Search results for: vector calculus

Authors: Stephen Kirkup

Abstract:

This paper discusses the implementation of the boundary element method (BEM) on an Excel spreadsheet and how it can be used in teaching vector calculus and simulation. There are two separate spreadheets, within which Laplace equation is solved by the BEM in two dimensions (LIBEM2) and axisymmetric three dimensions (LBEMA). The main algorithms are implemented in the associated programming language within Excel, Visual Basic for Applications (VBA). The BEM only requires a boundary mesh and hence it is a relatively accessible method. The BEM in the open spreadsheet environment is demonstrated as being useful as an aid to teaching and learning. The application of the BEM implemented on a spreadsheet for educational purposes in introductory vector calculus and simulation is explored. The development of assignment work is discussed, and sample results from student work are given. The spreadsheets were found to be useful tools in developing the students’ understanding of vector calculus and in simulating heat conduction.

Keywords: Boundary element method, laplace equation, vector calculus, simulation, education.

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Authors: Troy L. Story

Abstract:

Mathematical models of dynamics employing exterior calculus are mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of differential equations and a characteristic tangent vector (vortex vector) which define transformations of the system. Using this principle, a mathematical model for economic growth is constructed by proposing a characteristic differential one-form for economic growth dynamics (analogous to the action in Hamiltonian dynamics), then generating a pair of characteristic differential equations and solving these equations for the rate of economic growth as a function of labor and capital. By contracting the characteristic differential one-form with the vortex vector, the Lagrangian for economic growth dynamics is obtained.

Keywords: Differential geometry, exterior calculus, Hamiltonian geometry, mathematical economics.

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Authors: Troy L. Story

Abstract:

A mathematical model for the Dynamics of Economic Profit is constructed by proposing a characteristic differential oneform for this dynamics (analogous to the action in Hamiltonian dynamics). After processing this form with exterior calculus, a pair of characteristic differential equations is generated and solved for the rate of change of profit P as a function of revenue R (t) and cost C (t). By contracting the characteristic differential one-form with a vortex vector, the Lagrangian is obtained for the Dynamics of Economic Profit.

Keywords: Differential geometry, exterior calculus, Hamiltonian geometry, mathematical economics, economic functions, and dynamics

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Authors: Asia Majeed

Abstract:

This paper will investigate a form of learning mathematics by integrating technology in mathematics specifically for the university first-year calculus class to support students’ engagement in learning which influences students' conceptual and procedural understanding of the calculus content in a better way. The students with good grades in high school calculus generally struggle in first-year university calculus classes in learning mathematical analysis concepts. This problem has to be addressed. If this problem is not resolved, then most likely students with less ability to do mathematics might not able to complete their degrees. In this work, MATLAB is used to help students in learning and in improving calculus concepts.

Keywords: Calculus, first-year university students, teaching strategies, MATLAB.

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Authors: Xuan Qi, Changzhi Zhao

Abstract:

In this paper, a uniform calculus-based approach for synthesizing monitors checking correctness properties specified by a large variety of logics at runtime is provided, including future and past time logics, interval logics, state machine and parameterized temporal logics. We present a calculus mechanism to synthesize monitors from the logical specification for the incremental analysis of execution traces during test and real run. The monitor detects both good and bad prefix of a particular kind, namely those that are informative for the property under investigation. We elaborate the procedure of calculus as monitors.

Keywords: calculus, eagle logic, monitor synthesis, runtime verification

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Authors: M. De la Sen

Abstract:

This paper establishes some closed formulas for Riemann- Liouville impulsive fractional integral calculus and also for Riemann- Liouville and Caputo impulsive fractional derivatives.

Keywords: Rimann- Liouville fractional calculus, Caputofractional derivative, Dirac delta, Distributional derivatives, Highorderdistributional derivatives.

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Authors: M. Zarebnia, N. Aliniya

Abstract:

In this paper, a numerical solution based on sinc functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.

Keywords: Calculus of variation; Sinc functions; Galerkin; Numerical method

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Authors: M. Zarebnia, M. Hoshyar, M. Sedaghati

Abstract:

In this paper, a numerical solution based on nonpolynomial cubic spline functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.

