A New Gauss Sum and Its Recursion Properties

In this paper, we introduce a new Gauss sum, and then we use the elementary and analytic methods to study its various properties and prove several interesting three-order linear recursion formulae for it.

Parallel Tiled Code for Computing General Linear Recurrence Equations

In this article, we present a technique that allows us to generate parallel tiled code to calculate general linear recursion equations (GLRE). That code deals with multidimensional data and it is computing-intensive. We demonstrate that data dependencies available in an original code computing GLREs do not allow us to generate any parallel code because there is only one solution to the time partition constraints built for that program. We show how to transform the original code to another one that exposes dependencies such that there are two linear distinct solutions to the time partition restrictions derived from these dependencies. This allows us to generate parallel 2D tiled code computing GLREs. The wavefront technique is used to achieve parallelism, and the generated code conforms to the OpenMP C/C++ standard. The experiments that we conducted with the resulting parallel 2D tiled code show that this code is much more efficient than the original serial code computing GLREs. Code performance improvement is achieved by allowing parallelism and better locality of the target code.

Noun phrase in Enets

The paper presents a corpus-based description of the noun phrase structure in Enets dealing with both Enets dialects – Forest Enets and Tundra Enets. An Enets noun phrase has six slots for modifiers: determiner, relative clause, possessor NP, numeral, adjective phrase, apposed NP. Determiners, relative clauses, and adjective phrases are subject to linear recursion, other modifiers are not. All modifiers precede the head NP. In Enets, there is no agreement between head noun and modifiers, but numerals have different patterns in the choice of head noun number form. Kokkuvõte. Andrej Šluinski: Noomenifraas eenetsi keeles. Artikkel esitab korpuspõhise kirjelduse eenetsi keele noomenifraasi struktuurist mõlemas eenetsi keele murdes – metsaeenetsi ja tundraeenetsi. Eenetsi noomenifraasil on kuus täiendikohta: määratleja, relatiivlause, omajat väljendav NP, numeraal, omadussõnafraas, appositsiooniline NP. Määratlejad, relatiivlaused ja omadussõnafraasid alluvad lineaarsele rekursioonile, teised täiendid mitte. Kõik täiendid eelnevad põhisõnale. Eenetsi keeles puudub põhisõna ja täiendi ühilduvus, kui numeraalid nõuavad noomenifraasi põhisõnalt erinevaid arvuvorme. Аннотация. Андрей Шлуинский: Именная группа в энецком языке. В статье представлено выполненное на материале корпуса текстов описание структуры именной группы в обоих диалектах энецкого языка – лесном тундровом. Энецкая именная группа содержит шесть позиций для модификаторов вершинного существительного: детерминатор, относительное предложение, именная группа посессора, числительное, группа прилагательного, соположенная именная группа. Детерминаторы, относительные предложения и группы прилагательного подлежат линейной рекурсии, в отличие от других модификаторов. Все модификаторы предшествуют вершинному существительному. В энецком языке отсутствует согласование между вершинным существительным и модификаторами, но представлены разные модели выбора числовой формы вершинного существительного в именных группах с числительными.

Recurrences for the genus polynomials of linear sequences of graphs

Given a finite graph H, the nth member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in Gn−1 by adding edges or identifying vertices, always in the same way. The genus polynomial ΓG(z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials ΓGn(z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, …, k with polynomials $\begin{array}{} \Gamma_{G_n}^j \end{array}$ (z), for j = 1, 2, …, k; from these polynomials, we form a column vector $\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ that satisfies a recursion Vn(z) = M(z)Vn−1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a kth degree linear recursion for Γn(z), allowing us to avoid the partitioning, thereby yielding a reduction from k2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

Fluctuation Theory for Upwards Skip-Free Lévy Chains

A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free Lévy chains, i.e., for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of Lévy processes—several results, however, can be made more explicit/exhaustive in the compound Poisson setting. Importantly, the scale functions admit a linear recursion, of constant order when the support of the jump measure is bounded, by means of which they can be calculated—some examples are presented. An application to the modeling of an insurance company’s aggregate capital process is briefly considered.

Ontology-Mediated Querying with the Description Logic EL: Trichotomy and Linear Datalog Rewritability

We consider ontology-mediated queries (OMQs) based on an EL ontology and an atomic query (AQ), provide an ultimately fine-grained analysis of data complexity and study rewritability into linear Datalog-aiming to capture linear recursion in SQL. Our main results are that every such OMQ is in AC0, NL-complete or PTime-complete, and that containment in NL coincides with rewritability into linear Datalog (whereas containment in AC0 coincides with rewritability into first-order logic). We establish natural characterizations of the three cases, show that deciding linear Datalog rewritability (as well as the mentioned complexities) is ExpTime-complete, give a way to construct linear Datalog rewritings when they exist, and prove that there is no constant bound on the arity of IDB relations in linear Datalog rewritings.