In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and...In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and operator splitting techniques,the resulting saddle-point problem is solved by a serial of subproblems.To tackle the nonlinear constraints arising in the model,a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution.We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic,real-world and medical images for image denoising,image inpainting and image zooming problems.展开更多
Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construc...Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.展开更多
Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna...Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.展开更多
For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime...For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.展开更多
In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and dedu...In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.展开更多
A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to...A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to micro-electromechanical sensors. Although the bending of a uniform rod is well studied and gives rise to mathematical shapes described by elliptic integrals, no exact closed form solution to the nonlinear differential equations of static equilibrium is known for the deflection of a tapered rod. We report in this paper a comprehensive numerical analysis and experimental test of the exact theory of bending deformation of a tapered rod. Given the rod geometry and elastic modulus, the theory yields virtually all the geometric and physical features that an analyst, experimenter, or instrument designer might want as a function of impressed load, such as the exact curve of deformation (termed the elastica), maximum tip displacement, maximum tip deflection angle, distribution of curvature, and distribution of bending moment. Applied experimentally, the theory permits rapid estimation of the elastic modulus of a rod, which is not easily obtainable by other means. We have tested the theory by photographing the shapes of a set of flexible rods of different lengths and tapers subject to a range of impressed loads and using digital image analysis to extract the coordinates of the elastica curves. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the equations of equilibrium or neglect tapering of the rod. Agreement between the measured deflection curves and the exact theoretical predictions was excellent in all but several cases. In these exceptional cases, the nature of the anomalies provided important information regarding the deviation of the rods from an ideal Euler-Bernoulli cantilever, which thereby permitted us to model the deformation of the rods more accurately.展开更多
The depth from extreme points(DEXP)method can be used for estimating source depths and providing a rough image as a starting model for inversion.However,the application of the DEXP method is limited by the lack of pri...The depth from extreme points(DEXP)method can be used for estimating source depths and providing a rough image as a starting model for inversion.However,the application of the DEXP method is limited by the lack of prior information regarding the structural index.Herein,we describe an automatic DEXP method derived from Euler’s Homogeneity equation,and we call it the Euler–DEXP method.We prove that its scaling field is independent of structural indices,and the scaling exponent is a constant for any potential field or its derivative.Therefore,we can simultaneously estimate source depths with diff erent geometries in one DEXP image.The implementation of the Euler–DEXP method is fully automatic.The structural index can be subsequently determined by utilizing the estimated depth.This method has been tested using synthetic cases with single and multiple sources.All estimated solutions are in accordance with theoretical source parameters.We demonstrate the practicability of the Euler–DEXP method with the gravity field data of the Hastings Salt Dome.The results ultimately represent a better understanding of the geometry and depth of the salt dome.展开更多
Solutions to the differential equation in Smith’s Prize Examination taken by Maxwell are discussed. It was a competitive examination using which skill full students were identified and James Clerk Maxwell was one of ...Solutions to the differential equation in Smith’s Prize Examination taken by Maxwell are discussed. It was a competitive examination using which skill full students were identified and James Clerk Maxwell was one of them. He later formulated the theory of Electromagnetism and predicted the light speed & its value was subsequently confirmed by experiments. Light travel in a direction perpendicular to oscillating electric and magnetic field through a vacuum from sun. In the same exam paper, Maxwell answered the question related to Stokes Theorem of vector calculus which was used in the formalism of Electromagnetic theory.展开更多
As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the fi...As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.展开更多
In this work, we apply the Zhou’s method [1] or differential transformation method (DTM) for solving the Euler equidimensional equation. The Zhou’s method may be considered as alternative and efficient for finding t...In this work, we apply the Zhou’s method [1] or differential transformation method (DTM) for solving the Euler equidimensional equation. The Zhou’s method may be considered as alternative and efficient for finding the approximate solutions of initial values problems. We prove superiority of this method by applying them on the some Euler type equation, in this case of order 2 and 3 [2]. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equations. The results agreed with the exact solution obtained via transformation to a constant coefficient equation.展开更多
The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable ...The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10<sup>-</sup><sup>16</sup>. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.展开更多
文摘In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and operator splitting techniques,the resulting saddle-point problem is solved by a serial of subproblems.To tackle the nonlinear constraints arising in the model,a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution.We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic,real-world and medical images for image denoising,image inpainting and image zooming problems.
基金supported by the NNSF of China grants 11526110,11271069,61362036 and 61461032,the 863 Program of China grant 2015AA01A302the Open Research Fund of Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing(2016WICSIP013)the Youth Foundation of Nanchang Institute of Technology(2014KJ021).
文摘Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.
文摘Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.
