Weak geodesic topology and fixed finite subgraph theorems in infinite partial cubes I. Topologies and the geodesic convexity

AbstractThe weak geodesic topology on the vertex set of a partial cube G is the finest weak topology on V(G) endowed with the geodesic convexity. We prove the equivalence of the following properties: (i) the space V(G) is compact; (ii) V(G) is weakly countably compact; (iii) the vertex set of any ray of G has a limit point; (iv) any concentrated subset of V(G) (i.e. a set A such that any two infinite subsets of A cannot be separated by deleting finitely many vertices) has a finite positive number of limit points. Moreover, if V(G) is compact, then it is scattered. We characterize the partial cubes for which the weak geodesic topology and the geodesic topology (see [N. Polat, Graphs without isometric rays and invariant subgraph properties I. J. Graph Theory27 (1998), 99–109]) coincide, and we show that the class of these particular partial cubes is closed under Cartesian products, retracts and gated amalgams

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.

A Hierarchical Shape Representation by Convexities and Concavities and Its Application to Shape Matching.

Shape contains information. The identification and extraction of this information is not straightforward and is the main problem of Shape Analysis. The current trend of manipulating visual information, makes this problem more important. The abundant work published about shape analysis can be classified into two main approaches: statistical shape analysis and structural shape analysis. The structural approach was proposed around thirty years ago by K. S. Fu. The large amount of works published since then, prove the difficulty of defining a universal set of primitives. The structural description of shape is based on the assumption that shape recognition is a hierarchical process. Nevertheless, no effective general mechanism that captures hierarchical description has been found, and the existing representations may be applied to restricted applications. We propose a new structural representation of shape using convexity. Instead of using a predefined set of primitives, we use two basic components to decompose any shape: convexity and concavity. The decomposition obtained results in a natural hierarchy, of these basic components. We represent the decomposition by a new shape descriptor: the Convexity-Concavity Tree (CCT), which is a binary tree. The CCT representation is used for matching the shapes of two objects. The matching of two CCTs is represented by a binary tree, that we call the Matching Tree (MT). The Matching Tree represents the location and magnitude of the mismatch between corresponding convexities-concavities of the two shapes. Two shapes match if their corresponding CCTs match. Some of the advantages of our representation method are: (1) it is information preserving, (2) it has the desired properties of a good description method: invariance, uniqueness and stability, (3) it is economical, (4) it is robust in the presence of noise. Our matching method, based on convexity representation is superior to other methods in terms of simplicity, ability to explain and measure mismatches and also it may be used with other well known methods

Continuities and discontinuities during the late Middle Palaeolithic at the Oscurusciuto rock shelter (southern Italy). An integrated study of lithic manufacture in the strata SU 15, SU 14, SU 13 and SU 11.

