The minimal widths of three bounded subsets of the unit sphere associated to a unit vector in a normed linear space are studied,and three related geometric constants are introduced.New characterizations of inner produ...The minimal widths of three bounded subsets of the unit sphere associated to a unit vector in a normed linear space are studied,and three related geometric constants are introduced.New characterizations of inner product spaces are also presented.From the perspective of minimal width,strongε-symmetry of Birkhoff orthogonality is introduced,and its relation toε-symmetry of Birkhoff orthogonality is shown.Unlike most of the existing parameters of the underlying space,these new constants are full dimensional in nature.展开更多
In this paper,we introduce a new geometric constant R_(X)(κ)based on isosceles orthogonality.First,we explore some basic properties of this new constant and then provide several examples to estimate its exact values ...In this paper,we introduce a new geometric constant R_(X)(κ)based on isosceles orthogonality.First,we explore some basic properties of this new constant and then provide several examples to estimate its exact values in certain specific Banach spaces.Next,we investigate the relationships between this new constant and other classical constants.Specifically,we establish an inequality relationship between it and the J(X)constant,as well as an identity relationship between it and theρX(t)constant.Furthermore,we characterize some geometric properties of Banach spaces by means of this new constant.Finally,by restricting the above-mentioned constant to the unit sphere,we introduce another new constant,calculate its upper and lower bounds,and present a relevant example.展开更多
Within the framework of the first-order Born approximation, the triple differential cross sections （TDCSs） for simultaneous ionization and excitation of helium are calculated. The wave function of the ejected electr...Within the framework of the first-order Born approximation, the triple differential cross sections （TDCSs） for simultaneous ionization and excitation of helium are calculated. The wave function of the ejected electron is chosen to be orthogonal or non-orthogonal to the wave function of the bound electron before ionization. It is found that the orthogonality has a strong effect on the TDCS, especially when plane waves and Coulomb waves are used to describe the projectile and the ejected electron.展开更多
The orthogonality catastrophe(OC)of quantum many-body systems is an important phenomenon in condensed matter physics.Recently,an interesting relationship between the OC and the quantum speed limit(QSL)was shown(Fogart...The orthogonality catastrophe(OC)of quantum many-body systems is an important phenomenon in condensed matter physics.Recently,an interesting relationship between the OC and the quantum speed limit(QSL)was shown(Fogarty 2020 Phys.Rev.Lett.124110601).Inspired by the remarkable feature,we provide a quantitative version of the quantum average speed as another different method to investigate the measure of how it is close to the OC dynamics.We analyze the properties of an impurity qubit embedded into an isotropic Lipkin-Meshkov-Glick spin model,and show that the OC dynamics can also be characterized by the average speed of the evolution state.Furthermore,a similar behavior of the actual speed of quantum evolution and the theoretical maximal rate is shown which can provide an alternative speed-up protocol allowing us to understand some universal properties characterized by the QSL.展开更多
Round inductosyn is widely used in inertial navigation test equipment, and its accuracy has significant effect on the general accuracy of the equipment. Four main errors of round inductosyn,i.e. the first-order long-p...Round inductosyn is widely used in inertial navigation test equipment, and its accuracy has significant effect on the general accuracy of the equipment. Four main errors of round inductosyn,i.e. the first-order long-period (360°) harmonic error, the second-order long-period harmonic error, the first-order short-period harmonic error and the second-order short-period harmonic error, are described, and the orthogonality of these four kinds of errors is studied. An error separating technology is proposed to separate these four kinds of errors, and in the process of separating the short-period harmonic errors, the arrangement in the order of decimal part of the angle pitch number can be omitted. The effectiveness of the technology proposed is proved through measuring and adjusting the angular errors.展开更多
This paper introduced a dynamical system (neural networks) algorithm for solving a least squares problem with orthogonality constraints, which has wide applications in computer vision and signal processing. A rigorous...This paper introduced a dynamical system (neural networks) algorithm for solving a least squares problem with orthogonality constraints, which has wide applications in computer vision and signal processing. A rigorous analysis for the convergence and stability of the algorithm was provided. Moreover, a so called zero extension technique was presented to keep the algorithm always convergent to the needed result for any randomly chosen initial data. Numerical experiments illustrate the effectiveness and efficiency of the algorithm.展开更多
