Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 416

# Search results for: abstract algebra

##### 416 From Equations to Structures: Linking Abstract Algebra and High-School Algebra for Secondary School Teachers

Authors: J. Shamash

Abstract:

The high-school curriculum in algebra deals mainly with the solution of different types of equations. However, modern algebra has a completely different viewpoint and is concerned with algebraic structures and operations. A question then arises: What might be the relevance and contribution of an abstract algebra course for developing expertise and mathematical perspective in secondary school mathematics instruction? This is the focus of this paper. The course Algebra: From Equations to Structures is a carefully designed abstract algebra course for Israeli secondary school mathematics teachers. The course provides an introduction to algebraic structures and modern abstract algebra, and links abstract algebra to the high-school curriculum in algebra. It follows the historical attempts of mathematicians to solve polynomial equations of higher degrees, attempts which resulted in the development of group theory and field theory by Galois and Abel. In other words, algebraic structures grew out of a need to solve certain problems, and proved to be a much more fruitful way of viewing them. This theorems in both group theory and field theory. Along the historical ‘journey’, many other major results in algebra in the past 150 years are introduced, and recent directions that current research in algebra is taking are highlighted. This course is part of a unique master’s program – the Rothschild-Weizmann Program – offered by the Weizmann Institute of Science, especially designed for practicing Israeli secondary school teachers. A major component of the program comprises mathematical studies tailored for the students at the program. The rationale and structure of the course Algebra: From Equations to Structures are described, and its relevance to teaching school algebra is examined by analyzing three kinds of data sources. The first are position papers written by the participating teachers regarding the relevance of advanced mathematics studies to expertise in classroom instruction. The second data source are didactic materials designed by the participating teachers in which they connected the mathematics learned in the mathematics courses to the school curriculum and teaching. The third date source are final projects carried out by the teachers based on material learned in the course. Downloads 81
##### 415 Quantum Algebra from Generalized Q-Algebra

Authors: Muna Tabuni

Abstract:

The paper contains an investigation of the notion of Q algebras. A brief introduction to quantum mechanics is given, in that systems the state defined by a vector in a complex vector space H which have Hermitian inner product property. H may be finite or infinite-dimensional. In quantum mechanics, operators must be hermitian. These facts are saved by Lie algebra operators but not by those of quantum algebras. A Hilbert space H consists of a set of vectors and a set of scalars. Lie group is a differentiable topological space with group laws given by differentiable maps. A Lie algebra has been introduced. Q-algebra has been defined. A brief introduction to BCI-algebra is given. A BCI sub algebra is introduced. A brief introduction to BCK=BCH-algebra is given. Every BCI-algebra is a BCH-algebra. Homomorphism maps meanings are introduced. Homomorphism maps between two BCK algebras are defined. The mathematical formulations of quantum mechanics can be expressed using the theory of unitary group representations. A generalization of Q algebras has been introduced, and their properties have been considered. The Q- quantum algebra has been studied, and various examples have been given.

Keywords: Q-algebras, BCI, BCK, BCH-algebra, quantum mechanics

##### 414 Conspicuous and Significant Learner Errors in Algebra

Authors: Michael Lousis

Abstract:

The kind of the most important and conspicuous errors the students made during the three-years of testing of their progress in Algebra are presented in this article. The way these students’ errors changed over three-years of school Algebra learning also is shown. The sample is comprised of two hundred (200) English students and one hundred and fifty (150) Greek students, who were purposefully culled according to their participation in each occasion of testing in the development of the three-year Kassel Project in England and Greece, in both domains at once of Arithmetic and Algebra. Hence, for each of these English and Greek students, six test-scripts were available and corresponded to the three occasions of testing in both Arithmetic and Algebra respectively.

