The main aim of this article is to establish analogues of Landau’s theorem for solutions to the $\overline{\unicode[STIX]{x2202}}$-equation in Dirichlet-type spaces.

The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$-pure subgroups of the additive group of the ring of $p$-adic integers are investigated using only elementary methods.

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.

A proof is given that the improper Riemann integral of ζ(s, a) with respect to the real parameter a, taken over the interval (0, 1], vanishes for all complex s with ℜ(s) < 1. The integral does not exist (as a finite real number) when ℜ(s) ≥ 1.

We give a straightforward proof that divergence-form elliptic operators of order m on a d-dimensional Lie group with m ≥ d have Hölder continuous kernels satisfying Gaussian bounds.

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

The authors prove two theorems. The first theorem generalizes theorems due to T. Singh and O.P. Varshney, concerning absolute Nörlund summability of Fourier series of functions of bounded variation. The second theorem generalizes theorems of L.S. Bosanquet and H.P. Dikshit.

The concept of uniformly strongly prime (usp) is introduced for Γ-ring, and a usp radical τ(M) is defined for a Γ-ring M. If M has left and right unities, then τ(L)+ = τ(M) = τ(R)*, where L and R denote, respectively, the left and right operator rings of M, and τ(·) denotes the usp radical of a ring. If m, n are positive integers, then τ(Mmn) = (τ(M))mn, where Mmn is the matrix Γnm-ring. τ is shown to be a special radical in the variety of Γ-rings. τ1 is the upper radical determined by the class of usp Γ-rings of bound 1. τ ⊆ τ1, but the reverse inclusion does not hold in general. The place of τ and τ1 in the hierarchy of radicals for Γ-rings is shown.

In this paper, we show that the $\gamma $-vectors of Coxeter complexes (of types A and B) and associahedrons (of types A and B) can be obtained by using derivative polynomials of the tangent and secant functions. We provide a unified grammatical approach to generate these $\gamma $-vectors and the coefficient arrays of Narayana polynomials, Legendre polynomials and Chebyshev polynomials of both kinds.

We prove a characterisation, in terms of regularity and cancellation conditions, of the convolution kernels of Marcinkiewicz multiplier operators m (𝔏1,…,𝔏n, iT) on the Heisenberg group ℍn, where 𝔏1,…,𝔏n are the n partial sub-Laplacians. The necessity of these regularity and cancellation conditions was established by Fraser (2001); here, we prove their sufficiency.

We establish some supercongruences for the truncated $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series involving the $p$-adic gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated $_{3}F_{2}$ hypergeometric series. Related supercongruences modulo $p^{3}$ are proposed as conjectures.

We prove ${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is ${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of ${\it\delta}$ on ${\it\epsilon}$ within the class of uniformly convex hypersurfaces with bounded volume.

Infinite families of amicable orthogonal designs are constructed with

(i) both of type (1, q) in order q + 1 when q ≡ 3 (mod 4) is a prime power,

(ii) both of type (1, q) in order 2(q + 1) where q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix,

(iii) both of type (2, 2q) in order 2(q + 1) when q ≡ 1 (mod 4) is a prime power or q + 1 is the order of a conference matrix.

A theorem on Nemitsky operators between Sobolev spaces, contained in a 1973 paper of Marcus and Mizel, is extended to Orlicz-Sobolev spaces.

A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

In this paper we show that (i) l∞ does not admit an equivalent weak mid-point locally uniformly rotund norm and (ii) l∞ / c0 does not admit an equivalent rotund norm.

We prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.

This note exhibits a family of m – (ν, k, λ) supplementary difference sets with parameters ν = 2mf + 1, where ν = p is a prime and f is odd, k = mf and λ = m(mf−1)/2. These sets are composed of a union of cosets of 2m-th power residues of p.

In this paper, we deal with the multiplicity of solutions for a fourth-order impulsive differential equation with a parameter. Using variational methods and a ‘three critical points’ theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. An example is also given in order to illustrate the main results.

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:

(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all s≥r. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:

(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all s ≥ r whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).