This article investigates the second overtone thickness-extensional(TE2)vibrations and associated mode-coupling behaviors in ZnO piezoelectric film bulk acoustic resonator(FBAR),utilizing its wave dispersion relation ...This article investigates the second overtone thickness-extensional(TE2)vibrations and associated mode-coupling behaviors in ZnO piezoelectric film bulk acoustic resonator(FBAR),utilizing its wave dispersion relation and the higher-order stress balance principle.By superimposing the general wave solutions of multiple eigenmodes within the frequency range of the TE2 mode,mode-coupling solutions for ZnO FBAR are constructed.The substitution of these mode-coupling solutions into the higher-order stress balance principle,as laterally weak boundary conditions,leads to the frequency spectrogram equation,determining the relationship between resonance frequency and plate length-to-thickness ratio.A modified algorithm that combines the bisection method and the complex modulus ratio method is developed to solve the dispersion equation and frequency spectrogram equation(namely a kind of 2D complex transcendental equations)accurately and efficiently.The obtained results indicate that the operational TE2 mode may couple to unwanted 3^(rd)thickness-shear,fundamental thickness-shear,and flexural modes.Moreover,the mode-coupling behaviors depend strongly on resonance frequencies and plate length-to-thickness ratio.The displacement distributions of total displacement components,alongside the main displacement com-ponents of all considered eigenmodes,clearly demonstrate the variety of coupling behaviors.According to the obtained frequency spectrograms,the desirable values of plate length-to-thickness ratio for a clean operating mode with very weak coupling intensity are determined.These findings are of vital importance for the understanding of the mode-coupling me-chanism in overtone thickness-extensional FBARs,which will facilitate the structural design and optimization of FBAR devices.展开更多
Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is...Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .展开更多
Emily Dickinson is one of the most widely read and well-known American poets. Although there were just seven poems published during her lifetime, she was brought to many people's attention. Some people believe tha...Emily Dickinson is one of the most widely read and well-known American poets. Although there were just seven poems published during her lifetime, she was brought to many people's attention. Some people believe that Dickinson is a real romantic but others think she is a transcendentalist. However, both the factors of romanticism and transcendentalism can be seen in her character indeed. Romantic factors include individualism in her personality and her escaping from society. Transcendental factors are oversoul and mysticalness. If all the factors are taken into account, Emily Dickinson can fall into the category of both romanticism and transcendentalism.展开更多
A penetrating analysis into Hawthorne's writings of different periods and different angles brings us not only an astonishment at his unique understanding of evil, but also a definite awareness of his opposite but ...A penetrating analysis into Hawthorne's writings of different periods and different angles brings us not only an astonishment at his unique understanding of evil, but also a definite awareness of his opposite but staunch position as a pessimist.Meanwhile, following the discussions about Melville and other writers, his influences of pessimistic ideas and belief in evil allow of no doubt. Subsequently, through an inquisition to the roots of this tragic attitude, his standpoint is found to be inevitable.展开更多
In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional dec...In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional decagonal quasicrystal, we obtain the analytic solutions of modesⅠand Ⅱ using the transcendental function conformal mapping technique. Our calculation results provide an accurate estimate of the stress intensity factors K_Ⅰ and K_Ⅱ, which can be expressed in a quite simple form and are essential in the fracture theory of quasicrystals. Meanwhile, we suggest a generalized cohesive force model for the configuration to a two-dimensional decagonal quasicrystal. The results may provide theoretical guidance for the fracture theory of two-dimensional decagonal quasicrystals.展开更多
In this paper,we will prove that the system of differential-difference equations{((f(z)f′(z))^(n)+p^(2)_(1)(z)g m(z+η)=Q1(z),(g(z)g′(z))^(n)+p^(2)_(2)(z)f m(z+η)=Q_(2)(z),has no transcendental entire solution(f(z)...In this paper,we will prove that the system of differential-difference equations{((f(z)f′(z))^(n)+p^(2)_(1)(z)g m(z+η)=Q1(z),(g(z)g′(z))^(n)+p^(2)_(2)(z)f m(z+η)=Q_(2)(z),has no transcendental entire solution(f(z),g(z))withρ(f,g)<∞such thatλ(f)<ρ(f)andλ(g)<ρ(g),where P_(1()z),Q_(1)(z),P_(2)(z)and Q_(2)(z)are non-vanishing polynomials.展开更多