Keywords: algebra, errors, Kassel Project, progress of learning

##### 413 Explicit Chain Homotopic Function to Compute Hochschild Homology of the Polynomial Algebra

Authors: Zuhier Altawallbeh

Abstract:

In this paper, an explicit homotopic function is constructed to compute the Hochschild homology of a finite dimensional free k-module V. Because the polynomial algebra is of course fundamental in the computation of the Hochschild homology HH and the cyclic homology CH of commutative algebras, we concentrate our work to compute HH of the polynomial algebra.by providing certain homotopic function. Downloads 331
##### 412 Fuzzy Implicative Pseudo-Ideals of Pesudo-BCK Algebras

Authors: Alireza Gilani

Abstract:

In this paper, we consider the fuzzification of implicative pseudo-ideal in a pseudo-BCK algebra, and then we investigate some of their properties. We prove that the family of fuzzy implicative pseudo-ideal is completely distributive lattice. Downloads 265
##### 411 Non Commutative Lᵖ Spaces as Hilbert Modules

Authors: Salvatore Triolo

Abstract:

We discuss the possibility of extending the well-known Gelfand-Naimark-Segal representation to modules over a C*algebra. We focus our attention on the case of Hilbert modules. We consider, in particular, the problem of the existence of a faithful representation. Non-commutative Lᵖ-spaces are shown to constitute examples of a class of CQ*-algebras. Finally, we have shown that any semisimple proper CQ*-algebra (X, A#), with A# a W*-algebra can be represented as a CQ*-algebra of measurable operators in Segal’s sense. Downloads 96
##### 410 The Weights of Distinguished sl2-Subalgebras in Dn

Authors: Yassir I. Dinar

Abstract:

We computed the weights of the adjoint action of distinguished sl2-triples in Lie algebra of type Dn using mathematical induction. Downloads 234
##### 409 On Lie-Central Derivations and Almost Inner Lie-Derivations of Leibniz Algebras

Authors: Natalia Pacheco Rego

Abstract:

The Liezation functor is a map from the category of Leibniz algebras to the category of Lie algebras, which assigns a Leibniz algebra to the Lie algebra given by the quotient of the Leibniz algebra by the ideal spanned by the square elements of the Leibniz algebra. This functor is left adjoint to the inclusion functor that considers a Lie algebra as a Leibniz algebra. This environment fits in the framework of central extensions and commutators in semi-abelian categories with respect to a Birkhoff subcategory, where classical or absolute notions are relative to the abelianization functor. Classical properties of Leibniz algebras (properties relative to the abelianization functor) were adapted to the relative setting (with respect to the Liezation functor); in general, absolute properties have the corresponding relative ones, but not all absolute properties immediately hold in the relative case, so new requirements are needed. Following this line of research, it was conducted an analysis of central derivations of Leibniz algebras relative to the Liezation functor, called as Lie-derivations, and a characterization of Lie-stem Leibniz algebras by their Lie-central derivations was obtained. In this paper, we present an overview of these results, and we analyze some new properties concerning Lie-central derivations and almost inner Lie-derivations. Namely, a Leibniz algebra is a vector space equipped with a bilinear bracket operation satisfying the Leibniz identity. We define the Lie-bracket by [x, y]lie = [x, y] + [y, x] , for all x, y . The Lie-center of a Leibniz algebra is the two-sided ideal of elements that annihilate all the elements in the Leibniz algebra through the Lie-bracket. A Lie-derivation is a linear map which acts as a derivative with respect to the Lie-bracket. Obviously, usual derivations are Lie-derivations, but the converse is not true in general. A Lie-derivation is called a Lie-central derivation if its image is contained in the Lie-center. A Lie-derivation is called an almost inner Lie-derivation if the image of an element x is contained in the Lie-commutator of x and the Leibniz algebra. The main results we present in this talk refer to the conditions under which Lie-central derivation and almost inner Lie-derivations coincide. Downloads 69
##### 408 Classification of Tropical Semi-Modules

Authors: Wagneur Edouard

Abstract:

Tropical algebra is the algebra constructed over an idempotent semifield S. We show here that every m-dimensional tropical module M over S with strongly independent basis can be embedded into Sm, and provide an algebraic invariant -the Γ-matrix of M- which characterises the isomorphy class of M. The strong independence condition also yields a significant improvement to the Whitney embedding for tropical torsion modules published earlier We also show that the strong independence of the basis of M is equivalent to the unique representation of elements of M. Numerous examples illustrate our results. Downloads 308
##### 407 Mixed Number Algebra and Its Application

Authors: Md. Shah Alam

Abstract:

Mushfiq Ahmad has defined a Mixed Number, which is the sum of a scalar and a Cartesian vector. He has also defined the elementary group operations of Mixed numbers i.e. the norm of Mixed numbers, the product of two Mixed numbers, the identity element and the inverse. It has been observed that Mixed Number is consistent with Pauli matrix algebra and a handy tool to work with Dirac electron theory. Its use as a mathematical method in Physics has been studied. (1) We have applied Mixed number in Quantum Mechanics: Mixed Number version of Displacement operator, Vector differential operator, and Angular momentum operator has been developed. Mixed Number method has also been applied to Klein-Gordon equation. (2) We have applied Mixed number in Electrodynamics: Mixed Number version of Maxwell’s equation, the Electric and Magnetic field quantities and Lorentz Force has been found. (3) An associative transformation of Mixed Number numbers fulfilling Lorentz invariance requirement is developed. (4) We have applied Mixed number algebra as an extension of Complex number. Mixed numbers and the Quaternions have isomorphic correspondence, but they are different in algebraic details. The multiplication of unit Mixed number and the multiplication of unit Quaternions are different. Since Mixed Number has properties similar to those of Pauli matrix algebra, Mixed Number algebra is a more convenient tool to deal with Dirac equation. Downloads 275
##### 406 Formex Algebra Adaptation into Parametric Design Tools: Dome Structures

Authors: Réka Sárközi, Péter Iványi, Attila B. Széll

Abstract:

The aim of this paper is to present the adaptation of the dome construction tool for formex algebra to the parametric design software Grasshopper. Formex algebra is a mathematical system, primarily used for planning structural systems such like truss-grid domes and vaults, together with the programming language Formian. The goal of the research is to allow architects to plan truss-grid structures easily with parametric design tools based on the versatile formex algebra mathematical system. To produce regular structures, coordinate system transformations are used and the dome structures are defined in spherical coordinate system. Owing to the abilities of the parametric design software, it is possible to apply further modifications on the structures and gain special forms. The paper covers the basic dome types, and also additional dome-based structures using special coordinate-system solutions based on spherical coordinate systems. It also contains additional structural possibilities like making double layer grids in all geometry forms. The adaptation of formex algebra and the parametric workflow of Grasshopper together give the possibility of quick and easy design and optimization of special truss-grid domes. Downloads 175
##### 405 Extension and Closure of a Field for Engineering Purpose

Authors: Shouji Yujiro, Memei Dukovic, Mist Yakubu

Abstract:

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below. When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except possibly for commutativity, is today called a division ring ordivision algebra or sometimes a skew field. Also non-commutative field is still widely used. In French, fields are called corps (literally, body), generally regardless of their commutativity. When necessary, a (commutative) field is called corps commutative and a skew field-corps gauche. The German word for body is Körper and this word is used to denote fields; hence the use of the blackboard bold to denote a field. The concept of fields was first (implicitly) used to prove that there is no general formula expressing in terms of radicals the roots of a polynomial with rational coefficients of degree 5 or higher. An extension of a field k is just a field K containing k as a subfield. One distinguishes between extensions having various qualities. For example, an extension K of a field k is called algebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is called transcendental. The aim of Galois Theory is the study of algebraic extensions of a field. Given a field k, various kinds of closures of k may be introduced. For example, the algebraic closure, the separable closure, the cyclic closure et cetera. The idea is always the same: If P is a property of fields, then a P-closure of k is a field K containing k, having property, and which is minimal in the sense that no proper subfield of K that contains k has property P. For example if we take P (K) to be the property ‘every non-constant polynomial f in K[t] has a root in K’, then a P-closure of k is just an algebraic closure of k. In general, if P-closures exist for some property P and field k, they are all isomorphic. However, there is in general no preferable isomorphism between two closures.

