Showing papers in "Banach Center Publications in 2003"
291 citations
67 citations
TL;DR: The theory of matched pairs of groups is surveyed with special interest in applications to braided groups and quasitriangular structures on the bi-smash product Hopf algebra as mentioned in this paper.
Abstract: The theory of matched pairs of groups is surveyed with special interest in applications to braided groups and quasitriangular structures on the bi-smash product Hopf algebra. Introduction. In 1981, I introduced the notion of a matched pair of groups (S,B) and defined the bi-smash product Hopf algebra (kS)∗ ⊗ kB over a field k, when S is a finite group [T], motivated by W. M. Singer’s work [Si] on Hopf algebra extensions. Later, S. Majid [Mj] extended and applied this work in the study of quantum groups. Further, the cohomology theory of matched pairs of Hopf algebras was studied by I. Hofstetter [H] and A. Masuoka [Ms1], [Ms2]. During this research, Masuoka found out that the notion of a matched pair of groups had been obtained essentially by G. I. Kac [K] in 1968 and that the cohomology theory of matched pairs of groups had been studied there. Recently, some interesting progress in the study of matched pairs of groups has been made by two groups of algebraists, ESS (Etingof, Schedler, Soloviev) and LYZ (Lu, Yan, Zhu), during their study of set-theoretical solutions of the Yang-Baxter equation. Especially, the group ESS [ESS] has obtained some deep results on classification of finite braided sets. In this survey, I give an elementary and diagrammatic approach to the introductory part of the ESS-LYZ theory. This will enable the reader to appreciate their deeper results (which are not discussed here) more easily. On the other hand, the latter group LYZ has obtained some interesting construction of quasitriangular structures on the bi-smash product Hopf algebra H(S,B) = (kS)∗ ⊗ kB [LYZ2]. I also give a natural categorical explanation of this construction, based on Masuoka’s observation [Ms3]. In §1, we review the notion of a matched pair of groups (S,B) and matched product B ./ S, and explain the matched pair condition by means of diagrams of the form 2000 Mathematics Subject Classification: 16W35, 18D10. The paper is in final form and no version of it will be published elsewhere.
52 citations
TL;DR: In this paper, the authors introduce the notion of equivariant spectral triples with Hopf algebras as isometries of non-commutative manifolds.
Abstract: We present the review of noncommutative symmetries applied to Connes’ formulation of spectral triples. We introduce the notion of equivariant spectral triples with Hopf algebras as isometries of noncommutative manifolds, relate it to other elements of theory (equivariant K-theory, homology, equivariant differential algebras) and provide several examples of spectral triples with their isometries: isospectral (twisted) deformations (including noncommutative torus) and finite spectral triples.
49 citations
TL;DR: In this paper, a new geometrical setting for classical field theories is introduced and a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists.
Abstract: A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists. The main advantage of this algorithm is that the second order condition is automatically included.
40 citations
TL;DR: In this article, a detailed comparison between weak Hopf algebras and quantum groupoid algebroid is given, and the comparison is extended to cover module and comodule theory.
Abstract: We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Vainerman), and that of a $\times_R$-bialgebra due to Takeuchi (and also called a bialgebroid or quantum (semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a $\times_R$-bialgebra in which $R$ is Frobenius-separable. We extend the comparison to cover module and comodule theory, duality, and the question when a bialgebroid should be called a Hopf algebroid.
37 citations
TL;DR: In this article, the role of time delays in solid avascular tumour growth is considered and the model is formulated in terms of a reaction-diffusion equation and mass conservation law.
Abstract: The role of time delays in solid avascular tumour growth is considered. The model is formulated in terms of a reaction-diffusion equation and mass conservation law. Two main processes are taken into account—proliferation and apoptosis. We introduce time delay first in underlying apoptosis only and then in both processes. In the absence of necrosis the model reduces to one ordinary differential equation with one discrete delay which describes the changes of tumour radius. Basic properties of the model depending on the magnitude of delay are studied. Nonnegativity of solutions is investigated. Steady state and the Hopf bifurcation analysis are presented. The results are illustrated by computer simulations.
33 citations
TL;DR: In this article, an abstract semilinear parabolic equation in a Banach space X is considered and the problem of generating a bounded dissipative semigroup on X is shown to generate an (X − Z)-global attractor A that is closed, bounded, invariant in X, and attracts bounded subsets of X in a weaker topology of an auxiliary Banach topology Z.
