**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30

# Search results for: trigonometric B-spline

##### 30 Cubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two

**Authors:**
Nur Nadiah Abd Hamid ,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail

**Abstract:**

**Keywords:**
trigonometric B-spline,
two-point boundary valueproblem,
spline interpolation,
cubic spline

##### 29 Monotone Rational Trigonometric Interpolation

**Authors:**
Uzma Bashir,
Jamaludin Md. Ali

**Abstract:**

This study is concerned with the visualization of monotone data using a piecewise C1 rational trigonometric interpolating scheme. Four positive shape parameters are incorporated in the structure of rational trigonometric spline. Conditions on two of these parameters are derived to attain the monotonicity of monotone data and othertwo are leftfree. Figures are used widely to exhibit that the proposed scheme produces graphically smooth monotone curves.

**Keywords:**
Trigonometric splines,
Monotone data,
Shape preserving,
C1 monotone interpolant.

##### 28 Circular Approximation by Trigonometric Bézier Curves

**Authors:**
Maria Hussin,
Malik Zawwar Hussain,
Mubashrah Saddiqa

**Abstract:**

We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques.

**Keywords:**
Control points,
rational trigonometric Bézier curves,
radius error,
shape measure,
weight functions.

##### 27 Adaptive Motion Planning for 6-DOF Robots Based on Trigonometric Functions

**Authors:**
Jincan Li,
Mingyu Gao,
Zhiwei He,
Yuxiang Yang,
Zhongfei Yu,
Yuanyuan Liu

**Abstract:**

Building an appropriate motion model is crucial for trajectory planning of robots and determines the operational quality directly. An adaptive acceleration and deceleration motion planning based on trigonometric functions for the end-effector of 6-DOF robots in Cartesian coordinate system is proposed in this paper. This method not only achieves the smooth translation motion and rotation motion by constructing a continuous jerk model, but also automatically adjusts the parameters of trigonometric functions according to the variable inputs and the kinematic constraints. The results of computer simulation show that this method is correct and effective to achieve the adaptive motion planning for linear trajectories.

**Keywords:**
6-DOF robots,
motion planning,
trigonometric function,
kinematic constraints

##### 26 Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation

**Authors:**
Shazalina Mat Zin,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail,
Muhammad Abbas

**Abstract:**

The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

**Keywords:**
Collocation method,
Cubic trigonometric B-spline,
Finite difference,
Wave equation.

##### 25 L1-Convergence of Modified Trigonometric Sums

**Authors:**
Sandeep Kaur Chouhan,
Jatinderdeep Kaur,
S. S. Bhatia

**Abstract:**

**Keywords:**
Conjugate Dirichlet kernel,
Dirichlet kernel,
L1-convergence,
modified sums.

##### 24 A Thought on Exotic Statistical Distributions

**Authors:**
R K Sinha

**Abstract:**

**Keywords:**
Exotic Statistical Distributions,
Kurtosis,
Mixture
Distributions,
Multi-modal

##### 23 Flexure of Cantilever Thick Beams Using Trigonometric Shear Deformation Theory

**Authors:**
Yuwaraj M. Ghugal,
Ajay G. Dahake

**Abstract:**

A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick cantilever isotropic beams are considered for the numerical studies to demonstrate the efficiency of the. Results obtained are discussed critically with those of other theories.

**Keywords:**
Trigonometric shear deformation,
thick beam,
flexure,
principle of virtual work,
equilibrium equations,
stress.

##### 22 A Comparison of Some Splines-Based Methods for the One-dimensional Heat Equation

**Authors:**
Joan Goh,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail

**Abstract:**

**Keywords:**
Heat equation,
Collocation based,
Cubic Bspline,
Extended cubic uniform B-spline.

##### 21 Cubic B-spline Collocation Method for Numerical Solution of the Benjamin-Bona-Mahony-Burgers Equation

**Authors:**
M. Zarebnia,
R. Parvaz

**Abstract:**

In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.

