We study the geometry of non-reductive four-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

Let R be a ring with identity, and let C be a small, nonempty category. We denote the category of right R-modules by AbR and the category of contravariant functors C → AbR by AbRC*. The limit functor

is left exact, and its kth right derived functor is denoted by colimk. The R-cohomological dimension of C is defined by

If there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .

We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

A pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].

If n is a non-negative integer, define pr(n) as the coefficient of xn in

;

otherwise define pr(n) as 0. In a recent paper (2) the author established the following congruence:

Let r = 4, 6, 8, 10, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p2 − l)/24.

It is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the representation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable injective representations are described in terms of the injective indecomposable $R$-modules and the injective indecomposable $R[\text{x},\,{{\text{x}}^{-1}}]$ -modules.

We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

Let be a complex Banach space and the algebra of bounded linear operators on . In this paper we study functions from the complex numbers to of the form

Soit Dn = ﹛z = (z1, . . . , zn) ∈ Cn, |zi| <> 1﹜ le polydisque de Cn et ƛn la mesure de Lebesgue de Cn normalisée sur Dn. Pour b > 0, .on définit les espaces de Bergman Ap(ƛn) de la manière suivante:

Ap(ƛn) est l'espace des fonctions analytiques dans Dn telles que:

The good weights for the one-sided Hardy-Littlewood operators have been characterized by conditions . In this paper we introduce a new condition which is analogous to A∞. We show several characterizations of . For example, we prove that the class of weights is the union of classes. We also give a new characterization of weights. Finally, as an application of condition, we characterize the weights for one-sided fractional integrals and one-sided fractional maximal operators.

Let F be a non-singular metric vector space, that is, a vector space over a field F not of characteristic two, upon which is defined a non-singular symmetric bilinear form ( , ). For any a ϵ V, we define the cone with vertex a to be the set

Together with Vern Paulsen we were able to show that the elementary theory of tensor norms of Banach spaces carries over to operator spaces. We suggested that the Grothendieck tensor norm program, which was of course enormously important in the development of Banach space theory, be carried out for operator spaces. Some of this has been done by the authors mentioned above, and by Effros and Ruan. We give alternative developments of some of this work, and otherwise continue the tensor norm program.

Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L, that is, the Lie algebra of all derivations of L. Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases.

Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L). However, there is still some intimate connection between the structure of D(L) and that of L.

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.

Let M be the space of all bounded regular complex-valued Borel measures defined on I = [0, ∞). M is a Banach space with ‖μ‖ = ∫d|μ|(x) (μ ∈ M). (Integrals in this paper extend over all of I unless otherwise specified.) Let v be a fixed real number no smaller than and let if z ≠ 0 and , where Jv, is the Bessel function of the first kind of order v and cv =[2vΓ(v + 1)]–1; is an entire function, as can be seen from the power series definition of

The Hankel-Stieltjes transform of order v is given by .

In the paper we find, for certain values of the parameters, the spaces of multipliers (H(p, q, α), H(s, t, β) and (H(p, q, α), ls), where H(p, q, α) denotes the space of analytic functions on the unit disc such that . As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces.

Denote by Fα,β the multiplicative group generated by the two matrices where α and β are complex numbers. Sanov (3) proved that Fα,β is free, and Brenner (1) showed that Fm,m is free if m ⩾ 2.

This paper continues a study appearing in (5) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and 1's. Let the sum of row i of A be denoted by ri and let the sum of column i of A be noted by st. We call R = (r1, … , rm) the row sum vector and S = (s1, … , sn) the column sum vector of A. The vectors R and S determine a class consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. Simple arithmetic properties of R and S are necessary and sufficient for the existence of a class (1 ; 5).

In complex affine n-space with a unitary metric, a reflection is a congruent transformation leaving invariant all the points of a hyperplane. Thus the characteristic roots of a unitary reflection of period p consist of a primitive pth root of unity and n – 1 unities. A group generated by n reflections is conveniently represented by a graph having a node for each generator and a branch for each pair of non-commutative generators.