Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document compares JPEG and JPEG2000 image compression techniques using objective and perceptual quality measures. JPEG2000 provides higher PSNR values at all bitrates but JPEG has better picture quality scale (PQS) scores, a perceptual measure, at moderate and high bitrates. At very low bitrates below 0.5 bpp, JPEG2000 produces higher quality images according to PQS due to its wavelet-based compression method. The study uses four test images with different spatial and frequency characteristics to evaluate the compression methods.
The document presents a new block cipher that blends concepts from the modified Feistel cipher and advanced Hill cipher. The cipher uses an involutory key matrix K to encrypt plaintext matrices P and Q through iterative applications of mixing, permutation, and XOR operations per equations 1.1 and 1.2. Cryptanalysis shows the cipher is strong as the encryption equations are nonlinear and functions like Shift() and Mix() cause diffusion in each round. The encryption and decryption processes are illustrated through flowcharts and algorithms.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Neural network precept diagnosis on petrochemical pipelines for quality maint...Alexander Decker
This document describes a proposed neural network model for predicting degradation in petrochemical pipelines. It begins with background on pipelines and fatigue crack propagation based on Paris' law. It then discusses stresses in cylindrical pipelines under internal pressure. The model represents crack growth as a function of stress intensity factor and uses a recurrent formula to calculate cumulative damage over time. The goal is to develop a prognostic tool for quality maintenance in pipeline systems.
The document outlines an approach to summarize stability margins for multivariable feedback systems. It begins by introducing the problem of defining meaningful stability margins for multivariable systems. Next, it proposes using a PID controller of the form K(s) = K1 + K2/s + K3s with scalar values for each term. The problem is then defined as finding the ranges of these scalar values that ensure closed-loop stability. Finally, it proposes definitions for common and individual loop gain margins based on the stabilizing ranges of the scalar values. The approach aims to generalize stability margin concepts from single-input single-output systems to multivariable systems.
New multi step runge kutta method for solving fuzzy differential equationsAlexander Decker
This document presents a new multi-step Runge-Kutta method of order two for solving fuzzy differential equations. The method uses harmonic mean of the parameters in the main formula to increase the accuracy of the solution compared to existing Runge-Kutta methods. The paper defines fuzzy numbers and fuzzy derivatives. It presents the fuzzy Cauchy problem and its unique solution. The new second order Runge-Kutta method with harmonic mean is derived. Numerical examples are provided to illustrate the accuracy and efficiency of the proposed method compared to other methods.
This document compares JPEG and JPEG2000 image compression techniques using objective and perceptual quality measures. JPEG2000 provides higher PSNR values at all bitrates but JPEG has better picture quality scale (PQS) scores, a perceptual measure, at moderate and high bitrates. At very low bitrates below 0.5 bpp, JPEG2000 produces higher quality images according to PQS due to its wavelet-based compression method. The study uses four test images with different spatial and frequency characteristics to evaluate the compression methods.
The document presents a new block cipher that blends concepts from the modified Feistel cipher and advanced Hill cipher. The cipher uses an involutory key matrix K to encrypt plaintext matrices P and Q through iterative applications of mixing, permutation, and XOR operations per equations 1.1 and 1.2. Cryptanalysis shows the cipher is strong as the encryption equations are nonlinear and functions like Shift() and Mix() cause diffusion in each round. The encryption and decryption processes are illustrated through flowcharts and algorithms.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Neural network precept diagnosis on petrochemical pipelines for quality maint...Alexander Decker
This document describes a proposed neural network model for predicting degradation in petrochemical pipelines. It begins with background on pipelines and fatigue crack propagation based on Paris' law. It then discusses stresses in cylindrical pipelines under internal pressure. The model represents crack growth as a function of stress intensity factor and uses a recurrent formula to calculate cumulative damage over time. The goal is to develop a prognostic tool for quality maintenance in pipeline systems.
The document outlines an approach to summarize stability margins for multivariable feedback systems. It begins by introducing the problem of defining meaningful stability margins for multivariable systems. Next, it proposes using a PID controller of the form K(s) = K1 + K2/s + K3s with scalar values for each term. The problem is then defined as finding the ranges of these scalar values that ensure closed-loop stability. Finally, it proposes definitions for common and individual loop gain margins based on the stabilizing ranges of the scalar values. The approach aims to generalize stability margin concepts from single-input single-output systems to multivariable systems.
New multi step runge kutta method for solving fuzzy differential equationsAlexander Decker
This document presents a new multi-step Runge-Kutta method of order two for solving fuzzy differential equations. The method uses harmonic mean of the parameters in the main formula to increase the accuracy of the solution compared to existing Runge-Kutta methods. The paper defines fuzzy numbers and fuzzy derivatives. It presents the fuzzy Cauchy problem and its unique solution. The new second order Runge-Kutta method with harmonic mean is derived. Numerical examples are provided to illustrate the accuracy and efficiency of the proposed method compared to other methods.
11.new multi step runge kutta method for solving fuzzy differential equationsAlexander Decker
This document presents a new multi-step Runge-Kutta method of order two for solving fuzzy differential equations. The method uses harmonic mean of parameter values in the main formula to increase the accuracy of the solution compared to existing Runge-Kutta methods. The paper defines fuzzy numbers and fuzzy derivatives. It also establishes existence and uniqueness of solutions to fuzzy initial value problems. The new Runge-Kutta method is then derived and error analysis is provided to show it converges to the exact solution. An example is solved numerically to illustrate the method and compare its accuracy to other approaches.
Numerical Solutions of Stiff Initial Value Problems Using Modified Extended B...IOSR Journals
This document presents numerical solutions for stiff initial value problems using a modified extended backward differentiation formula (MEBDF). The MEBDF method is developed based on linear multi-step methods. Three stages of the two-step MEBDF are constructed and used to solve a sample stiff initial value problem. The numerical solutions from each stage are compared to the exact solution and each other to determine which stage provides the most accurate solutions.
