A multiplier rule is given as a necessary optimality condition for proper minimality in set-valued optimisation. We use derivatives in the sense of the lower Dini derivative for the objective set-valued map and the set-valued maps defining the constraints.

We present some new results on third Engel groups which are motivated by computer calculations but are not dependent on them. They include:

• for n > 2 every n-generator third Engel group is nilpotent of class at most 2n – 1;

• the fifth term of the lower central series of a third Engel group has exponent dividing 20;

• the subgroup generated by fifth powers of elements in a third Engel group is nearly centre-by-metabeliami;

and a normal form theorem for freest third Engel groups without elements of order 2.

Bilinear fractal interpolation surfaces were introduced by Ruan and Xu in 2015. In this paper, we present the formula for their box dimension under certain constraint conditions.

Let a, b, c be relatively prime positive integers such that a2 + b2 = c2. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an)x + (bn)y = (en)z in positive integers is x = y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > 1, c = b + 1 and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples (a, b, c) = (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) and (11, 60, 61).

The following system is considered in this paper: The primary goal is to establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) and a forcing term p(t) for all solutions to converge to the origin (0,0) as . Here, the zero solution of the corresponding homogeneous linear system is assumed to be neither uniformly stable nor uniformly attractive. Sufficient conditions are given for asymptotic stability of the zero solution of the nonlinear perturbed system under the assumption that q(t,0,0)=0.

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) where 1<α≤2, η∈(0,1),β∈ℝ=(−∞,+∞), βηα−1≠1, Dα is the Riemann–Liouville differential operator of order α, and f:[0,1]×ℝ→ℝ is continuous, q(t):[0,1]→[0,+∞) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity on f which was essential for the technique used in almost all existed literature.

In this note we give an example of an orthocomplemented modular lattice whose completion by cuts is not orthomodular. This solves negatively Problem 36, p. 131, in G. Birkhoff: Lattice theory (3rd edition).

In this note we shall prove the existence of an uncountable subset of Liouville numbers (which we call the set of ultra-Liouville numbers) for which there exist uncountably many transcendental analytic functions mapping the subset into itself.

New reverses of the Schwarz inequality in inner product spaces that incorporate the classical Klamkin-McLenaghan result for the case of positive n-tuples are given. Applications for Lebesgue integrals are also provided.

Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.

In this paper the structure of finite groups which are the product of two totally permutable subgroups is studied. In fact we can obtain the -residual, where is a formation, -projectors and -normalisers, where is a saturated formation, of the group from the corresponding subgroups of the factor subgroups.

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 a variant of Banach's contraction principle is established. By using the properties of the spectral radius of a bounded linear operator A defined in a suitable Banach space, we conclude that another operator A has exactly one fixed point in this space. In the second part of this paper some applications are given.

The Splitting Theorem says that any given Hamel basis for a (Hausdorff) barrelled space E determines topologically complementary subspaces Ec and ED, and that Ec is flat, that is, contains no proper dense subspace. By use of this device it was shown that if E is non-flat it must contain a dense subspace of codimension at least ℵ0; here we maximally increase the estimate to ℵ1 under the assumption that the dominating cardinal ∂ equals ℵ1 [strictly weaker than the Continuum Hypothesis (CH)]. A related assumption strictly weaker than the Generalised CH allows us to prove that ED is fit, that is, contains a dense subspace whose codimension in ED is dim (ED), the largest imaginable. Thus the two components are extreme opposites, and E itself is fit if and only if dim (ED) ≥ dim (Ec), in which case there is a choice of basis for which ED = E. Morover, E is non-flat (if and) only if the codimension of E′ is at least in E*. These results ensure latitude in the search for certain subspaces of E* transverse to E′, as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.

It is shown that if a variety V of (universal) algebras, defined by a set Σ of identities, is closed under the construction of power algebras then V can be defined by the subset Σ′ of Σ consisting of those identities ν = w of Σ such that every variable in ν = w occurs exactly once on both sides.

In a recent paper (which is to appear in J. Algebra Appl.) we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to ℤ2 and hence the unit sum number of a nonzero right self-injective ring is 2, ω or ∞. In this paper we characterise right self-injective rings with unit sum numbers ω and ∞. We prove that the unit sum number of a right self-injective ring R is ω if and only if R has a factor ring isomorphic to ℤ2 but no factor ring isomorphic to ℤ2 × ℤ2, and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is ∞ precisely when R has a factor ring isomorphic to ℤ2 × ℤ2. We also answer a question of Henriksen (which appeared in J. Algebra, Question E, page 192), by giving a large class of regular right self-injective rings having the unit sum number ω in which not all non-invertible elements are the sum of two units.

This paper extends some fundamental concepts of qualitative matrix analysis from sign pattern classes of real matrices to sign pattern classes of complex matrices. A complex sign pattern and its corresponding sign pattern class are defined in such a way that they generalize the definitions of a (real) sign pattern and its corresponding sign pattern class. A survey of several qualitative results on complex sign patterns is presented. In particular, sign nonsingular complex patterns are investigated. The type of region in the complex plane representing the distribution of the determinants of the matrices in the sign pattern class of a sign nonsingular complex pattern is identified. Cyclically nonnegative complex patterns and complex patterns that are signature similar to nonnegative patterns are characterized. Extensions of sign stable and sign semistable patterns from the real to the complex case are given. Results on ray patterns are also obtained. Finally, many open questions are mentioned.

A classification of two-generator 3-groups of second maximal class and low order is presented. All such groups with orders up to 38 are described, and in some cases with orders up to 310. The classification is based on computer aided computations. A description of the computations and their results are presented, together with an indication of their significance.

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.

1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.