Nonlinear transforms have significantly advanced learned image compression(LIC),particularly using residual blocks.This transform enhances the nonlinear expression ability and obtain compact feature representation by ...Nonlinear transforms have significantly advanced learned image compression(LIC),particularly using residual blocks.This transform enhances the nonlinear expression ability and obtain compact feature representation by enlarging the receptive field,which indicates how the convolution process extracts features in a high dimensional feature space.However,its functionality is restricted to the spatial dimension and network depth,limiting further improvements in network performance due to insufficient information interaction and representation.Crucially,the potential of high dimensional feature space in the channel dimension and the exploration of network width/resolution remain largely untapped.In this paper,we consider nonlinear transforms from the perspective of feature space,defining high-dimensional feature spaces in different dimensions and investigating the specific effects.Firstly,we introduce the dimension increasing and decreasing transforms in both channel and spatial dimensions to obtain high dimensional feature space and achieve better feature extraction.Secondly,we design a channel-spatial fusion residual transform(CSR),which incorporates multi-dimensional transforms for a more effective representation.Furthermore,we simplify the proposed fusion transform to obtain a slim architecture(CSR-sm),balancing network complexity and compression performance.Finally,we build the overall network with stacked CSR transforms to achieve better compression and reconstruction.Experimental results demonstrate that the proposed method can achieve superior ratedistortion performance compared to the existing LIC methods and traditional codecs.Specifically,our proposed method achieves 9.38%BD-rate reduction over VVC on Kodak dataset.展开更多
The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion pr...The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion process up to an independent exponential time e_(q)for 0<a<b.The results are expressed in terms of solutions to the differential equations associated with the diffusion generator.Applying these results,we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.展开更多
Agile Transformations are challenging processes for organizations that look to extend the benefits of Agile philosophy and methods beyond software engineering.Despite the impact of these transformations on orga-nizati...Agile Transformations are challenging processes for organizations that look to extend the benefits of Agile philosophy and methods beyond software engineering.Despite the impact of these transformations on orga-nizations,they have not been extensively studied in academia.We conducted a study grounded in workshops and interviews with 99 participants from 30 organizations,including organizations undergoing transformations(“final organizations”)and companies supporting these processes(“consultants”).The study aims to understand the motivations,objectives,and factors driving and challenging these transformations.Over 700 responses were collected to the question and categorized into 32 objectives.The findings show that organizations primarily aim to achieve customer centricity and adaptability,both with 8%of the mentions.Other primary important objectives,with above 4%of mentions,include alignment of goals,lean delivery,sustainable processes,and a flatter,more team-based organizational structure.We also detect discrepancies in perspectives between the objectives identified by the two kinds of organizations and the existing agile literature and models.This misalignment highlights the need for practitioners to understand with the practical realities the organizations face.展开更多
In this editorial we comment on the article by Ji et al.We focus specifically on the EGFR tyrosine kinase inhibitor(EGFR-TKI)treatment and the development of drug resistance to EGFR-TKIs.
Combining wavelet transforms with conventional log differential curves is used to identify fractured sections is a new idea.In this paper,we first compute the mother wavelet transform of conventional logs and the wave...Combining wavelet transforms with conventional log differential curves is used to identify fractured sections is a new idea.In this paper,we first compute the mother wavelet transform of conventional logs and the wavelet decomposed signals are compared with fractures identified from image logs to determine the fracture-matched mother wavelet.Then the mother wavelet-based decomposed signal combined with the differential curves of conventional well logs create a fracture indicator curve,identifying the fractured zone.Finally the fracture density can be precisely evaluated by the linear relationship of the indicator curve and image log fracture density.This method has been successfully used to evaluate igneous reservoir fractures in the southern Songnan basin and the calculated density from the indicator curve and density from image logs are both basically consistent.展开更多
