Let ð be a ring and ð, ð be an endomorphisms of ð , in this paper we will present and study the
concepts of higher (ð, ð)-centralizer, Jordan higher(ð, ð)-centralizer and Jordan triple higher (ð, ð)-
centralizer and their generalization on the ring. The main results are prove that every Jordan higher (ð, ð)-
centralizer of prime ring ð is higher (ð, ð)-centralizer of ð and we prove let ð be a 2-torsion free ring,ð ððð ð
are commutative endomorphism then every Jordan higher (ð, ð)-centralizer is Jordan triple higher (ð, ð)-
centralizer.
This document presents research on generalized Bernstein polynomials. It defines a new polynomial operator Unr that approximates Lebesgue integrable functions on the interval [0,1+r/n] in the L1 norm. The operator is a modification of the Bernstein operator to work on a finite interval. The document proves a generalization of Voronowskaja's theorem for this new operator, showing that the difference between the function value and the polynomial approximation converges to 1/2 times the second derivative of the function as n+r goes to infinity, under certain conditions. It also presents three technical lemmas used in the proof.
The document proposes a new hybrid conjugate gradient method for unconstrained optimization problems. It calculates the parameter βk in the conjugate gradient direction formula as a convex combination of βHS (the Hestenes-Stiefel formula) and βCD (the conjugate descent formula). The new method aims to avoid "jamming" seen in other conjugate gradient methods while maintaining good computational performance. It proves that if the parameter Ξk is chosen as described, the new search direction dk+1 will be a descent direction, indicating the method may converge.
International Refereed Journal of Engineering and Science (IRJES)irjes
Â
1) The document presents a class of p-valent functions defined using the Hadamard product and involving the generalized hypergeometric function.
2) It introduces a linear operator Lpka,b,c involving the generalized hypergeometric function and Hadamard product.
3) Theorems regarding coefficient bounds for functions in this class to be p-valent starlike or convex of order ÎŽ are provided. Distortion theorems and fractional differential operators for these functions are also obtained.
This document presents research on a new class of univalent functions defined using the Wright generalized hypergeometric function. Some key points:
- A new class of functions called T*(γ) is defined based on the Wright generalized hypergeometric function.
- Characterization properties are obtained for functions in this new T*(γ) class. Specifically, a necessary and sufficient condition is proved involving the coefficients of the function.
- Distortion theorems and radii of starlikeness and convexity are also obtained for the new class of functions.
- The results generalize previous work on characterization properties and distortion theorems for classes of uniformly convex and starlike functions.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
Â
In this paper we define the generalized Cesaro sequence spaces ííí (í, í, í ). We prove the space ííí (í, í, í ) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map ííí í, í, í to íâ and ííí (í, í, í ) to c, where íâ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
1. The open loop transfer function is G(s)H(s) = KKbb/(JJJJ + CC).
2. The feedforward transfer function is G(s) = KKbb/(JJJJ + CC).
3. The control ratio is C(s)/R(s) = G(s)/(1 + G(s)H(s)) = KKbb/(KKbbss + JJJJ + CC).
4. The feedback ratio is H(s) = 1.
5. The error ratio is E(s)/R(s) = 1/(1 + G(s)H(s)) = (KKbbss + JJJ
Backpropagation: Understanding How to Update ANNs Weights Step-by-StepAhmed Gad
Â
This presentation explains how the backpropagation algorithm is useful in updating the artificial neural networks (ANNs) weights using two examples step by step. Readers should have a basic understanding of how ANNs work, partial derivatives, and multivariate chain rule.
This presentation won`t dive directly into the details of the algorithm but will start by training a very simple network. This is because the backpropagation algorithm is meant to be applied over a network after training. So, we should train the network before applying it to catch the benefits of backpropagation algorithm and how to use it.
Some approximation properties of modified baskakov stancu operatorseSAT Journals
Â
Abstract In the present paper, we prove a global direct theorem for the modified Baskakovstancu operators in terms of Ditzian-Totik modulus of smoothness. Here, we have modified our operators by taking weight function of Beta operators and then generalizing it as stancu type generalized operators. We will also see that taking weight function of Beta operators will give better approximation. We study a global direct theorem using simultaneous approximation for ourstancu type generalized operator in í¿í 0,â . Here, first we estimate recurrence relation for moments and then develop some global direct results by making our stancu type generalized operators positive using differential and integral operators. In this paper, our effort is to give better global approximation for our stancu type generalized operator than the earlier integral modifications of Baskakov operators studied by various authors. Here, we will extend our results for the whole interval 0,â . In this paper, we will also make use of the fact that second modulus of smoothness introduced by Ditzian-Totik is equivalent to modified k-functional and í¿íí[0,â) is not contained in í¿1í[0,â) for obtaining results. Here, Riesz-Thorin theorem and Leibnitz theorem is used extensively for doing simultaneous approximation. We have also used Fubiniâs theorem for obtaining results. Key words: Stancu type generalization, simultaneous approximation, modulus of smoothness.
This document presents research on generalized Bernstein polynomials. It defines a new polynomial operator Unr that approximates Lebesgue integrable functions on the interval [0,1+r/n] in the L1 norm. The operator is a modification of the Bernstein operator to work on a finite interval. The document proves a generalization of Voronowskaja's theorem for this new operator, showing that the difference between the function value and the polynomial approximation converges to 1/2 times the second derivative of the function as n+r goes to infinity, under certain conditions. It also presents three technical lemmas used in the proof.
The document proposes a new hybrid conjugate gradient method for unconstrained optimization problems. It calculates the parameter βk in the conjugate gradient direction formula as a convex combination of βHS (the Hestenes-Stiefel formula) and βCD (the conjugate descent formula). The new method aims to avoid "jamming" seen in other conjugate gradient methods while maintaining good computational performance. It proves that if the parameter Ξk is chosen as described, the new search direction dk+1 will be a descent direction, indicating the method may converge.
International Refereed Journal of Engineering and Science (IRJES)irjes
Â
1) The document presents a class of p-valent functions defined using the Hadamard product and involving the generalized hypergeometric function.
2) It introduces a linear operator Lpka,b,c involving the generalized hypergeometric function and Hadamard product.
3) Theorems regarding coefficient bounds for functions in this class to be p-valent starlike or convex of order ÎŽ are provided. Distortion theorems and fractional differential operators for these functions are also obtained.
This document presents research on a new class of univalent functions defined using the Wright generalized hypergeometric function. Some key points:
- A new class of functions called T*(γ) is defined based on the Wright generalized hypergeometric function.
- Characterization properties are obtained for functions in this new T*(γ) class. Specifically, a necessary and sufficient condition is proved involving the coefficients of the function.
- Distortion theorems and radii of starlikeness and convexity are also obtained for the new class of functions.
- The results generalize previous work on characterization properties and distortion theorems for classes of uniformly convex and starlike functions.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
Â
In this paper we define the generalized Cesaro sequence spaces ííí (í, í, í ). We prove the space ííí (í, í, í ) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map ííí í, í, í to íâ and ííí (í, í, í ) to c, where íâ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
1. The open loop transfer function is G(s)H(s) = KKbb/(JJJJ + CC).
2. The feedforward transfer function is G(s) = KKbb/(JJJJ + CC).
3. The control ratio is C(s)/R(s) = G(s)/(1 + G(s)H(s)) = KKbb/(KKbbss + JJJJ + CC).
4. The feedback ratio is H(s) = 1.
5. The error ratio is E(s)/R(s) = 1/(1 + G(s)H(s)) = (KKbbss + JJJ
Backpropagation: Understanding How to Update ANNs Weights Step-by-StepAhmed Gad
Â
This presentation explains how the backpropagation algorithm is useful in updating the artificial neural networks (ANNs) weights using two examples step by step. Readers should have a basic understanding of how ANNs work, partial derivatives, and multivariate chain rule.
This presentation won`t dive directly into the details of the algorithm but will start by training a very simple network. This is because the backpropagation algorithm is meant to be applied over a network after training. So, we should train the network before applying it to catch the benefits of backpropagation algorithm and how to use it.
