By introducing the function 1/(min{x, y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case.

A method of embedding nk configurations into projective space of k–1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a “complementary” “theorem” about projective space (over a field or skew-field F) from any nk theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues 103 (also in 3d-space), Möbius 84 (in 3d-space), and the resulting 74 in 3d-space, 96 in 5d-space, and 107 in 6d-space. (The Möbius configuration is self-complementary.) There are some nk configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.

In this paper, we study two-point boundary value problems for the nonlinear second order difference equation We establish the relationship between the number of sign-variation of f on {0,…, T + 2} and the one of the solution u of the above problem.

It is shown that a spacelike maximal surface in the three dimensional Lorentz-Minkowski space can be extended analytically if it meets a spacelike plane at a constant hyperbolic angle.

Let G be a locally compact group and be the Banach space of all essentially bounded measurable functions on G vansihing an infinity. Here, we study some families of right completely continuous elements in the Banach algebra equipped with an Arens type product. As the main result, we show that has a certain right completely continuous element if and only if G is compact.

The S-rank (where ‘S’ abbreviates ‘sandwich’) of a right congruence ρ on a semigroup S is the Cantor-Bendixson rank of ρ in the lattice of right congruences ℛ of S with respect to a topology we call the finite type topology. If every ρ ϵ ℛ possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup 0(G, I) is ranked if and only if G is ranked and I is finite.

We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) ≅ E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic -class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.

Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

Some new integral conditions characterising the embedding ∧p(v) ↪ Γq(w), 0 < p, q ≤ ∞ are presented, including proofs also for the cases (i) p = ∞, 0 < q < ∞, (ii) q = ∞, I < p < ∞ and (iii) p = q = ∞. Only one condition is necessary for each case which means that our conditions are different from and simpler than other corresponding conditions in the literature. We even prove our results in a more general frame namely when the space Γq(w) is replaced by the more general space . In our proof we use a technique of discretisation and anti-discretisation developed by A. Gogatishvili and L. Pick, where they considered the opposite embedding.

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed modulo arbitrary polynomials.

Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.

Let p be a prime number ≥ 5, and n a positive integer > 1. This note is concerned with the diophantine equation x4 − y4 = nzp. We prove that, under certain conditions on n, this equation has no non-trivial solution in Z if p ≥ C(n), where C(n) is an effective constant.

Nelsen [J. Algebra 298 (2006) 134–141] asked whether there is a natural class of McCoy rings which includes all reversible rings and all rings R such that R[X] is semi-commutative. In this paper, some new equivalent conditions of McCoy rings are given. One of them is used to answer this question in the affirmative. Finally, an example is given which is McCoy and semi-commutative, but it is not reversible and does not have the property that R[x] is semi-commutative.

We show that linear operators from a Banach space into itself which satisfy some relaxed strong accretivity conditions are invertible. Moreover, we characterise a particular class of such operators in the Hilbert space case. By doing so we manage to answer a problem posed by B. Ricceri, concerning a linear second order partial differential operator.

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.