Search results for: object 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

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Authors: Lavinia Ciungu

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In the last decades, developing algebras related to the logical foundations of quantum mechanics became a central topic of research. Generally known as quantum structures, these algebras serve as models for the formalism of quantum mechanics. In this work, we introduce the notion of quantum-Wajsberg algebras by redefining the quantum-MV algebras starting from involutive BE algebras. We give a characterization of quantum-Wajsberg algebras, investigate their properties, and show that, in general, quantum-Wajsberg algebras are not (commutative) quantum-B algebras. We also define the ∨-commutative quantum-Wajsberg algebras and study their properties. Furthermore, we prove that any Wajsberg algebra (bounded ∨-commutative BCK algebra) is a quantum-Wajsberg algebra, and we give a condition for a quantum-Wajsberg algebra to be a Wajsberg algebra. We prove that Wajsberg algebras are both quantum-Wajsberg algebras and commutative quantum-B algebras. We establish the connection between quantum-Wajsberg algebras and quantum-MV algebras, proving that the quantum-Wajsberg algebras are term equivalent to quantum-MV algebras. We show that, in general, the quantum-Wajsberg algebras are not commutative quantum-B algebras and if a quantum-Wajsberg algebra is self-distributive, then the corresponding quantum-MV algebra is an MV algebra. Our study could be a starting point for the development of other implicative counterparts of certain existing algebraic quantum structures.

Keywords: quantum-Wajsberg algebra, quantum-MV algebra, MV algebra, Wajsberg algebra, BE algebra, quantum-B algebra

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Authors: Janusz Getta, Zhaoxi Pan

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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.

Keywords: conceptual schema, implementation schema, logical schema, object algebra, performance evaluation, query processing

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Authors: Yingxu Wang, Jianhua Lu, Jun Peng, Jiawei Zhang

Abstract:

The fundamental demand in contemporary intelligent science towards Autonomous AI (AI*) is the creation of unprecedented formal means of Intelligent Mathematics (IM). It is discovered that natural intelligence is inductively created rather than exhaustively trained. Therefore, IM is a family of algebraic and denotational mathematics encompassing Inference Algebra, Real-Time Process Algebra, Concept Algebra, Semantic Algebra, Visual Frame Algebra, etc., developed in our labs. IM plays indispensable roles in training-free AI* theories and systems beyond traditional empirical data-driven technologies. A set of applications of IM-driven AI* systems will be demonstrated in contemporary intelligence science, AI*, and cognitive computers.

Keywords: intelligence mathematics, foundations of intelligent science, autonomous AI, cognitive computers, inference algebra, real-time process algebra, concept algebra, semantic algebra, applications

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Authors: Michael Lousis

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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

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Authors: Zuhier Altawallbeh

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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.

Keywords: hochschild homology, homotopic function, free and projective modules, free resolution, exterior algebra, symmetric algebra

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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.

Keywords: BCK-algebra, pseudo-BCK algebra, pseudo-ideal, implicative pseudo-ideal

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Authors: Salvatore Triolo

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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.

Keywords: Gelfand-Naimark-Segal representation, CQ*-algebras, faithful representation, non-commutative Lᵖ-spaces, operator in Hilbert spaces

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Authors: Yassir I. Dinar

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We computed the weights of the adjoint action of distinguished sl2-triples in Lie algebra of type Dn using mathematical induction.

Keywords: lie algebra, root systems, representation theory, nilpotent orbits

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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.

Keywords: abstract algebra , linking abstract algebra and school mathematics, school algebra, secondary school mathematics, teacher professional development

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Authors: Liliana O. Martínez, Juan E. González, Manuel Ramírez-Aranda, Ana Cervantes-Herrera

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In this work, the use of a collection of mathematical challenges and puzzles aimed at students who are starting in algebra is proposed. The selected challenges and puzzles are intended to arouse students' interest in this area of mathematics, in addition to facilitating the teaching-learning process through challenges such as riddles, crossword puzzles, and board games, all in everyday situations that allow them to build themselves the learning. For this, it is proposed to carry out a "Math Rally: algebra" divided into four sections: mathematical reasoning, a hierarchy of operations, fractions, and algebraic equations.

Keywords: algebra, algebraic challenge, algebraic puzzle, math rally

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Authors: Natalia Pacheco Rego

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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.

