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
A Modified Decoupled Semi-Analytical Approach Based On SBFEM for Solving 2D Elastodynamic Problems
In this paper, a new trend for improvement in semianalytical
method based on scale boundaries in order to solve the 2D
elastodynamic problems is provided. In this regard, only the
boundaries of the problem domain discretization are by specific subparametric
elements. Mapping functions are uses as a class of higherorder
Lagrange polynomials, special shape functions, Gauss-Lobatto-
Legendre numerical integration, and the integral form of the weighted
residual method, the matrix is diagonal coefficients in the equations
of elastodynamic issues. Differences between study conducted and
prior research in this paper is in geometry production procedure of
the interpolation function and integration of the different is selected.
Validity and accuracy of the present method are fully demonstrated
through two benchmark problems which are successfully modeled
using a few numbers of DOFs. The numerical results agree very well
with the analytical solutions and the results from other 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.
Edge Detection in Low Contrast Images
The edges of low contrast images are not clearly
distinguishable to human eye. It is difficult to find the edges and
boundaries in it. The present work encompasses a new approach for
low contrast images. The Chebyshev polynomial based fractional
order filter has been used for filtering operation on an image. The
preprocessing has been performed by this filter on the input image.
Laplacian of Gaussian method has been applied on preprocessed
image for edge detection. The algorithm has been tested on two test
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.
Bernstein-Galerkin Approach for Perturbed Constant-Coefficient Differential Equations, One-Dimensional Analysis
A numerical approach for solving constant-coefficient differential equations whose solutions exhibit boundary layer structure is built by inserting Bernstein Partition of Unity into Galerkin variational weak form. Due to the reproduction capability of Bernstein basis, such implementation shows excellent accuracy at boundaries and is able to capture sharp gradients of the field variable by p-refinement using regular distributions of equi-spaced evaluation points. The approximation is subjected to convergence experimentation and a procedure to assemble the discrete equations without a background integration mesh is proposed.
Numerical Solution of Volterra Integro-differential Equations of Fractional Order by Laplace Decomposition Method
In this paper the Laplace Decomposition method is developed to solve linear and nonlinear fractional integro- differential equations of Volterra type.The fractional derivative is described in the Caputo sense.The Laplace decomposition method is found to be fast and accurate.Illustrative examples are included to demonstrate the validity and applicability of presented technique and comparasion is made with exacting results.
A Study of Thermal Convection in Two Porous Layers Governed by Brinkman's Model in Upper Layer and Darcy's Model in Lower Layer
This work examines thermal convection in two porous
layers. Flow in the upper layer is governed by Brinkman-s equations
model and in the lower layer is governed by Darcy-s model.
Legendre polynomials are used to obtain numerical solution when the
lower layer is heated from below.
Laplace Adomian Decomposition Method Applied to a Two-Dimensional Viscous Flow with Shrinking Sheet
Our aim in this piece of work is to demonstrate the
power of the Laplace Adomian decomposition method (LADM) in
approximating the solutions of nonlinear differential equations
governing the two-dimensional viscous flow induced by a shrinking
Hybrid Function Method for Solving Nonlinear Fredholm Integral Equations of the Second Kind
A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm type equations which have many applications in mathematical physics are then considered. The method is based on hybrid function approximations. The properties of hybrid of block-pulse functions and Chebyshev polynomials are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.
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.
System Overflow/Blocking Transients For Queues with Batch Arrivals Using a Family of Polynomials Resembling Chebyshev Polynomials
The paper shows that in the analysis of a queuing system with fixed-size batch arrivals, there emerges a set of polynomials which are a generalization of Chebyshev polynomials of the second kind. The paper uses these polynomials in assessing the transient behaviour of the overflow (equivalently call blocking) probability in the system. A key figure to note is the proportion of the overflow (or blocking) probability resident in the transient component, which is shown in the results to be more significant at the beginning of the transient and naturally decays to zero in the limit of large t. The results also show that the significance of transients is more pronounced in cases of lighter loads, but lasts longer for heavier loads.
Near-Lossless Image Coding based on Orthogonal Polynomials
In this paper, a near lossless image coding scheme
based on Orthogonal Polynomials Transform (OPT) has been
presented. The polynomial operators and polynomials basis operators
are obtained from set of orthogonal polynomials functions for the
proposed transform coding. The image is partitioned into a number of
distinct square blocks and the proposed transform coding is applied to
each of these individually. After applying the proposed transform
coding, the transformed coefficients are rearranged into a sub-band
structure. The Embedded Zerotree (EZ) coding algorithm is then
employed to quantize the coefficients. The proposed transform is
implemented for various block sizes and the performance is
compared with existing Discrete Cosine Transform (DCT) transform
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.
Discrete Polynomial Moments and Savitzky-Golay Smoothing
This paper presents unified theory for local (Savitzky-
Golay) and global polynomial smoothing. The algebraic framework
can represent any polynomial approximation and is seamless from
low degree local, to high degree global approximations. The representation
of the smoothing operator as a projection onto orthonormal
basis functions enables the computation of: the covariance matrix
for noise propagation through the filter; the noise gain and; the
frequency response of the polynomial filters. A virtually perfect Gram
polynomial basis is synthesized, whereby polynomials of degree
d = 1000 can be synthesized without significant errors. The perfect
basis ensures that the filters are strictly polynomial preserving. Given
n points and a support length ls = 2m + 1 then the smoothing
operator is strictly linear phase for the points xi, i = m+1. . . n-m.
