Concatenation Product Research Articles

Iterated integrals with values in Azumaya algebras

We define Chen’s iterated integrals of forms with values in an Azumaya algebra and show that in the flat case, these iterated integrals provide chain maps, and more generally, the curved structure is preserved. The appropriate trace maps are then defined and their corresponding geometric meanings are studied. We also define and study the wedge product of forms and the concatenation product of paths for these iterated integrals.

Ordered covering arrays and upper bounds on covering codes

AbstractThis work shows several direct and recursive constructions of ordered covering arrays (OCAs) using projection, fusion, column augmentation, derivation, concatenation, and Cartesian product. Upper bounds on covering codes in Niederreiter–Rosenbloom–Tsfasman (shorten by NRT) spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.

Outer product enhanced heterogeneous information network embedding for recommendation

With the rapid development of the internet, more and more sophisticated data can be utilized by recommendation systems to improve their performance. Such data consist of heterogeneous information networks (HINs) made up of multiple nodes and link types. A critical challenge is how to effectively extract and apply the useful HIN information. In particular, the embedding-based recommendation approach has been widely used, as it can extract affluent semantic and structural information from HINs. However, the existing HIN embedding for recommendation methods only combine user embedding and item embedding through a simple concatenation or elementwise product, which does not suffer for an efficient recommendation model. In order to extract and utilize more comprehensive and subtle information from the embedding for recommendation, we propose Outer Product Enhanced Heterogeneous Information Network Embedding for Recommendation, called HopRec. The main idea is to utilize the outer product to model the pairwise relationship between user HIN embedding and item HIN embedding. Specifically, by performing an outer product between user HIN embedding and item HIN embedding, we can obtain a two-dimensional interaction matrix. Subsequently, we can obtain a rating prediction function by integrating matrix factorization (MF), user HIN embedding, item HIN embedding and interaction matrix. The results of experiments conducted on three open benchmark datasets show that HopRec significantly outperforms the state-of-the-art methods.

Computation tool for the q -deformed quasi-shuffle algebras and representations of structure of MZVs

We would like to introduce our package using Maple to compute within the q -deformed 1 quasi-shuffle algebras and to represent structure of multiple zeta values (MZVs). For this package, we can define an arbitrary alphabet from which the letters associated with indices totally ordered and then carry out computations on words, that will complement functions for the package StringTools in Maple. In the vector space of (noncommutative) polynomials which is equipped q -deformed quasi-shuffle products and concatenation product [1, 2], we compute the bases in duality and express an arbitrary homogeneous polynomial in terms of these bases. Moreover, due to our algorithms, we can represent structure of MZVs on the transcendence bases in terms of irreducible elements [4]. We used this package to compute all examples and verify the results in the paper [4] which was present at the conference ISSAC 2015.

The Series Product and Its Application to Quantum Feedforward and Feedback Networks

The purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks. These methods build on and generalize existing methods for series (or cascade) connections by allowing for more general interfaces, and by introducing an efficient algebraic tool, the series product. We also introduce another product, which we call the concatenation product, that is useful for assembling and representing systems without necessarily having connections. We show how the concatenation and series products can be used to describe feedforward and feedback networks. A selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology.

The Miyazawa polynomial for long virtual knots

We introduce a two-variable polynomial invariant of a long virtual knot, which dominates the Kauffman f-polynomial and the Miyazawa polynomial of the closure. Our invariant satisfies a product formula for the concatenation product of long virtual knots. It describes a formula of the Miyazawa polynomial of a ‘connected sum’ of two virtual knots. It also gives lower bounds for the real crossing number and the virtual crossing number of a long virtual knot.

Deterministic Concatenation Product of Languages Recognized by Finite Idempotent Monoids

We give explicit descriptions of varieties of finite monoids as well as of varieties of finite ordered monoids corresponding to the left deterministic, right deterministic and bideterministic products of languages recognized by finite idempotent monoids.

Local rule substitutions and stepped surfaces

Substitutions on words, i.e., non-erasing morphisms of the free monoid, are simple combinatorial objects which produce infinite words by iteratively replacing letters by words. This paper introduces a notion of substitution acting on multi-dimensional words, namely local rule substitutions. Roughly speaking, local rules play for multi-dimensional words the role played by the concatenation product for substitutions on words. We then particularly focus on the local rule substitutions which act on the two-dimensional words coding stepped surfaces, and we show that a wide class of them can be derived from generalized substitutions.

Two algebraic approaches to variants of the concatenation product

We extend an existing approach of the bideterministic concatenation product of languages aiming at the study of three other variants: unambiguous, left deterministic and right deterministic. Such an approach is based on monoid expansions. The proofs are purely algebraic and use another approach, based on properties on the kernel category of a monoid relational morphism, without going through the languages. This gives a unified fashion to deal with all these variants and allows us to better understand the connections between these two approaches. Finally, we show that local finiteness of an M -variety is transferred to the M -varieties corresponding to these variants and apply the general results to the M -variety of idempotent and commutative monoids.

Actions, wreath products of [formula omitted]-varieties and concatenation product

The framework of C -varieties, introduced by the third author, extends the scope of Eilenberg's variety theory to new classes of languages. In this paper, we first define C -varieties of actions, which are closely related to automata, and prove their equivalence with the original definition of C -varieties of stamps. Next, we complete the study of the wreath product initiated by Ésik and Ito by extending its definition to C -varieties in two different ways, which are proved to be equivalent. We also state an extension of the wreath product principle, a standard tool of language theory. Finally, our main result generalizes to C -varieties the algebraic characterization of the closure under product of a variety of languages.

