Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Inverse systems of groupoids,with applications to groupoid C⁎-algebras

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connections to noncommutative CW-complexes and interval algebras. We prove that an inverse system of groupoids with Haar system preserving bonding maps has a limit, and that we get a corresponding direct system of groupoid $C^{?}$-algebras. An explicit construction of an inverse system of groupoids is used to approximate a σ-compact groupoid G by second countable groupoids; if G is equipped with a Haar system and 2-cocycle then so are the approximation groupoids, and the maps in the inverse system are Haar system preserving. As an application of this construction, we show how to easily extend the Maximal Equivalence Theorem of Jean Renault to σ-compact groupoids.     

Well-posedness and stability of the repairable system with N failure modes and one standby unit

The well-posedness and stability of the repairable system with N failure modes and one standby unit were discussed by applying the c₀ semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive c₀ semigroups T(t) and so the well-posedness of the system was obtained. Finally, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. At the same time the dynamic solution exponentially converges to the steady-state solution.     

A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair

In this paper, a simple repairable system (i.e. a one-component repairable system with one repairman) with preventive repair and failure repair is studied. Assume that the preventive repair is adopted before the system fails, when the system reliability drops to an undetermined constant R  , the work will be interrupted and the preventive repair is executed at once. And assume that the preventive repair of the system is “as good as new” while the failure repair of the system is not, and the deterioration of the system is stochastic. Under these assumptions, by using geometric process, we present a bivariate mixed policy (R,N)$(R\text{,}N)$, respectively based on a scale of the system reliability and the failure-number of the system. Our aim is to determine an optimal mixed policy (R,N)^∗$(R\text{,}N{)}^{\ast }$ such that the long-run average cost per unit time (i.e. the average cost rate) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal mixed policy can be determined analytically or numerically. Finally, a numerical example is given where the working time of the system yields a Weibull distribution. Some comparisons with a certain existing policy are also discussed by numerical methods.     

Kernels and best approximations related to the system of ultraspherical polynomials

We study the uniformly bounded orthonormal system of functions

where is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of L^p-norms of the kernels of the system . These estimates enable us to prove Nikol'skiı˘-type inequalities for -polynomials. Next, we prove directly that is a basis in each , where w is an arbitrary A_p-weight function. Finally, we apply these results to get sharp inequalities for the best -approximations in L^q in terms of the best -approximations in . For the trigonometric system such inequalities have been already known.     

Eight octads suffice

The following question is answered: What is the minimum number of octads required to define a Steiner system $\text{l}$(5, 8, 24)? A set of octads is said to define a particular Steiner system if it is a subset of the special octads of that system, but of no other. As the title suggests, it is possible to produce a set of eight octads which defines the system, whereas any collection of seven octads is a subset of no Steiner system or of more than one. Note that, given an element of S₂₄, one is now able to decide whether it is in M₂₄ by simply applying it to a defining set of eight octads. If the images are in the system the element must be in M₂₄.     

The holonomic rank of the Fisher–Bingham system of differential equations

The Fisher–Bingham system is a system of linear partial differential equations satisfied by the Fisher–Bingham integral for the n  -dimensional sphere Sⁿ$S^n$. The system is given in [4, Theorem 2] and it is shown that it is a holonomic system [1]. We show that the holonomic rank of the system is equal to 2n+2$2n+2$.     

Exact analysis of asymmetric random polling systems with single buffers and correlated input process

We introduce a simple approach for modeling and analyzing asymmetric random polling systems with single buffers and correlated input process. We consider two variations of single buffers system: the conventional system and the buffer relaxation system. In the conventional system, at most one customer may be resided in any queue at any time. In the buffer relaxation system, a buffer becomes available to new customers as soon as the current customer is being served. Previous studies concentrate on conventional single buffer system with independent Poisson process input process. It has been shown that the asymmetric system requires the solution ofm 2 ^m –1) linear equations; and the symmetric system requires the solution of 2 ^m–1–1 linear equations, wherem is the number of stations in the system. For both the conventional system and the buffer relaxation system, we give the exact solution to the more general case and show that our analysis requires the solution of 2 ^m –1 linear equations. For the symmetric case, we obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, loss probability, throughput, and the expected delay observed by a customer.     

Contact Systems and Corank One Involutive Subdistributions

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on J ⁿ (R,R ^m ). We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on J ⁿ (R,R ^m ) in a way analogous to that how the Kumpera–Ruiz normal form generalizes the canonical contact system on J ⁿ (R,R), which is also called the Goursat normal form.     

The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, ν. The eigenvalues of the linearized system are complex for ν<0 and pure imaginary for ν>0. Thus, for ν<0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν>0 these stable and unstable manifolds are gone. If the sign of a certain term in the normal form is positive then for small negative ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set.     

Joint availability of systems modelled by finite semi–markov processes

Consider a repairable system at the time instants t and t + x, where t, x ≥0. The joint availability of the system at these time instants is defined as the probability of the system being functional in both t and t + x. A set of integral equations is derived for the joint availability of a system modelled by a finite semi–Markov process. The result is applied to the semi–Markov model of a two–unit system with sequential preventive maintenance. The method used for the numerical solution of the resulting system of integral equations is a two–point trapezoidal rule. The system of implementation is the matrix computation package MATLAB on the Apple Macintosh SE/30. The numerical results obtained by this method are shown to be in good agreement with those from simulation.     