Keywords: field theory, mechanic maths, supertech, rolltech

##### 404 Learners’ Reactions to Writing Activities in an Elementary Algebra Classroom

Abstract:

Various research has shown that writing allows students to engage in metacognition and provides them with a venue to communicate their disposition towards what they are learning. However, few studies have explored students’ feelings about the incorporation of such writing activities in their mathematics classes. Through reflection sheets, group discussions, and interviews, this mixed-methods study explored students’ perceptions and insights on supplementary writing activities in their Elementary Algebra class. Findings revealed that while students generally have a positive regard for writing activities, they have conflicting views about how writing activities can help them in their learning. A big majority contend that writing activities can enhance the learning of mathematical content and attitudes towards mathematics if they allow students to explore and synthesize what they have learned and reflected on their emotional disposition towards mathematics. Also, gender does not appear to play a significant role in students’ reactions to writing activities. Downloads 365
##### 403 Mathematical Anxiety and Misconceptions in Algebra of Grade Vii Students in General Emilio Aguinaldo National High School

Authors: Nessa-Amie T. Peñaflor, Antonio Cinto

Abstract:

This is a descriptive research on the level of math anxiety and mathematics misconceptions in algebra. This research is composed of four parts: (1) analysis of the level of anxiety of the respondents; (2) analysis of the common mathematical misconceptions in algebra; (3) relationship of socio-demographic profile in math anxiety and mathematical misconceptions and (4) analysis of the relationship of math anxiety and misconceptions in algebra. Through the demographic profile questionnaire it was found out that most of the respondents were female. Majority had ages that ranged from 13-15. Most of them had parents who finished secondary education. The biggest portion of Grade Seven students where from families with annual family income ranging from PhP 100, 000 to PhP 299, 999. Most of them came from public school. Mathematics Anxiety Scale for Secondary and Senior Secondary School Students (MAS) and set of 10 open-ended algebraic expressions and polynomials were also administered to determine the anxiety level and the common misconceptions in algebra. Data analysis revealed that respondents had high anxiety in mathematics. Likewise, the common mathematical misconceptions of the Grade Seven students were: combining unlike terms; multiplying the base and exponents; regarding the variable x as 0; squaring the first and second terms only in product of two binomials; wrong meaning attached to brackets; writing the terms next to each other but not simplifying in using the FOIL Method; writing the literal coefficient even if the numerical coefficient is 0; and dividing the denominator by the numerator when the numerical coefficient in the numerator is smaller than the numerical coefficient of the denominator. Results of the study show that the socio-demographic characteristics were not related to mathematics anxiety and misconceptions. Furthermore, students from higher section had high anxiety than those students on the lower section. Thus, belonging to higher or lower section may affect the mathematical misconceptions of the respondents. Downloads 340
##### 402 Methodological Aspect of Emergy Accounting in Co-Production Branching Systems

Authors: Keshab Shrestha, Hung-Suck Park

Abstract:

