Related papers: Multi-indice B-series
Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split:…
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 extend the maximal unitarity method to amplitude contributions whose cuts define multidimensional algebraic varieties. The technique is valid to all orders and is explicitly demonstrated at three loops in gauge theories with any number…
In this work, we propose trait-based merge trees a generalization of merge trees to feature level sets, targeting the analysis of tensor field or general multi-variate data. For this, we employ the notion of traits defined in attribute…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…
In a multi-index model with $k$ index vectors, the input variables are transformed by taking inner products with the index vectors. A transfer function $f: \mathbb{R}^k \to \mathbb{R}$ is applied to these inner products to generate the…
An efficient method for finding all real roots of a univariate function in a given bounded domain is formulated. The proposed method uses adaptive mesh refinement to locate bracketing intervals based on bisection criterion for root finding.…
We give an introduction to a theory of b-functions, i.e. Bernstein-Sato polynomials. After reviewing some facts from D-modules, we introduce b-functions including the one for arbitrary ideals of the structure sheaf. We explain the relation…
We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…
Tree balance has received considerable attention in recent years, both in phylogenetics and in other areas. Numerous (im)balance indices have been proposed to quantify the (im)balance of rooted trees. A recent comprehensive survey…
Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary,…
Multivariate time series classification is a task with increasing importance due to the proliferation of new problems in various fields (economy, health, energy, transport, crops, etc.) where a large number of information sources are…
In these notes, we provide an introduction to a new regularity structure used for solving rough mean-field equations. The index set of this regularity structure is described a collection of novel objects which we refer to as Lions trees.…
Many scientific fields and applications require compact representations of multivariate functions. For this problem, decoupling methods are powerful techniques for representing the multivariate functions as a combination of linear…
We face a need of discovering a pattern in locations of a great number of points in a high-dimensional space. Goal is to group the close points together. We are interested in a hierarchical structure, like a B-tree. B-Trees are…
Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is…
In this paper, we introduce the concept of $b$-branchings in digraphs, which is a generalization of branchings serving as a counterpart of $b$-matchings. Here $b$ is a positive integer vector on the vertex set of a digraph, and a…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…