Keywords: Calculus of variation; Non-polynomial spline functions; Numerical method

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Authors: Chahid K. Ghaddar

Abstract:

The spreadsheet engine is exploited via a non-conventional mechanism to enable novel worksheet solver functions for computational calculus. The solver functions bypass inherent restrictions on built-in math and user defined functions by taking variable formulas as a new type of argument while retaining purity and recursion properties. The enabling mechanism permits integration of numerical algorithms into worksheet functions for solving virtually any computational problem that can be modelled by formulas and variables. Several examples are presented for computing integrals, derivatives, and systems of deferential-algebraic equations. Incorporation of the worksheet solver functions with the ubiquitous spreadsheet extend the utility of the latter as a powerful tool for computational mathematics.

Keywords: Calculus functions, nonlinear systems, differential algebraic equations, solvers, spreadsheet.

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Authors: S. Kherchaoui, A. Houacine

Abstract:

This paper presents a facial expression recognition system. It performs identification and classification of the seven basic expressions; happy, surprise, fear, disgust, sadness, anger, and neutral states. It consists of three main parts. The first one is the detection of a face and the corresponding facial features to extract the most expressive portion of the face, followed by a normalization of the region of interest. Then calculus of curvelet coefficients is performed with dimensionality reduction through principal component analysis. The resulting coefficients are combined with two ratios; mouth ratio and face edge ratio to constitute the whole feature vector. The third step is the classification of the emotional state using the SVM method in the feature space.

Keywords: Facial expression identification, curvelet coefficients, support vector machine (SVM).

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Authors: Haohao Wang

Abstract:

Technology, multimedia in Open Educational Resources, can contribute positively to student performance in an online instructional environment. Student performance data of past four years were obtained from an online course entitled Applied Calculus (MA139). This paper examined the data to determine whether multimedia (independent variable) had any impact on student performance (dependent variable) in online math learning, and how students felt about the value of the technology. Two groups of student data were analyzed, group 1 (control) from the online applied calculus course that did not use multimedia instructional materials, and group 2 (treatment) of the same online applied calculus course that used multimedia instructional materials. For the MA139 class, results indicate a statistically significant difference (p = .001) between the two groups, where group 1 had a final score mean of 56.36 (out of 100), group 2 of 70.68. Additionally, student testimonials were discussed in which students shared their experience in learning applied calculus online with multimedia instructional materials.

Keywords: Online learning, Open Educational Resources, Multimedia, Technology.

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Authors: M. Satya Sai Ram, P. Siddaiah, M. Madhavi Latha

Abstract:

Vector quantization is a powerful tool for speech coding applications. This paper deals with LPC Coding of speech signals which uses a new technique called Multi Switched Split Vector Quantization, This is a hybrid of two product code vector quantization techniques namely the Multi stage vector quantization technique, and Switched split vector quantization technique,. Multi Switched Split Vector Quantization technique quantizes the linear predictive coefficients in terms of line spectral frequencies. From results it is proved that Multi Switched Split Vector Quantization provides better trade off between bitrate and spectral distortion performance, computational complexity and memory requirements when compared to Switched Split Vector Quantization, Multi stage vector quantization, and Split Vector Quantization techniques. By employing the switching technique at each stage of the vector quantizer the spectral distortion, computational complexity and memory requirements were greatly reduced. Spectral distortion was measured in dB, Computational complexity was measured in floating point operations (flops), and memory requirements was measured in (floats).

Keywords: Unconstrained vector quantization, Linear predictiveCoding, Split vector quantization, Multi stage vector quantization, Switched Split vector quantization, Line Spectral Frequencies.

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Authors: Erna T. Herdiani, Maman A. Djauhari

Abstract:

In recent years, the use of vector variance as a measure of multivariate variability has received much attention in wide range of statistics. This paper deals with a more economic measure of multivariate variability, defined as vector variance minus all duplication elements. For high dimensional data, this will increase the computational efficiency almost 50 % compared to the original vector variance. Its sampling distribution will be investigated to make its applications possible.

Keywords: Asymptotic distribution, covariance matrix, likelihood ratio test, vector variance.

