### Working paper

## On generating series of finitely presented operads, v.3

This paper has been started as a particular application of the method of resolutions via Grobner bases we suggested here. We introduce a notion of a shuffle algebra. A shuffle algebra is a Z+-graded vector space V=∪∞i=1 such that for any pair (i,j) there exists a collection of operations ∗σ:Vi⊗Vj→Vi+j numbered by (i,j)-shuffle permutations σ∈Si+j (i.e. σ preserves the order of the first i elements and the order of the last j elements) yielding the natural associativity conditions. Enumerative problems for monomial shuffle algebras are in one-to-one correspondence with the pattern avoidance problems for permutations. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations. Both results generalizes the classical results for associative algebras. The first homological result is a generalization of the Golod-Shafarevich theorem and the second one generalizes the theory of Anick chains. It seems that most of particular applications we discuss are known to specialists but the general method was definitely not known. We hope that it will simplify a lot of work in this area. It is not hard to see that shuffle algebras form an interesting class of binary shuffle operads and illustrates quite well the importance of the latter notion.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

The main goal of this paper is to present a way to compute Quillen homology of a shuffle operad with a known Grobner basis. Similar to the strategy taken in a celebrated paper of David Anick, our approach goes in several steps. We define a combinatorial resolution for the ``monomial replacement'' of a shuffle operad, explain how to ``deform'' the differential to handle the general case, and find explicit representatives of Quillen homology for a large class of operads with monomial relations. We present various applications, including a new proof of Hoffbeck's PBW criterion, a proof of Koszulness for a class of operads coming from commutative algebras, and a homology computation for the operads of Batalin--Vilkovisky algebras and of Rota--Baxter algebras. The method of writing a resolution presented in this paper is very general. Namely, whenever you have a category which admits a theory of monomials (including Grobner bases and Buchberger algorithm) you can do the same procedure: First, take an object with a chosen Grobner basis. Second, define a resolution for an object with monomial relations (the monomial replacement of the starting object). Third, lower terms of relations will affect additional summands in the description of the differential in the resolution.

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras. © 2013 Springer-Verlag Berlin Heidelberg.

We consider varieties of linear multioperator algebras, that is, classes of algebras with several multilinear operations satisfying certain identities. To each such a variety one can assign a numerical sequence called a sequence of codimensions. The n-th codimension is equal to the dimension of the vector space of all n-linear operations in the free algebra of the variety. In recent decades, a new approach to such a sequence has appeared based on the fact that the union of the above vector spaces carries the structure of algebraic operad, so that the generating function of the codimension sequence is equal to the generating series of the operad. We show that in general there does not exist an algorithm to decide whether the growth exponent of the codimension sequence of the variety defined by given finite sets of operations and identities is equal to a given rational number. In particular, we solve negatively a recent conjecture by Bremner and Dotsenko by showing that the set of codimension sequences of varieties defined by a bounded number and degrees of operations and identities is infinite. Then we discuss algorithms which in many cases calculate the generating functions of the codimension series in the form of a defining algebraic or differential equation. For a more general class of varieties, these algorithms give upper and lower bounds for the codimensions in terms of generating functions. The upper bound is just a formal power series satisfying an algebraic equation defined effectively by the generators and the identities of the variety. The first stage of an algorithm for the lower bound is the construction of a Groebner basis of the operad. If the Groebner basis happens to be finite and satisfies mild restrictions, a recent theorem by the author and Anton Khoroshkin guarantees that the desired generating function is either algebraic or differential algebraic. We describe algorithms producing such equations. In the case of infinite Groebner basis, these algorithms applied to its finite subsets give lower bounds for the generating function of the codimension sequence.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.