Search results for: algebraic method

Authors: Rosy Joseph

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From an algebraic point of view, Semirings provide the most natural generalization of group theory and ring theory. In the absence of additive inverse in a semiring, one had to impose a weaker condition on the semiring, i.e., the additive cancellative law to study interesting structural properties. In many practical situations, fuzzy numbers are used to model imprecise observations derived from uncertain measurements or linguistic assessments. In this connection, a special class of fuzzy numbers whose shape is symmetric with respect to a vertical line called the symmetric fuzzy numbers i.e., for α ∈ (0, 1] the α − cuts will have a constant mid-point and the upper end of the interval will be a non-increasing function of α, the lower end will be the image of this function, is suitable. Based on this description, arithmetic operations and a ranking technique to order the symmetric fuzzy numbers were dealt with in detail. Wherein it was observed that the structure of the class of symmetric fuzzy numbers forms a commutative semigroup with cancellative property. Also, it forms a multiplicative monoid satisfying sub-distributive property.In this paper, we introduce the algebraic structure, sub-distributive semiring and discuss its various properties viz., ideals and prime ideals of sub-distributive semiring, sub-distributive ring of difference etc. in detail. Symmetric fuzzy numbers are visualized as an illustration.

Keywords: semirings, subdistributive ring of difference, subdistributive semiring, symmetric fuzzy numbers

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Authors: Peter Jean-Paul, Shanaz Wahid

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The problem of division by zero can be stated as: “what is the value of 0 x 1/0?” This expression has been considered undefined by mathematicians because it can have two equally valid solutions either 0 or 1. Recently semi-structured complex number set was invented to solve “division by zero”. However, whilst the number set had some merits it was considered to have a poor theoretical foundation and did not provide a quality solution to “division by zero”. Moreover, the set lacked consistency in simple algebraic calculations producing contradictory results when dividing by zero. To overcome these issues this research starts by treating the expression " 0 x 1/0" as a quantum mechanical system that produces two tangled results 0 and 1. Dirac Notation (a tool from quantum mechanics) was then used to redefine the unstructured unit p in semi-structured complex numbers so that p represents the superposition of two results (0 and 1) and collapses into a single value when used in algebraic expressions. In the process, this paper describes a new number set called Quantum Semi-structured Complex Numbers that provides a valid solution to the problem of “division by zero”. This research shows that this new set (1) forms a “Field”, (2) can produce consistent results when solving division by zero problems, (3) can be used to accurately describe systems whose mathematical descriptions involve division by zero. This research served to provide a firm foundation for Quantum Semi-structured Complex Numbers and support their practical use.

Keywords: division by zero, semi-structured complex numbers, quantum mechanics, Hilbert space, Euclidean space

Procedia PDF Downloads 157

Authors: Alexander N. Savostyanov, Tatyana A. Dolgorukova, Elena A. Esipenko, Mikhail S. Zaleshin, Margherita Malanchini, Anna V. Budakova, Alexander E. Saprygin, Yulia V. Kovas

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The present study compared spectral-power indexes and cortical topography of brain activity in a sample characterized by different levels of trait and mathematical anxiety. 52 healthy Russian-speakers (age 17-32; 30 males) participated in the study. Participants solved an error recognition task under 3 conditions: A lexical condition (simple sentences in Russian), and two numerical conditions (simple arithmetic and complicated algebraic problems). Trait and mathematical anxiety were measured using self-repot questionnaires. EEG activity was recorded simultaneously during task execution. Event-related spectral perturbations (ERSP) were used to analyze spectral-power changes in brain activity. Additionally, sLORETA was applied in order to localize the sources of brain activity. When exploring EEG activity recorded after tasks onset during lexical conditions, sLORETA revealed increased activation in frontal and left temporal cortical areas, mainly in the alpha/beta frequency ranges. When examining the EEG activity recorded after task onset during arithmetic and algebraic conditions, additional activation in delta/theta band in the right parietal cortex was observed. The ERSP plots reveled alpha/beta desynchronizations within a 500-3000 ms interval after task onset and slow-wave synchronization within an interval of 150-350 ms. Amplitudes of these intervals reflected the accuracy of error recognition, and were differently associated with the three (lexical, arithmetic and algebraic) conditions. The level of trait anxiety was positively correlated with the amplitude of alpha/beta desynchronization. The level of mathematical anxiety was negatively correlated with the amplitude of theta synchronization and of alpha/beta desynchronization. Overall, trait anxiety was related with an increase in brain activation during task execution, whereas mathematical anxiety was associated with increased inhibitory-related activity. We gratefully acknowledge the support from the №11.G34.31.0043 grant from the Government of the Russian Federation.

