The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor
and immune cells. The model consists of differential equations with piecewise constant arguments and
based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is
obtained a system of difference equations from the system of differential equations. In order to get local
and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion
and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a
consequence of Neimark-Sacker bifurcation.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
NONSTATIONARY RELAXED MULTISPLITTING METHODS FOR SOLVING LINEAR COMPLEMENTARI...ijcsa
In this paper we consider some non stationary relaxed synchronous and asynchronous
multisplitting methods for solving the linear complementarity problems with their coefficient
matrices being H−matrices. The convergence theorems of the methods are given,and the efficiency
is shown by numerical tests.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
Optimal Finite-Difference Grids for Elliptic Problem
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be and important problem. We develop an approach to grid optimization for finite-difference scheme for elliptic problem. Using this approach we are able to achieve exponential convergence of the boundary Neumann-to-Dirichlet map when applied to the bounded domains. It increases the convergence order without increasing the stencil size of the finite-difference scheme and without losing stability.
Stereographic Circular Normal Moment Distributionmathsjournal
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
Optimal Finite-Difference Grids for Elliptic Problem
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be and important problem. We develop an approach to grid optimization for finite-difference scheme for elliptic problem. Using this approach we are able to achieve exponential convergence of the boundary Neumann-to-Dirichlet map when applied to the bounded domains. It increases the convergence order without increasing the stencil size of the finite-difference scheme and without losing stability.
Stereographic Circular Normal Moment Distributionmathsjournal
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
Our efforts are mostly concentrated on improving the convergence rate of the numerical procedures both from the viewpoint of cost-efficiency and accuracy by handling the parametrization of the shape to be optimized. We employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The algorithms mimick classical multigrid strategies and are found very effective in terms of convergence acceleration. In this paper, we analyse and demonstrate the efficiency of the two-level correction algorithm which is the basic block of a more general miltilevel strategy.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC PANG AND HYPERCHAOTIC WANG-CHEN SYSTEMS ...ijctcm
Hyperchaotic systems are chaotic systems having more than one positive Lyapunov exponent and they have
important applications in secure data transmission and communication. This paper applies active control
method for the synchronization of identical and different hyperchaotic Pang systems (2011) and
hyperchaotic Wang-Chen systems (2008). Main results are proved with the stability theorems of Lypuanov
stability theory and numerical simulations are plotted using MATLAB to show the synchronization of
hyperchaotic systems addressed in this paper.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
https://utilitasmathematica.com/index.php/Index
Our journal has Ensuring that the statistics profession reflects the diversity of society itself is essential. Representation of underrepresented groups, including women and minorities, not only provides role models but also broadens the pool of talent and expertise in the field. Utilitas Mathematica Journal recognizes that achieving justice, equity, diversity, and inclusion is an ongoing journey that requires continuous dedication.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵mathsjournal
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
Modernization of radar technology and improved signal processing techniques are necessary to improve detection systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints
Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Qua...mathsjournal
Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by thegenetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.
An Application of Assignment Problem in Laptop Selection Problem Using MATLABmathsjournal
The assignment – selection problem used to find one-to- one match of given “Users” to “Laptops”, the main objective is to minimize the cost as per user requirement. This paper presents satisfactory solution for real assignment – Laptop selection problem using MATLAB coding.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
Cubic Response Surface Designs Using Bibd in Four Dimensionsmathsjournal
Response Surface Methodology (RSM) has applications in Chemical, Physical, Meteorological, Industrial and Biological fields. The estimation of slope response surface occurs frequently in practical situations for the experimenter. The rates of change of the response surface, like rates of change in the yield of crop to various fertilizers, to estimate the rates of change in chemical experiments etc. are of
interest. If the fit of second order response is inadequate for the design points, we continue the
experiment so as to fit a third order response surface. Higher order response surface designs are sometimes needed in Industrial and Meteorological applications. Gardiner et al (1959) introduced third order rotatable designs for exploring response surface. Anjaneyulu et al (1994-1995) constructed third order slope rotatable designs using doubly balanced incomplete block designs. Anjaneyulu et al (2001)
introduced third order slope rotatable designs using central composite type design points. Seshu babu et al (2011) studied modified construction of third order slope rotatable designs using central composite
designs. Seshu babu et al (2014) constructed TOSRD using BIBD. In view of wide applicability of third
order models in RSM and importance of slope rotatability, we introduce A Cubic Slope Rotatable Designs Using BIBD in four dimensions.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Table of Contents - September 2022, Volume 9, Number 2/3mathsjournal
Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
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DISCRETIZATION OF A MATHEMATICAL MODEL FOR TUMOR-IMMUNE SYSTEM INTERACTION WITH PIECEWISE CONSTANT ARGUMENTS
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
57
DISCRETIZATION OF A MATHEMATICAL MODEL FOR
TUMOR-IMMUNE SYSTEM INTERACTION WITH
PIECEWISE CONSTANT ARGUMENTS
Senol Kartal1
and Fuat Gurcan2
1
Department of Mathematics, Nevsehir Hacı Bektas Veli University, Nevsehir, Turkey
2
Department of Mathematics, Erciyes University, Kayseri, Turkey
2
Faculty of Engineering and Natural Sciences, International University of Sarajevo,
Hrasnicka cesta 15, 71000, Sarejevo, BIH
ABSTRACT
The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor
and immune cells. The model consists of differential equations with piecewise constant arguments and
based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is
obtained a system of difference equations from the system of differential equations. In order to get local
and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion
and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a
consequence of Neimark-Sacker bifurcation.