Some approximation properties of modified baskakov stancu operatorseSAT Journals
Â
Abstract In the present paper, we prove a global direct theorem for the modified Baskakovstancu operators in terms of Ditzian-Totik modulus of smoothness. Here, we have modified our operators by taking weight function of Beta operators and then generalizing it as stancu type generalized operators. We will also see that taking weight function of Beta operators will give better approximation. We study a global direct theorem using simultaneous approximation for ourstancu type generalized operator in í¿í 0,â . Here, first we estimate recurrence relation for moments and then develop some global direct results by making our stancu type generalized operators positive using differential and integral operators. In this paper, our effort is to give better global approximation for our stancu type generalized operator than the earlier integral modifications of Baskakov operators studied by various authors. Here, we will extend our results for the whole interval 0,â . In this paper, we will also make use of the fact that second modulus of smoothness introduced by Ditzian-Totik is equivalent to modified k-functional and í¿íí[0,â) is not contained in í¿1í[0,â) for obtaining results. Here, Riesz-Thorin theorem and Leibnitz theorem is used extensively for doing simultaneous approximation. We have also used Fubiniâs theorem for obtaining results. Key words: Stancu type generalization, simultaneous approximation, modulus of smoothness.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Maximum Likelihood Estimation of BeetleLiang Kai Hu
Â
- The document describes maximum likelihood estimation (MLE) of species parameters from beetle mass, length, and other character data.
- It derives EM steps to estimate species means (ÎŒ, Îœ), proportions (Ï), and priors (α) in the presence of missing species data.
- Running the EM algorithm for 120 iterations estimates the parameters and converges the log likelihood to 14 digits of precision with a convergence rate of approximately 1.
- It also derives steps for Gibbs sampling to estimate the missing species indicators and parameter values based on their posterior distributions.
1. The document discusses vector optimization problems and presents definitions and concepts related to nondominated solutions.
2. It introduces the concept of Ξ-ordering between solutions and defines what it means for one solution to be better than another based on their Ξ-ordering.
3. Formulas and properties are presented for calculating the Ξ-value of solutions based on the objective function values.
The document defines the derivative of a function as the limit of the average rate of change of the function over an interval as the interval approaches zero. It provides examples of calculating the derivative of various functions, including the velocity and acceleration of the function s(t)=t^3 - 2t^2. The derivative of s(t) is 3t^2 - 4t, the velocity is 3t^2 - 4t, and the acceleration is 6t - 4. Formulas are provided for taking the derivative of various functions.
The document describes a method for summarizing the essential information of a document in 3 sentences or less. It begins by providing definitions for key terms used in the method such as sets, functions, and ordering relationships. It then provides an example application of the method to a specific problem instance, calculating an ordering relationship over subsets of a set based on a given valuation function.
Intro to Quantitative Investment (Lecture 4 of 6)Adrian Aley
Â
This document discusses optimal trading strategies using stochastic control. It provides background on the speaker, Haksun Li, and references related work on optimal pairs trading and trend following rules. It then describes modeling the difference between two asset returns as a mean-reverting Ornstein-Uhlenbeck process. It formulates the optimal trading problem using dynamic programming and stochastic control to maximize expected utility of the trading portfolio value over time.
On ranges and null spaces of a special type of operator named ð â ððððððð. â ...IJMER
Â
In this article, ð â ðððð¡ððð has been introduced which is a generalization of trijection
operator as introduced in P.Chandraâs Ph. D. thesis titled âInvestigation into the theory of operators
and linear spacesâ (Patna University,1977). We obtain relation between ranges and null spaces of two
given ð â ðððð¡ðððð under suitable conditions.
On generating functions of biorthogonal polynomialseSAT Journals
Â
Abstract In this paper, we have obtained some novel generating functions (both bilateral and mixed trilateral) involving modified bi-orthogonal polynomials íí+ííŒ+í í¥;í , by group-theoretic method. As particular cases, we obtain the corresponding results on generalized Laguerre polynomials. Key words: AMS-2000 Classification Code :33C45,33C47, Biorthogonal polynomials, Laguerre polynomials, generating functions.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The document discusses the Gram-Schmidt process and related linear algebra concepts. It begins by defining orthogonal and orthonormal sets and bases. It then discusses projection theory and how to construct an orthonormal set from an orthogonal set using Gram-Schmidt. Examples are provided to illustrate orthogonalization and finding coordinates of a vector with respect to an orthogonal basis. The document concludes by providing an example of applying Gram-Schmidt to transform an orthogonal basis into an orthonormal basis.
The document discusses backpropagation, an algorithm used to train neural networks. It begins with background on perceptron learning and the need for an algorithm that can train multilayer perceptrons to perform nonlinear classification. It then describes the development of backpropagation, from early work in the 1970s to its popularization in the 1980s. The document provides examples of using backpropagation to design networks for binary classification and multi-class problems. It also outlines the generalized mathematical expressions and steps involved in backpropagation, including calculating the error derivative with respect to weights and updating weights to minimize loss.
The document analyzes linear transformations in R2 and R3. It determines whether three given functions define linear transformations based on whether they satisfy the property T(αu + βv) = αT(u) + βT(v).
The first function f(x,y) = (3x - y, x + y) is determined to define a linear transformation in R2 since it satisfies the property.
The second function f(x,y,z) = (x,y,z2) is determined not to define a linear transformation in R3 since z2 is not a linear term.
The third function f(x,y,z) = (x +
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
Jurnal Study of Anisotropy Superconductor using Time-Dependent Ginzburg-Landa...Fuad Anwar
Â
This document summarizes a study that used numerical simulations based on the time-dependent Ginzburg-Landau equation to study the properties of anisotropic superconductors. The study models an anisotropic superconductor with different effective masses for Cooper pairs along different crystal axes. It describes the theoretical formalism and numerical methods used to solve the time-dependent Ginzburg-Landau equations for an anisotropic superconductor. The results of the numerical simulations show that the anisotropic properties of superconductors can cause the critical field to be either lower or higher compared to isotropic superconductors.
This document provides information about neural networks from Parveen Malik, an Assistant Professor at KIIT University. It defines a neural network as a massively parallel distributed processor made up of simple processing units that has a natural ability to store experiential knowledge and make it available for use, similar to the human brain. Neural networks can be used for applications like object detection, image captioning, time series modelling, and more. The document also discusses the structure and function of biological neurons and their equivalence to artificial neurons in neural networks.
1) The document presents the use of the Homotopy Perturbation Method (HPM) to obtain exact solutions to the Burgers-Huxley nonlinear differential equation.
2) As a case study, the HPM is applied to solve a specific Burgers-Huxley equation with given initial conditions. Approximate solutions are obtained as a power series in the embedding parameter.
3) The solution is shown to rapidly converge to the exact solution, with a maximum relative error of less than 0.0000058% achieved, demonstrating the effectiveness of using HPM to solve this type of nonlinear differential equation.
1) The document presents a wavelet collocation method for numerically solving nth order Volterra integro-differential equations. It expands the unknown function as a series of Chebyshev wavelets of the second kind with unknown coefficients.
2) It states and proves a uniform convergence theorem that establishes the convergence of approximating the solution using truncated Chebyshev wavelet series expansions.
3) The paper demonstrates the validity and applicability of the proposed method through some illustrative examples of solving integro-differential equations using the Chebyshev wavelet collocation approach.
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
The document discusses numeric partitions and graphs involving "Eugênio Numbers" (EN), "Krishna Numbers" (KN), and functions related to the number e and other constants. It defines EN and KN sets and describes partitioning them into "generations" based on digit counts. Equations are given for EN functions involving operations like addition, multiplication and logarithms. Graphs are proposed plotting the ratios of "Aleph Red" to "Aleph Omega" for different values of variables like â.
Effect of glycine as an impurity on the properties of Epsomite single crystalsIOSR Journals
Â
Epsomite (MgSO4.7H2O) is a hydrogen bonded crystal having a wide range of applications in various fields. Pure and glycine added Epsomite (a total of six) single crystals have been grown and characterized in order to investigate the effect of glycine as an impurity on the properties of Epsomite crystals.The densities and lattice parameters observed indicate that the impurity molecules have entered into the crystal matrix. The grown crystals exhibit good optical transparency in the wavelength range 210-1100 nm. Second harmonic generation measurements indicate that they are nonlinear optically active. Results obtained through microhardness measurements follow the normal indentation size effect. Electrical (AC and DC) measurements indicate that all the six crystals grown in the present study exhibit a normal dielectric behavior and the electrical conduction is understood to be due to the protonic movement. Moreover, glycine addition is found to reduce εr value significantly indicating the possibility of Epsomite crystal becoming a low -εr value dielectric material.
