For an orientable surface S of finite topological type with genus g ≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph $\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger, J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger, Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

For X and Y Banach spaces, let X⊗εY, be the injective tensor product. If Z is also a Banach space and U ∊ L(X⊗εY,Z) we consider the operator

We prove that if U ∊ PI(X⊗εY, Z), then U# ∊ I(X, PI(Y,Z)). This result is then applied in the case of operators defined on the space of all X-valued continuous functions on the compact Hausdorff space T. We obtain also an affirmative answer to a problem of J. Diestel and J. J. Uhl about the RNP property for the space of all nuclear operators; namely if X* and Y have the RNP and Y can be complemented in its bidual, then N(X, Y) has the RNP.

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p, q), 2 < p < q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realised as semi-regular pq-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order pq. The general t-dimensional Frobenius problem (t ≥ 3) is NP-hard, and it may be that our restricted problem retains this property.

The classical generalizations (already investigated in the second half of last century) of the modular group SL(2, ℤ) are the groups ГK = SL(2, o)(o the principal order of a totally real number field K, [K:ℚ]=n), operating, originally, on a product of n upper half-planes or, for n=2, on the product 1×− of an upper and a lower half-plane by

(where v(i), for v∈K, denotes the jth conjugate of v), and Гn = Sp(n, ℤ), operating on n={Z∣Z=X+iY∈ℂ(n,n),tZ=Z, Y>0} by

Nowadays ГK is called Hilbert's modular group of K and Гn Siegel's modular group of degree (or genus) n. For n=1 we have Гℚ=Г1= SL(2, ℤ). The functions corresponding to modular forms and modular functions for SL(2, ℤ) and its subgroups are holomorphic (or meromorphic) functions with an invariance property of the form

J(L, t) for fixed L (or J(M, Z) for fixed M) denoting a holomorphic function without zeros on ) (or on n). A function J;, defined on ℤK×or ℤn×n to be able to appear in (1.3) with f≢0, has to satisfy certain functional equations (see below, (2.3)–(2.5) for ГK, (5.7)–(5.9) for Гn) and is called an automorphic factor (AF) then. In close analogy to the case n=1, mainly AFs of the following kind have been used:

with a complex number r, the weight of J, and complex numbers v(L), v(M). AFs of this kind are called classical automorphic factors (CAP) in the sequel. If r∉ℤ, the values of the function v on ГK (or Гn) depend on the branch of (…)r. For a fixed choice of the branch (for each L∈ГK or M∈Гn) the functional equations for J, by (1.4), (1.5), correspond to functional equations for v. A function v satisfying those equations is called a multiplier system (MS) of weight r for ГK (or Гn).

We define the nth cyclotomic polynomial Φn(z) by the equation

and we write

where ϕ is Euler's function.

Erdös and Vaughan [3] have shown that

uniformly in n as m-→∞, where

and that for every large m

Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory.

In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories.

In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R}, and has some connections with the “pure-semisimple conjecture”.

1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

We relate the coefficients of the probabilistic zeta function of a finite monolithic group to those of an almost simple group.

In this paper, we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally, we show that there is no surjective *-homomorphism from a SAW*-algebra onto C*-tensor product of two infinite dimensional C*-algebras.

It is the purpose of this note to discuss the solution of the pair of series

where F(x) and G(x) are given and the coefficients an are to be determined.

In a recent paper [4] Bennett and Kalton characterized dense, barrelled subspaces of an arbitrary FK space, E. In this note, it is shown that if E is assumed to be an AK space, then the characterization assumes a simpler and more explicit form.

In this paper we examine when a twisted group ring, Rγ(G), has a semi-simple, artinian quotient ring. In §1 we assemble results and definitions concerning quotient rings, Ore sets and Goldie rings and then, in §2, we define Rγ(G). We prove a useful theorem for constructing a twisted group ring of a factor group and establish an analogue of a theorem of Passman. Twisted polynomial rings are discussed in §3 and I am indebted to the referee for informing me of the existence of [4]. These are used as a tool in proving results in §4.

In this paper, we deal with complete hypersurfaces immersed with bounded higher order mean curvatures in steady state-type spacetimes and in hyperbolic-type spaces. By applying a generalised maximum principle for the Yau's square operator [11], we obtain uniqueness results in each of these ambient spaces.

Let G be a connected graph of order n, and let k ≥ 2 and m ≥ 0 be two integers. In this paper, we show that G is a fractional (k, m)-deleted graph if and for each pair of non-adjacent vertices x, y of G. This result is an extension of the previous result of Zhou [11].

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is

1. to study the lattice not with values in $\mathbb{C}$ but in the Banach algebra ${\cal L}$ of bounded operators and

2. to derive solutions of the original lattice ($\mathbb{C}$-solutions) by applying a functional $\tau$ to the ${\cal L}$-solutions constructed in 1.

In the second part we focus on applications. We start by rederiving line-solitons and briefly discuss discrete resonance phenomena. Moreover, we are able to find conditions under which it is possible to superpose even countably many line-solitons.

In the theory of Hilbert space operators an important question is whether an operator is subnormal [3], [4], [7], [8]. A densely defined linear operator S in a complex Hilbert space H is subnormal if there exists a normal operator N in a complex Hilbert space K ⊃ H such that S ⊂ N.

We use the principle of symmetric criticality to connect the Willmore variational problem for surfaces in a warped product space with base a circle, and the free elastica variational problem for curves on its fiber. In addition we obtain a rational oneparameter family of closed helices in the anti De Sitter 3-space which are critical points of the total squared curvature functional. This means they are free elasticae. Also they are spacelike; this allows us to construct a corresponding family of spacelike Willmore tori in a certain kind of spacetime close to the Robertson-Walker spaces.

We show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.

May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].

In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.

Let Γ be a totally ordered abelian group and I an order ideal in Γ. We prove a theorem which relates the structure of the Toeplitz algebra T(Γ) to the structure of the Toeplitz algebras T(I) and T(Γ/I). We then describe the primitive ideal space of the Toeplitz algebra T(Γ) when the set Σ(Γ) of order ideals in Γ is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure of T(Γ) when 0→ I→ Γ→ Γ/I→ 0 is a non-trivial group extension.

For a subfunction u, associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we generalise the classical Phragmén-Lindelöf theorem by making an a-harmonic majorant of u.