rss_2.0Mathematics FeedSciendo RSS Feed for Mathematics Feed methods for a system of coupled Cahn-Hilliard equations<abstract> <title style='display:none'>Abstract</title> <p>In this work, we consider a system of coupled Cahn-Hilliard equations describing the phase separation of a copolymer and a homopolymer blend. We propose some numerical methods to approximate the solution of the system which are based on suitable combinations of existing schemes for the single Cahn-Hilliard equation. As a verification for our experimental approach, we present some tests and a detailed description of the numerical solutions’ behaviour obtained by varying the values of the system’s characteristic parameters.</p> </abstract>ARTICLE2021-03-30T00:00:00.000+00:00Functional Sequence in Norm Space<abstract> <title style='display:none'>Summary</title> <p>In this article, we formalize in Mizar [1], [2] functional sequences and basic operations on functional sequences in norm space based on [5]. In the first section, we define functional sequence in norm space. In the second section, we define pointwise convergence and prove some related theorems. In the last section we define uniform convergence and limit of functional sequence.</p> </abstract>ARTICLE2021-05-21T00:00:00.000+00:00General Theory and Tools for Proving Algorithms in Nominative Data Systems<abstract> <title style='display:none'>Summary</title> <p>In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future.</p> <p>This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16].</p> <p>The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and postconditions [17], [19], [7], [5].</p> </abstract>ARTICLE2021-05-21T00:00:00.000+00:00Partial Correctness of an Algorithm Computing Lucas Sequences<abstract> <title style='display:none'>Summary</title> <p>In this paper we define some properties about finite sequences and verify the partial correctness of an algorithm computing <italic>n</italic>-th element of Lucas sequence [23], [20] with given <italic>P</italic> and <italic>Q</italic> coefficients as well as two first elements (<italic>x</italic> and <italic>y</italic>). The algorithm is encoded in nominative data language [22] in the Mizar system [3], [1].</p> <p><disp-quote><p><monospace>i := 0</monospace></p> <p><monospace>s := x</monospace></p> <p><monospace>b := y</monospace></p> <p><monospace>c := x</monospace></p> <p><monospace>while (i &lt;&gt; n)</monospace></p> <p> <monospace>c := s</monospace></p> <p> <monospace>s := b</monospace></p> <p> <monospace>ps := p*s</monospace></p> <p> <monospace>qc := q*c</monospace></p> <p> <monospace>b := ps − qc</monospace></p> <p> <monospace>i := i + j</monospace></p> <p><monospace>return s</monospace></p></disp-quote></p> <p>This paper continues verification of algorithms [10], [14], [12], [15], [13] written in terms of simple-named complex-valued nominative data [6], [8], [19], [11], [16], [17]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and post-conditions [18], [21], [7], [5].</p> </abstract>ARTICLE2021-05-21T00:00:00.000+00:00Introducing Fully Up-Semigroups<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UP-semigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right UP-semigroup, a right-right UP-semigroup, a fully-left UP-semigroup, a fully-right UP-semigroup, a left-fully UP-semigroup, a right-fully UP-semigroup, a fully-fully UP-semigroup, and find their examples.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Conrad’s Partial Order on P.Q.-Baer *-Rings<abstract> <title style='display:none'>Abstract</title> <p>We prove that a p.q.-Baer *-ring forms a pseudo lattice with Conrad’s partial order and also characterize p.q.-Baer *-rings which are lattices. The initial segments of a p.q.-Baer *-ring with the Conrad’s partial order are shown to be an orthomodular posets.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Commutativity of Prime Rings with Symmetric Biderivations<abstract> <title style='display:none'>Abstract</title> <p>The present paper shows some results on the commutativity of <italic>R</italic>: Let <italic>R</italic> be a prime ring and for any nonzero ideal <italic>I</italic> of <italic>R</italic>, if <italic>R</italic> admits a biderivation <italic>B</italic> such that it satisfies any one of the following properties (i) <italic>B</italic>([<italic>x, y</italic>], <italic>z</italic>) = [<italic>x, y</italic>], (ii) <italic>B</italic>([<italic>x, y</italic>], <italic>m</italic>) + [<italic>x, y</italic>] = 0, (iii) <italic>B</italic>(<italic>xoy, z</italic>) = <italic>xoy,</italic> (iv) <italic>B</italic>(<italic>xoy, z</italic>) + <italic>xoy</italic> = 0, (v) <italic>B</italic>(<italic>x, y</italic>)<italic>oB</italic>(<italic>y, z</italic>) = 0, (vi)<italic>B</italic>(<italic>x, y</italic>)<italic>oB</italic>(<italic>y, z</italic>) = <italic>xoz</italic>, (vii) <italic>B</italic>(<italic>x, y</italic>)<italic>oB</italic>(<italic>y, z</italic>) + <italic>xoy</italic> = 0, for all <italic>x, y, z</italic> ∈ <italic>R</italic>, then <italic>R</italic> is a commutative ring.