International Science Index
Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves
Newton-Lagrange Interpolations are widely used in
numerical analysis. However, it requires a quadratic computational
time for their constructions. In computer aided geometric design
(CAGD), there are some polynomial curves: Wang-Ball, DP and
Dejdumrong curves, which have linear time complexity algorithms.
Thus, the computational time for Newton-Lagrange Interpolations
can be reduced by applying the algorithms of Wang-Ball, DP and
Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong
algorithms, first, it is necessary to convert Newton-Lagrange
polynomials into Wang-Ball, DP or Dejdumrong polynomials. In
this work, the algorithms for converting from both uniform and
non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and
Dejdumrong polynomials are investigated. Thus, the computational
time for representing Newton-Lagrange polynomials can be reduced
into linear complexity. In addition, the other utilizations of using
CAGD curves to modify the Newton-Lagrange curves can be taken.
An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon
In this work, we present an efficient approach for
solving variable-order time-fractional partial differential equations,
which are based on Legendre and Laguerre polynomials. First, we
introduced the pseudo-operational matrices of integer and variable
fractional order of integration by use of some properties of
Riemann-Liouville fractional integral. Then, applied together with
collocation method and Legendre-Laguerre functions for solving
variable-order time-fractional partial differential equations. Also, an
estimation of the error is presented. At last, we investigate numerical
examples which arise in physics to demonstrate the accuracy of the
present method. In comparison results obtained by the present method
with the exact solution and the other methods reveals that the method
is very effective.
Some Results on the Generalized Higher Rank Numerical Ranges
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
Є > 0, the notion of Birkhoff-James approximate orthogonality
sets for Є−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
On a New Inverse Polynomial Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations
This paper presents the development, analysis and
implementation of an inverse polynomial numerical method which is
well suitable for solving initial value problems in first order ordinary
differential equations with applications to sample problems. We also
present some basic concepts and fundamental theories which are vital
to the analysis of the scheme. We analyzed the consistency,
convergence, and stability properties of the scheme. Numerical
experiments were carried out and the results compared with the
theoretical or exact solution and the algorithm was later coded using
MATLAB programming language.
Single-Crystal Kerfless 2D Array Transducer for Volumetric Medical Imaging: Theoretical Study
The aim of this work is to present a theoretical analysis of a 2D ultrasound transducer comprised of crossed arrays of metal strips placed on both sides of thin piezoelectric layer (a). Such a structure is capable of electronic beam-steering of generated wavebeam both in elevation and azimuth. In this paper a semi-analytical model of the considered transducer is developed. It is based on generalization of the well-known BIS-expansion method. Specifically, applying the electrostatic approximation, the electric field components on the surface of the layer are expanded into fast converging series of double periodic spatial harmonics with corresponding amplitudes represented by the properly chosen Legendre polynomials. The problem is reduced to numerical solving of certain system of linear equations for unknown expansion coefficients.
A Contribution to the Polynomial Eigen Problem
The relationship between eigenstructure (eigenvalues
and eigenvectors) and latent structure (latent roots and latent vectors)
is established. In control theory eigenstructure is associated with
the state space description of a dynamic multi-variable system and
a latent structure is associated with its matrix fraction description.
Beginning with block controller and block observer state space forms
and moving on to any general state space form, we develop the
identities that relate eigenvectors and latent vectors in either direction.
Numerical examples illustrate this result. A brief discussion of the
potential of these identities in linear control system design follows.
Additionally, we present a consequent result: a quick and easy
method to solve the polynomial eigenvalue problem for regular matrix
Bilinear and Bilateral Generating Functions for the Gauss’ Hypergeometric Polynomials
The object of the present paper is to investigate several
general families of bilinear and bilateral generating functions with
different argument for the Gauss’ hypergeometric polynomials.
An Interval Type-2 Dual Fuzzy Polynomial Equations and Ranking Method of Fuzzy Numbers
According to fuzzy arithmetic, dual fuzzy polynomials cannot be replaced by fuzzy polynomials. Hence, the concept of ranking method is used to find real roots of dual fuzzy polynomial equations. Therefore, in this study we want to propose an interval type-2 dual fuzzy polynomial equation (IT2 DFPE). Then, the concept of ranking method also is used to find real roots of IT2 DFPE (if exists). We transform IT2 DFPE to system of crisp IT2 DFPE. This transformation performed with ranking method of fuzzy numbers based on three parameters namely value, ambiguity and fuzziness. At the end, we illustrate our approach by two numerical examples.
