A two-level defect–correction method for the steady-state Navier–Stokes equations with a high Reynolds number is considered in this paper. The defect step is accomplished in a coarse-level subspace Hm by solving the standard Galerkin equation with an artificial viscosity parameter σ as a stability factor, and the correction step is performed in a fine-level subspace HM by solving a linear equation. H1 error estimates are derived for this two-level defect–correction method. Moreover, some numerical examples are presented to show that the two-level defect–correction method can reach the same accuracy as the standard Galerkin method in fine-level subspace HM. However, the two-level method will involve much less work than the one-level method.

We prove sharp estimates for the dilation operator f(x)⟼f(λx), when acting on Wiener amalgam spaces W(Lp,Lq). Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces Mp,q, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.

In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

It is shown that the set of chaotic self-maps of the unit interval contains an open dense subset of the space of all continuous self-maps of the unit interval. Other aspects of chaotic behaviour are also considered together with some illustrative examples.

A new proof is given of the following result: Let m and d be positive integers, and let a set of md + m − d points be given in d-dimensional space. Then the set can be partitioned into m sets such that the m convex polytopes spanned by the sets have a non-empty intersection.

Usual duality results are proved for Wolfe and Mond-Weir type multiobjective symmetric dual problems without nonnegativity constraints under invexity/generalised invexity assumptions. Moreover, assuming the kernel function to be skew symmetric, the multiobjective problems are exhibited to be self duals.

A continuous function mapping a compact interval of the real line into itself is called chaotic if the difference equation defined in terms of it behaves chaotically in the sense of Li and Yorke. The set of all such chaotic functions is shown to toe a dense subset of the space of continuous mappings of that interval into itself with the max norm. This result indicates the structural instability of nonchaotic difference equations with respect to chaotic behaviour.

A ring is called a commutator ring if every element is a sum of additive commutators. In this note we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a nonempty set Ω, EndR(⌖ΩN) and EndR(ΠΩN) are commutator rings if and only if either Ω is infinite or EndR(N) is itself a commutator ring. We also prove that over any ring, a matrix having trace zero can be expressed as a sum of two commutators.

A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is minimal if no other cover contains fewer members. We term minimised a minimal cover with the property that substituting for a member of the cover by a proper subgroup of that member produces a collection which is no longer a cover. We here describe the minimised covers for the groups GL2 (q), SL2 (q), PSL2 (q) and PGL2(q).

The aim of this paper is to extend a recent result of S.A. Husain and V.M. Sehgal [Bull. Austral. Math. Soc. 13 (1975), 261–267]. The condition that the function should be continuous in Husain and Sehgal, op. cit., is replaced by a semicontinuity condition. Moreover, a different proof is given, the last part of which requires no continuity or semicontinuity condition whatsoever.

The radial distribution of Julia sets and non-existence of unbounded Fatou components of transcendental meromorphic functions are investigated in this paper.

Let n, s, t be integers with s > t > 1 and n > (t+2)2S−t−l We prove that if n subsets of a set S with s elements have union S then some t of them have union S. The result is best possible.

The existence of metrizable (LF)-spaces was announced by Stephen A. Saxon (“Metrizable generalized (LF)-spaces”, 701–46–14), in Notices Amer. Math. Soc. 20 (1973), A–143. Elsewhere, the authors have discovered an abundant existence of metrizable and normable (generalized) (LF)-spaces, while observing that an (LF)-space is metrizable if and only if it is Baire-like. Recently, W. Robertson, I. Tweddle and F.E. Yeomans introduced the class of locally convex spaces E having the property

(db) if E is the union of an increasing sequence (En) of vector subspaces, then some En is dense and barrelled.

We give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

In this paper we extend the concept of invexity to set-valued maps and study vector optimisation problems with invex set-valued data. Necessary and sufficient optimality conditions are established in terms of contingent derivatives. Wolfe type dual problems are constructed via two recently developed approaches which guarantee the zero-gap duality property.

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We givn an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

We give a construction for Hadamard arrays and exhibit the arrays of orders 4t, t ∈ {1, 3, 5, 7, …, 19}. This gives seventeen new Hadamard matrices of order less than 4000.

Let B0 ⊂ B1 ⊂ Bn ⊂ … ⊂: Bw be all the non-trivial varieties of distributive pseudocomplemented lattices (L; ∨, ∧, *, 0, 1) considered as algebras of type (2, 2, 1, 0, 0). A subset J of such an algebra L is a congruence-kernel if and only if it is a lattice-ideal and x** ∈ J for each x ∈ J. The smallest congruence having J as its kernel is Θ(J), where a ≡ b (Θ(J)), (a, b ∈ L) if and only if a ∧ c* = b ∧ c* for some c ∈ J. For given 0 ≤ n ≤ w, let Σn (J) be the smallest congruence having J as its kernel and such that the associated quotient algebra is in Bn. Of course, Σw (J) = Θ(J) and the main result of this paper shows that for 1 ≤ n < w, Σn(J) = ∩{Θ(P1 ∩ P2 ∩ … ∩ Pn): J ⊆ P1, P2, …, Pn ⊂ L are minimal prime ideals}.

Lots of formulas for series of zeta function have been developed in many ways. We show how we can apply the theory of the double gamma function, which has recently been revived according to the study of determinants of Laplacians, to evaluate some series involving the Riemann zeta function.

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.