In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which ...In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.展开更多
Objective: To study the relationship between Ca 2+ intake,Ca 2+ absorption and Ca 2+ retention. Methods; Thirty-six weaning rats were firstly fed with Ca-poor diet for a 3-week Ca 2+ depletion period,and ...Objective: To study the relationship between Ca 2+ intake,Ca 2+ absorption and Ca 2+ retention. Methods; Thirty-six weaning rats were firstly fed with Ca-poor diet for a 3-week Ca 2+ depletion period,and then randomly divided into 3 groups to receive diet with low,adequate and high Ca 2+ contents.The experiment lasted 12 weeks,with a 3-day metabolism experiment conducted in the 5th week. Results:Supplement with adequate amount of Ca 2+ increased bone Ca 2+,bone mineral density(BMD),bone mineral content(BMC)and the weight of femur(P<0.05).Higher doses of Ca 2+ failed to further improve the efficiency.Analysis showed that Ca 2+ intake was positively correlated to bone Ca 2+,BND and BMC,and negatively correlated to the rates of Ca 2+ absorption and retention.Dose-effect study revealed that the Ca 2+ absorption rate()was a hyperbola function of Ca 2+ intake(X),i.e.=e 6.8068×X -0.8821+20(F=47.3154,P<0.0001).It was found from the results that bone Ca 2+ content(determined by atom absorption spectrophotometer)had a linear correlation to BMC(determined with uni-photon bone mineral densitometer). Conclusion:The rate of Ca 2+ absorption is a hyperbola function of Ca 2+ intake.The increment of Ca 2+ absorption will be minimal when Ca 2+ intake exceeds 3d 250 mg in rats.BMC,better than BMD,may substitute for the ideal bone Ca 2+ content as the index to assess the efficacy of Ca 2+ supplement.展开更多
The non-rectangular hyperbola(NRH)equation is the most popular method that plots the photosynthetic light-response(PLR)curve and helps to identify plant photosynthetic capability.However,the PLR curve can't be plo...The non-rectangular hyperbola(NRH)equation is the most popular method that plots the photosynthetic light-response(PLR)curve and helps to identify plant photosynthetic capability.However,the PLR curve can't be plotted well by the NRH equation at different plant growth phases due to the variations of plant development.Recently,plant physiological parameters have been considered into the NRH equation to establish the modified NRH equation,but plant height(H),an important parameter in plant growth phases,is not taken into account.In this study,H was incorporated into the NRH equation to establish the modified NRH equation,which could be used to estimate photosynthetic capability of herbage at different growth phases.To explore photosynthetic capability of herbage,we selected the dominant herbage species Potentilla anserina L.and Elymus nutans Griseb.in the Heihe River Basin,Northwest China as the research materials.Totally,twenty-four PLR curves and H at different growth phases were measured during the growing season in 2016.Results showed that the maximum net photosynthetic rate and the initial slope of PLR curve linearly increased with H.The modified NRH equation,which is established by introducing H and an H-based adjustment factor into the NRH equation,described better the PLR curves of P.anserina and E.nutans than the original ones.The results may provide an effective method to estimate the net primary productivity of grasslands in the study area.展开更多
An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be f...An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.展开更多
To study calculating method of settlement on top of extra-long large-diameter pile, the relevant research results were summarized. The hyperbola model, a nonlinear load transfer function, was introduced to establish t...To study calculating method of settlement on top of extra-long large-diameter pile, the relevant research results were summarized. The hyperbola model, a nonlinear load transfer function, was introduced to establish the basic differential equation with load transfer method. Assumed that the displacement of pile shaft was the high order power series of buried depth, through merging the same orthometric items and arranging the relevant coefficients, the solution which could take the nonlinear pile-soil interaction and stratum properties of soil into account was solved by power series. On the basis of the solution, by determining the load transfer depth with criterion of settlement on pile tip, the method by making boundary conditions compatible was advised to solve the load-settlement curve of pile. The relevant flow chart and mathematic expressions of boundary conditions were also listed. Lastly, the load transfer methods based on both two-broken-line model and hyperbola model were applied to analyzing a real project. The related coefficients of fitting curves by hyperbola were not less than 0.96, which shows that the hyperbola model is truthfulness, and is propitious to avoid personal error. The calculating value of load-settlement curve agrees well with the measured one, which indicates that it can be applied in engineering practice and making the theory that limits the design bearing capacity by settlement on pile top comes true.展开更多
A nonlinear transformation and some multi-solition solutions for the (2+1 )-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear tran...A nonlinear transformation and some multi-solition solutions for the (2+1 )-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear transformation, we reduce the (2+ 1)-dimensional GBK system to a simple linear evolution equation. Solving this equation,we can obtain some new explicit exact solutions of the original equations by means of the extended hyperbola function method.展开更多