KEYWORDS
piecewise constant arguments; difference equation; stability; bifurcation
1. INTRODUCTION
In population dynamics, the simplest and most widely used model describing the competition of
two species is of the Lotka-Volterra type. In addition, there exist numerous extensions and
generalizations of this type model in tumor growth model [1-8]. In 1995, Gatenby [1] used Lotka-
Volterra competition model describing competition between tumor cells and normal cells for
space and other resources in an arbitrarily small volume of tissue within an organ. On the other
hand, Onofrio [2] has presented a general class of Lotka-Volterra competition model as
follows:
x.
= x(f(x) − ϕ(x)y),
y.
= β(x)y − μ(x)y + σq(x) + θ(t).
(1)
Here x and y denote tumor cell and effector cell sizes respectively. The function f(x) represents
tumor growth rates and there are many versions of this term. For example, in Gompertz model:
f(x) = αLog(A/x) [3], the logistic model: f(x) = α(1 − x/A) [4].
The metamodel (1) also includes following exponential model which has been constructed by
Stepanova [6].
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
58
x.
= μ x(t) − γx(t)y(t),
y.
= μ (x(t) − βx(t) )y(t) − δy(t) + κ,
(2)
where x and y denote tumor and T-cell densities respectively. In this model, μ is the
multiplication rate of tumors, γ is the rate of elimination of cancer cells by activity of T-cells, μ
represents the production of T-cells which are stimulated by tumor cells, β denotes the
saturation density up from which the immunological system is suppressed, δ is the natural death
rate of T cell and κ is the natural rate of influx of T cells from the primary organs [3].
Recently, it has been observed that the differential equations with piecewise constant arguments
play an important role in modeling of biological problems. By using a first-order linear
differential equation with piecewise constant arguments, Busenberg and Cooke [9] presented a
model to investigate vertically transmitted. Following this work, using the method of reduction to
discrete equations, many authors have analyzed various types of differential equations with
piecewise constant arguments [10-19]. The local and global behavior of differential equation
dx(t)
dt
= rx(t){1 − αx(t) − β x([t]) − β x([t − 1])} (3)
has been analyzed by Gurcan and Bozkurt [10]. Using the equation (3), Ozturk et al [11] have
modeled a population density of a bacteria species in a microcosm. Stability and oscillatory
characteristics of difference solutions of the equation
dx(t)
dt
= x(t) r 1 − αx(t) − β x([t]) − β x([t − 1]) + γ x([t]) + γ x([t − 1]) (4)
has been investigated in [12]. This equation has also been used for modeling an early brain tumor
growth by Bozkurt [13].
In the present paper, we have modified model (2) by adding piecewise constant arguments such
as
x.
= μ x(t) − γx(t)y([t]),
y.
= μ (x([t]) − βx([t]) )y(t) − δy(t) + κ,
(5)
where [t] denotes the integer part of t ϵ [0, ∞) and all these parameters are positive.
2. STABILITY ANALYSIS
In this section, we investigate local and global stability behavior of the system (5). The system
can be written in the interval t ϵ [n, n + 1) as
⎩
⎨
⎧
dx
x(t)
= μ − γy(n) d(t),
dy
dt
+ βμ x(n) + δ − μ x(n) y(t) = κ.
(6)
Integrating each equations of system (6) with respect to t on [n, t) and letting t → n + 1, one can
obtain a system of difference equations
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
59
x(n + 1) = x(n)eμ γ ( )
,
y(n + 1) =
e μ ( ) βμ ( ) δ
βμ x(n) y(n) + δy(n) − μ x(n)y(n) − κ + κ
βμ x(n) + δ − μ x(n)
.
(7)
Computations give us that the positive equilibrium point of the system is
(x, y) =
⎝
⎜
⎜
⎜
⎛1 −
4βγκ + −4βδ + μ μ
μ μ
2β
,
μ
γ
⎠
⎟
⎟
⎟
⎞
.