IRJET- Analytic Evaluation of the Head Injury Criterion (HIC) within the Fram...IRJET Journal
This document presents an analytic evaluation of the Head Injury Criterion (HIC) within the framework of constrained optimization theory. The HIC is a weighted impulse function used to predict the probability of closed head injury based on measured head acceleration. Previous work analyzed the unclipped HIC function, but the clipped HIC formulation used in practice limits the evaluation window duration. The author develops analytic relationships for determining the window initiation and termination points to maximize the clipped HIC function. Example applications illustrate the general solutions for when head acceleration is defined by a single function or composite functions over the evaluation domain.
Solvability of Fractionl q -Difference Equations of Order 2 3 Involving ...journal ijrtem
This document summarizes a research paper that studies the existence of solutions for nonlinear fractional q-difference equations involving the p-Laplacian operator with boundary conditions where α is between 3/2 and 2. The paper represents the solution of the boundary value problem as an integral equation and proves, under certain conditions, that the nonlinear fractional boundary value problem has a unique solution using the Banach contraction mapping principle. It also provides relevant definitions and preliminary results from fractional q-calculus that are used in the analysis.
This document presents a new modified F-expansion method to obtain traveling wave solutions of the Benjamin-Bona-Mahony (BBM) equation and modified BBM equation. The method is applied to these nonlinear partial differential equations. Specifically:
1) The traveling wave solutions of the BBM equation are considered by substituting a transformation.
2) The solution is assumed to have the form of a polynomial in F(ξ) and its derivatives, where F(ξ) satisfies a Riccati equation.
3) Three explicit solutions for the BBM equation are obtained in terms of hyperbolic and trigonometric functions.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document outlines the key topics covered in Chapter 2 of the book "Linear Algebra" by Jin Ho Lee. The chapter introduces fundamental concepts in linear algebra including scalars, vectors, matrices, and tensors. It describes operations on these objects such as matrix multiplication and vector dot products. Important matrix properties and special types of matrices like identity, inverse, diagonal, and symmetric matrices are defined. Linear dependence, spans, and vector spaces are discussed. Various vector and matrix norms are also introduced.
The document discusses Hidden Markov Models (HMMs). It defines HMMs as a popular statistical tool that can model time series data through an underlying probabilistic process. HMMs have been successfully applied to natural language processing tasks like part-of-speech tagging. The document provides formal definitions of HMMs and describes algorithms like the forward algorithm that allow evaluating the probability of an observation sequence given an HMM model.
The document presents a three-level mathematical model of a well-stirred catalytic reactor with bidispersed catalysts. The model describes (1) the completely stirred fluid phase on the first level, (2) diffusion in the macropores on the second level, and (3) diffusion and reaction in the micropores on the third level. The model equations are solved using multilevel collocation to investigate transient phenomena in the reactor via computer simulation.
S-CUBE LP: The Chemical Computing model and HOCL Programmingvirtual-campus
This document provides an overview of the Chemical Computing model and the Higher Order Chemical Language (HOCL). It describes the vision of chemical computing using multiset rewriting to express inherently parallel problems. The Gamma language is presented as the first to capture chemical programming. The γ-calculus improved on Gamma by making it higher order and modeling reaction rules as active molecules. HOCL is then presented as a language based on γ-calculus, allowing active molecules to capture and produce other active molecules. Examples are given to demonstrate the chemical approach.
Motion Control and Dynamic Load Carrying Capacity of Mobile Robot via Nonline...IDES Editor
In this paper, two methods are presented for solving closed loop optimal control problem and finding dynamic load carrying capacity (DLCC) for fixed and mobile manipulators. These control laws are based on the numerical solution to nonlinear Hamilton-Jacobi Bellman (HJB) equation. First approach is the Successive Approximation (SA) for finding
solution of HJB equation in the closed loop form and second approach is based on solving state-dependent Riccati equation (SDRE) that is an extension of algebraic Riccati equation for nonlinear systems. Afterward dynamic load carrying capacity of manipulators is computed using these controllers. The DLCC is calculated by considering tracking error and limits of torque’s joints. Finally, results are presented for two cases, a two-link planar manipulator mounted on a differentially driven mobile base and a 6DOF articulated manipulator (6R). The simulation results are verified with the experimental test for the 6R manipulator. The simulation and experimental results demonstrate that these methods are convenient for finding nonlinear optimal control laws in state feedback form and finding the maximum allowable load on a given trajectory.
The document summarizes a lecture on packet routing algorithms for hypercubes, including analyzing the expected time for a random routing algorithm to route packets from source to destination in two phases. It then discusses primal-dual algorithms for solving multi-commodity flow problems on networks and how they maintain constraints for both the primal and dual optimization problems through an iterative process of adjusting primal and dual variables.
Unsupervised multispectral image Classification By fuzzy hidden Markov chains...CSCJournals
This paper deals with unsupervised classification of multi-spectral images, we propose to use a new vectorial fuzzy version of Hidden Markov Chains (HMC). The main characteristic of the proposed model is to allow the coexistence of crisp pixels (obtained with the uncertainty measure of the model) and fuzzy pixels (obtained with the fuzzy measure of the model) in the same image. Crisp and fuzzy multi-dimensional densities can then be estimated in the classification process, according to the assumption considered to model the statistical links between the layers of the multi-band image. The efficiency of the proposed method is illustrated with a Synthetic and real SPOTHRV images in the region of Rabat. The comparisons of two methods: fuzzy HMC and HMC are also provided. The classification results show the interest of the fuzzy HMC method.
Uncertainty Problem in Control & Decision TheorySSA KPI
AACIMP 2010 Summer School lecture by Viktor Ivanenko. "Applied Mathematics" stream. "On the Models of Uncertainty in Decision and Control Problems" course. Part 1.
