ]4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r [ges ] 2, the 4-Engel verbal subgroup of the r-generator Burnside group B(r, 4) is irreducible as an [ ]2GL(r, 2)-module. It is observed that the variety of all 4-Engel groups of exponent 4 is insoluble, but not minimal insoluble.