Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defin...Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.展开更多
This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight ...This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient 1 of the least deviation from zero in L_(p,ω)[-1,1]are optimal for 1≤p<∞.We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.展开更多
This study focuses on the three courtyards located in the eastern,central,and western sections of the Tongzhou Campus of Renmin University of China.Adopting a functional differentiation perspective,the research system...This study focuses on the three courtyards located in the eastern,central,and western sections of the Tongzhou Campus of Renmin University of China.Adopting a functional differentiation perspective,the research systematically analyzes the patterns of plant diversity within courtyards characterized by distinct functional orientations.This analysis employs various plant species diversity indices,including the Patrick richness index,Simpson dominance index,Shannon-Wiener diversity index,and Pielou evenness index,alongside a classification of functional plant types,namely ornamental,ecological regulation,spatial shaping,and recreational assistance.The results indicate that the east courtyard presents the highest Patrick species richness(S=42),predominantly comprising spatial shaping and recreational assistance plants,which are wellsuited for recreational and passage functions.Conversely,the central courtyard exhibits the lowest Patrick species richness(S=19),characterized by a balanced distribution of functional types,with an emphasis on public display and traffic guidance.The west courtyard demonstrates the greatest stability in the tree layer(D=0.87),featuring a combination of shade-tolerant and ornamental plants that fulfill the requirements for a tranquil and naturalistic environment.One-way analysis of variance reveals that only Patrick species richness differs significantly among the courtyards(P=0.007),whereas the diversity index does not show a significant difference.This finding suggests that functional requirements precisely regulate diversity through microhabitat heterogeneity and plant configuration strategies.This study offers both a theoretical foundation and practical guidance for the plant configuration and functional optimization of small-scale courtyards on campus.展开更多
In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel function...In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z（Bκ＋1^c f（z））′= κBκ^c f（z）-（κ- 1）Bκ＋1^c f（z）,where b, c, p ∈ C and κ = p ＋（b ＋ 1）/2 ∈ C ／ Z0^-（Z0^-= {0,-1,-2, … }）. The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.展开更多
This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of...This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).展开更多
In a structural system reliability analysis that lacks probabilistic information, calculating the numerical characteristics of the state functions, especially the first four moments of the state functions, is necessar...In a structural system reliability analysis that lacks probabilistic information, calculating the numerical characteristics of the state functions, especially the first four moments of the state functions, is necessary. Based on that, the structural system reliability is analyzed with a fourth-order moment method. The reliability sensitivity is required to conduct the differential operation of the numerical characteristic functions. A reliability sensitivity analysis formula is then derived in combination with the relation of the differential operation. Based on the matrix theory and Kronecker algebra, this paper systematically derives a matrix expression of the first four moments of the state functions, and establishes the matrix relation between the first four moments of the state functions and those of the basic random variables. On this basis, a differential operation formula of the first four moments of the state functions is further derived against the first four moments of the basic random variables. The vector relation between the state functions and the multidimensional basic random variables is described by means of the matrix operation to extend the operation method. Finally, a concise and intuitive formula is obtained to explore the inherent essential relation between the numerical characteristics of the state functions and those of the basic random variables, leading to a universal equation for the two kinds of numerical characteristics.展开更多
Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P&...Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +展开更多
Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asym...Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.展开更多
In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
The autotetraploid Carassius auratus(4nRR,4n=200,RRRR)is derived from whole-genome duplication of Carassius auratus red var.(RCC,2n=100,RR).In the current study,we demonstrated that chromatophores and pigment changes ...The autotetraploid Carassius auratus(4nRR,4n=200,RRRR)is derived from whole-genome duplication of Carassius auratus red var.(RCC,2n=100,RR).In the current study,we demonstrated that chromatophores and pigment changes directly caused the coloration and variation of 4nRR skin(red in RCC,brownish-yellow in4nRR).To further explore the molecular mechanisms underlying coloration formation and variation in 4nRR,we performed transcriptome profiling and molecular functional verification in RCC and 4nRR.Results revealed that scarb1,associated with carotenoid metabolism,underwent significant down-regulation in 4nRR.Efficient editing of this candidate pigment gene provided clear evidence of its significant role in RCC coloration.Subsequently,we identified four divergent scarb1 homeologs in 4nRR:two original scarb1 homeologs from RCC and two duplicated ones.Notably,three of these homeologs possessed two highly conserved alleles,exhibiting biased and allelespecific expression in the skin.Remarkably,after precise editing of both the original and duplicated scarb1homeologs and/or alleles,4nRR individuals,whether singly or multiply mutated,displayed a transition from brownishyellow skin to a cyan-gray phenotype.Concurrently,the proportional areas of the cyan-gray regions displayed a gene-dose correlation.These findings illustrate the subfunctionalization of duplicated scarb1,with all scarb1genes synergistically and equally contributing to the pigmentation of 4nRR.This is the first report concerning the functional differentiation of duplicated homeologs in an autopolyploidfish,substantiallyenrichingour understanding of coloration formation and change within this group of organisms.展开更多
