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The practical purpose of this research is to propose a candidate for post-quantum signature standard that is free of significant drawback of the finalists of the NIST world competition, which consists in the large size of the signature and the public key. The practical purpose is to propose a fundamentally new method for development of algebraic digital signature algorithms.

The proposed method is distinguished by the use of two different finite commutative associative algebras as a single algebraic support of the digital signature scheme and setting two different verification equation for a single signature. A single public key is computed as the first and the second public keys, elements of which are computed exponentiating two different generators of cyclic groups in each of the algebras.

Additionally, a scalar multiplication by a private integer is performed as final step of calculation of every element of the public key. The same powers and the same scalar values are used to compute the first and the second public keys by the same mathematic formulas. Due to such design, the said generators are kept in secret, providing resistance to quantum attacks. Two new finite commutative associative algebras, multiplicative group of which possesses four-dimensional cyclicity, have been proposed as a suitable algebraic support.

The introduced method is novel and includes new techniques for designing algebraic signature schemes that resist quantum attacks. On its base, a new practical post-quantum signature scheme with relatively small size of signature and public key is developed.

The purpose of this paper is to survey briefly how harmonic analyis started and developed throughout the centuries to reach its modern status and its surprisingly wide range of applications.

The author traces applications of harmonic analysis back to Mesopotamia, ancient Egypt and the Indus Valley, showing how the Greeks have applied trigonometry and influenced its birth, then the important developments in India in the sixth century laying the first brick to modern trigonometry with the definition of the sinus, then medieval India founding modern mathematical analysis. Trigonometry was developed further by the Arabs until the fourteenth century, then by the Europeans. The eighteenth century in France was particularly important when Bernoulli solved, with an infinite trigonometric series, the vibrating string problem, then Fourier, who studied these series extensively. The author goes on to harmonic analysis on locally compact groups, and ends up with a quick personal view on harmonic analysis nowadays. The last section of the paper presents some of the modern applications. Harmonic analysis is, of course, still used for navigation but also has many other very surprising applications such as signal processing, quantum mechanics, neuroscience, tomography, etc.

The power of harmonic analysis lies in giving the solutions to various problems as infinite series of basic functions, so to be able to produce algorithms for FFT boxes, it must be understood how these series came about and the convergence of these series.

The review should be useful to people interested in studying and/or applying harmonic analysis.

The authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.

The method is theoretical.

The authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.

The paper is original.

In this paper we will try to reach a twofold goal. First we will give an analysis of mathematical terminology in order to give practical hints for assigning subject headings to a book. Secondly, we will propose a method which could be helpful for the subject analysis of a given document. The main basis for this method is the connection between a special classification scheme, the Subject Classification Scheme of the American Mathematical Society, and the task of indexing books by subject headings. Examples of this method are given, and they are compared with Library of Congress Subject Headings and PRECIS entries. With both the study of the terminology and the proposed method, it should be possible to increase quality and consistency of the library indexing results for mathematical books. A thesaurus for mathematics with entries along the lines of the PRECIS rules and connected with the subject classification scheme of the American Mathematical Society, would be desirable, as would the printing of appropriate classification data, assigned to them by their authors, inside the books.

The purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.

The methods used are from the topic multiplicative ideal theory from commutative ring theory.

Let S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.

Motivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

The aim of this paper is to investigate the relationship between the ring structure of the twisted partial skew generalized power series ring RG,≤;Θ and the corresponding structure of its zero-divisor graph Γ̅RG,≤;Θ.

The authors first introduce the history and motivation of this paper. Secondly, the authors give a brief exposition of twisted partial skew generalized power series ring, in addition to presenting some properties of such structure, for instance, a-rigid ring, a-compatible ring and (G,a)-McCoy ring. Finally, the study’s main results are stated and proved.

The authors establish the relation between the diameter and girth of the zero-divisor graph of twisted partial skew generalized power series ring RG,≤;Θ and the zero-divisor graph of the ground ring R. The authors also provide counterexamples to demonstrate that some conditions of the results are not redundant. As well the authors indicate that some conditions of recent results can be omitted.

The results of the twisted partial skew generalized power series ring embrace a wide range of results of classical ring theoretic extensions, including Laurent (skew Laurent) polynomial ring, Laurent (skew Laurent) power series ring and group (skew group) ring and of course their partial skew versions.

Studies the concept of the Cartesian composition of fuzzy finite state machines. Shows that fuzzy finite state machines and their Cartesian composition share many structural properties. Some of these properties are singly generated; retrievability, connectedness, strong connectedness, commutativity, perfection and state independence. Thus a fuzzy finite state machine which is a Cartesian composition of submachines can be studied in terms of smaller machines.

This paper aims to concentrate on reviewing articles on open source software (OSS), found on the internet, that were published in the year 2016.

A critical look is taken at OSS generally as found in the literature and also on the commonly used OSS in libraries and information science work or research work.

It is hoped that this review will assist in pointing out the trends and aid in informed decision-making, as it concerns OSS.

Akintomide (2016) described OSS as a part of human existence which is exhibited in mobile devices that run on the android operating system, to computers, laptops and servers that run on the Linux operating system, to other software (Library management software inclusive) that are open access.

The openness of it makes the users (libraries and scholars) leave documentation that steers how the open software technologies are implemented and by whom. They are free, have used communities, are vetted by users and are constantly in development.

Discusses the relationship between computer proof and human proof. These issues are discussed both in general and specifically regarding the recent solution of the Robbins problem via a proof generated by computer. The Robbins problem was a long‐standing open problem about axioms for Boolean algebra. One point of this paper is to show that the proof of the Robbins conjecture, generated by a computer, can be filled in and understood by human beings. We accomplish this aim in the present paper by presenting a notational reformulation of Boolean algebra and the Robbins problem. The notational/linguistic issue developed here is of cybernetic, linguistic and semiotic interest. It is our contention that mathematics can behave non‐trivially under change of notation. Change of notation can be as significant as change of language. In the present case the change of language afforded by an appropriate change of notation makes a mathematical domain accessible to human beings that has heretofore been only accessible to computers.