Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

On the stability of the Motzkin representation of closed convex sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple (C,D) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization

This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements—normals and/or subdifferentials.

Cyclically-arranged, storm-controlled, prograding lithosomes in Messinian terrigenous shelves (Bajo Segura Basin, western Mediterranean)

This work focuses on a Messinian shallow-marine terrigenous unit, termed the La Virgen Formation, which forms part of the sedimentary infill of the Bajo Segura Basin (Betic margin of the western Mediterranean). This formation was deposited during a high sea level phase prior to the onset of the Messinian Salinity Crisis. Stratigraphically, it comprises a prograding stack of sandstone lithosomes alternating with marly intervals (1st-order cyclicity). These lithosomes are characterized by a homoclinal geometry that tapers distally, and interfinger with pelagic sediments rich in planktonic and benthic microfauna (Torremendo Formation). An analysis of sedimentary facies of each lithosome reveals a repetitive succession of sandy storm beds (tempestites), occasionally amalgamated, which are separated by thin marly layers (2nd-order cyclicity). Each storm bed contains internal erosional surfaces (3rd-order cyclicity) that delimit sets of laminae. Two categories of storm beds have been differentiated. The first one includes layers formed below storm wave base (SWB), characterized by traction structures associated to unidirectional flows (scoured base, planar lamination, and parting lineation). The second category consists of layers deposited above the SWB which display typical high regime oscillatory flow structures (swaley and hummocky cross lamination). In both cases, the ichnological record is characterized by an oligotypic association of Ophiomorpha nodosa, which can be interpreted as the result of allochthonous tracemakers (crustaceans) transported during storm events together with the sediment. The benthic microfauna in the marly intervals that separate the sandstone lithosomes (1st-order cyclicity) indicates that the storm ebb surges were deposited at depths ranging from those of inner shelf settings (with Elphidium spp. and Cibicides lobatulus) to those of outer shelf (with Valvulineria complanata and Uvigerina cylindrica). At the distal end of the sandstone lithosomes, the planktonic microfauna is characterized by a high content of taxa indicative of warm-oligotrophic waters (Globigerinoides obliquus and Globigerinoides bulloideus). In contrast, in the marly intervals, the microfauna is dominated by species typical of cold-eutrophic waters (Globigerina and Neogloboquadrina). This alternation of planktic foraminiferal assemblages is interpreted as being the expression of climatic cycles, in which every episode of progradation of tempestite-dominated lithosomes corresponds to maximum insolation and warm waters, whereas episodes of marly deposition correspond to minimal insolation and cold waters. The 1st-order cyclicity recorded in the La Virgen Formation, in a context of terrigenous storm-dominated shelf, corresponds to sapropel/homogeneous marl cycles formed in a pelagic basin (Torremendo Fm). These cycles in pelagic sediments are commonplace throughout the Mediterranean during the Messinian and reflect precession orbital changes: repeated periods of maximum insolation – minimum precession (sapropels) and minimal insolation – maximum precession (homogeneous marls). The fact that the example of terrigenous unit studied herein is coetaneous with the well-developed reef complexes in the Mediterranean basins points out the importance of sediment supply in the formation of large-scale sandy lithosomes. This is a crucial aspect to understanding reservoir genesis as well as lateral stratigraphic relationships with potential seal and/or source rocks.

Pitfalls in the Evaluation of the Thermodynamic Consistency of Experimental VLE Data Sets

The thermodynamic consistency of almost 90 VLE data series, including isothermal and isobaric conditions for systems of both total and partial miscibility in the liquid phase, has been examined by means of the area and point-to-point tests. In addition, the Gibbs energy of mixing function calculated from these experimental data has been inspected, with some rather surprising results: certain data sets exhibiting high dispersion or leading to Gibbs energy of mixing curves inconsistent with the total or partial miscibility of the liquid phase, surprisingly, pass the tests. Several possible inconsistencies in the tests themselves or in their application are discussed. Related to this is a very interesting and ambitious initiative that arose within the NIST organization: the development of an algorithm to assess the quality of experimental VLE data. The present paper questions the applicability of two of the five tests that are combined in the algorithm. It further shows that the deviation of the experimental VLE data from the correlation obtained by a given model, the basis of some point-to-point tests, should not be used to evaluate the quality of these data.

The minimal overlap rule: restrictions on mergers for creditors’ consensus

As it is known, there is no rule satisfying additivity in the complete domain of bankruptcy problems. This paper proposes a notion of partial additivity in this context, to be called μ-additivity. We find out that this property, together with two quite compelling axioms, equal treatment of equals and continuity, identify the minimal overlap rule, introduced by O’Neill (Math. Soc. Sci. 2:345–371, 1982).

Algorithm for the detection of outliers based on the theory of rough sets

Outliers are objects that show abnormal behavior with respect to their context or that have unexpected values in some of their parameters. In decision-making processes, information quality is of the utmost importance. In specific applications, an outlying data element may represent an important deviation in a production process or a damaged sensor. Therefore, the ability to detect these elements could make the difference between making a correct and an incorrect decision. This task is complicated by the large sizes of typical databases. Due to their importance in search processes in large volumes of data, researchers pay special attention to the development of efficient outlier detection techniques. This article presents a computationally efficient algorithm for the detection of outliers in large volumes of information. This proposal is based on an extension of the mathematical framework upon which the basic theory of detection of outliers, founded on Rough Set Theory, has been constructed. From this starting point, current problems are analyzed; a detection method is proposed, along with a computational algorithm that allows the performance of outlier detection tasks with an almost-linear complexity. To illustrate its viability, the results of the application of the outlier-detection algorithm to the concrete example of a large database are presented.