基金Foundation item： Supported by the National Natural Science Foundation of China（10671056）
文摘For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.
基金Supported by the PCSIRT of Education of China(IRT0621)Supported by the Innovation Program of Shanghai Municipal Education Committee of China(08ZZ24)Supported by the Henan Innovation Project for University Prominent Research Talents of China(2007KYCX0021)
文摘In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.
文摘A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to micro-electromechanical sensors. Although the bending of a uniform rod is well studied and gives rise to mathematical shapes described by elliptic integrals, no exact closed form solution to the nonlinear differential equations of static equilibrium is known for the deflection of a tapered rod. We report in this paper a comprehensive numerical analysis and experimental test of the exact theory of bending deformation of a tapered rod. Given the rod geometry and elastic modulus, the theory yields virtually all the geometric and physical features that an analyst, experimenter, or instrument designer might want as a function of impressed load, such as the exact curve of deformation (termed the elastica), maximum tip displacement, maximum tip deflection angle, distribution of curvature, and distribution of bending moment. Applied experimentally, the theory permits rapid estimation of the elastic modulus of a rod, which is not easily obtainable by other means. We have tested the theory by photographing the shapes of a set of flexible rods of different lengths and tapers subject to a range of impressed loads and using digital image analysis to extract the coordinates of the elastica curves. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the equations of equilibrium or neglect tapering of the rod. Agreement between the measured deflection curves and the exact theoretical predictions was excellent in all but several cases. In these exceptional cases, the nature of the anomalies provided important information regarding the deviation of the rods from an ideal Euler-Bernoulli cantilever, which thereby permitted us to model the deformation of the rods more accurately.
基金supported by the National Natural Science Foundation of China (Grant No.42176186).
文摘The depth from extreme points(DEXP)method can be used for estimating source depths and providing a rough image as a starting model for inversion.However,the application of the DEXP method is limited by the lack of prior information regarding the structural index.Herein,we describe an automatic DEXP method derived from Euler’s Homogeneity equation,and we call it the Euler–DEXP method.We prove that its scaling field is independent of structural indices,and the scaling exponent is a constant for any potential field or its derivative.Therefore,we can simultaneously estimate source depths with diff erent geometries in one DEXP image.The implementation of the Euler–DEXP method is fully automatic.The structural index can be subsequently determined by utilizing the estimated depth.This method has been tested using synthetic cases with single and multiple sources.All estimated solutions are in accordance with theoretical source parameters.We demonstrate the practicability of the Euler–DEXP method with the gravity field data of the Hastings Salt Dome.The results ultimately represent a better understanding of the geometry and depth of the salt dome.
文摘Solutions to the differential equation in Smith’s Prize Examination taken by Maxwell are discussed. It was a competitive examination using which skill full students were identified and James Clerk Maxwell was one of them. He later formulated the theory of Electromagnetism and predicted the light speed & its value was subsequently confirmed by experiments. Light travel in a direction perpendicular to oscillating electric and magnetic field through a vacuum from sun. In the same exam paper, Maxwell answered the question related to Stokes Theorem of vector calculus which was used in the formalism of Electromagnetic theory.
基金Supported by the National Natural Science Foundation Fujian province of China(2016J01032).
文摘As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.
文摘In this work, we apply the Zhou’s method [1] or differential transformation method (DTM) for solving the Euler equidimensional equation. The Zhou’s method may be considered as alternative and efficient for finding the approximate solutions of initial values problems. We prove superiority of this method by applying them on the some Euler type equation, in this case of order 2 and 3 [2]. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equations. The results agreed with the exact solution obtained via transformation to a constant coefficient equation.
文摘The Owen’s T function is presented in four new ways, one of them as a series similar to the Euler’s arctangent series divided by 2π, which is its majorant series. All possibilities enable numerically stable and fast convergent computation of the bivariate normal integral with simple recursion. When tested computation on a random sample of one million parameter triplets with uniformly distributed components and using double precision arithmetic, the maximum absolute error was 3.45 × 10<sup>-</sup><sup>16</sup>. In additional testing, focusing on cases with correlation coefficients close to one in absolute value, when the computation may be very sensitive to small rounding errors, the accuracy was retained. In rare potentially critical cases, a simple adjustment to the computation procedure was performed—one potentially critical computation was replaced with two equivalent non-critical ones. All new series are suitable for vector and high-precision computation, assuming they are supplemented with appropriate efficient and accurate computation of the arctangent and standard normal cumulative distribution functions. They are implemented by the R package Phi2rho, available on CRAN. Its functions allow vector arguments and are ready to work with the Rmpfr package, which enables the use of arbitrary precision instead of double precision numbers. A special test with up to 1024-bit precision computation is also presented.