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Authors: Bhupendra C. S. Chauhan, P. S. Bisht, O. P. S. Negi

Abstract:

According to celebrated Hurwitz theorem, there exists four division algebras consisting of R (real numbers), C (complex numbers), H (quaternions) and O (octonions). Keeping in view the utility of octonion variable we have tried to extend the three dimensional vector analysis to seven dimensional one. Starting with the scalar and vector product in seven dimensions, we have redefined the gradient, divergence and curl in seven dimension. It is shown that the identity n(n - 1)(n - 3)(n - 7) = 0 is satisfied only for 0, 1, 3 and 7 dimensional vectors. We have tried to write all the vector inequalities and formulas in terms of seven dimensions and it is shown that same formulas loose their meaning in seven dimensions due to non-associativity of octonions. The vector formulas are retained only if we put certain restrictions on octonions and split octonions.

Keywords: Octonions, Vector Space and seven dimensions

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Authors: Süha Yılmaz, Emin Özyılmaz, Melih Turgut, Şuur Nizamoğlu

Abstract:

In this paper, position vector of a partially null unit speed curve with respect to standard frame of Minkowski space-time is studied. First, it is proven that position vector of every partially null unit speed curve satisfies a vector differential equation of fourth order. In terms of solution of the differential equation, position vector of a partially null unit speed curve is expressed.

Keywords: Frenet Equations, Partially Null Curves, Minkowski Space-time, Vector Differential Equation.

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Authors: U.K. Saha, L.K. Arora

Abstract:

The multiindex Mittag-Leffler (M-L) function and the multiindex Dzrbashjan-Gelfond-Leontiev (D-G-L) differentiation and integration play a very pivotal role in the theory and applications of generalized fractional calculus. The object of this paper is to investigate the relations that exist between the Riemann-Liouville fractional calculus and multiindex Dzrbashjan-Gelfond-Leontiev differentiation and integration with multiindex Mittag-Leffler function.

Keywords: Multiindex Mittag-Leffler function, Multiindex Dzrbashjan-Gelfond-Leontiev differentiation and integration, Riemann-Liouville fractional integrals and derivatives.

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Authors: Nisa Özge Önal, Kamil Karaçuha, Göksu Hazar Erdinç, Banu Bahar Karaçuha, Ertuğrul Karaçuha

Abstract:

The study aims to use a mathematical approach with the fractional calculus which is developed to have the ability to continuously analyze the factors related to the children’s height development. Until now, tracking the development of the child is getting more important and meaningful. Knowing and determining the factors related to the physical development of the child any desired time would provide better, reliable and accurate results for childcare. In this frame, 7 groups for height percentile curve (3th, 10th, 25th, 50th, 75th, 90th, and 97th) of Turkey are used. By using discrete height data of 0-18 years old children and the least squares method, a continuous curve is developed valid for any time interval. By doing so, in any desired instant, it is possible to find the percentage and location of the child in Percentage Chart. Here, with the help of the fractional calculus theory, a mathematical model is developed. The outcomes of the proposed approach are quite promising compared to the linear and the polynomial method. The approach also yields to predict the expected values of children in the sense of height.

Keywords: Children growth percentile, children physical development, fractional calculus, linear and polynomial model.

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Authors: C. Patrascioiu, V. Matei, N. Nicolae

Abstract:

The paper describes the experiments and the kinetic parameters calculus of the gasoil hydrofining. They are presented experimental results of gasoil hidrofining using Mo and promoted with Ni on aluminum support catalyst. The authors have adapted a kinetic model gasoil hydrofining. Using this proposed kinetic model and the experimental data they have calculated the parameters of the model. The numerical calculus is based on minimizing the difference between the experimental sulf concentration and kinetic model estimation.

Keywords: Hydrofining, kinetic, modeling, optimization.

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Authors: Truong Nguyen Luan Vu, Le Hieu Giang, Le Linh

Abstract:

The fractional–order proportional integral (FOPI) controller tuning rules based on the fractional calculus for the cascade control system are systematically proposed in this paper. Accordingly, the ideal controller is obtained by using internal model control (IMC) approach for both the inner and outer loops, which gives the desired closed-loop responses. On the basis of the fractional calculus, the analytical tuning rules of FOPI controller for the inner loop can be established in the frequency domain. Besides, the outer loop is tuned by using any integer PI/PID controller tuning rules in the literature. The simulation study is considered for the stable process model and the results demonstrate the simplicity, flexibility, and effectiveness of the proposed method for the cascade control system in compared with the other methods.