More info at http://summerschool.ssa.org.ua
Gauge Invariance Of The Action Principle For Gauge Systems With Noncanonical ...guest9fa195
This document discusses gauge invariance of the action principle for gauge systems with noncanonical symplectic structures. It shows that for such systems, the complete set of commuting observables at the time boundary is now fixed by the boundary term and the symplectic structure, rather than just the canonical symplectic structure. The theory is applied to two nontrivial models with SL(2,R) and SU(2) gauge symmetries whose phase spaces have new interactions due to noncanonical symplectic structures.
This document discusses cluster analysis and its various techniques. It begins by defining cluster analysis and outlining the major categories of clustering methods, including partitioning, hierarchical, density-based, grid-based, and model-based methods. It then discusses the types of data that can be used for cluster analysis and how to measure similarity and dissimilarity between data objects. The document also covers considerations for different data types, such as how to handle binary, nominal, ordinal, and ratio-scaled variables. It concludes by discussing what constitutes good clustering and requirements for clustering in data mining.
The document provides instructions for an online aerospace engineering examination. It states that the exam has 65 multiple-choice questions worth a total of 100 marks. Questions are either worth 1 or 2 marks depending on the question number. There is no negative marking for numerical answer questions but negative marking for multiple choice questions. Calculators are allowed but no other materials. The exam is timed for 3 hours.
1) The document proposes a cardinality-constrained k-means clustering approach to address practical challenges with standard k-means, such as skewed clustering and sensitivity to outliers.
2) It formulates the problem as a mixed integer nonlinear program (MINLP) and provides a convex relaxation to the problem using semidefinite programming (SDP).
3) The approach provides optimality guarantees and a rounding algorithm to recover an integer feasible solution. Numerical experiments demonstrate competitive performance versus heuristics.
This document describes unbiased Markov chain Monte Carlo (MCMC) methods using coupled Markov chains. It begins by discussing how standard MCMC estimators are biased due to initialization and finite simulation length. It then introduces the idea of running two coupled Markov chains such that they meet and become equal after some meeting time τ. The difference in function values between the chains can then be used to construct an unbiased estimator. Several methods for designing coupled chains that meet this criterion are described, including couplings of popular MCMC algorithms like Metropolis-Hastings. Conditions under which the resulting estimators are guaranteed to be unbiased and have good statistical properties are also outlined.
This document proposes a theoretical framework for analyzing the probability of successful decoding in single-relay networks using network coding. It defines key terms like random linear network coding and presents two theorems:
1) The probability that two randomly generated coding matrices at a source and relay are simultaneously full rank is given by a formula involving the dimensions and number of common rows of the matrices.
2) The probability of successful decoding at two destinations in a network defined by certain parameters is calculated as the sum of probabilities involving the coding matrices and dimensions at each stage of transmission through the source, relay, and destinations.
Numerical results are presented to validate the theoretical analysis.
Solvability of Fractionl q -Difference Equations of Order 2 3 Involving ...IJRTEMJOURNAL
In this paper, we study the existence of solutions for non-linear fractional q-difference
equations of order
2 3
involving the p-Laplacian operator with various boundary value conditions. By
using the Banach contraction mapping principle, we prove that, under certain conditions, the suggested
non-linear fractional boundary value problem involving the p-Laplacian operator has a unique solution. Finally,
we illustrate our results with some examples.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
11.new multi step runge kutta method for solving fuzzy differential equationsAlexander Decker
This document presents a new multi-step Runge-Kutta method of order two for solving fuzzy differential equations. The method uses harmonic mean of parameter values in the main formula to increase the accuracy of the solution compared to existing Runge-Kutta methods. The paper defines fuzzy numbers and fuzzy derivatives. It also establishes existence and uniqueness of solutions to fuzzy initial value problems. The new Runge-Kutta method is then derived and error analysis is provided to show it converges to the exact solution. An example is solved numerically to illustrate the method and compare its accuracy to other approaches.
Numerical Solutions of Stiff Initial Value Problems Using Modified Extended B...IOSR Journals
This document presents numerical solutions for stiff initial value problems using a modified extended backward differentiation formula (MEBDF). The MEBDF method is developed based on linear multi-step methods. Three stages of the two-step MEBDF are constructed and used to solve a sample stiff initial value problem. The numerical solutions from each stage are compared to the exact solution and each other to determine which stage provides the most accurate solutions.
IRJET- Analytic Evaluation of the Head Injury Criterion (HIC) within the Fram...IRJET Journal
This document presents an analytic evaluation of the Head Injury Criterion (HIC) within the framework of constrained optimization theory. The HIC is a weighted impulse function used to predict the probability of closed head injury based on measured head acceleration. Previous work analyzed the unclipped HIC function, but the clipped HIC formulation used in practice limits the evaluation window duration. The author develops analytic relationships for determining the window initiation and termination points to maximize the clipped HIC function. Example applications illustrate the general solutions for when head acceleration is defined by a single function or composite functions over the evaluation domain.
Solvability of Fractionl q -Difference Equations of Order 2 3 Involving ...journal ijrtem
This document summarizes a research paper that studies the existence of solutions for nonlinear fractional q-difference equations involving the p-Laplacian operator with boundary conditions where α is between 3/2 and 2. The paper represents the solution of the boundary value problem as an integral equation and proves, under certain conditions, that the nonlinear fractional boundary value problem has a unique solution using the Banach contraction mapping principle. It also provides relevant definitions and preliminary results from fractional q-calculus that are used in the analysis.
This document presents a new modified F-expansion method to obtain traveling wave solutions of the Benjamin-Bona-Mahony (BBM) equation and modified BBM equation. The method is applied to these nonlinear partial differential equations. Specifically:
1) The traveling wave solutions of the BBM equation are considered by substituting a transformation.
