Adaptive age replacement strategies based on nonparametric predictive inference

We consider an age replacement problem using nonparametric predictive inference (NPI) for the lifetime of a future unit. Based on n observed failure times, NPI provides lower and upper bounds for the survival function for a future lifetime X_n+1, which are lower and upper survival functions in the theory of interval probability, and which lead to upper and lower cost functions, respectively, for age replacement based on the renewal reward theorem. Optimal age replacement times for X_n+1 follow by minimizing these cost functions. Although the renewal reward theorem implicitly assumes that the corresponding optimal strategy will be used for a long period, we study the effect on this strategy when the observed value for X_n+1, which is either an observed failure time or a right-censored observation, becomes available. This is possible due to the fully adaptive nature of our nonparametric approach, and the next optimal strategy will be for X_n+2. We compare the optimal strategies for X_n+1 and X_n+2 both analytically and via simulation studies. Our NPI-based approach is fully adaptive to the data, to which it adds only few structural assumptions. We discuss the possible use of this approach, and indeed the wider importance of the conclusions of this study to situations where one wishes to combine the statistical aspects of estimating a lifetime distribution with the more traditional operational research approach of determining optimal replacement strategies for lifetime distributions that are assumed to be known.     

General Theorem on a Finite Support of Mixed Strategy in the Theory of Zero-Sum Games

A theorem related to the theory of zero-sum games is proved. Rather general assumptions on the payoff function are found that are sufficient for an optimal strategy of one of the players to be chosen in the class of mixed strategies concentrated in at most m + 1 points if the opponent chooses a pure strategy in a finite-dimensional convex compact set and m is its dimension. This theorem generalizes results of several authors, starting from Bohnenblust, Karlin, and Shapley (1950).     

A Note on the Joint Replenishment Problem under Constant Demand

This note considers the joint replenishment inventory problem for N items under constant demand. The frequently-used cyclic strategy (T; k₁, …, k _N ) is investigated: a family replenishment is made every T time units and item i is included in each k _i th replenishment. Goyal proposed a solution to find the global optimum within the class of cyclic strategies. However, we will show that the algorithm of Goyal does not always lead to the optimal cyclic strategy. A simple correction is suggested.     

Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

The paper studies the global convergence of the Jacobi method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A ^[t+3]) ≤ γ S(A ^[t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A ^[t] stands for the matrix obtained from A after t full cycles of the Jacobi method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi method.     

Optimal inspection intervals for safety systems with partial inspections

The introduction of International Standard IEC 61508 and its industry-specific derivatives sets demanding requirements for the definition and implementation of life-cycle strategies for safety systems. Compliance with the Standard is important for human safety and environmental perspectives as well as for potential adverse economic effects (eg, damage to critical downstream equipment or a clause for an insurance or warranty contract). This situation encourages the use of reliability models to attain the recommended safety integrity levels using credible assumptions. During the operation phase of the safety system life cycle, a key decision is the definition of an inspection programme, namely its frequency and the maintenance activities to be performed. These may vary from minimal checks to complete renewals. This work presents a model (which we called ρβ model) to find optimal inspection intervals for a safety system, considering that it degrades in time, even when it is inspected at regular intervals. Such situation occurs because most inspections are partial, that is, not all potential failure modes are observable through inspections. Possible reasons for this are the nature and the extent of the inspection, or potential risks generated by the inspection itself. The optimization criterion considered here is the mean overall availability A _o , but also taking into account the requirements for the safety availability A _s . We consider several conditions that ensure coherent modelling for these systems: sub-systems decomposition, k-out-of-n architectures, diagnostics coverage (observable/total amount of failure modes), dependent and independent failures, and non-negligible inspection times. The model requires an estimation for the coverage and dependent-failure ratios for each component, global failure rates, and inspection times. We illustrate its use through case studies and compare results with those obtained by applying previously published methodologies.     

Multi-Component Ginzburg-Landau Theory: Microscopic Derivation and Examples

This paper consists of three parts. In part I, we microscopically derive Ginzburg–Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator \({K_{T_c}+V}\) to be n-fold degenerate and the resulting GL theory then couples n order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure d-wave order parameters and (b) mixed (s + d)-wave order parameters, in two and three-dimensions. In part III, we present explicit choices of spherically symmetric interactions V which produce the examples in part II. In fact, we find interactions V which produce ground state sectors of \({K_{T_c}+V}\) of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schrödinger operators \({-\nabla^2+V}\), for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions.     