An implicit algorithm of Bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow problems is proposed in this paper. A Newton fluid flow with two kinds of artificial v...An implicit algorithm of Bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow problems is proposed in this paper. A Newton fluid flow with two kinds of artificial viscosity subjected to the inequality constraint is introduced to approximate the Bingham fluid flow. This approach can effectively simulate the Bingham fluid flow with floating rigid cores or fixing rigid cores.展开更多
Based on empirical likelihood method and QR decomposition technique, an orthogonality empirical likelihood based estimation method for the fixed effects in linear mixed effects models is proposed. Under some regularit...Based on empirical likelihood method and QR decomposition technique, an orthogonality empirical likelihood based estimation method for the fixed effects in linear mixed effects models is proposed. Under some regularity conditions, the proposed empirical log-likelihood ratio is proved to be asymptotically chi-squared, and then the confidence intervals for the fixed effects are constructed. The proposed estimation procedure is not affected by the random effects,and then the resulting estimator is more effective. Some simulations and a real data application are conducted for further illustrating the performances of the proposed method.展开更多
Let X be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle Aq（x, y） between two vectors x and y in X, such that x is Birkhoff orthogonal to y i...Let X be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle Aq（x, y） between two vectors x and y in X, such that x is Birkhoff orthogonal to y if and only if Aq（x,y）=π/2. Some other properties of this angle are also discussed.展开更多
In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid- ered. Results on orthogonality of the range and the kern...In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid- ered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.展开更多
In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and find that the method could be regarded as the me...In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref. [1], and then give the orthogonality conditions (i.e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i.e. under uniform load, under compound toad, with different boundary conditiors), we deduce the specific expressions and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kai-yuan and Hwang Chien in Refs. [1 - 4J.展开更多
Abstract. Let {L（Ln^（A,λ）（x）}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0,∞） byLn^（A,λ）（x）=n!/（-λ）^n ∑nk-0（-λ）^k/k!（n-k）!（A＋I）n[（A＋I）k]^-1x^k,where A ∈ C^r×...Abstract. Let {L（Ln^（A,λ）（x）}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0,∞） byLn^（A,λ）（x）=n!/（-λ）^n ∑nk-0（-λ）^k/k!（n-k）!（A＋I）n[（A＋I）k]^-1x^k,where A ∈ C^r×r. It is known that {Ln^（A,λ）（x）}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re（z） 〉 -1 for every z E or（a）. In this note we show that forA such that σ（A） does not contain negative integers, the Laguerre matrix polynomials Ln^（A,λ）（x） are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases： the above matrix case and the known scalar case.展开更多
For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabili...For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.展开更多
It is a valuable challenge to design an integrated material system with high information security and diversity.Herein,we report a viable solution by designing novel supramolecular composites capable of undergoing ort...It is a valuable challenge to design an integrated material system with high information security and diversity.Herein,we report a viable solution by designing novel supramolecular composites capable of undergoing orthogonal photoreactions.Excitingly,the first case of three-dimensional(3D)+two-dimensional(2D)image encryption has been demonstrated.The 3D images are holographic,whose colors are set as blue,green,and red,respectively,using Denisyuk recording geometry through visible laser photopolymerization.The 2D images are photoluminescent,whose color are tuned in a broad range from pale-blue to red by varying the contents of supramolecular Pt(II)complexes.The complexes are decorated with stilbene moiety that can undergo UV photocycloaddition,leading to a broad-range blueshift of emissions and an augmentation of quantum yield by up to 17.2 times.Mechanistically,the emission transformation is ascribed to the weakening of both intramolecular through-space conjugation and intermolecular Pt…Pt andπ-πinteractions.The 2D photoluminescent images exhibit facile thermoresponse while the 3D images are unchanged when varying temperatures,further boosting the information security and diversity.This research is expected to pave a new way to design high-security level materials for combating ever-increasing counterfeiting.展开更多
The two-dimensional grating serves as a critical component in plane grating interferometers for achieving high-precision multidimensional displacement measurements.The calibration of grating groove density and orthogo...The two-dimensional grating serves as a critical component in plane grating interferometers for achieving high-precision multidimensional displacement measurements.The calibration of grating groove density and orthogonality error of grating grooves not only improves the positioning accuracy of grating interferometers but also provides essential feedback for optimizing two-dimensional grating fabrication.This study proposes a method for simultaneous calibration of these parameters using orthogonal heterodyne laser interferometry.A two-dimensional grating interferometer is built with the grating to be measured,and a biaxial laser interferometer provides a displacement reference for it.The phase mapping