Integer Generalized Splines on Cycles

Handschy, Madeline; Melnick, Julie; Reinders, Stephanie

doi:10.48550/arxiv.1409.1481

Cited by 5 publications

(11 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When we first considered flow-up bases in Section 3.1, we said that geometers typically require the flow-up class p v restricted to the vertex v to be the product of labels on edges directed out of v. There are examples of flow-up bases for generalized splines for which this condition does not hold (see [24], who give an example over the integers). If X is a GKM space then Definition 3.1 (and the GKM conditions) implies the product condition on p v v .…”

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

“…Bowden-Philbin-Swift-Tammaro chose a particular m and showed explicitly that the minimal representatives of the basis for splines over Z/mZ also forms a basis for splines on (G, α) over Z [12]. • Explicit bases for splines are known for several families of graphs, including trees (using arbitrary rings) [19] and cycles (using Z) [24]. Handschy-Melnick-Reinders gave an explicit formula for the smallest value of each element of their flow-up basis [24].…”

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

“…• Explicit bases for splines are known for several families of graphs, including trees (using arbitrary rings) [19] and cycles (using Z) [24]. Handschy-Melnick-Reinders gave an explicit formula for the smallest value of each element of their flow-up basis [24]. Bowden-Hagen-King-Reinders gave an explicit formula for each entry of each basis element of splines on cycles over Z under the hypothesis that one pair of adjacent edge-labels is relatively prime [11].…”

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

See 2 more Smart Citations

Splines in geometry and topology

Tymoczko

2016

Computer Aided Geometric Design

View full text Add to dashboard Cite

Abstract. This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.The goal of this survey paper is to describe how splines arise in geometry and topology. Geometric splines usually appear under the name GKM theory after Goresky-Kottwitz-MacPherson, who developed them to compute what is called the cohomology ring of a geometric object. Geometers and analysts ask many of the same questions about splines: what is their dimension? can we identify a basis? can we find explicit formulas for the elements of the basis? However geometric constraints can change the tone of these questions: the splines may satisfy various symmetries or have a basis satisfying certain conditions. And some questions are specific to geometric splines: geometers particularly care about the multiplication table with respect to a given basis.In Section 1 we discuss GKM theory, followed in Section 2 by some important families of geometric examples, some of which are well-known to students of analytic splines and some of which may be useful in future. Section 3 sketches some techniques that are natural from the perspective of a geometer/topologist, including Morse flows and symmetries that come from geometric representation theory. Finally in Section 4 we generalize splines to a more abstract ring setting, both as a useful conceptual framework and because it provides new combinatorial tools. This paper is targeted at researchers in geometric design, especially those with an analytic background. Our aim is to give an overview of theoretical tools and techniques from geometry and topology; we often illustrate concepts by example and refer to the literature for details on technical aspects. A reader with a different mathematical perspective may be interested in surveys like [27,35,36]. GKM theoryCohomology is an algebraic gadget associated to a geometric object X that encodes various properties of X. Among other things, the cohomology of X indicates the dimension of X, the number of connected components (how many separate pieces X has), how many holes X has, whether X has singularities, and how different subspaces of X intersect.We could treat very general kinds of geometric objects but for simplicity in this survey we take X to be a compact complex manifold. When we say that cohomology is an "algebraic gadget" the most general interpretation is that cohomology is a ring. In the cases of interest here, the cohomology ring is actually an algebra, namely a vector space in which one can multiply vectors. The technical condition we assume is that cohomology has coefficients in Q, R, or especially C.In fact we will consider an enhanced version of cohomology called T -equivariant cohomology. Equivariant cohomology has strictly more information than ordinary cohomology yet surprisingly can be easier to compute. In our case T is a torus, ...

show abstract

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

Section: Geometric and Topological Tools For Computing With Splinesmentioning

confidence: 99%

See 1 more Smart Citation

Splines in geometry and topology

Tymoczko

2016

Computer Aided Geometric Design

View full text Add to dashboard Cite

show abstract

“…Splines occur throughout mathematics: applied mathematicians use them to approximate complicated functions or isolated data points by relatively manageable functions; analysts classify splines in low dimensions or with other fixed parameters [2,3,31,30,1]; algebraists study algebraic invariants of spline modules [5,6,7,10,11,20,27,28,33]; and geometers and topologists use splines to describe the equivariant cohomology ring of well-behaved geometric objects [19,25,4,29]. Recent work generalizes splines to a more abstract algebraic and combinatorial setting, of which the definition in the previous paragraph is still just a special case (see also Section 2.1) [17,22,8].…”

Section: Contentsmentioning

confidence: 99%

“…In this section we give the general definition of splines, as well as the special case of the definition used in this manuscript. We also describe an analogue of uppertriangular bases for splines, which are called flow-up splines because of geometric applications in which the elements are defined by certain torus flows [18,23,32,22]. The section ends by using the structure theorem for finite abelian groups to give conditions for when flow-up splines also form a minimum generating set.…”

Section: Notation and Backgroundmentioning

confidence: 99%

Splines mod m

Bowden¹,

Tymoczko²

2015

Preprint

View full text Add to dashboard Cite

Given a graph whose edges are labeled by ideals in a ring, a generalized spline is a labeling of each vertex by a ring element so that adjacent vertices differ by an element of the ideal associated to the edge. We study splines over the ring Z{mZ. Previous work considered splines over domains, in which very different phenomena occur. For instance when the ring is the integers, the elements of bases for spline modules are indexed by the vertices of the graph. However we prove that over Z{mZ spline modules can essentially have any rank between 1 and n. Using the classification of finite Z-modules, we begin the work of classifying splines over Z{mZ and produce minimum generating sets for splines on cycles over Z{mZ. We close with many open questions.

show abstract

Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles

Bowden,

Hagen,

King

et al. 2015

Preprint

Self Cite

View full text Add to dashboard Cite

An integer generalized spline is a set of vertex labels on an edgelabeled graph that satisfy the condition that if two vertices are joined by an edge, the vertex labels are congruent modulo the edge label. Foundational work on these objects comes from Gilbert, Polster, and Tymoczko, who generalize ideas from geometry/topology (equivariant cohomology rings) and algebra (algebraic splines) to develop the notion of generalized splines. Gilbert, Polster, and Tymoczko prove that the ring of splines on a graph can be decomposed in terms of splines on its subgraphs (in particular, on trees and cycles), and then fully analyze splines on trees. Following Handschy-Melnick-Reinders and Rose, we analyze splines on cycles, in our case integer generalized splines.The primary goal of this paper is to establish two new bases for the module of integer generalized splines on cycles: the triangulation basis and the King basis. Unlike bases in previous work, we are able to characterize each basis element completely in terms of the edge labels of the underlying cycle. As an application we explicitly construct the multiplication table for the ring of integer generalized splines in terms of the King basis.

show abstract

Integer Generalized Splines on Cycles

Cited by 5 publications

References 17 publications

Splines in geometry and topology

Splines in geometry and topology

Splines mod m

Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles

Contact Info

Product

Resources

About