On the notion of measurability

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I wish to record my thanks to Dr. Kummer for his constant encouragement and many helpful suggestions in preparing this paper.     

On the notion of derivo-periodicity

A De Blasi-like differentiable multivalued function is shown to have a periodic derivative (i.e., to be derivo-periodic) if and only if it is a sum of a function of a continuous (single-valued) periodic function, linear function and a bounded interval (a multivalued constant). At the same time, the single-valued part is derivo-periodic a.e. in the usual sense. In the single-valued case, a characterization of a more general class of derivo-periodic ACG_∗-functions is given. Derivo-periodicity in terms of the Clarke subdifferentials and an impossibility of an almost-periodic analogy are also discussed. The obtained results are finally applied to differential equations and inclusions.     

The dynamics of perturbations of the contracting Lorenz attractor

We show here that by modifying the eigenvalues ₂ < ₃ < 0 < ₁ of the geometric Lorenz attractor, replacing the usualexpanding condition ₃+₁ > 0 by acontracting condition ₃+₁ < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.     

On the notion of infinite Hamiltonian graph

Based on the terms “end” and “cofinal spanning subtree” a general notion of Hamiltonicity of infinite graphs is developed. It is shown that the cube of every connected locally finite graph is Hamiltonian in this generalized sense.     

On the notion of perfect Bayesian equilibrium

Often, perfect Bayesian equilibrium is loosely defined by stating that players should be sequentially rational given some beliefs in which Bayes rule is applied “whenever possible.” We argue that there are situations in which it is not clear what “whenever possible” means. Then we provide an elementary definition of perfect Bayesian equilibrium for general extensive games that refines both weak perfect Bayesian equilibrium and subgame perfect equilibrium.     

On the Hencl's notion of absolute continuity

We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in , J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].     

On the notion of space in geometry

The development of the notion of space in geometry is traced from the early axiomatization in Euclid’s Elements over the discovery of non-Euclidean geometries to geometry of manifolds in relativity theory and in gauge and string theories in contemporary physics. The notion of space is considered in a historic-philosophical perspective including a short discussion of the contributions of artists to visualization of spatial objects.     

On the practical value of the notion ofBN-stability

The oldest concept of unconditional stability of numerical integration methods for ordinary differential systems is that ofA-stability. This concept is related to linear systems having constant coefficients and has been introduced by Dahlquist in 1963. More recently, since another contribution of Dahlquist in 1975, there has been much interest in unconditional stability properties of numerical integration methods when applied to non-linear dissipative systems (G-stability,BN-stability,A-contractivity). Various classes of implicit Runge-Kutta methods have already been shown to beBN-stable. However, contrary to the property ofA-stability, when implementing such a method for practical use this unconditional stability property may be lost. The present note clarifies this for a class of diagonally implicit methods and shows at the same time that Rosenbrock's method is notBN-stable.     

On generalizations of the notion of a frame

Some approaches to generalizations of the notion of a frame are discussed. Spatial measurable frames with one-dimensional components are considered in more detail. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 121–131     

On the notion of coexistence in quantum mechanics

The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of observables and of the statistical and individual aspects of quantum measurements.     

Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor

In this paper, a new Lorenz-like chaotic system is reported. Nonlinear characteristic and basic dynamic properties of the three-dimensional autonomous system are studied by means of nonlinear dynamics theory. The chaotic system is not only demonstrated by theoretical analysis and numerical simulation, such as equilibria stability analysis, Lyapunov-exponent spectra, Lyapunov dimension, bifurcation diagrams, but also implemented via an electronic circuit.     

On the notion of bimodel for functorial semantics

We discuss the notion of bimodel in order to obtain a classification of the equivalences between categories of models in the sense of functorial semantics.Research supported by NATO grant 900959 and FNRS grant 1.5.181.90F     

On the notion of cobordism for bounded manifolds

The usual concept of cobordism is concerned with closed manifolds. In this paper, we generalise the concept of cobordism to the extent of bounded manifolds and obtain a homotopy theorem similar to the theorem of Thom. Supported by the National Natural Science Foundation of China and partially supported by a grant of Zhejiang Natural Science Foundation.     

On the equivalence of three types of blocking in non-Markovian tandem queues

Definitions of blocking types for tandem queueing systems are given where each stage consists of multiple servers. Some of them are shown to be identical with respect to service initiation epochs at each stage. Such identical relationships are proved using sample path correspondence, and are therefore valid irrespective of the arrival process and the service time distributions.