On a quasilinear system involving the operator curl

This paper concerns a quasilinear system involving the operator curl. This system is an approximation of the anisotropic Ginzburg–Landau system which describes the Meissner state of type II superconductors. The existence of the weak solutions of the quasilinear system is proved by applying a variational method to a modified functional, and the C ^2+α regularity of the weak solutions H is established without assuming the boundedness of curl H.     

Asymptotic Results and a Markovian Approximation for the M(n)/M(n)/s+GI System

In this paper for the M(n)/M(n)/s+GI system, i.e. for a s-server queueing system where the calls in the queue may leave the system due to impatience, we present new asymptotic results for the intensities of calls leaving the system due to impatience and a Markovian system approximation where these results are applied. Furthermore, we present a new proof for the formulae of the conditional density of the virtual waiting time distributions, recently given by Movaghar for the less general M(n)/M/s+GI system. Also we obtain new explicit expressions for refined virtual waiting time characteristics as a byproduct.     

Small embeddings for partial 5‐cycle systems

It has been conjectured that any partial 5‐cycle system of order u can be embedded in a 5‐cycle system of order v whenever v ≥ 3 u/ 2+1 and v ≡ 1 , 5 (mod 10) . The smallest known embeddings for any partial 5‐cycle system of order u is 10 u +5 . In this paper we significantly improve this result by proving that for any partial 5‐cycle system of order u ≥ 255 , there exists a 5‐cycle system of order at most (9 u +146) / 4 into which the partial 5‐cycle system of order u can be embedded. © 2011 Wiley Periodicals, Inc. J Combin Designs     

Hypergeometric functions of two variables

Summary The systematic investigation of contour integrals satisfying the system of partial differential equations associated with Appell's hypergeometric functionF ₁ leads to new solutions of that system. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. The transformation theory of the solutions reveals connections between the system under consideration and other hypergeometric systems of partial differential equations. Presently it is discovered that any hypergeometric system of partial differential equations of the second order (with two independent variables) which has only three linearly independent solutions can be transformed into the system ofF ₁ or into a particular or limiting case of this system. There are also other hypergeometric systems (with four linearly independent solutions) the integration of which can be reduced to the integration of the system ofF ₁.     

The conservation laws and group properties of the equations of gas dynamics with zero velocity of sound

All the conservation laws of zero order are obtained by the method of A-operators for a system of n-dimensional (n ≥ 1) equations of gas dynamics with zero velocity of sound. A group subdivision is carried out of this system with respect to an infinite subgroup, which is a normal divider of its main Lie group of transformations; the main group of the resolving system is obtained. First-order non-local symmetries are obtained for the initial system. A special choice of the mass Lagrange variables enables this system to be converted to a reduced system equivalent to it, containing n - 1 spatial variables, which, for n = 2, is written in the form of a one-dimensional complex heat-conduction equation using complex dependent and independent variables.     

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0$t>0$ with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

Nonlinear dynamics of a novel fractional-order Francis hydro-turbine governing system with time delay

This paper focuses on the stability of a hydropower station. First, we established a novel nonlinear mathematical model of a Francis hydro-turbine governing system considering both fractional-order derivative and time delay. The fractional-order α, which is introduced into the penstock system, in the range from 0.82 to 1.00 is on the left side of the model in a incommensurate manner in increment of 0.03 to provide an adjustable degree of system memory. The time delay τ, which exists between the signal and response in the hydraulic servo system, in the range from 0 s to 0.26 s is inserted on the right side of the model in increment of 0.04 s. Utilizing the principle of statistical physics, we respectively explored the effects of the fractional-order α and the time delay τ on the stable region of the system. Furthermore, we exhaustively investigated the nonlinear dynamic behaviors of the system with different governor parameters by using bifurcation diagrams, time waveforms and power spectrums, finding that only under the condition of reasonable collocation of governor parameters the system can maintain stable operation. Finally, all of the above numerical experiments supply new methods for studying the stability of a hydropower station.     

General system of -maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms

In this paper, a new system of nonlinear (set-valued) variational inclusions involving (A,η)-maximal relaxed monotone and relative (A,η)-maximal monotone mappings in Hilbert spaces is introduced and its approximation solvability is examined. The notion of (A,η)-maximal relaxed monotonicity generalizes the notion of general η-maximal monotonicity, including (H,η)-maximal monotonicity (also referred to as (H,η)-monotonicity in literature). Using the general (A,η)-resolvent operator method, approximation solvability of this system based on a generalized hybrid iterative algorithm is investigated. Furthermore, for the nonlinear variational inclusion system on hand, corresponding nonlinear Yosida regularization inclusion system and nonlinear Yosida approximations are introduced, and as a result, it turns out that the solution set for the nonlinear variational inclusion system coincides with that of the corresponding Yosida regularization inclusion system. Approximation solvability of the Yosida regularization inclusion system is based on an existence theorem and related Yosida approximations. The obtained results are general in nature.