Keywords: Fractional calculus, fractional–order proportional integral controller, cascade control system, internal model control approach.

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Authors: Mikhail Bouniaev

Abstract:

The paper reviews design and implementation of a Calculus Course required for the Biomedical Competency Based Program developed as a joint project between The University of Texas Rio Grande Valley, and the University of Texas’ Institute for Transformational Learning, from the theoretical perspective as presented in scholarly work on active learning, formative assessment, and on-line teaching. Following a four stage curriculum development process (objective, content, delivery, and assessment), and theoretical recommendations that guarantee effectiveness and efficiency of assessment in active learning, we discuss the practical recommendations on how to incorporate a strong formative assessment component to address disciplines’ needs, and students’ major needs. In design and implementation of this project, we used Constructivism and Stage-by-Stage Development of Mental Actions Theory recommendations.

Keywords: Active learning, assessment, Calculus, cognitive demand, constructivism, mathematics, Stage-by-Stage Development of Mental Action Theory.

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Authors: Prof. Chokri SLIM

Abstract:

A Novel fuzzy neural network combining with support vector learning mechanism called support-vector-based fuzzy neural networks (SVBFNN) is proposed. The SVBFNN combine the capability of minimizing the empirical risk (training error) and expected risk (testing error) of support vector learning in high dimensional data spaces and the efficient human-like reasoning of FNN.

Keywords: Neural network, fuzzy inference, machine learning, fuzzy modeling and rule extraction, support vector regression.

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Authors: Mei Li, Fahui Zhai

Abstract:

In this paper, the definitions of the quaternion measure and the quaternion vector measure are introduced. The relation between the quaternion measure and the complex vector measure as well as the relation between the quaternion linear functional and the complex linear functional are discussed respectively. By using these relations, the necessary and sufficient condition to determine the quaternion vector measure is given.

Keywords: Quaternion, Quaternion measure, Quaternion vector measure, Quaternion Banach space, Quaternion linear functional.

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Authors: Emin Özyılmaz

Abstract:

In this work, position vector of a time-like dual curve according to standard frame of D31 is investigated. First, it is proven that position vector of a time-like dual curve satisfies a dual vector differential equation of fourth order. The general solution of this dual vector differential equation has not yet been found. Due to this, in terms of special solutions, position vectors of some special time-like dual curves with respect to standard frame of D31 are presented.

Keywords: Classical Differential Geometry, Dual Numbers, DualFrenet Equations, Time-like Dual Curve, Position Vector, DualLorentzian Space.

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Authors: Ju-Hong Lee, Ching-Wei Liao, Kun-Che Lee

Abstract:

This paper deals with the problem of using antenna sensors for adaptive beamforming in the presence of random steering mismatch. We present an efficient adaptive array beamformer with robustness to deal with the considered problem. The robustness of the proposed beamformer comes from the efficient designation of the steering vector. Using the received array data vector, we construct an appropriate correlation matrix associated with the received array data vector and a correlation matrix associated with signal sources. Then, the eigenvector associated with the largest eigenvalue of the constructed signal correlation matrix is designated as an appropriate estimate of the steering vector. Finally, the adaptive weight vector required for adaptive beamforming is obtained by using the estimated steering vector and the constructed correlation matrix of the array data vector. Simulation results confirm the effectiveness of the proposed method.

Keywords: Adaptive beamforming, antenna array, linearly constrained minimum variance, robustness, steering vector.

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Authors: Athanasios T. Alexiou, Maria M. Psiha, John A. Rekkas, Panayiotis M. Vlamos

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Mitochondria are dynamic organelles, capable to interact with each other. While the number of mitochondria in a cell varies, their quality and functionality depends on the operation of fusion, fission, motility and mitophagy. Nowadays, several researches declare as an important factor in neurogenerative diseases the disruptions in the regulation of mitochondrial dynamics. In this paper a stochastic model in BioAmbients calculus is presented, concerning mitochondrial fusion and its distribution in the renewal of mitochondrial population in a cell. This model describes the successive and dependent stages of protein synthesis, protein-s activation and merging of two independent mitochondria.

Keywords: Mitochondrial Dynamics, P-Calculus, StochasticModeling.

