For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

Let (M, G) be a pair of groups, in which M is a normal subgroup of G such that G/M and M/Z(M, G) are of orders pm and pn. respectively. In 1998, Ellis proved that the commutator subgroup [M, G] has order at most pn(n + 2 m−1)/2.

In the present paper by assuming /[M, G] = pn(n+2m−1)/2, we determine the pair (M, G). An upper bound is obtained for the Schur multiplier of the pair (M, G), which generalizes the work of Green (1956).

Some properties of the singular ideal are established. In particular its behaviour when passing to one-sided ideals is studied. Obtained results are applied to study some radicals related to the singular ideal. In particular a radical S such that for every ring R, S(R) and R/S(R) are close to being a singular ring and a non-singular ring, respectively, is constructed.

A class of cyclically presented groups with n generators and n Fibonacci type relations is discussed. Conditions are given for these groups to be finite and metacyclic. With these conditions the presentations are reduced to the standard form for metacyclic groups with trivial Schur multiplicator. This enables certain ismorphisms between the groups to be found.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 F 05.

We make use of the Beylkin-Coifman-Rokhlin wavelet decomposition algorithm on the Calderón-Zygmund kernel to obtain some fine estimates on the operator and prove the T(l) theorem on Besov and Triebel-Lizorkin spaces. This extends previous results of Frazier et al., and Han and Hofmann.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

We consider some unconventional partition problems in which the parts of the partition are restricted by divisibility conditions, for example, partitions n = a1+…+ak into positive integers a1, …, ak such that a1 ∣ a2 ∣ … ∣ ak. Some rather weak estimates for the various partition functions are obtained.

The purpose of this paper is to derive asymptotic relations giving the error of a Gauss type quadrature, applied to analytic functions, in terms of certain coefficients in the orthogonal expansion of the integrand. The Fourier expansions of the integrand we consider here are those in terms of the Legendre and the Chebyshev polynomials. In Section 3 we obtain the error of the Gauss-Legendre quadrature expressed in terms of the Legendre-Fourier coefficients of the integrand. In Section 4 the errors of Gauss-Legendre, Lobatto and Radau quadrature formulas are obtained, for large n, expressed in terms of the Chebyshev-Fourier coefficients of the integrand. In deriving these estimates we have used complex variable methods restricting ourselves to the class of analytic integrands; this allows us to obtain simple contour integral representations for the errors of these quadratures for large values of n. However, the form of the estimates obtained indicate that these are applicable to a much wider class of functions.

A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.

The main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

The quadrature formula of order m using values of derivatives up to the m — 1st order with the best possible bound in is derived. Using certain properties of the polynomials minimal in Lq norm, it is proved that the optimal formula not use the derivatives of m — 1st order if m is even.

A great deal of attention has been given in the literature to the various properties of the simple binomial random walk. Explicit expressions are available for first passage times, absorption probabilities, average duration of the walk up to absorption and other quantities of interest. One aspect of the behaviour of this work which has, however, attracted little attention is the form of the distribution of occupation totals. This paper is devoted to the derivation of an explict expression for the joint probility generating function of the occupation totals up to absorption, for the binomial random walk in the presence of two absorbing points. The appropriate marginal form of this p.g.f. yields the distribution of the occupation total, and expected occupation total, at any particular lattice point. The limiting forms of these results provide explicit expressions for the corresponding quatities in the case of a binomial random walk having a single absorbing point and, where relevent, in the case of the unrestricted binomial random walk.

We adapt the Toeplitz operator proof of Bott periodicity to give a short direct proof of Bott periodicity for the representable K-theory of σ-C*-algebras. We further show how the use of this proof and the right definitions simplifies the derivation of the basic properties of representable K-theory.

A classical theorem states that if a polynomial with integral coefficients is an in mth power for every integral value of its argument, then it is the mth power of a polynomial with integral coefficients.

In this paper we deal with analogous problems concerning functions which arise as solutions of recurrence equations with constant coefficients.

A regular semigroup S is said to be locally inverse if each local submonoid eSe, with e an idempotent, is an inverse semigroup. In this paper we apply known covering theorems for inverse semigroups and a covering theorem for locally inverse semigroups due to the author to obtain some covering theorems for locally inverse semigroups. The techniques developed here permit us to give an alternative proof for, and sligbt strengthening of, an important covering theorem for locally inverse semigroups due to F. Pastijn.

We obtain the enumerator by node-valencies of planted plane trees, whose square gives the enumerator of rooted plane trees. We also study the enumeration by number of nodes and black-node-valencies of bichromatic rooted plane trees, encountering a remarkably simple inversion formula. Finally, we remark that these bichromatic trees are in 1–1 correspondence with solutions to the weak lead ballot problem.

It is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

We prove that when $(a, m)= 1$ and $a$ is a quadratic residue $\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$, there are infinitely many Carmichael numbers in the arithmetic progression $a\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$. Indeed the number of them up to $x$ is at least ${x}^{1/ 5} $ when $x$ is large enough (depending on $m$).

Let S be a sub-Markovian semigroup on L2(ℝd) generated by a self-adjoint, second-order, divergence-form, elliptic operator H with W1,∞(ℝd) coefficients ckl, and let Ω be an open subset of ℝd. We prove that if either C∞c(ℝd) is a core of the semigroup generator of the consistent semigroup on Lp(ℝd)for some p∈[1,∞]  or Ω has a locally Lipschitz boundary, then S leaves L2 (Ω)invariant if and only if it is invariant under the flows generated by the vector fields ∑ dl=1ckl∂l for all k. Further, for all p∈[1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.

Using elementary methods we give a new proof of a result concerning the special form of the character of the bounded peripheral element which arises at an end of a curve component of the character variety of a knot complement.