In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have con...In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have considered also hyperbola-preserving conformal transformation and the relevant physical sense. And therefore obtained other invariant properties under the illustrious concircular transformation.展开更多
More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance m...More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance method.展开更多
In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions ...In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.展开更多
We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only ...We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only one. Secondly, a special boundary value problem of lower triangular matrix is presented and transformed into four related boundary value problems. Finally, Liouville theorem and Painlevé theorem and pseudo-orthogonal polynomials are used to give solutions.展开更多
Hyperbola is one of the important knowledge points in high school mathematics, and it is also a very common question type in the final question of college entrance examination mathematics. For the current hyperbola te...Hyperbola is one of the important knowledge points in high school mathematics, and it is also a very common question type in the final question of college entrance examination mathematics. For the current hyperbola teaching method, there is too single problem. It is not enough to only pay attention to the explanation of classroom knowledge. In the long run, students will lack practical application ability, and the cultivation of students' overall thinking will be ignored. This paper mainly analyzes some common learning mistakes in hyperbola teaching, and puts forward how to improve these mistakes, explores specific strategies to improve hyperbola teaching methods and teaching ideas, focusing on how to improve students' ability of mathematical analysis and application in senior high school.展开更多
Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-...Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems.For ideal free-surface flow with zero surface tension and gravity,the authors review existing works that describe"splash singularities",singular hyperbolic solutions related to jet formation and"flip-through",and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure.The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation.Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.展开更多
Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bé...Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.展开更多
The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstr...The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstruction method at a cell interface.The iteration method is used to determine the velocity of fluid interface,based on the force balance between both sides due to the incidence and bounce back of particles at the interface.An improved Aitken method is proposed with a simple hybrid of the modified Aitken method(Aitken-Chen)and the Steffensen method.Numerical tests validate its efficiency with significantly less calls to the function not only for the average number but also for the maximum.The new reconstruction is based on the tangent of hyperbola for interface capturing(THINC)but applied only to the volume fraction,which is very simple to be implemented under the stratified frame-work and capable of resolving fluid interface in mixture.Furthermore,the directional splitting is adopted rather than the previous quasi-one-dimensional method.Typical numerical tests,including several watergas shock tube flows,and the shock-water cylinder interaction flow show that the improved gas-kinetic scheme can capture fluid interfaces much sharper,while preserving the advantages of the original one.展开更多
This article puts forward a novel smooth rotated hyperbola model for support vector machine( RHSSVM) for classification. As is well known,the support vector machine( SVM) is based on statistical learning theory( SLT)a...This article puts forward a novel smooth rotated hyperbola model for support vector machine( RHSSVM) for classification. As is well known,the support vector machine( SVM) is based on statistical learning theory( SLT)and performs its high precision on data classification. However,the objective function is non-differentiable at the zero point. Therefore the fast algorithms cannot be used to train and test the SVM. To deal with it,the proposed method is based on the approximation property of the hyperbola to its asymptotic lines. Firstly,we describe the development of RHSSVM from the basic linear SVM optimization programming. Then we extend the linear model to non-linear model. We prove the solution of RHSSVM is convergent,unique,and global optimal. We show how RHSSVM can be practically implemented. At last,the theoretical analysis illustrates that compared with other three typical models,the rotated hyperbola model has the least error on approximating the plus function. Meanwhile,computer simulations show that the RHSSVM can reduce the consuming time at most 54. 6% and can efficiently handle large scale and high dimensional programming.展开更多
基金Supported by the National Key Basic Research Development Project of China(1998030600)Supported by the National Natural Science Foudation of China(10072013)Supported by the Educational Commmittee of Liaoning Province(990421093)
文摘In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.