Aquesta investigació té com a objectiu interpretar les variacions en la producció Neandertal, així com la funcionalitat de les eines de pedra mitjançant l'anàlisi de conjunts lítics dins l'últim Paleolític Mitjà del sud d'Itàlia. L'abric Oscurusciuto és clau per a la comprensió del comportament dels neandertals, ja que ofereix un dipòsit llarg i fiable, d'uns 6 m de profunditat, format per diversos nivells que oscil•len entre 42,724 ± 716 cal BP i 55 ± 2 kyrs (40Ar/39Ar). L’objectiu principal d'aquesta investigació és realitzar un estudi del conjunt d’eines de lítica, present a la secció inferior de la sèrie investigada fins ara en aquest jaciment: SU 15, SU 14, SU 13 i SU 11. Volem individualitzar, des d'un punt de vista diacrònic, les continuïtats i discontinuïtats d'aquests tecno-complexos lítics. Tenim la intenció d'aconseguir aquest objectiu a través d'un estudi integrat del material lític que permetrà una descripció del comportament econòmic relacionat amb l'explotació de les fonts de matèria primera lítica, seguit d'una descripció detallada de les fases de la seqüència de reducció, així com una definició de conceptes, mètodes, dinàmiques i objectius de la talla. Aquesta anàlisi tecnològica fonamental s'aplica als quatre nivells: SU 15, SU 14, SU 13 i SU 11. Basant-nos en les característiques del material recollit, s'han dut a terme diversos estudis: d’unitats de matèria primera (RMU) i de remuntatges per SU 13 i SU 14, per tal de comprendre millor la fragmentació de la cadena operativa. A més, s'ha desenvolupat un protocol tecno-funcional sobre un grup seleccionat de peces de la SEU 13, per tal d'aprofundir la nostra comprensió sobre la potencialitat les eines. Els resultats demostren que aquestes unitats estratigràfiques mostren peculiaritats específiques en termes de sistemes de producció lítica, elements estructurals, distribució espacial i tipus d'ocupació.Esta investigación tiene como objetivo interpretar las variaciones en la producción y la funcionalidad de herramientas de piedra de los Neandertales a través del análisis de los conjuntos líticos referibles a la última parte del Paleolítico Medio del sur de Italia. Un sitio clave para este propósito es el abrigo Oscurusciuto. Este es esencial para la comprensión del comportamiento de los Neandertales porque ofrece un depósito largo y confiable, de unos 6 m de profundidad. El mismo se encuentra formado por varios niveles que oscilan entre 42,724 ± 716 cal BP y 55 ± 2 kyrs (40Ar/39Ar). El propósito explícito de esta investigación es realizar un estudio integrado de las herramientas líticas presentes en la sección inferior de la serie hasta ahora investigada del abrigo Oscurusciuto: SU 15, SU 14, SU 13 y SU 11. Se pretende individualizar, desde un punto de vista diacrónico, las continuidades y discontinuidades de estos tecnocomplejos líticos. Para lograr este propósito se realizará un estudio integrado del material lítico, que permita una descripción del comportamiento económico relacionado con la explotación de las fuentes de materias primas líticas, una descripción detallada de las fases de la secuencia de reducción, así como también una definición de conceptos, métodos, dinámicas y objetivos del debitage. Este análisis tecnológico fundamental se aplica a los cuatro niveles. Considerando las características del material recogido, se han llevado a cabo estudios adicionales, como Unidades de Materia Prima (RMU) y remontajes para SU 13 y SU 14, con el fin de comprender mejor la fragmentación de la cadena operativa. Además, se ha desarrollado un protocolo tecno-funcional sobre un grupo seleccionado de piezas de SU 13, con el fin de profundizar nuestra comprensión de la potencialidad de las herramientas. Los resultados demuestran que estas unidades estratigráficas muestran peculiaridades específicas en términos de sistemas de producción lítica, elementos estructurales, manejo espacial y tipo de ocupación.This research aims to interpret the variations in the Neanderthal production and functionality of stone tools through analyses of lithic assemblages referable to the last Middle Palaeolithic of southern Italy. A key site for this purpose is the Oscurusciuto rock shelter, which is essential to the understanding of Neanderthal behaviour, as it offers a long, reliable deposit, about 6 m in depth, made up of several levels ranging between 42,724 ± 716 cal BP and 55 ± 2 kyrs (40Ar/39Ar). The explicit purpose of this research is to perform an integrated study of the lithic manufacture present in the lower section of the series so far investigated the Oscurusciuto rock shelter: SU 15, SU 14, SU 13 and SU 11. We want to individuate, from a diachronic point of view, the continuities and discontinuities of these lithic techno-complexes. We intend to achieve this purpose through an integrated study of the lithic material, which in turn allows for a description of the economic behaviour involved in the exploitation of the sources of lithic raw material, followed by a detailed description of the phases in the reduction sequence, as well as a definition of concepts, methods, dynamics and objectives of the debitage. This fundamental technological analysis is applied to all four levels: SU 15, SU 14, SU 13, and SU 11 (48,382 items). Based on the characteristics of the collected material, further studies have been undertaken, such as Raw Material Units plus refitting studies for SU 13 and SU 14, in order to better understand the fragmentation of the operative chain. Moreover, a techno-functional protocol has been developed regarding a selected group of pieces from SU 13, in order to deepen our comprehension of productional vs. functional tools. The results demonstrate that these stratigraphic units show specific peculiarities in terms of lithic production systems, structural elements, spatial management, and type of occupation

