随机康托集的概率性质

本文讨论了两类随机康托集。关于一类随机康托集，证明它们相交等价于稳定从属过程；关于另一类随机康托集，该文证明了由它们产生的随机过程的密度函数与稳定从属过程的密度函数有很大的相似性。     

Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type

We consider solitary-wave solutions of equations of Benjamin-Bona-Mahony type. We show that for a large class of equations of BBM type, there do exist stable sets consisting of solitary-wave profile functions. In the case of generalized BBM equations, we found that there are profile functions of stable solitary waves that are not the minimizers of the associated variational problem. Such a phenomenon is not known to exist for equations of Korteweg-de Vries type.     

Improved Exponential Convergence Rates by Oversampling Near the Boundary

Sampling inequalities for smooth functions bound a continuous norm in terms of a discretized norm and an error term that tends to zero exponentially as the discrete data set becomes dense. Improved estimates are derived for discrete point sets that cluster near the boundary, in particular for scattered point sets that are distributed quadratically in a boundary layer, and for tensorized Chebyshev grids. If applied to residuals of stable reconstruction processes, such inequalities yield exponential convergence orders. Our results agree with the observation that exponential deterministic approximation rates are often improved globally if the data sets are distributed more densely near the boundary.     

Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations

Sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations. The investigations are carried out by means of piecewise continuous functions of Lyapunov type and by using Markoff’s sets. We provide an example to demonstrate the effectiveness of our results.     

System-theoretic and algebraic aspects of the rings of stable and proper stable rational functions

A new approach in transfer-function methods for solving a variety of control-theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices. The rings of stable and proper stable rational functions are well known to be Euclidean domains. These rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained. This major difficulty in extending the idea of polynomial models of Fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form. A natural realization theory is thus developed for matrix fraction representations of transfer-function matrices over these rings. As an application of the new theory developed, linear matrix equations (QX+RY=T and QX+YR=T) over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field.     

Active and stable project scheduling

The paper proposes a new classification of schedules for resource-constrained project scheduling problems with minimum and maximum time lags between project activities and regular and different types of nonregular objective functions. The feasible region of the scheduling problems represents a (generally disconnected) union of polytopes. In addition to the well-known concepts of active and semiactive schedules, pseudoactive and quasiactive as well as stable, semistable, pseudostable, and quasistable schedules are introduced. The (quasi-, pseudo-, semi-)active schedules are related to different types of left-shifts of sets of activities and correspond to minimal points of certain subsets of the feasible region. The (quasi-, pseudo-, semi-)stable schedules do not allow oppositely directed shifts and correspond to extreme points of certain subsets of the feasible region. The different sets of schedules contain optimal schedules for project scheduling problems which differ in their objective functions. The correspondence between those sets of schedules and vertices of specific polyhedral subsets of the feasible region can be exploited for analyzing schedule generation schemes which have been developed recently for finding solutions to the different classes of project scheduling problems.     

Modeling optimal social choice: matrix-vector representation of various solution concepts based on majority rule

Various Condorcet consistent social choice functions based on majority rule (tournament solutions) are considered in the general case, when ties are allowed: the core, the weak and strong top cycle sets, versions of the uncovered and minimal weakly stable sets, the uncaptured set, the untrapped set, classes of k-stable alternatives and k-stable sets. The main focus of the paper is to construct a unified matrix-vector representation of a tournament solution in order to get a convenient algorithm for its calculation. New versions of some solutions are also proposed.     

Quasicrystals are sets of stable sampling

Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Betti numbers in multidimensional persistent homology are stable functions

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of scalar‐valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, that is, the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector‐valued filtering functions, we can consider the multidimensional analog of persistent Betti numbers. Varying the lower level sets, we obtain that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non‐negative integers. In this paper, we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector‐valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector‐valued filtering functions and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalar‐valued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers, we obtain a lower bound for the natural pseudo‐distance. Copyright © 2013 John Wiley & Sons, Ltd.     

Symmetry and <Emphasis Type="Italic">p</Emphasis>-Stability

A symmetry of a game is a permutation of the player set and their strategy sets that leaves the payoff functions invariant. In this paper we introduce and discuss two relatively mild symmetry properties for set-valued solution concepts (that are equivalent when the solution concepts are single-valued) and show using examples that stable sets satisfy neither version. These examples also show that for every integer q, there exists a game with an equilibrium component of index q.Received February 2002/Revised November 2003Supported by an EPSRC doctoral grant.     

Near-optimal data-independent point locations for radial basis function interpolation

The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples. AMS subject classification 41A05, 41063, 41065, 65D05, 65D15This work has been done with the support of the Vigoni CRUI-DAAD programme, for the years 2001/2002, between the Universities of Verona and Göttingen.     

Trivariate Cr Polynomial Macroelements

Trivariate C^r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1,2, these spaces reduce to well-known macroelement spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.     

Stable generalization of convex functions

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.     

Sub-topical functions and plus-co-radiant sets

In this paper, we study sub-topical functions in the framework of abstract convexity and examine the relevant properties such as support sets, polar sets and sub-differentials for these functions. Plus-radiant and plus-co-radiant sets, and their relations with sub-topical functions are studied. Applying sub-topical functions, we present some separation theorems for both plus-radiant and plus-co-radiant sets.     

Construction of functions with uniform sublevel sets

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.     

MARS: selecting basis functions and knots with an empirical Bayes method

An empirical Bayes method to select basis functions and knots in multivariate adaptive regression spline (MARS) is proposed, which takes both advantages of frequentist model selection approaches and Bayesian approaches. A penalized likelihood is maximized to estimate regression coefficients for selected basis functions, and an approximated marginal likelihood is maximized to select knots and variables involved in basis functions. Moreover, the Akaike Bayes information criterion (ABIC) is used to determine the number of basis functions. It is shown that the proposed method gives estimation of regression structure that is relatively parsimonious and more stable for some example data sets.     

Reflections on and of minor-closed classes of multisorted operations

The minor relation of functions is generalized to multisorted functions. Pippenger’s Galois theory for minor-closed sets of functions is extended to multisorted functions and multisorted relation pairs. Reflections of minor-closed sets are again minor-closed, and the effect of reflections on the invariant relation pairs of minor-closed sets of multisorted functions is described.     

Stable Border Bases for Ideals of Numerical Cartesian Sets

In this paper, we discuss a special class of sets of bivariate empirical points, namely, numerical cartesian sets. We find that the stable quotient bases for numerical cartesian sets are unique if they exist. Furthermore, the corresponding border bases are the unique stable border bases for the vanishing ideals of numerical cartesian sets.     

Additive functions and their application to uncertain information

For additive functions over ordered sets, a minimum representation is given in case the ordered set has a particular deviation property. Additive functions are known to be special supermodular functions on power sets and in probability calculus, they are related to, for example, capacities and distribution functions of random sets. Their capability to encode vague information is stressed.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.