We characterise singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

A ring is called clean if every element is the sum of an idempotent and a unit. It is an open question whether the tensor products of two clean algebras over a field is clean. In this note we study the tensor product of clean algebras over a field and we provide some examples to show that the tensor product of two clean algebras over a field need not be clean.

We introduce the concept of an E-valued function algebra, a type of Banach algebra that consists of continuous E-valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E-valued function algebras. We give some specific examples.

Let be the semigroup of all total one-to-one transformations of an infinite set X. For an ƒ ∈ let the defect of ƒ def ƒ, be the cardinality of X – R(ƒ), where R(ƒ) = ƒ(X) is the range of ƒ. Then is a disjoint union of the symmetric group x on X, the semigroup S of all transformations in with finite non-zero defects and the semigroup Ā of all transformations in S with infinite defects, such that S U Ā and Ā are ideals of . The properties of x and Ā have been investigated by a number of authors (for the latter it was done via Baer-Levi semigroups, see [2], [3], [5], [6], [7], [8], [9], [10] and note that Ā decomposes into a union of Baer–Levi semigroups). Our aim here is to study the semigroup S. It is not difficult to see that S is left cancellative (we compose functions ƒ, g in S as ƒg(x) = ƒ(g(x)), for x ∈ X) and idempotent-free. All automorphisms of S are inner [4], that is of the form ƒ → hƒhfh-1 ƒ ∈ S, h ∈ x.

We present Bäcklund transformations for the non-commutative anti-self-dual Yang–Mills equations where the gauge group is G = GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants and belong to a non-commutative version of the Atiyah–Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard.

In this paper, we investigate various “arithmetical” functions associated with the factorisation of polynomials in GF[q, X1, …, Xk], where k ≥ 1 and GF[q]is the finite field of order q. We shall assume throughout that all polynomials discussed are non-zero and have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. The constant polynomial will be denoted by 1. With this normalisation, GF[q, X1, …, Xk] becomes a unique factorisation domain. When k = 1, normalisation is achieved by considering only monic polynomials. By the degree of a polynomial A(X1, …, Xk) will be understood the ordered set (m1, …, mk), where m1 is the degree of A(X1, …, Xk) in X1,(i = 1, …, k).

In this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

In this paper we consider differential equations of the form

where the coefficients Ai are holomorphic functions in a domain G1 × G2 ⊂ C × C. We restrict our attention to those equations for which it is possible to represent the solutions in the form

where g1(z1) and g2(z2) are arbitrary holomorphic functions in G1 and G2 respectively. The coefficients a1, k and a2, k depend on the given differential equation. Within the last ten years a number of publications have been devoted to this kind of representation of solutions.

In [3] the authors (M. Borodzik and S. Friedl, Unknotting number and classical invariants (preprint 2012)) associated to a knot K ⊂ S3 an invariant nℝ(K), which is defined using the Blanchfield form and which gives a lower bound on the unknotting number. In this paper, we express nℝ(K) in terms of the Levine-Tristram signatures and nullities of K. We also show in the proof that the Blanchfield form for any knot K is diagonalisable over ℝ[t±1].

The joint spectrum for a commuting n-tuple in functional analysis has its origin in functional calculus which appeared in J. L. Taylor's epoch-making paper [19] in 1970. Since then, many papers have been published on commuting n-tuples of operators on Hilbert spaces (for example, [3], [4], [5], [8], [9], [10], [21], [22]).

In this paper, we classify 3-dimensional Lagrangian Willmore submanifolds of the nearly kaehler 6-sphere $S^6(1)$ with constant scalar curvature.

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.

Let the closed orientable surface Fh of genus h be a covering of F2 and let and f be the fundamental groups respectively. Then is a subgroup of f of index n = h − 1. A covering is called regular if is normal in f.

Conversely, let be a normal subgroup of f of finite index. Then there is a uniquely determined regular covering Fh such that is the fundamental group of Fh. The covering Fh is an orientable surface. Since the index n of in f is supposed to be finite, Fh is closed, and its genus is given by n = h − 1.

The fundamental group f can be defined by

.

The α-regular classes of any finite group G are important since they are those classes on which the projective characters of G with factor set α take non-zero value, and thus a knowledge of the α-regular classes gives the number of irreducible projective representations of G with factor set α (see [4]). Here we look at the particular case of the generalized symmetric group Cm wr Sl. The analogous problem of constructing the irreducible projective representations of Cm wr Sl has been dealt with in [6] by generalizing Clifford's theory of inducing from normal subgroups, but unfortunately, it is not in general possible to determine the irreducible projective characters (and hence the α-regular classes) by this method.

In this paper, we continue the work initiated by Morris [5] and Saeed-ul-Islam [6,7] and determine complete sets of inequivalent irreducible projective representations (which we shall write as i.p.r.) of finite Abelian groups with respect to some additional factor sets.