relationship between grating interference and laser interference is established.The interference phase information obtained by any two displacements can simultaneously solve the above three parameters and obtain the grating installation error.The feasibility of the proposed method is verified by using a 1200 gr/mm two-dimensional grating.The standard deviation of the grating groove density in the X and Y directions is 0.012 gr/mm and 0.014 gr/mm,respectively.The standard deviation of the orthogonality error of grating grooves is 0.004°,and the standard deviation of the installation error is 0.002°.Compared with the atomic force microscope method,the consistency of the grating groove density in the X and Y directions is better than 0.03 gr/mm and 0.06 gr/mm,and the orthogonality error of grating grooves is better than 0.008°.The experimental results show that the proposed method can be simply and efficiently applied to the calibration of the grating line parameters of the two-dimensional grating.展开更多
Space-filling designs with superior low-dimensional properties are highly required in computer experiments.Strong orthogonal arrays(SOAs)represent a class of such designs that outperform ordinary orthogonal arrays in ...Space-filling designs with superior low-dimensional properties are highly required in computer experiments.Strong orthogonal arrays(SOAs)represent a class of such designs that outperform ordinary orthogonal arrays in their stratification properties within low dimensions.Nevertheless,current methods for constructing high-strength SOAs are rare,and they typically rely on regular designs,thereby limiting the number of runs in the final arrays to prime powers.This study presents new construction methods for three types of SOAs:SOAs of strength three,column-orthogonal SOAs(OSOAs)of strength three and three minus.The resulting designs have run sizes of twice an odd prime power without replications,filling the gaps in run sizes left by existing constructions.The projection properties of Addelman–Kempthorne orthogonal arrays are instrumental in the development of these construction methods.展开更多
The thought how dual vectors are constructed in a new orthogonality relationship for theory of elasticity is generalized into orthotropic thin plate bending problems by using the analogy theory between plane elasticit...The thought how dual vectors are constructed in a new orthogonality relationship for theory of elasticity is generalized into orthotropic thin plate bending problems by using the analogy theory between plane elasticity problems and plate bending problems.Dual differential equations are directly obtained by using a mixed variables method.A dual differential matrix to be derived possesses a peculiarity of which principal diagonal sub-matrixes are zero matrixes.Two independently and symmetrically orthogonality sub-relationships are discovered.By using the integral form for elastic bending theory of orthotropic thin plate the orthogonality relationship is demonstrated.By selecting felicitous dual vectors a new orthogonality relationship for theory of elasticity can be generalized into elastic bending theory of orthotropic thin plate.By using the integral form a variational principle which is relative to differential form and a whole function expression are proposed.展开更多
基金supported by the National Natural Science Foundation of China(12071444,12201581)the Fundamental Research Program of Shanxi Province of China(202103021223191).
文摘The minimal widths of three bounded subsets of the unit sphere associated to a unit vector in a normed linear space are studied,and three related geometric constants are introduced.New characterizations of inner product spaces are also presented.From the perspective of minimal width,strongε-symmetry of Birkhoff orthogonality is introduced,and its relation toε-symmetry of Birkhoff orthogonality is shown.Unlike most of the existing parameters of the underlying space,these new constants are full dimensional in nature.
基金Supported by the Higher Education Science Research Project(Natural Science)of Anhui Province(Grant No.2023AH050487)。
文摘In this paper,we introduce a new geometric constant R_(X)(κ)based on isosceles orthogonality.First,we explore some basic properties of this new constant and then provide several examples to estimate its exact values in certain specific Banach spaces.Next,we investigate the relationships between this new constant and other classical constants.Specifically,we establish an inequality relationship between it and the J(X)constant,as well as an identity relationship between it and theρX(t)constant.Furthermore,we characterize some geometric properties of Banach spaces by means of this new constant.Finally,by restricting the above-mentioned constant to the unit sphere,we introduce another new constant,calculate its upper and lower bounds,and present a relevant example.
基金Project supported by the Anhui University Doctoral Research Starting Foundation,China(Grant Nos.02303319 and 33190203)the National Natural Science Foundation of China(Grant No.11274219)
文摘Within the framework of the first-order Born approximation, the triple differential cross sections （TDCSs） for simultaneous ionization and excitation of helium are calculated. The wave function of the ejected electron is chosen to be orthogonal or non-orthogonal to the wave function of the bound electron before ionization. It is found that the orthogonality has a strong effect on the TDCS, especially when plane waves and Coulomb waves are used to describe the projectile and the ejected electron.
