The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(...The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(mx-k)+∑p-kφ(k-mx) where m ≥ 2 is an integer. Based on the positive mask {pk+} and negative mask {p-k}, the conditions that guarantee the above equation has compactly distributional solutions or L2-stable solutions are established. Furthermore, the condition that the L2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of φ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.展开更多
Assuming the material properties varying with an exponential law both in the thick- ness and radial directions, axisymmetric bending of two-directional functionally graded circular and annular plates is studied using ...Assuming the material properties varying with an exponential law both in the thick- ness and radial directions, axisymmetric bending of two-directional functionally graded circular and annular plates is studied using the semi-analytical numerical method in this paper. The deflections and stresses of the plates are presented. Numerical results show the well accuracy and convergence of the method. Compared with the finite element method, the semi-analytical nu- merical method is with great advantage in the computational efficiency. Moreover, study on ax- isymmetric bending of two-directional functionally graded annular plate shows that such plates have better performance than those made of isotropic homogeneous materials or one-directional functionally graded materials. Two-directional functionally graded material is a potential alternative to the one-directional functionally graded material. And the integrated design of materials and structures can really be achieved in two-directional functionally graded materials.展开更多
This article aims at studying two-direction refinable functions and two-direction wavelets in the setting R^s, s 〉 1. We give a sufficient condition for a two-direction refinable function belonging to L^2（R^s）. The...This article aims at studying two-direction refinable functions and two-direction wavelets in the setting R^s, s 〉 1. We give a sufficient condition for a two-direction refinable function belonging to L^2（R^s）. Then, two theorems are given for constructing biorthogonal （orthogonal） two-direction refinable functions in L^2（R^s） and their biorthogonal （orthogonal） two-direction wavelets, respectively. From the constructed biorthogonal （orthogonal） two-direction wavelets, symmetric biorthogonal （orthogonal） multiwaveles in L^2（R^s） can be obtained easily. Applying the projection method to biorthogonal （orthogonal） two-direction wavelets in L^2（R^s）, we can get dual （tight） two-direction wavelet frames in L^2（R^m）, where m ≤ s. From the projected dual （tight） two-direction wavelet frames in L^2（R^m）, symmetric dual （tight） frames in L^2（R^m） can be obtained easily. In the end, an example is given to illustrate theoretical results.展开更多
In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded sandwich plate. Unlike any other theory, the number of unknown functions involve...In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded sandwich plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Two common types of functionally graded sandwich plates, namely, the sandwich with fimctionally graded facesheet and homogeneous core and the sandwich with homogeneous facesheet and functionally graded core, are considered. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to sinu- soidal loading has been obtained by using the Navier method. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static bending behavior of functionally graded sandwich plates.展开更多
Suppose that f（x）=（f<sub>1</sub>（x）,....f<sub>r</sub>（x））<sup>T</sup>, x∈R<sup>d</sup> is a vector-valued function satisfying the refinement equation f（x）=∑&...Suppose that f（x）=（f<sub>1</sub>（x）,....f<sub>r</sub>（x））<sup>T</sup>, x∈R<sup>d</sup> is a vector-valued function satisfying the refinement equation f（x）=∑<sub> </sub>c<sub>k</sub> f（2x-k） with finite set of Z<sup>d</sup> and some r×r matricex c<sub>k</sub>. The requirements for f to have accuracy p are given in terms of the symbol function m（ξ）.展开更多
This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate ...This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.展开更多
This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-...This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.展开更多
An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties ...An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties of MFG porous plates change according to the length,width,and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology.Especially,new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work.Applying Hamilton's principle and the refined higher-order shear deformation plate theory,the governing equation of motion of an MFG porous rectangular plate in a fluid medium(the fluid-plate system)is obtained.The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass.The GalerkinVlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium.The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature.The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail.These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium.Thereby,they offer appropriate operational approaches for the structure when employed in various mediums.展开更多
Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven mo...Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven model has been recently developed.In this paper,for higher-order shear deformation beams,the ill-posed issue(i.e.,excessive mandatory boundary conditions(BCs)cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones.The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D)models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG)materials.The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions.By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically.The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.展开更多
