Classification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

Let d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).

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.

Let be a group-theoretic property. We say a group has a finite covering by -subgroups if it is the set-theoretic union of finitely many -subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups, n-abelian subgroups or 2-central subgroups.

R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC (G)] finite implies the existence of a finite covering by subgroups of nilpotency class c, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, but Z2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.

Peller [4, 5] has proved that a Hankel operator S on the Hardy space H2 is in the trace class if and only if with h analytic on the open unit disc Dand with its second derivative belonging to the Bergman space L1a. This theorem does not include an estimate for the trace class norm ∥S∥1, of the operator in terms of the symbol function. In fact it is clear that cannot give an estimate for since the first two terms in the coefficient sequence of the Hankel operator have been removed by differentiation.

Let G be a finite linear group of degree n; that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients). We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [5].

If q(≥2) is a fixed integer it is well known that every positive integer k may be expressed uniquely in the form

We introduce the ‘sum of digits’ function

Both the above sums are of course finite. Although the behaviour of α(q, k) is somewhat erratic, its average behaviour is more regular and has been widely studied.

Let Q(S) denote the maximal right quotient semigroup of the semigroup S as defined in [4]. In this paper, we initiate a study of Q(S) when S is a semilattice of groups. A structure theorem for such semigroups is given by Theorem 4.11 of [2].

We prove that the norm $\Vert\,{\cdot}\,\Vert_{n}$ of the space $T[\mathcal{S}_{n},\theta]$ and the norm $\Vert\,{\cdot}\,\Vert_{n}^{M}$ of its modified version $T_{M}[\mathcal{S}_{n},\theta]$ are 3-equivalent. As a consequence, using the results of E. Odell and N. Tomczak-Jaegermann, we obtain that there exists a $K\,{<}\,\infty$ such that for all $n$, $\Vert\cdot\Vert_{n}^{M}$ does not $K-$ distort any subspace of Tsirelson's space $T$.

Page 188 in Ramanujan's lost notebook is devoted to a certain class of infinite series connected with Euler's pentagonal number theorem. These series are represented in terms of Ramanujan's famous Eisenstein series $P, Q$, and $R$. The purpose of this paper is to prove all the formulas on page 188 and to show that one of them leads to an interesting, new recurrence formula for $\sigma (n)$, the sum of the positive divisors of $n$.

An element r in a ring R is clean if r is a sum of a unit and an idempotent. Camillo and Yu showed that unit regular rings are clean and in a very surprising development Nicholson and Varadarajan showed that linear transformations on countable dimension vector spaces over division rings are clean. These rings are very far from being unit regular.

Here we note that an idempotent is just a root of g(x)=x^{2}-x. For any g(x) we say R is g(x)-clean if every r in R is a sum of a root of g(x) and a unit. We show that if V is a countable dimensional vector space and over a division ring D and g(x) is any polynomial with coefficients in <formtex>K={\text Center}D and two distinct roots in K, then {\text End}V_D is g(x)-clean.

We study orthogonal projections of generic embedded hypersurfaces in ℝ4 with boundary to 2-spaces. Therefore, we classify simple map germs from ℝ3 to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi–Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module up to an explicit normalising monomial factor.

In the present paper, for a large family of topological semigroups namely, compactly cancellative and right cancellative foundation semigroups S, we study the topological centers of the Banach algebrasLUC(S) * and M_a(S)**. We also give a generalization of a known result of Lau and Lorsert by showing that for such topological semigroups the topological center of LUC(S)* (M_a(S)**, respectively) is the same as M(S) (M_a(S), respectively).

The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.

A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the following

Theorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.

It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].

1. For functions f ∈ LLoc[0, ∞) the Riemann-Liouville operator of fractional integration I∞ is defined by

and its adjoint operator, the Weyl operator Kα, is defined by

for functions f ∈ LLoc[0, ∞) having a suitable behaviour at infinity.

This note discusses the determination of the coefficients an in the dual trigonometrical series

where p = ± 1 and F(x), G(x) are prescribed functions of X. It is shown that this problem and the corresponding one in which the sines in equations (1) are replaced by cosines are easily reduced to a form in which the results I have recently given in this journal [1] may be applied.

As with my previous paper on this subject, the analysis is purely formal and no attempt is made to give precise conditions for which the solution is valid.

Any extension of a group A by a group B can be embedded in their wreath product A Wr B. Here we consider generalizations of this result for inverse semigroups.

Suppose S is an inverse semigroup and ρ0 is a congruence on S. We put T = S/ρ0 and denote the natural map from S to T by ρ. The kernel of ρ is the inverse image ETρ−1 of the semilattice ET of idempotents of T. First we show that if each ρ0-class of idempotents of S is inversely well-ordered, then S can be embedded in K Wr T, the standard wreath product of K and T. In general, not all elements of K Wr T have inverses. However, we can define a wreath product W(K, T) which is an inverse semigroup and which contains S when the previous condition holds. If ρ0 is idempotent-separating and S is 0-bisimple, K is the union of zero and a family of isomorphic groups. In this case, we can replace K by a single component group G of K, augmented by zero, and show that S can be embedded in W(G0, T). These results are analogous to the extension theories of D'Alarcao [1] and Munn [3] and they give conditions under which all inverse semigroup extensions of an inverse semigroup A by an inverse semigroup T are contained in a semigroup with structure depending only on A and T.

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.