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Folding Theory of Implicative and Obstinate Ideals in Bl-Algebras<abstract> <title style='display:none'>Abstract</title> <p>In this paper, the concepts of <italic>n</italic>-fold implicative ideals and <italic>n</italic>-fold obstinate ideals in <italic>BL</italic>-algebras are introduced. With respect to this concepts, some related results are given. In particular, it is proved that an ideal is an <italic>n</italic>-fold implicative ideal if and only if is an <italic>n</italic>-fold Boolean ideal. Also, it is shown that a <italic>BL</italic>-algebra is an <italic>n</italic>-fold integral <italic>BL</italic>-algebra if and only if trivial ideal {0} is an <italic>n</italic>-fold obstinate ideal. Moreover, the relation between <italic>n</italic>-fold obstinate ideals and <italic>n</italic>-fold (integral) obstinate filters in <italic>BL</italic>-algebras are studied by using the set of complement elements. Finally, it is proved that ideal <italic>I</italic> of <italic>BL</italic>-algebra <italic>L</italic> is an <italic>n</italic>-fold obstinate ideal if and only if <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_dmgaa.1295_eq_001.png"/> <mml:math xmlns:mml="" display="inline"><mml:mrow><mml:mfrac><mml:mi>L</mml:mi><mml:mi>T</mml:mi></mml:mfrac></mml:mrow></mml:math> <tex-math>${L \over T}$</tex-math> </alternatives> </inline-formula> is an <italic>n</italic>-fold obstinate <italic>BL</italic>-algebra.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Nondistributive Rings and Their Öre Localizations<abstract> <title style='display:none'>Abstract</title> <p>In the paper, we introduce the notion of a nondistributive ring <italic>N</italic> as a generalization of the notion of an associative ring with unit, in which the addition needs not be abelian and the distributive law is replaced by <italic>n</italic>0 = 0<italic>n</italic> = 0 for every element <italic>n</italic> of <italic>N</italic>. For a nondistributive ring <italic>N</italic>, we introduce the notion of a nondistributive ring of left quotients <italic>S</italic><sup>−1</sup><italic>N</italic> with respect to a multiplicatively closed set <italic>S</italic> ⊆ <italic>N</italic>, and determine necessary and sufficient conditions for the existence of <italic>S</italic><sup>−1</sup><italic>N</italic>.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Bi-Interior Ideals of Γ-Semirings<abstract> <title style='display:none'>Abstract</title> <p>In this paper, as a further generalization of ideals, we introduce the notion of bi-interior ideal as a generalization of quasi ideal, bi-ideal and interior ideal of Γ-semiring and study the properties of bi-interior ideals of Γ-semiring. We prove that if <italic>M</italic> is a field Γ-semiring, then <italic>M</italic> is a bi-interior simple Γ-semiring.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00The Armendariz Graph of a Ring<abstract> <title style='display:none'>Abstract</title> <p>In this paper we initiate the study of Armendariz graph of a commutative ring <italic>R</italic> and investigate the basic properties of this graph such as diameter, girth, domination number, etc. The Armendariz graph of a ring <italic>R</italic>, denoted by <italic>A</italic>(<italic>R</italic>), is an undirected graph with nonzero zero-divisors of <italic>R</italic>[<italic>x</italic>] (i.e., <italic>Z</italic>(<italic>R</italic>[<italic>x</italic>])*) as the vertex set, and two distinct vertices <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_dmgaa.1292_eq_001.png"/> <mml:math xmlns:mml="" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msup></mml:mrow></mml:math> <tex-math>$f(x) = \sum\nolimits_{i = 0}^n {{a_i}{x^i}}$</tex-math> </alternatives> </inline-formula> and <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_dmgaa.1292_eq_002.png"/> <mml:math xmlns:mml="" display="inline"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mrow></mml:math> <tex-math>$g(x) = \sum\nolimits_{j = 0}^m {{b_j}{x^j}}$</tex-math> </alternatives> </inline-formula> are adjacent if and only if <italic>a<sub>i</sub>b<sub>j</sub></italic> = 0, for all <italic>i, j</italic>. It is shown that <italic>A</italic>(<italic>R</italic>), a subgraph of Γ(<italic>R</italic>[<italic>x</italic>]), the zero divisor graph of the polynomial ring <italic>R</italic>[<italic>x</italic>], have many graph properties in common with Γ(<italic>R</italic>[<italic>x</italic>]).</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Aggregating Fuzzy Binary Relations and Fuzzy Filters<abstract> <title style='display:none'>Abstract</title> <p>The main goal of this paper is to investigate the aggregation of diverse families of binary fuzzy relations, fuzzy filters, and fuzzy lattices. Some links between these families and their images via an aggregation are explored.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Local Cohomology Modules and Relative Cohen-Macaulayness<abstract> <title style='display:none'>Abstract</title> <p>Let (<italic>R</italic>, 𝔪) denote a commutative Noetherian local ring and let <italic>M</italic> be a finite <italic>R</italic>-module. In this paper, we study relative Cohen-Macaulay rings with respect to a proper ideal 𝔞 of <italic>R</italic> and give some results on such rings in relation with Artinianness, Non-Artinianness of local cohomology modules and Lyubeznik numbers. We also present some related examples to this issue.