We study real-valued valuations on the space of Lipschitz functions over Euclidean unit sphere S n ? 1 . After introducing an appropriate notion convergence, we show that continuous are bounded sets which with respect to norm. This fact, in combination measure theoretical arguments, will yield integral representation for and rotation invariant 1-dimensional sphere.

The notion of generalized locally bounded $I$-topological vectorspaces is introduced. Some of their important properties arestudied. The relationship between this kind of spaces and thelocally bounded $I$-topological vector spaces introduced by Wu andFang [Boundedness and locally bounded fuzzy topological vectorspaces, Fuzzy Math. 5 (4) (1985) 87$-$94] is discussed. Moreover, wealso use the fam...

Abstract. For the standard map the homotopically non-trivial invariant curves of rotation number ω satisfying the Bryuno condition are shown to be analytic in the perturbative parameter ε, provided |ε| is small enough. The radius of convergence ρ(ω) of the Lindstedt series – sometimes called critical function of the standard map – is studied and the relation with the Bryuno function B(ω) is der...

We define the notion of rotations on infinite binary trees, and construct an irrational tree rotation with bounded distortion. This lifts naturally to a Lipschitz circle homeomorphism having the middle-thirds Cantor set as its minimal set. This degree of smoothness is best possible, since it is known that no C1 circle diffeomorphism can have a linearly self-similar Cantor set as its minimal set.

We construct embedded ancient solutions to mean curvature flow related certain classes of unstable minimal hypersurfaces in \({\mathbb {R}}^{n+1}\) for \(n \ge 2\). These provide examples convex yet nonconvex that are not solitons, meaning they do evolve by rigid motions or homotheties. Moreover, we eternal \(O(n)\times O(1)\)-invariant, and nonconvex. They out the catenoid rotation a profile c...

We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely α-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based ...

We construct a well-defined regularized path integral for Lorentzian quantum gravity in terms of dynamically triangulated causal space-times. Each Lorentzian geometry and its action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time and, for finite lattice volume, the associated transfer matrix is self-adjoint, ...

First we prove an exponential decay result for solutions of the equations of linear, homogeneous, isotropic thermoelasticity in bounded regions in two or three space dimensions if the rotation of the displacement vanishes. As a consequence, we describe the decay in radially symmetrical situations, and in a cylinder in R3. Then we establish the global existence of solutions to the corresponding ...