This document proposes a new energy efficient MAC protocol called E-BMA for wireless sensor networks. It aims to minimize idle times during contention periods to improve energy efficiency, especially for low to medium traffic loads. The E-BMA protocol is simulated and its energy consumption is compared to TDMA, EA-TDMA, and BMA protocols. The results show that E-BMA performs better and consumes less energy than the other protocols under different traffic conditions.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Maximum Likelihood Estimation of BeetleLiang Kai Hu
Â
- The document describes maximum likelihood estimation (MLE) of species parameters from beetle mass, length, and other character data.
- It derives EM steps to estimate species means (ÎŒ, Îœ), proportions (Ï), and priors (α) in the presence of missing species data.
- Running the EM algorithm for 120 iterations estimates the parameters and converges the log likelihood to 14 digits of precision with a convergence rate of approximately 1.
- It also derives steps for Gibbs sampling to estimate the missing species indicators and parameter values based on their posterior distributions.
1. The document discusses vector optimization problems and presents definitions and concepts related to nondominated solutions.
2. It introduces the concept of Ξ-ordering between solutions and defines what it means for one solution to be better than another based on their Ξ-ordering.
3. Formulas and properties are presented for calculating the Ξ-value of solutions based on the objective function values.
The document defines the derivative of a function as the limit of the average rate of change of the function over an interval as the interval approaches zero. It provides examples of calculating the derivative of various functions, including the velocity and acceleration of the function s(t)=t^3 - 2t^2. The derivative of s(t) is 3t^2 - 4t, the velocity is 3t^2 - 4t, and the acceleration is 6t - 4. Formulas are provided for taking the derivative of various functions.
The document describes a method for summarizing the essential information of a document in 3 sentences or less. It begins by providing definitions for key terms used in the method such as sets, functions, and ordering relationships. It then provides an example application of the method to a specific problem instance, calculating an ordering relationship over subsets of a set based on a given valuation function.
Intro to Quantitative Investment (Lecture 4 of 6)Adrian Aley
Â
This document discusses optimal trading strategies using stochastic control. It provides background on the speaker, Haksun Li, and references related work on optimal pairs trading and trend following rules. It then describes modeling the difference between two asset returns as a mean-reverting Ornstein-Uhlenbeck process. It formulates the optimal trading problem using dynamic programming and stochastic control to maximize expected utility of the trading portfolio value over time.
On ranges and null spaces of a special type of operator named ð â ððððððð. â ...IJMER
Â
In this article, ð â ðððð¡ððð has been introduced which is a generalization of trijection
operator as introduced in P.Chandraâs Ph. D. thesis titled âInvestigation into the theory of operators
and linear spacesâ (Patna University,1977). We obtain relation between ranges and null spaces of two
given ð â ðððð¡ðððð under suitable conditions.
On generating functions of biorthogonal polynomialseSAT Journals
Â
Abstract In this paper, we have obtained some novel generating functions (both bilateral and mixed trilateral) involving modified bi-orthogonal polynomials íí+ííŒ+í í¥;í , by group-theoretic method. As particular cases, we obtain the corresponding results on generalized Laguerre polynomials. Key words: AMS-2000 Classification Code :33C45,33C47, Biorthogonal polynomials, Laguerre polynomials, generating functions.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The document discusses the Gram-Schmidt process and related linear algebra concepts. It begins by defining orthogonal and orthonormal sets and bases. It then discusses projection theory and how to construct an orthonormal set from an orthogonal set using Gram-Schmidt. Examples are provided to illustrate orthogonalization and finding coordinates of a vector with respect to an orthogonal basis. The document concludes by providing an example of applying Gram-Schmidt to transform an orthogonal basis into an orthonormal basis.
The document discusses backpropagation, an algorithm used to train neural networks. It begins with background on perceptron learning and the need for an algorithm that can train multilayer perceptrons to perform nonlinear classification. It then describes the development of backpropagation, from early work in the 1970s to its popularization in the 1980s. The document provides examples of using backpropagation to design networks for binary classification and multi-class problems. It also outlines the generalized mathematical expressions and steps involved in backpropagation, including calculating the error derivative with respect to weights and updating weights to minimize loss.
The document analyzes linear transformations in R2 and R3. It determines whether three given functions define linear transformations based on whether they satisfy the property T(αu + βv) = αT(u) + βT(v).
The first function f(x,y) = (3x - y, x + y) is determined to define a linear transformation in R2 since it satisfies the property.
The second function f(x,y,z) = (x,y,z2) is determined not to define a linear transformation in R3 since z2 is not a linear term.
The third function f(x,y,z) = (x +
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
Jurnal Study of Anisotropy Superconductor using Time-Dependent Ginzburg-Landa...Fuad Anwar
Â
This document summarizes a study that used numerical simulations based on the time-dependent Ginzburg-Landau equation to study the properties of anisotropic superconductors. The study models an anisotropic superconductor with different effective masses for Cooper pairs along different crystal axes. It describes the theoretical formalism and numerical methods used to solve the time-dependent Ginzburg-Landau equations for an anisotropic superconductor. The results of the numerical simulations show that the anisotropic properties of superconductors can cause the critical field to be either lower or higher compared to isotropic superconductors.
This document provides information about neural networks from Parveen Malik, an Assistant Professor at KIIT University. It defines a neural network as a massively parallel distributed processor made up of simple processing units that has a natural ability to store experiential knowledge and make it available for use, similar to the human brain. Neural networks can be used for applications like object detection, image captioning, time series modelling, and more. The document also discusses the structure and function of biological neurons and their equivalence to artificial neurons in neural networks.
1) The document presents the use of the Homotopy Perturbation Method (HPM) to obtain exact solutions to the Burgers-Huxley nonlinear differential equation.
2) As a case study, the HPM is applied to solve a specific Burgers-Huxley equation with given initial conditions. Approximate solutions are obtained as a power series in the embedding parameter.
3) The solution is shown to rapidly converge to the exact solution, with a maximum relative error of less than 0.0000058% achieved, demonstrating the effectiveness of using HPM to solve this type of nonlinear differential equation.
1) The document presents a wavelet collocation method for numerically solving nth order Volterra integro-differential equations. It expands the unknown function as a series of Chebyshev wavelets of the second kind with unknown coefficients.
2) It states and proves a uniform convergence theorem that establishes the convergence of approximating the solution using truncated Chebyshev wavelet series expansions.
3) The paper demonstrates the validity and applicability of the proposed method through some illustrative examples of solving integro-differential equations using the Chebyshev wavelet collocation approach.
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
The document discusses numeric partitions and graphs involving "Eugênio Numbers" (EN), "Krishna Numbers" (KN), and functions related to the number e and other constants. It defines EN and KN sets and describes partitioning them into "generations" based on digit counts. Equations are given for EN functions involving operations like addition, multiplication and logarithms. Graphs are proposed plotting the ratios of "Aleph Red" to "Aleph Omega" for different values of variables like â.
Effect of glycine as an impurity on the properties of Epsomite single crystalsIOSR Journals
Â
Epsomite (MgSO4.7H2O) is a hydrogen bonded crystal having a wide range of applications in various fields. Pure and glycine added Epsomite (a total of six) single crystals have been grown and characterized in order to investigate the effect of glycine as an impurity on the properties of Epsomite crystals.The densities and lattice parameters observed indicate that the impurity molecules have entered into the crystal matrix. The grown crystals exhibit good optical transparency in the wavelength range 210-1100 nm. Second harmonic generation measurements indicate that they are nonlinear optically active. Results obtained through microhardness measurements follow the normal indentation size effect. Electrical (AC and DC) measurements indicate that all the six crystals grown in the present study exhibit a normal dielectric behavior and the electrical conduction is understood to be due to the protonic movement. Moreover, glycine addition is found to reduce εr value significantly indicating the possibility of Epsomite crystal becoming a low -εr value dielectric material.
This document proposes a new energy efficient MAC protocol called E-BMA for wireless sensor networks. It aims to minimize idle times during contention periods to improve energy efficiency, especially for low to medium traffic loads. The E-BMA protocol is simulated and its energy consumption is compared to TDMA, EA-TDMA, and BMA protocols. The results show that E-BMA performs better and consumes less energy than the other protocols under different traffic conditions.
