Abstract: We consider a class of piecewise C 2 Lozi-like maps and prove the existence of invariant measures with absolutely continuous conditional mea­sures on unstable manifolds

Abstract: The concept of coefficient shift matrix is introduced to represent delay variables in block pulse series. The optimal control of a linear delay system with quadratic performance index is then studied via block pulse functions, which convert the problems into the minimization of a quadratic form with linear algebraic equation constraints. The solution of the two-point boundary-value problem with both delay and advanced arguments is circumvented. The control variable obtained is piecewise constant.

Abstract: This paper concerns the optimal stopping time problem for a piecewise deterministic process. The process has deterministic dynamics between random jumps. The as¬sociated dynamic programming equation is a variational inequality with integral and (first order) differential terms. Our main results are W4,00-existence results and probabilistic representations for the solutions of the optimal stopping time problem in bounded domains and in R. We also generalize these results to the case when the state space is “countable folds” of Euclidean space

Abstract: The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment.

Abstract: A process for signal matching. The process is general and can be applied to matching signals of arbitrary dimension. To implement the process, a suitable discretized description of the two signals to be matched is defined. Such descriptions can be the local signal extrema or any other qualitative signal significant points of interest or features. Allowable feature matches and values for matches are defined, and determined for all potential matches. Matches are confined to the features within defined matching windows and are mapped for each significant point. Within a defined similarity disparity window, a neighborhood of potentially interacting matches are evaluated. Matches within a neighborhood contribute to the decision about the appropriate match for each significant point to determine a composite similarity weighted best value match for each point. Mapping is piecewise continuous. The two signals are matched with disparities therebetween resolved and removed responsive to the best match values. Following the coherence process, the overall process of the present invention provides a superposition of several such piecewise transformations. In an illustrated application domain, a stereo correspondence technique provides for matching of figurally similar three dimensional images, according to a simple no iterative, parallel and local process that can successfully detect disparities of opaque as well as transparent surfaces.

Abstract: Fritsch and Carlson [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] developed an algorithm which produces a monotone $C^1 $ piecewise cubic interpolant to a monotone function. We show that their algorithm yields a third-order approximation, while a modification is fourth-order accurate.

Abstract: We consider a finite element method for the Stokes problem on a rectangular domain based on piecewise bilinear velocities and piecewise constant pressures on a uniform rectangular grid. It is shown that by a simple stabilization strategy the method can be implemented in a convergent multigrid procedure. Introduction. One of the most natural ways of discretizing the Stokes equations on a two-dimensional rectangular domain is to apply the finite element techniques with continuous piecewise bilinear velocities and piecewise constant pressures on a rectangular grid. This method is easily extended to more general quadrilateral meshes and it has proved to be quite effective in practice. The convergence analysis of the method was first carried out in (10) on a rectangular domain. In (12) the estimates of (10) are improved and extended to more general quadrilateral meshes. So far, the above method has been used mainly in connection with direct band solvers, usually combined with penalty/perturbation techniques to eliminate the pressure (9). In this paper we consider an iterative variant of the method based on multigrid techniques. Such an algorithm seems attractive, especially since the un- derlying finite-difference equations are relatively simple. Multigrid methods for the Stokes problem and for more general elliptic systems were considered previously by Hackbusch (8) and Verfurth (14). In (14) it is shown that a convergent multigrid algorithm for the Stokes problem can be constructed by appropriately scaling the variables, provided that the underlying finite element scheme satisfies the Babu?ka-Brezzi stability condition (2), (5). In our case, however, the finite element method is not stable in this sense, and we have not been able to show that the straightforward application of the algorithm proposed in (14) yields a convergent process. Therefore, we suggest first modifying the finite element method in such a way that it becomes stable in the ordinary sense. The usual way of stabilizing an unstable mixed method is to add more velocity (or primary) variables. A classical example of this is the quadratic/linear velocity-pres- sure element of Crouzeix and Raviart (6), where added "bubble" functions act as stabilizers in the velocity space. Increasing the dimension of the velocity space would be an alternative in our case. However, we propose another method which seems to yield a simpler algorithm. It is based on adding an extra stabilizing term into the

Abstract: : This document shows that for certain translation invariant spaces S, a necessary and sufficient condition for the eventual denseness of the corresponding scaled spaces S sub h is that S contain a stable and locally supported partition of unity. These results have been motivated by recent work on approximation by multivariate piecewise polynomials on regular meshes. (Author)

Abstract: In this paper we describe high-order accurate Godunov-type schemes for the computation of weak solutions of hyperbolic conservation laws that are essentially non-oscillatory We show that the problem of designing such schemes reduces to a problem in approximation of functions, namely that of reconstructing a piecewise smooth function from its given cell averages to high order accuracy and without introducing large spurious oscillatons To solve this reconstruction problem we introduce a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy wherever the function is smooth but avoids having a Gibbs-phenomenon at discontinuities