Hereafter,
γ <
δμ
κ
and β ≤
μ μ
−4γκ + 4δμ
. (8)
The linearized system of (7) about the positive equilibrium point is w(n + 1) = Aw(n), where A
is a matrix as;
A =
⎝
⎜
⎜
⎜
⎛
1 −
γ(1 −
4βγκ + (−4βδ + μ )μ
μ μ
)
2β
e
γκ
μ
(−1 + e
γκ
μ
) μ μ
/
4βγκ + (−4βδ + μ )μ
γ κ
e
γκ
μ
⎠
⎟
⎟
⎟
⎞
. (9)
The characteristic equation of the matrix A is
p(λ) = λ + λ −1 − e
γκ
μ
+ e
γκ
μ
−
e
γκ
μ
(−1 + e
γκ
μ
)μ 4βγκ + (−4βδ + μ )μ (− μ μ + 4βγκ + (−4βδ + μ )μ )
2βγκ
. (10)
Now we can determine the stability conditions of system (7) with the characteristic equation (10).
Hence, we use following theorem that is called Schur-Chon criterion.
Theorem A ([20]). The characteristic polynomial
p(λ) = λ + p λ + p (11)
has all its roots inside the unit open disk (|λ| < 1) if and only if
(a) p(1) = 1 + p + p > 0,
(b) p(−1) = 1 − p + p > 0,
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
60
(c) D = 1 + p > 0,
(d) D = 1 − p > 0.
Theorem 1. The positive equilibrium point (x, y) of system (7) is local asymptotically stable if
δμ
κ + κμ
< γ <
δμ
κ
and β ≤
μ μ
−4γκ + 4δμ
.
Proof. From characteristic equations (10), we have
p = −1 − e
γκ
μ
,
p = e
γκ
μ
−
e
γκ
μ
(−1 + e
γκ
μ
)μ 4βγκ + (−4βδ + μ )μ (− μ μ + 4βγκ + (−4βδ + μ )μ )
2βγκ
.
From Theorem A/a we get
p(1) =
2βγκ − (−1 + e
γκ
μ
)μ 4βγκ + (−4βδ + μ )μ (− μ μ + 4βγκ + (−4βδ + μ )μ )
2βγκ
.
It can be shown that if
− μ μ + 4βγκ + (−4βδ + μ )μ < 0, (12)
then p(1) > 0. On the other hand, the inequality (12) always holds under the condition (8). When
we consider Theorem A/b and Theorem A/c with the fact (12), we have respectively
p(−1) = 2 + 2e
γκ
μ
−
e
γκ
μ
(−1 + e
γκ
μ
)μ 4βγκ + (−4βδ + μ )μ (− μ μ + 4βγκ + (−4βδ + μ )μ )
2βγκ
> 0
And
D = 1 + e
γκ
μ
−
e
γκ
μ
(−1 + e
γκ
μ
)μ 4βγκ + (−4βδ + μ )μ (− μ μ + 4βγκ + (−4βδ + μ )μ )
2βγκ
> 0.
From Theorem A/d, we get
D = e
γκ
μ
(−1 + e
γκ
μ
)(2βγκ + 4βγκμ + (−4βδ + μ )μ − μ μ 4βγκ + −4βδ + μ μ ).
By using the conditions of Theorem 1, we can also see that D > 0. This completes the proof.
Now we can use parameters value in Table 1 for the testing the conditions of Theorem 1. Using
these parameter values, it is observed that the positive equilibrium
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
61
point (x, y) = (7.41019,0.5599) is local asymptotically stable where blue and red graphs
represent x(n) and y(n) population densities respectively (see Figure 1).
Table 1. Parameters values used for numerical analysis
Parameters Numerical Values Ref
μ tumor growth parameter 0.5549 [8]
γ interaction rate 1 [8]
μ tumor stimulated proliferation rate 0.00484 [8]
β inverse threshold for tumor suppression 0.00264 [8]
δ death rate 0.37451 [8]
κ rate of influx 0.19
Figure 1. Graph of the iteration solution of x(n) and y(n), where x(1) = y(1) = 1
Theorem 2. Let {x(n), y(n)}∞
be a positive solution of the system. Suppose that
μ − γy(n) < 0, βx(n) − 1 > 0 and βμ x(n) y(n) + δy(n) − μ x(n)y(n) − κ < 0 for n =
0,1,2,3 …. Then every solution of (7) is bounded, that is,
x(n) ∈ (0, x(0)) and y(n) ∈ 0,
κ
δ
.
Proof. Since {x(n), y(n)}∞
> 0 and μ − γy(n) < 0, we have
x(n + 1) = x(n)eμ γ ( )
< x(n).