Abstract: An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on X. This semigroup possesses an (X − Z)-global attractor A that is closed, bounded, invariant in X, and attracts bounded subsets of X in a ‘weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in R and to the partly dissipative system. 1. Introductory notes. The theory of global attractors was a very active field of studies through the last 20 years. One of the limitation inside the classical setting of this theory ([Ha], [La]) was the requirement of compactness (or, at least, asymptotic compactness) of the semigroup in considered phase space. This assumption cannot be however satisfied when the space variable x belongs to large unbounded domains, in particular, to the whole of R. To overcome this difficulty, it was necessary to generalize the idea of an attractor allowing the convergence to the attractor in a weaker topology than the topology of the phase space in which the semigroup acts. Such generalization was introduced in [B-V] and further developed in [M] and [M-S]. We have already joined these studies in [Ch-Dl1] and [Ca-Dl]. The aim of the present note is to describe a general approach to such bi-spaces attractors for sectorial equations. We propose an abstract existence result (Theorem 2.6) and discuss its applications to two specific problems; bistable reaction-diffusion equation and FitzHugh-Nagumo system (Examples 3.1 and 3.6). Both these problems are of special interest because of their nontrivial dynamics, which is partially connected with the existence of the families of travelling waves (or relaxation Supported partially by the State Committee for Scientific Research (KBN) grant 2 P03A 035 18. 2000 Mathematics Subject Classification: Primary 35B40, 35B41, 35K15, 35K45.
31 citations
30 citations
TL;DR: In this article, the authors apply the theory of linear and nonlinear semigroups of operators to models of structured populations dynamics and apply it to analyze such population behaviors as extinction, growth, stabilization, oscillation, and chaos.
Abstract: The objective of these lectures is to apply the theory of linear and nonlinear semigroups of operators to models of structured populations dynamics. The mathematical models of structured populations are typically partial differential equations with variables corresponding to such properties of individual as age, size, maturity, proliferative state, quiescent state, phenotype expression, or other physical properties. The main goal is to connect behavior at the individual level to behavior at the population level. Theoretical results from semigroup theory are applied to analyze such population behaviors as extinction, growth, stabilization, oscillation, and chaos. 1. General theory of operator semigroups in Banach spaces. In this section we provide basic definitions and theorems in the theory of semigroups of operators in Banach spaces and illustrate the concepts with some examples relevant to structured populations. Definition 1.1. Let X be a Banach space and let Y ⊂ X. A strongly continuous semigroup of operators in Y is a set of operators (linear or nonlinear) T (t), t ≥ 0 satisfying (i) T (t) is continuous from Y to Y ∀t ≥ 0, (ii) T (0)φ = φ ∀φ ∈ Y , (iii) T (t+ s)φ = T (t)T (s)φ ∀t, s ≥ 0 and φ ∈ Y , (iv) t 7→ T (t)φ is continuous ∀φ ∈ Y . We remark that if Y = X and T (t) ∈ B(X) ∀t ≥ 0, where B(X) is the Banach algebra of bounded linear operators in X, then T (t), t ≥ 0 is called a strongly continuous semigroup of bounded linear operators in X. Otherwise, T (t), t ≥ 0 is called a nonlinear semigroup in Y . Example 1.1. LetX = C[0, 1] or C0[0, 1], where C[0, 1] is the Banach space of continuous real-valued functions on [0, 1] with supremum norm and C0[0, 1] is the subspace of C[0, 1] 2000 Mathematics Subject Classification: Primary 92D25; Secondary 34G10, 34G20, 47D03. The paper is in final form and no version of it will be published elsewhere.
28 citations
TL;DR: In this article, it was shown that locality of a Loday bracket on sections of a one-dimensional vector bundle forces skew-symmetry, i.e., a local Lie algebra structure in the sense of A.A.Kirillov.
Abstract: Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply the properties which usually are additionally required. In particular, it is proved that locality of a Loday bracket on sections of a one-dimensional vector bundle forces skew-symmetry, i.e. a local Lie algebra structure in the sense of A.A.Kirillov.
TL;DR: In this article, an extension of the classical Fokker-Planck equation was considered on IR with L = −∆ and V = 1 2 |x|2, where L is a general operator describing the diffusion.
Abstract: We consider extensions of the classical Fokker–Planck equation ut + Lu = ∇·(u∇V (x)) on IR with L = −∆ and V (x) = 1 2 |x|2, where L is a general operator describing the diffusion and V is a suitable potential.
TL;DR: A list of known quantum spheres of dimension one, two and three is presented in this paper, where the authors aid the book-keeping of these newly emerged species by systematically comparing their basic properties.