A Modified Laplace Decomposition Algorithm Solution for Blasius’ Boundary Layer Equation of the Flat Plate in a Uniform Stream
In this work, we apply the Modified Laplace
decomposition algorithm in finding a numerical solution of Blasius’
boundary layer equation for the flat plate in a uniform stream. The
series solution is found by first applying the Laplace transform to the
differential equation and then decomposing the nonlinear term by the
use of Adomian polynomials. The resulting series, which is exactly the
same as that obtained by Weyl 1942a, was expressed as a rational
function by the use of diagonal padé approximant.
Conventional and PSO Based Approaches for Model Reduction of SISO Discrete Systems
Reduction of Single Input Single Output (SISO) discrete systems into lower order model, using a conventional and an evolutionary technique is presented in this paper. In the conventional technique, the mixed advantages of Modified Cauer Form (MCF) and differentiation are used. In this method the original discrete system is, first, converted into equivalent continuous system by applying bilinear transformation. The denominator of the equivalent continuous system and its reciprocal are differentiated successively, the reduced denominator of the desired order is obtained by combining the differentiated polynomials. The numerator is obtained by matching the quotients of MCF. The reduced continuous system is converted back into discrete system using inverse bilinear transformation. In the evolutionary technique method, Particle Swarm Optimization (PSO) is employed to reduce the higher order model. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical example.
Exact Solutions of the Helmholtz equation via the Nikiforov-Uvarov Method
The Helmholtz equation often arises in the study of physical problems involving partial differential equation. Many researchers have proposed numerous methods to find the analytic or approximate solutions for the proposed problems. In this work, the exact analytical solutions of the Helmholtz equation in spherical polar coordinates are presented using the Nikiforov-Uvarov (NU) method. It is found that the solution of the angular eigenfunction can be expressed by the associated-Legendre polynomial and radial eigenfunctions are obtained in terms of the Laguerre polynomials. The special case for k=0, which corresponds to the Laplace equation is also presented.
Numerical Solution of Riccati Differential Equations by Using Hybrid Functions and Tau Method
A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.
Orthogonal Functions Approach to LQG Control
In this paper a unified approach via block-pulse functions (BPFs) or shifted Legendre polynomials (SLPs) is presented to solve the linear-quadratic-Gaussian (LQG) control problem. Also a recursive algorithm is proposed to solve the above problem via BPFs. By using the elegant operational properties of orthogonal functions (BPFs or SLPs) these computationally attractive algorithms are developed. To demonstrate the validity of the proposed approaches a numerical example is included.
Orthogonal Polynomial Density Estimates: Alternative Representation and Degree Selection
The density estimates considered in this paper comprise
a base density and an adjustment component consisting of a linear
combination of orthogonal polynomials. It is shown that, in the
context of density approximation, the coefficients of the linear combination
can be determined either from a moment-matching technique
or a weighted least-squares approach. A kernel representation of
the corresponding density estimates is obtained. Additionally, two
refinements of the Kronmal-Tarter stopping criterion are proposed
for determining the degree of the polynomial adjustment. By way of
illustration, the density estimation methodology advocated herein is
applied to two data sets.
4D Flight Trajectory Optimization Based on Pseudospectral Methods
The optimization and control problem for 4D trajectories
is a subject rarely addressed in literature. In the 4D navigation
problem we define waypoints, for each mission, where the arrival
time is specified in each of them. One way to design trajectories for
achieving this kind of mission is to use the trajectory optimization
concepts. To solve a trajectory optimization problem we can use
the indirect or direct methods. The indirect methods are based on
maximum principle of Pontryagin, on the other hand, in the direct
methods it is necessary to transform into a nonlinear programming
problem. We propose an approach based on direct methods with a
pseudospectral integration scheme built on Chebyshev polynomials.
