scholarly journals Dissimilarity Vectors of Trees and Their Tropical Linear Spaces (Extended Abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Benjamin Iriarte Giraldo
Keyword(s):  
Algebraic Geometry ◽  
Linear Space ◽  
Point Of View ◽  
Weighted Tree ◽  
Linear Spaces ◽  
Weighted Trees ◽  
International Audience ◽  
Tropical Linear Space ◽  
Associated Family

International audience We study the combinatorics of weighted trees from the point of view of tropical algebraic geometry and tropical linear spaces. The set of dissimilarity vectors of weighted trees is contained in the tropical Grassmannian, so we describe here the tropical linear space of a dissimilarity vector and its associated family of matroids. This gives a family of complete flags of tropical linear spaces, where each flag is described by a weighted tree. Nous étudions les propriétés combinatoires des arbres pondérés avec le formalisme de la géométrie tropicale et des espaces linéaires tropicaux. L'ensemble de vecteurs de dissimilarité des arbres pondérés est contenu dans la grassmannienne tropicale, donc nous décrivons ici l'espace linéaire tropical d'un vecteur de dissimilarité et sa famille de matroïdes associée. Cela permet d'obtenir une famille de drapeaux complets d'espaces linéaires tropicaux, où chaque drapeau est décrit par un arbre pondéré.

scholarly journals $Star^1$-convex functions on tropical linear spaces of complete graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Complete Graph ◽  
Piecewise Linear ◽  
Negative Answer ◽  
Linear Functions ◽  
One Dimension ◽  
Complete Graphs ◽  
Piecewise Linear Functions ◽  
Linear Spaces ◽  
International Audience ◽  
Tropical Linear Space

International audience Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same. Soit $\Delta$ un fan, pour $\sigma \in \Delta$ nous définissons $star^1(\sigma )$ comme l'ensemble de cônes qui contiennent $\sigma$ dont la dimension est un de plus que la dimension de $\sigma$ . Nous étudions deux cônes d'applications linéaires par morceaux définis sur $\Delta$ : le cône de fonctions convexes sur$star^1(\sigma )$, où $\sigma \in \Delta$ est un cône quelconque, et le cône de fonctions convexes sur $star^1(σ\sigma)$ où σ est un cône de codimension 1. étant donnés deux structures sur l'espace tropical linéaire de graphes complets, nous donnons de beaux descriptions combinatoires des cônes décrits en haut. Pour le graphe complet $K_5$, on démontre que avec la subdivision en fans plus fine, les deux cônes sont différentes, mais avec la subdivision plus gros ils sont cônes sont les mêmes. Ce résultant réponde négativement une question de Gibney-Maclagan.

Linear geometries on the Moebius strip: A theorem of Skornyakov type

2005 ◽  
Vol 72 (1) ◽  
pp. 17-30
Author(s):  
Rainer Löwen ◽  
Burkard Polster
Keyword(s):  
Linear Space ◽  
Real Line ◽  
Open Disk ◽  
Stable Plane ◽  
Linear Spaces ◽  
Unique Line ◽  
The Real ◽  
Continuity Properties ◽  
Point Set ◽  
Moebius Strip

We show that the continuity properties of a stable plane are automatically satisfied if we have a linear space with point set a Moebius strip, provided that the lines are closed subsets homeomorphic to the real line or to the circle. In other words, existence of a unique line joining two distinct points implies continuity of join and intersection. For linear spaces with an open disk as point set, the same result was proved by Skornyakov.

scholarly journals Nearly directed subspaces of partially ordered linear spaces

1968 ◽  
Vol 16 (2) ◽  
pp. 135-144
Author(s):  
G. J. O. Jameson
Keyword(s):  
Linear Space ◽  
Transitive Relation ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Partially Ordered ◽  
Real Linear ◽  
Image Position ◽  
Ordered Linear Spaces

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

Intersections of Loci on an Algebraic V4

1937 ◽  
Vol 33 (4) ◽  
pp. 425-437 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Algebraic Geometry ◽  
Linear Space ◽  
Considerable Importance ◽  
Classical Paper ◽  
Residual Intersection ◽  
Pass Through ◽  
The Given

The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.

Fourier-Mukai Transforms in Algebraic Geometry

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Algebraic Geometry ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Point Of View ◽  
Basic Knowledge ◽  
Research Directions ◽  
Coherent Sheaves ◽  
Geometric Point ◽  
Singular Cohomology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

New Fixed Point Theorems via Contraction Mappings in Complete Intuitionistic Fuzzy Normed Linear Space

2018 ◽  
Vol 15 (01) ◽  
pp. 65-83
Author(s):  
Nabanita Konwar ◽  
Ayhan Esi ◽  
Pradip Debnath
Keyword(s):  
Fixed Point ◽  
Mathematical Models ◽  
Linear Space ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Existence Of A Solution ◽  
Linear Spaces ◽  
Contraction Mappings ◽  
Normed Linear Spaces ◽  
Intuitionistic Fuzzy

Contraction mappings provide us with one of the major sources of fixed point theorems. In many mathematical models, the existence of a solution may often be described by the existence of a fixed point for a suitable map. Therefore, study of such mappings and fixed point results becomes well motivated in the setting of intuitionistic fuzzy normed linear spaces (IFNLSs) as well. In this paper, we define some new contraction mappings and establish fixed point theorems in a complete IFNLS. Our results unify and generalize several classical results existing in the literature.

Gaps between Spheres in Normed Linear Spaces

1980 ◽  
Vol 23 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Normed Linear Spaces ◽  
Concentric Spheres ◽  
General Conditions

AbstractThe geometric notions of a gap and gap points between “concentric” spheres in a normed linear space are introduced and studied. The existence of gap points characterizes finitedimensional spaces. General conditions are given under which an infinite-dimensional normed linear space admits concentric spheres such that both these spheres and their dual spheres fail to have gap points.

On the calculation of the Dunkl-Williams constant of normed linear spaces

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Hiroyasu Mizuguchi ◽  
Kichi-Suke Saito ◽  
Ryotaro Tanaka
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Normal Structure ◽  
Linear Spaces ◽  
Normed Linear Spaces

AbstractRecently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.

Line-transitive point-imprimitive linear spaces with number of points being a product of two primes

2017 ◽  
Vol 16 (06) ◽  
pp. 1750110
Author(s):  
Haiyan Guan ◽  
Shenglin Zhou
Keyword(s):  
Linear Space ◽  
Automorphism Groups ◽  
Linear Spaces ◽  
Finite Linear Space ◽  
Transitive Point ◽  
Finite Linear

The work studies the line-transitive point-imprimitive automorphism groups of finite linear spaces, and is underway on the situation when the numbers of points are products of two primes. Let [Formula: see text] be a non-trivial finite linear space with [Formula: see text] points, where [Formula: see text] and [Formula: see text] are two primes. We prove that if [Formula: see text] is line-transitive point-imprimitive, then [Formula: see text] is solvable.

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