An Efficient implementation of PKI architecture based Digital Signature using...IOSR Journals
Â
Abstract: Digital Signature technique is widely being used to detect unauthorized modification to data and to
authenticate the identity of the signatory. It is essential for secure transaction over unsecure/ open networks.
Digital Signature schemes are mostly used in cryptographic protocols to provide services like entity
authentication, authenticated key transport and key agreement. The PKI (Public Key Infrastructure) based
digital signature architecture is related with RSA algorithm and secure Hash functions (MD5 &, SHA variants).
RSA digital signature algorithm is an asymmetric cryptographic method whose security is associated with
difficulty of factorization and hash function is applied to the message to yields a fixed-size message digest. This
paper explores the PKI architecture based digital signature and presents an efficient way of its implementation
and discusses various issues associated with signature schemes based upon RSA and hash functions. The results
show that signing and verification are much faster in the developed application.
Keywords: Digital Signature, MD5, RSA, SHA1, SHA2
This document discusses using genetic algorithms and particle swarm optimization to optimize software testing by finding the most error-prone paths in a program. It provides an overview of genetic algorithms and particle swarm optimization, describing how they can be applied to generate test cases to discover faults. The paper implements both genetic algorithms and particle swarm optimization on sample problems to find optimal solutions and compares the two approaches. It finds that while genetic algorithms can get trapped in local optima, particle swarm optimization tracks personal and global best positions to move toward global optima without getting stuck.
A comparative analysis on qos multicast routing protocols in MANETsIOSR Journals
Â
Abstract: Simultaneous transmission of data from one sender to multiple receivers is called multicasting.
Several widely used applications require multicasting at least at the logical level. Examples include audio video
teleconferencing, real time video streaming and the maintenance of distributed databases. In many cases it is
advantageous to implement multicasting at the level of the routing algorithm (other approaches would be oneto-all
unicast or the implementation of multicasting at the application layer). In this paper we are presenting a
comparative analysis on various multicast routing protocols in adhoc networks.
Keywords: multicasting, multicast protocols,dynamic core, performance evaluation,Qos Parameters
Factors Influencing the Choice of Promotional Practices of Agricultural Equip...IOSR Journals
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The purpose of this study is to identify the factors influencing the choice of promotional practices of agricultural equipment manufacturers and its influence on annual sales of agricultural equipments. The consumer focus, demand creation, exposure, familiarity, encouragement, reputation, existence and value creation are the factors determining the choice of promotional practices of agricultural equipments by the manufactures. Besides, the consumer focus, demand creation, familiarity, reputation and value creation are positively influencing the annual sales of agricultural equipments. The manufacturers should formulate the promotional measures not only attract the farmers and also dealers in order to increase the dealership and also sales. The manufacturers should create demand for their equipments through proper promotion mix strategies in order to make stakeholders familiar with their equipments and also keep their reputations.
This document discusses predicting optimized EDM machining parameters through thermo-mechanical analysis. It begins with an abstract that outlines the goal of determining optimal machining parameters to maximize material removal rate in EDM of OHNS die steel. It then reviews relevant literature on thermal modeling and optimization of EDM processes. The problem statement indicates the challenges of determining optimal pulse duration, peak current, and voltage to increase material removal rate cost-effectively. The objectives are presented as determining parameters to maximize material removal rate and improve surface integrity of machined workpieces. Finally, the theoretical analysis section outlines the numerical modeling approach using finite element analysis and Joshi and Pande's thermal model of EDM sparks.
Restoration of Images Corrupted by High Density Salt & Pepper Noise through A...IOSR Journals
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Abstract: In this paper an efficient algorithm is proposed for removal of salt & pepper noise from digital images. Salt and pepper noise in images is present due to bit errors in transmission or introduced during the signal acquisition stage. It represents itself as randomly occurring white and black pixels. This noise can be removed using standard Median Filter (SMF), Progressive Switched Median Filter (PSMF) under low density noise conditions. Decision Based Algorithm (DBA) and Modified Decision Based Unsymmetric Trimmed Median Filter (MDBUTMF) do not give better results at high noise density. So, in this project, this drawback will be overcome by using Adaptive Median based Modified Mean Filter (AMMF). This proposed algorithm shows better Peak Signal-to-Noise Ratio and clear image than the existing algorithm. Keywords- Median filter, Progressive Switched Median Filter, Decision Based Algorithm, Modified Decision Based Unsymmetric Trimmed Median Filter
Application of Langmuir-Hinshelwood Model to Bioregeneration of Activated Car...IOSR Journals
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Environmental pollution, high cost and high energy consumption associated with thermal regeneration of activated carbon polluted with hydrocarbon necessitated the search for a better way of regenerating activated carbon, bioregeneration. Spent granular activated carbon was regenerated having been initially characterized using cultured Pseudomonas Putida. The rate of bioregeneration was studied by varying the volume of bacteria from 10ml, 20ml, 30ml and 40ml. The regeneration temperature was also varied from 25oC to ambient temperature of 27oC, 35oC and further at 40 and 45oC over a period of 21 days. The experimental results showed clear correlation when validated using the Langmuir-Hinshelwood kinetic model. The experiment at ambient temperature showed a negative correlation due to the fluctuation in the ambient temperature unlike all other experiment where temperature was controlled in an autoclave machine.
Chitosan capped Silver nanoparticles used as Pressure sensorsIOSR Journals
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In the present work, we report the synthesis and characterization of silver nanoparticles, capped with chitosan (biopolymer ). The majority of the particles produced in this way had sizes around 18 nm. Composite films of capped silver nanoparticles and chitosan polymer were studied to understand the charge transport under different pressure. Films of different compositions were prepared to measure current voltage curves across the film thickness. The results reveal that these materials exhibit electrical conductivity as predicted by the âclassical theory of percolationâ. Pressure dependent electrical conductivity and these composites can be explored to develop low cost pressure sensors.
This document summarizes an experiment that used the Taguchi method to optimize weld bead geometry in submerged arc welds on EN24 steel alloy. The researchers conducted experiments using an orthogonal array to test different combinations of welding voltage, current, and travel speed. Analysis of variance showed that voltage had the greatest influence on weld penetration shape factor. The optimal parameters found were a voltage of 30V, current of 300A, and speed of 30m/hr, which produced the maximum penetration. In conclusion, voltage was identified as the most significant factor controlling weld bead geometry according to this experiment.
Global Perspective and Issues Relating to Product RecallIOSR Journals
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The paper aims to analyze the various issues related to a product recall in the globalized world where the market has become a single trading place. A recall acts like an alarm that makes the company realize that it is the time to take corrective action by reviewing and making changes in the required areas. It is very important to understand the various issues related to the product recall like the factors related to costing, legal framework, recall planning, and managementâs perspective and so on. The paper aims to discuss the issues identified with recall with the examples of global companies that have faced a recall. Further the paper also focuses on the recall framework with respect to India. The paper has a practical implication both for the academicians and for the readers in terms of their concern with the aspect issues regarding product recall.
This document presents an analysis of scheduling a combination of non-preemptive strict periodic tasks and preemptive sporadic tasks. It first reviews related work and defines the task models. It then shows that the schedule of strict periodic tasks has a transient phase and a permanent phase, and proves that the worst-case response times of sporadic tasks can be analyzed by only considering the release times in the permanent phase of strict periodic tasks. Finally, it provides a necessary and sufficient schedulability condition for the sporadic tasks by extending the classical response time analysis to account for the strict periodic tasks.
Role of Virtual Machine Live Migration in Cloud Load BalancingIOSR Journals
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Abstract: Cloud computing has touched almost every field of the life. Hence number of cloud application
consumers is increasing every day and so as the number of application request to the cloud provider. This leads
increment of workload in many of the cloud nodes. The motive to use load balancing concepts in cloud
environment is to efficiently utilize available resources keeping in mind that no any single system is heavily
loaded or not a single system is idle during the active phase of the request completion. Even though cloud
computing being a software facility most often, how does it actually performs well in heavily loaded
environment at processor level, is discussed in the paper. This paper aims to throw some light on what is cloud
load balancing and what is the role of Virtual machine migration in improving it.
Keywords: Cloud load balancing, Live Migration, Migration, Virtualization, Virtual machine.
