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.

The episcopal complex of Eio-El Tolmo de Minateda (Hellín, Albacete, Spain). Architecture and spatial organization. 7th to 8th centuries AD

The episcopal complex of Eio, located in El Tolmo de Minateda, was built between the end of the 6th century and the beginning of the 7th century, possibly as a political decision taken by the ecclesiastical authority in the capital of the Visigothic kingdom (Toletum). With the comprehensive study of the whole complex presented below (construction cycles, furniture, decoration and location of spaces), we can interpret the function of each space in the basilica and the domus episcopi, the liturgical and general movement routes, the existence of some hierarchical environments, and specify the chronological development of the buildings. After the Arab-Berber conquest of Hispania in the early 8th century, the whole complex will experience a series of transformations that will convert the religious and monumental public area into a private, residential and industrial Islamic quarter.

Mexico City, Mexico, 1913 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: Plano de la Ciudad de México. It was published by American Book and Printing C[o]. in 1913. Scale [ca. 1:21,000]. Covers a portion of Mexico City, Mexico. Map in Spanish.The image inside the map neatline is georeferenced to the surface of the earth and fit to the Universal Transverse Mercator (UTM Zone 14N, meters, WGS 1984) projected coordinate system. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. This map is an outline map showing features such as roads, parks, built-up areas with selected public buildings, city district boundaries, and more. Relief shown by hachures. Includes also inset map "Colonia Romero Rubio."This layer is part of a selection of digitally scanned and georeferenced historic maps from the Harvard Map Collection. These maps typically portray both natural and manmade features. The selection represents a range of originators, ground condition dates, scales, and map purposes.

Valle de México, Mexico, Hydrography, 1862 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: Carta hidrográfica del Valle de México, levantada de órden del Ministerio de Fomento por los ingenieros Miguel Iglesias, Ramon Almaraz, Mariano Santa-María y José Antonio de la Peña, bajo la direccion del Ingeniero Geógrafo Francisco Diaz Covarrubias antiguos alumnos del Colegio Nacional de Minería 1862 ; Ramón Almaraz delineo. It was published by La Sociedad in 1863. Scale 1:80,000. Covers the Valley of Mexico region. Map in Spanish. The image inside the map neatline is georeferenced to the surface of the earth and fit to the 'World Mercator' projection. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. This map shows features such as drainage, canals, aqueducts, cities, towns, and other human settlements, selected buildings, ranches and properties, built-up areas, roads, ground cover, and more. Relief shown by hachures. Depths shown by soundings. Includes 2 profiles: Plano de Comparacion -- Perfil de los acueductos with chart entitled "Tabla comparativa de alturas de los puntos principales del perfil." This layer is part of a selection of digitally scanned and georeferenced historic maps from the Harvard Map Collection and the Harvard University Library as part of the Open Collections Program at Harvard University project: Organizing Our World: Sponsored Exploration and Scientific Discovery in the Modern Age. Maps selected for the project correspond to various expeditions and represent a range of regions, originators, ground condition dates, scales, and purposes.

Chromatographic fractionation of bitumen from sediments and presence of glycerol ethers in immature sediments (Table 2)

Alkanes having unusual saturated isoprenoidal and methyl-branched structures have been isolated from the bitumen of several sediments. The methanogenic biomarkers 2,6,10,15,19-pentamethyleicosane and squalane were found in sediments which also contained bacteriogenic glycerol ethers. However, in one ether-containing sediment, 2,6,10,13,17,21-hexamethyldocosane was tentatively identified and this compound was found in place of the established alkane biomarkers. Other hydrocarbons found were regular C21 and C23 isoprenoid alkanes, compounds which cannot be derived from phytol; two isoprenoids of the type 3,7,11.-polymethylalkane, previously reported only in petroleums; 8-methylheptadecane, a probable biomarker for cyanobacteria and a number of its homologs and other methyl-branched alkanes. Ubiquitous branched-chain alkylbenzenes and a homology of trimethylalkylbenzenes were also isolated. Most, or all, of the compounds reported here are probably not catagenetic products but may represent direct algal or bacterial bioinputs.

(Table 1) Thermal conductivities of lithified core samples from ODP Leg 166 sites

The geothermal regime of the western margin of the Great Bahama Bank was examined using the bottom hole temperature and thermal conductivity measurements obtained during and after Ocean Drilling Program (ODP) Leg 166. This study focuses on the data from the drilling transect of Sites 1003 through 1007. These data reveal two important observational characteristics. First, temperature vs. cumulative thermal resistance profiles from all the drill sites show significant curvature in the depth range of 40 to 100 mbsf. They tend to be of concave-upward shape. Second, the conductive background heat-flow values for these five drill sites, determined from deep, linear parts of the geothermal profiles, show a systematic variation along the drilling transect. Heat flow is 43-45 mW/m**2 on the seafloor away from the bank and decreases upslope to ~35 mW/m**2. We examine three mechanisms as potential causes for the curved geothermal profiles. They are: (1) a recent increase in sedimentation rate, (2) influx of seawater into shallow sediments, and (3) temporal fluctuation of the bottom water temperature (BWT). Our analysis shows that the first mechanism is negligible. The second mechanism may explain the data from Sites 1004 and 1005. The temperature profile of Site 1006 is most easily explained by the third mechanism. We reconstruct the history of BWT at this site by solving the inverse heat conduction problem. The inversion result indicates gradual warming throughout this century by ~1°C and is agreeable to other hydrographic and climatic data from the western subtropic Atlantic. However, data from Sites 1003 and 1007 do not seem to show such trends. Therefore, none of the three mechanisms tested here explain the observations from all the drill sites. As for the lateral variation of the background heat flow along the drill transect, we believe that much of it is caused by the thermal effect of the topographic variation. We model this effect by obtaining a two-dimensional analytical solution. The model suggests that the background heat flow of this area is ~43 mW/m**2, a value similar to the background heat flow determined for the Gulf of Mexico in the opposite side of the Florida carbonate platform.