Tight Span Research Articles

Tight span of path connected subsets of the Manhattan plane

We show that the tight span of a path connected subset of the Manhattan plane can be constructed in a very simple way by “hatching” the subset in horizontal and vertical directions and then taking the closure of the resulting set.

A New Face Iterator for Polyhedra and for More General Finite Locally Branched Lattices

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, García-Sánchez, O’Neill, and Wilburne, we apply this algorithm to prove Wilf’s conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf’s conjecture.

Sparse graphs and an augmentation problem

For two integers k>0 and ell , a graph G=(V,E) is called (k,ell )-tight if |E|=k|V|-ell and i_G(X)le k|X|-ell for each Xsubseteq V for which i_G(X)ge 1, where i_G(X) denotes the number of induced edges by X. G is called (k,ell )-redundant if G-e has a spanning (k,ell )-tight subgraph for all ein E. We consider the following augmentation problem. Given a graph G=(V,E) that has a (k,ell )-tight spanning subgraph, find a graph H=(V,F) with the minimum number of edges, such that Gcup H is (k,ell )-redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is (k,ell )-tight. For general inputs, we give a polynomial algorithm when kge ell and show the NP-hardness of the problem when k<ell . Since (k,ell )-tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

Politopos en filogenética

We introduce mathematical notions used in phylogenetics and three sorts of phylogenetics polytopes. The Tight span and the Lipschitz polytope are both associated to finite metric spaces and can be connected to distance-preserving embeddings, while the balanced minimum evolution (BME) polytope is associated to natural numbers.&#x0D;

An algorithm for the construction of the tight span of finite subsets of the Manhattan plane

We give a simple algorithm to obtain the tight span of a finite subset of the Manhattan plane.

Infinitesimal Rigidity in Normed Planes

We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a $(2,2)$-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph $K_4$ by considering smoothness and strict convexity properties of the unit ball.

P103 Developing Social Content for Social Marketing: How? Why? What?

Objective Demonstrate how to develop relevant content ideas and concepts for use on social media to encourage and inspire behavior change. Content is king; whether on your smart phone, your computer, your tablet or your television, developing content that is attractive, relevant and engaging has never been more important. Budgets are tight and attention spans are short, so make what you develop be impactful and memorable. Use of Theory or Research Research among target audience engagement with posts and content that speaks to food consumption, budgeting food purchases and engaging in healthy activities confirms that reaching a low-income audience through these means is an efficient and effective method. Their use of social media is impactful when it comes to shopping decisions, meal planning, and being inspired to move more. Target Audience Low-income moms with children. Program Description Developing content to reach moms who are busy, over-stressed, low on cash and with poor literacy skills can seem daunting and difficult. We will share how the development of strategic content for social media can be easier than you think, sharing tips and tricks learned from our years of experience in working with SNAP-Ed clients. Evaluation Methods We evaluate every post against an identified list of goals. From sharing how many engagements a post elicited to how many clicks and, when possible, what behavior the respondent engaged with on the website are also measured. Results We have seen increased engagement and increased response to our calls to action as our content development has become more robust and tailors to the target audience of low-income moms. Quality content provides key opportunities to engage in two-way communication with our target audience. Conclusions The more that local nutrition educators can develop quality content, using what they know and understand about their audiences, the more opportunities that will arise to encourage and engage consumers in making healthy behaviors becomes easier to achieve. Funding None.

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

Trimming of metric spaces and the tight span

We use the trimming transformations to study the tight span of a metric space.

Algorithms for tight spans and tropical linear spaces

We describe a new method for computing tropical linear spaces and more general duals of polyhedral subdivisions. This is based on an algorithm of Ganter for finite closure systems.

Transient modelling of flexible belt drive dynamics using the equations of a deformable string with discontinuities

ABSTRACTDynamics of a belt drive is analysed using a non-linear model of an extensible string at contour motion, in which the trajectories of particles of the belt are predetermined. The equations of string dynamics at the tight and slack spans are considered in a fixed domain by transforming into a spatial frame. Assuming the absence of slip of the belt on the surface of the pulleys, we arrive at a new model with a discontinuous velocity field and concentrated contact forces. Finite difference discretization allows numerical analysis of the resulting system of partial differential equations with delays. Example solution for the acceleration of a belt drive and an investigation of its frequency response depending on the velocity are presented and discussed.