2) The solution is assumed to have the form of a polynomial in F(ξ) and its derivatives, where F(ξ) satisfies a Riccati equation.
3) Three explicit solutions for the BBM equation are obtained in terms of hyperbolic and trigonometric functions.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document outlines the key topics covered in Chapter 2 of the book "Linear Algebra" by Jin Ho Lee. The chapter introduces fundamental concepts in linear algebra including scalars, vectors, matrices, and tensors. It describes operations on these objects such as matrix multiplication and vector dot products. Important matrix properties and special types of matrices like identity, inverse, diagonal, and symmetric matrices are defined. Linear dependence, spans, and vector spaces are discussed. Various vector and matrix norms are also introduced.
The document discusses Hidden Markov Models (HMMs). It defines HMMs as a popular statistical tool that can model time series data through an underlying probabilistic process. HMMs have been successfully applied to natural language processing tasks like part-of-speech tagging. The document provides formal definitions of HMMs and describes algorithms like the forward algorithm that allow evaluating the probability of an observation sequence given an HMM model.
The document presents a three-level mathematical model of a well-stirred catalytic reactor with bidispersed catalysts. The model describes (1) the completely stirred fluid phase on the first level, (2) diffusion in the macropores on the second level, and (3) diffusion and reaction in the micropores on the third level. The model equations are solved using multilevel collocation to investigate transient phenomena in the reactor via computer simulation.
S-CUBE LP: The Chemical Computing model and HOCL Programmingvirtual-campus
This document provides an overview of the Chemical Computing model and the Higher Order Chemical Language (HOCL). It describes the vision of chemical computing using multiset rewriting to express inherently parallel problems. The Gamma language is presented as the first to capture chemical programming. The γ-calculus improved on Gamma by making it higher order and modeling reaction rules as active molecules. HOCL is then presented as a language based on γ-calculus, allowing active molecules to capture and produce other active molecules. Examples are given to demonstrate the chemical approach.
Motion Control and Dynamic Load Carrying Capacity of Mobile Robot via Nonline...IDES Editor
In this paper, two methods are presented for solving closed loop optimal control problem and finding dynamic load carrying capacity (DLCC) for fixed and mobile manipulators. These control laws are based on the numerical solution to nonlinear Hamilton-Jacobi Bellman (HJB) equation. First approach is the Successive Approximation (SA) for finding
solution of HJB equation in the closed loop form and second approach is based on solving state-dependent Riccati equation (SDRE) that is an extension of algebraic Riccati equation for nonlinear systems. Afterward dynamic load carrying capacity of manipulators is computed using these controllers. The DLCC is calculated by considering tracking error and limits of torque’s joints. Finally, results are presented for two cases, a two-link planar manipulator mounted on a differentially driven mobile base and a 6DOF articulated manipulator (6R). The simulation results are verified with the experimental test for the 6R manipulator. The simulation and experimental results demonstrate that these methods are convenient for finding nonlinear optimal control laws in state feedback form and finding the maximum allowable load on a given trajectory.
The document summarizes a lecture on packet routing algorithms for hypercubes, including analyzing the expected time for a random routing algorithm to route packets from source to destination in two phases. It then discusses primal-dual algorithms for solving multi-commodity flow problems on networks and how they maintain constraints for both the primal and dual optimization problems through an iterative process of adjusting primal and dual variables.
Unsupervised multispectral image Classification By fuzzy hidden Markov chains...CSCJournals
This paper deals with unsupervised classification of multi-spectral images, we propose to use a new vectorial fuzzy version of Hidden Markov Chains (HMC). The main characteristic of the proposed model is to allow the coexistence of crisp pixels (obtained with the uncertainty measure of the model) and fuzzy pixels (obtained with the fuzzy measure of the model) in the same image. Crisp and fuzzy multi-dimensional densities can then be estimated in the classification process, according to the assumption considered to model the statistical links between the layers of the multi-band image. The efficiency of the proposed method is illustrated with a Synthetic and real SPOTHRV images in the region of Rabat. The comparisons of two methods: fuzzy HMC and HMC are also provided. The classification results show the interest of the fuzzy HMC method.
Uncertainty Problem in Control & Decision TheorySSA KPI
AACIMP 2010 Summer School lecture by Viktor Ivanenko. "Applied Mathematics" stream. "On the Models of Uncertainty in Decision and Control Problems" course. Part 1.
More info at http://summerschool.ssa.org.ua
Gauge Invariance Of The Action Principle For Gauge Systems With Noncanonical ...guest9fa195
This document discusses gauge invariance of the action principle for gauge systems with noncanonical symplectic structures. It shows that for such systems, the complete set of commuting observables at the time boundary is now fixed by the boundary term and the symplectic structure, rather than just the canonical symplectic structure. The theory is applied to two nontrivial models with SL(2,R) and SU(2) gauge symmetries whose phase spaces have new interactions due to noncanonical symplectic structures.
This document discusses cluster analysis and its various techniques. It begins by defining cluster analysis and outlining the major categories of clustering methods, including partitioning, hierarchical, density-based, grid-based, and model-based methods. It then discusses the types of data that can be used for cluster analysis and how to measure similarity and dissimilarity between data objects. The document also covers considerations for different data types, such as how to handle binary, nominal, ordinal, and ratio-scaled variables. It concludes by discussing what constitutes good clustering and requirements for clustering in data mining.
The document provides instructions for an online aerospace engineering examination. It states that the exam has 65 multiple-choice questions worth a total of 100 marks. Questions are either worth 1 or 2 marks depending on the question number. There is no negative marking for numerical answer questions but negative marking for multiple choice questions. Calculators are allowed but no other materials. The exam is timed for 3 hours.
1) The document proposes a cardinality-constrained k-means clustering approach to address practical challenges with standard k-means, such as skewed clustering and sensitivity to outliers.
