In this paper, we show that if q(x) satisfies suitable conditions, then the Neumann problem -Δu+u = q(x)ⅠuⅠp−2u in Ω has at least two solutions of which one is positive and the other changes sign.

A necessary and sufficient condition for the existence of a supermanifold structure on a quotient defined by an equivalence relation is established. Furthermore, we show that an equivalence relation R on a Berezin-Leĭtes-Kostant supermanifold X determines a quotient supermanifold X/R if and only if the restriction R0 of R to the underlying smooth manifold X0 of X determines a quotient smooth manifold X0/R0.

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

In a Minkowski plane with unit circle E, the product of the positive circumference of a plane convex body K and that of its polar dual is greater than or equal to the square of the Euclidean length of the polar dual of E. Equality holds if and only if K is a Euclidean unit circle.

The concept of a tetradiffric function is introduced. This new scheme for defining discrete analytic functions is shown to retain the algebraic simplicity of monodiffric functions, while introducing to the theory a symmetry similar to the Schwarz Reflection Principle.

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

I discussed the sort of thing Bernhard had in mind concerning the BFC-problem in [3], an article written for his sixieth birthday. This vigesennial contribution deals with one of the “and others”, namely the zero deficiency problem. in his article [1] bearing the same title as this one, Bernhard gave 2-generator 2-relator presentations of some metacyclic groups; if one distils and formalises the process used there, one gets the following result, to be used later in constructing some zero-deficiency finite p-groups.

In this paper we give an extension of a curious combinatorial identity due to B. Sury. Our proof is very simple and elementary. As an application, we obtain two congruences for Fermat quotients modulo p3. Moreover, we prove an extension of a result by H. Pan that generalizes Carlitz’s congruence.

The classification of strongly subdirectly irreducible rings, which was begun in a previous paper, is completed by showing that a mixed group G is strongly subdirectly irreducible if and only if G ⋍ Z(p∞) ⊕ H, H a rank one, p-divisible, torsion free nil group.

Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction.

In this short note we present a common characterisation of the logarithmic function and the subspace of all trace zero elements in finite von Neumann factors.

The question whether there exist non-free projective groups of rank r in the variety has been answered in the affirmative for n ≥ 2, r ≥ 2, except for n = r = 2, by V.A. Artamonov. This paper consists in a proof that a projective group G of rank 2 in is free. If x and y are any two elements which generate G modulo , then the group F generated by x and y is free in , and the index of F in G is finite and not divisible by 2. One wishes to replace x by xu and y by yν, where u and vν lie in , so that 〈xu, yν〉 is the whole of G. This can be done: first, on general grounds, it is sufficient that 〈xu, yν〉 contain every C(a), where C(a) is the centralizer in the G/-module of an element a in (and moreover choices of u and ν for each C(a) can be combined to give a single choice good for all C(a)); second, for the particular small numbers involved, the structure of C(a) is sufficiently simple for one to pick suitable u and ν without trouble.

It is well known that an orientation-preserving homeomorphism of the plane without fixed points has trivial dynamics; that is, its non-wandering set is empty and all the orbits diverge to infinity. However, orbits can diverge to infinity in many different ways (or not) giving rise to fundamental regions of divergence. Such a map is topologically equivalent to a plane translation if and only if it has only one fundamental region. We consider the conservative, orientation-preserving and fixed point free Hénon map and prove that it has only one fundamental region of divergence. Actually, we prove that there exists an area-preserving homeomorphism of the plane that conjugates this Hénon map to a translation.

We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

Let a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the Diophantine equation has at most two positive integer solutions (m,n) satisfying min (m,n)>1.

An algorithm which has been developed to solve the problem of determining an optimal path of the hand of a robot is applied to various classical problems in the calculus of variations.

We present a characterisation of Banach spaces possessing the weak Radon–Nikodym property in terms of finitely additive interval functions whose McShane variational measures are absolutely continuous with respect to Lebesgue measure.

A Stieltjes-Volterra integral equation system

is firstly considered. Pointwise estimates and boundedness of its solutions are obtained under various conditions on the function K. To do this, the well-known Gronwall-Bellman integral inequality is generalized. For a particular choice of u, it is shown that the integral equation reduces to a difference equation. The problem of existence (and non-existence), uniqueness (and non-uniqueness) of the difference equation is discussed. Gronwall-Bellman inequality is further generalized to n linear terms and is subsequently applied to obtain sufficient conditions in order that a certain stability of the unperturbed Volterra system

implies the corresponding local stability of the (discontinuously) perturbed system