A sequential method for preventive maintenance and replacement of a repairable single-unit system

This paper is concerned with when to implement preventive maintenance (PM) and replacement for a repairable ‘single-unit’ system in use. Under the main assumption that a ‘single-unit’ system gradually deteriorates with time, a sequential method is proposed to determine an optimal PM and replacement strategy for the system based on minimising expected loss rate. According to this method, PM epochs are determined one after the other, and consequently we can make use of all previous information on the operation process of the system. Also the replacement epoch depends on the effective age of the system. A numerical example shows that the sequential method can be used to solve the PM and replacement problem of a ‘single-unit’ system efficiently. Some properties of the loss functions W(L? _n ,b? _n ) and W? _r (N) with respect to PM and replacement respectively are discussed in the appendix.     

On a system of piecewise linear functions with vanishing integrals on <Emphasis Type="Italic">R</Emphasis>

For an admissible sequence T we define an orthonormal system consisting of piecewise linear functions with vanishing integrals on R. Necessary and sufficient conditions on T are found for the corresponding system to be a basis in H¹(R).     

Isochronicity at infinity for a class of polynomial differential system

In this paper, we explore the problem of isochronicity at infinity for a class of polynomial differential system. The technique is based on taking infinity into the origin by means of a homeomorphism. Simultaneously, we derive a recursive algorithm to compute period constants at the origin of the transformed system. At the end, as an application of our algorithm, we study pseudo-isochronous center conditions at infinity for a class of septic system.     

Stability of Differential-Algebraic Equations under Uncertainty

We consider a linear time-invariant homogeneous system of first-order ordinary differential equations with a noninvertible matrix multiplying the derivative of the unknown vector function and with perturbed coefficients. We introduce a class of perturbations of the coefficient matrices of the system and determine conditions on the perturbations of this class under which they do not affect the internal structure of the system. We obtain sufficient conditions for the robust stability of the system under such perturbations.     

On unconditional basis property in the space <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">R</Emphasis>) of a system of Franklin functions with vanishing means

We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H ¹(R).     

Wandering Operators for Unitary Systems of Hilbert Spaces

In this paper, we introduce the concept of complete wandering operators for a system \({{\mathcal {U}}}\) of unitary operators acting on a Hilbert space, which can be viewed as an abstract mathematical model for \(g\)-orthonormal bases of Hilbert spaces and operator-valued wavelets for \(L^2(R)\). The idea comes from Dai and Larson’s work (Mem Am Math Soc 134:640 1998), where the wandering vectors for a unitary system are introduced as an abstract model for orthogonal wavelets. The topological and algebraical properties of the set \({{\mathcal {W}}}({{\mathcal {U}}})\) of all complete wandering operators for a unitary system \({{\mathcal {U}}}\) are studied. In particular, properties of the local commutant of \({{\mathcal {U}}}\) are established. A parametrization formula for \({{\mathcal {W}}}({{\mathcal {U}}})\) and some interesting algebraic properties of complete wandering operators for a unitary system are obtained. The special case of greatest interest is the wavelet system \(\{{{\mathcal {U}}}_{D,T}\}\). We pay certain attention on studying this more structured unitary system and some structural theorems are established. Lots of properties of the wandering vectors for a unitary system are extended to the more general case, i.e. the wandering operators for a unitary system. However, operator-valued case is more complicated. We also give some examples to illustrate our results. Our works show that wavelet theory and frame theory are deeply connected with operator theory.     

Parseval Frames Built Up from Generalized Shift-Invariant Systems

Wavelet systems, and many of its generalizations such as wavelet packets, shearlets, and composite dilation wavelets are generalized shift-invariant systems (GSI) in the sense of the work by Ron and Shen. It is well known that a wavelet system is never Z-shift invariant (SI). Nevertheless, one can modify it and construct a Z-SI system, called a quasi-affine system, which shares most of the frame properties of the wavelet system. The analogue of a quasi-affine system for a GSI system is called an oblique oversampling: it is shift invariant with respect to a fixed lattice. Assumptions on a GSI system X were given by Ron and Shen to ensure that any oblique oversampling is a Parseval frame for ${L^2(\mathbb{R}^n)}$ whenever X is. We show that these assumptions are not satisfied for some of the wavelet generalizations mentioned above and that elements implicit in their work provide other sufficient conditions on the system under which any oblique oversampling is a Parseval frame for ${L^2(\mathbb{R}^n)}$ (shift invariant with respect to a fixed lattice). Moreover, in the orthonormal setting it is shown that completeness yields a shift-invariant Parseval frame for suitable proper subspaces of ${L^2(\mathbb{R}^n)}$ , too.     