2) It formulates the problem as a mixed integer nonlinear program (MINLP) and provides a convex relaxation to the problem using semidefinite programming (SDP).
3) The approach provides optimality guarantees and a rounding algorithm to recover an integer feasible solution. Numerical experiments demonstrate competitive performance versus heuristics.
This document describes unbiased Markov chain Monte Carlo (MCMC) methods using coupled Markov chains. It begins by discussing how standard MCMC estimators are biased due to initialization and finite simulation length. It then introduces the idea of running two coupled Markov chains such that they meet and become equal after some meeting time τ. The difference in function values between the chains can then be used to construct an unbiased estimator. Several methods for designing coupled chains that meet this criterion are described, including couplings of popular MCMC algorithms like Metropolis-Hastings. Conditions under which the resulting estimators are guaranteed to be unbiased and have good statistical properties are also outlined.
This document proposes a theoretical framework for analyzing the probability of successful decoding in single-relay networks using network coding. It defines key terms like random linear network coding and presents two theorems:
1) The probability that two randomly generated coding matrices at a source and relay are simultaneously full rank is given by a formula involving the dimensions and number of common rows of the matrices.
2) The probability of successful decoding at two destinations in a network defined by certain parameters is calculated as the sum of probabilities involving the coding matrices and dimensions at each stage of transmission through the source, relay, and destinations.
Numerical results are presented to validate the theoretical analysis.
Solvability of Fractionl q -Difference Equations of Order 2 3 Involving ...IJRTEMJOURNAL
In this paper, we study the existence of solutions for non-linear fractional q-difference
equations of order
2 3
involving the p-Laplacian operator with various boundary value conditions. By
using the Banach contraction mapping principle, we prove that, under certain conditions, the suggested
non-linear fractional boundary value problem involving the p-Laplacian operator has a unique solution. Finally,
we illustrate our results with some examples.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
Wang-Landau Monte Carlo simulation is a method for calculating the density of states function which can then be used to calculate thermodynamic properties like the mean value of variables. It improves on traditional Monte Carlo methods which struggle at low temperatures due to complicated energy landscapes with many local minima separated by large barriers. The Wang-Landau algorithm calculates the density of states function directly rather than relying on sampling configurations, allowing it to overcome barriers and fully explore the configuration space even at low temperatures.
The document provides information about product codes and their properties. It defines a product code C constructed from two component codes C1 and C2 by encoding rows of information bits with C1 and then encoding columns with C2. It proves several properties of product codes, including that C is a linear block code with parameters (n1n2, k1k2, d1d2) where ni, ki, di are the parameters of Ci. It also discusses encoding and decoding methods for product codes.
This document discusses the analysis and formation of an optimal questionnaire to monitor the connection between re-engineering economic parameters in small and medium enterprises. It presents an algorithm for transforming an initial questionnaire into an optimal questionnaire through arranging the questions and associated weights and values. It then applies this methodology to analyze the connection between economic parameters for a re-engineering model, presenting an initial questionnaire with groups of parameters, events, and associated weights and values.
On the Odd Gracefulness of Cyclic Snakes With Pendant EdgesGiselleginaGloria
Graceful and odd gracefulness of a graph are two entirely different concepts. A graph may posses one or both of these or neither. We present four new families of odd graceful graphs. In particular we show an odd graceful labeling of the linear 4 1 kC snake mK − e and therefore we introduce the odd graceful labeling of 4 1 kC snake mK − e ( for the general case ). We prove that the subdivision of linear 3 kC snake − is odd graceful. We also prove that the subdivision of linear 3 kC snake − with m-pendant edges is odd graceful. Finally, we present an odd graceful labeling of the crown graph P mK n 1 e .
Kekre’s hybrid wavelet transform technique with dct, walsh, hartley and kekre’sIAEME Publication
The document discusses Kekre's hybrid wavelet transform technique for image fusion. It begins by introducing Kekre's transform matrix and how hybrid wavelet matrices can be generated by combining two orthogonal transform matrices like DCT, Walsh, Hartley, and Kekre's. The proposed method applies different hybrid transforms to input images, averages the components, and performs inverse transforms on the fused image. Results on color and gray images show that different hybrid techniques produce varied performance based on measures like entropy, mean, standard deviation, and mutual information. The Kekre Hartley hybrid performed best for most measures and datasets.
A New Hybrid Inversion Method For 2D Nuclear Magnetic Resonance Combining TSV...Pedro Craggett
This paper presents a new hybrid method for inverting 2D nuclear magnetic resonance (NMR) data that combines truncated singular value decomposition (TSVD) and Tikhonov regularization. The method computes the exact TSVD of the kernel matrix using its Kronecker product structure, avoiding approximations. It then solves a Tikhonov-like optimization problem using the truncated kernel. The paper also proposes using the Discrete Picard Condition to automatically select both the TSVD truncation index and Tikhonov regularization parameter. The performance of the new hybrid method is evaluated on simulated and real NMR data.
ANALYTICAL SOLUTIONS OF THE MODIFIED COULOMB POTENTIAL USING THE FACTORIZATIO...ijrap
This document presents analytical solutions to the Schrödinger equation with a modified Coulomb potential using the factorization method. The energy levels and wave functions are obtained in terms of associated Laguerre polynomials. Energy eigenvalues are computed for selected elements like hydrogen, lithium, sodium, potassium and copper for various values of n and l. The results show the expected degeneracies and reduce to the Coulomb energy solution when appropriate limits are taken.
1. The document analyzes the dynamics of a satellite with an elastic tether system. It develops mathematical models to describe the oscillations of the satellite caused by changes in the magnitude and direction of the tether force.
2. Equations of motion are derived for the rotating tethered satellite system using Lagrange's equations. Approximate analytical solutions are also obtained for oscillations of the satellite under the influence of the elastic tether.