A newalgorithm, called Magnitude Cut, to recover a signal from its phase in the transform domain, is proposed.First, the recovery problem is converted to an equivalent convex optimization problem, and then it is solve...A newalgorithm, called Magnitude Cut, to recover a signal from its phase in the transform domain, is proposed.First, the recovery problem is converted to an equivalent convex optimization problem, and then it is solved by the block coordinate descent（ BCD） algorithm and the interior point algorithm. Finally, the one-dimensional and twodimensional signal reconstructions are implemented and the reconstruction results under the Fourier transform with a Gaussian random mask（ FTGM）, the Cauchy wavelets transform（ CWT）, the Fourier transform with a binary random mask（ FTBM） and the Gaussian random transform（ GRT） are also comparatively analyzed. The analysis results reveal that the M agnitude Cut method can reconstruct the original signal with the phase information of different transforms; and it needs less phase information to recover the signal from the phase of the FTGM or GRT than that of FTBM or CWT under the same reconstruction error.展开更多
In this article, we study LP-boundedness properties of the oscillation and vari- ation operators for the heat and Poissson semigroup and Riesz transforms in the Laguerre settings. Also, we characterize Hardy spaces as...In this article, we study LP-boundedness properties of the oscillation and vari- ation operators for the heat and Poissson semigroup and Riesz transforms in the Laguerre settings. Also, we characterize Hardy spaces associated to Laguerre operators by using the variation operator of the heat semigroup.展开更多
Quantum watermarking is a technique to embed specific information, usually the owner's identification,into quantum cover data such for copyright protection purposes. In this paper, a new scheme for quantum waterma...Quantum watermarking is a technique to embed specific information, usually the owner's identification,into quantum cover data such for copyright protection purposes. In this paper, a new scheme for quantum watermarking based on quantum wavelet transforms is proposed which includes scrambling, embedding and extracting procedures. The invisibility and robustness performances of the proposed watermarking method is confirmed by simulation technique.The invisibility of the scheme is examined by the peak-signal-to-noise ratio(PSNR) and the histogram calculation.Furthermore the robustness of the scheme is analyzed by the Bit Error Rate(BER) and the Correlation Two-Dimensional(Corr 2-D) calculation. The simulation results indicate that the proposed watermarking scheme indicate not only acceptable visual quality but also a good resistance against different types of attack.展开更多
The modified atomic transformations are constructed and proved. On their basis the new complex analytic wavelets are obtained. The proof of the Fourier transforms existence in L~ and L2 on the basis of the theory of a...The modified atomic transformations are constructed and proved. On their basis the new complex analytic wavelets are obtained. The proof of the Fourier transforms existence in L~ and L2 on the basis of the theory of atomic functions （AF） are presented. The numerical experiments of digital time series processing and physical analysis of the results confirm the efficiency of the proposed transforms.展开更多
In this paper we proved the A_(p)-weighted inequalities for martingale transforms and differential subordinations of Banach-space-valued regular maringales.We discussed the relations between the weighted inequalities,...In this paper we proved the A_(p)-weighted inequalities for martingale transforms and differential subordinations of Banach-space-valued regular maringales.We discussed the relations between the weighted inequalities,A_(p)-weight functions and the Banach spaces which has the UMD property or are isomorphic to Hilbert space.展开更多
The problem of estimating an image corrupted by additive white Gaussian noise has been of interest for practical reasons. Non-linear denoising methods based on wavelets, have become popular but Multiwavelets outperfor...The problem of estimating an image corrupted by additive white Gaussian noise has been of interest for practical reasons. Non-linear denoising methods based on wavelets, have become popular but Multiwavelets outperform wavelets in image denoising. Multiwavelets are wavelets with several scaling and wavelet functions, offer simultaneously Orthogonality, Symmetry, Short support and Vanishing moments, which is not possible with ordinary (scalar) wavelets. These properties make Multiwavelets promising for image processing applications, such as image denoising. The aim of this paper is to apply various non-linear thresholding techniques such as hard, soft, universal, modified universal, fixed and multivariate thresholding in Multiwavelet transform domain such as Discrete Multiwavelet Transform, Symmetric Asymmetric (SA4), Chui Lian (CL), and Bi-Hermite (Bih52S) for different Multiwavelets at different levels, to denoise an image and determine the best one out of it. The performance of denoising algorithms and various thresholding are measured using quantitative performance measures such as, Mean Square Error (MSE), and Root Mean Square Error (RMSE), Signal-to-Noise Ratio (SNR), Peak Signal-to-Noise Ratio (PSNR). It is found that CL Multiwavelet transform in combination with modified universal thresholding has given best results.展开更多
Assume that 0<p<∞ and that B is a connected nonempty open set in R^(n),and that A^(p)(B)is the vector space of all holomorphic functions F in the tubular domains R^(n)+iB such that for any compact set K⊂B,‖ y...Assume that 0<p<∞ and that B is a connected nonempty open set in R^(n),and that A^(p)(B)is the vector space of all holomorphic functions F in the tubular domains R^(n)+iB such that for any compact set K⊂B,‖ y →‖x →F(x+iy)‖Lp(R^(n))‖ L(K)<∞,so A^(p)(B)is a Frechet space with the Heine-Borel property,its topology is induced by a complete invariant metric,is not locally bounded,and hence is not normal.Furthermore,if 1≤p≤2,then the element F of A^(p)(B)can be written as a Laplace transform of some function f∈L(R^(n)).展开更多
