Monotone Set Research Articles

On the optimal designs for the prediction of Ornstein–Uhlenbeck sheets

Computer simulations are often used to replace physical experiments for exploring the complex relationships between input and output variables. We study the optimal design problem for the prediction of a stationary Ornstein–Uhlenbeck sheet on a monotonic set with respect to the integrated mean square prediction error criterion and the entropy criterion. We show that there is a substantial difference between the shapes of optimal designs for Ornstein–Uhlenbeck processes and sheets. In particular, we show that the optimal prediction based on the integrated mean square prediction error does not necessarily lead to space-filling designs.

Rational splines for Hermite interpolation with shape constraints

This paper is concerned in shape-preserving Hermite interpolation of a given function f at the endpoints of an interval using rational functions. After a brief presentation of the general Hermite problem, we investigate two cases. In the first one, f and f′ are given and it is proved that for any monotonic set of data, it is always possible to construct a monotonic rational function of type [3/2] interpolating those data. Positive and convex interpolants can be computed by a similar method. In the second case, results are proved using rational function of type [5/4] for interpolating the data coming from f, f′ and f″ with the goal of constructing positive, monotonic or convex interpolants. Error estimates are given and numerical examples illustrate the algorithms.

Rank-based tapering estimation of bandable correlation matrices

The nonparanormal model assumes that variables follow a multivariate normal distribution after a set of unknown monotone increasing transformations. It is a flexible generalization of the normal model but retains the nice interpretabil- ity of the latter. In this paper we propose a rank-based tapering estimator for estimating the correlation matrix in the nonparanormal model in which the vari- ables have a natural order. The rank-based tapering estimator does not require knowing or estimating the monotone transformation functions. We establish the rates of convergence of the rank-based tapering under Frobenius and matrix oper- ator norms, where the dimension is allowed to grow at a nearly exponential rate relative to the sample size. Monte Carlo simulation is used to demonstrate the finite performance of the rank-based tapering estimator. A data example is used to illustrate the nonparanormal model and the efficacy of the proposed rank-based tapering estimator.

Hardy–Littlewoodʼs inequalities in the case of a capacity

Hardy–Littlewoodʼs inequalities, well known in the case of a probability measure, are extended to the case of a monotone (but not necessarily additive) set function, called a capacity. The upper inequality is established in the case of a capacity assumed to be continuous and submodular, the lower — under assumptions of continuity and supermodularity.

Abstract Regular Null-Null-Additive Set Multifunctions in Hausdorff Topology

Abstract In this paper we introduce and study various types of abstract regularity in Hausdorff topology for monotone set multifunctions, with direct applications in wellknown situations such as the Borel σ-algebra of a Hausdorff space and/or the Borel (Baire, respectively) σ-ring or σ-ring of a locally compact Hausdorff space. Some relationships among abstract regularities are investigated.

Set-Valued Lebesgue and Riesz Type Theorems

Abstract In this paper, we continue a previous study concerning different types of pseudo-convergences of sequences of measurable functions with respect to set-valued nonadditive monotonic set functions and we establish some pseudo-versions of Lebesgue and Riesz theorems in the set-valued case. We also characterize some important structural properties of fuzzy multimeasures.

Lagrangian cobordism. I

We discuss Lagrangian cobordism between Lagrangian submanifolds (in the monotone setting) and we show that Lagrangian Floer and quantum homologies are rigid with respect to this type of cobordism. We also discuss obstructions to cobordism based on properties of Lagrangian quantum homology, relations to Lagrangian surgery, as well as examples of non-isotopic but cobordant Lagrangians. Finally, Lagrangian cobordism is used to structure the respective class of Lagrangians as a category and the results of the paper are interpreted as indicating the existence of a functor defined on this category with values in an appropriate (derived) Fukaya category and which is compatible with the triangulated structure of the target.

An Extension of Fuzzy Measures to Multisets and Its Relation to Distorted Probabilities

Fuzzy measures are monotonic set functions on a reference set; they generalize probabilities replacing the additivity condition by monotonicity. The typical application of these measures is with fuzzy integrals. Fuzzy integrals integrate a function with respect to a fuzzy measure, and they can be used to aggregate information from a set of sources (opinions from experts or criteria in a multicriteria decision-making problem). In this context, background knowledge on the sources is represented by means of the fuzzy measures. For example, interactions between criteria are represented by means of nonadditive measures. In this paper, we introduce fuzzy measures on multisets. We propose a general definition, and we then introduce a family of fuzzy measures for multisets which we show to be equivalent to distorted probabilities when the multisets are restricted to proper sets.

Generalized positive sets and abstract monotonicity

The theory of q-positive sets on SSD spaces has been introduced by Simons (J Convex Anal, 14:297–317, 2007; From Hahn–Banach to monotonicity, Springer, Berlin, 2008). Monotone sets can be considered as special case of q-positive sets. In this paper, we develop a theory of q-positive sets in the framework of abstract monotonicity. We use generalized Fenchel’s duality theorem and give some criteria for maximality of abstract q-positive sets. Finally, we investigate the relation between abstract q-positive sets and abstract convex functions.