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Authors: M. Satya Sai Ram, P. Siddaiah, M. Madhavi Latha

Abstract:

Vector quantization is a powerful tool for speech coding applications. This paper deals with LPC Coding of speech signals which uses a new technique called Multi Switched Split Vector Quantization (MSSVQ), which is a hybrid of Multi, switched, split vector quantization techniques. The spectral distortion performance, computational complexity, and memory requirements of MSSVQ are compared to split vector quantization (SVQ), multi stage vector quantization(MSVQ) and switched split vector quantization (SSVQ) techniques. It has been proved from results that MSSVQ has better spectral distortion performance, lower computational complexity and lower memory requirements when compared to all the above mentioned product code vector quantization techniques. Computational complexity is measured in floating point operations (flops), and memory requirements is measured in (floats).

Keywords: Linear predictive Coding, Multi stage vectorquantization, Switched Split vector quantization, Split vectorquantization, Line Spectral Frequencies (LSF).

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Authors: S.Esakkirajan, T. Veerakumar, V. Senthil Murugan, P.Navaneethan

Abstract:

In this paper, image compression using hybrid vector quantization scheme such as Multistage Vector Quantization (MSVQ) and Pyramid Vector Quantization (PVQ) are introduced. A combined MSVQ and PVQ are utilized to take advantages provided by both of them. In the wavelet decomposition of the image, most of the information often resides in the lowest frequency subband. MSVQ is applied to significant low frequency coefficients. PVQ is utilized to quantize the coefficients of other high frequency subbands. The wavelet coefficients are derived using lifting scheme. The main aim of the proposed scheme is to achieve high compression ratio without much compromise in the image quality. The results are compared with the existing image compression scheme using MSVQ.

Keywords: Lifting Scheme, Multistage Vector Quantization and Pyramid Vector Quantization.

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Authors: Nagi Mekhiel

Abstract:

Many applications use vector operations by applying single instruction to multiple data that map to different locations in conventional memory. Transferring data from memory is limited by access latency and bandwidth affecting the performance gain of vector processing. We present a memory system that makes all of its content available to processors in time so that processors need not to access the memory, we force each location to be available to all processors at a specific time. The data move in different orbits to become available to other processors in higher orbits at different time. We use this memory to apply parallel vector operations to data streams at first orbit level. Data processed in the first level move to upper orbit one data element at a time, allowing a processor in that orbit to apply another vector operation to deal with serial code limitations inherited in all parallel applications and interleaved it with lower level vector operations.

Keywords: Memory organization, parallel processors, serial code, vector processing.

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Authors: Mehrnaz Najafi, Hassan Haghighi

Abstract:

Morgan-s refinement calculus (MRC) is one of the well-known methods allowing the formality presented in the program specification to be continued all the way to code. On the other hand, Object-Z (OZ) is an extension of Z adding support for classes and objects. There are a number of methods for obtaining code from OZ specifications that can be categorized into refinement and animation methods. As far as we know, only one refinement method exists which refines OZ specifications into code. However, this method does not have fine-grained refinement rules and thus cannot be automated. On the other hand, existing animation methods do not present mapping rules formally and do not support the mapping of several important constructs of OZ, such as all cases of operation expressions and most of constructs in global paragraph. In this paper, with the aim of providing an automatic path from OZ specifications to code, we propose an approach to map OZ specifications into their counterparts in MRC in order to use fine-grained refinement rules of MRC. In this way, having counterparts of our specifications in MRC, we can refine them into code automatically using MRC tools such as RED. Other advantages of our work pertain to proposing mapping rules formally, supporting the mapping of all important constructs of Object-Z, and considering dynamic instantiation of objects while OZ itself does not cover this facility.

Keywords: Formal method, Formal specification, Formalprogram development, Morgan's Refinement Calculus, Object-Z

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Authors: Bo Meng, Wei Wang

Abstract:

Resistance of denial of service attacks is a key security requirement in voting protocols. Acquisti protocol plays an important role in development of internet voting protocols and claims its security without strong physical assumptions. In this study firstly Acquisti protocol is modeled in extended applied pi calculus, and then resistance of denial of service attacks is proved with ProVerif. The result is that it is not resistance of denial of service attacks because two denial of service attacks are found. Finally we give the method against the denial of service attacks.

Keywords: Applied pi calculus, protocol state, symbolic model, availability.

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