We consider an Abelian group

which will be referred to as an Abelian group of type (a1, …, am).

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

Let $\rho:G\hookrightarrow {\rm GL}(n, {\bb F})$ be a representation of a finite group over the field ${\bb F}, V = {\bb F}^n$ the corresponding $G$-module, and ${\bb F}[V\hspace*{1.5pt}]$ the algebra of polynomial functions on $V$. The action of $G$ on $V$ extends to ${\bb F}[V\hspace*{1.5pt}]$, and ${\bb F}[V\hspace*{1.5pt}]^G$, respectively ${\bb F}[V\hspace*{1.5pt}]_G$, denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when ${\bb F} = {\bb C}$, dim$_{\bb C}$ (Tot$({\bb C}[V\hspace*{1.5pt}]_G))=|G|$ if and only if $G$ is a complex reflection group. Here Tot$({\bb F}[V\hspace*{1.5pt}]_G)$ denotes the direct sum of all the homogeneous components of the graded algebra ${\bb F}[V\hspace*{1.5pt}]_G$ and $|G|$ is the order of $G$. Chevalley's theorem tells us that the ring of invariants of a complex pseudoreflection representation $G\hookrightarrow {\rm GL}(n, {\bb C})$ is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim$_{\bb F}$(Tot$({\bb C}[V\hspace*{1.5pt}]_G) = |G|$ if and only if ${\bb C}[V\hspace*{1.5pt}]^G$ is a polynomial algebra. The purpose of this note is to show that the two conditions

(i) dim$_{\bb F}$(Tot(${\bb F}[V\hspace*{1.5pt}]_G))\,{=}\,|G|$,

(ii) ${\bb F}[V\hspace*{1.5pt}]^G$ is a polynomial algebra

are equivalent regardless of the ground field; in particular in the modular case.

Given N a finite separable normal extension of a field F, it is well known that the Brauer group Br(N/F) of classes of central simple F-algebras split by N is isomorphic with Ext(N*, G), the classes of group extensions of N* by the Galois group G of N over F. In the construction of this isomorphism, a key role is played by the Skolem-Noether Theorem which extends automorphisms to inner automorphisms in central simple algebras.

Let Fr denote the free group of rank r and Out Fr: = AutFr/Inn Fr the outer automorphism group of Fr (automorphisms modulo inner automorphisms). In [10] we determined the maximal order 2rr! (for r > 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r > 3, is unique up to conjugation. In the present paper we determine all maximal finite subgroups (that is not contained in a larger finite subgroup) of Out F3, up to conjugation (Theorem 2 in Section 3). Here the considered case r = 3 serves as a model case: our method can be applied for other small values of r (in principle for any value of r) but the computations become considerably longer and are more apt for a computer then; the method can also be applied to determine the maximal finite subgroups of the automorphism group Aut Fr of Fr. Note that the canonical projection Aut Fr ⃗ Out Fr is injective on finite subgroups of Aut Fr; however, not all finite subgroups of Out Fr lift to finite subgroups of Aut Fr.

We define algebraically for each map germ f:Kn,0→Kp, 0 and for each Boardman symbol i=(i1,…,ik) a number ci(f) which is -invariant. If f is finitely determined, this number is the generalization of the Milnor number of f when p = 1, the number of cusps of f when n = p = 2, or the number of cross caps when n = 2, p = 3. We study some properties of this number and prove that, in some particular cases, this number can be interpreted geometrically as the number of Σi points that appear in a generic deformation of f. In the last part, we compute this number in the case that the map germ is a projection and give some applications to catastrophe map germs.

Consider any group G. A [G, 2]-complex is a connected 2-dimensional CW-complex with fundamental group G. If X is a [G, 2]-complex and L is a subgroup of G, let XL denote the covering complex of X corresponding to the subgroup L. We say that a [G, 2]-complex is L-Cockcroft if the Hurewicz map hL:π2(X)→;H2(XL) is trivial. In case L = G we call X Cockcroft. There are interesting classes of 2-complexes that have the Cockcroft property. A [G, 2]-complex X is aspherical if π2(X) = 0. It was observed in [4] that a subcomplex of an aspherical 2-complex is Cockcroft. The Cockcroft property is of interest to group theorists as well. Let X be a [G, 2]-complex modelled on a presentation (〈S; R〉 of the group G. If it can be shown that X is Cockcroft, then it follows from Hopf's theorem (see [2, p. 31]) that H2(G) is isomorphic to H2(X). In particular H2(G) is free abelian. For a survey on the Cockcroft property see Dyer [5]. A collection {Gα: α ∈ Ώ} of subgroups of a group G that is totally ordered by inclusion is called a chain of subgroups of G. Denning β ≤ α if and only if Gα ≤ Gβ makes Ώ into a totally ordered set. The main result of this paper is the following theorem.