The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the self- affine measu...The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the self- affine measures have some surprising connections with a number of areas in mathematics, and have been received much attention in recent years. In the present paper, we shall determine the spectrality and non-spectrality of a class of self-aiffine measures with decomposable digit sets. We present a method to deal with such case, and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.展开更多
In this paper, we investigate complex homogeneous and non-homogeneous higher order linear differential equations with meromorphic coefficients. We obtain several results concerning the iterated order of meromorphic so...In this paper, we investigate complex homogeneous and non-homogeneous higher order linear differential equations with meromorphic coefficients. We obtain several results concerning the iterated order of meromorphic solutions, and the iterated convergence exponent of the zeros of meromorphic solutions.展开更多
The self-affine measure μM,D associated with an expanding matrix M ∈ Mn(Z) and a finite digit set D ? Znis uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functio...The self-affine measure μM,D associated with an expanding matrix M ∈ Mn(Z) and a finite digit set D ? Znis uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functions E(Λ) := {e2πiλ,x : λ∈Λ} in the Hilbert space L2(μM,D) is simply called μM,D-orthogonal exponentials. We consider in this paper the finiteness of μM,D-orthogonality. A necessary and sufficient condition is obtained for the set E(Λ) to be a finite μM,D-orthogonal exponentials. The research here is closely connected with the non-spectrality of self-affine measures.展开更多
Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results,since without this nonlinearity,the results of the network will be less accurate.Nonlinearity...Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results,since without this nonlinearity,the results of the network will be less accurate.Nonlinearity is the mission of all nonlinear functions,except for polynomials.The activation function must be dif-ferentiable for backpropagation learning.This study’s objective is to determine the best activation functions for the approximation of each fractal image.Different results have been attained using Matlab and Visual Basic programs,which indi-cate that the bounded function is more helpful than other functions.The non-lin-earity of the activation function is important when using neural networks for coding fractal images because the coefficients of the Iterated Function System are different according to the different types of fractals.The most commonly cho-sen activation function is the sigmoidal function,which produces a positive value.Other functions,such as tansh or arctan,whose values can be positive or negative depending on the network input,tend to train neural networks faster.The coding speed of the fractal image is different depending on the appropriate activation function chosen for each fractal shape.In this paper,we have provided the appro-priate activation functions for each type of system of iterated functions that help the network to identify the transactions of the system.展开更多
In this paper, we investigate the complex oscillation of higher order homogenous and non- homogeneous linear differential equations with meromorphic coefficients of iterated order, and obtain some results which improv...In this paper, we investigate the complex oscillation of higher order homogenous and non- homogeneous linear differential equations with meromorphic coefficients of iterated order, and obtain some results which improve and extend those given by Z. X. Chen, L. Kinnunen, etc.展开更多
The authors derive laws of the iterated logarithm for kernel estimator of regression function based on directional data. The results are distribution free in the sense that they are true for all distributions of desig...The authors derive laws of the iterated logarithm for kernel estimator of regression function based on directional data. The results are distribution free in the sense that they are true for all distributions of design variable.展开更多
For an expanding integer matrix M ∈ Mn（Z） and two finite digit sets D, S C Z^n with O∈ D ∩ S, we shall investigate and study the possible conditions on the spectral pair （μM,D,A（M, S）） associated with the it...For an expanding integer matrix M ∈ Mn（Z） and two finite digit sets D, S C Z^n with O∈ D ∩ S, we shall investigate and study the possible conditions on the spectral pair （μM,D,A（M, S）） associated with the iterated function systems {Фd（X） = M^-1（x ＋ d）}d∈D and {φs（x） = M^＊x ＋ s}s∈S in the case when ｜D｜= ｜S｜ = ｜ det（M）｜. Under the condition that （M^-1D, S） is a compatible pair, we obtain a series of necessary and sufficient conditions for （μM,D, A（M, S）） to be a spectral pair. These conditions include how to characterize the invariant sets A（M, S） and T（M, D） such that A（M, S） = Zn and μL（T（M, D）） = 1 which play an important role in the number system research and in the construction of Haar-type orthogonal wavelet basis respectively.展开更多
In this paper we study the estimation of the regression function.We establish a law ofthe iterated logarithm for the random window-width kernel estimator and,as an application,fora nearest neighbor estimator.These res...In this paper we study the estimation of the regression function.We establish a law ofthe iterated logarithm for the random window-width kernel estimator and,as an application,fora nearest neighbor estimator.These results give sharp pointwise rates of strong consistency ofthese estimators.展开更多
