The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

This paper treats the L2-discrepancy of digital (0, 1)-sequences over ℤ2, and gives conditions on the generator matrix of such a sequence which guarantee minimal possible order of L2-discrepancy of the generated sequence. The existence is proved for the first time of digital (0; 1)-sequences over ℤ2 with L2-discrepancy of order . This order is best possible by a result of K. Roth. The existence proof is constructive.

Let be a finite-dimensional vector space over a square-root closed ordered field (this restriction permits an inner product with corresponding norm to be imposed on ). Many properties of the family :=() of convex polytopes in can be expressed in terms of valuations (or finitely additive measures). Valuations such as volume, surface area and the Euler characteristic are translation invariant, but others, such as the moment vector and inertia tensor, display a polynomial behaviour under translation. The common framework for such valuations is the polytope (or Minkowski) ring Π:=Π(), and its quotients under various powers of the ideal T of Π which is naturally associated with translations. A central result in the theory is that, in all but one trivial respect, the ring Π/T is actually a graded algebra over . Unfortunately, while the quotients Π/Tk+1 are still graded rings for k > 1, they now only possess a rational algebra structure; to obtain an algebra over , some (weak) continuity assumptions have to be made, although these can be achieved algebraically, by factoring out a further ideal A, the algebra ideal.

We prove that the reflection coefficient of one-dimensional Schrödinger operators with potentials supported on a half-line can be represented in the upper half-plane as the quotient of a contractive analytic function and a properly regularized Blaschke product. We apply this fact to obtain a new trace formula and trace inequality for the reflection coefficient that yields a description of the Weyl m-function of Dirichlet half-line Schrödinger operators with slowly decaying potentials q subject to Among others, we also refine the 3/2-Lieb-Thirring inequality.

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the sum of the pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of ℓq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given of the volume of Zp in terms of the volume of Z1, as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

We establish a new Liouville-type comparison principle for entire weak solutions of quasilinear elliptic partial differential inequalities of the form A(u) ≤ A(v) on Rn, n ≥ 2. Typical examples of the operator A(w) are the p-Laplacian and its well-known modifications for 1 < p ≤ 2.

In this paper we analyse the methodology of the theory of differential inclusions. First, we emphasize that any sequence of piecewise affine functions with successive elements obtained by perturbations of preceding functions in the sets of their affinity converges strongly, together with the gradients. This gives a simple algorithm with which to construct sequences of approximate solutions that converge to exact solutions (neither the specific choice suggested by ‘the method of convex integration for Lipschitz functions’ nor Baire category methodology is required). We then suggest a functional that is defined in the set of admissible functions and measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given function. This functional can be used to prove that the set of stable solutions is dense in the weak topology in the closure of the set of admissible functions either via the Baire category lemma or via a specific choice of strictly convergent sequences.

We explain how the above-mentioned methods of finding solutions to differential inclusions are connected to earlier results on weak–strong convergence, i.e. to results on stability, in the calculus of variations and in differential inclusions.

We also include information on developments in the subject in the three years after the results of this work were obtained.

A tensor-type integral formula for intrinsic volumes is used to define a further variant of directed projection functions and show that these determine a convex body uniquely. Averages of directed projection functions are then studied, and the connections between the resulting operators and previously considered spherical transforms discussed.

We describe Poincaré–Birkhoff–Witt bases for the two-parameter quantum groups U = Ur,s(sln) following Kharchenko and show that the positive part of U has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on U, which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to determine the centre of U.

A modification of the implicit function theorem is advanced for cases where the continuity of the derivative fails. It is applied to a superposition principle for periodicpartial differential equations. The assumption of the principle, that there should exist a non-degenerate solution, is studied and instances of it realized using perturbation arguments and scaling. The positivity of solutions is considered.

In this paper a notion of difference function Δf is introduced for real-valued, non-negative and log-concave functions f defined in Rn. The difference function represents a functional analogue of the difference body K + (−K) of a convex body K. The main result is a sharp inequality which bounds the integral of Δf from above in terms of the integral of f. Equality conditions are characterized. The investigation is extended to an analogous notion of difference function for α-concave functions, with α < 0. In this case also an upper bound for the integral of the α-difference function of f in terms of the integral of f is proved. The bound is sharp in the case α = −∞ and in the one-dimensional case.

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝd, and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝd. If the union B ∪ W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W, provided that A is regular enough (e.g., polyconvex). An analogous formula is obtained for the case when the conditions B ⊂ A and W ⊂ AC are replaced by prescribed threshold volumes of B in A and W in AC. Applications in stochastic geometry are discussed. First, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B. An analogous result holds for the hit-or-miss function. Second, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.

It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

We study topological properties of the correspondence of prime spectra associated with a non-commutative ring homomorphism $R\rightarrow S$. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski closed subsets of $\spec R$ and $\spec S$; these functors arise naturally from restriction and extension of scalars. When $R$ and $S$ are left Noetherian, adjointness occurs only for centralizing and ‘nearly centralizing’ homomorphisms.

We study algebraicity and transcendency of certain basic special values of the double sine functions due to Hölder and Shintani by employing the zeta regularized product expressions.

We solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space $W^{s-1}_2(0,1)$, $s\in[0,1]$. The potential is recovered from two spectra or from one spectrum and the norming constants. Necessary and sufficient conditions for the spectral data to correspond to a potential in $W^{s-1}_2(0,1)$ are established.

We construct moduli curves of polarized supersingular K3 surfaces in characteristic 2 with Artin invariant 2. As an application, we detect a ‘jump’ phenomenon in a family of automorphism groups of supersingular K3 surfaces with a constant Néron–Severi lattice.

Given a left Noetherian ring $R$, we give a necessary and sufficient condition in order that a complex of $R$-modules be DG-injective. Using this result we prove that if $(K_i)_{i\in I}$ is a family of DG-injective complexes of left $R$-modules and $K$ is the $\aleph_1$-product of $(K_i)_{i\in I}$ (i.e. $K\subset\prod_{i\in I}K_i$ is such that, for each $n$, $K^n\subset\prod_{i\in I}K_i^n$ consists of all $(x_i)_{i\in I}$ such that $\{i\mid x_i\neq0\}$ is at most countable), then $K$ is DG-injective.

We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface $S^*$ obtained by deleting a closed set from one original surface $S$. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.

We find positive solutions to a nonlinear equation of Klein–Gordon type. Our analysis is carried out by truncating the related functional and estimating mountain pass solutions by Moser’s iterative scheme.