基金supported by the National Natural Science Foundation of China under Grant Nos.11875086 and11775019。
文摘The orthogonality catastrophe(OC)of quantum many-body systems is an important phenomenon in condensed matter physics.Recently,an interesting relationship between the OC and the quantum speed limit(QSL)was shown(Fogarty 2020 Phys.Rev.Lett.124110601).Inspired by the remarkable feature,we provide a quantitative version of the quantum average speed as another different method to investigate the measure of how it is close to the OC dynamics.We analyze the properties of an impurity qubit embedded into an isotropic Lipkin-Meshkov-Glick spin model,and show that the OC dynamics can also be characterized by the average speed of the evolution state.Furthermore,a similar behavior of the actual speed of quantum evolution and the theoretical maximal rate is shown which can provide an alternative speed-up protocol allowing us to understand some universal properties characterized by the QSL.
文摘Round inductosyn is widely used in inertial navigation test equipment, and its accuracy has significant effect on the general accuracy of the equipment. Four main errors of round inductosyn,i.e. the first-order long-period (360°) harmonic error, the second-order long-period harmonic error, the first-order short-period harmonic error and the second-order short-period harmonic error, are described, and the orthogonality of these four kinds of errors is studied. An error separating technology is proposed to separate these four kinds of errors, and in the process of separating the short-period harmonic errors, the arrangement in the order of decimal part of the angle pitch number can be omitted. The effectiveness of the technology proposed is proved through measuring and adjusting the angular errors.
基金National Natural Science Foundation of China (No. 1990 10 18)
文摘This paper introduced a dynamical system (neural networks) algorithm for solving a least squares problem with orthogonality constraints, which has wide applications in computer vision and signal processing. A rigorous analysis for the convergence and stability of the algorithm was provided. Moreover, a so called zero extension technique was presented to keep the algorithm always convergent to the needed result for any randomly chosen initial data. Numerical experiments illustrate the effectiveness and efficiency of the algorithm.
文摘An implicit algorithm of Bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow problems is proposed in this paper. A Newton fluid flow with two kinds of artificial viscosity subjected to the inequality constraint is introduced to approximate the Bingham fluid flow. This approach can effectively simulate the Bingham fluid flow with floating rigid cores or fixing rigid cores.
基金Supported by the National Social Science Foundation of China(Grant No.18BTJ035).
文摘Based on empirical likelihood method and QR decomposition technique, an orthogonality empirical likelihood based estimation method for the fixed effects in linear mixed effects models is proposed. Under some regularity conditions, the proposed empirical log-likelihood ratio is proved to be asymptotically chi-squared, and then the confidence intervals for the fixed effects are constructed. The proposed estimation procedure is not affected by the random effects,and then the resulting estimator is more effective. Some simulations and a real data application are conducted for further illustrating the performances of the proposed method.
文摘Let X be a Minkowski plane, i.e., a real two dimensional normed linear space. We use projections to give a definition of the angle Aq（x, y） between two vectors x and y in X, such that x is Birkhoff orthogonal to y if and only if Aq（x,y）=π/2. Some other properties of this angle are also discussed.
基金National Council for Science and Technology (NCST) of KenyaDAAD-Germany for the financial support
文摘In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid- ered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.
基金Project supported by the National Natural Science Foundation of China
文摘In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref. [1], and then give the orthogonality conditions (i.e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i.e. under uniform load, under compound toad, with different boundary conditiors), we deduce the specific expressions and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kai-yuan and Hwang Chien in Refs. [1 - 4J.
基金Supported by the National Natural Science Foundation of China(No.10571122)the Beijing Natural Science Foundation(No.1052006)+1 种基金the Project of Excellent Young Teachersthe Doctoral Programme Foundation of National Education Ministry of China
文摘Abstract. Let {L（Ln^（A,λ）（x）}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0,∞） byLn^（A,λ）（x）=n!/（-λ）^n ∑nk-0（-λ）^k/k!（n-k）!（A＋I）n[（A＋I）k]^-1x^k,where A ∈ C^r×r. It is known that {Ln^（A,λ）（x）}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re（z） 〉 -1 for every z E or（a）. In this note we show that forA such that σ（A） does not contain negative integers, the Laguerre matrix polynomials Ln^（A,λ）（x） are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases： the above matrix case and the known scalar case.
文摘For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.
基金support from the Innovation and Talent Recruitment Base of New Energy Chemistry and Device(B21003)HUST Analytical&Testing Center,and Core Facilities of Life Sciences.