Keywords: almost inner Lie-derivation, Lie-center, Lie-central derivation, Lie-derivation

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Authors: Safiah Ahmed Madkhali

Abstract:

The grammatical relation object has not attracted the same attention in the literature as subject has. Where there is a clearly monotransitive verb such as kick, the criteria for identifying the grammatical relation may converge. However, the term object is also used to refer to phenomena that do not subsume all, or even most, of the recognized properties of the canonical object. Instances of such phenomena include non-canonical objects such as the ones in the so-called double-object construction i.e. the indirect object and the direct object as in (He bought his dog a new collar). In this paper, it is demonstrated how criteria of identifying the grammatical relation object that are found in the theoretical and typological literature can be applied to Arabic. Also, further language-specific criteria are here derived from the regularities of the canonical object in the language. The criteria established in this way are then applied to the non-canonical objects to demonstrate how far they conform to, or diverge from, the canonical object. Contrary to the claim that the direct object is more similar to the canonical object than is the indirect object, it was found that it is, in fact, the indirect object rather than the direct object that shares most of the aspects of the canonical object in monotransitive clauses.

Keywords: canonical objects, double-object constructions, cognate object constructions, non-canonical objects

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Authors: Wagneur Edouard

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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.

Keywords: classification, idempotent semi-modules, strong independence, tropical algebra

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Authors: Ranjit Biswas

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The existing subject Algebra is incomplete to support mathematical computations being done by scientists of all areas: Mathematics, Physics, Statistics, Chemistry, Space Science, Cosmology etc. even starting from the era of great Einstein. A huge hidden gap in the subject ‘Algebra’ is unearthed. All the scientists today, including mathematicians, physicists, chemists, statisticians, cosmologists, space scientists, and economists, even starting from the great Einstein, are lucky that they got results without facing any contradictions or without facing computational errors. Most surprising is that the results of all scientists, including Nobel Prize winners, were proved by them by doing experiments too. But in this paper, it is rigorously justified that they all are lucky. An algebraist can define an infinite number of new algebraic structures. The objective of the work in this paper is not just for the sake of defining a distinct algebraic structure, but to recognize and identify a major gap of the subject ‘Algebra’ lying hidden so far in the existing vast literature of it. The objective of this work is to fix the unearthed gap. Consequently, a different algebraic structure called ‘Region’ has been introduced, and its properties are studied.

Keywords: region, ROR, RORR, region algebra

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Authors: Md. Shah Alam

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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.

Keywords: mixed number, special relativity, quantum mechanics, electrodynamics, pauli matrix

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

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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.

Keywords: parametric design, structural morphology, space structures, spherical coordinate system

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Authors: Early Sol A. Gadong, Lourdes C. Zamora, Jonny B. Pornel, Aurora Fe C. Bautista

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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.

Keywords: writing in math, metacognition, affective factors in learning, elementary algebra classroom

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Authors: Nessa-Amie T. Peñaflor, Antonio Cinto

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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.

Keywords: algebra, grade 7 math, math anxiety, math misconceptions

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Authors: Roni Naor-Hofri

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This paper shows how self-inflicted pain enabled the expression of love for God among Christian monastic ascetics in medieval central Europe. As scholars have shown, being in a state of pain leads to a change in or destruction of language, an essential feature of the self. The author argues that this transformation allows the self to transcend its boundaries as an object, even if only temporarily and in part. The epistemic achievement of love for God, a non-object, would not otherwise have been possible. To substantiate her argument, the author shows that the self’s transformation into a non-object enables the imitation of God: not solely in the sense of imitatio Christi, of physical and visual representations of God incarnate in the flesh of His son Christ, but also in the sense of the self’s experience of being a non-object, just like God, the target of the self’s love.

Keywords: love for God , pain, philosophy, religion

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Authors: Bingquan Shen

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Convolutional Neural Networks (CNN) have demonstrated their effectiveness in synthesizing 3D views of object instances at various viewpoints. Given the problem where one have limited viewpoints of a particular object for classification, we present a pose normalization architecture to transform the object to existing viewpoints in the training dataset before classification to yield better classification performance. We have demonstrated that this Pose Normalization Network (PNN) can capture the style of the target object and is able to re-render it to a desired viewpoint. Moreover, we have shown that the PNN improves the classification result for the 3D chairs dataset and ShapeNet airplanes dataset when given only images at limited viewpoint, as compared to a CNN baseline.

Keywords: convolutional neural networks, object classification, pose normalization, viewpoint invariant

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Authors: Shih-Yu Lo

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Stereotypes exist in almost every society, affecting how people interact with each other. However, to our knowledge, the influence of stereotypes was rarely explored in the context of basic perceptual processes. This study aims to explore how the gender stereotype affects object recognition. Participants were presented with a series of scene pictures, followed by a target display with a man or a woman, holding a weapon or a non-weapon object. The task was to identify whether the object in the target display was a weapon or not. Although the gender of the object holder could not predict whether he or she held a weapon, and was irrelevant to the task goal, the participant nevertheless tended to identify the object as a weapon when the object holder was a man than a woman. The analysis based on the signal detection theory showed that the stereotype effect on object recognition mainly resulted from the participant’s bias to make a 'weapon' response when a man was in the scene instead of a woman in the scene. In addition, there was a trend that the participant’s sensitivity to differentiate a weapon from a non-threating object was higher when a woman was in the scene than a man was in the scene. The results of this study suggest that the irrelevant social cues implied in the visual scene can be very powerful that they can modulate the basic object recognition process.