New results on q-positivity

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.

A structure theorem for Boolean functions with small total influences

We show that on every product probability space, Boolean functions with small total inuences are essentially the ones that are almost measurable with respect to certain natural sub-sigma algebras. This theorem in particular describes the structure of monotone set properties that do not exhibit sharp thresholds. Our result generalizes the core of Friedgut’s seminal work on properties of random graphs to the setting of arbitrary Boolean functions on general product probability spaces and improves the result of Bourgain in his appendix to Friedgut’s paper.

Measures of assurance and opportunity in modeling uncertain information

Our focus is on the representation of uncertain information using set measures. We first discuss the basic properties of monotonic set measures. We then discuss the appropriateness of their use in modeling uncertain information. We look at some notable types of measures of uncertain information and investigate in considerable detail cardinality-based measures. We look at the Sugeno measure and provide a formulation of the underlying cardinality-based measures. We then look at quasi-additive uncertainty measures. We discuss the entropy and attitudinal character of an uncertainty measure. Finally, we introduce the ideas of the assurance and opportunity of the occurrence of an outcome. © 2012 Wiley Periodicals, Inc.

Entailment Principle for Measure-Based Uncertainty

We describe the use of monotonic set measures, which are sometimes referred to as fuzzy measures, in the representation of uncertain information about a variable. The representation of significant types of uncertain information, such as probabilistic, possibilistic, and others, is discussed. The concept of the dual of a measure is introduced and is used to define the measures of opportunity and assurance associated with the occurrence of an event. We observe that these generalize the ideas of possibility and necessity in possibility theory, as well as the ideas of plausibility and belief in Dempster-Shafer theory. Using the measures of opportunity and assurance, we are able to introduce a concept of entailment between two uncertainty representations, which generalizes Zadeh's idea of entailment between possibility distributions. We show, using the idea of entailment, how to relate possibilistic and probabilistic uncertainty and discuss a potential application in hard/soft information fusion. We, finally, consider the measure representation of joint variables.

A new iterative scheme with nonexpansive mappings for equilibrium problems

In this paper, we suggest a new iteration scheme for finding a common of the solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed points of a nonexpansive mapping. The scheme is based on both hybrid method and extragradient-type method. We obtain a strong convergence theorem for the sequences generated by these processes in a real Hilbert space. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature. AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C25, 90 C33.

Geometric influences

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn–Kalai–Linial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small “influence sum” are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in ℝn of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on ℝn and the class of sets invariant under transitive permutation group of the coordinates.

Measure based representation of uncertain information

We discuss the use of monotonic set measures for the representation of uncertain information. We look at some important examples of measure-based uncertainty, specifically probability and possibility and necessity. Others types of uncertainty such as cardinality based and quasi-additive measures are discussed. We consider the problem of determining the representative value of a variable whose uncertain value is formalized using a monotonic set measure. We note the central role that averaging and particularly weighted averaging operations play in obtaining these representative values. We investigate the use of various integrals such as the Choquet and Sugeno for obtaining these required averages. We suggest ways of extending a measure defined on a set to the case of fuzzy sets and the power sets of the original set. We briefly consider the problem of question answering under uncertain knowledge.

On Prioritized Multiple-Criteria Aggregation

We describe multicriteria aggregation and discuss its central role in many modern applications. The concept of aggregation imperative is introduced to indicate the description of how the individual criteria satisfactions should be combined to obtain the overall score. We focus on a particular type of aggregation imperative called prioritized aggregation that is characteristic of situations where lack of satisfaction to criteria denoted as higher priority cannot be compensated by increased satisfaction by those denoted as lower priority. We discuss two approaches to the formulation of this type of aggregation process. One of these uses the prioritized aggregation operator, and the second is based on an integral-type aggregation using a monotonic set measure to convey the prioritized imperative.

Set-valued Lusin type theorem for null–null-additive set multifunctions

In a previous paper, we presented a set-valued Egoroff type theorem for monotone set multifunctions. In this paper, a new version of this set-valued Egoroff type theorem is obtained. It enables a set-valued Lusin type theorem for null–null-additive monotone continuous set multifunctions to be established. Several applications of this Lusin type theorem are given. Pointing out some relationships among μ-measurability, A-measurability and totally measurability of real functions, we also compare our new Lusin type theorem with Lusin's theorem previously obtained by the second author.

Cardinal invariants of monotone and porous sets

AbstractA metric space (X, d) ismonotoneif there is a linear order < onXand a constantcsuch thatd(x, y)≤c d(x, z)for allx<y<zinX. We investigate cardinal invariants of theσ-idealMongenerated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are also investigated. In particular, we show that non(Mon) ≥mσ-linked, but non(Mon) <mσ-centeredis consistent. Also cov(Mon) <cand cof (N) < cov(Mon) are consistent.

Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities

We introduce a new iteration method and prove strong convergence theorems for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of monotone and Lipschitz-type continuous Ky Fan inequality. Under certain conditions on parameters, we show that the iteration sequences generated by this method converge strongly to the common element in a real Hilbert space. Some preliminary computational experiences are reported.