3. The dynamics of the elastic tether itself are also modeled through equations that describe vibrations of the tether near the local vertical.
This document describes a quadratic assignment problem (QAP) involving assigning 358 constraints and 50 variables. It provides an example of a QAP with 3 facilities and 3 locations. The QAP aims to assign facilities to locations in a way that minimizes total cost, which is a function of the flow between facilities and the distance between locations. Several applications of QAP are discussed, including facility location, scheduling, and ergonomic design problems.
This document discusses Jordan decomposition via the Z-transform. It begins by introducing Jordan decomposition and proving a theorem about representing any matrix power Ak as a unique sum involving projection and nilpotent matrices. It then provides background on the Z-transform and certain important functions. The document gives examples of applying the Z-transform approach to find the Jordan decomposition of specific matrices. It demonstrates rewriting the matrix equation Ak+l = AkAl in terms of the Z-transform to extract properties of the projection and nilpotent matrices.
The document summarizes the key results of a research paper on representation rings of cyclic groups over algebraically closed fields. It calculates the Jordan form of the tensor product of invertible Jordan block matrices whose pth power is the identity, for both characteristic zero and positive characteristic p fields.
It shows that for tensor products of Jordan blocks of size less than or equal to p, over a field of characteristic p, the Jordan form is a direct sum of indecomposable representations of the cyclic group of order p. Explicit formulas for various tensor product cases are provided, such as J2 ⊗ Jn and Jm ⊗ Jp for p ≥ m. Applications to arithmetic geometry, K-theory, cryptography and
On the Mathematical Structure of the Fundamental Forces of NatureRamin (A.) Zahedi
The main idea of this article is based on my previous articles (references [1], [2], [3]). In this work by introducing a new mathematical approach based on the algebraic structure of integers (the domain of integers), and assuming the “discreteness” of physical quantities such as the components of the relativistic n-momentum, we derive all the mathematical laws governing the fundamental forces of nature. These obtained laws that are unique, distinct and in the form of the complex tensor equations, represent the force of gravity, the electromagnetic (including electroweak) force, and the (strong) nuclear force (and only these three kinds of forces, for all dimensions D ≥2). Each derived tensor equation contains the term of the mass m_0 (as the invariant mass of the supposed force carrier particle), as well as the term of the external current (as the external source of the force field). In some special cases, these tensor equations are turned into the wave equations that are similar to the Pauli and Dirac equations. In fact, the mathematical laws obtained in this paper, are the corrected and generalized forms of the current field equations including Maxwell equations, Yang-Mils equations and Einstein equations, as well as (in some special conditions) Pauli equation, Dirac equation, and so on. A direct proof of the absence of magnetic monopoles in nature is one of the outcomes of this research, according to the unique formulations of the laws of the fundamental forces that we have derived.
Keywords: Foundations of Physics, Ontology, Discrete Physics, Discrete Mathematics, The Fundamental Forces of Nature.
Comments: 51 Pages. Expanded version of my previous articles:
Ramin (A.) Zahedi, "Linearization Method in the Ring Theory," Bulletin of the Lebedev Physics Institute, Springer-Verlag, No. 5-6, 1997;
Ramin (A.) Zahedi, "On the Connection Between Methods of the Ring Theory and the Group Approach", Bulletin of the Lebedev Physics Institute, Springer-Verlag, No. 7-8, 1997.
PACS Classifications: 04.20.Cv, 04.50.Kd, 04.90.+e, 04.62.+v, 02.10.Hh, 02.10.Yn, 02.20.Bb, 02.90.+p, 03.50.-z, 03.65.Fd, 03.65.Pm, 03.50.Kk, 12.40.-y, 12.60.-i, 12.10.Dm, 12.10.-g.
External URL: http://arXiv.org/abs/1501.01373. (arXiv:1501.01373 [physics.gen-ph])
Copyright: CC Attribution-NonCommercial-NoDerivs 4.0 International
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
Exact Matrix Completion via Convex Optimization Slide (PPT)Joonyoung Yi
Slide of the paper "Exact Matrix Completion via Convex Optimization" of Emmanuel J. Candès and Benjamin Recht. We presented this slide in KAIST CS592 Class, April 2018.
- Code: https://github.com/JoonyoungYi/MCCO-numpy
- Abstract of the paper: We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys
𝑚≥𝐶𝑛1.2𝑟log𝑛
for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
This document describes two algorithms for calculating ideal solution chemical equilibrium in multiphase systems. Both algorithms utilize a duality transformation of the Gibbs energy function to formulate the problem. The Lagrange-Newton method finds a stationary point of the Lagrangian using Newton's method. The multiplier penalty method replaces the constrained optimization with sequential unconstrained optimizations. Both methods were tested on a system with 10 gaseous compounds and up to 6 solid phases, and were able to reliably calculate the phase equilibria over a range of pressures.
Ijcatr03051008Implementation of Matrix based Mapping Method Using Elliptic Cu...Editor IJCATR
Elliptic Curve Cryptography (ECC) gained a lot of attention in industry. The key attraction of ECC over RSA is that it
offers equal security even for smaller bit size, thus reducing the processing complexity. ECC Encryption and Decryption methods can
only perform encrypt and decrypt operations on the curve but not on the message. This paper presents a fast mapping method based on
matrix approach for ECC, which offers high security for the encrypted message. First, the alphabetic message is mapped on to the
points on an elliptic curve. Later encode those points using Elgamal encryption method with the use of a non-singular matrix. And the
encoded message can be decrypted by Elgamal decryption technique and to get back the original message, the matrix obtained from
decoding is multiplied with the inverse of non-singular matrix. The coding is done using Verilog. The design is simulated and
synthesized using FPGA.
