On the shape of the mean residual lifetime function

Some general properties of the mean residual life (MRL) function are studied. The analysis is based on the shape of the corresponding failure rate. The conditions under which the failure rate and the reciprocal to the MRL function have asymptotically equivalent behaviour as t→∞ are discussed. The simplest non‐monotone shapes of the functions under consideration (bathtub and upside down bathtub) are also considered. The MRL functions for mixtures of distributions are described via the corresponding conditional probability density functions. The direct proportional model of mixing is characterized and some asymptotic results on the shape of the mixture MRL are obtained. Some simple examples are given. Copyright © 2002 John Wiley & Sons, Ltd.     

Connections between Size Functions and Critical Points

A mathematical approach to the concept of shape of a submanifold ℳ︁ of a Euclidean space had previously been given by means of ‘measuring functions’ (e.g. diameter or volume) and of the derived ‘size functions’. This paper relates the study and the computation of any such size function to the structure of critical points of the associated measuring function.     

Contracting convex hypersurfaces by curvature

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where this does not occur, non-uniformly convex regions and singular parts in the initial hypersurface may persist, including flat sides, ridges of infinite curvature, or ‘cylindrical’ regions where some of the principal curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in such cases the hypersurface may even collapse to a line segment or higher-dimensional disc rather than to a point. We provide sufficient conditions for these various disasters to occur, and by avoiding these arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become smooth and uniformly convex and contract to points.     

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??The mean residual life (MRL) function plays a very important role in the area of reliability engineering, survival analysis, and many other fields. In this paper, we introduce and study a new stochastic order which gives stochastic comparison for mean residual life of strictly increasing concave function of two random variables. We show that this new stochastic order lies between the hazard rate and mean residual life orders. The preservation properties under mixtures are presented here. Finally, we give some applications of this new order in reliability theory.     

Large sets in intuitionistic set theory

We consider properties of sets in an intuitionistic setting corresponding to large cardinals in classical set theory. Adding such ‘large set axioms’ to intuitionistic ZF set theory does not violate well-know metamathematical properties of intuitionistic systems. Moreover, we consider statements in constructive analysis equivalent to the consistency of such ‘large set axioms’.     

Ray equations of a weak shock in a hyperbolic system of conservation laws in multi-dimensions

In this paper we give a complete proof of a theorem, which states that ‘for a weak shock, the shock ray velocity is equal to the mean of the ray velocities of nonlinear wavefronts just ahead and just behind the shock, provided we take the wavefronts ahead and behind to be instantaneously coincident with the shock front. Similarly, the rate of turning of the shock front is also equal to the mean of the rates of turning of such wavefronts just ahead and just behind the shock’. A particular case of this theorem for shock propagation in gasdynamics has been used extensively in applications. Since it is useful also in other physical systems, we present here the theorem in its most general form.     

Harmonic Properties of the Logarithmic Potential and the Computability of Elliptic Fekete Points

We investigate the properties of the function sending each N-tuple of points to minus the logarithm of the product of their mutual distances. We prove that, as a function defined on the product of N spheres, this function is subharmonic, and indeed its (Riemannian) Laplacian is constant. We also prove a mean value equality and an upper bound on the derivative of the function. We use these results to get sharp upper bounds for the precision needed to describe an approximation to elliptic Fekete points (in the sense demanded by Smale’s 7th problem). We also conclude that Smale’s 7th problem has solutions given by rational spherical points of bounded (small) bit length, proving that there exists an exponential running time algorithm which solves it on the Turing machine model.     

On the Immersion Relation and an Embedding Problem for Infinite Graphs

Recent results on the immersion relation for graphs are applied to show that there exist (uncountable) graphs A and G such that G contains n disjoint copies of A for each positive integer n, but the graph N₀A consisting of infinitely many disjoint copies of A does not embed in G, where by ‘embedding’ we mean ‘immersion in the weak sense’.     

Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation

We present a pathfollowing strategy based on pseudo-arclength parametrization for the solution of parameter-dependent boundary value problems for ordinary differential equations. We formulate criteria which ensure the successful application of this method for the computation of solution branches with turning points for problems with an essential singularity. The advantages of our approach result from the possibility to use efficient mesh selection, and a favorable conditioning even for problems posed on a semi-infinite interval and subsequently transformed to an essentially singular problem. This is demonstrated by a Matlab implementation of the solution method based on an adaptive collocation scheme which is well suited to solve problems of practical relevance. As one example, we compute solution branches for the complex Ginzburg-Landau equation which start from non-monotone ‘multi-bump’ solutions of the nonlinear Schrödinger equation. Following the branches around turning points, real-valued solutions of the nonlinear Schrödinger equation can easily be computed.     

Study of the plant recolonization process in a burnt heathland using a method of hierarchical classification

The reinstallation of different plant-species and their evolution during a ten-year period in a heathland after a fire accident has been studied by using the algorithm of heirarchical classification based on correlation (ABC) introduced by Tallur.¹ Classification under contiguity restraint of the set of observation points (subintervals of an experimental observation line) enables one to determine the ‘patches’ having uniform vegetation structure. Lerman's ‘local’ and ‘global’ statistics are used to condense the classification tree to its significant nodes and to choose the most significant partition. x² statistics are proposed to test whether a given patch has a ‘significant’ vegetation structure and if the association between a patch and a plant species is significant. Evolution of the horizontal structure of vegetation is studied by comparing the sets of patches obtained at successive observation dates and the corresponding dominant species.     

