The purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian down-up algebras. We will show that the Noetherian down-up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are locally Artinian provided the roots of X2 − αX − β are distinct roots of unity or both equal to 1.

An exact solution of triple trigonometrical equations is obtained by using the finiteHilbert transform. The solution of these equations is used to solve a two-dimensional electrostatic problem. The problem of determining the electrostatic potential due to two parallel coplanar strips of equal length, charged to equal and opposite potentials, each parallel to and equidistant from an earthed strip, is considered. Both the charged strips lie along the x-axis and they are equally spaced with respect to the y-axis. Finally the expression for the surface charge density (per unit depth) of the strip is derived

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equation

The eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce to

aN = bN = N2

when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namely

Suppose first that N is an odd integer. Then there is an expansion

where

These functions π satisfy

and

On Substituting (3) in (1), one obtains the algebraic equation

where

Explicitly,

{11} = q

{lm} = 0 otherwise.

Suppose that $A$ and $B$ are ‘isoloid’ operators acting on a complex Banach space, that is, every isolated point of their spectra is an eigenvalue. In this note it is shown that if Weyl's theorem holds for both $A$ and $B$ then it holds for $A\otimes B$.

§ 1. Introductory. The formula to be proved is

where b>0.

In this paper, we discuss implications of the results obtained in [5]. It was shown there that eigenvectors of the Bethe algebra of the quantum N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [5] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.

We denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of a normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.

In [2] I described the deformation of bending a hyperbolic manifold along an embedded totally geodesic hypersurface. As I remarked there, the deformation is particularly interesting in the case of a surface, because a surface contains many embedded totally geodesic hypersurfaces, namely simple closed curves, along which it is possible to bend. Furthermore, for a surface it is possible to extend the definition of bending to the case of a geodesic lamination, by using the fact that the set of simple closed geodesies is dense in the space of geodesic laminations. This direction has been developed by Epstein and Marden in [1].

All rings in this paper are associative but not necessarily with an identity. The ring R with an identity adjoined will be denoted by R#.

To denote that I is an ideal (right ideal, left ideal) of a ring R we write I ◃ R (I <rR, I <1R).

A ring R is called right (left) fully idempotent if for every I <rR (I<1R), I = I2.

At the conference “Methoden der Modul und Ringtheorie” in Oberwolfach, Germany in 1993, J. Clark raised the question as to whether every right fully idempotent ring is left fully idempotent (see also [3]). A similar question was raised by S. S. Page in [5]. In this note we answer the questions in the negative.

We start with some general observations most of which are perhaps well known. We include their simple proofs for completeness.

In this paper we investigate existence and nonexistence of positive solutions for inhomogeneous semilinear elliptic systems in $R^n$. A criteria of existence and nonexistence of positive solutions is given by the exponents of the rate of nonlinear terms at infinity.

Two coupled bilinear equations are considered, and then two new coupled differential-difference systems are found. Also two special reductions of these two systems are studied. By using Hirota's method, Bäcklund transformation and superposition formulae, soliton solutions to these equations are presented.

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi-étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with pg = q = 1 and Albanese fibre of genus bigger than K2.

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

Let H ≤ K be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever ag ∈ K, where a ∈ H, g ∈ G, then ag ∈ H. In this paper, we investigate the structure of a group G under the assumption that every subgroup of order 2m (and 4 if m = 1) of a 2-Sylow subgroup S of G is strongly closed in S with respect to G. Some results related to 2-nilpotence and supersolvability of a group G are obtained. This is a complement to Guo and Wei (J. Group Theory13(2) (2010), 267–276).

We show that there exists a locally compact separable metrizable space $L$ such that $C_{0}(L)$, the Banach space of all continuous complex-valued functions vanishing at infinity with the supremum norm, is almost transitive. Due to a result of Greim and Rajagopalan [3], this implies the existence of a locally compact Hausdorff space $\tilde L$ such that $C_{0}(\tilde L)$ is transitive, disproving a conjecture of Wood [9]. We totally owe our construction to a topological characterization due to Sánches [8].

Let G be a group and K be a complete ultrametric valued field. Let AP(G, K) be the algebra of the generalized almost periodic functions of G in K. We have shown in a previous paper that when AP(G, K) has an invariant mean, then any almost periodic linear representation is quasi-reducible. Here, we show that with the same hypothesis, any topologically irreducible almost periodic linear representation is finite dimensional; also, any almost periodic linear representation is the topological sum of irreducible representations. Furthermore, we obtain a Peter-Weyl theorem for the algebra AP(G, K).

We use the technical tools of Hopf algebra theory.

Let ∧ denote a basic artin ring and r its radical. In most of this paper we assume that r2 = 0 and that Λ is a trivial extension Λ/r ⋉ r (see Section 1 for definition). Let P1 …, Pn be the non-isomorphic indecomposable projective (left) Λ-modules, and consider triples (Pi, Mi, ui), where the Mi, are (left) Λ-modules and ui:rPi → Mi/rMi isomorphisms. From this data we construct a new ring Г, which in “nice cases” has the property that r′3 = 0, Г/r′2 ≅ Λ, and r′ Qi ≅ Mi as (left) Λ-modules, where the Qi are the indecomposable projective (left) Г-modules and r′ is the radical of Г.

A group is called a T-group if all its subnormal subgroups are normal. It is proved here that if G is a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then either G is a T-group or it has finite rank. It follows that if G is an arbitrary group whose Fitting subgroup has infinite rank, then G has the property T if and only if all its subnormal subgroups of infinite rank are normal.

By the group G(2 cos π/q) we mean the group of linear fractional transformations of the complex plane onto itself, generated by V(z)= — 1/z and U(z) = z+λq, where λq = 2 cos (π/q), qbeing a positive integer greater than 2. In this paper we shall be concerned only with the group given by q = 5, and we shall therefore omit the subscript 5 on the λ. We note that λ = λ5 satisfies the equation

x2–x–l=0;(1)

henceλ = (l + 5½)/2.