Related papers: Growth problems in diagram categories
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
The density of a subgroupoid with respect to a free groupoid is defined as the asymptotic ratio of their growths. This notion can be interpreted as a generalisation of the index's inverse for groups or as the probability of an element…
We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A…
I compute several terms of the asymptotic expansion of the number of connected labelled graphs with n nodes and m edges, for small k=m-n.
We obtain asymptotic and exact formulae of growth functions for some families of $n$-valued coset groups. We also describe connections between the theory of $n$-valued groups and Symbolic Dynamics.
For any $k>1$, we find the asymptotics of the counting function of $k$-th power-free elements in an additive arithmetic semigroup with exponential growth of the abstract prime counting function. This paper continues the authors' earlier…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…
We realize any submultiplicative increasing function which is equivalent to a polynomial proportion of itself as the growth function of a finitely generated simple Lie algebra. As an application, we resolve two open problems posed by…
We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an asymptotic variant of relative submajorization, defined on…
In the talk at the workshop my aim was to demonstrate the usefulness of graph techniques for tackling problems that have been studied predominantly as problems on the term level: increasing sharing in functional programs, and addressing…
In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These…
The understanding of complex quantum many-body systems has been vastly boosted by tensor network (TN) methods. Among others, excitation spectrum and long-range interacting systems can be studied using TNs, where one however confronts the…
We give elementary applications of quasi-homomorphisms to growth problems in groups. A particular case concerns the number of torsion elements required to factorise a given element in the mapping class group of a surface.
We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with…
We use a version of the Trotter-Kato approximation theorem for strongly continuous semigroups in order to study flows on growing networks. For that reason we use the abstract notion of direct limits in the sense of category theory.
Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate…
The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and…
For two-loop two-point diagrams with arbitrary masses, an algorithm to derive the asymptotic expansion at large external momentum squared is constructed. By using a general theorem on asymptotic expansions of Feynman diagrams, the…
We exhibit a procedure to asymptotically enumerate monotone grid classes of permutations. This is then applied to compute the asymptotic number of permutations in any connected one-corner class. Our strategy consists of enumerating the…