Keywords: anxiety, EEG, lexical and numerical error-recognition tasks, alpha/beta desynchronization

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Authors: Walaa Obaidallah Alqarafi, Wafaa Mohammed Fakieh, Alaa Abdallah Altassan

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Algebraic graph theory is defined as a bridge between algebraic structures and graphs. It has several uses in many fields, including chemistry, physics, and computer science. The prime graph is a type of graph associated with a ring R, where the vertex set is the whole ring R, and two vertices x and y are adjacent if either xRy=0 or yRx=0. However, the investigation of the prime graph over rings remains relatively limited. The behavior of this graph in extended rings, like R[x] and R[[x]], where R is a non-commutative ring, deserves more attention because of the wider applicability in algebra and other mathematical fields. To study the prime graphs over polynomials and power series rings, we used a combination of ring-theoretic and graph-theoretic techniques. This paper focuses on two invariants: the diameter and the girth of these graphs. Furthermore, the work discusses how the graph structures change when passing from R to R[x] and R[[x]]. In our study, we found that the set of strong zero-divisors of ring R represents the set of vertices in prime graphs. Based on this discovery, we redefined the vertices of prime graphs using the definition of strong zero divisors. Additionally, our results show that although the prime graphs of R[x] and R[[x]] are comparable to the graph of R, they have different combinatorial characteristics since these extensions contain new strong zero-divisors. In particular, we find conditions in which the diameter and girth of the graphs, as they expand from R to R[x] and R[[x]], do not change or do change. In conclusion, this study shows how extending a non-commutative ring R to R[x] and R[[x]] affects the structure of their prime graphs, particularly in terms of diameter and girth. These findings enhance the understanding of the relationship between ring extensions and graph properties.

Keywords: prime graph, diameter, girth, polynomial ring, power series ring

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Authors: Abdorreza Rabiee, Shahla Mohammad Hoseini Mirzaei

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The advantages of distributed generations (DGs) based on renewable energy sources (RESs) leads to high penetration level of DGs in distribution network. With incorporation of DGs in distribution systems, the system reliability and security, as well as voltage profile, is improved. However, the protection of such systems is still challenging. In this paper, at first, the related papers are reviewed and then a practical scheme is proposed for coordination of OCRs in distribution system with DGs. The coordination problem is formulated as a nonlinear programming (NLP) optimization problem with the object function of minimizing total operating time of OCRs. The proposed method is studied based on a simple test system. The optimization problem is solved by General Algebraic Modeling System (GAMS) to calculate the optimal time dial setting (TDS) and also pickup current setting of OCRs. The results show the effectiveness of the proposed method and its applicability.

Keywords: distributed generation, DG, distribution network, over current relay, OCR, protection coordination, pickup current, time dial setting, TDS

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Authors: Shubham Jaiswal

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During modelling of many physical problems and engineering processes, fractional calculus plays an important role. Those are greatly described by fractional differential equations (FDEs). So a reliable and efficient technique to solve such types of FDEs is needed. In this article, a numerical solution of a class of fractional differential equations namely space fractional order reaction-advection dispersion equations subject to initial and boundary conditions is derived. In the proposed approach shifted Jacobi polynomials are used to approximate the solutions together with shifted Jacobi operational matrix of fractional order and spectral collocation method. The main advantage of this approach is that it converts such problems in the systems of algebraic equations which are easier to be solved. The proposed approach is effective to solve the linear as well as non-linear FDEs. To show the reliability, validity and high accuracy of proposed approach, the numerical results of some illustrative examples are reported, which are compared with the existing analytical results already reported in the literature. The error analysis for each case exhibited through graphs and tables confirms the exponential convergence rate of the proposed method.