We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms Wσ associated with the Dunk...We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms Wσ associated with the Dunkl operators, where a is a symbol in the Schwartz space S（Rd × Rd）. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we give criteria in terms of σ for boundedness and compactness of the transform Wσ.展开更多
In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic （continuous） functions is developed based on Dirichlet characters, Gauss sums and the generalized MSbius transform. ...In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic （continuous） functions is developed based on Dirichlet characters, Gauss sums and the generalized MSbius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.展开更多
This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study suc...This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μλ with λ = θ-1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μλ ahaost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.展开更多
The objective of this note is to provide some(potentially useful) integral transforms(for example, Euler, Laplace, Whittaker etc.) associated with the generalized k-Bessel function defined by Saiful and Nisar [3]. We ...The objective of this note is to provide some(potentially useful) integral transforms(for example, Euler, Laplace, Whittaker etc.) associated with the generalized k-Bessel function defined by Saiful and Nisar [3]. We have also discussed some other transforms as special cases of our main results.展开更多
The theorems concerning the summation of Fourier series with parameter were given by using Laplace transforms. By means of the known result of Laplace transforms, many new, important problems of summation of Fourier s...The theorems concerning the summation of Fourier series with parameter were given by using Laplace transforms. By means of the known result of Laplace transforms, many new, important problems of summation of Fourier series with parameter in mechanics can be solved.展开更多
In a preceding letter （2007 Opt. Lett. 32 554） we propose complex continuous wavelet transforms and found Laguerre-Gaussian mother wavelets family. In this work we present the inversion formula and Parseval theorem ...In a preceding letter （2007 Opt. Lett. 32 554） we propose complex continuous wavelet transforms and found Laguerre-Gaussian mother wavelets family. In this work we present the inversion formula and Parseval theorem for complex continuous wavelet transform by virtue of the entangled state representation, which makes the complex continuous wavelet transform theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.展开更多
Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + ...Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.展开更多
The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of...The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain,in this manuscript,an improved Heisenberg uncertainty principle is obtained in linear canonical trans-forms domain.展开更多
基金supported by the Key Program of the National Natural Science Foundation of China(Grant No.62031013)Guangdong Province Key Construction Discipline Scientific Research Capacity Improvement Project(Grant No.2022ZDJS117).
文摘Nonlinear transforms have significantly advanced learned image compression(LIC),particularly using residual blocks.This transform enhances the nonlinear expression ability and obtain compact feature representation by enlarging the receptive field,which indicates how the convolution process extracts features in a high dimensional feature space.However,its functionality is restricted to the spatial dimension and network depth,limiting further improvements in network performance due to insufficient information interaction and representation.Crucially,the potential of high dimensional feature space in the channel dimension and the exploration of network width/resolution remain largely untapped.In this paper,we consider nonlinear transforms from the perspective of feature space,defining high-dimensional feature spaces in different dimensions and investigating the specific effects.Firstly,we introduce the dimension increasing and decreasing transforms in both channel and spatial dimensions to obtain high dimensional feature space and achieve better feature extraction.Secondly,we design a channel-spatial fusion residual transform(CSR),which incorporates multi-dimensional transforms for a more effective representation.Furthermore,we simplify the proposed fusion transform to obtain a slim architecture(CSR-sm),balancing network complexity and compression performance.Finally,we build the overall network with stacked CSR transforms to achieve better compression and reconstruction.Experimental results demonstrate that the proposed method can achieve superior ratedistortion performance compared to the existing LIC methods and traditional codecs.Specifically,our proposed method achieves 9.38%BD-rate reduction over VVC on Kodak dataset.
基金Supported by the National Natural Science Foundation of China(12271062,11731012)by the Hunan Provincial National Natural Science Foundation of China(2019JJ50405)。
文摘The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion process up to an independent exponential time e_(q)for 0<a<b.The results are expressed in terms of solutions to the differential equations associated with the diffusion generator.Applying these results,we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.
基金funding from the European Commission for the Ruralities Project(grant agreement no.101060876).