By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-differen...By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-difference equations, two interesting results are obtained. And it extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods...For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .展开更多
In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The prop...In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.展开更多
In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines.In this paper we shall...In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines.In this paper we shall consider the distribution problem of Julia sets of meromorphic maps.We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines.Meanwhile,examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines.Moreover,we shall show that the Julia set of a transcendental analytic self-map of C*can neither contain a free Jordan arc nor be contained in any finite set of straight lines.展开更多
The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolati...The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.展开更多
Let q be a finite nonzero complex number,let the q-difference equation f(qz)f(z/p)=R(z,f(z))=P(z,f(z))/Q(z,f(z))=Σ_(j=0)^(p)=0aj(z)f^(j)(z)/Σ_(k=0)b_(k)(z)f^(k)(z)(+)admit a nonconstant meromorphic solution f,where ...Let q be a finite nonzero complex number,let the q-difference equation f(qz)f(z/p)=R(z,f(z))=P(z,f(z))/Q(z,f(z))=Σ_(j=0)^(p)=0aj(z)f^(j)(z)/Σ_(k=0)b_(k)(z)f^(k)(z)(+)admit a nonconstant meromorphic solution f,where p and q are nonnegative integers,a_(j)with 0≤j≤p and b_(k)with 0≤k≤q polynomials in zwith a_(p)≠0 and b_(q)≠0 such that P(z,f(z))and Q(z,f(z))are relatively prime polynomials in f(z)and let m=p-q≥3.Then,(+)has no transcendental meromorphic solution when|q|=1,and the lower bound of the lower order of f is obtained when m≥3 and |q|≠1.展开更多
This paper presents an inequality, by use of which some results about the value distribution of f nf (k) are proved, where n and k are two positive integers.
In this paper, a class of slightly perturbed equations of the form F(x)= ξ -x+αΦ(x) will be treated graphically and symbolically, where Φ(x) is an analytic function of x. For graphical developments, we set up a si...In this paper, a class of slightly perturbed equations of the form F(x)= ξ -x+αΦ(x) will be treated graphically and symbolically, where Φ(x) is an analytic function of x. For graphical developments, we set up a simple graphical method for the real roots of the equation F(x)=0 illustrated by four transcendental equations. In fact, the graphical solution usually provides excellent initial conditions for the iterative solution of the equation. A property avoiding the critical situations between divergent to very slow convergent solutions may exist in the iterative methods in which no good initial condition close to the root is available. For the analytical developments, literal analytical solutions are obtained for the most celebrated slightly perturbed equation which is Kepler’s equation of elliptic orbit. Moreover, the effect of the orbital eccentricity on the rate of convergence of the series is illustrated graphically.展开更多
Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated...Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated, that is, h · f = g · h, in this paper, we will show that z ∈ J（f） if and only if h（z） ∈ J（g） （ similarly, z F（f） if and only ifh（z） ∈ F（g））, and this extends a result of Bergweiler.展开更多
In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c...In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c（z）.展开更多
The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are s...The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are studied with the wave equation,as well as the kinetic energy of the oscillating mass M.The kinetic energy and potential energy of the spring,and total energy are numerically simulated for different ratios ms/M with considering the spring’s mass,which makes the property of energy of the oscillating system understood easily.展开更多
We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.
基金supported by the National Natural Science Foundation of China(Grant Nos.12192210,12192211,12102183,12302200,and 12402192)the Natural Science Foundation of Zhejiang Province(Grant No.LD21A020001)+2 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20230873)the National Postdoctoral Program for Innovation Talents(Grant No.BX2021261)supported by the specialized research projects of Huanjiang Laboratory,Zhuji,Zhejiang Province.