Keywords: space fractional order linear/nonlinear reaction-advection diffusion equation, shifted Jacobi polynomials, operational matrix, collocation method, Caputo derivative

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Authors: Nidal Ali

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Consider an algebraic number field K of degree n, A0 K is its ring of integers and a prime number p inert in K. Let F(u1, . . . , un, x) be the generic polynomial of integers of K. We will study in advance the stability of this polynomial and then, we will apply it in order to obtain all the monic irreducible polynomials in Fp[x] of degree d dividing n.

Keywords: generic polynomial, irreducibility, iteration, stability, inert prime, totally ramified

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Authors: W. Hamaidia, T. Zebbiche

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This paper presents the design of axisymmetric supersonic nozzles, in order to accelerate a supersonic flow to the desired Mach number and that having a small weight, in the same time gives a high thrust. The concerned nozzle gives a parallel and uniform flow at the exit section. The nozzle is divided into subsonic and supersonic regions. The supersonic portion is independent to the upstream conditions of the sonic line. The subsonic portion is used to give a sonic flow at the throat. In this case, nozzle gives a uniform and parallel flow at the exit section. It’s named by minimum length Nozzle. The study is done at high temperature, lower than the dissociation threshold of the molecules, in order to improve the aerodynamic performances. Our aim consists of improving the performances both by the increase of exit Mach number and the thrust coefficient and by reduction of the nozzle's mass. The variation of the specific heats with the temperature is considered. The design is made by the Method of Characteristics. The finite differences method with predictor-corrector algorithm is used to make the numerical resolution of the obtained nonlinear algebraic equations. The application is for air. All the obtained results depend on three parameters which are exit Mach number, the stagnation temperature, the chosen mesh in characteristics. A numerical simulation of nozzle through Computational Fluid Dynamics-FASTRAN was done to determine and to confirm the necessary design parameters.

Keywords: flux supersonic flow, axisymmetric minimum length nozzle, high temperature, method of characteristics, calorically imperfect gas, finite difference method, trust coefficient, mass of the nozzle, specific heat at constant pressure, air, error

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Authors: Akhilesh Kumar, Sathyanarayan G., Nirmala S.

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An unusual approach is developed to determine the orbit of satellites/space objects. The determination of orbits is considered a boundary value problem and has been solved using the finite difference method (FDM). Only positions of the satellites/space objects are known at two end times taken as boundary conditions. The technique of finite difference has been used to calculate the orbit between end times. In this approach, the governing equation is defined as the satellite's equation of motion with a perturbed acceleration. Using the finite difference method, the governing equations and boundary conditions are discretized. The resulting system of algebraic equations is solved using Tri Diagonal Matrix Algorithm (TDMA) until convergence is achieved. This methodology test and evaluation has been done using all GPS satellite orbits from National Geospatial-Intelligence Agency (NGA) precise product for Doy 125, 2023. Towards this, two hours of twelve sets have been taken into consideration. Only positions at the end times of each twelve sets are considered boundary conditions. This algorithm is applied to all GPS satellites. Results achieved using FDM compared with the results of NGA precise orbits. The maximum RSS error for the position is 0.48 [m] and the velocity is 0.43 [mm/sec]. Also, the present algorithm is applied on the IRNSS satellites for Doy 220, 2023. The maximum RSS error for the position is 0.49 [m], and for velocity is 0.28 [mm/sec]. Next, a simulation has been done for a Highly Elliptical orbit for DOY 63, 2023, for the duration of 6 hours. The RSS of difference in position is 0.92 [m] and velocity is 1.58 [mm/sec] for the orbital speed of more than 5km/sec. Whereas the RSS of difference in position is 0.13 [m] and velocity is 0.12 [mm/sec] for the orbital speed less than 5km/sec. Results show that the newly created method is reliable and accurate. Further applications of the developed methodology include missile and spacecraft targeting, orbit design (mission planning), space rendezvous and interception, space debris correlation, and navigation solutions.