Design of Adjustable Reconfigurable Wireless Single Core CORDIC based Rake Re...IOSR Journals
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In wireless communication system transmitted signals are subjected to multiple reflections,
diffractions and attenuation caused by obstacles such as buildings and hills, etc. At the receiver end, multiple
copies of the transmitted signal are received that arrive at clearly distinguishable time instants and are faded by
signal cancellation. Rake receiver is a technique to combine these so called multi-paths [2] by utilizing multiple
correlation receivers allocated to those delay positions on which the significant energy arrives which achieves a
significant improvement in the SNR of the output signal. This paper shows how the rake, including dispreading
and descrambling could be replaced by a receiver that can be implemented on a CORDIC based hardware
architecture. The performance in conjunction with the computational requirements of the receiver is widely
adjustable which is significantly better than that of the conventional rake receiver
Ensuring Distributed Accountability for Data Sharing Using Reversible Data Hi...IOSR Journals
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Recently, more and more attention is paid to reversible data hiding (RDH) in encrypted images,
since it maintains the excellent property that the original cover can be lossless recovered after embedded data is
extracted while protecting the image contentâs confidentiality. All previous methods embed data by reversibly
vacating room from the encrypted images, which may be subject to some errors on data extraction and/or image
restoration. In this paper, we propose a novel method by reserving room before encryption with a traditional
RDH algorithm, and thus it is easy for the data hider to reversibly embed data in the encrypted image. The
proposed method can achieve real reversibility, that is, data extraction and image recovery are free of any
error. A major feature of the centralized database services is that usersâ data are usually processed remotely in
unknown machines that users do not own or operate. While enjoying the convenience brought by this new
emerging technology, usersâ fears of losing control of their own data (particularly, financial and health data)
can become a significant barrier to the wide adoption of centralized database services. To address this problem,
in this paper, we propose a novel highly decentralized information accountability framework to keep track of the
actual usage of the userâs data in the cloud Over-lay Network. We leverage the LOG file create a dynamic and
traveling object, and to ensure that any access to usersâ data will trigger authentication and automated logging
local to the LOGs. To strengthen userâs control, we also provide distributed auditing mechanisms. We provide
extensive experimental studies that demonstrate the efficiency and effectiveness of the proposed approaches.
Index Terms : Reversible data hiding, image encryption, privacy protection, data sharing.
This document provides a review of the design of an intelligent restaurant system with a touch screen-based menu ordering system. It discusses using technologies like a multi-touch LCD screen connected to an Arduino mega microcontroller, an RF module for wireless connectivity, a database, and a line-following robot. The system allows customers to browse menus and place orders directly on the touchscreen. Their order is then sent to the database and kitchen display. A line-following robot delivers meals along a predetermined path. The document also reviews related works and discusses the hardware and software components of the proposed system in more detail.
The study examines partially replacing ordinary Portland cement with fly ash and granite powder in mortar mixtures to evaluate compressive strength. Mortar cubes were produced with cement replaced 5-20% by weight of fly ash or granite powder and tested at 7, 28, and 56 days. Results showed 10% replacement of cement with granite powder produced the highest compressive strength of 46.65 MPa at 28 days.
Exploring the Effect of Web Based Communications on Organizations Service Qua...IOSR Journals
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The paper aims to study the effect of web based communications on the service quality of organizations. Web based communications is used for a variety of reasons.The quality of a Web-based customer support system involves the information it supplies, the service it provides, andcharacteristics of the system itself; its effectiveness is reflected by the satisfaction of its users. This paper presents the results of astudy of quality and effectiveness in Web-based customer support systems. Data from a survey of 726 Internet users were used to test theoretically expected relationships. The results of this study indicate that information and system quality determine effectiveness while service quality has no impact. Practical implications for managers and designers are offered.The Internet is the latest in a long succession of communication technologies. The goal of this work is to draw lessons from the evolution of all these services. Little attention is paid to technology as such, since that has changed radically many times. Instead, the stress is on the steady growth in volume of communication, the evolution in the type of traffic sent, the qualitative change this growth produces in how people treat communication, and the evolution of pricing. The focus is on the user, and in particular on how quality and price differentiation have been used by service providers to influenceconsumer behavior, and how consumers have reacted.
âAn analytical study on medal tally of top ten countries in 2012 Olympic Game...IOSR Journals
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The origin of the ancient Olympic Games is lost the midst of pre-history, but for many centuries they
were only a festival of the Greek people. The Games were first held in honour of the Greek God, Zeus in 776
BC in the plain of kingdom of Elis, nestled in lush valley between the Alpheus River and Mount Kronion, 15
km from the Ionian Sea. The Olympiad celebrated that year was considered as the first and was used to date
subsequent historic events.
This document discusses three exercises involving linear transformations. The exercises ask the reader to determine if given functions define linear transformations and to determine the output of linear transformations given their behavior on sample inputs. The document provides the definitions, inputs, and step-by-step workings to solve each exercise. It concludes that exercises 1 and 3 define linear transformations while exercise 2 does not and determines the output of two other linear transformations.
BSC_COMPUTER _SCIENCE_UNIT-2_DISCRETE MATHEMATICSRai University
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This document provides an introduction to definite integration and its applications. It defines indefinite integration as finding the integral or primitive function F(x) of a function f(x). Definite integration involves finding the area under a curve defined by a function f(x) over a specified interval. Standard formulae for integrating common functions like polynomials, trigonometric functions, and exponentials are provided. Methods for integrating functions using substitution and integration by parts are described. Examples of applying these techniques to evaluate definite integrals are also given.
Recurrence relation of Bessel's and Legendre's functionPartho Ghosh
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This presentation tells about use recurrence relation in finding the solution of ordinary differential equations, with special emphasis on Bessel's and Legendre's Function.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
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Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrangeâs Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdfWasswaderrick3
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We look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the exponential and hyperbolic functions as derived in literature of conduction in fins. We use the technique of transforming the PDE into an integral equation. But in the case of radial heat flow, we have to multiply through by r the heat equation and then introduce integrals. We do this to avoid introducing integrals of the form of the exponential integral whose solutions cannot be expressed in the form of a simple mathematical function. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders do not reduce to those of semi-infinite hollow cylinders. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction.
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
BSC_Computer Science_Discrete Mathematics_Unit-IRai University
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This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
- The nth derivative of a function y with respect to x is denoted as d^n y/dx^n.
- Standard formulas are given for finding the nth derivative of functions such as x^m, e^ax, a^x, 1/(ax+b), (ax+b)^m, log(ax+b), sin(ax+b), and cos(ax+b).
- Examples demonstrate calculating specific high-order derivatives such as the 10th derivative of x
BSC_COMPUTER _SCIENCE_UNIT-1_DISCRETE MATHEMATICSRai University
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This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
- The nth derivative of a function y with respect to x is denoted as d^n y/dx^n.
- Formulas are given for finding the nth derivative of functions such as x^m, e^ax, a^x, 1/(ax+b), (ax+b)^m, log(ax+b), sin(ax+b), and cos(ax+b).
- Examples demonstrate calculating specific high-order derivatives, such as the 10th derivative of
This document discusses the metric tensor, which describes how distances are measured in different coordinate systems. It gives the general formula for the metric tensor gij in terms of partial derivatives of the coordinate transformations. As an example, it derives the specific form of the metric tensor in spherical polar coordinates, obtaining the known result that the line element ds2 is given by (dr)2 + r2(dΞ)2 + r2sin2Ξ(dÏ)2.
On Certain Classess of Multivalent Functions iosrjce
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In this we defined certain analytic p-valent function with negative type denoted by íí
. We obtained
sharp results concerning coefficient bounds, distortion theorem belonging to the class íí
.
This document contains mathematical equations and calculations related to signal processing and dynamic systems. It includes equations for:
- The number of levels in a quantization system based on the number of bits
- The height of each quantization level
- Signal to noise ratio based on the number of bits
- The period and frequency of a signal based on its angular frequency
- A transfer function relating input and output of a spring-mass-damper system
- Finding natural frequency and damping ratio from the system transfer function
- Deriving discrete-time transfer functions from continuous transfer functions
The document discusses integration and the definition of the definite integral. It provides 30 rules of integration involving trigonometric, logarithmic, exponential and other common functions. It also briefly discusses integration by substitution and defines the process of making a u-substitution to evaluate integrals that can be written in a particular form.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
B.tech ii unit-5 material vector integrationRai University
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This document discusses various vector integration topics:
1. It defines line, surface, and volume integrals and provides examples of evaluating each. Line integrals deal with vector fields along paths, surface integrals deal with vector fields over surfaces, and volume integrals deal with vector fields throughout a volume.