Let M be a d × d expansive matrix, and FL2（Ω） be a reducing subspace of L2（Rd）. This paper characterizes bounded measurable sets in Rd which are the supports of Fourier transforms of M-refinable frame functi...Let M be a d × d expansive matrix, and FL2（Ω） be a reducing subspace of L2（Rd）. This paper characterizes bounded measurable sets in Rd which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA （W-type FMRA） frame scaling functions and MRA （W-type MRA） scaling functions for FL2（Ω）, respectively. Some examples are also provided.展开更多
This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（...This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.展开更多
In this paper some properties of refinable functions and some relationships between the mask symbol and the refinable functions are studied. Especially, it is illustrated by examples that the linear spaces formed by t...In this paper some properties of refinable functions and some relationships between the mask symbol and the refinable functions are studied. Especially, it is illustrated by examples that the linear spaces formed by the translates over the lattice points of refinable functions may contain polynomial spaces of deg-ree higher than the smooth order of the corresponding refinable functions.展开更多
In this article, we show that the generalized logarithmic mean is strictly Schurconvex function for p 〉 2 and strictly Schur-concave function for p 〈 2 on R_＋^2. And then we give a refinement of an inequality for t...In this article, we show that the generalized logarithmic mean is strictly Schurconvex function for p 〉 2 and strictly Schur-concave function for p 〈 2 on R_＋^2. And then we give a refinement of an inequality for the generalized logarithmic mean inequality using a simple majoricotion relation of the vector.展开更多
As a kind of formal specification language, Z has gained a position in the field of software development, but there is still no standard way of transforming Z specification into executable code that is promising in in...As a kind of formal specification language, Z has gained a position in the field of software development, but there is still no standard way of transforming Z specification into executable code that is promising in increasing the quality, reusability and maintainability of software.With the automatic programming model of software engineering, through the analysis for Z specification language, a feasible semi-automatic way of refinement and transformation is proposed, and the correctness of the procedure is also discussed.展开更多
This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite(FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here,...This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite(FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube(CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton's principle. Navier's solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.展开更多
Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key proper...Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.展开更多
A three-dimensional(3D)thermomechanical vibration model is developed for rotating pre-twisted functionally graded(FG)microbeams according to the refined shear deformation theory(RSDT)and the modified couple stress the...A three-dimensional(3D)thermomechanical vibration model is developed for rotating pre-twisted functionally graded(FG)microbeams according to the refined shear deformation theory(RSDT)and the modified couple stress theory(MCST).The material properties are assumed to follow a power-law distribution along the chordwise direction.The model introduces one axial stretching variable and four transverse deflection variables including two pure bending components and two pure shear ones.The complex modal analysis and assumed mode methods are used to solve the governing equations of motion under different boundary conditions(BCs).Several examples are presented to verify the effectiveness of the developed model.By coupling the slenderness ratio,gradient index,rotation speed,and size effect with the pre-twisted angle,the effects of these factors on the thermomechanical vibration of the microbeam with different BCs are investigated.It is found that with the increase in the pre-twisted angle,the critical slenderness ratio and gradient index corresponding to the thermal instability of the microbeam increase,while the critical material length scale parameter(MLSP)and rotation speed decrease.The sensitivity of the fundamental frequency to temperature increases with the increasing slenderness ratio and gradient index,and decreases with the other increasing parameters.Moreover,the size effect can suppress the dynamic stiffening effect and enhance the Coriolis effect.Finally,the mode transition is quantitatively demonstrated by a modal assurance criterion(MAC).展开更多
基金This work was Supported by the Natural Science Foundation of Guangdong Province (Grant Nos.06105648,05008289,032038)the Doctoral Foundation of Guangdong Province (Grant No.04300917)
文摘The concept of two-direction refinable functions and two-direction wavelets is introduced.We investigate the existence of distributional(or L2-stable) solutions of the two-direction refinement equation: φ(x)=∑p+kφ(mx-k)+∑p-kφ(k-mx) where m ≥ 2 is an integer. Based on the positive mask {pk+} and negative mask {p-k}, the conditions that guarantee the above equation has compactly distributional solutions or L2-stable solutions are established. Furthermore, the condition that the L2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of φ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.