P-positions in modular extensions to Nim

In this paper, we consider a modular extension to the game of Nim, which we call m-Modular Nim, and explore its optimal strategy. In m-Modular Nim, a player can either make a standard Nim move or remove a multiple of m tokens in total. We develop a winning strategy for all m with 2 heaps and for odd m with any number of heaps.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

Game theory with translucent players

A traditional assumption in game theory is that players are opaque to one another—if a player changes strategies, then this change in strategies does not affect the choice of other players’ strategies. In many situations this is an unrealistic assumption. We develop a framework for reasoning about games where the players may be translucent to one another; in particular, a player may believe that if she were to change strategies, then the other player would also change strategies. Translucent players may achieve significantly more efficient outcomes than opaque ones. Our main result is a characterization of strategies consistent with appropriate analogues of common belief of rationality. Common Counterfactual Belief of Rationality (CCBR) holds if (1) everyone is rational, (2) everyone counterfactually believes that everyone else is rational (i.e., all players i believe that everyone else would still be rational even if i were to switch strategies), (3) everyone counterfactually believes that everyone else is rational, and counterfactually believes that everyone else is rational, and so on. CCBR characterizes the set of strategies surviving iterated removal of minimax-dominated strategies, where a strategy \(\sigma \) for player i is minimax dominated by \(\sigma '\) if the worst-case payoff for i using \(\sigma '\) is better than the best possible payoff using \(\sigma \).     

Boundary Crossing Probabilities for General Exponential Families

We consider parametric exponential families of dimension K on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension K. Formally, our result is a concentration inequality that bounds the probability that B ^ψ (θ? _n , θ*) ≥ f(t/n)/n, where θ* is the parameter of an unknown target distribution, θ? _n is the empirical parameter estimate built from n observations, ψ is the log-partition function of the exponential family and B ^ψ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function f is logarithmic, as it is enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when K = 1, while we provide results for arbitrary finite dimension K, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T. L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits.     

Markov decision processes with noise-corrupted and delayed state observations

We consider the partially observed Markov decision process with observations delayed by k time periods. We show that at stage t, a sufficient statistic is the probability distribution of the underlying system state at stage t - k and all actions taken from stage t - k through stage t - 1. We show that improved observation quality and/or reduced data delay will not decrease the optimal expected total discounted reward, and we explore the optimality conditions for three important special cases. We present a measure of the marginal value of receiving state observations delayed by (k - 1) stages rather than delayed by k stages. We show that in the limit as k →∞ the problem is equivalent to the completely unobserved case. We present numerical examples which illustrate the value of receiving state information delayed by k stages.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Pointwise multipliers for Besov spaces of dominating mixed smoothness-Ⅱ

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R~d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R~d).     

Nonparametric adaptive opportunity-based age replacement strategies

We consider opportunity-based age replacement (OAR) using nonparametric predictive inference (NPI) for the time to failure of a future unit. Based on n observed failure times, NPI provides lower and upper bounds for the survival function for the time to failure X_n+1 of a future unit which lead to upper and lower cost functions, respectively, for OAR based on the renewal reward theorem. Optimal OAR strategies for unit n+1 follow by minimizing these cost functions. Following this strategy, unit n+1 is correctively replaced upon failure, or preventively replaced upon the first opportunity after the optimal OAR threshold. We study the effect of this replacement information for unit n+1 on the optimal OAR strategy for unit n+2. We illustrate our method with examples and a simulation study. Our method is fully adaptive to available data, providing an alternative to the classical approach where the probability distribution of a unit's time to failure is assumed to be known. We discuss the possible use of our method and compare it with the classical approach, where we conclude that in most situations our adaptive method performs very well, but that counter-intuitive results can occur.     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

Axiomatisability problems for <Emphasis Type="Italic">S</Emphasis>-posets

Let \(\mathcal{C}\) be a class of ordered algebras of a given fixed type τ. Associated with the type is a first order language L _τ , which must also contain a binary predicate to be interpreted by the ordering in members of \(\mathcal{C}\). One can then ask the question, when is the class \(\mathcal{C}\) axiomatisable by sentences of L _τ ? In this paper we will be considering axiomatisability problems for classes of left S-posets over a pomonoid S (that is, a monoid S equipped with a partial order compatible with the binary operation). We aim to determine the pomonoids S such that certain categorically defined classes are axiomatisable. The classes we consider are the free S-posets, the projective S-posets and classes arising from flatness properties. Some of these cases have been studied in a recent article by Pervukhin and Stepanova. We present some general strategies to determine axiomatisability, from which their results for the classes of weakly po-flat and po-flat S-posets will follow. We also consider a number of classes not previously examined.     

Coordination games on graphs

We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equilibria is 1 in almost all of the special cases for which we have proven strong equilibria to exist. The PoA for pure Nash equilbria turns out to be unbounded, even when we fix the graph on which the coordination game is to be played. For the PoA for k-equilibria, we show that the price of anarchy is between \(2(n-1)/(k-1) - 1\) and \(2(n-1)/(k-1)\). The latter upper bound is tight for \(k=n\) (i.e., strong equilibria). Finally, we consider the problems of computing strong equilibria and of determining whether a joint strategy is a k-equilibrium or strong equilibrium. We prove that, given a coordination game, a joint strategy s, and a number k as input, it is co-NP complete to determine whether s is a k-equilibrium. On the positive side, we give polynomial time algorithms to compute strong equilibria for various special cases.     

On needed reals

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

On implicit function theorems at abnormal points

We consider the equation F(x, σ) = 0, x ∈ K, in which σ is a parameter and x is an unknown variable taking values in a specified convex cone K lying in a Banach space X. This equation is investigated in a neighborhood of a given solution (x*, σ*), where Robinson’s constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson’s constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone K coincides with the whole space X.