2. Green's theorem, Stokes' theorem, and Gauss's theorem are introduced as relationships between these types of integrals but their proofs are not shown.
3. Examples are provided to demonstrate evaluating line integrals of conservative and non-conservative vector fields, as well as a surface integral over a spherical surface.
The document discusses hyperbolic functions and their properties. It defines hyperbolic cosine and sine as cosh x = (ex + e-x)/2 and sinh x = (ex - e-x)/2. Some key relationships between hyperbolic and circular functions are given. Formulas for hyperbolic functions of 2x and 3x are provided. The document also discusses differentiation and integration of hyperbolic functions, product formulas, inverse hyperbolic functions, and integration formulas involving hyperbolic functions.
Integration is the reverse process of differentiation. The integral of a function F(x) with respect to x is the antiderivative or indefinite integral represented by the symbol â« F(x)dx = f(x) + c. Several common integral formulae are provided, including integrals involving constants, polynomials, exponentials, logarithms, trigonometric functions, and their inverses.
How do you calculate the particular integral of linear differential equations?
Learn this and much more by watching this video. Here, we learn how the inverse differential operator is used to find the particular integral of trigonometric, exponential, polynomial and inverse hyperbolic functions. Problems are explained with the relevant formulae.
This is useful for graduate students and engineering students learning Mathematics. For more videos, visit my page
https://www.mathmadeeasy.co/about-4
Subscribe to my channel for more videos.
Similar to Jordan Higher (ð, ð)-Centralizer on Prime Ring (20)
This document provides a technical review of secure banking using RSA and AES encryption methodologies. It discusses how RSA and AES are commonly used encryption standards for secure data transmission between ATMs and bank servers. The document first provides background on ATM security measures and risks of attacks. It then reviews related work analyzing encryption techniques. The document proposes using a one-time password in addition to a PIN for ATM authentication. It concludes that implementing encryption standards like RSA and AES can make transactions more secure and build trust in online banking.
This document analyzes the performance of various modulation schemes for achieving energy efficient communication over fading channels in wireless sensor networks. It finds that for long transmission distances, low-order modulations like BPSK are optimal due to their lower SNR requirements. However, as transmission distance decreases, higher-order modulations like 16-QAM and 64-QAM become more optimal since they can transmit more bits per symbol, outweighing their higher SNR needs. Simulations show lifetime extensions up to 550% are possible in short-range networks by using higher-order modulations instead of just BPSK. The optimal modulation depends on transmission distance and balancing the energy used by electronic components versus power amplifiers.
This document provides a review of mobility management techniques in vehicular ad hoc networks (VANETs). It discusses three modes of communication in VANETs: vehicle-to-infrastructure (V2I), vehicle-to-vehicle (V2V), and hybrid vehicle (HV) communication. For each communication mode, different mobility management schemes are required due to their unique characteristics. The document also discusses mobility management challenges in VANETs and outlines some open research issues in improving mobility management for seamless communication in these dynamic networks.
This document provides a review of different techniques for segmenting brain MRI images to detect tumors. It compares the K-means and Fuzzy C-means clustering algorithms. K-means is an exclusive clustering algorithm that groups data points into distinct clusters, while Fuzzy C-means is an overlapping clustering algorithm that allows data points to belong to multiple clusters. The document finds that Fuzzy C-means requires more time for brain tumor detection compared to other methods like hierarchical clustering or K-means. It also reviews related work applying these clustering algorithms to segment brain MRI images.
1) The document simulates and compares the performance of AODV and DSDV routing protocols in a mobile ad hoc network under three conditions: when users are fixed, when users move towards the base station, and when users move away from the base station.
2) The results show that both protocols have higher packet delivery and lower packet loss when users are either fixed or moving towards the base station, since signal strength is better in those scenarios. Performance degrades when users move away from the base station due to weaker signals.
3) AODV generally has better performance than DSDV, with higher throughput and packet delivery rates observed across the different user mobility conditions.
This document describes the design and implementation of 4-bit QPSK and 256-bit QAM modulation techniques using MATLAB. It compares the two techniques based on SNR, BER, and efficiency. The key steps of implementing each technique in MATLAB are outlined, including generating random bits, modulation, adding noise, and measuring BER. Simulation results show scatter plots and eye diagrams of the modulated signals. A table compares the results, showing that 256-bit QAM provides better performance than 4-bit QPSK. The document concludes that QAM modulation is more effective for digital transmission systems.
The document proposes a hybrid technique using Anisotropic Scale Invariant Feature Transform (A-SIFT) and Robust Ensemble Support Vector Machine (RESVM) to accurately identify faces in images. A-SIFT improves upon traditional SIFT by applying anisotropic scaling to extract richer directional keypoints. Keypoints are processed with RESVM and hypothesis testing to increase accuracy above 95% by repeatedly reprocessing images until the threshold is met. The technique was tested on similar and different facial images and achieved better results than SIFT in retrieval time and reduced keypoints.
This document studies the effects of dielectric superstrate thickness on microstrip patch antenna parameters. Three types of probes-fed patch antennas (rectangular, circular, and square) were designed to operate at 2.4 GHz using Arlondiclad 880 substrate. The antennas were tested with and without an Arlondiclad 880 superstrate of varying thicknesses. It was found that adding a superstrate slightly degraded performance by lowering the resonant frequency and increasing return loss and VSWR, while decreasing bandwidth and gain. Specifically, increasing the superstrate thickness or dielectric constant resulted in greater changes to the antenna parameters.
This document describes a wireless environment monitoring system that utilizes soil energy as a sustainable power source for wireless sensors. The system uses a microbial fuel cell to generate electricity from the microbial activity in soil. Two microbial fuel cells were created using different soil types and various additives to produce different current and voltage outputs. An electronic circuit was designed on a printed circuit board with components like a microcontroller and ZigBee transceiver. Sensors for temperature and humidity were connected to the circuit to monitor the environment wirelessly. The system provides a low-cost way to power remote sensors without needing battery replacement and avoids the high costs of wiring a power source.
1) The document proposes a model for a frequency tunable inverted-F antenna that uses ferrite material.
2) The resonant frequency of the antenna can be significantly shifted from 2.41GHz to 3.15GHz, a 31% shift, by increasing the static magnetic field placed on the ferrite material.
3) Altering the permeability of the ferrite allows tuning of the antenna's resonant frequency without changing the physical dimensions, providing flexibility to operate over a wide frequency range.
This document summarizes a research paper that presents a speech enhancement method using stationary wavelet transform. The method first classifies speech into voiced, unvoiced, and silence regions based on short-time energy. It then applies different thresholding techniques to the wavelet coefficients of each region - modified hard thresholding for voiced speech, semi-soft thresholding for unvoiced speech, and setting coefficients to zero for silence. Experimental results using speech from the TIMIT database corrupted with white Gaussian noise at various SNR levels show improved performance over other popular denoising methods.
This document reviews the design of an energy-optimized wireless sensor node that encrypts data for transmission. It discusses how sensing schemes that group nodes into clusters and transmit aggregated data can reduce energy consumption compared to individual node transmissions. The proposed node design calculates the minimum transmission power needed based on received signal strength and uses a periodic sleep/wake cycle to optimize energy when not sensing or transmitting. It aims to encrypt data at both the node and network level to further optimize energy usage for wireless communication.
This document discusses group consumption modes. It analyzes factors that impact group consumption, including external environmental factors like technological developments enabling new forms of online and offline interactions, as well as internal motivational factors at both the group and individual level. The document then proposes that group consumption modes can be divided into four types based on two dimensions: vertical (group relationship intensity) and horizontal (consumption action period). These four types are instrument-oriented, information-oriented, enjoyment-oriented, and relationship-oriented consumption modes. Finally, the document notes that consumption modes are dynamic and can evolve over time.