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
Similar to Research Inventy : International Journal of Engineering and Science (20)
"Polyadic supersymmetry" by S. Duplij, arxiv 2406.02188
Research Inventy : International Journal of Engineering and Science
1. RESEARCH INVENTY: International Journal of Engineering and Science
ISBN: 2319-6483, ISSN: 2278-4721, Vol. 1, Issue 11
(December 2012), PP 25-30
www.researchinventy.com
New Kronecker product decompositions and its applications
Fuxiang Liu
Science College and Institute of Intelligent Vision and Image Information, China Three Gorges University,
Yichang, Hubei, 443002, PR China
Abstract: Firstly, two new kinds of Kronecker decompositions is developed, i.e. KPGD and KPID; Secondly,
the sufficient and necessary conditions and algorithms of Kronecker product(KPD), KPGD, and KPID are
discussed; At last, some useful properties of the rank of the sum of Kronecker product gemel decompositions are
obtained.
Keywords: Kronecker product decomposition, gemel decomposition, isomer decomposition, rank,
dimensionality reduction. MSC: 65L09,65Y04,15A69
I. Introduction
Because of its elegant algebraic properties, the Kronecker product is a useful tool to solve matrix
equations and the nearest kronecker product problems[1,2], do inference in multivariate analysis[3], and
construct fast and practical algorithms in signal processing, image processing, computer vision, semidefinite
programming, quantum computing, linear systems and stochastic automata networks etc. see[1,4,5,6,7,8,9,10]
and so on. Meantime, the applications in nearly all those areas are related to some certain kinds of Kronecker
product decompositions which in fact are the inverse problems of Kronecker product, see [3,13,14,18] etc.
Usually, Kronecker product decomposition(KPD) means that a matrixM can be transformed to the
ronecker product form of the other matrices A;B, i:e: M = A B; Kronecker product gemel
decomposition(KPGD) means the Kronecker product with the special case A = B; And Kronecker product
isomer decomposition(KPID) corresponds to the case M = A A′. Obviously, these decompositions often have
many solutions as well as the other inverse problems. In this direction, Eugene Tyrtyshnikov[14] has some
interesting work about Kronecker ranks; T.G.Kolda[12] has some meaningful work on orthogonal tensor
decompositions; DE Launey and Seberry[18] developed some properties and their applications on the strong
Kronecker product; In addition, Sadegh Jokar and Molker Mehrmann[11], Jun-e Feng, James Lam, Yimin
Wei[13] etc, have obtained some useful properties of the sum of Kronecker products. Undoubtedly,
these different decompositions are helpful for dimensionality reduction procedure which is very important key
for high dimensional image processing and gene analyzing.
Unfortunately, many natural questions about seemingly “simple”cases are still not answered in spite
of an ever increasing interest and some significant results with applications, such as the conditions and the rank
of the sum of these decompositions. In this paper, from the perspective of the inverse problem theory, we mainly
explore the sufficient and necessary conditions and algorithms of KPD, KPGD, KPID, and obtain some useful
properties of the rank of the sum of Kronecker product gemel decomposition. And our research works are
mainly motivated by doing multivariate statistical inference and huge dimensional statistical analysis, solving
high dimensional matrix equations and constructing the algorithm of image processing and computer vision.
25
2. New Kronecker product decompositions and its applications
II. KPD, KPGD And KPID Problems
Obviously, the essential preconditions of KPD(M = A B), KPGD(M = A A) and KPID(M =
A A′) means that the matrices M,A,B have the proper columns and rows. And this is easily verified, so, we
assume that all the matrices in the following discussion have the suitable column and row numbers.
2.1 The sufficient and necessary conditions of KPD
Let A (a1,, am ) Rnm with ai Rn ,1 i m , then denote vec( A) (a1 ', a2 ', , am ') ' .
Firstly, we explore the sufficient and necessary conditions of Kronecker product decomposition,
and give the elegant form of this result as follows.
Theorem 2.1. (KPD) For an arbitrary matrix M R mr ns ,
M11 M1n
M , (M ij R r s , i 1,, m, j 1,, n) , can be decomposed to the form
M M
m1 mn
M A B , where A R mn , B R r s ,m, n, r, s are some certain integers.
(equivalent to) rank vec(M11), vec(M12 ),, vec(M1n ),, vec(M mn ) =1.
Miraculously, the proof of this theorem is not difficult, so the details of the proof are omitted.
Remark 1. Generally speaking, the KPD of an arbitrary matrix is not unique, because of (kA) (lB) = A
B with kl = 1 for arbitrary constants k; l and matrices A,B.
The following algorithm describes the general program of KPD problem that includes whether a matrix can
be decomposed or not and how to get the results of KPD.
Algorithm 1(KPD):
step 1: input M , m, n, r , s , verify the size of M is mr ns and M! = 0
step 2: define M ij , i 1,, m , j 1,, n
step 3: calculate vec(Mij )
step 4: if rank{ vec( M11 ), vec( M12 ), , vec( M1n ), , vec( M mn ) } == 1 goto step 5
else output ”can not decomposition”; end
step 5: look for the first M ij ! = 0, define B = M ij
step 6: calculate aij : vec(Mij ) = aij vec( B) , i 1,, m , j 1,, n
step 7: define A (aij ) , i 1,, m , j 1,, n ; output A,B; end
26
3. New Kronecker product decompositions and its applications
2.2 The sufficient and necessary conditions of KPGD and KPID
With the similar inference, we can have the following conditions about KPGD.
2 2
Theorem 2.2. (KPGD) For an arbitrary matrix M R p q ( M 0 ),
M11 M1q
M , (M ij R pq , i 1,, p, j 1,, q) , can be decomposed to the form
M M
p1 pq
M A A , where A R pq , p, q are some certain integers.
(equivalent to) there exists a subblock Mij 0 , and miij j 0 where M ij (ms,t ) s 1,, p,t 1,, q ,
,
ij
if denote Mij / miij j (akl )k 1,, p,l 1,,q A or A , then for k , l , M kl akl A .