Mechanism functions and kinetic parameters of AlOOH（boehmite or diaspore） dissolving in sodium hydroxide solution were researched.The mixture of boehmite or diaspore and caustic solution was scanned by high-pressure...Mechanism functions and kinetic parameters of AlOOH（boehmite or diaspore） dissolving in sodium hydroxide solution were researched.The mixture of boehmite or diaspore and caustic solution was scanned by high-pressure differential scanning calorimetry（DSC） instrument with heating rate of 10 ℃/min,and differential equation method was used to analyse the DSC curves,combining with iterative method and linear least square method.The most probable mechanism functions for both boehmite or diaspore and caustic solution reactions were logically selected from 30 types of non-isothermal kinetics differential equations,according to the calculated results obtained by Matlab program.The most probable differential mechanism function of boehmite dissolving in caustic solution is f（α）=1-α,which reveals the first-order reaction with apparent activation energy of 79.178 kJ/mol and the preexponential constant 1.031×108 s-1.The function,f（α）=2（1-α）3/2,can describe the dissolution of diaspore sample in sodium hydroxide solution.The calculated results of kinetic parameters are apparent activation energy of 73.858 kJ/mol,preexponential constant of 5.752×107 s-1 and reaction order of 1.5.展开更多
Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present wor...Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “<em>The point and the straight line are one and the same</em>”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: <em>ab</em> = <em>ba</em> and (<em>ab</em>)(<em>ac</em>) = <em>a</em>, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.展开更多
The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are ...The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are established for the existence of periodic solutions and some previous results are extended.展开更多
The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the ...The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.展开更多
By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(...By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].展开更多
This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several differen...This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.展开更多
文摘Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.
基金Supported by the National Natural Science Foundation of China(Grant No.11871006).
文摘This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient 1 of the least deviation from zero in L_(p,ω)[-1,1]are optimal for 1≤p<∞.We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.
基金Sponsored by 2025 Undergraduate Innovation and Entrepreneurship Training Program Project.
文摘This study focuses on the three courtyards located in the eastern,central,and western sections of the Tongzhou Campus of Renmin University of China.Adopting a functional differentiation perspective,the research systematically analyzes the patterns of plant diversity within courtyards characterized by distinct functional orientations.This analysis employs various plant species diversity indices,including the Patrick richness index,Simpson dominance index,Shannon-Wiener diversity index,and Pielou evenness index,alongside a classification of functional plant types,namely ornamental,ecological regulation,spatial shaping,and recreational assistance.The results indicate that the east courtyard presents the highest Patrick species richness(S=42),predominantly comprising spatial shaping and recreational assistance plants,which are wellsuited for recreational and passage functions.Conversely,the central courtyard exhibits the lowest Patrick species richness(S=19),characterized by a balanced distribution of functional types,with an emphasis on public display and traffic guidance.The west courtyard demonstrates the greatest stability in the tree layer(D=0.87),featuring a combination of shade-tolerant and ornamental plants that fulfill the requirements for a tranquil and naturalistic environment.One-way analysis of variance reveals that only Patrick species richness differs significantly among the courtyards(P=0.007),whereas the diversity index does not show a significant difference.This finding suggests that functional requirements precisely regulate diversity through microhabitat heterogeneity and plant configuration strategies.This study offers both a theoretical foundation and practical guidance for the plant configuration and functional optimization of small-scale courtyards on campus.
基金partly supported by the Natural Science Foundation of China(11271045)the Higher School Doctoral Foundation of China(20100003110004)+2 种基金the Natural Science Foundation of Inner Mongolia of China(2010MS0117)athe Higher School Foundation of Inner Mongolia of China(NJZY13298)the Commission for the Scientific Research Projects of Kafkas Univertsity(2012-FEF-30)
文摘In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z（Bκ＋1^c f（z））′= κBκ^c f（z）-（κ- 1）Bκ＋1^c f（z）,where b, c, p ∈ C and κ = p ＋（b ＋ 1）/2 ∈ C ／ Z0^-（Z0^-= {0,-1,-2, … }）. The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.
基金The second named author was supported in part by an NSERC Postdoctoral Fellowship,Canada and a CR F Grant,University of Alberta
文摘This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).