Abstract: The rapid convergence characteristics and high accuracy of the three-dimensional piecewise continuous kernel function method are tested on a nonplanar configuration. The nature of the singularity of the three-dimensional kernel function for the nonplanar configuration is examined and anomalies associated with almost adjoining lifting surfaces are explained. Applications are made using the standard AGARD wing-tail configuration and the interference aerodynamic forces are compared with results obtained using other numerical methods. 7 to compute aerodynamic forces on a nonplanar configuration is de- scribed. The nature of the singularity of the nonplanar kernel function is examined and its characteristics are taken into con- sideration while developing the nonplanar version of the PCKFM. The numerical results obtained using the PCKFM for the above-mentioned AGARD examples are compared with results obtained using other methods. analysis as such. All other pressure singularities are ignored during the analysis, and their consideration is limited to the determination of the boundaries between the different boxes. The problems associated with the basic three-dimensional PCKFM were treated in Refs. 5-7. Additional problems which arise from the three-dimensional nonplanar flow configura- tions and which require the formulation of numerical tech- niques for the successful application of the method are ad- dressed in this paper. An example of a wing with geometrical discontinuities divided into boxes is given in Fig. 1. The pressure distribution in each of the boxes formed by the PCKFM can, therefore, be represented in general terms by the following expression:

Abstract: The ground state of a one-dimensional model with two sublattices which is a variation of the piecewise parabola Frenkel-Kontorova model. is explicitly calculated. This model involves an additional parameter (the electric field) which breaks a (non-symmorphic) symmetry element when it is non-zero. At a fixed commensurability ratio :. it is shown that the polarisation curve as a function of the electric field is the sum of a linear part and a staircase, This staircase is either a harmless staircase (: rational) with true first-order transitions or a Devil's staircase ( :irrational). The plateaus of these staircases are obtained when an unusual condition (called subcommensurability condition) which involves the relative phase shift between the two sublattices is fulfilled. The phase diagram of this model as a function of the chemical potential and the electric field is explicitly calculated. It is entirely filled by the domain of stability of phases which are characterised both by a rational commensurability ratio and a polarisation fulfilling the subcommensurability condition. These domains are polygons with an infinite number of edges. At fixed electric field. the commensurability ratio {varies as a function of the chemical potential as a Devil's staircase with plateaus at rational values of :but these ones have widths submitted to selection rules related to the model symmetry. The corresponding curve for the variation of the polarisation is a new 'devilish' function called a Manhattan profile. This curve exhibits infinitely many plateaus but unlike a Devil's staircase is alternatively increasing and decreasing infinitely many times by discontinuities. The results obtained on this model are favorably compared with experimental results in thiourea. Predictions are also given and in particular a scintil- lating variation of the polarisation when the temperature or the pressure varies (which is the consequence of the Manhattan profile). Finally. it is noted that the method used for the exact solution of this model can be extended to a wider class of other models: (1) with several sublattices. (2) with long-range interactions. (3) with lattices at any dimension. which should in the future allow more complex models, which approach more closely the real systems. to be solved exactly.

Abstract: We study a one-step method for delay differential equations, which is equivalent to an implicit Runge-Kutta method. It approximates the solution in the whole interval with a piecewise polynomial of fixed degree n. For an appropiate choice of the mesh points, it provides uniform convergence 0(hn+1) and the superconvergence 0(h2n) at the nodes.

Abstract: SUMMARY This paper presents two families of curves for interpolating and smoothing sequences of points on the surface of the unit sphere. One family is a natural analogue of the usual splines for curve-fitting in the plane; the other is slightly less optimal but more convenient from a computational viewpoint. Data which may be regarded as time-ordered sequences of points on the surface of the sphere arise in a variety of situations in the Earth Sciences. An interesting example of such data is a set of virtual geomagnetic pole positions calculated from rock specimens of different ages, for a single continent. The chronological sequence of these pole positions is known as the Apparent Polar Wander Path. Comparison of such paths for two continents, over the same time period, is some- times used to decide whether the two continents have moved relative to each other over this period. Various authors have discussed the problem of fitting curves piecewise to sequences

Abstract: LetS be a triangulation of ℂ andf(z) = zd +ad−1zd−1+⋯+a0, a complex polynomial. LetF be the piecewise linear approximation off determined byS. For certainS, we establish an upper bound on the complexity of an algorithm which finds zeros ofF. This bound is a polynomial in terms ofn, max{∥ai∥}i, and measures of the sizes of simplices inS.

Abstract: This paper presents a new numerical method for computing the elastic normal surface displacement field caused by a given normal pressure distribution. The pressure function is approximated by a piecewise biquadratic polynomial on the whole domain analyzed, and the deformation of every node is expressed as a linear combination of the nodal pressures whose coefficients can be combined into a deformation matrix. Consequently, the iterative calculation of elastic deformation is simplified and the amount of work is greatly reduced. It has been proved, in addition, that the numerical accuracy of the new method is higher than that of some others.