The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered [Formula: see text]-graphs. A [Formula: see text]-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text]. An ordered [Formula: see text]-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We also show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We provide lower bounds for ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones. For ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered [Formula: see text]-graphs have worse spanning ratios than unordered [Formula: see text]-graphs. Finally, we show that, unlike their unordered counterparts, the ordered [Formula: see text]-graphs with 4, 5, and 6 cones are not spanners.

Tight span of subsets of the plane with the maximum metric

We prove that a nonempty closed and geodesically convex subset of the l∞ plane R∞2 is hyperconvex and we characterize the tight spans of arbitrary subsets of R∞2 via this property: Given any nonempty X⊆R∞2, a closed, geodesically convex and minimal subset Y⊆R∞2 containing X is isometric to the tight span T(X) of X.

The Maximum Multiflow Problems with Bounded Fractionality

We consider the weighted maximum multiflow problem with respect to a terminal weight. We show that if the dimension of the tight span associated with the weight is at most 2, then this problem has a 1/12-integral optimal multiflow for every Eulerian supply graph. This result solves a weighted generalization of Karzanov's conjecture for classifying commodity graphs with finite fractionality. In addition, our proof technique proves the existence of an integral or half-integrality optimal multiflow for a large class of multiflow maximization problems, and it gives a polynomial time algorithm.

Diversities, hyperconvexity and fixed points

Diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated. In this work we take up a number of questions about hyperconvexity, diversities and fixed points of nonexpansive mappings. Most of these questions are motivated by the study of the connection between a hyperconvex diversity and its induced metric space for which we provide some answers. Examples are given, for instance, showing that such a metric space need not be hyperconvex but still we prove, as our main result, that they enjoy the fixed point property for nonexpansive mappings provided the diversity is bounded and that this boundedness condition cannot be transferred from the diversity to the induced metric space.

Searching for realizations of finite metric spaces in tight spans

An important problem that commonly arises in areas such as internet traffic-flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs. Such a graph is called an optimal realization and finding such realizations is known to be NP-hard. Recently Varone presented a heuristic greedy algorithm for computing optimal realizations. Here we present an alternative heuristic that exploits the relationship between realizations of the metric D and its so-called tight span TD. The tight span TD is a canonical polytopal complex that can be associated to D, and our approach explores parts of TD for realizations in a way that is similar to the classical simplex algorithm. We also provide computational results illustrating the performance of our approach for different types of metrics, including l1-distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.

Metric stability of trees and tight spans

We prove optimal extension results for roughly isometric relations between metric (\({\mathbb{R}}\)-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov–Hausdorff (GH) distance, for certain metric spanning constructions: the GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.

Obtaining splits from cut sets of tight spans

To any metric D on a finite set X, one can associate a metric space T(D) known as its tight span. Properties of T(D) often reveal salient properties of D. For example, cut sets of T(D), i.e., subsets of T(D) whose removal disconnect T(D), can help to identify clusters suggested by D and indicate how T(D) (and hence D) may be decomposed into simpler components. Given a bipartition or split S of X, we introduce in this paper a real-valued index ε(D,S) that comes about by considering cut sets of T(D). We also show that this index is intimately related to another, more easily computable index δ(D,S) whose definition does not directly depend on T(D). In addition, we provide an illustration for how these two new indices could help to extend and complement current distance-based methods for phylogenetic network construction such as split decomposition and NeighborNet.

On the Topological and Uniform Structure of Diversities

Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note, we consider the analytical properties of diversities, in particular the generalizations of uniform continuity, uniform convergence, Cauchy sequences, and completeness to diversities. We develop conformities, a diversity analogue of uniform spaces, which abstract these concepts in the metric case. We show that much of the theory of uniform spaces admits a natural analogue in this new structure; for example, conformities can be defined either axiomatically or in terms of uniformly continuous pseudodiversities. Just as diversities can be restricted to metrics, conformities can be restricted to uniformities. We find that these two notions of restriction, which are functors in the appropriate categories, are related by a natural transformation.

Hyperconvexity and tight-span theory for diversities

The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications to metric classification and data visualisation. Here we introduce a generalisation of metrics, called diversities, and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity.