Showing papers in "Acta Applicandae Mathematicae in 1984"
TL;DR: In this article, the authors provide an axiomatic framework in order to define the concept of value function for risky operations for which there is no market, and construct a portfolio of these assets which will mimic the risks involved in the operation.
Abstract: The objective of this article is to provide an axiomatic framework in order to define the concept of value function for risky operations for which there is no market. There is a market for assets, whose prices are characterized as stochastic processes. The method consists of constructing a portfolio of these assets which will mimic the risks involved in the operation. We follow the terminology of the theory of options although the set-up goes beyond that particular problem.
321 citations
TL;DR: In this article, the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures are explained and the cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of PDE.
Abstract: Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures Roughly speaking, we explain what analogs of ‘higher KdV equations’ are for an arbitrary system of partial differential equations and also how one can find and use them The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations In particular, it is shown that ‘symmetry’ and ‘conservation law’ are, in some sense, the ‘dual’ conceptions which coincides in the ‘self-dual’ case, namely, for Euler-Lagrange equations Training examples are also given
185 citations
TL;DR: In this article, the notion of covering σ ≥ 0 is introduced for partial differential equations and nonlocal symmetries are defined as transformations of σ ∼ 0 to preserve the underlying contact structure.
Abstract: For a systemY of partial differential equations, the notion of a coveringŶ
∞→Y
∞ is introduced whereY
∞ is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations ofŶ
∞ which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.
147 citations
TL;DR: In this paper, the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces is described, and the review is in two parts, in two stages.
Abstract: In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.
99 citations
TL;DR: For an arbitrary uniformly continuous completely positive semigroup (ℑ t : t≥ 0) on the space B(ϵ) of bounded operators on a Hilbert space ǫ 0, the authors constructed a family of unitary operators U(t): t ≥ 0, which satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion.
Abstract: For an arbitrary uniformly continuous completely positive semigroup (ℑ t : t≥ 0) on the space B(ɧ0) of bounded operators on a Hilbert space ɧ0, we construct a family (U(t): t ≥ 0) of unitary operators on a Hilbert space ℌ0 = ɧ0 ⊗ ℌ and a conditional expectation E0 from B(ℌ0) to B(ℌ0), such that, for arbitrary t ≥0, X ∈ B(ɧ0) ℑ t (X) = E0[U(t)X ⊗ IU(t)†]. The unitary operators U(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.
70 citations
TL;DR: In this paper, the authors present methods using positive semigroups and perturbation theory in the application to the linear Boltzmann equation, and also present generalizations of known results and develops known methods in a more abstract setting.
Abstract: We present methods using positive semigroups and perturbation theory in the application to the linear Boltzmann equation. Besides being a review, this paper also presents generalizations of known results and develops known methods in a more abstract setting.
52 citations
TL;DR: In this paper, an extension of Lawvere's Theorem showing that all classical results on limitations stem from the same underlying connection between self-referentiality and fixed points is presented.
Abstract: We consider an extension of Lawvere's Theorem showing that all classical results on limitations (i.e. Cantor, Russel, Godel, etc.) stem from the same underlying connection between self-referentiality and fixed points. We first prove an even stronger version of this result. Secondly, we investigate the Theorem's converse, and we are led to the conjecture that any structure with the fixed point property is a retract of a higher reflexive domain, from which this property is inherited. This is proved here for the category of chain complete posets with continuous morphisms. The relevance of these results for computer science and biology is briefly considered.
45 citations
TL;DR: In this article, the Perron-Frobenius spectral theory for positive semigroups on Banach lattices is surveyed and applied to stability theory of retarded differential equations and quasi-periodic flows.
Abstract: In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.
42 citations
TL;DR: In this article, a new notion of stability, called pointwise stability, is defined and is shown to be generically equivalent to uniform asymptotic stability independent of delay.
Abstract: Feedback control of linear neutral (and retarded) time-delay systems with one or more non-commensurate time delays is studied. A new (algebraic) notion of stability, called pointwise stability, is defined and is shown to be generically equivalent to uniform asymptotic stability independent of delay. Necessary and sufficient conditions are then given for regulability, that is, for the existence of a dynamic output feedback compensator with pure delays such that the closed-loop system is internally pointwise stable (and thus stable independent of delay). Necessary and sufficient conditions involving matrix-fraction descriptions are also given for the existence of a state realization which is regulable. Finally, the problem of stabilization using nondynamic state feedback is briefly considered in the case when the system's input matrix has constant rank.
32 citations
TL;DR: A survey of the recent work on the infinitesimal generators of one-parameter semigroups of positivity preserving maps on operator algebras, in the presence of compact symmetry groups or flows can be found in this paper.
Abstract: A survey of the recent work on the infinitesimal generators of one-parameter semigroups of positivity preserving maps on operator algebras, in the presence of compact symmetry groups or flows.
15 citations
TL;DR: In this paper, the authors consider families of vector fields which exhibit a cusp singularity for some values of the parameters and obtain algebraic conditions for these families to have versal properties, which enables the topologically distinct, structurally stable types of phase portrait which occur in the family near to the bifurcation point to be predicted.
Abstract: We consider families of vector fields which exhibit a cusp singularity for some values of the parameters. Algebraic conditions are obtained for these families to have versal properties. This enables the topologically distinct, structurally stable types of phase portrait which occur in the family near to the bifurcation point to be predicted. Applications of these results in population dynamics and oscillation theory are discussed.
TL;DR: A review of the field of positive semigroups of operators with special emphasis on new developments and applications can be found in this paper, where the editors have attempted to present a state-of-the-art review.
Abstract: In this collection of papers the editors have attempted to present a state of the art review of the field of positive semigroups of operators with special emphasis on new (and recent) research developments and applications.
TL;DR: In this article, a theory of discrete mechanics based on the results of D. Greenspan is developed and the consistency, stability, and convergence of the methods are studied and some numerical examples presented.
Abstract: A theory of discrete mechanics is developed based on the results of D. Greenspan. Discrete dynamical equations in an inertial frame, in a coordinate system related to some material point, and in a rotating frame are given and the consistency, stability, and convergence of the methods are studied and some numerical examples presented.
TL;DR: In this paper, a simple proof is given showing the existence of semigroups of operators on the space of bounded Borel measurable functions for nonnegative continuous attractive spin rates along with a proof of invariant measures for the semigroup so constructed.
Abstract: Methods of constructing semigroups of operators describing interacting particle systems are reviewed. A simple proof is given showing the existence of semigroups of operators on the space of bounded Borel measurable functions for nonnegative continuous attractive spin rates along with a proof of the existence of invariant measures for the semigroups so constructed.