On nonatomicity for non-additive functions

The notion of nonatomicity for set functions plays a key role in classical measure theory and its applications. For classical measures taking values in finite dimensional Banach spaces, it guarantees the connectedness of range. Even just replacing σ-additivity with finite additivity for measures requires some stronger nonatomicity property for the same conclusion to hold. In the present paper, we deal with non-additive functions – called ‘s-outer’ and ‘quasi-triangular’ – defined on rings and taking values in Hausdorff topological spaces. No algebraic structure is required on their target spaces. In this context, we make use of a notion of strong nonatomicity involving just the behavior of functions on ultrafilters of their underlying Boolean domains. This notion is proved to be equivalent to that proposed in earlier contributions concerning Lyapunov-types theorems in additive and non-additive frameworks. Thus, in particular, our analysis allows to generalize, improve and unify several known results on this topic.     

Additive representations of non-additive measures and the choquet integral

This paper studies some new properties of set functions (and, in particular, “non-additive probabilities” or “capacities”) and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results:

the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand;

the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures;

under fairly general conditions one may define a “Radon-Nikodym derivative” of one capacity with respect to another;

the “optimistic” pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space.

We also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.     

Allocation of risks and equilibrium in markets with finitely many traders

The optimal risk allocation problem, equivalently the optimal risk sharing problem, in a market with n traders endowed with risk measures ?₁,…,?_n is a classical problem in insurance and mathematical finance. This problem however only makes sense under a condition motivated from game theory which is called Pareto equilibrium. There are many situations of practical interest, where this condition does not hold. This is the case if the risk measures are based on essential different views towards risk. In this paper we introduce and analyze a meaningful extension of the optimal risk allocation (risk sharing) problem without assuming the equilibrium condition. The main point of this is to introduce a suitable and well motivated restriction on the class of admissible allocations which prevents effects of artificial ‘risk arbitrage’. As a result we obtain a new coherent risk measure which describes the inherent risk which remains after using admissible risk exchange in an optimal way.     

Planar maximal covering location problem under block norm distance measure

This paper introduces a new model for the planar maximal covering location problem (PMCLP) under different block norms. The problem involves locating g facilities anywhere on the plane in order to cover the maximum number of n given demand points. The generalization, in this paper, is that the distance measures assigned to facilities are block norms of different types and different proximity measures. First, the PMCLP under different block norms is modelled as a maximum clique partition problem on an equivalent multi-interval graph. Then, the equivalent graph problem is modelled as an unconstrained binary quadratic problem (UQP). Both the maximum clique partition problem and the UQP are NP-hard problems; therefore, we solve the UQP format through a genetic algorithm heuristic. Computational examples are given.     

Lebesgue decomposition by σ-null-additive set functions using ideals

Using V. Ficker’s and P. Capek’s algebraic approach by ideals a Lebesgue decomposition theorem for a wide class of non-additive set functions called null-additive set functions is obtained. In special cases decompositions theorems for ⊕-decomposable measures andk-triangular set functions are obtained.     

Products of Capacities Derived from Additive Measures

One of the unanswered questions in non-additive measure theory is how to define product of non-additive measures. Most of the approaches that have already been presented only work for discrete measures. In this paper a new approach is presented for not necessarily discrete non-additive measures that are in a certain relation with additive measures, usually this means that they are somehow derived from the additive measures.     

Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral

Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach.     

A generalization of local divergence measures

In this paper we propose a generalization of the concept of the local property for divergence measures. These new measures will be called g-local divergence measures, and we study some of their properties. Once this family is defined, a characterization based on Ling’s theorem is given. From this result, we obtain the general form of g-local divergence measures as a function of the divergence in each element of the reference set; this study is divided in three parts according to the cardinality of the reference set: finite, infinite countable or non-countable. Finally, we study the problem of componible divergence measures as a dual concept of g-local divergence measures.     

Randomized vs. deterministic distance query strategies for point location on the line

Suppose that n points are located at n mutually distinct but unknown positions on the line, and we can measure their pairwise distances. How many measurements are needed to determine their relative positions uniquely? The problem is motivated by DNA mapping techniques based on pairwise distance measures. It is also interesting by itself for its own and surprisingly deep. Continuing our earlier work on this problem, we give a simple randomized two-round strategy that needs, with high probability, only (1+o(1))n measurements. We show that deterministic strategies cannot manage the task in two rounds with (1+o(1))n measurements in the worst case. We improve an earlier deterministic bound to roughly 4n/3 measurements.     