Keywords: finite difference method, grid generation, NavIC system, orbit perturbation

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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: Serges Mendomo Meye, Li Guowei, Shen Zhenzhong

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Earthquake is one of the main loads threatening dam safety. Once the dam is damaged, it will bring huge losses of life and property to the country and people. Therefore, it is very important to research the seismic safety of the dam. Due to the complex foundation conditions, high fortification intensity, and high scientific and technological content, it is necessary to adopt reasonable methods to evaluate the seismic safety performance of concrete arch dams built and under construction in strong earthquake areas. Structural seismic vulnerability analysis can predict the probability of structural failure at all levels under different intensity earthquakes, which can provide a scientific basis for reasonable seismic safety evaluation and decision-making. In this paper, the response surface method (RSM) is applied to the seismic vulnerability analysis of arch dams, which improves the efficiency of vulnerability analysis. Based on the central composite test design method, the material-seismic intensity samples are established. The response surface model (RSM) with arch crown displacement as performance index is obtained by finite element (FE) calculation of the samples, and then the accuracy of the response surface model (RSM) is verified. To obtain the seismic vulnerability curves, the seismic intensity measure ??(?1) is chosen to be 0.1~1.2g, with an interval of 0.1g and a total of 12 intensity levels. For each seismic intensity level, the arch crown displacement corresponding to 100 sets of different material samples can be calculated by algebraic operation of the response surface model (RSM), which avoids 1200 times of nonlinear dynamic calculation of arch dam; thus, the efficiency of vulnerability analysis is improved greatly.

Keywords: high concrete arch dam, performance index, response surface method, seismic vulnerability analysis, vector-valued intensity measure

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Authors: Nan Tang, Bin Wu, Cunfu He

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The trapezoid rods are widely used in civil engineering as load –carrying members. Ultrasonic guided wave is one of the most popular techniques in analyzing the propagation of elastic guided wave. The goal of this paper is to investigate the propagation of elastic waves in the isotropic bar with trapezoid cross-section. Dispersion curves that describe the relationship between the frequency and velocity provide the fundamental information to describe the propagation of elastic waves through a structure. Based on the SAFE (semi-analytical finite element) a linear algebraic system of equations is obtained. By using numerical methods, dispersion curves solved for the rods with the trapezoid cross-section. These fundamental information plays an important role in applying ultrasonic guided waves to NTD for structures with trapezoid cross section.

Keywords: guided wave, dispersion, finite element method, trapezoid rod

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Authors: Khawar Ali

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We present the numerical study of unsteady hydromagnetic (MHD) flow and heat transfer characteristics of a viscous incompressible electrically conducting water-based hybrid metallic nanofluid (containing Cu-Au/ H₂O nanoparticles) between two orthogonally moving porous coaxial disks with suction. Different from the classical shooting methodology, we employ a combination of a direct and an iterative method (SOR with optimal relaxation parameter) for solving the sparse systems of linear algebraic equations arising from the FD discretization of the linearized self similar nonlinear ODEs. Effects of the governing parameters on the flow and heat transfer are discussed and presented through tables and graphs. The findings of the present investigation may be beneficial for the electronic industry in maintaining the electronic components under effectiveand safe operational conditions.

