Cut Graphs Research Articles

The diagrammatic coaction beyond one loop

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

Causality and unitarity via the tree-loop duality relation

The tree-loop duality relation is used as a starting point to derive the constraints of causality and unitarity. Specifically, the Bogoliubov causality condition is ab initio derived at the individual graph level. It leads to a representation of a graph in terms of lower order cut graphs. Extracting the absorptive part gives then the general unitarity relation (Cutkosky rule). The derivation, being carried out directly in momentum space, holds for any local (polynomial) hermitian interaction vertices. This is in contrast to the technical difficulties arising from contact terms in the spacetime approach based on the largest time equation.

Range-valued fuzzy colouring of interval-valued fuzzy graphs

In this paper, we observed that range-valued fuzzy graphs are more popular than fuzzy graphs following the level of participation of nodes and branches in the range [0, 1] other than at a point in fuzzy graphs. Colouring interval-valued fuzzy graphs has a few general applications. In this paper, another idea of using colouring for interval-valued fuzzy graphs is presented. Additionally, a few vital terms, such as power cut graphs, interval-valued fuzzy colour, and chromatic numbers of interval-valued fuzzy graphs, are depicted. Some important theorems are discussed. This procedure is utilized to colour the India political map, revealing the power of the relationship within the nation. Additionally, another sort of traffic signal system is analysed.

Consistent quadrangulation for shape collections via feature line co-extraction

This paper presents a method that takes a collection of 3D surface shapes, and produces a consistent and individually feature preserving quadrangulation of each shape. By exploring the correspondence among shapes within a collection, we coherently extract a set of representative feature lines as the key characteristics for the given shapes. Then we compute a smooth cross-field interpolating sparsely distributed directional constraints induced from the feature lines and apply the mixed-integer quadrangulation to generate the quad meshes. We develop a greedy algorithm to extract aligned cut graphs across the shape collection so that the meshes can be aligned in a common parametric domain. Computational results demonstrate that our approach not only produces consistent quad meshes across the entire collection with significant geometry variation but also achieves a trade-off between global structural simplicity for the collection and local geometry fidelity for each shape.

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki (Graph structure theory. Contemporary Mathematics, vol. 147, 1993), a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length $$O(g^{3/2}n^{1/2})$$O(g3/2n1/2) for any triangulated combinatorial surface of genus $$g$$g with $$n$$n triangles, and describe an $$O(gn)$$O(gn)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.

Fuzzy colouring of fuzzy graphs

Colouring of fuzzy graphs has several applications in real world. In this paper, a new concept of colouring of fuzzy graphs has been introduced. Also, some important terms like strength cut graphs, fuzzy colour, chromatic number of fuzzy graphs have been described. Some relevant results are proved. This technique is used to colour world political map mentioning the strength of relationship among the countries. Also, a new kind of traffic light system has been proposed.

Finding and defining the generators of genus-n objects for constructing topological and cut graphs

The topology of an object is commonly represented through a topological graph or a cut graph (polygonal scheme). Over the past few years, many studies have focused on extracting the topological and cut graphs of complex freeform objects that are represented by meshes. For an object with genus-n, the topological graph has n cycles, while the cut graph contains 2n cycles. These loops, however, do not always explicitly represent the holes in the objects. That is, a cycle in the graph can be a cycle around a solid (meridian), a cycle around a hole (longitude), or almost any combination of the two. The task of classifying the cycles (generators) as cycles around holes (longitude) and cycles around solids (meridians) on the mesh is not straightforward. Every closed orientable 2-manifold with genus-n can be seen as a collection of n toruses stitched together, so that each hole in the object can be referred to as a torus with two generators. This paper proposes a method that extracts the generators from which the longitudes and the meridians are found. The topological graph is defined by the longitudes and by a spanning tree constructed between them. The cut graph is constructed using the same concept. The advantage of the proposed method over other methods is that each loop in the topological graph explicitly represents a hole in the object.

Cutting Rules at Finite Temperature

We discuss the cutting rules in the real-time approach to finite temperature field theory and show the existence of cancellations among classes of cut graphs which allows a physical interpretation of the imaginary part of the relevant amplitude in terms of the underlying microscopic processes. Furthermore, with these cancellations, any calculation of the imaginary part of an amplitude becomes much easier and completely parallel to the zero temperature case.

Infrared and mass regularization in AF field theories (II). QCD

We determine the non-leading QCD corrections to deep inelastic (DI) scattering and to the Drell-Yan (DY) process by analysing the order g 2 parton cross sections. Using the insight and the techniques developed in the more simple environment of π 6 3, we regularize and subsequently remove the mass (M), infrared (IR), and ultraviolet (UV) divergences of the (double) cut graphs with the methods exposed in paper I. The QCD calculations are technically more involved. New M divergences are found, IR singularities occur, and overlapping M and IR singularities exist leading to double poles if n-dimensional regularization is used. Gluon gauge invariance must be satisfied. Although the off-shell method leads to gauge-dependent parton cross sections, the correction terms extracted via M factorization remain gauge invariant. The off-shell mass assignment was ill-defined in the past and did not satisfy the double cut rule; we explain the implications. The correction terms are explicitly shown to be independent of any regularization method. Defining the scale-dependent momentum distribution to consist of log Q 2 terms only, their influence in DI scattering is moderate, but considerable in DY massive μμ pair production for Q 2 ⩽ 100 GeV 2.