文摘Objective: To study the relationship between Ca 2+ intake,Ca 2+ absorption and Ca 2+ retention. Methods; Thirty-six weaning rats were firstly fed with Ca-poor diet for a 3-week Ca 2+ depletion period,and then randomly divided into 3 groups to receive diet with low,adequate and high Ca 2+ contents.The experiment lasted 12 weeks,with a 3-day metabolism experiment conducted in the 5th week. Results:Supplement with adequate amount of Ca 2+ increased bone Ca 2+,bone mineral density(BMD),bone mineral content(BMC)and the weight of femur(P<0.05).Higher doses of Ca 2+ failed to further improve the efficiency.Analysis showed that Ca 2+ intake was positively correlated to bone Ca 2+,BND and BMC,and negatively correlated to the rates of Ca 2+ absorption and retention.Dose-effect study revealed that the Ca 2+ absorption rate()was a hyperbola function of Ca 2+ intake(X),i.e.=e 6.8068×X -0.8821+20(F=47.3154,P<0.0001).It was found from the results that bone Ca 2+ content(determined by atom absorption spectrophotometer)had a linear correlation to BMC(determined with uni-photon bone mineral densitometer). Conclusion:The rate of Ca 2+ absorption is a hyperbola function of Ca 2+ intake.The increment of Ca 2+ absorption will be minimal when Ca 2+ intake exceeds 3d 250 mg in rats.BMC,better than BMD,may substitute for the ideal bone Ca 2+ content as the index to assess the efficacy of Ca 2+ supplement.
基金funded by the National Natural Science Foundation of China(91025015,51178209)the Project of Arid Meteorological Science Research Foundation of China Meteorological Administration(IAM201608)
文摘The non-rectangular hyperbola(NRH)equation is the most popular method that plots the photosynthetic light-response(PLR)curve and helps to identify plant photosynthetic capability.However,the PLR curve can't be plotted well by the NRH equation at different plant growth phases due to the variations of plant development.Recently,plant physiological parameters have been considered into the NRH equation to establish the modified NRH equation,but plant height(H),an important parameter in plant growth phases,is not taken into account.In this study,H was incorporated into the NRH equation to establish the modified NRH equation,which could be used to estimate photosynthetic capability of herbage at different growth phases.To explore photosynthetic capability of herbage,we selected the dominant herbage species Potentilla anserina L.and Elymus nutans Griseb.in the Heihe River Basin,Northwest China as the research materials.Totally,twenty-four PLR curves and H at different growth phases were measured during the growing season in 2016.Results showed that the maximum net photosynthetic rate and the initial slope of PLR curve linearly increased with H.The modified NRH equation,which is established by introducing H and an H-based adjustment factor into the NRH equation,described better the PLR curves of P.anserina and E.nutans than the original ones.The results may provide an effective method to estimate the net primary productivity of grasslands in the study area.
文摘An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave.The new method can be applied to other nonlinear equations in mathematical physics.
基金Project(1004178) supported by Talented Person Recommend Foundation of Changsha University of Science and Technology, China
文摘To study calculating method of settlement on top of extra-long large-diameter pile, the relevant research results were summarized. The hyperbola model, a nonlinear load transfer function, was introduced to establish the basic differential equation with load transfer method. Assumed that the displacement of pile shaft was the high order power series of buried depth, through merging the same orthometric items and arranging the relevant coefficients, the solution which could take the nonlinear pile-soil interaction and stratum properties of soil into account was solved by power series. On the basis of the solution, by determining the load transfer depth with criterion of settlement on pile tip, the method by making boundary conditions compatible was advised to solve the load-settlement curve of pile. The relevant flow chart and mathematic expressions of boundary conditions were also listed. Lastly, the load transfer methods based on both two-broken-line model and hyperbola model were applied to analyzing a real project. The related coefficients of fitting curves by hyperbola were not less than 0.96, which shows that the hyperbola model is truthfulness, and is propitious to avoid personal error. The calculating value of load-settlement curve agrees well with the measured one, which indicates that it can be applied in engineering practice and making the theory that limits the design bearing capacity by settlement on pile top comes true.
文摘A nonlinear transformation and some multi-solition solutions for the (2+1 )-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear transformation, we reduce the (2+ 1)-dimensional GBK system to a simple linear evolution equation. Solving this equation,we can obtain some new explicit exact solutions of the original equations by means of the extended hyperbola function method.