基金Project supported by the National Natural Science Foundation of China (No.10432030).
文摘Assuming the material properties varying with an exponential law both in the thick- ness and radial directions, axisymmetric bending of two-directional functionally graded circular and annular plates is studied using the semi-analytical numerical method in this paper. The deflections and stresses of the plates are presented. Numerical results show the well accuracy and convergence of the method. Compared with the finite element method, the semi-analytical nu- merical method is with great advantage in the computational efficiency. Moreover, study on ax- isymmetric bending of two-directional functionally graded annular plate shows that such plates have better performance than those made of isotropic homogeneous materials or one-directional functionally graded materials. Two-directional functionally graded material is a potential alternative to the one-directional functionally graded material. And the integrated design of materials and structures can really be achieved in two-directional functionally graded materials.
基金supported by the Natural Science Foundation China(11126343)Guangxi Natural Science Foundation(2013GXNSFBA019010)+1 种基金supported by Natural Science Foundation China(11071152)Natural Science Foundation of Guangdong Province(10151503101000025,S2011010004511)
文摘This article aims at studying two-direction refinable functions and two-direction wavelets in the setting R^s, s 〉 1. We give a sufficient condition for a two-direction refinable function belonging to L^2（R^s）. Then, two theorems are given for constructing biorthogonal （orthogonal） two-direction refinable functions in L^2（R^s） and their biorthogonal （orthogonal） two-direction wavelets, respectively. From the constructed biorthogonal （orthogonal） two-direction wavelets, symmetric biorthogonal （orthogonal） multiwaveles in L^2（R^s） can be obtained easily. Applying the projection method to biorthogonal （orthogonal） two-direction wavelets in L^2（R^s）, we can get dual （tight） two-direction wavelet frames in L^2（R^m）, where m ≤ s. From the projected dual （tight） two-direction wavelet frames in L^2（R^m）, symmetric dual （tight） frames in L^2（R^m） can be obtained easily. In the end, an example is given to illustrate theoretical results.
文摘In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded sandwich plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Two common types of functionally graded sandwich plates, namely, the sandwich with fimctionally graded facesheet and homogeneous core and the sandwich with homogeneous facesheet and functionally graded core, are considered. Governing equations are derived from the principle of virtual displacements. The closed-form solution of a simply supported rectangular plate subjected to sinu- soidal loading has been obtained by using the Navier method. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static bending behavior of functionally graded sandwich plates.
文摘Suppose that f（x）=（f<sub>1</sub>（x）,....f<sub>r</sub>（x））<sup>T</sup>, x∈R<sup>d</sup> is a vector-valued function satisfying the refinement equation f（x）=∑<sub> </sub>c<sub>k</sub> f（2x-k） with finite set of Z<sup>d</sup> and some r×r matricex c<sub>k</sub>. The requirements for f to have accuracy p are given in terms of the symbol function m（ξ）.
基金This research is funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under Grant number 107.02-2019.330.
文摘This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.
基金Supported by the National Natural Science Foundation of China(60933008 and 61272300)
文摘This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.
文摘An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties of MFG porous plates change according to the length,width,and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology.Especially,new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work.Applying Hamilton's principle and the refined higher-order shear deformation plate theory,the governing equation of motion of an MFG porous rectangular plate in a fluid medium(the fluid-plate system)is obtained.The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass.The GalerkinVlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium.The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature.The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail.These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium.Thereby,they offer appropriate operational approaches for the structure when employed in various mediums.
