Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the form

If y′ is always affine in y, i.e.

where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.

Let f denote a function, meromorphic in C. The question of when a deficient value of f, in the sense of Nevanlinna, is an asymptotic value has recently received some attention (see e.g. Hayman [6]). We assume acquaintance with the standard notation of the Nevanlinna theory ([[5] Chapter I) which we use without further mention. The following two theorems are known ([1] Theorem 4, and [6] Corollary 2).

The integral equation of the title is

It was studied in [4], though h(x) was written as x-1g(x-1) there, and using a method involving orthogonal Watson transformations, it was shown there that if h ∈ L2(0, ∞), then the equation has a solution f ∈ L2(0, ∞), and that / is given by

In this paper, using the techniques of [3], we shall show that the equation can be solved for ℎ in the space ℒμ, p of [3] for 1 ≤ p < ∘, μ > 0, and that for these spaces, which include L2(0, ∘), f is given by the simpler formula

We shall further show that these results can be extended to the spaces ℒw, μ, p of [3]. This forms the content of our theorem below.

Let G be a finite additive abelian group, and suppose that A and B are subsets of G. We say that G = A⊕B if every element g ∈ G can be uniquely written in the form g = a + b, where a ∈ A, b ∈ B. The study of such decompositions (usually called factorizations in the literature) was initiated by G. Hájos [3] in connection with his solution to a problem of Minkowski in the geometry of numbers.

A Hadamard matrix H is an orthogonal square matrix of order m all the entries of which are either + 1 or - 1; i. e.

where H′ denotes the transpose of H and Im is the identity matrix of order m. For such a matrix to exist it is necessary [1] that

It has been conjectured, but not yet proved, that this condition is also sufficient. However, many values of m have been found for which a Hadamard matrix of order m can be constructed. The following is a list of such m (p denotes an odd prime).

In the following let Ω be the set of irrational numbers in the interval [0,1] and let λ be Lebesgue measure restricted to Ω. For any real number x, let {x} = x - [x] be the fractional part of x. Let N be anatural number and let α e Ω. Then

is known as the discrepancy of the sequence (nα)n>1 modulo 1; here c[x, y) denotes the characteristic function of the interval [x, y).

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.

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].

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.

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.

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.

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.

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

We show that if R is a ring such that each minimal left ideal is essential in a (direct) summand of RR, then the dual of each simple right R-module is simple if and only if R is semiperfect with Soc(RR)=Soc(RR) and Soc(Re) is simple and essential for every local idempotent e of R. We also show that R is left CS and right Kasch if and only if R is a semiperfect left continuous ring with Soc(RR)⊆eRR. As a particular case of both results we obtain that R is a ring such that every (essential) closure of a minimal left ideal is summand (R is then said to be left strongly min-CS) and the dual of each simple right R-module is simple if and only if R is a semiperfect left continuous ring with Soc(RR)=Soc(RR)⊆eRR. Moreover, in this case R is also left Kasch, Soc(eR)≠0 for every local idempotent e of R, and R admits a (Nakayama) permutation of a basic set of primitive idempotents. As a consequence of this result we characterise left PF rings in terms of simple modules over the 2×2 matrix ring by showing that R is left PF if and only if M2(R) is a left strongly min-CS ring such that the dual of every simple right module is simple.

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 consider the divisibility of the class numbers of imaginary quadratic fields , where q is an odd prime number, k and n are positive integers. Suppose that k ≡ 1 mod 2 or n ≢ 3 mod 6. We show that the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 3 mod 8. This is a generalization of the result of Kishi for imaginary quadratic fields when k ≡ 1 mod 2 or n ≢ 3 mod 6. We also show that the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 1 mod 4 and the class numbers of imaginary quadratic fields ≠ are divisible by n for q ≡ 7 mod 8.

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).

The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.

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.

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.