Keywords: heat transfer enhancement, hybrid metallic nanofluid, viscous dissipation and joule heating effect , Two dimensional flow

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Authors: Arka Roy, Chandra Prakash Dubey

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Inverse problem generally used for recovering hidden information from outside available data. Vertical component of gravity field we will be going to use for underneath density structure calculation. Ill-posing nature is main obstacle for any inverse problem. Linear regularization using Tikhonov formulation are used for appropriate choice of SVD and GSVD components. For real time data handle, signal to noise ratios should have to be less for reliable solution. In our study, 2D and 3D synthetic model with rectangular grid are used for gravity field calculation and its corresponding inversion for density reconstruction. Fine grid also we have considered to hold any irregular structure. Keeping in mind of algebraic ambiguity factor number of observation point should be more than that of number of data point. Picard plot is represented here for choosing appropriate or main controlling Eigenvalues for a regularized solution. Another important study is depth resolution plot (DRP). DRP are generally used for studying how the inversion is influenced by regularizing or discretizing. Our further study involves real time gravity data inversion of Vredeforte Dome South Africa. We apply our method to this data. The results include density structure is in good agreement with known formation in that region, which puts an additional support of our method.

Keywords: depth resolution plot, gravity inversion, Picard plot, SVD, Tikhonov formulation

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Authors: R. J. Chang

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A cyclostationary Gaussian linearization method is formulated for investigating the time average response of nonlinear system under sinusoidal signal and white noise excitation. The quantitative measure of cyclostationary mean, variance, spectrum of mean amplitude, and mean power spectral density of noise is analyzed. The qualitative response behavior of stochastic jump and bifurcation are investigated. The validity of the present approach in predicting the quantitative and qualitative statistical responses is supported by utilizing Monte Carlo simulations. The present analysis without imposing restrictive analytical conditions can be directly derived by solving non-linear algebraic equations. The analytical solution gives reliable quantitative and qualitative prediction of mean and noise response for the Duffing system subjected to both sinusoidal signal and white noise excitation.

Keywords: cyclostationary, duffing system, Gaussian linearization, sinusoidal, white noise

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Authors: Adriano Valdes-Gomez, Francisco Javier Sevilla

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We devise a numerical algorithm to simulate the diffusion of a Brownian particle restricted to the surface of a three-dimensional sphere when the particle is under the effects of an external potential that is coupled linearly. It is obtained using elementary geometry, yet, it converges, in the weak sense, to the solutions to the Smoluchowski equation. Rotations on the sphere, which are the analogs of linear displacements in euclidean spaces, are calculated using algebraic operations and then by a proper scaling, which makes the algorithm efficient and quite simple, especially to what may be the short-time propagator approach. Our findings prove that the global effects of curvature are taken into account in both dynamic and stationary processes, and it is not restricted to work in configuration space, neither restricted to the overdamped limit. We have generalized it successfully to simulate the Kramers or the Ornstein-Uhlenbeck process, where it is necessary to work directly in phase space, and it may be adapted to other two dimensional surfaces with non-constant curvature.

Keywords: diffusion on the sphere, Fokker-Planck equation on the sphere, non equilibrium processes on the sphere, numerical methods for diffusion on the sphere

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Authors: Fanous Mohammadi, Majid H. Tangestani, Mohammad H. Tayebi

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This research aims to evaluate and compare Geographical Information Systems (GIS)-based fuzzy models for producing orogenic gold prospectivity maps in the Saqqez area, NW of Iran. Gold occurrences are hosted in sericite schist and mafic to felsic meta-volcanic rocks in this area and are associated with hydrothermal alterations that extend over ductile to brittle shear zones. The predictor maps, which represent the Pre-(Source/Trigger/Pathway), syn-(deposition/physical/chemical traps) and post-mineralization (preservation/distribution of indicator minerals) subsystems for gold mineralization, were generated using empirical understandings of the specifications of known orogenic gold deposits and gold mineral systems and were then pre-processed and integrated to produce mineral prospectivity maps. Five fuzzy logic operators, including AND, OR, Fuzzy Algebraic Product (FAP), Fuzzy Algebraic Sum (FAS), and GAMMA, were applied to the predictor maps in order to find the most efficient prediction model. Prediction-Area (P-A) plots and field observations were used to assess and evaluate the accuracy of prediction models. Mineral prospectivity maps generated by AND, OR, FAP, and FAS operators were inaccurate and, therefore, unable to pinpoint the exact location of discovered gold occurrences. The GAMMA operator, on the other hand, produced acceptable results and identified potentially economic target sites. The P-A plot revealed that 68 percent of known orogenic gold deposits are found in high and very high potential regions. The GAMMA operator was shown to be useful in predicting and defining cost-effective target sites for orogenic gold deposits, as well as optimizing mineral deposit exploitation.