The document summarizes a study of different microstrip patch antenna configurations with slotted ground planes. Three antenna designs were proposed and their performance evaluated through simulation: a conventional square patch, an elliptical patch, and a star-shaped patch. All antennas were mounted on an FR4 substrate. The effects of adding different slot patterns to the ground plane on resonance frequency, bandwidth, gain and efficiency were analyzed parametrically. Key findings were that reshaping the patch and adding slots increased bandwidth and shifted resonance frequency. The elliptical and star patches in particular performed better than the conventional design. Three antenna configurations were selected for fabrication and measurement based on the simulations: a conventional patch with a slot under the patch, an elliptical patch with slots
1) The document describes a study conducted to improve call drop rates in a GSM network through RF optimization.
2) Drive testing was performed before and after optimization using TEMS software to record network parameters like RxLevel, RxQuality, and events.
3) Analysis found call drops were occurring due to issues like handover failures between sectors, interference from adjacent channels, and overshooting due to antenna tilt.
4) Corrective actions taken included defining neighbors between sectors, adjusting frequencies to reduce interference, and lowering the mechanical tilt of an antenna.
5) Post-optimization drive testing showed improvements in RxLevel, RxQuality, and a reduction in dropped calls.
This document describes the design of an intelligent autonomous wheeled robot that uses RF transmission for communication. The robot has two modes - automatic mode where it can make its own decisions, and user control mode where a user can control it remotely. It is designed using a microcontroller and can perform tasks like object recognition using computer vision and color detection in MATLAB, as well as wall painting using pneumatic systems. The robot's movement is controlled by DC motors and it uses sensors like ultrasonic sensors and gas sensors to navigate autonomously. RF transmission allows communication between the robot and a remote control unit. The overall aim is to develop a low-cost robotic system for industrial applications like material handling.
This document reviews cryptography techniques to secure the Ad-hoc On-Demand Distance Vector (AODV) routing protocol in mobile ad-hoc networks. It discusses various types of attacks on AODV like impersonation, denial of service, eavesdropping, black hole attacks, wormhole attacks, and Sybil attacks. It then proposes using the RC6 cryptography algorithm to secure AODV by encrypting data packets and detecting and removing malicious nodes launching black hole attacks. Simulation results show that after applying RC6, the packet delivery ratio and throughput of AODV increase while delay decreases, improving the security and performance of the network under attack.
The document describes a proposed modification to the conventional Booth multiplier that aims to increase its speed by applying concepts from Vedic mathematics. Specifically, it utilizes the Urdhva Tiryakbhyam formula to generate all partial products concurrently rather than sequentially. The proposed 8x8 bit multiplier was coded in VHDL, simulated, and found to have a path delay 44.35% lower than a conventional Booth multiplier, demonstrating its potential for higher speed.
This document discusses image deblurring techniques. It begins by introducing image restoration and focusing on image deblurring. It then discusses challenges with image deblurring being an ill-posed problem. It reviews existing approaches to screen image deconvolution including estimating point spread functions and iteratively estimating blur kernels and sharp images. The document also discusses handling spatially variant blur and summarizes the relationship between the proposed method and previous work for different blur types. It proposes using color filters in the aperture to exploit parallax cues for segmentation and blur estimation. Finally, it proposes moving the image sensor circularly during exposure to prevent high frequency attenuation from motion blur.
This document describes modeling an adaptive controller for an aircraft roll control system using PID, fuzzy-PID, and genetic algorithm. It begins by introducing the aircraft roll control system and motivation for developing an adaptive controller to minimize errors from noisy analog sensor signals. It then provides the mathematical model of aircraft roll dynamics and describes modeling the real-time flight control system in MATLAB/Simulink. The document evaluates PID, fuzzy-PID, and PID-GA (genetic algorithm) controllers for aircraft roll control and finds that the PID-GA controller delivers the best performance.
We present the JWST discovery of SNâ2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
â
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SNâIa, SNâ2023adsy is both fairly red (
ᅵ
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(
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â
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)
âŒ
0.9
) despite a host galaxy with low-extinction and has a high CaâII velocity (
19
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000
±
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000
km/s) compared to the general population of SNeâIa. While these characteristics are consistent with some Ca-rich SNeâIa, particularly SNâ2016hnk, SNâ2023adsy is intrinsically brighter than the low-
ᅵ
Ca-rich population. Although such an object is too red for any low-
ᅵ
cosmological sample, we apply a fiducial standardization approach to SNâ2023adsy and find that the SNâ2023adsy luminosity distance measurement is in excellent agreement (
â²
1
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) with
Î
CDM. Therefore unlike low-
ᅵ
Ca-rich SNeâIa, SNâ2023adsy is standardizable and gives no indication that SNâIa standardized luminosities change significantly with redshift. A larger sample of distant SNeâIa is required to determine if SNâIa population characteristics at high-
ᅵ
truly diverge from their low-
ᅵ
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
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Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
PPT on Alternate Wetting and Drying presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)eitps1506
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Description:
Dive into the fascinating realm of solid-state physics with our meticulously crafted online PowerPoint presentation. This immersive educational resource offers a comprehensive exploration of the fundamental concepts, theories, and applications within the realm of solid-state physics.
From crystalline structures to semiconductor devices, this presentation delves into the intricate principles governing the behavior of solids, providing clear explanations and illustrative examples to enhance understanding. Whether you're a student delving into the subject for the first time or a seasoned researcher seeking to deepen your knowledge, our presentation offers valuable insights and in-depth analyses to cater to various levels of expertise.
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Crystal Structures: Unravel the mysteries of crystalline arrangements and their significance in determining material properties.
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Semiconductor Physics: Delve into the behavior of semiconductors, including doping, carrier transport, and device applications.
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With visually engaging slides, informative content, and interactive elements, our online PowerPoint presentation serves as a valuable resource for students, educators, and enthusiasts alike, facilitating a deeper understanding of the captivating world of solid-state physics. Explore the intricacies of solid-state materials and unlock the secrets behind their remarkable properties with our comprehensive presentation.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
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Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ⌠106Mâ black hole (BH) that is currently in the process of âturning onâ. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1âW2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ⌠3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ⌠106Mâ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: activeâ accretion, accretion discsâ galaxies: individual: SDSS J133519.91+072807.4
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
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Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of GÀvle, Sweden.
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptxshubhijain836
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Centrifugation is a powerful technique used in laboratories to separate components of a heterogeneous mixture based on their density. This process utilizes centrifugal force to rapidly spin samples, causing denser particles to migrate outward more quickly than lighter ones. As a result, distinct layers form within the sample tube, allowing for easy isolation and purification of target substances.
The cost of acquiring information by natural selectionCarl Bergstrom
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This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Jordan Higher (ð, ð)-Centralizer on Prime Ring
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 6 (May. - Jun. 2013), PP 05-11
www.iosrjournals.org
www.iosrjournals.org 5 | Page
Jordan Higher (ð, ð)-Centralizer on Prime Ring
Salah M.Salih, Marwa M.Shaapan
Department of Mathematics, College of Education, Al-Mustansiriya University.
Abstract: Let ð be a ring and ð, ð be an endomorphisms of ð , in this paper we will present and study the
concepts of higher (ð, ð)-centralizer, Jordan higher(ð, ð)-centralizer and Jordan triple higher (ð, ð)-
centralizer and their generalization on the ring. The main results are prove that every Jordan higher (ð, ð)-
centralizer of prime ring ð is higher (ð, ð)-centralizer of ð and we prove let ð be a 2-torsion free ring,ð ððð ð
are commutative endomorphism then every Jordan higher (ð, ð)-centralizer is Jordan triple higher (ð, ð)-
centralizer.
Mathematics Subject Classification: 16A12,16N60,16W25,16Y99.
Keywords: higher (ð, ð)-centralizer, Jordan higher (ð, ð)-cenralizer, Jordan triple higher (ð, ð)-centralizr
I. Introcation
Throughout this paper, ð is a ring and ð is called prime if ðð ð = (0) implies ð = 0 ðð ð = 0 and ð is
semiprime if ðð ð = (0) implies ð = 0. A mapping ð: ð â ð is called derivation if ð ð¥ðŠ = ð ð¥ ðŠ + ð¥ð(ðŠ)
holds for all ð¥, ðŠðð . A left(right) centralizer of ð is an additive mapping ð: ð â ð which satisfies ð ð¥ðŠ =
ð ð¥ ðŠ (ð ð¥ðŠ = ð¥ð(ðŠ) for all ð¥, ðŠðð .
