Tight contact structures on solid tori

In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori.

We prove the existence of a -valued and a -valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic.

At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori.

At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.

     

Curvature properties of normal almost contact manifolds with B-metric

Curvature properties of normal almost contact manifolds with B-metric are studied. Relations involving scalar invariants on such manifolds are obtained. Necessary and sufficient conditions for a normal almost contact manifold with B-metric to be of isotropic Kähler-type are given. An example illustrating some of the obtained results is constructed on a Lie algebra.     

Local equivalence relations

In this paper, we investigate the concept of local equivalence relation, a notion suggested by Grothendieck. A local equivalence relation on a topological space X is a global section of the sheaf of germs of equivalence relations on X. We investigate the extent to which a local equivalence relation can be described by a global one and analogously when can a global equivalence relation be recovered from its associated local one. We also look at the notion of a fiber map, which sheds further light on these concepts. A motivating example is that of a foliation on a manifold.     

An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction

The aim of this Note is to propose an example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction in linear elasticity. To cite this article: P. Hild, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Eigenvector method,consistency test and inconsistency repairing for an incomplete fuzzy preference relation

In this paper, we extend the eigenvector method (EM) to priority for an incomplete fuzzy preference relation. We give a reasonable definition of multiplicative consistency for an incomplete fuzzy preference relation. We also give an approach to judge whether an incomplete fuzzy relation is acceptable or not. We develop the acceptable consistency ratio for an incomplete multiplicative fuzzy preference relation, which is simple and similar to Saaty’s consistency ratio (CR) for the multiplicative preference relation. If the incomplete fuzzy preference relation is not of acceptable consistency, we define a criterion to find the unusual and false element (UFE) in the preference relation, and present an algorithm to repair an inconsistent fuzzy preference relation until its consistency is satisfied with the consistency ratio. As a result, our improvement method cannot only satisfy the consistency requirement, but also preserve the initial preference information as much as possible. Finally, an example is illustrated to show that our method is simple, efficiency, and can be performed on computer easily.     

Matrix Riemann-Hilbert problems and factorization on Riemann surfaces

The Wiener-Hopf factorization of 2×2 matrix functions and its close relation to scalar Riemann-Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q₁,Q₂) is defined. To each class C(Q₁,Q₂) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q₁,Q₂) is reduced to solving a scalar Riemann-Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

The strong Brauer group of a cocommutative coalgebra

Using strong equivalences for coalgebras we define the strong Brauer group of a cocommutative coalgebra C, which is a subgroup of the Brauer group of C. In general there is not a good relation between the Brauer group of a coalgebra and the Brauer group of the dual algebra C∗, the former is not even a torsion group. We find that this subgroups embeds in the Brauer group of C∗. A key tool in this result is the use of techniques from torsion theory. Some cases where both subgroups coincide are shown, for example, C being coreflexive.     

Risk averse decision making under catastrophic risk

A nonstandard probabilistic setting for modeling of the risk of catastrophic events is presented. It allows random variables to take on infinitely large negative values with non-zero probability, which correspond to catastrophic consequences unmeasurable in monetary terms, e.g. loss of human lives. Thanks to this extension, the safety-first principle is proved to be consistent with traditional axioms on a preference relation, such as monotonicity, continuity, and risk aversion. Also, a robust preference relation is introduced, and an example of a monotone robust preference relation, sensitive to catastrophic events in the sense of Chichilnisky (2002), is provided. The suggested setting is demonstrated in evaluating nuclear power plant projects when the probability of a catastrophe is itself a random variable.     

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical ? ? ? ^n?1 structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.     

D-Homothetic Transforms of φ-Symmetric Spaces

We present a step towards the solution of an open problem in contact Riemannian geometry: whether there exists an example of non-Sasakian (strongly) locally φ-symmetric spaces other than the so-called (κ, μ)-spaces. The main theorem in the present paper says that if such examples exist, they are not D-homothetic to other locally φ-symmetric contact metric spaces.     

Distributions on almost contact manifolds

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to determine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G₂-structure.     

Lattice valued relations and automata

A lattice-valued relation, lvr for short, from a set X to a set Y is a function from the Cartesian product of X and Y to a lattice. This concept is a generalization of other structures, notably tolerance spaces, nets and automata, separately investigated by the authors elsewhere. It is adequate to admit a natural definition of homogeneity and a classification of homogeneous lvr's by their isomorphism groups. The main result of the present paper is a proof of this classification. The application of this to automata, also interpretable as lvr's, is described, and an example given. We conclude with a brief discussion of the lvr theory of fuzzy and stochastic automata.     

The desirability relation of simple games

We prove some properties of simple games with a complete desirability relation, and investigate the relationships between the desirability of a simple game G and that of some simple games that are derived from G. We also provide an example of a proper simple game that has a complete and acyclic desirability relation but is not a weighted majority game.     

Anisotropic contact processes on homogeneous trees

Sufficient conditions are given for the existence of a weak survival phase in a homogeneous but not necessarily isotropic contact process on a homogeneous tree. These require that the contact process be homogeneous, that is, for any two vertices x,y of the tree there is an automorphism mapping x to y leaving the infection rates invariant; and that the contact process be weakly symmetric, that is, for each vertex there should be at least two incident edges with the same infection rate.     

Singularities in the dynamics of systems with non-ideal constraints

The fundamental problem of dynamics involving determining the generalised accelerations and reactions of constraints as a function of the applied forces is considered for mechanical systems with k ⩾ 1 non-ideal geometrical constraints. A relation is established between this problem and the analysis of the singularities of piecewise-smooth mappings of a space R^k into itself. For Coulomb-type friction, a criterion for there to be no paradoxes is obtained and it is shown that when k = 1 possible singularities are convolutions, while when k = 2 they are subdivided into a fold, a casp and a double fold. The well-known Painlevé-Klein example is considered in detail for cases of bilateral and unilateral contacts; a complete list of possible paradoxical situations is presented for the first time.     

Eigen fuzzy sets and fuzzy relations

When R is a fuzzy relation between the elements of a finite set X, the fuzzy sets A of X such that R ° A = A (MAX-MIN composition) are called eigen fuzzy sets. The main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example. We then state that the greatest eigen fuzzy set associated with $\text{R}\text{?}$, the transitive closure of R, is exactly the one associated with R. Finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set.     

Contact quasiconformal immersions

Contact immersions of contact manifolds endowed with the associated Carnot-Carathéodory (CC) metric (for example, immersions of the Heisenberg group H ³ ～ ? _CC ³ in itself) are considered. It is assumed that the manifolds have the same dimension and the immersions are quasiconformal with respect to the CC metric. The main assertion is as follows: A quasiconformal immersion of the Heisenberg group in itself, just as a quasiconformal immersion of any contact manifold of conformally parabolic type in a simply connected contact manifold, is globally injective; i.e., such an immersion is an embedding, which, in addition, is surjective in the case of the Heisenberg group. Thus, the global homeomorphism theorem, which is well known in the space theory of quasiconformal mappings, also holds in the contact case.     

Some remarks on reachability of infinite-dimensional linear systems

In this paper we shall be concerned with the question of reachability when allowing distribution inputs. We show that a certain class of systems accept distribution inputs, but, in general, they cannot be exactly reachable. We shall also consider the problem of the uniqueness of canonical realizations in relation to exact reachability, and show that Matsuo's result on uniqueness (Ph.D. dissertation, in preparation) does not apply to the example given in Baras, Brockett, and Fuhrmann [IEEE Trans. Automatic ControlAC-19 (1974), 693–700].