A description of n-ary semigroups polynomial-derived from integral domains

We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing G{\l}azek and Gleichgewicht's classification of the corresponding ternary semigroups

Associative polynomial functions over bounded distributive lattices

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n>=1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of polynomial functions over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same.Comment: Final versio

Some functional equations related to the characterizations of information measures and their stability

The main purpose of this paper is to investigate the stability problem of some functional equations that appear in the characterization problem of information measures.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1307.0657, arXiv:1307.0631, arXiv:1307.0664, arXiv:1307.065

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

Solution of distributive-like quasigroup functional equations

summary:We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259--1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup functional equations up to parastrophic equivalence [Sokhatsky F.M.: On classification of distributive-like functional equations, Book of Abstracts of the $8^{th}$ International Algebraic Conference in Ukraine, July 5--12 (2011), Lugansk, Ukraine, p. 79]. In the article, we find the solution sets of four generalized distributive-like quasigroup functional equations of different classes. In consequence, we solve one of the equations on topological quasigroup operations, defined on arbitrary topological space as well as on the space of real numbers with the natural topology. The fifth class contains the generalized left distributivity functional equation. V.D. Belousov [Some remarks on the functional equation of generalized distributivity, Aequationes Math. 1 (1968), no. 1--2, 54--65] described only a subset of its solution set. The set of all solutions still remains an open problem in the quasigroup theory and in the functional equation theory

Responsible Source Multicasting

Part 4: Protocols and PerformanceInternational audienceThere is no effective method to support IP level Internet wide multisource multicast sessions, that can be easily used from almost every ISP There are several protocols implementing the necessary functionality, but the penetration of them is really low recently. The most obvious work all-round is using SSM – Source Specific Multicasting, in which, the IP multicast session is identified by the multicast group address and the source’s unicast IP address. SSM allows using all the SSM address range for every source IP addresses and limits the address allocation problem inside the host of the source; however, its significant drawback is that the SSM has no native support to create multicast sessions with more than one source; it uses separate source specific distribution trees for every single source therefore it needs more resources on the router side. The alternative solution for supporting multisource multicast session is the ASM – Any Source Multicasting. However, its significant drawback is the lack of Internet wide dynamic address allocation. To address the recent problems of the Internet wide multisource multicast session a novel IP multicast service model, the Responsible Source Multicasting - RSM is introduced in this paper. RSM uses shared distribution trees like ASM; however, builds a reverse path tree towards an appropriate well-known unicast IP address like SSM. The paper demonstrates that this novel multicast routing protocol handles Internet wide multisource multicast sessions. The paper also shortly presents the DAMA – Dynamic Address Allocation of Multicast Addresses protocol for dynamic multicast IP address allocation, which works in a strong collaboration with the RSM