Rigid cohomology via the tilting equivalence

We define a de Rham cohomology theory for analytic varieties over a valued field $K^{?}$ of equal characteristic p with coefficients in a chosen untilt of the perfection of $K^{?}$ by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field, also in mixed characteristic.     

Deformation cohomology of Lie algebroids and Morita equivalence

Let $A?M$ be a Lie algebroid. In this short note, we prove that a pull-back of A along a fibration with homologically m-connected fibers shares the same deformation cohomology of A up to degree m.     

Orbit equivalence of p-adic transformations and their iterates

Orbit equivalence is a weaker notion of equivalence than isomorphism for measurable systems. We examine \(p\) -adic transformations in various orbit equivalence classes, including an example that preserves an infinite measure. Although \(p\) -adic transformations have been studied with respect to Haar measure, we use other i.i.d. product measures to see examples of different orbit equivalence classes. Since translation by an integer is an iterate of translation by 1, we gain an understanding of the possible behaviors of orbit equivalence classes under iteration.     

Group cohomology and gauge equivalence of some twisted quantum doubles

We study the module category associated to the quantum double of a finite abelian group twisted by a 3-cocycle, which is known to be a braided monoidal category, and investigate the question of when two such categories are equivalent. We base our discussion on an exact sequence which interweaves the ordinary and Eilenberg-Mac Lane cohomology of . Roughly speaking, this reveals that the data provided by such module categories is equivalent to (among other things) a finite quadratic space equipped with a metabolizer, and also a pair of rational lattices with self-dual and integral.

     

Orbit equivalence and permutation groups defined by unordered relations

For a set Ω an unordered relation on Ω is a family R of subsets of Ω. If R is such a relation we let G(R)\mathcal{G}(R) be the group of all permutations on Ω that preserve R, that is g belongs to G(R)\mathcal{G}(R) if and only if x∈R implies x ^g ∈R. We are interested in permutation groups which can be represented as G=G(R)G=\mathcal{G}(R) for a suitable unordered relation R on Ω. When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group ≠Alt(Ω) and of degree ≥11 is a relation group. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprimitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation.     

Cohomology of partially ordered sets and local cohomology of section rings

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley-Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster's decomposition of local cohomology of Stanley-Reisner rings.     

Existential equivalence of ordered abelian groups with parameters

Summary In [GK], Gurevich and Kokorin proved that any two non-trivial ordered abelian groups (o-groups, for short) satisfy the same existential sentences. Let nowG, H be non-trivialo-groups with a commono-subgroupG ₀. We determine whetherG andH are existentially equivalent overG ₀. As a corollary, we obtain algebraic criteria for deciding, whether ano-subgroupG is existentially closed in ano-groupH. Corresponding results are proved foro-groups in which congruences are regarded as atomic relations.     

On the cohomology of incidence algebras of partially ordered sets

We show how to construct a representation of the Lyons sporadic simple group in 651 dimensions over G F(3), which was apparently not previously known. This is the smallest faithful representation in charac-teristic 3.     

Conditions for strong Morita equivalence of partially ordered semigroups

We investigate when a partially ordered semigroup (with various types of local units) is strongly Morita equivalent to a posemigroup from a given class of partially ordered semigroups. Necessary and sufficient conditions for such equivalence are obtained for a series of well-known classes of posemigroups. A number of sufficient conditions for several classes of naturally ordered posemigroups are also provided.     

Strong Morita equivalence for ordered semigroups with local units

We prove that partially ordered semigroups with local units are strongly Morita equivalent if and only if they have a joint enlargement, which in turn happens if and only if the Cauchy completions of the semigroups are equivalent.     

Orbit equivalence of global attractors of semilinear parabolic differential equations

We consider global attractors of dissipative parabolic equations

on the unit interval with Neumann boundary conditions. A permutation is defined by the two orderings of the set of (hyperbolic) equilibrium solutions according to their respective values at the two boundary points and We prove that two global attractors, and , are globally orbit equivalent, if their equilibrium permutations and coincide. In other words, some discrete information on the ordinary differential equation boundary value problem characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.

     

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.      

Orbit equivalence of Cantor minimal systems: A survey and a new proof

We give a new proof of the classification, up to topological orbit equivalence, of minimal AF-equivalence relations and minimal actions of the group of integers on the Cantor set. This proof relies heavily on the structure of AF-equivalence relations and the theory of dimension groups; we give a short survey of these topics.     

Stochastic equivalence of ordered random variables with applications in reliability theory

This paper presents some conditions for stochastic equality of (univariate or multivariate) random variables under various stochastic orderings. The main results provide generalizations of several known results. Applications to stochastic equivalence and characterization problems associated with multivariate NBU (NBUE) distributions are also included.     

Online scheduling of ordered flow shops

We consider online as well as offline scheduling of ordered flow shops with the makespan as objective. In an online flow shop scheduling problem, jobs are revealed to a decisionmaker one by one going down a list. When a job is revealed to the decision maker, its operations have to be scheduled irrevocably without having any information regarding jobs that will be revealed afterwards. We consider for the online setting the so-called Greedy Algorithm which generates permutation schedules in which the jobs on the machines are at all times processed without any unnecessary delays. We focus on ordered flow shops, in particular proportionate flow shops with different speeds and proportionate flow shops with different setup times. We analyze the competitive ratio of the Greedy Algorithm for such flow shops in the online setting. For several cases, we derive lower bounds on the competitive ratios.     

A certain synchronizing property of subshifts and flow equivalence

We will study a certain synchronizing property of subshifts called λ-synchronization. The λ-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the λ-synchronization is invariant under flow equivalence of subshifts. The λ-synchronizing K-groups and the λ-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of λ-synchronizing subshifts. They are new flow equivalence invariants for λ-synchronizing subshifts.     

Motivic cohomology and unramified cohomology of quadrics

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension ≤4 and ≥11. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics. For most of the paper we have to assume that the ground field has characteristic 0, because we use Voevodsky’s motivic cohomology. Received August 18, 1999 / final version received December 10, 1999?Published online April 19, 2000     

Singular gradient flow of the distance function and homotopy equivalence

Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$ . It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$ , from the boundary of $\varOmega $ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $ .     

Algorithm for verifying the equivalence of linear unary recursive programs on ordered semigroup scales

A new method for solving the problem of equivalence of linear unary recursive programs is proposed. The main idea behind this method is to reduce the equivalence problem to known problems on graphs and to problems in group theory. A class of program semantics is singled out for which the problem of the equivalence of the programs in question is solvable in polynomial time by the proposed method.