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Quadratic Approximation of Generalized Tribonacci Sequences<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we give quadratic approximation of generalized Tribonacci sequence {<italic>V<sub>n</sub></italic>}<italic><sub>n</sub></italic><sub>≥0</sub> defined by <italic>V<sub>n</sub></italic> = <italic>rV<sub>n−1</sub></italic> + <italic>sV <sub>n−2</sub></italic> + <italic>tV <sub>n−3</sub></italic> (<italic>n</italic> ≥ 3) and use this result to give the matrix form of the <italic>n</italic>-th power of a companion matrix of {<italic>V<sub>n</sub></italic>}<italic><sub>n</sub></italic><sub>≥0</sub>. Then we re-prove the cubic identity or Cassini-type formula for {<italic>Vn</italic>} <italic><sub>n</sub></italic><sub>≥0</sub> and the Binet’s formula of the generalized Tribonacci quaternions.</p> </abstract>ARTICLE2018-11-21T00:00:00.000+00:00Design of software-defined network experimental teaching scheme based on virtualised Environment<abstract><title style='display:none'>Abstract</title><p>Aiming to address the shortage of experimental resources, the high cost of large-scale deployment of hardware experimental environment and the difficulty for students to get started in the software-defined network (SDN) course, this article proposes an SDN experimental teaching scheme based on the virtualised environment, and gives a specific experimental scheme design. The scheme utilises virtualisation technology to build a SDN experimental environment quickly, uses a lightweight network simulation platform – that goes by the name of Mininet – to build the SDN network and uses open-source controller Floodlight for centralised control of the SDN network. The scheme is mainly divided into three phases: basic, improvement and synthesis. In the basic phase, experimental projects mainly include the study of SDN basic concepts and the use of relevant tools; in the improvement phase, experimental projects mainly include the use of SDN flow table, group table, etc; in the synthetic phase, we design two innovative experimental projects that use computational intelligence technology to achieve efficient load balancing and accurate malicious attack detection. The difficulty of each phase is increasing. The constantly evolving levels of difficulty allow the individual needs of students with different levels to be met, thereby improving the effect of SDN experimental teaching and cultivating innovative SDN talents.</p></abstract>ARTICLE2021-05-25T00:00:00.000+00:00The Comprehensive Diagnostic Method Combining Rough Sets and Evidence Theory<abstract><title style='display:none'>Abstract</title><p>To solve the difficulties in practice caused by the subjectivity, relativity and evidence combination focus element explosion during the process of solving the uncertain problems of fault diagnosis with evidence theory, this paper proposes a fault diagnosis inference strategy by integrating rough sets with evidence theory along with the theories of information fusion and mete-synthesis. By using rough sets, redundancy of characteristic data is removed and the unrelated essential characteristics are extracted, the objective way of basic probability assignment is proposed, and an evidence synthetic method is put forward to solve high conflict evidence. The method put forward in this paper can improve the accuracy rate of fault diagnosis with the redundant and complementary information of various faults by synthesizing all evidences with the rule of the composition of evidence theory. Besides, this paper proves the feasibility and validity of experiments and the efficiency in improving fault diagnosis.</p></abstract>ARTICLE2021-05-25T00:00:00.000+00:00Economic analysis of rice production by small-holder women farmers in Adamawa State, Nigeria<abstract> <title style='display:none'>Abstract</title> <p>This study is focused on analysing the economics of rice production by the small-holder female rice-farmers in Adamawa State, Nigeria. Specifically, the study described female rice-farmers’ socio-economic characteristics, analysed their technical and allocative efficiencies, and also assessed their economic efficiency in the area. Primary data were collected from 180 randomly selected female rice-farmers in Adamawa State using a semi-structured questionnaire. Descriptive statistics and stochastic frontier function were used analysing the data collected. Findings of the study revealed that the technical in-efficiency of the farmers is influenced by farming experience, education, and access to credit facilities. Also, the allocative in-efficiency of female farmers was influenced by education, household size, and access to credits. The result showed that education and access to credit facilities were common factors affecting the technical and allocative efficiency of female rice farmers in the area. The mean economic efficiency of the female farmers was 0.6. Therefore, the study recommends that the government and other actors in the agricultural sector should facilitate women’s access to credit facilities and other agricultural information relevant to rice production.