In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u＋bu=Λρu,inΩRn,u｜Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper boun...In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u＋bu=Λρu,inΩRn,u｜Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the （k＋1）-th eigenvalueΛ_k＋1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.展开更多
We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l2 space, we obt...We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l2 space, we obtain some new characterizations of quasiconformal extendablity and asymptotic conformality for univalent functions.展开更多
In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are of...In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.展开更多
Let （y, d, dλ） be （Rn, ｜·｜,μ）, where ｜·｜ is the Euclidean distance, μ is a nonnegative Radon measure on Rn satisfying the polynomial growth condition, or the Gauss measure metric space （Rn, ｜...Let （y, d, dλ） be （Rn, ｜·｜,μ）, where ｜·｜ is the Euclidean distance, μ is a nonnegative Radon measure on Rn satisfying the polynomial growth condition, or the Gauss measure metric space （Rn, ｜·｜,dγ）, or the space （S, d, p）, where S - Rn×R＋ is the （ax ＋ b）-group, d is the left-invariant Riemannian metric and p is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces {Xs（Y））0〈s≤∞ and the BM0-type spaces {BM0（y, s）}0〈s≤∞. Let Hi（Y） be the known atomic Hardy space and L01（y） the subspace of f ∈ L1（Y） with integral 0. The authors prove that the dual space of Xs（Y） is SM0（Y,s） when s∈ （0, ∞）, Xs（Y） = H1（Y） when s ∈ （0, 1], and X∞（y） = L01（Y） （or L1（Y））. As applications, the authors show that if T is a linear operator bounded from H1 （Y） to L1 （Y） and from L1（y） to L1,∞（Y）, then for all r ∈ （1, ∞） and s ∈ （r, ∞], T is bounded from Xr（y） to the Lorentz space L1,8（y）, which applies to the Calderon-Zygmund operator on （Rn, ｜·｜,μ）, the imaginary powers of the 0rnstein-Uhlenbeck operator on （Rn, ｜·｜,dγ） and the spectral operator associated with the spectral multiplier on （S, d, p）. All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.展开更多
We study the joint distributions of four random variables h(ω) (stopping time or optional time), location x(h), l(ω) (co-optional time) and location x(l) for Markov processes. Some distributions for d(≥3) -dimensio...We study the joint distributions of four random variables h(ω) (stopping time or optional time), location x(h), l(ω) (co-optional time) and location x(l) for Markov processes. Some distributions for d(≥3) -dimensional Brownian motion and the joint distribution of first exit and last exit locations for symmetric stable process are found.展开更多
Subpixel accuracy for V-groove center in robot welding is researched and a software measure to increase the accuracy of seam tracking by laser is presented. LOG （ Laplacian of Gaussian ） operator is adopted to detec...Subpixel accuracy for V-groove center in robot welding is researched and a software measure to increase the accuracy of seam tracking by laser is presented. LOG （ Laplacian of Gaussian ） operator is adopted to detect image edge. Vgroove center is extracted by corner detection of extremum curvature. Subpixel position is obtained by Lagarange polynomial interpolation algorithm. Experiment results show that the method is brief and applied, and is sufficient for the real time of robot welding by laser sensors.展开更多
Using an operator ordering method for some commutative superposition operators,we introduce two new multi-variable special polynomials and their generating functions,and present some new operator identities and integr...Using an operator ordering method for some commutative superposition operators,we introduce two new multi-variable special polynomials and their generating functions,and present some new operator identities and integral formulas involving the two special polynomials.Instead of calculating compli-cated partial differential,we use the special polynomials and their generating functions to concsely address the normalzation,photoount distributions and Wigner distributions of several quantum states that can be realized physically,the rsults of which provide real convenience for further investigating the properties and applications of these states.展开更多
In this paper, we present several expansions of the symbolic operator （1 ＋E）^x. Moreover, we derive some series transforms formulas and the Newton generating functions of {f（k）}.