基金Project supported by the National Natural Science Foundation of China(Nos.51135003 and U1234208)the Major State Basic Research Development Program of China(973 Program)(No.2014CB046303)
文摘In a structural system reliability analysis that lacks probabilistic information, calculating the numerical characteristics of the state functions, especially the first four moments of the state functions, is necessary. Based on that, the structural system reliability is analyzed with a fourth-order moment method. The reliability sensitivity is required to conduct the differential operation of the numerical characteristic functions. A reliability sensitivity analysis formula is then derived in combination with the relation of the differential operation. Based on the matrix theory and Kronecker algebra, this paper systematically derives a matrix expression of the first four moments of the state functions, and establishes the matrix relation between the first four moments of the state functions and those of the basic random variables. On this basis, a differential operation formula of the first four moments of the state functions is further derived against the first four moments of the basic random variables. The vector relation between the state functions and the multidimensional basic random variables is described by means of the matrix operation to extend the operation method. Finally, a concise and intuitive formula is obtained to explore the inherent essential relation between the numerical characteristics of the state functions and those of the basic random variables, leading to a universal equation for the two kinds of numerical characteristics.
文摘Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +
基金Supported by the National Natural Science Foundation of China(Grant No.11171014)
文摘Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.
文摘In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
基金supported by the National Natural Science Foundation of China (32172972,U19A2040)Science and Technology Innovation Program of Hunan Province (2021RC4028)+4 种基金Earmarked Fund for China Agriculture Research System (CARS-45)Hunan Provincial Science and Technology Department (2019RS5001)Special Funds for Construction of Innovative Provinces in Hunan Province (2021NK1010)Special Science Found of Nansha-South China Agricultural University Fishery Research Institute,Guangzhou (NSYYKY202305,NSYYKY202306)Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province。
文摘The autotetraploid Carassius auratus(4nRR,4n=200,RRRR)is derived from whole-genome duplication of Carassius auratus red var.(RCC,2n=100,RR).In the current study,we demonstrated that chromatophores and pigment changes directly caused the coloration and variation of 4nRR skin(red in RCC,brownish-yellow in4nRR).To further explore the molecular mechanisms underlying coloration formation and variation in 4nRR,we performed transcriptome profiling and molecular functional verification in RCC and 4nRR.Results revealed that scarb1,associated with carotenoid metabolism,underwent significant down-regulation in 4nRR.Efficient editing of this candidate pigment gene provided clear evidence of its significant role in RCC coloration.Subsequently,we identified four divergent scarb1 homeologs in 4nRR:two original scarb1 homeologs from RCC and two duplicated ones.Notably,three of these homeologs possessed two highly conserved alleles,exhibiting biased and allelespecific expression in the skin.Remarkably,after precise editing of both the original and duplicated scarb1homeologs and/or alleles,4nRR individuals,whether singly or multiply mutated,displayed a transition from brownishyellow skin to a cyan-gray phenotype.Concurrently,the proportional areas of the cyan-gray regions displayed a gene-dose correlation.These findings illustrate the subfunctionalization of duplicated scarb1,with all scarb1genes synergistically and equally contributing to the pigmentation of 4nRR.This is the first report concerning the functional differentiation of duplicated homeologs in an autopolyploidfish,substantiallyenrichingour understanding of coloration formation and change within this group of organisms.
基金Project(2007BC13504)supported by the National Basic Research Program of ChinaProject(20050145029)supported by Research Fund for the Doctoral Program of Higher EducationProject(2005221012)supported by the Science and Technology Talents Fund for Excellent Youth of Liaoning Province,China
文摘Mechanism functions and kinetic parameters of AlOOH（boehmite or diaspore） dissolving in sodium hydroxide solution were researched.The mixture of boehmite or diaspore and caustic solution was scanned by high-pressure differential scanning calorimetry（DSC） instrument with heating rate of 10 ℃/min,and differential equation method was used to analyse the DSC curves,combining with iterative method and linear least square method.The most probable mechanism functions for both boehmite or diaspore and caustic solution reactions were logically selected from 30 types of non-isothermal kinetics differential equations,according to the calculated results obtained by Matlab program.The most probable differential mechanism function of boehmite dissolving in caustic solution is f（α）=1-α,which reveals the first-order reaction with apparent activation energy of 79.178 kJ/mol and the preexponential constant 1.031×108 s-1.The function,f（α）=2（1-α）3/2,can describe the dissolution of diaspore sample in sodium hydroxide solution.The calculated results of kinetic parameters are apparent activation energy of 73.858 kJ/mol,preexponential constant of 5.752×107 s-1 and reaction order of 1.5.
文摘Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “<em>The point and the straight line are one and the same</em>”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: <em>ab</em> = <em>ba</em> and (<em>ab</em>)(<em>ac</em>) = <em>a</em>, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.
文摘The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are established for the existence of periodic solutions and some previous results are extended.
基金Supported by the National Natural Science Foundation of China(11171298)the Zhejiang Natural Science Foundation of China(Y6110425)
文摘The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.
基金National Natural Science Foundation of China( 198710 0 5 )
文摘By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].
基金Supported by NSFC (11001091)Chinese UniversityResearch Foundation (2010MS129)
文摘This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.