基金Foundation item: Supported by the Natural Science foundation of Henan Education Committee (20021100002)
文摘In this paper, we have considered a class curves with some geometric properties in a higher dimensional space and obtained the differential equation of such a class curves, which are called the hyperbolas. We have considered also hyperbola-preserving conformal transformation and the relevant physical sense. And therefore obtained other invariant properties under the illustrious concircular transformation.
文摘More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance method.
文摘In this work, we apply a hyperbola function method to solve the nonlinear family of third order Korteweg-de Vries equations. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions and trigonometric functions. The method used is a promising method to solve other nonlinear evaluation equations.
文摘We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only one. Secondly, a special boundary value problem of lower triangular matrix is presented and transformed into four related boundary value problems. Finally, Liouville theorem and Painlevé theorem and pseudo-orthogonal polynomials are used to give solutions.
文摘Hyperbola is one of the important knowledge points in high school mathematics, and it is also a very common question type in the final question of college entrance examination mathematics. For the current hyperbola teaching method, there is too single problem. It is not enough to only pay attention to the explanation of classroom knowledge. In the long run, students will lack practical application ability, and the cultivation of students' overall thinking will be ignored. This paper mainly analyzes some common learning mistakes in hyperbola teaching, and puts forward how to improve these mistakes, explores specific strategies to improve hyperbola teaching methods and teaching ideas, focusing on how to improve students' ability of mathematical analysis and application in senior high school.
基金supported by the National Science Foundation under NSF Research Network Grant RNMS11-07444(KI-Net)the NSF Grants DMS-1514826,DMS-1812573,DMS-1515400,DMS-1812609the Simons Foundation under Grant 395796
文摘Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems.For ideal free-surface flow with zero surface tension and gravity,the authors review existing works that describe"splash singularities",singular hyperbolic solutions related to jet formation and"flip-through",and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure.The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation.Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.
基金supported by the Foundation of State Key Basic Research 973 Item(Grant No.2004CB719400)the National Natural Science Foundation of China(Grant Nos.60373033&60333010)National Natural Science Foundation for Innovative Research Groups(Grant No.60021201).
文摘Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.
基金supported by Science Challenge Project(TZ2016001)National Natural Science Foundation of China(U1430235)Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund(the second phase).
文摘The efficiency of recently developed gas-kinetic scheme for multimaterial flows is increased through the adoption of a new iteration method in the kinetic non-mixing Riemann solver and an interface sharpening reconstruction method at a cell interface.The iteration method is used to determine the velocity of fluid interface,based on the force balance between both sides due to the incidence and bounce back of particles at the interface.An improved Aitken method is proposed with a simple hybrid of the modified Aitken method(Aitken-Chen)and the Steffensen method.Numerical tests validate its efficiency with significantly less calls to the function not only for the average number but also for the maximum.The new reconstruction is based on the tangent of hyperbola for interface capturing(THINC)but applied only to the volume fraction,which is very simple to be implemented under the stratified frame-work and capable of resolving fluid interface in mixture.Furthermore,the directional splitting is adopted rather than the previous quasi-one-dimensional method.Typical numerical tests,including several watergas shock tube flows,and the shock-water cylinder interaction flow show that the improved gas-kinetic scheme can capture fluid interfaces much sharper,while preserving the advantages of the original one.
基金supported by the National Nature Science Foundation of China under Grant ( 61100165, 61100231, 61472307 )Natural Science Foundation of Shaanxi Province ( 2016JM6004)
文摘This article puts forward a novel smooth rotated hyperbola model for support vector machine( RHSSVM) for classification. As is well known,the support vector machine( SVM) is based on statistical learning theory( SLT)and performs its high precision on data classification. However,the objective function is non-differentiable at the zero point. Therefore the fast algorithms cannot be used to train and test the SVM. To deal with it,the proposed method is based on the approximation property of the hyperbola to its asymptotic lines. Firstly,we describe the development of RHSSVM from the basic linear SVM optimization programming. Then we extend the linear model to non-linear model. We prove the solution of RHSSVM is convergent,unique,and global optimal. We show how RHSSVM can be practically implemented. At last,the theoretical analysis illustrates that compared with other three typical models,the rotated hyperbola model has the least error on approximating the plus function. Meanwhile,computer simulations show that the RHSSVM can reduce the consuming time at most 54. 6% and can efficiently handle large scale and high dimensional programming.