Emergy accounting of the systems networks is guided by a definite rule called ‘emergy algebra’. The systems networks consist of two types of branching. These are the co-product branching and split branching. The emergy accounting procedure for both the branching types is different. According to the emergy algebra, each branch in the co-product branching has different transformity values whereas the split branching has the same transformity value. After the transformity value of each branch is determined, the emergy is calculated by multiplying this with the energy. The aim of this research is to solve the problems in determining the transformity values in the co-product branching through the introduction of a new methodology, the modified physical quantity method. Initially, the existing methodologies for emergy accounting in the co-product branching is discussed and later, the modified physical quantity method is introduced with a case study of the Eucalyptus pulp production. The existing emergy accounting methodologies in the co-product branching has wrong interpretations with incorrect emergy calculations. The modified physical quantity method solves those problems of emergy accounting in the co-product branching systems. The transformity value calculated for each branch is different and also applicable in the emergy calculations. The methodology also strictly follows the emergy algebra rules. This new modified physical quantity methodology is a valid approach in emergy accounting particularly in the multi-production systems networks. Downloads 221
##### 401 Generalized π-Armendariz Authentication Cryptosystem

Abstract:

Algebra is one of the important fields of mathematics. It concerns with the study and manipulation of mathematical symbols. It also concerns with the study of abstractions such as groups, rings, and fields. Due to the development of these abstractions, it is extended to consider other structures, such as vectors, matrices, and polynomials, which are non-numerical objects. Computer algebra is the implementation of algebraic methods as algorithms and computer programs. Recently, many algebraic cryptosystem protocols are based on non-commutative algebraic structures, such as authentication, key exchange, and encryption-decryption processes are adopted. Cryptography is the science that aimed at sending the information through public channels in such a way that only an authorized recipient can read it. Ring theory is the most attractive category of algebra in the area of cryptography. In this paper, we employ the algebraic structure called skew -Armendariz rings to design a neoteric algorithm for zero knowledge proof. The proposed protocol is established and illustrated through numerical example, and its soundness and completeness are proved. Downloads 128
##### 400 Effect of Digital Technology on Students Interest, Achievement and Retention in Algebra in Abia State College of Education (Technical) Arochukwu

Authors: Stephen O. Amaraihu

Abstract:

This research investigated the effect of Computer Based Instruction on Students’ interest, achievement, and retention in Algebra in Abia State College of Education (Technical), Arochukwu. Three research questions and two hypotheses guided the study. Two instruments, Maths Achievement Test (MAT) and Maths Interest Inventory were employed, to test a population of three hundred and sixteen (316) NCE 1 students in algebra. It is expected that this research will lead to the improvement of students’ performance and enhance their interest and retention of basic algebraic concept. It was found that the majority of students in the college are not proficient in the use of ICT as a result of a lack of trained personnel. It was concluded that the state government was not ready to implement the usage of mathematics in Abia State College of Education. The paper recommends, amongst others, the employment of mathematics Lectures with competent skills in ICT and the training of lecturers of mathematics. Downloads 103
##### 399 Semirings of Graphs: An Approach Towards the Algebra of Graphs

Authors: Gete Umbrey, Saifur Rahman

Abstract:

Graphs are found to be most capable in computing, and its abstract structures have been applied in some specific computations and algorithms like in phase encoding controller, processor microcontroller, and synthesis of a CMOS switching network, etc. Being motivated by these works, we develop an independent approach to study semiring structures and various properties by defining the binary operations which in fact, seems analogous to an existing definition in some sense but with a different approach. This work emphasizes specifically on the construction of semigroup and semiring structures on the set of undirected graphs, and their properties are investigated therein. It is expected that the investigation done here may have some interesting applications in theoretical computer science, networking and decision making, and also on joining of two network systems. Downloads 65
##### 398 Performance-Based Quality Evaluation of Database Conceptual Schemas

Authors: Janusz Getta, Zhaoxi Pan

Abstract:

Performance-based quality evaluation of database conceptual schemas is an important aspect of database design process. It is evident that different conceptual schemas provide different logical schemas and performance of user applications strongly depends on logical and physical database structures. This work presents the entire process of performance-based quality evaluation of conceptual schemas. First, we show format. Then, the paper proposes a new specification of object algebra for representation of conceptual level database applications. Transformation of conceptual schemas and expression of object algebra into implementation schema and implementation in a particular database system allows for precise estimation of the processing costs of database applications and as a consequence for precise evaluation of performance-based quality of conceptual schemas. Then we describe an experiment as a proof of concept for the evaluation procedure presented in the paper. Downloads 225
##### 397 The Influence of Concreteness on English Compound Noun Processing: Modulation of Constituent Transparency

