On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions

Symbolic ultrametrics define edge-colored complete graphs K_n and yield a simple tree representation of K_n. We discuss, under which conditions this idea can be generalized to find a symbolic ultrametric that, in addition, distinguishes between edges and non-edges of arbitrary graphs G=(V,E) and thus, yielding a simple tree representation of G. We prove that such a symbolic ultrametric can only be defined for G if and only if G is a so-called cograph. A cograph is uniquely determined by a so-called cotree. As not all graphs are cographs, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of G. The latter problem is equivalent to find an optimal cograph edge k-decomposition {E_1,...,E_k} of E so that each subgraph (V,E_i) of G is a cograph. An upper bound for the integer k is derived and it is shown that determining whether a graph has a cograph 2-decomposition, resp., 2-partition is NP-complete

C2 popunjavanje praznina pomoću konveksne kombinacije ploha pod rubnim ograničenjima

Two surface generation methods are presented, one for connecting two surfaces with C2 continuity while matching also two prescribed border lines on the free sides of the gap, and one for G1 filling a three-sided hole in a special case. The surfaces are generated as convex combination of surface and curve constituents with an appropriate correction function, and are represented in parametric form.Dane su dvije metode za izvođenje ploha. Jedna za povezivanje dviju ploha sa C2 neprekinutošću koja odgovara i dvjema graničnim linijama, a druga za G1 popunjavanje posebnog slučaja trostrane rupe. Plohe se izvode kao konveksna kombinacija plošnih i krivuljnih sastavnih dijelova sa odgovarajućom korektivnom funkcijom, a dane su u parametarskom obliku

C2 popunjavanje praznina pomoću konveksne kombinacije ploha pod rubnim ograničenjima

Two surface generation methods are presented, one for connecting two surfaces with C2 continuity while matching also two prescribed border lines on the free sides of the gap, and one for G1 filling a three-sided hole in a special case. The surfaces are generated as convex combination of surface and curve constituents with an appropriate correction function, and are represented in parametric form.Dane su dvije metode za izvođenje ploha. Jedna za povezivanje dviju ploha sa C2 neprekinutošću koja odgovara i dvjema graničnim linijama, a druga za G1 popunjavanje posebnog slučaja trostrane rupe. Plohe se izvode kao konveksna kombinacija plošnih i krivuljnih sastavnih dijelova sa odgovarajućom korektivnom funkcijom, a dane su u parametarskom obliku

High-Resolution Optical Functional Mapping of the Human Somatosensory Cortex

Non-invasive optical imaging of brain function has been promoted in a number of fields in which functional magnetic resonance imaging (fMRI) is limited due to constraints induced by the scanning environment. Beyond physiological and psychological research, bedside monitoring and neurorehabilitation may be relevant clinical applications that are yet little explored. A major obstacle to advocate the tool in clinical research is insufficient spatial resolution. Based on a multi-distance high-density optical imaging setup, we here demonstrate a dramatic increase in sensitivity of the method. We show that optical imaging allows for the differentiation between activations of single finger representations in the primary somatosensory cortex (SI). Methodologically our findings confirm results in a pioneering study by Zeff et al. (2007) and extend them to the homuncular organization of SI. After performing a motor task, eight subjects underwent vibrotactile stimulation of the little finger and the thumb. We used a high-density diffuse-optical sensing array in conjunction with optical tomographic reconstruction. Optical imaging disclosed three discrete activation foci one for motor and two discrete foci for vibrotactile stimulation of the first and fifth finger, respectively. The results were co-registered to the individual anatomical brain anatomy (MRI) which confirmed the localization in the expected cortical gyri in four subjects. This advance in spatial resolution opens new perspectives to apply optical imaging in the research on plasticity notably in patients undergoing neurorehabilitation

Injective split systems

A split system $\mathcal S$ on a finite set $X$, $|X|\ge3$, is a set of bipartitions or splits of $X$ which contains all splits of the form $\{x,X-\{x\}\}$, $x \in X$. To any such split system $\mathcal S$ we can associate the Buneman graph $\mathcal B(\mathcal S)$ which is essentially a median graph with leaf-set $X$ that displays the splits in $\mathcal S$. In this paper, we consider properties of injective split systems, that is, split systems $\mathcal S$ with the property that $\mathrm{med}_{\mathcal B(\mathcal S)}(Y) \neq \mathrm{med}_{\mathrm B(\mathcal S)}(Y')$ for any 3-subsets $Y,Y'$ in $X$, where $\mathrm {med}_{\mathcal B(\mathcal S)}(Y)$ denotes the median in $\mathcal B(\mathcal S)$ of the three elements in $Y$ considered as leaves in $\mathcal B(\mathcal S)$. In particular, we show that for any set $X$ there always exists an injective split system on $X$, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph $\mathcal B(\mathcal S)$ needs to become in order for a split system $\mathcal S$ on $X$ to be injective. We do this by introducing a quantity for $|X|$ which we call the injective dimension for $|X|$, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on $X$ can be represented using Buneman graphs.Comment: 22 pages, 3 figure

Preparation And Characterization Of Composite Hollow Fiber Reverse Osmosis Membranes By Plasma Polymerization. 1. Design Of Plasma Reactor And Operational Parameters

Composite hollow fiber reverse osmosis membranes were prepared by depositing a thin layer (10-50 nm) of plasma polymers on hollow fibers with porous walls (made of polysulfone). The coating was carried out in a semicontinuous manner with six strands of substrate fibers. Operational parameters which influence reverse osmosis characteristics of composite membranes were investigated. © 1984, American Chemical Society. All rights reserved

Quadrilateral-octagon coordinates for almost normal surfaces

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from 10n to 6n. As an application, we show that quadrilateral-octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using cohomology, plus other minor changes; v3: Minor housekeepin