The method is demonstrated on geometric surfaces data lying on an
invariant 2D lattice.
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.
An Adaptive Mammographic Image Enhancement in Orthogonal Polynomials Domain
X-ray mammography is the most effective method for
the early detection of breast diseases. However, the typical diagnostic
signs such as microcalcifications and masses are difficult to detect
because mammograms are of low-contrast and noisy. In this paper, a
new algorithm for image denoising and enhancement in Orthogonal
Polynomials Transformation (OPT) is proposed for radiologists to
screen mammograms. In this method, a set of OPT edge coefficients
are scaled to a new set by a scale factor called OPT scale factor. The
new set of coefficients is then inverse transformed resulting in
contrast improved image. Applications of the proposed method to
mammograms with subtle lesions are shown. To validate the
effectiveness of the proposed method, we compare the results to
those obtained by the Histogram Equalization (HE) and the Unsharp
Masking (UM) methods. Our preliminary results strongly suggest
that the proposed method offers considerably improved enhancement
capability over the HE and UM methods.
Robust Control Synthesis for an Unmanned Underwater Vehicle
The control design for unmanned underwater vehicles (UUVs) is challenging due to the uncertainties in the complex dynamic modeling of the vehicle as well as its unstructured operational environment. To cope with these difficulties, a practical robust control is therefore desirable. The paper deals with the application of coefficient diagram method (CDM) for a robust control design of an autonomous underwater vehicle. The CDM is an algebraic approach in which the characteristic polynomial and the controller are synthesized simultaneously. Particularly, a coefficient diagram (comparable to Bode diagram) is used effectively to convey pertinent design information and as a measure of trade-off between stability, response speed and robustness. In the polynomial ring, Kharitonov polynomials are employed to analyze the robustness of the controller due to parametric uncertainties.
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.
Codebook Generation for Vector Quantization on Orthogonal Polynomials based Transform Coding
In this paper, a new algorithm for generating codebook is proposed for vector quantization (VQ) in image coding. The significant features of the training image vectors are extracted by using the proposed Orthogonal Polynomials based transformation. We propose to generate the codebook by partitioning these feature vectors into a binary tree. Each feature vector at a non-terminal node of the binary tree is directed to one of the two descendants by comparing a single feature associated with that node to a threshold. The binary tree codebook is used for encoding and decoding the feature vectors. In the decoding process the feature vectors are subjected to inverse transformation with the help of basis functions of the proposed Orthogonal Polynomials based transformation to get back the approximated input image training vectors. The results of the proposed coding are compared with the VQ using Discrete Cosine Transform (DCT) and Pairwise Nearest Neighbor (PNN) algorithm. The new algorithm results in a considerable reduction in computation time and provides better reconstructed picture quality.
Image Adaptive Watermarking with Visual Model in Orthogonal Polynomials based Transformation Domain
In this paper, an image adaptive, invisible digital
watermarking algorithm with Orthogonal Polynomials based
Transformation (OPT) is proposed, for copyright protection of digital
images. The proposed algorithm utilizes a visual model to determine
the watermarking strength necessary to invisibly embed the
watermark in the mid frequency AC coefficients of the cover image,
chosen with a secret key. The visual model is designed to generate a
Just Noticeable Distortion mask (JND) by analyzing the low level
image characteristics such as textures, edges and luminance of the
cover image in the orthogonal polynomials based transformation
domain. Since the secret key is required for both embedding and
extraction of watermark, it is not possible for an unauthorized user to
extract the embedded watermark. The proposed scheme is robust to
common image processing distortions like filtering, JPEG
compression and additive noise. Experimental results show that the
quality of OPT domain watermarked images is better than its DCT
A Note on the Numerical Solution of Singular Integral Equations of Cauchy Type
This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebyshev polynomials of the first kind. It is shown that the numerical solution of characteristic singular integral equation is identical with the exact solution, when the force function is a cubic function. Moreover, it also shown that this numerical method gives exact solution for other singular integral equations with degenerate kernels.
Numerical Solution of Linear Ordinary Differential Equations in Quantum Chemistry by Clenshaw Method
As we know, most differential equations concerning
physical phenomenon could not be solved by analytical method. Even if we use Series Method, some times we need an appropriate change of variable, and even when we can, their closed form solution may be
so complicated that using it to obtain an image or to examine the structure of the system is impossible. For example, if we consider Schrodinger equation, i.e.,
We come to a three-term recursion relations, which work with it takes, at least, a little bit time to get a series solution. For this
reason we use a change of variable such as or when we consider the orbital angular momentum, it will be
necessary to solve. As we can observe, working with this equation is tedious. In this paper, after introducing Clenshaw method, which is a kind of Spectral method, we try to solve some of such equations.
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.