Optimal Control of a Linear Distributed Parameter System via Shifted Legendre Polynomials
The optimal control problem of a linear distributed
parameter system is studied via shifted Legendre polynomials (SLPs)
in this paper. The partial differential equation, representing the
linear distributed parameter system, is decomposed into an n - set
of ordinary differential equations, the optimal control problem is
transformed into a two-point boundary value problem, and the twopoint
boundary value problem is reduced to an initial value problem
by using SLPs. A recursive algorithm for evaluating optimal control
input and output trajectory is developed. The proposed algorithm is
computationally simple. An illustrative example is given to show the
simplicity of the proposed approach.
System Identification Based on Stepwise Regression for Dynamic Market Representation
A system for market identification (SMI) is presented.
The resulting representations are multivariable dynamic demand
models. The market specifics are analyzed. Appropriate models and
identification techniques are chosen. Multivariate static and dynamic
models are used to represent the market behavior. The steps of the
first stage of SMI, named data preprocessing, are mentioned. Next,
the second stage, which is the model estimation, is considered in more
details. Stepwise linear regression (SWR) is used to determine the
significant cross-effects and the orders of the model polynomials. The
estimates of the model parameters are obtained by a numerically stable
estimator. Real market data is used to analyze SMI performance.
The main conclusion is related to the applicability of multivariate
dynamic models for representation of market systems.
Fuzzy Fingerprint Vault using Multiple Polynomials
Fuzzy fingerprint vault is a recently developed cryptographic construct based on the polynomial reconstruction problem to secure critical data with the fingerprint data. However, the previous researches are not applicable to the fingerprint having a few minutiae since they use a fixed degree of the polynomial without considering the number of fingerprint minutiae. To solve this problem, we use an adaptive degree of the polynomial considering the number of minutiae extracted from each user. Also, we apply multiple polynomials to avoid the possible degradation of the security of a simple solution(i.e., using a low-degree polynomial). Based on the experimental results, our method can make the possible attack difficult 2192 times more than using a low-degree polynomial as well as verify the users having a few minutiae.
Secure Data Aggregation Using Clusters in Sensor Networks
Wireless sensor network can be applied to both abominable
and military environments. A primary goal in the design of
wireless sensor networks is lifetime maximization, constrained by
the energy capacity of batteries. One well-known method to reduce
energy consumption in such networks is data aggregation. Providing
efcient data aggregation while preserving data privacy is a challenging
problem in wireless sensor networks research. In this paper,
we present privacy-preserving data aggregation scheme for additive
aggregation functions. The Cluster-based Private Data Aggregation
(CPDA)leverages clustering protocol and algebraic properties of
polynomials. It has the advantage of incurring less communication
overhead. The goal of our work is to bridge the gap between
collaborative data collection by wireless sensor networks and data
privacy. We present simulation results of our schemes and compare
their performance to a typical data aggregation scheme TAG, where
no data privacy protection is provided. Results show the efficacy and
efficiency of our schemes.
On Generalized New Class of Matrix Polynomial Set
New generalization of the new class matrix polynomial set have been obtained. An explicit representation and an expansion of the matrix exponential in a series of these matrix are given for these matrix polynomials.
Holistic Face Recognition using Multivariate Approximation, Genetic Algorithms and AdaBoost Classifier: Preliminary Results
Several works regarding facial recognition have dealt with methods which identify isolated characteristics of the face or with templates which encompass several regions of it. In this paper a new technique which approaches the problem holistically dispensing with the need to identify geometrical characteristics or regions of the face is introduced. The characterization of a face is achieved by randomly sampling selected attributes of the pixels of its image. From this information we construct a set of data, which correspond to the values of low frequencies, gradient, entropy and another several characteristics of pixel of the image. Generating a set of “p" variables. The multivariate data set with different polynomials minimizing the data fitness error in the minimax sense (L∞ - Norm) is approximated. With the use of a Genetic Algorithm (GA) it is able to circumvent the problem of dimensionality inherent to higher degree polynomial approximations. The GA yields the degree and values of a set of coefficients of the polynomials approximating of the image of a face. By finding a family of characteristic polynomials from several variables (pixel characteristics) for each face (say Fi ) in the data base through a resampling process the system in use, is trained. A face (say F ) is recognized by finding its characteristic polynomials and using an AdaBoost Classifier from F -s polynomials to each of the Fi -s polynomials. The winner is the polynomial family closer to F -s corresponding to target face in data base.