文摘Agile Transformations are challenging processes for organizations that look to extend the benefits of Agile philosophy and methods beyond software engineering.Despite the impact of these transformations on orga-nizations,they have not been extensively studied in academia.We conducted a study grounded in workshops and interviews with 99 participants from 30 organizations,including organizations undergoing transformations(“final organizations”)and companies supporting these processes(“consultants”).The study aims to understand the motivations,objectives,and factors driving and challenging these transformations.Over 700 responses were collected to the question and categorized into 32 objectives.The findings show that organizations primarily aim to achieve customer centricity and adaptability,both with 8%of the mentions.Other primary important objectives,with above 4%of mentions,include alignment of goals,lean delivery,sustainable processes,and a flatter,more team-based organizational structure.We also detect discrepancies in perspectives between the objectives identified by the two kinds of organizations and the existing agile literature and models.This misalignment highlights the need for practitioners to understand with the practical realities the organizations face.
文摘In this editorial we comment on the article by Ji et al.We focus specifically on the EGFR tyrosine kinase inhibitor(EGFR-TKI)treatment and the development of drug resistance to EGFR-TKIs.
基金sponsored by National Science and Technology Major Project of China (No. 2008 ZX 05009-001)
文摘Combining wavelet transforms with conventional log differential curves is used to identify fractured sections is a new idea.In this paper,we first compute the mother wavelet transform of conventional logs and the wavelet decomposed signals are compared with fractures identified from image logs to determine the fracture-matched mother wavelet.Then the mother wavelet-based decomposed signal combined with the differential curves of conventional well logs create a fracture indicator curve,identifying the fractured zone.Finally the fracture density can be precisely evaluated by the linear relationship of the indicator curve and image log fracture density.This method has been successfully used to evaluate igneous reservoir fractures in the southern Songnan basin and the calculated density from the indicator curve and density from image logs are both basically consistent.
基金The National Natural Science Foundation of China(No.6120134461271312+7 种基金11301074)the Specialized Research Fund for the Doctoral Program of Higher Education(No.2011009211002320120092120036)the Program for Special Talents in Six Fields of Jiangsu Province(No.DZXX-031)the Natural Science Foundation of Jiangsu Province(No.BK2012329BK2012743)the United Creative Foundation of Jiangsu Province(No.BY2014127-11)the"333"Project(No.BRA2015288)
文摘A newalgorithm, called Magnitude Cut, to recover a signal from its phase in the transform domain, is proposed.First, the recovery problem is converted to an equivalent convex optimization problem, and then it is solved by the block coordinate descent（ BCD） algorithm and the interior point algorithm. Finally, the one-dimensional and twodimensional signal reconstructions are implemented and the reconstruction results under the Fourier transform with a Gaussian random mask（ FTGM）, the Cauchy wavelets transform（ CWT）, the Fourier transform with a binary random mask（ FTBM） and the Gaussian random transform（ GRT） are also comparatively analyzed. The analysis results reveal that the M agnitude Cut method can reconstruct the original signal with the phase information of different transforms; and it needs less phase information to recover the signal from the phase of the FTGM or GRT than that of FTBM or CWT under the same reconstruction error.
基金supported by Ministerio de Educación y Ciencia (Spain),grant MTM 2007-65609supported by Ministerio de Educacióon y Ciencia (Spain),grant MTM 2008-06621-C02supported by Universidad Nacional del Comahue (Argentina) and Ministerio de Educación y Ciencia (Spain) grant PCI 2006-A7-0670
文摘In this article, we study LP-boundedness properties of the oscillation and vari- ation operators for the heat and Poissson semigroup and Riesz transforms in the Laguerre settings. Also, we characterize Hardy spaces associated to Laguerre operators by using the variation operator of the heat semigroup.
基金Supported by Kermanshah Branch,Islamic Azad University,Kermanshah,Iran
文摘Quantum watermarking is a technique to embed specific information, usually the owner's identification,into quantum cover data such for copyright protection purposes. In this paper, a new scheme for quantum watermarking based on quantum wavelet transforms is proposed which includes scrambling, embedding and extracting procedures. The invisibility and robustness performances of the proposed watermarking method is confirmed by simulation technique.The invisibility of the scheme is examined by the peak-signal-to-noise ratio(PSNR) and the histogram calculation.Furthermore the robustness of the scheme is analyzed by the Bit Error Rate(BER) and the Correlation Two-Dimensional(Corr 2-D) calculation. The simulation results indicate that the proposed watermarking scheme indicate not only acceptable visual quality but also a good resistance against different types of attack.