文摘This article investigates the second overtone thickness-extensional(TE2)vibrations and associated mode-coupling behaviors in ZnO piezoelectric film bulk acoustic resonator(FBAR),utilizing its wave dispersion relation and the higher-order stress balance principle.By superimposing the general wave solutions of multiple eigenmodes within the frequency range of the TE2 mode,mode-coupling solutions for ZnO FBAR are constructed.The substitution of these mode-coupling solutions into the higher-order stress balance principle,as laterally weak boundary conditions,leads to the frequency spectrogram equation,determining the relationship between resonance frequency and plate length-to-thickness ratio.A modified algorithm that combines the bisection method and the complex modulus ratio method is developed to solve the dispersion equation and frequency spectrogram equation(namely a kind of 2D complex transcendental equations)accurately and efficiently.The obtained results indicate that the operational TE2 mode may couple to unwanted 3^(rd)thickness-shear,fundamental thickness-shear,and flexural modes.Moreover,the mode-coupling behaviors depend strongly on resonance frequencies and plate length-to-thickness ratio.The displacement distributions of total displacement components,alongside the main displacement com-ponents of all considered eigenmodes,clearly demonstrate the variety of coupling behaviors.According to the obtained frequency spectrograms,the desirable values of plate length-to-thickness ratio for a clean operating mode with very weak coupling intensity are determined.These findings are of vital importance for the understanding of the mode-coupling me-chanism in overtone thickness-extensional FBARs,which will facilitate the structural design and optimization of FBAR devices.
文摘Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .
文摘Emily Dickinson is one of the most widely read and well-known American poets. Although there were just seven poems published during her lifetime, she was brought to many people's attention. Some people believe that Dickinson is a real romantic but others think she is a transcendentalist. However, both the factors of romanticism and transcendentalism can be seen in her character indeed. Romantic factors include individualism in her personality and her escaping from society. Transcendental factors are oversoul and mysticalness. If all the factors are taken into account, Emily Dickinson can fall into the category of both romanticism and transcendentalism.
文摘A penetrating analysis into Hawthorne's writings of different periods and different angles brings us not only an astonishment at his unique understanding of evil, but also a definite awareness of his opposite but staunch position as a pessimist.Meanwhile, following the discussions about Melville and other writers, his influences of pessimistic ideas and belief in evil allow of no doubt. Subsequently, through an inquisition to the roots of this tragic attitude, his standpoint is found to be inevitable.
基金supported by the National Natural Science Foundation of China(Grant No.10802043)
文摘In this paper, we investigate the well-known problem of a finite width strip with a single edge crack, which is useful in basic engineering and material science. By extending the configuration to a two-dimensional decagonal quasicrystal, we obtain the analytic solutions of modesⅠand Ⅱ using the transcendental function conformal mapping technique. Our calculation results provide an accurate estimate of the stress intensity factors K_Ⅰ and K_Ⅱ, which can be expressed in a quite simple form and are essential in the fracture theory of quasicrystals. Meanwhile, we suggest a generalized cohesive force model for the configuration to a two-dimensional decagonal quasicrystal. The results may provide theoretical guidance for the fracture theory of two-dimensional decagonal quasicrystals.
基金Supported by the National Natural Science Foundation of China(Grant No.11971344).
文摘In this paper,we will prove that the system of differential-difference equations{((f(z)f′(z))^(n)+p^(2)_(1)(z)g m(z+η)=Q1(z),(g(z)g′(z))^(n)+p^(2)_(2)(z)f m(z+η)=Q_(2)(z),has no transcendental entire solution(f(z),g(z))withρ(f,g)<∞such thatλ(f)<ρ(f)andλ(g)<ρ(g),where P_(1()z),Q_(1)(z),P_(2)(z)and Q_(2)(z)are non-vanishing polynomials.