Bucolic Complexes

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.Comment: 45 pages, 4 figure

Dynamics of long term fluvial response in postglacial catchments of the Ladakh Batholith, Northwest Indian Himalaya

Upland rivers control the large-scale topographic form of mountain belts, allow coupling of climate and tectonics at the earth’s surface and are responsible for large scale redistribution of sediment from source areas to sinks. However, the details of how these rivers behave when perturbed by changes to their boundary conditions are not well understood. I have used a combination of fieldwork, remotely sensed data, mathematical analysis and computer modelling to investigate the response of channels to well constrained changes in the forcings upon them, focussing in particular on the effects of glacial remoulding of the catchments draining the south flank of the Ladakh batholith, northwest Indian Himalaya. The last glacial maximum for these catchments is atypically old (~100 ka), and this allows investigation of the response to glaciation on a timescale not usually available. The geomorphology of the catchments is divided into three distinct domains on the basis of the behaviour of the trunk stream – an upper domain where the channel neither aggrades above or incises into the valley form previously carved by glacial abrasion, a middle domain where the channel incises a gorge down into glacial sediments which mantle the valley floor, and a lower domain where the channel aggrades above this postglacial sediment surface. This landscape provides a framework in which to analyze the processes and timescales of fluvial response to glacial modification. The dimensions of the gorge and the known dates of glacial retreat record a time averaged peak river incision rate of approximately 0.5 mm/y; the timescale for the river long profile to recover to a smooth, concave up form must exceed 1 Ma. These values are comparable with those from similarly sized catchments that have been transiently perturbed by changing tectonics, but have never been quoted for a glacially forced basin-scale response. I have also demonstrated that lowering of the upper reaches of the Ladakh channel long profiles by glacial processes can systematically and nonlinearly perturb the slope-area (concavity) scaling of the channel downstream of the resulting profile convexities, or knickzones. The concavity values are elevated significantly above the expected equilibrium values of 0.3-0.6, with the magnitude controlled by the relative position of the knickzone within the catchment, and thus also by the degree of glacial modification of the fluvial system. This work also documents the existence of very similar trends in measured concavities downstream of long profile convexities in other transiently responding river systems in different tectonoclimatic settings, including those responding to changes in relative channel uplift. This previously unrecognised unity of response across a wide variety of different environments argues that such a trend is an intrinsic property of river response to perturbation. Importantly, it is consistent with the scaling expected from variation in incision efficiency driven by evolving sediment flux downstream of knickzones. The pervasive nature of this altered scaling, and its implications for fluvial erosion laws in perturbed settings, have significant consequences for efforts to interpret past changes in forcings acting on river systems from modern topography. I follow this by examining in detail the channel hydraulics of the Ladakh streams as they incise in response to the glacial perturbation. I present a new framework under which the style of erosion of a natural channel can be characterized as either detachment- or transport-limited based upon comparison of the downstream distribution of shear stress with the resulting magnitude of incision. This framework also allows assessment of the importance of sediment flux driven effects in studied channels. This approach is then used to demonstrate that fluvial erosion and deposition in the Ladakh catchments is best modelled as a sediment flux dependent, thresholded, detachment-limited system. The exceptional quality of the incision record in this landscape enables an unprecedented calibration of the sediment flux function within this incision law for three different trunk streams. The resulting curves are not compatible with the theoretically-derived parabolic form of this relation, instead showing nonzero erosion rates at zero sediment flux, a rapid rise and peak at relative sediment fluxes of less than 0.5 and a quasi exponential decrease in erosional efficiency beyond this. The position of the erosional efficiency peak in relative sediment flux space and the magnitude of the curve are shown to be both variable between the catchments explored and also correlated with absolute sediment flux in the streams