Algebraic tools for the concatenation product

This paper is a contribution to the algebraic study of the concatenation product. In the first part of the paper, we extend to the ordered case standard algebraic tools related to the concatenation product, like the Schützenberger product and the relational morphisms. We show in a precise way how the ordered Schützenberger product corresponds to polynomial operations on languages. In the second part of the paper, we apply these results to establish a bridge between the three standard concatenation hierarchies, namely the Straubing–Thérien hierarchy, the Brzozowski's (or dot-depth) hierarchy and the group hierarchy.

A conjecture on the concatenation product

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing–Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.

Quasi-Shuffle Products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication a on the set of noncommutative polynomials in A which we call a quasi-shuffle products it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, a, Δ)s in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle products we give an isomorphism exp of the shuffle Hopf algebra (U, III, Δ) onto (U, a, Δ) the set L of Lyndon words on A and their images l exp(w) m w e Lr freely generate the algebra (U, a). We also consider the graded dual of (U, a, Δ). We define a deformation a_q of a that coincides with a when q e 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.

Knowledge Discovery in Complex Objects

Learning concepts and rules from structured (complex) objects is a quite challenging but very relevant problem in the area of machine learning and knowledge discovery. In order to take into account and exploit the semantic relationships that hold between atomic components of structured objects, we propose a knowledge discovery process, which starts from a set of complex objects to produce a set of related atomic objects (called contexts). The second step of the process makes use of the concatenation product to get a global context in which binary relations of individual contexts coexist with relations produced by the application of some operators to individual contexts. The last step permits the discovery of concepts and implication rules using the concept lattice as a framework in order to discover and interpret nontrivial concepts and rules that may relate different components of complex objects. This paper focuses on two main steps of the knowledge discovery process, namely data mining and interpretation.

Trees, congruences and varieties of finite semigroups

A classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups ♦ u( V) is associated to each tree u. In this paper, starting with the congruence γ A generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence  u ( γA) in such a way to generate ♦ u( V) for A. We give partial results on the problem of comparing the congruences  u ( γA) or the pseudovarieties ♦ u( V) . We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties.

On the Pin-Thérien expansion of idempotent monoids

The objective of this paper is to pesent a description of the varieties\(\bar V\) of finite monoids introduced in the Pin and Therien’s paper “On the bideterministic concatenation product” [4] in the case thatV is any variety of finite monoids.

THE BIDETERMINISTIC CONCATENATION PRODUCT

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, [Formula: see text] the corresponding variety of languages and [Formula: see text] the smallest variety containing [Formula: see text] and the bideterministic products of two languages of [Formula: see text]. We give an algebraic description of the variety of monoids [Formula: see text] corresponding to [Formula: see text]. For instance, we compute [Formula: see text] when V is one of the following varieties: the variety of idempotent and commutative monoids, the variety of monoids which are semilattices of groups of a given variety of groups, the variety of ℛ-trivial and idemptotent monoids. In particular, we show that the smallest variety of languages closed under bideterministic product and containing the language {1}, corresponds to the variety of [Formula: see text]-trivial monoids with commuting idempotents. Similar results were known for the other variants of the concatenation product, but the corresponding algebraic operations on varieties of monoids were based on variants of the semidirect product and of the Malcev product. Here the operation [Formula: see text] makes use of a construction which associates to any finite monoid M an expansion [Formula: see text] with the following properties: (1) M is a quotient of [Formula: see text] (2) the morphism [Formula: see text] induces an isomorphism between the submonoids of [Formula: see text] and of M generated by the regular elements and (3) the inverse image under π of an idempotent of M is a 2-nilpotent semigroup.

Un modèle asynchrone pour les systèmes distribués

Many researchers are engaged in the field of distributed systems modelization. We present a new model inspired by the language theory. Its particularity within this theory is the rejection of interleaving for the representation of concurrency. Here, the main ideas are born from languages such as CSP or ESTELLE. The point is mainly the real lack of dependency between processes apart from synchronisations, which are rendez-vous. Once the model is defined, we introduce, as in the infinitary free monoïd, notions such as length, concatenation, prefix order, upper bound and infinite product. So the distributed model is provided with all the operations necessary to develop semantics. Then, we prove that the finitary part of the distributed model is isomorphic to the free partially commutative monoïd (fpcm). Finally, we settle the bases of an infinitary extension of the fpcm.

Locally trivial categories and unambiguous concatenation

We use the recently developed theory of finite categories and the two-sided kernel to study the effect of the unambiguous concatenation product of recognizable languages on the syntactic monoids of the languages involved. As a result of this study we obtain an algebraic characterization (originally due to Pin) of the closure of a variety of languages under boolean operations and unambiguous concatenation, and a new proof of a theorem of Straubing characterizing the closure of a variety of languages under boolean operations and concatenation. We also note some connections to the study of the dot-depth hierarchy.

Hiérarchies de concaténation

Is it well-known that there exists a one-to-one correspondence between varieties of recognizable languages and varieties of finite semigroups. New hierarchies of varieties of languages (based on the concatenation product) are defined and an algebraic description of the corresponding hierarchies of varieties of semigroups is given. Various well-known hierarchies are obtained as particular cases. The construction is based on the following result: if a language L is recognized by the Schutzenberger product of the monoids M_0, ..., M_n, then L belongs to the Boolean closure of the set of languages of the form L_{i_0}a_1L_{i_1} ... a_rL_{i_r} (0 ≤ i_0 < i_1 < ... < i_r ≤ n) where the ak are letters and the Lik are recognized by M_{i_k} (0 ≤ k ≤ r). Decidability and inclusion problems are also discussed.