文摘By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-difference equations, two interesting results are obtained. And it extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
文摘For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .
基金supported partly by the National Natural Science Foundation of China(11926201,12171050)the National Science Foundation of Guangdong Province(2018A030313508)。
文摘In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.
文摘In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines.In this paper we shall consider the distribution problem of Julia sets of meromorphic maps.We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines.Meanwhile,examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines.Moreover,we shall show that the Julia set of a transcendental analytic self-map of C*can neither contain a free Jordan arc nor be contained in any finite set of straight lines.
基金supported by the National Natural Science Foundation of China(Nos.12172154 and 11925204)the 111 Project of China(No.B14044)the National Key Project of China(No.GJXM92579)。
文摘The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1226102311861023)the Foundation of Science and Technology Project of Guizhou Province of China(Grant No.[2018]5769-05).
文摘Let q be a finite nonzero complex number,let the q-difference equation f(qz)f(z/p)=R(z,f(z))=P(z,f(z))/Q(z,f(z))=Σ_(j=0)^(p)=0aj(z)f^(j)(z)/Σ_(k=0)b_(k)(z)f^(k)(z)(+)admit a nonconstant meromorphic solution f,where p and q are nonnegative integers,a_(j)with 0≤j≤p and b_(k)with 0≤k≤q polynomials in zwith a_(p)≠0 and b_(q)≠0 such that P(z,f(z))and Q(z,f(z))are relatively prime polynomials in f(z)and let m=p-q≥3.Then,(+)has no transcendental meromorphic solution when|q|=1,and the lower bound of the lower order of f is obtained when m≥3 and |q|≠1.
文摘This paper presents an inequality, by use of which some results about the value distribution of f nf (k) are proved, where n and k are two positive integers.
文摘In this paper, a class of slightly perturbed equations of the form F(x)= ξ -x+αΦ(x) will be treated graphically and symbolically, where Φ(x) is an analytic function of x. For graphical developments, we set up a simple graphical method for the real roots of the equation F(x)=0 illustrated by four transcendental equations. In fact, the graphical solution usually provides excellent initial conditions for the iterative solution of the equation. A property avoiding the critical situations between divergent to very slow convergent solutions may exist in the iterative methods in which no good initial condition close to the root is available. For the analytical developments, literal analytical solutions are obtained for the most celebrated slightly perturbed equation which is Kepler’s equation of elliptic orbit. Moreover, the effect of the orbital eccentricity on the rate of convergence of the series is illustrated graphically.
文摘Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated, that is, h · f = g · h, in this paper, we will show that z ∈ J（f） if and only if h（z） ∈ J（g） （ similarly, z F（f） if and only ifh（z） ∈ F（g））, and this extends a result of Bergweiler.
基金The NSF (06C417) of Hunan Provincethe QNF (04QN10) of Hunan AgricultureUniversity
文摘In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c（z）.
基金supported by the program for Higher educational Quality engineering projects of Anhui Province(2018zygc062)Anhui Provincial Natural Science Foundation(1808085MA20 and 1708085MA10)+3 种基金Excellent Young Talents in University of Anhui Province(gxyq2017027)the key Scientific Research Foundation of Anhui Provincial Education Department(KJ2019A0564,KJ2019A0580,KJ2018A0366,Wdxy2018jyxm008 and Wdxy2018jyxm009)Excellent course of Anhui Provincial Education Department(2017kfk061)Wisdom classroom of Anqing Normal University(2018aqnuzhkt008).
文摘The energy of a spring with a well-distributed mass ms is theoretically studied in this paper.The solution of the wave equation is derived in detail,and then the kinetic energy and potential energy of the spring are studied with the wave equation,as well as the kinetic energy of the oscillating mass M.The kinetic energy and potential energy of the spring,and total energy are numerically simulated for different ratios ms/M with considering the spring’s mass,which makes the property of energy of the oscillating system understood easily.
文摘We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.