**Keywords:**
Benjamin-Bona-Mahony-Burgers equation,
Cubic Bspline,
Collocation method,
Finite difference.

##### 20 Flexure of Simply Supported Thick Beams Using Refined Shear Deformation Theory

**Authors:**
Yuwaraj M. Ghugal,
Ajay G. Dahake

**Abstract:**

A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick simply supported isotropic beams are considered for the numerical studies to demonstrate the efficiency of the results obtained is discussed critically with those of other theories.

**Keywords:**
Trigonometric shear deformation,
thick beam,
flexure,
principle of virtual work,
equilibrium equations,
stress.

##### 19 Application of Generalized NAUT B-Spline Curveon Circular Domain to Generate Circle Involute

**Authors:**
Ashok Ganguly,
Pranjali Arondekar

**Abstract:**

In the present paper, we use generalized B-Spline curve in trigonometric form on circular domain, to capture the transcendental nature of circle involute curve and uncertainty characteristic of design. The required involute curve get generated within the given tolerance limit and is useful in gear design.

**Keywords:**
Bézier,
Circle Involute,
NAUT B-Spline,
Spur Gear.

##### 18 Applications of Trigonometic Measures of Fuzzy Entropy to Geometry

**Authors:**
Om Parkash,
C.P.Gandhi

**Abstract:**

**Keywords:**
Entropy,
Uncertainty,
Fuzzy Entropy,
Concavity,
Symmetry.

##### 17 Analytical Modeling of Globular Protein-Ferritin in α-Helical Conformation: A White Noise Functional Approach

**Authors:**
Vernie C. Convicto,
Henry P. Aringa,
Wilson I. Barredo

**Abstract:**

**Keywords:**
Globular protein,
modulating function,
white noise,
winding probability.

##### 16 Coefficients of Some Double Trigonometric Cosine and Sine Series

**Authors:**
Jatinderdeep Kaur

**Abstract:**

**Keywords:**
Conjugate Dirichlet kernel,
conjugate Fejer kernel,
Fourier series,
Semi-convexity.

##### 15 Efficient Alias-free Level Crossing Sampling

**Authors:**
Negar Riazifar,
Nigel G. Stocks

**Abstract:**

This paper proposes strategies in level crossing (LC) sampling and reconstruction that provide alias-free high-fidelity signal reconstruction for speech signals without exponentially increasing sample number with increasing bit-depth. We introduce methods in LC sampling that reduce the sampling rate close to the Nyquist frequency even for large bit-depth. The results indicate that larger variation in the sampling intervals leads to alias-free sampling scheme; this is achieved by either reducing the bit-depth or adding a jitter to the system for high bit-depths. In conjunction with windowing, the signal is reconstructed from the LC samples using an efficient Toeplitz reconstruction algorithm.

**Keywords:**
Alias-free,
level crossing sampling,
spectrum,
trigonometric polynomial.

##### 14 Traveling Wave Solutions for Shallow Water Wave Equation by (G'/G)-Expansion Method

**Authors:**
Anjali Verma,
Ram Jiwari,
Jitender Kumar

**Abstract:**

This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

**Keywords:**
Shallow water wave equation,
Exact solutions,
(G'/G) expansion method.

##### 13 On Symmetry Analysis and Exact Wave Solutions of New Modified Novikov Equation

**Authors:**
Anupma Bansal,
R. K. Gupta

**Abstract:**

In this paper, we study a new modified Novikov equation for its classical and nonclassical symmetries and use the symmetries to reduce it to a nonlinear ordinary differential equation (ODE). With the aid of solutions of the nonlinear ODE by using the modified (G/G)-expansion method proposed recently, multiple exact traveling wave solutions are obtained and the traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.

**Keywords:**
New Modified Novikov Equation,
Lie Classical Method,
Nonclassical Method,
Modified (G'/G)-Expansion Method,
Traveling Wave Solutions.

