We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

It is known that there are no real hypersurfaces with parallel structure Jacobi operators in a nonflat complex space form. In this paper, we classify real hypersurfaces in a nonflat complex space form whose structure Jacobi operator is cyclic-parallel.

It is well known that an exponential real function, which is Lebesgue measurable (Baire measurable, respectively) or bounded on a set of positive Lebesgue measure (of the second category with the Baire property, respectively), is continuous. Here we consider bounded on ‘big’ set solutions of an equation generalizing the exponential equation as well as the Goła̧b–Schinzel equation. Moreover, we unify results into a more general and abstract case.

In this paper, using a scalarization method, we obtain sufficient conditions for the lower semicontinuity and continuity of the solution mapping to a parametric generalized weak vector equilibrium problem with set-valued mappings.

A hermitian algebra is a unital associative ℂ-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ℝ. In the case of an algebra 𝒜 endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra 𝒜 is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups Un(𝒜) on each flag manifold is transitive.

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.

If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, have between them found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs of universality.

For three points , and in the n-dimensional space 𝔽nq over the finite field 𝔽q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points define acute angle triangles. A similar question in the real space ℛn dates back to P. Erdős and has been studied by several authors.

In this paper, some existence and uniqueness results for generalized solutions to a periodic-Dirichlet problem for semilinear wave equations are given, using a global inverse function theorem. These results extend those known in the literature.

Using potential theory, we prove the existence of filling disks of an algebroid function of finite order defined in the unit disk.

In this paper, compositions of a natural number are studied. The number of restricted compositions is given in a closed form, and some applications are presented.

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

In this paper, we give an analogue of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jørgensen’s inequality in the two-dimensional Möbius group of the above case.

In this paper we discuss the simplicity criteria of (−1,−1)-Freudenthal Kantor triple systems and give examples of such triple systems, from which we can construct some Lie superalgebras. We also show that we can associate a Jordan triple system to any (ε,δ)-Freudenthal Kantor triple system. Further, we introduce the notion of δ-structurable algebras and connect them to (−1,δ)-Freudenthal Kantor triple systems and the corresponding Lie (super)algebra construction.

A link L in S3 possibly admits an involution of the exterior E(L) with fixed point set a closed surface, which is not extendable to an involution of S3. In this paper, we focus on the case of graph links and show that the genus of the surface provides a lower estimate of the number of link components.

Let G be a graph, and k a positive integer. Let h:E(G)→[0,1] be a function. If ∑ e∋xh(e)=k holds for each x∈V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G)∣h(e)>0}. In this paper we use neighbourhoods to obtain a new sufficient condition for a graph to have a fractional k-factor. Furthermore, this result is shown to be best possible in some sense.

We find necessary and sufficient conditions for a Legendre curve in a Sasakian manifold to have: (i) a pseudo-Hermitian parallel mean curvature vector field; (ii) a pseudo-Hermitian proper mean curvature vector field in the normal bundle.

In this paper, four new discreteness criteria for isometric groups on complex hyperbolic spaces are proved, one of which shows that the Condition C hypothesis in Cao [‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc.78 (2008), 211–224, Theorem 1.4] is removable; another shows that the parabolic condition hypothesis in Li and Wang [‘Discreteness criteria for Möbius groups acting on II’, Bull. Aust. Math. Soc.80 (2009), 275–290, Theorem 3.1] is not necessary.

A group G is called morphic if every endomorphism α:G→G for which Gα◃G satisfies G/Gα≅ker (α). Call an endomorphism α∈end(G) regular if αβα=α for some β∈end(G), and call α unit regular if β can be chosen to be an automorphism of G. The main purpose of this paper is to prove the following group-theoretic analogue of a theorem of Ehrlich: if G is a morphic group, an endomorphism α:G→G for which Gα◃G is unit regular if and only if it is regular. As an application, a cancellation theorem is proved that characterizes the morphic groups among those with regular endomorphism monoids.

Browkin [‘Some new kinds of pseudoprimes’, Math. Comp. 73 (2004), 1031–1037] gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprimes to two bases only, where p=2 or 3. In contrast to Browkin’s examples, Zhang [‘Notes on some new kinds of pseudoprimes’, Math. Comp. 75 (2006), 451–460] gave facts and examples which are unfavorable for Browkin’s observation on detecting compositeness of odd composite numbers. In particular, Zhang gave a Sylowp-pseudoprime (with 27 decimal digits) to the first 6 prime bases for all the first 6 primes p, and conjectured that for any k≥1, there would exist Sylow p-pseudoprimes to the first k prime bases for all the first k primes p. In this paper we tabulate 27 Sylow p-pseudoprimes less than 1036 to the first 7 prime bases for all the first 7 primes p (two of which are Sylow p-pseudoprimes to the first 7 prime bases for all the first 8 primes p). We describe the procedure for finding these numbers. The main tools used in our method are the cubic residue characters and the Chinese remainder theorem.