Abstract: A list of known quantum spheres of dimension one, two and three is presented. M.S.C. : 81R60, 81R50, 20G42, 58B34, 58B32, 17B37. Key words and phrases : Noncommutative geometry, quantum spheres.Ref. SISSA 79/2001/FM 0. Introduction. Recently, examples of quantum spheres cropped in abundance in the literature. The goal ofthis note is to aid the book-keeping of these newly emerged species by systematically comparingtheir basic properties.As is customary in noncommutative geometry, these quantum spaces are described and stud-ied in terms of certain noncommutative algebras, generalizing the usual correspondence betweenspaces and function algebras. Here I am concerned mainly with ‘deformations’ of the ∗-algebra ofpolynomials on the sphere S n and their enveloping C ∗ -algebras. The ∗-algebras are usually givenin terms of generators and relations. Some of these relations can be regarded as deformationsof the commutation relations and some as deformations of the sphere relationP n+1j
TL;DR: It is shown that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, and may be up to the replacement of the involved Lie group by a central extension of it.
Abstract: Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems.
We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the involved Lie group by a central extension of it.
The geometric techniques developed for dealing with Lie systems are also used in problems of control theory. Specifically, we will study some examples of control systems on Lie groups and homogeneous spaces.
TL;DR: In this article, an affine Cartan calculus is developed and the concepts of affine bundles and affine duality are introduced, and the canonical isomorphisms for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.
Abstract: An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.
TL;DR: The relation between simple algebraic groups and simple singularities was studied in this article, where the simple singularity appeared as the generic singularity in codimension two of the unipotent variety of simple algebra groups.
Abstract: There is a well known relation between simple algebraic groups and simple singularities, cf. [5], [28]. The simple singularities appear as the generic singularity in codimension two of the unipotent variety of simple algebraic groups. Furthermore, the semi-universal deformation and the simultaneous resolution of the singularity can be constructed in terms of the algebraic group. The aim of these notes is to extend this kind of relation to loop groups and simple elliptic singularities. It is the successful completion of work begun long ago, [29], cf. also [31], [30], [32] for more general situations. 1. Simple groups and simple singularities. (For details cf. [28].) Let G be a simple and simply connected algebraic group over C. Then, G acts by conjugation on itself and the ring of invariant functions on G is a polynomial ring in the fundamental characters of G C[G] = C[χ1, . . . , χ`]. The induced map G −−→ SpecC[χ1, . . . , χ`] ' C 2000 Mathematics Subject Classification: Primary: 14B07, 14L32. Secondary: 14H52, 14H60, 32S25.
TL;DR: In this paper, it was shown that the vacuum cannot be generated in finite time, and that if the vacuum is present for some positive time then it must be present in the initial data, in a precise sense which is given.
Abstract: We consider the equations of isentropic gas dynamics in Lagrangian coordinates. We are interested in global interactions of large waves, and their relation to global solvability and well-posedness for large data. One of the main difficulties in this program is the possible occurrence of a vacuum, in which the specific volume is infinite. In this paper we show that the vacuum cannot be generated in finite time. More precisely, if the vacuum is present for some positive time, then it must be present in the initial data, in a precise sense which is given. We also discuss the annihilation of vacuums that are present in the initial data.
TL;DR: In this paper, a localization approach to non-affine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras is proposed.
Abstract: Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. In the quantum case, calculations with quantum minors yield the structure theorems. Notation. Ground field is k and we assume it is of characteristic zero. If we deal just with one k-Hopf algebra, say B, the comultiplication is ∆ : B → B ⊗ B, unit map η : k → B, counit : B → k, multiplication μ : B ⊗ B → B, and antipode (coinverse) is S : B → B. Warning: letter S often stands for a generic Ore set. We use [56, 49, 38, 74] Sweedler notation ∆(h) = ∑ h(1)⊗h(2) with or without explicit summation sign, as well as its extension for coactions: ρ(v) = ∑ v(0) ⊗ v(1), where the zero-component is in the comodule and nonzero component(s) in the coalgebra. An entry symbol and name of a matrix will match, except for upper vs. lower case, e.g. G = (g j); and G I J will be a submatrix with row multilabel I = (i1, . . . , ik) and column multilabel J = (j1, . . . , jk). As a rule, row labels are placed as superscripts and column labels as subscripts. 1991 Mathematics Subject Classification: Primary 14A22; Secondary 16W30,14L30,58B32.
TL;DR: In this article, the authors proved the C∞-well posedness of the Cauchy problem for quasi-linear hyperbolic equations with coefficients non-Lipschitz in the time variable and smooth in spatial variables.