##### 12 RBF- based Meshless Method for Free Vibration Analysis of Laminated Composite Plates

**Authors:**
Jeeoot Singh,
Sandeep Singh,
K. K. Shukla

**Abstract:**

**Keywords:**
Composite plates,
Meshfree method,
free vibration,
Shear deformation,
RBFs

##### 11 Applying Half-Circle Fuzzy Numbers to Control System: A Preliminary Study on Development of Intelligent System on Marine Environment and Engineering

**Authors:**
Chen-Yuan Chen,
Wan-I Lee,
Yi-Chaio Sui,
Cheng-Wu Chen

**Abstract:**

This study focuses on the development of triangular fuzzy numbers, the revising of triangular fuzzy numbers, and the constructing of a HCFN (half-circle fuzzy number) model which can be utilized to perform more plural operations. They are further transformed for trigonometric functions and polar coordinates. From half-circle fuzzy numbers we can conceive cylindrical fuzzy numbers, which work better in algebraic operations. An example of fuzzy control is given in a simulation to show the applicability of the proposed half-circle fuzzy numbers.

**Keywords:**
triangular fuzzy number,
half-circle fuzzy numbers,
predictions,
polar coordinates,
Lyapunov method

##### 10 Image Rotation Using an Augmented 2-Step Shear Transform

**Authors:**
Hee-Choul Kwon,
Heeyong Kwon

**Abstract:**

**Keywords:**
High speed rotation operation,
image rotation,
transform matrix,
image processing,
pattern recognition.

##### 9 Solution of S3 Problem of Deformation Mechanics for a Definite Condition and Resulting Modifications of Important Failure Theories

**Authors:**
Ranajay Bhowmick

**Abstract:**

Analysis of stresses for an infinitesimal tetrahedron leads to a situation where we obtain a cubic equation consisting of three stress invariants. This cubic equation, when solved for a definite condition, gives the principal stresses directly without requiring any cumbersome and time-consuming trial and error methods or iterative numerical procedures. Since the failure criterion of different materials are generally expressed as functions of principal stresses, an attempt has been made in this study to incorporate the solutions of the cubic equation in the form of principal stresses, obtained for a definite condition, into some of the established failure theories to determine their modified descriptions. It has been observed that the failure theories can be represented using the quadratic stress invariant and the orientation of the principal plane.

**Keywords:**
Cubic equation,
stress invariant,
trigonometric,
explicit solution,
principal stress,
failure criterion.

##### 8 2-Dimensional Finger Gesture Based Mobile Robot Control Using Touch Screen

**Authors:**
O. Ejale,
N.B. Siddique,
R. Seals

**Abstract:**

The purpose of this study was to present a reliable mean for human-computer interfacing based on finger gestures made in two dimensions, which could be interpreted and adequately used in controlling a remote robot's movement. The gestures were captured and interpreted using an algorithm based on trigonometric functions, in calculating the angular displacement from one point of touch to another as the user-s finger moved within a time interval; thereby allowing for pattern spotting of the captured gesture. In this paper the design and implementation of such a gesture based user interface was presented, utilizing the aforementioned algorithm. These techniques were then used to control a remote mobile robot's movement. A resistive touch screen was selected as the gesture sensor, then utilizing a programmed microcontroller to interpret them respectively.

**Keywords:**
2-Dimensional interface,
finger gesture,
mobile
robot control,
touch screen.

##### 7 On the Exact Solution of Non-Uniform Torsion for Beams with Axial Symmetric Cross-Section

**Authors:**
A.Campanile,
M. Mandarino,
V. Piscopo,
A. Pranzitelli

**Abstract:**

**Keywords:**
Non-uniform torsion,
Axial symmetric cross-section,
Fourier series,
Helmholtz equation,
FE method.

##### 6 Efficient High Fidelity Signal Reconstruction Based on Level Crossing Sampling

**Authors:**
Negar Riazifar,
Nigel G. Stocks

**Abstract:**

This paper proposes strategies in level crossing (LC) sampling and reconstruction that provide high fidelity signal reconstruction for speech signals; these strategies circumvent the problem of exponentially increasing number of samples as the bit-depth is increased and hence are highly efficient. Specifically, the results indicate that the distribution of the intervals between samples is one of the key factors in the quality of signal reconstruction; including samples with short intervals does not improve the accuracy of the signal reconstruction, whilst samples with large intervals lead to numerical instability. The proposed sampling method, termed reduced conventional level crossing (RCLC) sampling, exploits redundancy between samples to improve the efficiency of the sampling without compromising performance. A reconstruction technique is also proposed that enhances the numerical stability through linear interpolation of samples separated by large intervals. Interpolation is demonstrated to improve the accuracy of the signal reconstruction in addition to the numerical stability. We further demonstrate that the RCLC and interpolation methods can give useful levels of signal recovery even if the average sampling rate is less than the Nyquist rate.

**Keywords:**
Level crossing sampling,
numerical stability,
speech processing,
trigonometric polynomial.

##### 5 Analytical Formulae for the Approach Velocity Head Coefficient

**Authors:**
Abdulrahman Abdulrahman

**Abstract:**

Critical depth meters, such as abroad crested weir, Venture Flume and combined control flume are standard devices for measuring flow in open channels. The discharge relation for these devices cannot be solved directly, but it needs iteration process to account for the approach velocity head. In this paper, analytical solution was developed to calculate the discharge in a combined critical depth-meter namely, a hump combined with lateral contraction in rectangular channel with subcritical approach flow including energy losses. Also analytical formulae were derived for approach velocity head coefficient for different types of critical depth meters. The solution was derived by solving a standard cubic equation considering energy loss on the base of trigonometric identity. The advantage of this technique is to avoid iteration process adopted in measuring flow by these devices. Numerical examples are chosen for demonstration of the proposed solution.

**Keywords:**
Broad crested weir,
combined control meter,
control structures,
critical flow,
discharge measurement,
flow control,
hydraulic engineering,
hydraulic structures,
open channel flow.

##### 4 Jeffrey's Prior for Unknown Sinusoidal Noise Model via Cramer-Rao Lower Bound

**Authors:**
Samuel A. Phillips,
Emmanuel A. Ayanlowo,
Rasaki O. Olanrewaju,
Olayode Fatoki

**Abstract:**

This paper employs the Jeffrey's prior technique in the process of estimating the periodograms and frequency of sinusoidal model for unknown noisy time variants or oscillating events (data) in a Bayesian setting. The non-informative Jeffrey's prior was adopted for the posterior trigonometric function of the sinusoidal model such that Cramer-Rao Lower Bound (CRLB) inference was used in carving-out the minimum variance needed to curb the invariance structure effect for unknown noisy time observational and repeated circular patterns. An average monthly oscillating temperature series measured in degree Celsius (0C) from 1901 to 2014 was subjected to the posterior solution of the unknown noisy events of the sinusoidal model via Markov Chain Monte Carlo (MCMC). It was not only deduced that two minutes period is required before completing a cycle of changing temperature from one particular degree Celsius to another but also that the sinusoidal model via the CRLB-Jeffrey's prior for unknown noisy events produced a miniature posterior Maximum A Posteriori (MAP) compare to a known noisy events.

**Keywords:**
Cramer-Rao Lower Bound (CRLB),
Jeffrey's prior,
Sinusoidal,
Maximum A Posteriori (MAP),
Markov Chain Monte
Carlo (MCMC),
Periodograms.