Authors: Turgut Coskun

Abstract:

'Concreteness effect' refers to faster processing of concrete words and 'compound facilitation' refers to faster response to compounds. In this study, our main goal was to investigate the interaction between compound facilitation and concreteness effect. The latter might modulate compound processing basing on constituents’ transparency patterns. To evaluate these, we created lists for compound and monomorphemic words, sub-categorized them into concrete and abstract words, and further sub-categorized them basing on their transparency. The transparency conditions were opaque-opaque (OO), transparent-opaque (TO), and transparent-transparent (TT). We used RT data from English Lexicon Project (ELP) for our comparisons. The results showed the importance of concreteness factor (facilitation) in both compound and monomorphemic processing. Important for our present concern, separate concrete and abstract compound analyses revealed different patterns for OO, TO, and TT compounds. Concrete TT and TO conditions were processed faster than Concrete OO, Abstract OO and Abstract TT compounds, however, they weren’t processed faster than Abstract TO compounds. These results may reflect on different representation patterns of concrete and abstract compounds. Downloads 72
##### 396 Computation of Natural Logarithm Using Abstract Chemical Reaction Networks

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Recent researches has focused on nucleic acids as a substrate for designing biomolecular circuits for in situ monitoring and control. A common approach is to express them by a set of idealised abstract chemical reaction networks (ACRNs). Here, we present new results on how abstract chemical reactions, viz., catalysis, annihilation and degradation, can be used to implement circuit that accurately computes logarithm function using the method of Arithmetic-Geometric Mean (AGM), which has not been previously used in conjunction with ACRNs. Downloads 80
##### 395 Learners’ Conspicuous and Significant Errors in Arithmetic

Authors: Michael Lousis

Abstract:

The systematic identification of the most conspicuous and significant errors made by learners during three-years of testing of their progress in learning Arithmetic are presented in this article. How these errors have changed over three-years of school instruction of Arithmetic also is shown. The sample is comprised of two hundred (200) English students and one hundred and fifty (150) Greek students. These students were purposefully selected according to their participation in each testing session in the development of the three-year Kassel Project in England and Greece, in both domains simultaneously in Arithmetic and Algebra. The data sample includes six test-scripts corresponding to three testing sessions in both Arithmetic and Algebra respectively. Downloads 185
##### 394 Integral Domains and Alexandroff Topology

Authors: Shai Sarussi

Abstract:

Let S be an integral domain which is not a field, let F be its field of fractions, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R ∩ F = S and F R = A. A topological space whose set of open sets is closed under arbitrary intersections is called an Alexandroff space. Inspired by the well-known Zariski-Riemann space and the Zariski topology on the set of prime ideals of a commutative ring, we define a topology on the set of all S-nice subalgebras of A. Consequently, we get an interplay between Algebra and topology, that gives us a better understanding of the S-nice subalgebras of A. It is shown that every irreducible subset of S-nice subalgebras of A has a supremum; and a characterization of the irreducible components is given, in terms of maximal S-nice subalgebras of A. Downloads 13
##### 393 Instructional Game in Teaching Algebra for High School Students: Basis for Instructional Intervention

Authors: Jhemson C. Elis, Alvin S. Magadia

Abstract:

