Volume 25, Issue 2

Abstract. We construct a non-Lipschitz flow in H s for the cubic nonlinear Schrödinger equation on the 2-torus of revolution with a Lipschitz or smooth metric. The non-Lipschitz property holds for all for a Lipschitz metric and for a smooth metric. Both coincide with the Sobolev exponents for uniform local well-posedness.

Abstract. Let be an eigenfunction with respect to the Dirichlet Laplacian on a compact Riemannian manifold N with boundary: in the interior of N and on the boundary of N . We show the following gradient estimate on : for every , there holds , where C is a positive constant depending only on N . In the proof, we use a basic geometrical property of nodal sets of eigenfunctions and elliptic a priori estimates.

Abstract. In 2001, J. Lin detected a non-trivial element in the stable homotopy group of the sphere spectrum S for at the prime greater than three. In the stable homotopy group of the Smith–Toda spectrum , X. Liu constructed an essential element for at the prime greater than three. Let denote the Spanier–Whitehead dual of the generator , which defines the -element . Let . In this paper, we show that the composite maps and are non-trivial, where and . In the Adams–Novikov spectral sequence, the maps and are represented by and , respectively. Here denotes the well-known element in .

Abstract. We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of -modules. Then entropy will be viewed as a function on modules over the polynomial ring extending L . In this framework we obtain the main results of this paper, namely that under some mild conditions the induced entropy is additive, thus entropy becomes an operator from the length functions on R -modules to length functions on -modules. Furthermore, if one requires that the induced length function satisfies two very natural conditions, then this process is uniquely determined. When R is Noetherian, we will deduce that, in this setting, entropy coincides with the multiplicity symbol as conjectured by the second named author. As an application we show that if R is also commutative, the L -entropy of the right Bernoulli shift on the infinite direct product of a module of finite positive length has value , generalizing a result proved for Abelian groups by A. Giordano Bruno.

Abstract. The paper investigates a significant part of the automorphic, in fact of the so-called Eisenstein cohomology of split odd orthogonal groups over . The main result provides a description of residual and regular Eisenstein cohomology classes for maximal parabolic -subgroups in case of generic cohomological cuspidal automorphic representations of their Levi subgroups. That is, such identifying necessary conditions on these latter representations as well as on the complex parameters in order for the associated Eisenstein series to possibly yield non-trivial classes in the automorphic cohomology.

Abstract. Let be a 4-dimensional almost-Hermitian manifold which satisfies the Kähler identity. We show that is complex Osserman if and only if has constant holomorphic sectional curvature. We also classify in arbitrary dimensions all the complex Osserman Kähler models which do not have three eigenvalues.

Abstract. Let S be a finitely generated standard multigraded algebra over an Artinian local ring A , and let M be a finitely generated multigraded S -module. This paper answers the question when mixed multiplicities of M are positive and characterizes them in terms of lengths of A -modules. As an application, we get interesting results on mixed multiplicities of ideals, and recover some early results.

Abstract. This article studies chain transitivity for semigroup actions on fiber bundles whose typical fibers are compact topological spaces. We discuss the number of maximal chain transitive sets and, as a consequence, we obtain conditions for the existence of a finest Morse decomposition. Some of the results obtained are applied to orthonormal frame and Stiefel manifolds. A description of the maximal chain transitive sets is provided in terms of the action of shadowing semigroups. This description is well known in the literature under the hypothesis of local transitivity. Here, we exclude the hypothesis of local transitivity when the state space is a compact quotient space of a topological group.

Abstract. Several improvements are made to the 2-primary homotopy theory of the double suspension. We refine Richter's factorization of the second-power map on through the loops on the double suspension, show that the 4 th -power map on  factors through the double suspension, and that the 4 th -power map on is null homotopic, where is the fiber of the double suspension. The latter two results are improvements by one loop.

Abstract. In his foundation of probability theory, Bruno de Finetti devised a betting scheme where a bookmaker offers bets on the outcome of events occurring in the future. He introduced a criterion for coherent bookmaking, and showed that coherent betting odds are given by some probability distribution. While de Finetti dealt with yes-no events and boolean propositional logic, Mundici generalized the theory to the continuous spectrum events formalized within Łukasiewicz logic. Both de Finetti and Mundici assume that the bookmaker/bettor roles can be interchanged. In this paper we deal with a more realistic situation, dropping the reversibility assumption. Working in the framework of Łukasiewicz logic, we introduce a coherence criterion for non-reversible bookmaking. Our main tool is given by `imprecise probabilities', which are formulated in terms either of compact convex sets of probabilities or equivalently in terms of suitable sublinear functionals. Our main result states that our coherence criterion arises from imprecise probabilities just as de Finetti's criterion arises from probabilities. Throughout, we will work with MV-algebras. They play the same role for Łukasiewicz logic as Boolean algebras play for classical logic. Unital abelian lattice-ordered groups will provide an intermediate structure: while being categorically equivalent to MV-algebras, they are more akin to the Banach space . Functional analytic methods are used for the proof of our main result.