文摘The modified atomic transformations are constructed and proved. On their basis the new complex analytic wavelets are obtained. The proof of the Fourier transforms existence in L~ and L2 on the basis of the theory of atomic functions （AF） are presented. The numerical experiments of digital time series processing and physical analysis of the results confirm the efficiency of the proposed transforms.
文摘In this paper we proved the A_(p)-weighted inequalities for martingale transforms and differential subordinations of Banach-space-valued regular maringales.We discussed the relations between the weighted inequalities,A_(p)-weight functions and the Banach spaces which has the UMD property or are isomorphic to Hilbert space.
文摘The problem of estimating an image corrupted by additive white Gaussian noise has been of interest for practical reasons. Non-linear denoising methods based on wavelets, have become popular but Multiwavelets outperform wavelets in image denoising. Multiwavelets are wavelets with several scaling and wavelet functions, offer simultaneously Orthogonality, Symmetry, Short support and Vanishing moments, which is not possible with ordinary (scalar) wavelets. These properties make Multiwavelets promising for image processing applications, such as image denoising. The aim of this paper is to apply various non-linear thresholding techniques such as hard, soft, universal, modified universal, fixed and multivariate thresholding in Multiwavelet transform domain such as Discrete Multiwavelet Transform, Symmetric Asymmetric (SA4), Chui Lian (CL), and Bi-Hermite (Bih52S) for different Multiwavelets at different levels, to denoise an image and determine the best one out of it. The performance of denoising algorithms and various thresholding are measured using quantitative performance measures such as, Mean Square Error (MSE), and Root Mean Square Error (RMSE), Signal-to-Noise Ratio (SNR), Peak Signal-to-Noise Ratio (PSNR). It is found that CL Multiwavelet transform in combination with modified universal thresholding has given best results.
基金This work was partially supported by NSFC(11971045,12071035 and 11971063).
文摘Assume that 0<p<∞ and that B is a connected nonempty open set in R^(n),and that A^(p)(B)is the vector space of all holomorphic functions F in the tubular domains R^(n)+iB such that for any compact set K⊂B,‖ y →‖x →F(x+iy)‖Lp(R^(n))‖ L(K)<∞,so A^(p)(B)is a Frechet space with the Heine-Borel property,its topology is induced by a complete invariant metric,is not locally bounded,and hence is not normal.Furthermore,if 1≤p≤2,then the element F of A^(p)(B)can be written as a Laplace transform of some function f∈L(R^(n)).
基金Supported by the DGRST Research Project LR11ES11 and CMCU Program 10G/1503
文摘We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms Wσ associated with the Dunkl operators, where a is a symbol in the Schwartz space S（Rd × Rd）. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we give criteria in terms of σ for boundedness and compactness of the transform Wσ.
基金Supported by National Natural Science Foundation of China(11201370)the Science and Technology Program of Shaanxi Province of China(2013JM1017,2014JM1007,2014KJXX-61)the Natural Science Foundation of the Education Department of Shaanxi Province of China(2013JK0558)
文摘In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic （continuous） functions is developed based on Dirichlet characters, Gauss sums and the generalized MSbius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.
文摘This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μλ with λ = θ-1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μλ ahaost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.
文摘The objective of this note is to provide some(potentially useful) integral transforms(for example, Euler, Laplace, Whittaker etc.) associated with the generalized k-Bessel function defined by Saiful and Nisar [3]. We have also discussed some other transforms as special cases of our main results.
文摘The theorems concerning the summation of Fourier series with parameter were given by using Laplace transforms. By means of the known result of Laplace transforms, many new, important problems of summation of Fourier series with parameter in mechanics can be solved.
基金supported by the National Natural Science Foundation of China (Grant No. 10775097)the Research Foundation of the Education Department of Jiangxi Province of China (Grant No. GJJ10097)
文摘In a preceding letter （2007 Opt. Lett. 32 554） we propose complex continuous wavelet transforms and found Laguerre-Gaussian mother wavelets family. In this work we present the inversion formula and Parseval theorem for complex continuous wavelet transform by virtue of the entangled state representation, which makes the complex continuous wavelet transform theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.
文摘Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.
文摘The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain,in this manuscript,an improved Heisenberg uncertainty principle is obtained in linear canonical trans-forms domain.