,
2 2
Corollary 2.3. Denote M R p q ( M 0 ), M (Mij ), i 1,, p, j 1,, q , M ij R pq ,
ij
and M ij (ms,t ) s 1,, p,t 1,, q , then the matrix M can be carried out KPGD (M A A)
rank vec(M11), vec(M12 ),, vec(M1q ),, vec(M pq ) =1.
The following algorithm describes the general program of KPGD problem that includes whether a matrix
can be decomposed or not and how to get the results of KPGD.
Algorithm 2(KPGD):
step 1: input M , p, q , verify the size of M is p2 q2 and M ! = 0
step 2: define flag=0, M ij , i 1,, p, j 1,, q
step 3: for (i, j ) , i 1,, p, j 1,, q
if: M ij == 0, continue;
else if: M ij ! = 0&& miij j 0
, flag=1; break;
else: define B M ij / miij j
, flag=2; break;
step 4: if flag==2&&M == B B A = B; output A; end
else output ”can not gemel decomposition”; end
2 2
Theorem 2.4. (KPID) For an arbitrary matrix M R p q ( M 0 ),
M11 M1q
M
, (M ij R pq , i 1,, p, j 1,, q) , can be decomposed to the form
M
p1 M pq
27
4. New Kronecker product decompositions and its applications
M A A ' , where A R pq , p, q are some certain integers.
(equivalent to) there exists a subblock Mij 0 , and mij,i 0 where M ij (ms,t ) s 1,, p,t 1,, q ,
j
ij
such that k , l , M kl akl A ' , where A (akl )k 1,, p,l 1,,q , and A ' M ij / miij j
, or
A ' Mij / miij j .
,
i 1,, k , j 1,, n , then
III. The Rank Of The Sum Of KPGD
i 1 Ai Ai ).
k
In this section, we discuss some properties of the rank of the sum of KPGD(
Lemma 3.1. Let A and B be n × n real symmetric matrices.
(i) There exists a real orthogonal matrix Q such that Q′AQ and Q′BQ are both diagonal if and
only if AB = BA (that is AB is symmetric).
(ii) The previous result holds for more than two matrices. A set of real symmetric matrices
are simultaneously diagonalizable by the same orthogonal matrix Q if and only if they commute pairwise.
see George A. F. Seber[17] for more details about matrices simultaneous diagonalization.
Theorem 3.2. Let A1, ,A k ( k
2 ) be n n real symmetric and positive definite(> 0)
i 1 Ai Ai
k
or negative definite(< 0) matrices. If they commute pairwise, then rank( )
= rank ( A1)2 n2 .
Proof: By the result of Lemma 3.1, there exists an orthogonal matrix Q, such that
QAi Q ' diag (i1, , in ) , i 1,, k , where i1, , in are the eigenvalues of Ai , and ij 0 ,
(Q Q)(i 1 Ai Ai )(Q Q) i 1 diag (i1,, in ) diag (i1,, in )
k k
i 1 diag (i1,, in ) diag (i1,, in )) n2 , then the proof is completed.
k
Obviously, the rank (
With similar discussions, we have the following two theorems.
Theorem 3.3. Let A1, ,A k ( k
2 ) be n n real symmetric matrices, there at least a positive
i 1 Ai Ai )
k
definite(> 0) or negative definite(< 0) matrices. If they commute pairwise, then rank(
= rank ( A1)2 n2 .
Theorem 3.4. Let A1, ,A k ( k
2 ) be n n real symmetric and positive definite(> 0)
or negative definite(< 0) matrices. If they commute pairwise, then
i 1 Ai Ai ) max rank ( A1 A1),, rank ( Ak Ak ) .
k
rank(
28
5. New Kronecker product decompositions and its applications
These results will be helpful to study the solutions of the following general Sylvester matrix
equation problem[19, 20]:
A1 XA1 Ak XAk C k
A Ai vec( X ) vec(C )
i 1 i
Theorem 3.5. Let A, B be n n real symmetric matrices with AB BA , the eigenvalues of A are
1, , n , then the eigenvalues of B are 1, , n , define the vector (1, , n ) ' ,
( 1, , n ) ' , (1, , n ) ' , where 1, , n is an arbitrary permutation of the elements
( 1,, n ) ' , Q (i, j ) : i j i j 0, i, j 1, , n , the number of Q , then
n2 max rank ( A A B B) n2 min
Proof: There exists an orthogonal matrix Q, such that QAQ ' diag (1, , n ) and
QBQ ' diag (1,, n ) . Thus,
(Q Q)( A A B B)(Q ' Q ') diag (1,, n ) diag (1,, n )
diag (1, , n ) diag (1, , n ) . Calculate the number of nonzero elements of
i j i j , i, j 1,, n , and the result is obtained.
IV. Discussion
In this paper, we discuss the sufficient and necessary conditions and algorithms of KPD, KPGD and
KPID problems, which play a great role in all kinds of Kronecker product application areas, and obtain some
i 1 Ai Ai ) in simultaneous diagonalization situation
k
useful properties of the rank of the sum of KPGD(
which performs some wonderful algebra advantages. More interesting work in the future maybe include the
i 1 Ai Bi
k
following aspects: the conditions that a matrix can be decomposed to the form which is a
meaningful work especially in sparse matrices cases[11, 21], decomposing program, and the properties of the
rank of the sum of decomposition in more general cases. Also, those Kronecker product decomposition
properties maybe associated with seeking the sufficient and necessary conditions under which the following
i 1 Ai ( I X ) Bi C has
n
matrix equation a uniquely solution, where Ai , Bi and Ci are known
matrices, and X is an unknown matrix.
V. Acknowledge
This work is supported by National Natural Science Foundation of China (NSFC) Grants 60972162.
29
6. New Kronecker product decompositions and its applications
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