We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is ob...We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition recently introduced by Ngai and Wang.展开更多
The design, analysis and parallel implementation of particle filter(PF) were investigated. Firstly, to tackle the particle degeneracy problem in the PF, an iterated importance density function(IIDF) was proposed, wher...The design, analysis and parallel implementation of particle filter(PF) were investigated. Firstly, to tackle the particle degeneracy problem in the PF, an iterated importance density function(IIDF) was proposed, where a new term associating with the current measurement information(CMI) was introduced into the expression of the sampled particles. Through the repeated use of the least squares estimate, the CMI can be integrated into the sampling stage in an iterative manner, conducing to the greatly improved sampling quality. By running the IIDF, an iterated PF(IPF) can be obtained. Subsequently, a parallel resampling(PR) was proposed for the purpose of parallel implementation of IPF, whose main idea was the same as systematic resampling(SR) but performed differently. The PR directly used the integral part of the product of the particle weight and particle number as the number of times that a particle was replicated, and it simultaneously eliminated the particles with the smallest weights, which are the two key differences from the SR. The detailed implementation procedures on the graphics processing unit of IPF based on the PR were presented at last. The performance of the IPF, PR and their parallel implementations are illustrated via one-dimensional numerical simulation and practical application of passive radar target tracking.展开更多
Mass distribution principle is one of important tools in studying Hausdorff dimension and Hausdorff measure. In this paper we will give a numerical approximate method of upper bound and lower bound of mass distributio...Mass distribution principle is one of important tools in studying Hausdorff dimension and Hausdorff measure. In this paper we will give a numerical approximate method of upper bound and lower bound of mass distribution function f（x）（it is a monotone increasing fractal function） and its some applications.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.10871123,11171201)
文摘The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the self- affine measures have some surprising connections with a number of areas in mathematics, and have been received much attention in recent years. In the present paper, we shall determine the spectrality and non-spectrality of a class of self-aiffine measures with decomposable digit sets. We present a method to deal with such case, and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.
基金This work is supported by the National Natural Science Foundation of China (No.10161006)the Natural Science Foundation of Jiangxi Province (No.0311043).
文摘In this paper, we investigate complex homogeneous and non-homogeneous higher order linear differential equations with meromorphic coefficients. We obtain several results concerning the iterated order of meromorphic solutions, and the iterated convergence exponent of the zeros of meromorphic solutions.
基金supported by National Natural Science Foundation of China(Grant No.11171201)the Fundamental Research Fund for the Central University(Grant No.GK201401004)
文摘The self-affine measure μM,D associated with an expanding matrix M ∈ Mn(Z) and a finite digit set D ? Znis uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functions E(Λ) := {e2πiλ,x : λ∈Λ} in the Hilbert space L2(μM,D) is simply called μM,D-orthogonal exponentials. We consider in this paper the finiteness of μM,D-orthogonality. A necessary and sufficient condition is obtained for the set E(Λ) to be a finite μM,D-orthogonal exponentials. The research here is closely connected with the non-spectrality of self-affine measures.
文摘Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results,since without this nonlinearity,the results of the network will be less accurate.Nonlinearity is the mission of all nonlinear functions,except for polynomials.The activation function must be dif-ferentiable for backpropagation learning.This study’s objective is to determine the best activation functions for the approximation of each fractal image.Different results have been attained using Matlab and Visual Basic programs,which indi-cate that the bounded function is more helpful than other functions.The non-lin-earity of the activation function is important when using neural networks for coding fractal images because the coefficients of the Iterated Function System are different according to the different types of fractals.The most commonly cho-sen activation function is the sigmoidal function,which produces a positive value.Other functions,such as tansh or arctan,whose values can be positive or negative depending on the network input,tend to train neural networks faster.The coding speed of the fractal image is different depending on the appropriate activation function chosen for each fractal shape.In this paper,we have provided the appro-priate activation functions for each type of system of iterated functions that help the network to identify the transactions of the system.