Values and cores of fuzzy games with infinitely many players

The problem of the existence ofvalues (FA-valued, linear, positive, symmetric and efficient operators) on symmetric spaces of “fuzzy games” (that is, ideal set functions of bounded variation) arises naturally from [8], [18], [23] and [2], [3], [4] where it is implicitely approached for technical purposes. In our present work, this problem is approached in itself for the main reason that it is essentially related with the problem of the existence of significant countable additive measures lying in the cores of the “market games”. In fact, it is shown here that there exists a continuous value on the closed subspacebv′ICA ofIBV spanned by thebv′ functions of “fuzzy probability measures” ([9]), this values is “diagonal” onpICA, the closed subspace ofbv′ICA spanned by the natural powers of the fuzzy measures and this is used to prove the main result stating that the cooperative markets contained inpICA have unique fuzzy measures in their cores which are exactly the corresponding diagonal values. This result is of interest because it is providing a tool of determiningCA measures lying in the cores of large classes of games which are not necessarily “non-atomic” and, specially, because it is opening a way toward a new approach of the “Value Equivalence Principle” for differentiable markets with a continuum of traders which are not “perfectly competitive”.     

An integer linear programming approach for approximate string comparison

We introduce a problem called maximum common characters in blocks (MCCB), which arises in applications of approximate string comparison, particularly in the unification of possibly erroneous textual data coming from different sources. We show that this problem is NP-complete, but can nevertheless be solved satisfactorily using integer linear programming for instances of practical interest. Two integer linear formulations are proposed and compared in terms of their linear relaxations. We also compare the results of the approximate matching with other known measures such as the Levenshtein (edit) distance.     

On the influence of the prior distribution in image reconstruction

Two measures of the influence of the prior distribution p(θ) in Bayes estimation are proposed. Both involve comparing with alternative prior distributions proportional to p(θ) ^s , for s  ≥  0. The first one, the influence curve for the prior distribution, is simply the curve of parameter values which are obtained as estimates when the estimation is made using p(θ) ^s instead of p(θ). It measures the overall influence of the prior. The second one, it the influence rate for the prior, is the derivative of this curve at s = 1, and quantifies the sensitivity to small changes or inaccuracies in the prior distribution. We give a simple formula for the influence rate in marginal posterior mean estimation, and discuss how the influence measures may be computed and used in image processing with Markov random field priors. The results are applied to an image reconstruction problem from visual field testing and to a stylized image analysis problem.     

K-Harmonic means data clustering with tabu-search method

Clustering is a popular data analysis and data mining technique. Since clustering problem have NP-complete nature, the larger the size of the problem, the harder to find the optimal solution and furthermore, the longer to reach a reasonable results. A popular technique for clustering is based on K-means such that the data is partitioned into K clusters. In this method, the number of clusters is predefined and the technique is highly dependent on the initial identification of elements that represent the clusters well. A large area of research in clustering has focused on improving the clustering process such that the clusters are not dependent on the initial identification of cluster representation. Another problem about clustering is local minimum problem. Although studies like K-Harmonic means clustering solves the initialization problem trapping to the local minima is still a problem of clustering. In this paper we develop a new algorithm for solving this problem based on a tabu search technique—Tabu K-Harmonic means (TabuKHM). The experiment results on the Iris and the other well known data, illustrate the robustness of the TabuKHM clustering algorithm.     

The problem of harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1-1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures.     

Positivity of Riesz functionals and solutions of quadratic and quartic moment problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y, we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K⊆Rn if and only if the associated Riesz functional Ly is K-positive. For a determining set K, we prove that if Ly is strictly K-positive, then y admits a representing measure supported in K. As a consequence, we are able to solve the truncated K-moment problem of degree k in the cases: (i) (n,k)=(2,4) and K=R²; (ii) n?1, k=2, and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.     

Hessian Measures III

In this paper, we continue previous investigations into the theory of Hessian measures. We extend our weak continuity result to the case of mixed k-Hessian measures associated with k-tuples of k-convex functions, on domains in Euclidean n-space, k=1,2,…,n. Applications are given to capacity, quasicontinuity, and the Dirichlet problem, with inhomogeneous terms, continuous with respect to capacity or combinations of Dirac measures.     

下模(上模)不可加测度的条件期望(英文)

本文研究了下模(上模)不可加测度的条件期望.利用下模不可加测度μ的Choquet积分的最大可加表示定理定义了下模(上模)不可加测度的条件期望, 并且证明了这种条件期望的相关性质.     

Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the (N ? 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N ? 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N ? 1 particles move approximately on a large circular orbit of the two body problems and the remaining N ? 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.