文摘It is a valuable challenge to design an integrated material system with high information security and diversity.Herein,we report a viable solution by designing novel supramolecular composites capable of undergoing orthogonal photoreactions.Excitingly,the first case of three-dimensional(3D)+two-dimensional(2D)image encryption has been demonstrated.The 3D images are holographic,whose colors are set as blue,green,and red,respectively,using Denisyuk recording geometry through visible laser photopolymerization.The 2D images are photoluminescent,whose color are tuned in a broad range from pale-blue to red by varying the contents of supramolecular Pt(II)complexes.The complexes are decorated with stilbene moiety that can undergo UV photocycloaddition,leading to a broad-range blueshift of emissions and an augmentation of quantum yield by up to 17.2 times.Mechanistically,the emission transformation is ascribed to the weakening of both intramolecular through-space conjugation and intermolecular Pt…Pt andπ-πinteractions.The 2D photoluminescent images exhibit facile thermoresponse while the 3D images are unchanged when varying temperatures,further boosting the information security and diversity.This research is expected to pave a new way to design high-security level materials for combating ever-increasing counterfeiting.
文摘The two-dimensional grating serves as a critical component in plane grating interferometers for achieving high-precision multidimensional displacement measurements.The calibration of grating groove density and orthogonality error of grating grooves not only improves the positioning accuracy of grating interferometers but also provides essential feedback for optimizing two-dimensional grating fabrication.This study proposes a method for simultaneous calibration of these parameters using orthogonal heterodyne laser interferometry.A two-dimensional grating interferometer is built with the grating to be measured,and a biaxial laser interferometer provides a displacement reference for it.The phase mapping relationship between grating interference and laser interference is established.The interference phase information obtained by any two displacements can simultaneously solve the above three parameters and obtain the grating installation error.The feasibility of the proposed method is verified by using a 1200 gr/mm two-dimensional grating.The standard deviation of the grating groove density in the X and Y directions is 0.012 gr/mm and 0.014 gr/mm,respectively.The standard deviation of the orthogonality error of grating grooves is 0.004°,and the standard deviation of the installation error is 0.002°.Compared with the atomic force microscope method,the consistency of the grating groove density in the X and Y directions is better than 0.03 gr/mm and 0.06 gr/mm,and the orthogonality error of grating grooves is better than 0.008°.The experimental results show that the proposed method can be simply and efficiently applied to the calibration of the grating line parameters of the two-dimensional grating.
基金supported by the Fundamental Research Funds for the Central Universities[grant number 2025JBZX013]the National Natural Science Foundation of China[grant numbers 12001036,12271166,32030063]+1 种基金Young Elite Scientists Sponsorship Program by CAST[grant number 2022QNRC001]National Key Research and Development Program of China[grant number 2024YFA1016200].
文摘Space-filling designs with superior low-dimensional properties are highly required in computer experiments.Strong orthogonal arrays(SOAs)represent a class of such designs that outperform ordinary orthogonal arrays in their stratification properties within low dimensions.Nevertheless,current methods for constructing high-strength SOAs are rare,and they typically rely on regular designs,thereby limiting the number of runs in the final arrays to prime powers.This study presents new construction methods for three types of SOAs:SOAs of strength three,column-orthogonal SOAs(OSOAs)of strength three and three minus.The resulting designs have run sizes of twice an odd prime power without replications,filling the gaps in run sizes left by existing constructions.The projection properties of Addelman–Kempthorne orthogonal arrays are instrumental in the development of these construction methods.
基金supported by the National Natural Science Foundation of China(Grant No.10272063)the Basic Science Research Foundation of Tsinghua University(JC2002003)+1 种基金the Special Scientific Foundation for Chinese Doctoral Education(20020003044)the Foundation for the Author of National Excellent Doctoral Dissertation of China(200242).
文摘The thought how dual vectors are constructed in a new orthogonality relationship for theory of elasticity is generalized into orthotropic thin plate bending problems by using the analogy theory between plane elasticity problems and plate bending problems.Dual differential equations are directly obtained by using a mixed variables method.A dual differential matrix to be derived possesses a peculiarity of which principal diagonal sub-matrixes are zero matrixes.Two independently and symmetrically orthogonality sub-relationships are discovered.By using the integral form for elastic bending theory of orthotropic thin plate the orthogonality relationship is demonstrated.By selecting felicitous dual vectors a new orthogonality relationship for theory of elasticity can be generalized into elastic bending theory of orthotropic thin plate.By using the integral form a variational principle which is relative to differential form and a whole function expression are proposed.
