In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author.

We use an intermediate value theorem for quasi-monotone increasing functions to prove the existence of the smallest and the greatest solution of the Dirichlet problem u″ + f(t, u) = 0, u(0) = α, u(1) = β between lower and upper solutions, where f:[0,1] × E → E is quasi-monotone increasing in its second variable with respect to a regular cone.

Let be an arbitrary strictly increasing infinite sequence of positive integers. For an integer n≥1, let . Let r>0 be a real number and q≥ 1 a given integer. Let be the eigenvalues of the reciprocal power LCM matrix having the reciprocal power of the least common multiple of xi and xj as its i, j-entry. We show that the sequence converges and . We show that the sequence converges if and . We show also that if r> 1, then the sequence converges and , where t and l are given positive integers such that t≤l−1.

In this note, we find all the solutions of the Diophantine equation x2 + 5a 13b = yn in positive integers x, y, a, b, n≥ 3 with x and y coprime.

The Arens products are the standard way of extending the product from a Banach algebra to its bidual ″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

Let and let us consider a del Pezzo surface of degree one given by the equation . In this paper we prove that if the set of rational points on the curve Ea,b : Y2 = X3 + 135(2a−15)X−1350(5a + 2b − 26) is infinite then the set of rational points on the surface ϵf is dense in the Zariski topology.

In this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Brézis and Nirenberg.

We consider a linear integrodifferential equation of second order in a Hilbert space and show that the solution tends to zero polynomially if the decay of the convolution kernel is polynomial. Both polynomials are of the same order.

The approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock–Kurzweil–Pettis and a Denjoy–Khintchine–Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock–Kurzweil–Pettis and Denjoy–Khintchine–Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.

Let Mm be a closed smooth manifold with an involution having fixed point set of the form Fn ∪ F2, where Fn and F2 are submanifolds with dimensions n and 2, respectively, where n ≥ 4 is even (n < m). Suppose that the normal bundle of F2 in Mm, μ → F2, does not bound, and denote by β the stable cobordism class of μ → F2. In this paper, we determine the upper bound for m in terms of the pair (n, β) for many such pairs. The similar question for n odd (n ≥ 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of Boardman, which establishes that, under the above hypotheses, m ≤ 5/2n.