Keywords: mineral prospectivity mapping, fuzzy logic, GIS, orogenic gold deposit, Saqqez, Iran

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Authors: Carlos A. Vega-Posada, Jeisson Alejandro Higuita-Villa, Julio C. Saldarriaga-Molina

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This paper presents a simplified analytical approach to conduct elastic stability and second-order lateral stiffness analyses of beam-column elements (i.e., piles) with generalized end-boundary conditions embedded on a homogeneous or non-homogeneous Pasternak foundation. The solution is derived using the well-known Differential Transformation Method (DTM), and it consists simply of solving a system of two linear algebraic equations. Using other conventional approaches to solve the governing differential equation of the proposed element can be cumbersome and the solution challenging to implement, especially when the non-homogeneity of the soil is considered. The proposed formulation includes the effects of i) any rotational or lateral transverse spring at the ends of the pile, ii) any external transverse load acting along the pile, iii) soil non-homogeneity, and iv) the second-parameter of the elastic foundation (i.e., shear layer connecting the springs at the top). A parametric study is conducted to investigate the effects of different modulus of subgrade reactions, degrees of non-homogeneities, and intermediate end-boundary conditions on the pile response. The same set of equations can be used to conduct both elastic stability and static analyses. Comprehensive examples are presented to show the simplicity and practicability of the proposed method.

Keywords: elastic stability, second-order lateral stiffness, soil-non-homogeneity, pile analysis

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Authors: Antonio Aguilar Aguilar, Eliezer Braun Guitler

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Using a previously developed procedure and with the aid of algebraic software, a two-dimensional generalized Ising model with a 4×2 unitary cell (UC), we obtain a Kagome Lattice with twelve different spin-spin values of interaction, in order to determine the partition function per spin L(T). From the partition function we can study the magnetic behavior of the system. Because of the competition phenomenon between spins, a very complex behavior among them in a variety of magnetic states can be observed.

Keywords: correlations, Ising, Kagome, exact functions

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Authors: Sajjad Izadpanah, Seyed Hadi Ghaderi, Morteza Sayah Irani, Mahdi Gerdooei

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In this research work, a sophisticated yield criterion known as BBC2003, capable of describing planar anisotropic behaviors of aluminum alloy sheets, was integrated into the commercial finite element code ABAQUS/Standard via a user subroutine. The complete formulation of the implementation process using a fully implicit integration scheme, i.e., the classic backward Euler method, is presented, and relevant aspects of the yield criterion are introduced. In order to solve nonlinear differential and algebraic equations, the line-search algorithm was adopted in the user-defined material subroutine (UMAT) to expand the convergence domain of the iterative Newton-Raphson method. The developed subroutine was used to simulate a challenging computational problem with complex stress states, i.e., deep drawing of an anisotropic aluminum alloy AA3105. The accuracy and stability of the developed subroutine were confirmed by comparing the numerically predicted earing and thickness variation profiles with the experimental results, which showed an excellent agreement between numerical and experimental earing and thickness profiles. The integration of the BBC2003 yield criterion into ABAQUS/Standard represents a significant contribution to the field of computational mechanics and provides a useful tool for analyzing the mechanical behavior of anisotropic materials subjected to complex loading conditions.