Our world is full of numbers, shapes, and figures that illustrate the wholeness of a thing. Indeed, this statement signifies that mathematics is everywhere. Mathematics in its broadest sense helps people in their everyday life that is why in education it is a must to be taken by the students as a subject. The study aims to determine the profile of the respondents in terms of gender and age, performance of the control and experimental groups in the pretest and posttest, impact of the instructional game used as instructional intervention in teaching algebra for high school students, significant difference between the level of performance of the two groups of respondents in their pre–test and post–test results, and the instructional intervention can be proposed. The descriptive method was also utilized in this study. The use of the certain approach was to that it corresponds to the main objective of this research that is to determine the effectiveness of the instructional game used as an instructional intervention in teaching algebra for high school students. There were 30 students served as respondents, having an equal size of the sample of 15 each while a greater number of female teacher respondents which totaled 7 or 70 percent and male were 3 or 30 percent. The study recommended that mathematics teacher should conceptualize instructional games for the students to learn mathematics with fun and enjoyment while learning. Mathematics education program supervisor should give training for teachers on how to conceptualize mathematics intervention for the students learning. Meaningful activities must be provided to sustain the student’s interest in learning. Students must be given time to have fun at the classroom through playing while learning since mathematics for them was considered as difficult. Future researcher must continue conceptualizing some mathematics intervention to suffice the needs of the students, and teachers should inculcate more educational games so that the discussion will be successful and joyful. Downloads 524
##### 392 Characterization of Number of Subgroups of Finite Groups

Authors: Khyati Sharma, A. Satyanarayana Reddy

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The topic of how many subgroups exist within a certain finite group naturally arises in the study of finite groups. Over the years, different researchers have investigated this issue from a variety of angles. The significant contributions of the key mathematicians over the time have been summarized in this article. To this end, we classify finite groups into three categories viz. (a) Groups for which the number of subgroups is less than |G|, (b) equals to |G|, and finally, (c) greater than |G|. Because every element of a finite group generates a cyclic subgroup, counting cyclic subgroups is the most important task in this endeavor. A brief survey on the number of cyclic subgroups of finite groups is also conducted by us. Furthermore, we also covered certain arithmetic relations between the order of a finite group |G| and the number of its distinct cyclic subgroups |C(G)|. In order to provide pertinent context and possibly reveal new novel areas of potential research within the field of research on finite groups, we finally pose and solicit a few open questions. Downloads 8
##### 391 A Proof of the N. Davydov Theorem for Douglis Algebra Valued Functions

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The classical Beltrami system of elliptic equations generalizes the Cauchy Riemann equation in the complex plane and offers the possibility to consider homogeneous system with no terms of zero order. The theory of Douglis-valued functions, called Hyper-analytic functions, is special case of the above situation. In this note, we prove an analogue of the N. Davydov theorem in the framework of the theory of hyperanalytic functions. The used methodology contemplates characteristic methods of the hypercomplex analysis as well as the singular integral operators and elliptic systems of the partial differential equations theories. Downloads 127
##### 390 Stem Covers of Leibniz n-Algebras

Authors: Natália Maria Rego

Abstract:

ALeibnizn-algebraGis aK-vector space endowed whit a n-linearbracket operation [-,…-] : GG … G→ Gsatisfying the fundamental identity, which can be expressed saying that the right multiplication map Ry2, …, ᵧₙ: Gn→ G, Rᵧ₂, …, ᵧₙn(ˣ¹, …, ₓₙ) = [[ˣ¹, …, ₓₙ], ᵧ₂, …, ᵧₙ], is a derivation. This structure, together with its skew-symmetric version, named as Lie n-algebra or Filippov algebra, arose in the setting of Nambumechanics, an n-ary generalization of the Hamiltonian mechanics. Thefirst goal of this work is to provide a characterization of various classes of central extensions of Leibniz n-algebras in terms of homological properties. Namely, Commutator extension, Quasi-commutator extension, Stem extension, and Stem cover. These kind of central extensions are characterized by means of the character of the map *(E): nHL1(G) → M provided by the five-term exact sequence in homology with trivial coefficients of Leibniz n-algebras associated to an extension E : 0 → M → K → G → 0. For a free presentation 0 →R→ F →G→ 0of a Leibniz n-algebra G,the term M(G) = (R[F,…n.., F])/[R, F,..n-1..,F] is called the Schur multiplier of G, which is a Baer invariant, i.e., it does not depend on the chosen free presentation, and it is isomorphic to the first Leibniz n-algebras homology with trivial coefficients of G. A central extension of Leibniz n-algebras is a short exact sequenceE : 0 →M→K→G→ 0such that [M, K,.. ⁿ⁻¹.., K]=0. It is said to be a stem extension if M⊆[G, .. n.., G]. Additionally, if the induced map M(K) → M(G) is the zero map, then the stem extension Eis said to be a stem cover. The second aim of this work is to analyze the interplay between stem covers of Leibniz n-algebras and the Schur multiplier. Concretely, in the case of finite-dimensional Leibniz n-algebras, we show the existence of coverings, and we prove that all stem covers with finite-dimensional Schur multiplier are isoclinic. Additionally, we characterize stem covers of perfect Leibniz n-algebras. Downloads 95
##### 389 Big Data Analytics and Data Security in the Cloud via Fully Homomorphic Encyption Scheme