Keywords: gender stereotype, object recognition, signal detection theory, weapon

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Authors: Keshab Shrestha, Hung-Suck Park

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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.

Keywords: co-product branching, emergy accounting, emergy algebra, modified physical quantity method, transformity value

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Authors: Jinsiang Shaw, Pik-Hoe Chen

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This paper aims to integrate human recognition, motion recognition, and object tracking technologies without requiring a pre-training database model for motion recognition or the unknown object itself. Furthermore, it can simultaneously track multiple users and multiple objects. Unlike other existing human motion recognition methods, our approach employs a rule-based condition method to determine if a user hand is approaching or departing an object. It uses a background subtraction method to separate the human and object from the background, and employs behavior features to effectively interpret human object-grabbing actions. With an object’s histogram characteristics, we are able to isolate and track it using back projection. Hence, a moving object trajectory can be recorded and the object itself can be located. This particular technique can be used in a camera surveillance system in a shopping area to perform real-time intelligent surveillance, thus preventing theft. Experimental results verify the validity of the developed surveillance algorithm with an accuracy of 83% for shoplifting detection.

Keywords: Automatic Tracking, Back Projection, Motion Recognition, Shoplifting

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Authors: In-Geun Lim, Sung-Woong Ra

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As high resolution satellite images can be used, lots of studies are carried out for exploiting these images in various fields. This paper proposes the method based on mathematical morphology for extracting the ‘horse's hoof shaped object’. This proposed method can make an automatic object detection system to track the meaningful object in a large satellite image rapidly. Mathematical morphology process can apply in binary image, so this method is very simple. Therefore this method can easily extract the ‘horse's hoof shaped object’ from any images which have indistinct edges of the tracking object and have different image qualities depending on filming location, filming time, and filming environment. Using the proposed method by which ‘horse's hoof shaped object’ can be rapidly extracted, the performance of the automatic object detection system can be improved dramatically.

Keywords: facility detection, satellite image, object, mathematical morphology

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Authors: Nastaran Ahmadpour Samani, Shahram Talebi

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Added mass is an important quantity in analysis of the motion of a submerged object ,which can be calculated by solving the equation of potential flow around the object . Here, we consider systems in which a square object is submerged in a channel of fluid and moves parallel to the wall. The corresponding added mass at a given distance from the wall d and for the object size s (which is the side of square object) is calculated via lattice Blotzmann simulation . By changing d and s separately, their effect on the added mass is studied systematically. The simulation results reveal that for the systems in which d > 4s, the distance does not influence the added mass any more. The added mass increases when the object approaches the wall and reaches its maximum value as it moves on the wall (d -- > 0). In this case, the added mass is about 73% larger than which of the case d=4s. In addition, it is observed that the added mass increases by increasing of the object size s and vice versa.

Keywords: Lattice Boltzmann simulation , added mass, square, variable size

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Authors: Areej M. Abduldaim, Nadia M. G. Al-Saidi

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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.

Keywords: cryptosystem, identification, skew π-Armendariz rings, skew polynomial rings, zero knowledge protocol

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Authors: Stephen O. Amaraihu

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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.

Keywords: achievement, computer based instruction, interest, retention

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Authors: Shaomei Li, Yawen Wang, Chao Gao

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To improve tracking drift which often occurs in adaptive tracking, an algorithm based on the fusion of tracking and detection is proposed in this paper. Firstly, object tracking is posed as a binary classification problem and is modeled by partial least squares (PLS) analysis. Secondly, tracking object frame by frame via particle filtering. Thirdly, validating the tracking reliability based on both positive and negative models matching. Finally, relocating the object based on SIFT features matching and voting when drift occurs. Object appearance model is updated at the same time. The algorithm cannot only sense tracking drift but also relocate the object whenever needed. Experimental results demonstrate that this algorithm outperforms state-of-the-art algorithms on many challenging sequences.

Keywords: object tracking, tracking drift, partial least squares analysis, positive and negative models matching

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Authors: Benjamin Lee Warren

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Many problems exist in additive number theory which is essential to determine the primes that are the sum of two elements from a given single-variable polynomial sequence, and most of them are unattackable in the present day. Here, we determine solutions for this problem to a few certain sequences (certain binomial coefficients and power sums) using only elementary algebra and some algebraic factoring methods (as well as Euclid’s Lemma and Faulhaber’s Formula). In particular, we show that there are finitely many primes as sums of two of these types of elements. Several cases are fully illustrated, and bounds are presented for the cases not fully illustrated.

Keywords: binomial coefficients, power sums, primes, algebra

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