For a polynomial p（z） of degree n which has no zeros in ｜z｜ 〈 1, Dewan et al., （K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, 38-41...For a polynomial p（z） of degree n which has no zeros in ｜z｜ 〈 1, Dewan et al., （K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, 38-41） establishedfor any complex number β with ｜β｜≤ and｜z｜ = 1. In this paper we consider the operator B, which carries a polynomial p（z） into展开更多
In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smoo...In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.展开更多
Let Pn be the class of polynomials of degree at most n. Rather and Shah [15] proved that if P∈Pn and P（z） 6=0 in｜z｜〈1, then for every R〉0 and 0≤q〈∞,｜B[P（Rz）]｜q≤｜RnB[zn]＋λ0｜q｜1＋zn｜q ｜P（z）｜q, w...Let Pn be the class of polynomials of degree at most n. Rather and Shah [15] proved that if P∈Pn and P（z） 6=0 in｜z｜〈1, then for every R〉0 and 0≤q〈∞,｜B[P（Rz）]｜q≤｜RnB[zn]＋λ0｜q｜1＋zn｜q ｜P（z）｜q, where B is a Bn-operator. In this paper, we prove some generalization of this result which in particular yield-s some known polynomial inequalities as special. We also consider an operator Dαwhich maps a polynomial P（z） into DαP（z）：=nP（z）＋（α-z）P′（z） and obtain exten-sions and generalizations of a number of well-known Lq inequalities.展开更多
The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such op...The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.展开更多
If X is an Asplund space, then every uniformly continuous function on Bx＊ which is holomorphic on the open unit ball, can be perturbed by a w＊ continuous and homogeneous polynomial on X＊ to obtain a norm attaining ...If X is an Asplund space, then every uniformly continuous function on Bx＊ which is holomorphic on the open unit ball, can be perturbed by a w＊ continuous and homogeneous polynomial on X＊ to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall＇s variational principle. We also show that the set of N-homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N-homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.展开更多
In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-ha...In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.展开更多
基金supported by the National Natural Science Foundation of China(11001130)the NUST Research Funding(2010ZYTS064)supported by China Postdoctoral Science Foundation(20080430351)
文摘In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u＋bu=Λρu,inΩRn,u｜Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the （k＋1）-th eigenvalueΛ_k＋1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.
基金supported by the Program for New Century Excellent Talents in University (Grant No. 06-0504)National Natural Science Foundation of China (Grant No. 10771153)
文摘We obtain some convergence properties concerning Faber polynomials and apply them to studying univalent functions with quasiconformal extensions. In particular, by introducing an operator on the usual l2 space, we obtain some new characterizations of quasiconformal extendablity and asymptotic conformality for univalent functions.
文摘In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.
基金Supported by National Natural Science Foundation of China （Grant No. 10871025）
文摘Let （y, d, dλ） be （Rn, ｜·｜,μ）, where ｜·｜ is the Euclidean distance, μ is a nonnegative Radon measure on Rn satisfying the polynomial growth condition, or the Gauss measure metric space （Rn, ｜·｜,dγ）, or the space （S, d, p）, where S - Rn×R＋ is the （ax ＋ b）-group, d is the left-invariant Riemannian metric and p is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces {Xs（Y））0〈s≤∞ and the BM0-type spaces {BM0（y, s）}0〈s≤∞. Let Hi（Y） be the known atomic Hardy space and L01（y） the subspace of f ∈ L1（Y） with integral 0. The authors prove that the dual space of Xs（Y） is SM0（Y,s） when s∈ （0, ∞）, Xs（Y） = H1（Y） when s ∈ （0, 1], and X∞（y） = L01（Y） （or L1（Y））. As applications, the authors show that if T is a linear operator bounded from H1 （Y） to L1 （Y） and from L1（y） to L1,∞（Y）, then for all r ∈ （1, ∞） and s ∈ （r, ∞], T is bounded from Xr（y） to the Lorentz space L1,8（y）, which applies to the Calderon-Zygmund operator on （Rn, ｜·｜,μ）, the imaginary powers of the 0rnstein-Uhlenbeck operator on （Rn, ｜·｜,dγ） and the spectral operator associated with the spectral multiplier on （S, d, p）. All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.