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This paper describes the problem of building secure computational services for encrypted information in the Cloud. Computing without decrypting the encrypted data; therefore, it meets the yearning of computational encryption algorithmic aspiration model that could enhance the security of big data for privacy or confidentiality, availability and integrity of the data and user’s security. The cryptographic model applied for the computational process of the encrypted data is the Fully Homomorphic Encryption Scheme. We contribute a theoretical presentations in a high-level computational processes that are based on number theory that is derivable from abstract algebra which can easily be integrated and leveraged in the Cloud computing interface with detail theoretic mathematical concepts to the fully homomorphic encryption models. This contribution enhances the full implementation of big data analytics based on cryptographic security algorithm. Downloads 389
##### 388 Importance of Mathematical Modeling in Teaching Mathematics

Authors: Selahattin Gultekin

Abstract:

Today, in engineering departments, mathematics courses such as calculus, linear algebra and differential equations are generally taught by mathematicians. Therefore, during mathematicians’ classroom teaching there are few or no applications of the concepts to real world problems at all. Most of the times, students do not know whether the concepts or rules taught in these courses will be used extensively in their majors or not. This situation holds true of for all engineering and science disciplines. The general trend toward these mathematic courses is not good. The real-life application of mathematics will be appreciated by students when mathematical modeling of real-world problems are tackled. So, students do not like abstract mathematics, rather they prefer a solid application of the concepts to our daily life problems. The author highly recommends that mathematical modeling is to be taught starting in high schools all over the world In this paper, some mathematical concepts such as limit, derivative, integral, Taylor Series, differential equations and mean-value-theorem are chosen and their applications with graphical representations to real problems are emphasized. Downloads 238
##### 387 An Experimental Quantitative Case Study of Competency-Based Learning in Online Mathematics Education

Authors: Pascal Roubides

Abstract:

The presentation proposed herein describes a research case study of a hybrid application of the competency-based education model best exemplified by Western Governor’s University, within the general temporal confines of an accelerated (8-week) term of a College Algebra course at the author’s institution. A competency-based model was applied to an accelerated online College Algebra course, built as an Open Educational Resources (OER) course, seeking quantifiable evidence of any differences in the academic achievement of students enrolled in the competency-based course and the academic achievement of the current delivery of the same course. Competency-based learning has been gaining in support in recent times and the author’s institution has also been involved in its own efforts to design and develop courses based on this approach. However, it is unknown whether there had been any research conducted to quantify evidence of the effect of this approach against traditional approaches prior to the author’s case study. The research question sought to answer in this experimental quantitative study was whether the online College Algebra curriculum at the author’s institution delivered via an OER-based competency-based model can produce statistically significant improvement in retention and success rates against the current delivery of the same course. Results obtained in this study showed that there is no statistical difference in the retention rate of the two groups. However, there was a statistically significant difference found between the rates of successful completion of students in the experimental group versus those in the control group. Downloads 71