Keywords: BBC2003 yield function, plastic anisotropy, fully implicit integration scheme, line search algorithm, explicit and implicit integration schemes

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Authors: Sibawu Witness Siyepu

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This study explores the effects of instructional design using blended learning in the learning of radian measures among Engineering students. Blended learning is an education programme that combines online digital media with traditional classroom methods. It requires the physical presence of both lecturer and student in a mathematics computer laboratory. Blended learning provides element of class control over time, place, path or pace. The focus was on the use of Khan Academy to supplement traditional classroom interactions. Khan Academy is a non-profit educational organisation created by educator Salman Khan with a goal of creating an accessible place for students to learn through watching videos in a computer assisted computer. The researcher who is an also lecturer in mathematics support programme collected data through instructing students to watch Khan Academy videos on radian measures, and by supplying students with traditional classroom activities. Classroom activities entails radian measure activities extracted from the Internet. Students were given an opportunity to engage in class discussions, social interactions and collaborations. These activities necessitated students to write formative assessments tests. The purpose of formative assessments tests was to find out about the students’ understanding of radian measures, including errors and misconceptions they displayed in their calculations. Identification of errors and misconceptions serve as pointers of students’ weaknesses and strengths in their learning of radian measures. At the end of data collection, semi-structure interviews were administered to a purposefully sampled group to explore their perceptions and feedback regarding the use of blended learning approach in teaching and learning of radian measures. The study employed Algebraic Insight Framework to analyse data collected. Algebraic Insight Framework is a subset of symbol sense which allows a student to correctly enter expressions into a computer assisted systems efficiently. This study offers students opportunities to enter topics and subtopics on radian measures into a computer through the lens of Khan Academy. Khan academy demonstrates procedures followed to reach solutions of mathematical problems. The researcher performed the task of explaining mathematical concepts and facilitated the process of reinvention of rules and formulae in the learning of radian measures. Lastly, activities that reinforce students’ understanding of radian were distributed. Results showed that this study enthused the students in their learning of radian measures. Learning through videos prompted the students to ask questions which brought about clarity and sense making to the classroom discussions. Data revealed that sense making through reinvention of rules and formulae assisted the students in enhancing their learning of radian measures. This study recommends the use of Khan Academy in blended learning to be introduced as a socialisation programme to all first year students. This will prepare students that are computer illiterate to become conversant with the use of Khan Academy as a powerful tool in the learning of mathematics. Khan Academy is a key technological tool that is pivotal for the development of students’ autonomy in the learning of mathematics and that promotes collaboration with lecturers and peers.

Keywords: algebraic insight framework, blended learning, Khan Academy, radian measures

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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: 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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Authors: M. H. M. Rashid

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A Banach space operator T obeys property (gm) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T − λI is a left Drazin invertible. In this article, we study the stability of property (gm), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, by quasi-nilpotent operators, or more generally by algebraic operators commuting with T.

Keywords: Weyl's Theorem, Weyl Spectrum, Polaroid operators, property (gm)

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Authors: Yousef Bakhbakhi

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Crystallization with supercritical fluids (SCFs), as a developed technology to produce particles of micron and sub-micron size with narrow size distribution, has found appreciable importance as an environmentally friendly technology. Particle synthesis using SCFs can be achieved employing a number of special processes involving solvent and antisolvent mechanisms. In this study, the compressed antisolvent (PCA) process is utilized as a model to analyze the theoretical complexity of crystallization with supercritical fluids. The population balance approach has proven to be an effectual technique to simulate and predict the particle size and size distribution. The nucleation and growth mechanisms of the particles formation in the PCA process is investigated using the population balance equation, which describes the evolution of the particle through coalescence and breakup levels with time. The employed mathematical population balance model contains a set of the partial differential equation with algebraic constraints, which demands a rigorous numerical approach. The combined Collocation and Galerkin finite element method are proposed as a high-resolution technique to solve the dynamics of the PCA process.