</p> </abstract>ARTICLE2021-05-29T00:00:00.000+00:00Remittance in Niger: effects on economic growth and on migrants’ left behind<abstract> <title style='display:none'>Abstract</title> <p>Remittances have long been an important source of revenue for many people in the Republic of Niger. In order to fight poverty, young people choose to migrate. In 2019, a total of 293 million U.S. dollars was sent by migrants to their relatives in Niger; that is 3% of Niger Gross Domestic Product (GDP). The objective of this study is to analyze the effects of remittances on economic growth in Niger and the significance of its contribution in improving the living condition of migrants’ left behind families. The study applies a three-step econometric procedure followed by a survey on the usage of the remittances in the city of Tahoua (Republic of Niger). The study also performed some tests on the residuals for the accuracy of the prediction of the model. The empirical results showed no long run relationship between remittances, economic growth and gross fixed capital formation in Niger. However, in the short-run, the study revealed the existence of causal effect between remittances and economic growth. On the other hand, the results of the conducted survey in the city of Tahoua showed that 45.7% of the received remittance is used in food expenditure, 19.3% in education expenditure, 10.36% in health expenditure and 5.4% is allocated to house rent. The survey also revealed the importance of the remittances for the left behind. It indicates that 14% of the respondent left behind wish to see another family member engage in migration.</p> </abstract>ARTICLE2021-05-29T00:00:00.000+00:00Auction theory and a note on game mechanisms<abstract> <title style='display:none'>Abstract</title> <p>This paper will review important topics on the subject of auction theory and mechanism design, these include: efficiency first and foremost, also revenue comparison between different types of auctions and the issue of incentive compatibility, individual rationality with the general idea and proof that bilateral trade is inefficient. Mechanism design theory tells us that if buyers and sellers both have private information full efficiency is impossible, however Vickrey auction (single unit auction) will be efficient i.e. will put the goods in the hands of the buyers that value them most. However, the conclusion from this paper is that because of overvaluation of bidders the main result is inefficient, i.e. bids are too high. When weak and strong bidders are compared the main conclusion is that strong bidders’ expected payoff is higher in second price auction (SPA), while weak bidder prefers first price auction (FPA) bid.</p> </abstract>ARTICLE2021-05-29T00:00:00.000+00:00Business and Economics Students’ Self-Reports on Academic Reading in English as L2: A Factor Analysis<abstract> <title style='display:none'>Abstract</title> <p>Adequate reading behaviour is vital for text comprehension across fields. In today’s professional environment, a well-developed reading skill is also expected in English as a second language (EL2), which happens already in college, although transition to college-level reading may be difficult even in the first language. It is therefore useful to analyse students’ use of reading strategies to facilitate their academic progress. This study investigates the reading behaviour of junior students of business/economics when reading academic texts in EL2. We conducted an exploratory factor analysis (N=134) of a 45-item questionnaire about students’ awareness of reading strategies and their reading confidence (i.e. self-perceived competence in text retelling). The majority of the items were based on self-reports found in the literature (Kolić-Vehovec, Bajšanski, 2001; Mokhtari, Reichard, 2002; Taraban, Kerr, Rynearson, 2004) and several items were added to the questionnaire (e.g. questions related to note taking). Five factors were interpretable: four factors related to Metacognitive Awareness of Reading Strategies (Repeated Reading, Monitoring/Regulation, Note Taking and Elaboration) and the fifth factor covering self-perceived competence in text retelling (Reading Confidence). Internal consistency of the factors indicated by standardized Cronbach’s alphas were 0.83 (Repeated Reading), 0.80 (Monitoring and Regulation), 0.77 (Note Taking), 0.63 (Elaboration) and 0.75 (Reading Confidence). Three strategies positively correlated with each other (Repeated Reading, Monitoring/Regulation and Elaboration), while negative correlation was found between Note Taking and Reading Confidence. The results provide valuable information on the patterns in student reading as a baseline for further analysis of L2 text comprehension in college.</p> </abstract>ARTICLE2021-05-29T00:00:00.000+00:00en-us-1