Abstract: In this paper we prove the C∞-well posedness of the Cauchy problem for quasilinear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0, T ] and smooth in x ∈ R. 0. Introduction. In this paper we consider Cauchy problems for quasi-linear hyperbolic equations with coefficients non-Lipschitz in the time variable and smooth in spatial variables. Our goal is to prove C∞-well posedness for the Cauchy problem (CP1) Pu[u] = utt − n ∑ j,k=1 ajk(t, x;u, ut,∇u)uxjxk + ρ(t, x;u, ut,∇u) = 0 for (t, x) ∈ (0, T )× R, u = φ(x), ut = ψ(x) at t = 0. (0.1) The paper [2] is devoted to the study of Cauchy problems for second order hyperbolic equations with coefficients depending on the time variable of the form utt − n ∑ j,k=1 ajk(t)uxjxk + n ∑ j=1 bj(t)uxj + c(t)u = 0, (0.2) Research is supported by DFG No. 446 JAP 17/1/01. 2000 Mathematics Subject Classification: Primary 35L80; Secondary 35L15. The paper is in final form and no version of it will be published elsewhere.
TL;DR: The cleftness of hopf algebras is studied in this article, where it is shown that a hopf algebra A is not only cleft as a right H-comodule algebra, but also cocleft as a left K-module coalgebra, and hence is described as a bicrossed product of K and H. In this paper, we give a new approach from the cleft results to the theorems by Nagata, by Takeuchi [T1], and by Sullivan [Su], which all are directly connected to the theory
Abstract: Introduction. The introductory part of the beautiful lectures given by H. -J. Schneider1 at the conference reminded us of one of the original motivations of Hopf algebra theory, that is, to simplify and generalize (or quantize, in more recent language) the theory of affine group schemes, by studying Hopf algebras. One of the most successful results in Hopf algebra theory can be found in the study of cleft comodule algebras, or in other words Hopf-crossed products. In this paper we give a new approach from the cleftness results to the theorems by Nagata, by Takeuchi [T1], and by Sullivan [Su], which all are directly connected to the theory of affine group schemes; see [A, Chap. 4]. For a Hopf algebra H, a right H-comodule algebra A is said to be cleft [Sw1] if there is a convolution-invertible H-colinear map H → A. Such an algebra A is characterized as a Hopf-Galois extension with normal basis, and also as a Hopf-crossed product [DT; BCM]; the crossed product is given by a measuring action and a non-abelian Hopf 2-cocycle. Therefore, cleft comodule algebras are studied, regarded as an important, special class of Hopf-Galois extensions [DT; BM], as a generalized object unifying various constructions of rings such as group-crossed products or Ore extensions [BCM], and as an interesting example of non-abelian cohomology [D3]. A sequence K → A → H of Hopf algebras is called a cleft extension [S1], if there is a left K-linear and right H-colinear isomorphism K ⊗H ∼= A. Then A is not only cleft as a right H-comodule algebra, but also cocleft, in the dual situation, as a left K-module coalgebra, and hence is described as a bicrossed product of K and H; see Section 1 below. Such extensions are studied in the abelian situation from the view-point of abelian cohomology [S1; Si; H; M2], and also in the non-abelian situation toward applications to quantum groups [Mj1; AD].
TL;DR: In this paper, extended Hopf algebras are defined in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebraic structures.
Abstract: We introduce the concept of {\it extended Hopf algebras} and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, including the Connes-Moscovici algebra $\mathcal{H}_{FM}$, are extended Hopf algebras.
TL;DR: In this article, a generalisation of the Cuntz-Quillen theorem relating existence of connections in a module to projectivity of this module is proven, and a correspondence between categories of comodules and flat connections is established.
Abstract: Characteristic properties of corings with a grouplike element are analysed. Associated differential graded rings are studied. A correspondence between categories of comodules and flat connections is established. A generalisation of the Cuntz-Quillen theorem relating existence of connections in a module to projectivity of this module is proven.
TL;DR: In this article, it was shown that the symmetric function defining the Bethe equations can be interpreted as the generating function of the map sending a pair of complex polynomials into their Wronski determinant: the critical orbits determine the preimage of a given polynomial under this map.
Abstract: We consider the Gaudin model associated to a point z in C^n with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl_2-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector.
In [ReV], it was shown that for generic z the Bethe vectors span the space of singular vectors, i.e. that the number of critical orbits is bounded from below by the dimension of the space of singular vectors. The upper bound by the same number is one of the main results of [SV].