Hypersonic flow past a sphere

The effects of dissociation or ionization of air on the analytical solution for hypersonic flow past a sphere are considered here, under certain assumptions. It has been assumed that the shock wave is in the shape of a sphere, that the density ratio across the shock is constant, that the flow behind the shock is at constant density and that dissociation or ionization only occurs behind the shock wave. Thus the effects of the compressibility of the air, variation of density ratio along the shock, and the department of the shock shape from being circular are not taken into account. Here the velocity, pressure, temperature, pressure coefficient and vorticity, etc., at any point between the shock and the surface of the sphere in the presence of dissociation or ionization are obtained. In addition, shock detachment distance, drag coefficient, stagnation point velocity gradient and sonic points on the shock and the surface have also been obtained. The results have been compared with the corresponding results obtained in the case when dissociation or ionization does not occur behind the shock.     

Conditioning in fuzzy environment

The concepts of ‘conditional fuzzy algebra’ and ‘conditional evaluation of possibility’ are introduced when the conditioning event is either in the initial algebra or does not belong to it. In the latter case a ‘wider’ fuzzy environment in which ‘old’ and ‘new’ elements could be considered together is constructed, and a new evaluation of possibility which allows to evaluate the elements of such new environment is defined.     

Stochastic Comparisons of Symmetric Sampling Designs

We compare estimators of the integral of a monotone function f that can be observed only at a sample of points in its domain, possibly with error. Most of the standard literature considers sampling designs ordered by refinements and compares them in terms of mean square error or, as in Goldstein et al. (2011), the stronger convex order. In this paper we compare sampling designs in the convex order without using partition refinements. Instead we order two sampling designs based on partitions of the sample space, where a fixed number of points is allocated at random to each partition element. We show that if the two random vectors whose components correspond to the number allocated to each partition element are ordered by stochastic majorization, then the corresponding estimators are likewise convexly ordered. If the function f is not monotone, then we show that the convex order comparison does not hold in general, but a weaker variance comparison does.     

On some measures and distances for positive random variables

A number of conventional measures of risk as real‐valued functions on the space of positive random variables are considered: the expected shortfall, the mean excess over the threshold, the stop‐loss and some others. Ordering of risks, based on these measures and the distances between corresponding distribution functions, are described. The perturbed measures, describing the effect of changing environment, are discussed. These measures are defined by the accelerated life and proportional hazards models widely used in reliability and survival analysis. The case of a random environment is of a prime interest in the paper. The main result states that if, for instance, the stochastic environment is ‘neutral in expectation’ with respect to the baseline one, the distance between the corresponding distribution functions can be still sufficiently large. Copyright © 2003 John Wiley & Sons, Ltd.     

Class invariants from a new kind of Weber-like modular equation

A technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely ‘algebraic’ method. They are obtained by means of specialising certain modular equations related to Weber’s modular equations of ‘irrational type’. The technique works for certain eta quotients evaluated at points in an imaginary quadratic field with discriminant \(d \equiv 1 \pmod {8}\).     

Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework

Standard second order sufficient conditions in optimal control theory provide not only the information that an extremum is a weak local minimizer, but also tell us that the extremum is locally unique. It follows that such conditions will never cover problems in which the extremum is continuously embedded in a family of constant cost extrema. Such problems arise in periodic control, when the cost is invariant under time translations, in shape optimization, where the cost is invariant under Euclidean transformations (translations and rotations of the extremal shape), and other areas where the domain of the optimization problem does not really comprise elements in a linear space, but rather an equivalence class of such elements. We supply a set of sufficient conditions for minimizers that are not locally unique, tailored to problems of this nature. The sufficient conditions are in the spirit of earlier conditions for ‘non-isolated’ minima, in the context of general infinite dimensional nonlinear programming problems provided by Bonnans, Ioffe and Shapiro, and require coercivity of the second variation in directions orthogonal to the constant cost set. The emphasis in this paper is on the derivation of directly verifiable sufficient conditions for a narrower class of infinite dimensional optimization problems of special interest. The role of the conditions in providing easy-to-use tests of local optimality of a non-isolated minimum, obtained by numerical methods, is illustrated by an example in optimal control.     

The directed polymer in a random environment

A continuous space/time approximation of the well known ‘directed polymer’ problem is considered. Connection between the ‘Helmholtz Free Energy’ and the ‘Two Walker problem’ is shown. Rigorous proof of the superdiffusive mean squared displacement exponent of 4/3 is given when there is one space dimension and one time dimension. Asymptotically diffusive behaviour of c(k)tis shown when there are one ‘time’ and two ‘space’ dimensions. For higher dimensions, the behaviour is diffusive and the mean squared displacement is asymptotically t d. These results hold for all temperature, because the phase transition in the discrete model is no longer present in the continuous model; the renormalization procedure has set the transition temperature to k _crit =0The joint distribution is also shown to be asymptotically sub-Gaussian for all dimensions and all temperatures (in the sense that the p ^thmoments as a function of pincrease more slowly than the moments of a Gaussian distribution). The ‘Helmholtz Free Energy’ is also calculated for this model and the quenched and annealed free energies are shown to be identical for all temperature     

Ternary Operations as Primitive Notions for Constructive Plane Geometry IV

In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols (to be interpreted as ‘points’). We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries. Mathematics Subject Classification: 03F65, 51M05, 51M15, 03B30.     

Dense sets and the projection theorem for acoustic harmonic waves in a homogeneous finite depth ocean

The scattering of a ‘plane wave’ off a submerged body situated in an ocean of finite depth is investigated. The index of refraction is considered to be depth-independent. It is shown that the far field is not unique; hence, the problem of determining the shape of an object from its far field is not well-posed. If solutions are sought among a restricted class of problems the ‘dense set’ property implies that the problem can be made well-posed.