KMS states and complex multiplication

We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the Dedekind zeta function as partition function and the idèle class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. The geometric notion underlying the construction is that of commensurability of K-lattices.     

Synthesis of minimum-order robust inverters

In the present paper, we consider the inversion problem for dynamical systems, that is, the problem of reconstruction of the unknown input signal ξ(t) of a given system on the basis of known information (about either the complete phase vector or a measurable output of the system). An auxiliary dynamical system forming the desired estimate of the signal ξ(t) is called an inverter.In earlier papers of the authors, attention was mainly paid to the possibility of inversion of a dynamical system in different cases in principle. In this relation, a model of dynamical systems with some stabilizing control was used as an inverter for the solution of the problem; moreover, this control was often designed with the use of an additional dynamical system, an observer of the phase vector of the original system or the system in deviations. Thus, a dynamical system whose dimension either coincides with the dimension of the original system or exceeds it was considered as an inverter.In the solution of practical problems, it is often required to synthesize inverters of minimal order. (This requirement is related to constraints on the complexity, cost, and operation speed of automated control systems.) In the present paper, we consider the problem on the possible reduction of the order of the inverter in various cases and the problem on the construction of inverters of minimal order.     

Prevalence of locally parameter identifiable systems

The problem of local parameter identifiability of an input–output system is considered. A set lf of systems is studied for which the property of local parameter identifiability holds for almost all values of input signals and parameters in both topological and metric senses. Sufficient conditions are pointed out under which the set LI contains a prevalent subset. The proof is based on the prevalent transversality theorem proved by Kaloshin. Systems are considered that are characterized by a family of (structural) parameters a and a control block. It is shown that if the dimension of the set of parameters a is large enough (the structure of the system is rich enough), then, generically, a system f _a belongs to the class lf for a set of parameters a having full measure.     

Tocher's Principle of Modelling: Practical Significance and Mathematical Implications

This paper describes an important principle of modelling put forward by the late K. D. Tocher, namely that a clear distinction should be made between a system modelled and problems about the system. An example illustrates the many different practical problems one may be led to solve about a given economic system, e.g. an industrial firm. The example also shows that problems often result from the solutions to other problems, and thus cannot all be simultaneously anticipated. This suggests the need for a modelling system which, given a model of a system, may be used to solve any problem about the system.The overall problem can be described as that of solving an underdetermined system of equations. The precise meaning of the problem is defined for the case of sparse systems. Finally, the main features of a computer program based on Tocher's philosopy are outlined.     

Towards a normative model for inventory cost management in a generalized ABC classification system

This paper establishes a general ABC inventory classification system as the foundation for a normative model of the maintenance cost structure and stock turnover characteristics of a large, multi-item inventory system with constant demand. For any specified number of inventory classes, the model allows expression of the overall system combined ordering and holding cost in terms of (i) the re-ordering frequencies for the items in each inventory class and (ii) the inventory class structure, that is, the proportion of the total system's items that are in each inventory class. The model yields a minimum total maintenance cost function, which reflects the effect of class structure on inventory maintenance costs and turnover. If the Pareto curve (a.k.a. Distribution-by-value function) for the inventory system can be expressed (or approximated) analytically, the model can also be used to determine an optimal class structure, as well as an appropriate number of inventory classes. A special case of the model produces a simply structured, class-based ordering policy for minimizing total inventory maintenance costs. Using real data, the cost characteristics of this policy are compared to those of a heuristic, commonly used by managers of multi-item inventory systems. This cost comparison, expressed graphically, underscores the need for normative modelling approaches to the problem of inventory cost management in large, multi-item systems.     

A hybrid maintenance model with imperfect inspection for a system with deterioration and Poisson failure

A new maintenance model for a system with both deterioration and Poisson failures is proposed. In this model, at any time-instant G _S and when the system is operating, one of the following decisions may be taken: (1) stop the system to perform a scheduled minimal maintenance; (2) stop the system to perform an inspection; and (3) no action and allow the system to go on with its operation. Following an inspection, based on the deterioration condition of the system, one of the following decisions may be taken: (a) if the system is in a ‘good’ condition, no maintenance action is taken and a number of periodic minimal maintenance activities are scheduled, starting T1 later; (b) if the system is in an ‘intermediate’ condition, a minimal maintenance is performed and an inspection is scheduled for T2 later (T2 < T1); and (c) if the system is in a ‘bad’ condition, a major maintenance is performed and a number of periodic minimal maintenances are scheduled, starting T1 later. In addition, a deterioration failure is restored by a major repair and a Poisson failure is restored by a minimal repair. Generalised stochastic Petri nets are used to represent and analyse the model, which represents a ‘composite’ maintenance strategy. Based on maximisation of the throughput of the system the benefit of this model compared to (1) an equivalent periodic inspection model and (2) an equivalent planned scheduled maintenance model, is demonstrated. This study presents a new hybrid model with a general framework for incorporating various types of maintenance policies. Also by incorporation of a number of features, this model will be more applicable to real world technical systems (complex systems), although it can be applied to individual components that are part of a complex system.     

