Related papers: Multiple logarithms, algebraic cycles and trees

Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…

We develop the Tree-Loop Duality Relation for two- and three-loop integrals with multiple identical propagators (multiple poles). This is the extension of the Duality Relation for single poles and multiloop integrals derived in previous…

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…

We develop a canonical pairing between trees and graphs, which passes to their quotients by Jacobi identities. This pairing is an effective and simple tool for understanding the Lie and Poisson operads, providing canonical duals. In the…

Here we briefly describe some topics along the lines of projective spaces and related geometric constructions connected to linear algebra, which provide fundamental examples in classical geometry and analysis.

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral…

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we…

We investigate hierarchical structure in various complex systems according to Minimum Spanning Tree methods. Firstly, we investigate stock markets where the graphis obtained from the matrix of correlations coefficient computed between all…

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…

A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power…

In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we…

We present an extension of the duality theorem, previously defined by S. Catani et al. on the one-loop level, to higher loop orders. The duality theorem provides a relation between loop integrals and tree-level phase-space integrals. Here,…

This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras,…

We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…

We discuss the duality theorem, which provides a relation between loop integrals and phase space integrals. We rederive the duality relation for the one-loop case and extend it to two and higher-order loops. We explicitly show its…

We introduce a class of functions which constitutes an obvious elliptic generalization of multiple polylogarithms. A subset of these functions appears naturally in the \epsilon-expansion of the imaginary part of the two-loop massive sunrise…

In this paper, we provide new insights and analysis for the two elementary tree-based data structures - the AVL tree and binary heap. We presented two simple properties that gives a more direct way of relating the size of an AVL tree and…