基金Project supported by the National Natural Science Foundation of China(No.11672131)。
文摘Due to the conflict between equilibrium and constitutive requirements,Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest.As an alternative,the stress-driven model has been recently developed.In this paper,for higher-order shear deformation beams,the ill-posed issue(i.e.,excessive mandatory boundary conditions(BCs)cannot be met simultaneously)exists not only in strain-driven nonlocal models but also in stress-driven ones.The well-posedness of both the strain-and stress-driven two-phase nonlocal(TPN-Strain D and TPN-Stress D)models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded(FG)materials.The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions.By using the generalized differential quadrature method(GDQM),the coupling governing equations are solved numerically.The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.
基金Supported by Beijing Natural Science Foundation (No.1122008)the Scientific Research Common Programof Beijing Municipal Commission of Education (No.KM201110005030)
文摘Let M be a d × d expansive matrix, and FL2（Ω） be a reducing subspace of L2（Rd）. This paper characterizes bounded measurable sets in Rd which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA （W-type FMRA） frame scaling functions and MRA （W-type MRA） scaling functions for FL2（Ω）, respectively. Some examples are also provided.
基金Supported by the National Natural Science Foundation of China (10071071)
文摘This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.
文摘In this paper some properties of refinable functions and some relationships between the mask symbol and the refinable functions are studied. Especially, it is illustrated by examples that the linear spaces formed by the translates over the lattice points of refinable functions may contain polynomial spaces of deg-ree higher than the smooth order of the corresponding refinable functions.
基金Foundation item： Supported by the Scientific Research Common Program of Beijing Municipal Commission of Education of China（Km200611417009） Suppoted by the Natural Science Foundation of Fujian Province Education Department of China（JA05324）
文摘In this article, we show that the generalized logarithmic mean is strictly Schurconvex function for p 〉 2 and strictly Schur-concave function for p 〈 2 on R_＋^2. And then we give a refinement of an inequality for the generalized logarithmic mean inequality using a simple majoricotion relation of the vector.
文摘As a kind of formal specification language, Z has gained a position in the field of software development, but there is still no standard way of transforming Z specification into executable code that is promising in increasing the quality, reusability and maintainability of software.With the automatic programming model of software engineering, through the analysis for Z specification language, a feasible semi-automatic way of refinement and transformation is proposed, and the correctness of the procedure is also discussed.
基金Project supported by the Foundation for Science and Technology Development of National University of Civil Engineering-Ha Noi-Vietnam (No. 27-2020/KHXD-TD)。
文摘This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite(FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube(CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton's principle. Navier's solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.
基金supported by the Natural Sciences and Engineering Research Council of Canada(NSERC)(Grant No.RGPIN-2019-04276)。
文摘Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.
基金the National Natural Science Foundation of China(Nos.11602204 and 12102373)the Fundamental Research Funds for the Central Universities of China(Nos.2682022ZTPY081 and 2682022CX056)the Natural Science Foundation of Sichuan Province of China(Nos.2023NSFSC0849,2023NSFSC1300,2022NSFSC1938,and 2022NSFSC2003)。
文摘A three-dimensional(3D)thermomechanical vibration model is developed for rotating pre-twisted functionally graded(FG)microbeams according to the refined shear deformation theory(RSDT)and the modified couple stress theory(MCST).The material properties are assumed to follow a power-law distribution along the chordwise direction.The model introduces one axial stretching variable and four transverse deflection variables including two pure bending components and two pure shear ones.The complex modal analysis and assumed mode methods are used to solve the governing equations of motion under different boundary conditions(BCs).Several examples are presented to verify the effectiveness of the developed model.By coupling the slenderness ratio,gradient index,rotation speed,and size effect with the pre-twisted angle,the effects of these factors on the thermomechanical vibration of the microbeam with different BCs are investigated.It is found that with the increase in the pre-twisted angle,the critical slenderness ratio and gradient index corresponding to the thermal instability of the microbeam increase,while the critical material length scale parameter(MLSP)and rotation speed decrease.The sensitivity of the fundamental frequency to temperature increases with the increasing slenderness ratio and gradient index,and decreases with the other increasing parameters.Moreover,the size effect can suppress the dynamic stiffening effect and enhance the Coriolis effect.Finally,the mode transition is quantitatively demonstrated by a modal assurance criterion(MAC).