An Approach to Control Design for Nonlinear Systems via Two-stage Formal Linearization and Two-type LQ Controls
In this paper we consider a nonlinear control design for
nonlinear systems by using two-stage formal linearization and twotype
LQ controls. The ordinary LQ control is designed on almost
linear region around the steady state point. On the other region,
another control is derived as follows. This derivation is based on
coordinate transformation twice with respect to linearization functions
which are defined by polynomials. The linearized systems can be
made up by using Taylor expansion considered up to the higher order.
To the resulting formal linear system, the LQ control theory is applied
to obtain another LQ control. Finally these two-type LQ controls
are smoothly united to form a single nonlinear control. Numerical
experiments indicate that this control show remarkable performances
for a nonlinear system.
GridNtru: High Performance PKCS
Cryptographic algorithms play a crucial role in the
information society by providing protection from unauthorized
access to sensitive data. It is clear that information technology will
become increasingly pervasive, Hence we can expect the emergence
of ubiquitous or pervasive computing, ambient intelligence. These
new environments and applications will present new security
challenges, and there is no doubt that cryptographic algorithms and
protocols will form a part of the solution. The efficiency of a public
key cryptosystem is mainly measured in computational overheads,
key size and bandwidth. In particular the RSA algorithm is used in
many applications for providing the security. Although the security
of RSA is beyond doubt, the evolution in computing power has
caused a growth in the necessary key length. The fact that most chips
on smart cards can-t process key extending 1024 bit shows that there
is need for alternative. NTRU is such an alternative and it is a
collection of mathematical algorithm based on manipulating lists of
very small integers and polynomials. This allows NTRU to high
speeds with the use of minimal computing power. NTRU (Nth degree
Truncated Polynomial Ring Unit) is the first secure public key
cryptosystem not based on factorization or discrete logarithm
problem. This means that given sufficient computational resources
and time, an adversary, should not be able to break the key. The
multi-party communication and requirement of optimal resource
utilization necessitated the need for the present day demand of
applications that need security enforcement technique .and can be
enhanced with high-end computing. This has promoted us to develop
high-performance NTRU schemes using approaches such as the use
of high-end computing hardware. Peer-to-peer (P2P) or enterprise
grids are proven as one of the approaches for developing high-end
computing systems. By utilizing them one can improve the
performance of NTRU through parallel execution. In this paper we
propose and develop an application for NTRU using enterprise grid
middleware called Alchemi. An analysis and comparison of its
performance for various text files is presented.
An Approach to Polynomial Curve Comparison in Geometric Object Database
In image processing and visualization, comparing two
bitmapped images needs to be compared from their pixels by matching
pixel-by-pixel. Consequently, it takes a lot of computational time
while the comparison of two vector-based images is significantly
faster. Sometimes these raster graphics images can be approximately
converted into the vector-based images by various techniques. After
conversion, the problem of comparing two raster graphics images
can be reduced to the problem of comparing vector graphics images.
Hence, the problem of comparing pixel-by-pixel can be reduced to
the problem of polynomial comparisons. In computer aided geometric
design (CAGD), the vector graphics images are the composition of
curves and surfaces. Curves are defined by a sequence of control
points and their polynomials. In this paper, the control points will be
considerably used to compare curves. The same curves after relocated
or rotated are treated to be equivalent while two curves after different
scaled are considered to be similar curves. This paper proposed an
algorithm for comparing the polynomial curves by using the control
points for equivalence and similarity. In addition, the geometric
object-oriented database used to keep the curve information has also
been defined in XML format for further used in curve comparisons.
On Bounds For The Zeros of Univariate Polynomial
Problems on algebraical polynomials appear in many fields of mathematics and computer science. Especially the task of determining the roots of polynomials has been frequently investigated.Nonetheless, the task of locating the zeros of complex polynomials is still challenging. In this paper we deal with the location of zeros of univariate complex polynomials. We prove some novel upper bounds for the moduli of the zeros of complex polynomials. That means, we provide disks in the complex plane where all zeros of a complex polynomial are situated. Such bounds are extremely useful for obtaining a priori assertations regarding the location of zeros of polynomials. Based on the proven bounds and a test set of polynomials, we present an experimental study to examine which bound is optimal.