In [6] B.Zalar worked on centralizer of semiprime rings and defined that let R be a semiprimering. A left
(right) centralizer of R is an additive mapping ð: ð â ð satisfying ð ð¥ðŠ = ð ð¥ ðŠ ð ð¥ðŠ = ð¥ð ðŠ for all
ð¥, ðŠ â ð . If ð is a left and a right centralizer then ð is a centralizer. An additive mapping ð: ð â ð is called a
Jordan centralizer if ð satisfies ð ð¥ðŠ + ðŠð¥ = ð ð¥ ðŠ + ðŠð ð¥ = ð ðŠ ð¥ + ð¥ð ðŠ for all ð¥, ðŠ â ð . A left (right)
Jordan centralizer of R is an additive mapping ð: ð â ð such that ð ð¥2
= ð ð¥ ð¥ (ð ð¥2
= ð¥ð ð¥ ) for all
ð¥ â ð .
Joso Vakman [3,4,5] developed some Remarks able results using centralizers on prime and semiprime rings.
E.Albas [1] developed some remarks able results using ð-centralizer of semiprime rings.
W.Cortes and G.Haetinger [2] defined a left(resp.right) Jordan ð-centralizer and proved any
left(resp.right) Jordan ð-centralizer of 2-torsion free ring is a left(resp.right) ð-centralizer.
In this paper, we present the concept of higher (ð, ð)-centralizer, Jordan higher(ð, ð)-centralizer and Jordan
triple higher (ð, ð)-centralizer and We have also prove that every Jordan higher (ð, ð)-centralizer of prime ring
is higher (ð, ð)-centralizer Also prove that every Jordan higher (ð, ð)-centralizer of 2-torsion free ring is Jordan
triple higher (ð, ð)-centralizer.
II. Higher(Ï,Ï )-Centralizer
Now we will the definition of higher (ð, ð)-centralizer, Jordan higher(ð, ð)-centralizer and Jordan triple
higher (ð, ð)-centralizer on the ring ð and other concepts which be used in our work.
We start work with the following definition:-
Definition ( 2.1 ):
let R be a ring, Ï and Ï are endomorphism of R and ð = (ð¡ð)ðâð be a family of additive mappings of
ð then ð is said to be higher (Ï, Ï)- centralizer of ð if
ð¡ ð (ð¥ðŠ + ð§ð¥) = ð¡ð ð¥
ð
ð=1
ð ð
(ðŠ) + ð ð
(ð§)ð¡ð(ð¥)
for all ð¥, ðŠ, ð§ðð and ðð ð
the following is an example of higher (Ï, Ï)-centralizer .
Example ( 2.2 ):
Let ð = {
ð¥ ð
ð ð¥
: ð¥ ð ðŒ} be a ring, ð and ð are endomorphism of ð such that
ð ð ð¥ ð
ð ð¥
=
ð¥
ð
ð
ð ð
ððð ð ð ð¥ ð
ð ð¥
=
ð¥
ð
ð
ð ð
we use the usual addition and multiplication on matrices of ð ,and let
ð¡ ð ⶠð â ð be additive mapping defined by
2. Jordan Higher (ð, ð)-Centralizer On Prime Ring
www.iosrjournals.org 6 | Page
ð¡ ð
ð¥ ð
ð ð¥
=
ðð¥ 0
0 0
ðð
ð¥ 0
0 ð¥
â ð ððð ð ⥠1.
it is clear that ð = (ð¡ð)ðâð is higher (ð, ð)-centralizer
Definition ( 2.3 ):
Let ð be a ring, ð and ð are endomorphism of ð and ð = (ð¡ð)ðâð be a family of additive mappings of ð
then ð is said to be Jordan higher (ð, ð)-centralizer of ð if for every ð¥, ðŠ ð ð ððð ðð ð
ð¡ ð ð¥ðŠ + ðŠð¥ = ð¡ð ð¥ ð ð
ðŠ + ð ð
ðŠ ð¡ð ð¥
ð
ð=1
It is clear that every higher (ð, ð)-centralizer of a ring ð is Jordan higher (ð, ð)-centralizer of ð , but the
converse is not true in general as shown by the following example .
Example ( 2.4):
Let ð be a ring , ð¡ = (ð¡ð)ðâð be a higher (ð, ð)-centralizer and ð,ð are endomorphism of ð define ð 1 =
ð¥, ð¥ : ð¥ â ð , ð1 ððð ð1 are endomorphism of ð 1 defined by ð1
ð
ð¥, ð¥ = ðð ð¥ , ðð ð¥ ððð ð1
ð
ð¥, ð¥ =
(ðð ð¥ , ðð ð¥ ) such that ð¥1 ð¥2 â ð¥2 ð¥1 but ð¥1 ð¥3 = ð¥3 ð¥1 for all ð¥1 ð¥2 ð¥3 â ð . Let the operation of addition and
multiplication on R defined by
ð¥1, ð¥1 + ð¥2, ð¥2 = (ð¥1 + ð¥2, ð¥1 + ð¥2)
ð¥1, ð¥1 ð¥2, ð¥2 = (ð¥1 ð¥2, ð¥1 ð¥2) , for every ð¥1, ð¥2 â ð . let ð¡ ð : ð â ð be a higher (ð, ð)-centralizer mapping, and
let ðð : ð 1 â ð 1 be a higher (ð, ð)-centralizer mapping defined as ðð ð¥, ð¥ = (ð¡ ð (ð¥), ð¡ ð (ð¥)) for ððð . then ðð is
a Jordan higher (ð1, ð1)-centralizer which is not higher (ð1, ð1)-centralizer of ð 1.
Definition ( 2.5 ):
let ð be a ring, ð and ð be endomorphism of ð and ð = (ð¡ð)ðâð be a family of additive mappings of ð
then ð is said to be a Jordan triple higher (ð, ð)-centralizer of ð if for every ð¥ , ðŠ ð ð and ðð ð
ð¡ ð (ð¥ðŠð¥) = ð¡ð(ð¥)
ð
ð=1
ð ð
(ðŠ)ðð(ð¥)
Now, we give some properties of higher (ð, ð)-centralizer of ð .
lemma ( 2.6 ):
Let ð be a ring, ð and ð are endomorphism of ð and ð = (ð¡ð)ðâð be higher (ð, ð)-centralizer of ð then for
all ð¥, ðŠ, ð§, ð, ð€ðð and ð , the following statements holds
ð ð¡ ð ð¥ðŠ + ð§ð€ = ð¡ð ð¥ ð ð
(ðŠ)
ð
ð=1
+ ð ð
ð§ ð¡ð(ð€)
ðð In particular ð(ð ) is commutative
ð¡ ð ð¥ðŠð¥ + ð¥ðŠð¥ = ð¡ð ð¥ ð ð
ðŠ ð ð
ð¥ + ð ð
ð¥ ð ð
ðŠ ð¡ð ð¥
ð
ð=1
ððð In particular ð¡ð ð§ ð ð
ðŠ ð ð
ð¥ = ð ð
ð§ ð ð
ðŠ ð¡ð(ð¥)
ð¡ ð ð¥ðŠð§ + ð§ðŠð¥ = ð¡ð ð¥ ð ð
ðŠ ð ð
ð§ + ð ð
ð§ ð ð
ðŠ ð¡ð ð¥
ð
ð=1
ðð£ In particular ð(ð ) is commutative
ð¡ ð ð¥ðŠð + ð§ð€ð¥ = ð¡ð ð¥ ð ð
ðŠ ð ð
ð + ð ð
ð§ ð ð
ð€ ð¡ð ð¥
ð
ð=1
Proof:
ð Replace ð¥ + ð€ for ð¥ and ðŠ + ð for ðŠ and ð§ + ð for ð§ in definition (2.1) we get
ð¡ ð ( ð¥ + ð€ ðŠ + ð + ð§ + ð ð¥ + ð€ )
= ð¡ð ð¥ + ð€ ð ð
ðŠ + ð + ð ð
ð§ + ð ð¡ð ð¥ + ð€
ð
ð=1
= ð¡ð ð¥ ð ð
ðŠ + ð¡ð ð¥ ð ð
ð + ð¡ð ð€ ð ð
ðŠ + ð¡ð ð€ ð ð
ð +
ð
ð=1
ð ð
ð§ ð¡ð ð¥ + ð ð
ð§ ð¡ð ð€ + ð ð
ð ð¡ð ð¥ + ð ð
ð ð¡ð ð€
...(1)
On the other hand