基金Project supported by the Chinese National Tian Yuan Science Foundation.
文摘We study the joint distributions of four random variables h(ω) (stopping time or optional time), location x(h), l(ω) (co-optional time) and location x(l) for Markov processes. Some distributions for d(≥3) -dimensional Brownian motion and the joint distribution of first exit and last exit locations for symmetric stable process are found.
基金This work is financially supported by National Nature Science Foundation of China (Grant No. 50175027).
文摘Subpixel accuracy for V-groove center in robot welding is researched and a software measure to increase the accuracy of seam tracking by laser is presented. LOG （ Laplacian of Gaussian ） operator is adopted to detect image edge. Vgroove center is extracted by corner detection of extremum curvature. Subpixel position is obtained by Lagarange polynomial interpolation algorithm. Experiment results show that the method is brief and applied, and is sufficient for the real time of robot welding by laser sensors.
基金the National Natural Science Foundation of China(Grant No.11347026)the Natural Science Foundation of Shandong Province(Grant Nos.ZR2016AM03 and ZR2017M A011).
文摘Using an operator ordering method for some commutative superposition operators,we introduce two new multi-variable special polynomials and their generating functions,and present some new operator identities and integral formulas involving the two special polynomials.Instead of calculating compli-cated partial differential,we use the special polynomials and their generating functions to concsely address the normalzation,photoount distributions and Wigner distributions of several quantum states that can be realized physically,the rsults of which provide real convenience for further investigating the properties and applications of these states.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘In this paper, we present several expansions of the symbolic operator （1 ＋E）^x. Moreover, we derive some series transforms formulas and the Newton generating functions of {f（k）}.
文摘For a polynomial p（z） of degree n which has no zeros in ｜z｜ 〈 1, Dewan et al., （K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, 38-41） establishedfor any complex number β with ｜β｜≤ and｜z｜ = 1. In this paper we consider the operator B, which carries a polynomial p（z） into
基金Supported by Fundamental Research Funds for the Central Universities(Key Program)National Natural Science Foundation of China(Grant No.41076125)+1 种基金973 project(Grant No.2010CB950504)Polar Climate and Environment Key Laboratory
文摘In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jФjψj, where each Фj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each Фjψj is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.
文摘Let Pn be the class of polynomials of degree at most n. Rather and Shah [15] proved that if P∈Pn and P（z） 6=0 in｜z｜〈1, then for every R〉0 and 0≤q〈∞,｜B[P（Rz）]｜q≤｜RnB[zn]＋λ0｜q｜1＋zn｜q ｜P（z）｜q, where B is a Bn-operator. In this paper, we prove some generalization of this result which in particular yield-s some known polynomial inequalities as special. We also consider an operator Dαwhich maps a polynomial P（z） into DαP（z）：=nP（z）＋（α-z）P′（z） and obtain exten-sions and generalizations of a number of well-known Lq inequalities.
文摘The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.
基金supported by MEC Project MTM-2009-07498Junta de Andalucía "Proyecto de Excelencia" FQM-4911+1 种基金partially supported by MEC and FEDER Project MTM2008-03211supported by Prometeo 2008/101
文摘If X is an Asplund space, then every uniformly continuous function on Bx＊ which is holomorphic on the open unit ball, can be perturbed by a w＊ continuous and homogeneous polynomial on X＊ to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall＇s variational principle. We also show that the set of N-homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N-homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.
基金supported by National Natural Science Foundation of China(11401310)Natural Science Foundation of Jiangsu Province(BK20140965)+2 种基金High level talent research fund of Nanjing Forestry University(G2014022)supported by the overseas research program of Jiangsu Provincesponsored by Qing Lan Project of Jiangsu Province
文摘In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.