In the present paper we get this upper bound in another, ``less technical'', way. The crucial observation is that the symmetric function defining the Bethe equations can be interpreted as the generating function of the map sending a pair of complex polynomials into their Wronski determinant: the critical orbits determine the preimage of a given polynomial under this map. Within the framework of the Schubert calculus, the number of critical orbits can be estimated by the intersection number of special Schubert classes. Relations to the sl_2 representation theory [F] imply that this number is the dimension of the space of singular vectors.
We prove also that the spectrum of the Gaudin hamiltonians is simple for generic z.
TL;DR: In this paper, the integrability and superintegrability properties of the spherical Kepler (Schrödinger) potential and the spherical Schmid potential were investigated. But the authors focused on the properties of spherical (isotropic and nonisotropic) harmonic oscillators.
Abstract: The spherical version of the two-dimensional central harmonic oscillator, as well as the spherical Kepler (Schrödinger) potential, are superintegrable systems with quadratic constants of motion. They belong to two different spherical “Smorodinski-Winternitz” families of superintegrable potentials. A new superintegrable oscillator have been recently found in S2. It represents the spherical version of the nonisotropic 2:1 oscillator and it also belongs to a spherical family of quadratic superintegrable potentials. In the first part of the article, several properties related to the integrability and superintegrability of these spherical families of potentials are studied. The second part is devoted to the analysis of the properties of the spherical (isotropic and nonisotropic) harmonic oscillators.
TL;DR: In this paper, an infinite-dimensional version of the Amann-Conley-Zehnder reduction for boundary problems related to nonlinear perturbed elliptic operators with symmetric derivative was introduced.
Abstract: We introduce an infinite-dimensional version of the Amann-Conley-Zehnder reduction for a class of boundary problems related to nonlinear perturbed elliptic operators with symmetric derivative. We construct global generating functions with finite auxiliary parameters, describing the solutions as critical points in a finite-dimensional space.
TL;DR: In this article, two asexual density-dependent population dynamics models with age-dependent and child care are presented, one of them includes the random diffusion while in the other the population is assumed to be non-dispersing.
Abstract: Two asexual density-dependent population dynamics models with age-dependence and child care are presented. One of them includes the random diffusion while in the other the population is assumed to be non-dispersing. The population consists of the young (under maternal care), juvenile, and adult classes. Death moduli of the juvenile and adult classes in both models are decomposed into the sum of two terms. The first presents death rate by the natural causes while the other describes the environmental influence depending on the total density of the juvenile and adult individuals. An existence and uniqueness theorem is proved, a class of separable solutions is constructed, and the large time behavior of the general and separable solutions is given for the non-dispersing population with stationary vital rates. The steady-state and separable solutions are constructed and the large time behavior of the separable solutions is studied for the population with the spatial dispersal.
TL;DR: In this article, the authors provide a conceptual framework for variational formulations of classical physics and present an extended analysis of local equilibria for systems with configuration manifolds of finite dimensions.
Abstract: This note presents an attempt to provide a conceptual framework for variational formulations of classical physics. Variational principles of physics have all a common source in the {\it principle of virtual work} well known in statics of mechanical systems. This principle is presented here as the first step in characterizing local stable equilibria of static systems. An extended analysis of local equilibria is given for systems with configuration manifolds of finite dimensions. Numerous examples of the principle of virtual work and the Legendre transformation applied to static mechanical systems are provided. Configuration spaces for the dynamics of autonomous mechanical systems and for statics of continua are constructed in the final sections. These configuration spaces are not differential manifolds.
TL;DR: In this paper, a general class of Boltzmann-like bilinear integro-differential systems of equations (GKM, Generalized Kinetic Models) is considered.
Abstract: In this paper a general class of Boltzmann-like bilinear integro-differential systems of equations (GKM, Generalized Kinetic Models) is considered. It is shown that their solutions can be approximated by the solutions of appropriate systems describing the dynamics of individuals undergoing stochastic interactions (at the “microscopic level”). The rate of approximation can be controlled. On the other hand the GKM result in various models known in biomathematics (at the “macroscopic level”) including the “SIR” model, some competitive systems and the Smoluchowski coagulation model.
TL;DR: In this article, necessary and sufficient conditions for both conservativity and uniqueness of solutions to birth-and-death systems of equations using methods of semigroup theory were presented, which correspond to the uniqueness criteria due to Reuter, [10, 11, 1], that were derived in a different context by Markov processes' techniques.
Abstract: We shall present necessary and sufficient conditions for both conservativity and uniqueness of solutions to birth-and-death system of equations using methods of semigroup theory. The derived conditions correspond to the uniqueness criteria for forward and backward birth-and-death systems due to Reuter, [10, 11, 1], that were derived in a different context by Markov processes’ techniques.