Disintegration of invariant measures for hyperbolic skew products

We study hyperbolic skew products and the disintegration of the SRB measure into measures supported on local stable manifolds. Such a disintegration gives a method for passing from an observable v on the skew product to an observable \(\overline v \) on the system quotiented along stable manifolds. Under mild assumptions on the system we prove that the disintegration preserves the smoothness of v, first in the case where v is Hölder and second in the case where v is \({\mathcal{C}^1}\).     

NP-completeness conditions for consistency verification of some types of systems of linear Diophantine dis-equations

Three series of number-theoretic problems with explicitly defined parameters concerning systems of Diophantine dis-equations with solutions from a given domain are considered. Constraints on these parameters under which any problem of each series is NP-complete are proved. It is proved that for any m and m′ (m < m′) the consistency problem on the segment [m, m′] for a system of linear Diophantine dis-equations, each of which contains exactly three variables (even if the coefficients at these variables belong to {–1, 1}), is NP-complete. This problem admits a simple geometric interpretation of NP-completeness for the determination of whether there exists an integer-valued point inside a multidimensional cube that is not covered by given hyperplanes that cut off equal segments on three arbitrary axes and are parallel to all other axes. If in a system of linear Diophantine dis-equations each dis-equation contains exactly two variables, the problem remains NP-complete under the condition that the following inequality holds: m′–m > 2. It is also proved that if the solution to a system of linear Diophantine dis-equations, each containing exactly three variables, is sought in the domain given by a system of polynomial inequalities that contain an n-dimensional cube and are contained in an n-dimensional parallelepiped symmetric with respect to the point of origin, its consistency problem is NP-complete.     

Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with \(L^{\infty }\)-variable coefficients in \(L^2\)-based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in \(L^2\)-based Sobolev spaces by using the remarkable Nec?as–Babus?ka–Brezzi technique (see Babus?ka in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Nec?as in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in \(L^2\)-based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos? and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.     

A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System

A generalization of the block replacement policy (BRP) is proposed and analysed for a multi-unit system which has the specific multivariate distribution. Under such a policy an operating system is preventively replaced at times kT (k = 1, 2, 3,...), as in the ordinary BRP, and the replacement of the failed system at failure is not mandatory; instead, a minimal repair to the component of the system can be made. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the component at age y depends on the random part C(y) and the deterministic part C_i(y). The aim of the paper is to find the optimal block interval T which minimizes the long-run expected cost per unit time of the policy.     

A Three-Dimensional Optimization Problem in Water-Resources Engineering

The problem under consideration is related to a system consisting of two storage facilities in series. Part of the water reaching the upstream reservoir may be diverted for storage in the downstream facility. However, there is a certain lag between the time when water is released from the upper reservoir and the time when it is available for utilization in the lower one. Thus the vector describing the state of the system at any one time consists of three elements:

1. a)
   amount of water stored in the upstream reservoir;
    
2. b)
   amount of water stored in the downstream reservoir;
    
3. c)
   amount of water in transit.
    

An optimum operating policy for the system had to be derived.The method of dynamic programming was applied in the analysis of this problem. The algorithm obtained was programmed for solution on an IBM 7090 digital computer. Optimal policies were obtained for each admissible state of the system.     

Linking method applied to a class of a system of critical growth wave equations

We show the existence of at least two solutions for a class of a system of critical growth wave equations, with periodic condition on t and the Dirichlet boundary condition

     

The Cauchy Problem for the Average Field Equation Describing a Model of a Magnetic Solid

We study a model of a magnetic solid treated as a system of particles with mechanical moment $\vec s,\vec s \in S^2$ , and magnetic moment $\vec \mu ,\vec \mu = \vec s$ , interacting with one another via the magnetic field, which determines variations in the mechanical moment of each particle. We study the system of integro-differential equations describing the evolution of the one-particle distribution function for this system of particles. We prove existence and uniqueness theorems for the generalized and the classical solution of the Cauchy problem for this system of equations. We also prove that the generalized solution continuously depends on the initial conditions.