In addition, if we use βμ x(n) y(n) + δy(n) − μ x(n)y(n) − κ < 0 and βx(n) − 1 > 0, we have
y(n + 1) =
e μ ( ) βμ ( ) δ
y(n)(βμ x(n) + δ − μ x(n)) − κ + κ
μ x(n)(βx(n) − 1) + δ
0 50 100 150 200 250 300 350 400 450 500
0
1
2
3
4
5
6
7
n
x(n)
and
y(n)
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
62
<
κ
μ x(n)(βx(n) − 1) + δ
<
κ
δ
.
This completes the proof.
Theorem 3. Let the conditions of Theorem 1 hold and assume that
x <
1
2β
and y <
κ
2μ x(n)(βx(n) − 1) + 2δ
.
If
x(n) >
1
β
and y(n) >
κ
μ x(n)(βx(n) − 1) + δ
,
then the positive equilibrium point of the system is global asymptotically stable.
Proof. Let E = (x , y) is a positive equilibrium point of system (7) and we consider a Lyapunov
function V(n) defined by
V(n) = [E(n) − E] , n = 0,1,2 …
The change along the solutions of the system is
∆V(n) = V(n + 1) − V(n) = {E(n + 1) − E(n)}{E(n + 1) + E(n) − 2E}.
Let A = μ − γy(n) < 0 which gives us that y(n) >
μ
γ
= y. If we consider first equation in (7)
with the fact x(n) > 2x , we get
∆V (n) = {x(n + 1) − x(n)}{x(n + 1) + x(n) − 2x}
= x(n) e − 1 {x(n)e + x(n) − 2x} < 0.
Similarly, Suppose that A = βμ x(n) + δ − μ x(n) > 0 which yields x(n) >
β
. Computations
give us that if y(n) >
κ
and y(n) > 2 , we have
∆V (n) = {y(n + 1) − y(n)}{y(n + 1) + y(n) − 2y}
=
1 − e (κ − y(n)A )
A
y(n)A e + 1 + κ 1 − e − 2yA
A
< 0.
Under the conditions
x <
1
2β
and y <
κ
2μ x(n)(βx(n) − 1) + 2δ
,
we can write
x(n) >
1
β
> 2x and y(n) >
κ
A
=
κ
μ x(n)(βx(n) − 1) + δ
> 2y.
7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
63
As a result, we obtain ∆V(n) = (∆V (n), ∆V (n)) < 0.
3. NEIMARK-SACKER BIFURCATION ANALYSIS
In this section, we discuss the periodic solutions of the system through Neimark-Sacker
bifurcation. This bifurcation occurs of a closed invariant curve from a equilibrium point in
discrete dynamical systems, when the equilibrium point changes stability via a pair of complex
eigenvalues with unit modulus. These complex eigenvalues lead to periodic solution as a result
of limit cycle. In order to study Neimark-Sacker bifurcation we use the following theorem that is
called Schur-Cohn criterion.
Theorem B. ([20]) A pair of complex conjugate roots of equation (11) lie on the unit circle and
the other roots of equation (11) all lie inside the unit circle if and only if
(a) p(1) = 1 + p + p > 0,
(b) p(−1) = 1 − p + p > 0,
(c) D = 1 + p > 0,
(d) D = 1 − p = 0.
In stability analysis, we have shown that Theorem B/a, Theorem B/b and Theorem B/c always
holds. Therefore, to determine bifurcation point we can only analyze Theorem B/d. Solving
equation d of Theorem B, we have κ = 0.0635352. Furthermore, Figure 2 shows that κ is the
Neimark-Sacker bifurcation point of the system with eigenvalues
λ , = |0.945907 ± 0.324439i| = 1, where blue, and red graphs represent x(n) and
y(n) population densities respectively.
As seen in Figure 2, a stable limit cycle occurs around the positive equilibrium point at the
Neimark-Sacker bifurcation point. This limit cycle leads to periodic solution which means that
tumor and immune cell undergo oscillations (Figure 3). This oscillatory behavior has also
occurred in continuous biological model as a result of Hopf bifurcation and has observed
clinically.
Figure 2. Graph of Neimark-Sacker bifurcation of system (7) for κ = 0.0635352. Initial conditions and
other parameters are the same as Figure 1
0 20 40 60 80 100 120 140 160 180 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
x(n)
y(n)
8. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
64
Figure 3. Graph of the iteration solution of x(n) and y(n) for κ = 0.063535. Initial conditions and other
parameters are the same as Figure 1
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Authors
Senol Kartal is research assistant in Nevsehir Haci Bektas Veli University in Turkey. He is a
Phd student Department of Mathematics, University of Erciyes. His research interests
include issues related to dynamical systems in biology.
Fuat Gurcan received her PhD in Accounting at the University of Leeds, UK. He is a
Lecturer at the Department of Mathematics, University of Erciyes and International
University of Sarajevo. Her research interests are related to Bifurcation Theory, Fluid
Dynamics, Mathematical Biology, Computational Fluid Dynamics, Difference Equations and
Their Bifurcations. He has published research papers at national and international journals.