##### 3 Rayleigh-Bénard-Taylor Convection of Newtonian Nanoliquid

**Authors:**
P. G. Siddheshwar,
T. N. Sakshath

**Abstract:**

**Keywords:**
Nanoliquid,
rigid-rigid,
rotation,
single-phase.

##### 2 Price Prediction Line, Investment Signals and Limit Conditions Applied for the German Financial Market

**Authors:**
Cristian Păuna

**Abstract:**

In the first decades of the 21st century, in the electronic trading environment, algorithmic capital investments became the primary tool to make a profit by speculations in financial markets. A significant number of traders, private or institutional investors are participating in the capital markets every day using automated algorithms. The autonomous trading software is today a considerable part in the business intelligence system of any modern financial activity. The trading decisions and orders are made automatically by computers using different mathematical models. This paper will present one of these models called Price Prediction Line. A mathematical algorithm will be revealed to build a reliable trend line, which is the base for limit conditions and automated investment signals, the core for a computerized investment system. The paper will guide how to apply these tools to generate entry and exit investment signals, limit conditions to build a mathematical filter for the investment opportunities, and the methodology to integrate all of these in automated investment software. The paper will also present trading results obtained for the leading German financial market index with the presented methods to analyze and to compare different automated investment algorithms. It was found that a specific mathematical algorithm can be optimized and integrated into an automated trading system with good and sustained results for the leading German Market. Investment results will be compared in order to qualify the presented model. In conclusion, a 1:6.12 risk was obtained to reward ratio applying the trigonometric method to the DAX Deutscher Aktienindex on 24 months investment. These results are superior to those obtained with other similar models as this paper reveal. The general idea sustained by this paper is that the Price Prediction Line model presented is a reliable capital investment methodology that can be successfully applied to build an automated investment system with excellent results.

**Keywords:**
Algorithmic trading,
automated investment system,
DAX Deutscher Aktienindex.

##### 1 Holistic Approach to Teaching Mathematics in Secondary School as a Means of Improving Students’ Comprehension of Study Material

**Authors:**
Natalia Podkhodova,
Olga Sheremeteva,
Mariia Soldaeva

**Abstract:**

Creating favourable conditions for students’ comprehension of mathematical content is one of the primary problems in teaching mathematics in secondary school. The fact of comprehension includes the ability to build a working situational model and thus becomes an important means of solving mathematical problems. This paper describes a holistic approach to teaching mathematics designed to address the primary challenges of such teaching; specifically, the challenge of students’ comprehension. Essentially, this approach consists of (1) establishing links between the attributes of the notion: the sense, the meaning, and the term; (2) taking into account the components of student’s subjective experience—value-based emotions, contextual, procedural and communicative—during the educational process; (3) linking together different ways to present mathematical information; (4) identifying and leveraging the relationships between real, perceptual and conceptual (scientific) mathematical spaces by applying real-life situational modelling. The article describes approaches to the practical use of these foundational concepts. Identifying how proposed methods and techniques influence understanding of material used in teaching mathematics was the primary goal. The study included an experiment in which 256 secondary school students took part: 142 in the study group and 114 in the control group. All students in these groups had similar levels of achievement in math and studied math under the same curriculum. In the course of the experiment, comprehension of two topics — “Derivative” and “Trigonometric functions”—was evaluated. Control group participants were taught using traditional methods. Students in the study group were taught using the holistic method: under teacher’s guidance, they carried out assignments designed to establish linkages between notion’s characteristics, to convert information from one mode of presentation to another, as well as assignments that required the ability to operate with all modes of presentation. Identification, accounting for and transformation of subjective experience were associated with methods of stimulating the emotional value component of the studied mathematical content (discussions of lesson titles, assignments aimed to create study dominants, performing theme-related physical exercise ...) The use of techniques that forms inter-subject notions based on linkages between, perceptual real and mathematical conceptual spaces proved to be of special interest to the students. Results of the experiment were analysed by presenting students in each of the groups with a final test in each of the studied topics. The test included assignments that required building real situational models. Statistical analysis was used to aggregate test results. Pierson criterion x2 was used to reveal statistics significance of results (pass-fail the modelling test). Significant difference of results was revealed (p < 0.001), which allowed to conclude that students in the study group showed better comprehension of mathematical information than those in the control group. The total number of completed assignments of each student was analysed as well, with average results calculated for each group. Statistical significance of result differences against the quantitative criterion (number of completed assignments) was determined using Student’s t-test, which showed that students in the study group completed significantly more assignments than those in the control group (p = 0.0001). Authors thus come to the conclusion that suggested increase in the level of comprehension of study material took place as a result of applying implemented methods and techniques.

**Keywords:**
Comprehension of mathematical content,
holistic approach to teaching mathematics in secondary school,
subjective experience,
technology of the formation of inter-subject notions.