基金Supported by the National Natural Science Foundation of China (10471039)the Grant of Higher Schools' Natural Science Basic Research of Jiangsu Province of China (06KJD11017507KJB110115)
文摘The authors study the iterated commutators on the weighted Bergman spaces A2（φ）, and prove that Cnh is compact on A2（φ） if and only if h ∈ B0.
基金This research is supported by the Research Foundation of Doctor Points of China （No. 20060422049） and the National Natural Science Foundation of China （No. 10371065）.
文摘In this paper, we investigate the complex oscillation of higher order homogenous and non- homogeneous linear differential equations with meromorphic coefficients of iterated order, and obtain some results which improve and extend those given by Z. X. Chen, L. Kinnunen, etc.
基金Project supported by the National Natural Science Foundation of China (Nos. 19631040 19971085), the Doctoral Program Foundatio
文摘The authors derive laws of the iterated logarithm for kernel estimator of regression function based on directional data. The results are distribution free in the sense that they are true for all distributions of design variable.
基金supported by the Key Project of Chinese Ministry of Education (Grant No. 108117)National Natural Science Foundation of China (Grant No. 10871123, 11171201)
文摘For an expanding integer matrix M ∈ Mn（Z） and two finite digit sets D, S C Z^n with O∈ D ∩ S, we shall investigate and study the possible conditions on the spectral pair （μM,D,A（M, S）） associated with the iterated function systems {Фd（X） = M^-1（x ＋ d）}d∈D and {φs（x） = M^＊x ＋ s}s∈S in the case when ｜D｜= ｜S｜ = ｜ det（M）｜. Under the condition that （M^-1D, S） is a compatible pair, we obtain a series of necessary and sufficient conditions for （μM,D, A（M, S）） to be a spectral pair. These conditions include how to characterize the invariant sets A（M, S） and T（M, D） such that A（M, S） = Zn and μL（T（M, D）） = 1 which play an important role in the number system research and in the construction of Haar-type orthogonal wavelet basis respectively.
文摘In this paper we study the estimation of the regression function.We establish a law ofthe iterated logarithm for the random window-width kernel estimator and,as an application,fora nearest neighbor estimator.These results give sharp pointwise rates of strong consistency ofthese estimators.
文摘We provide a simple formula to compute the Hausdorff dimension of the attractor of an overlapping iterated function system of contractive similarities satisfying a certain collection of assumptions. This formula is obtained by associating a non-overlapping infinite iterated function system to an iterated function system satisfying our assumptions and using the results of Moran to compute the Hausdorff dimension of the attractor of this infinite iterated function system, thus showing that the Hausforff dimension of the attractor of this infinite iterated function system agrees with that of the attractor of the original iterated function system. Our methods are applicable to some iterated function systems that do not satisfy the finite type condition recently introduced by Ngai and Wang.
基金Project(61372136) supported by the National Natural Science Foundation of China
文摘The design, analysis and parallel implementation of particle filter(PF) were investigated. Firstly, to tackle the particle degeneracy problem in the PF, an iterated importance density function(IIDF) was proposed, where a new term associating with the current measurement information(CMI) was introduced into the expression of the sampled particles. Through the repeated use of the least squares estimate, the CMI can be integrated into the sampling stage in an iterative manner, conducing to the greatly improved sampling quality. By running the IIDF, an iterated PF(IPF) can be obtained. Subsequently, a parallel resampling(PR) was proposed for the purpose of parallel implementation of IPF, whose main idea was the same as systematic resampling(SR) but performed differently. The PR directly used the integral part of the product of the particle weight and particle number as the number of times that a particle was replicated, and it simultaneously eliminated the particles with the smallest weights, which are the two key differences from the SR. The detailed implementation procedures on the graphics processing unit of IPF based on the PR were presented at last. The performance of the IPF, PR and their parallel implementations are illustrated via one-dimensional numerical simulation and practical application of passive radar target tracking.
基金Foundation item： Supported by the Youth Science Foundation of Henan Normal University（521103）
文摘Mass distribution principle is one of important tools in studying Hausdorff dimension and Hausdorff measure. In this paper we will give a numerical approximate method of upper bound and lower bound of mass distribution function f（x）（it is a monotone increasing fractal function） and its some applications.