Keywords: particle formation, particle size and size distribution, PCA, supercritical carbon dioxide

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Authors: Mohsen Zahraei

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‎In this paper, ‎the notion of ‎rank-k numerical range of rectangular complex matrix polynomials‎ ‎are introduced. ‎Some algebraic and geometrical properties are investigated. ‎Moreover, ‎for ε>0 the notion of Birkhoff-James approximate orthogonality sets for ε-higher ‎rank numerical ranges of rectangular matrix polynomials is also introduced and studied. ‎The proposed definitions yield a natural generalization of the standard higher rank numerical ranges.

Keywords: ‎‎Rank-k numerical range‎, ‎isometry‎, ‎numerical range‎, ‎rectangular matrix polynomials

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Authors: Fola John Adeyeye

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In this paper, we propose a cryptosystem private and public key base on symmetric group Pn and validates its theoretical formulation. This proposed system benefits from the algebraic properties of Pn such as noncommutative high logical, computational speed and high flexibility in selecting key which makes the discrete permutation multiplier logic (DPML) resist to attack by any algorithm such as Pohlig-Hellman. One of the advantages of this scheme is that it explore all the possible triangular symmetries. Against these properties, the only disadvantage is that the law of permutation multiplicity only allow an operation from left to right. Many other cryptosystems can be transformed into their symmetric group.

Keywords: cryptosystem, private and public key, DPML, symmetric group Pn

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Authors: Mahmoud Enayati, Sirous Mohammadi

Abstract:

In this paper, a novel procedure to find the firing angles of the multilevel inverters of supply voltage and, consequently, to decline the total harmonic distortion (THD), has been presented. In order to eliminate more harmonics in the multilevel inverters, its number of levels can be lessened or pulse width modulation waveform, in which more than one switching occur in each level, be used. Both cases complicate the non-algebraic equations and their solution cannot be performed by the conventional methods for the numerical solution of nonlinear equations such as Newton-Raphson method. In this paper, Cuckoo algorithm is used to compute the optimal firing angle of the pulse width modulation voltage waveform in the multilevel inverter. These angles should be calculated in such a way that the voltage amplitude of the fundamental frequency be generated while the total harmonic distortion of the output voltage be small. The simulation and theoretical results for the 9-levels inverter offer the high applicability of the proposed algorithm to identify the suitable firing angles for declining the low order harmonics and generate a waveform whose total harmonic distortion is very small and it is almost a sinusoidal waveform.

Keywords: evolutionary algorithms, multilevel inverters, total harmonic content, Cuckoo Algorithm

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Authors: Puttaswamy, Anwar Alwardi, Nayaka S. R.

Abstract:

One of the algebraic representation of a graph is the graph polynomial. In this article, we introduce the paired-domination polynomial of a graph G. The paired-domination polynomial of a graph G of order n is the polynomial Dp(G, x) with the coefficients dp(G, i) where dp(G, i) denotes the number of paired dominating sets of G of cardinality i and γpd(G) denotes the paired-domination number of G. We obtain some properties of Dp(G, x) and its coefficients. Further, we compute this polynomial for some families of standard graphs. Further, we obtain some characterization for some specific graphs.

Keywords: domination polynomial, paired dominating set, paired domination number, paired domination polynomial

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Authors: Muhammad Jawwad, Riffat Iqbal

Abstract:

The Socratic method, also known as the dialectical method and elenctic method, has significant relevance in the contemporary educational system. It can be incorporated into modern-day educational systems theoretically as well as practically. Being interactive and dialogue-based in nature, this teaching approach is followed by critical thinking and innovation. The pragmatic value of the Dialectical Method has been discussed in this article, and the limitations of the Socratic method have also been highlighted. The interactive Method of Socrates can be used in many subjects for students of different grades. The Limitations and delimitations of the Method have also been discussed for its proper implementation. This article has attempted to elaborate and analyze the teaching method of Socrates with all its pre-suppositions and Epistemological character.

Keywords: Socratic method, dialectical method, knowledge, teaching, virtue

Procedia PDF Downloads 134