This paper covers the concept of a conservative vector field, and its application in vector physics and Newtonian mechanics. Conservative vector fields are defined as the gradient of a scalar-valued potential function...This paper covers the concept of a conservative vector field, and its application in vector physics and Newtonian mechanics. Conservative vector fields are defined as the gradient of a scalar-valued potential function. Gradient fields are irrotational, as in the curl in all conservative vector fields is zero, by Clairaut’s Theorem. Additionally, line integrals in conservative vector fields are path-independent, and line integrals over closed paths are always equal to zero, properties proved by the Gradient Theorem of multivariable calculus. Gradient fields represent conservative forces, and the associated potential function is analogous to potential energy associated with said conservative forces. The Intersect Rule provides a new, unique shortcut for determining if a vector field is conservative and deriving potential functions, by treating the indefinite integral as a set of infinitely many functions which satisfy the integral.展开更多
Guidance path-planning and following are two core technologies used for controlling un-manned aerial vehicles(UAVs)in both military and civilian applications.However,only a few approaches treat both the technologies s...Guidance path-planning and following are two core technologies used for controlling un-manned aerial vehicles(UAVs)in both military and civilian applications.However,only a few approaches treat both the technologies simultaneously.In this study,an innovative hybrid gradient vector fields for path-following guidance(HGVFs-PFG)algorithm is proposed to control fixed-wing UAVs to follow a generated guidance path and oriented target curves in three-dimensional space,which can be any combination of straight lines,arcs,and helixes as motion primitives.The algorithm aids the creation of vector fields(VFs)for these motion primitives as well as the design of an effective switching strategy to ensure that only one VF is activated at any time to ensure that the complex paths are followed completely.The strategies designed in earlier studies have flaws that prevent the UAV from following arcs that make its turning angle too large.The proposed switching strategy solves this problem by introducing the concept of the virtual way-points.Finally,the performance of the HGVFs-PFG algorithm is verified using a reducedorder autopilot and four representative simulation scenarios.The simulation considers the constraints of the aircraft,and its results indicate that the algorithm performs well in following both lateral and longitudinal control,particularly for curved paths.In general,the proposed technical method is practical and competitive.展开更多
This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from t...This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from the known Poincare estimates. The main ingredients of the proofs rely on the fractional maximal functions. These results evidently have applications to the regularity of subelliptic PDE.展开更多
1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). T...1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). The isotropy subgroup of a point展开更多
In this paper, the authors investigate the invariant cones of quadratic homoge- neous polynomial vector fields in three variables. Necessary and sufficient conditions for the existence of non-isolated invariant closed...In this paper, the authors investigate the invariant cones of quadratic homoge- neous polynomial vector fields in three variables. Necessary and sufficient conditions for the existence of non-isolated invariant closed cones are obtained by the algebraic expressions in terms of the coefficients of certain quadratic homogeneous polynomials.展开更多
For a n-dimensional vector fields preserving some n-form, the following conclusion is reached by the method of Lie group. That is, if it admits an one-parameter, n-form preserving symmetry group, a transformation inde...For a n-dimensional vector fields preserving some n-form, the following conclusion is reached by the method of Lie group. That is, if it admits an one-parameter, n-form preserving symmetry group, a transformation independent of the vector field is constructed explicitly, which can reduce not only dimesion of the vector field by one, but also make the reduced vector field preserve the corresponding ( n - 1)-form. In partic ular, while n = 3, an important result can be directly got which is given by Me,ie and Wiggins in 1994.展开更多
In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold wit...In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.展开更多
For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit...For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit sets for any m>1. Similar results arc extended to highel-dimensional polynomial homogeneous vector fields under certain conditions.展开更多
Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is s...Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable.In this article,the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn,and give the relationships between the minimal unit vector fields and the geodesic vector fields,the strongly normal unit vectors respectively.展开更多
In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for th...In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here.As a consequence,the Harnack inequality of weak solutions is also given.展开更多
In this paper,we study the conformal vector fields on Finsler warped product manifolds.We obtain a system of equivalent equations that the conformal vector fields on Finsler warped product manifolds satisfy and comple...In this paper,we study the conformal vector fields on Finsler warped product manifolds.We obtain a system of equivalent equations that the conformal vector fields on Finsler warped product manifolds satisfy and completely characterize conformal vector fields on such manifolds.Further,by solving the equation,we give the classification.And we also give some examples.展开更多
In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebe...In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander's condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander's condition, but they also hold for Grushin vector fields as well with obvious modifications.展开更多
We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into ...We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2+y2-1+y(ax+by+c),y=x(ax+by+c),and the ellipse becomes x2+y2=1.We prove that(i) this quadratic system is analytic integrable if and only if a=0;(ii) if x2+y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2+y2=1 is not a limit cycle;and(iii) if x2+y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.展开更多
In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of fin...In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.展开更多
In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials: uq/ t=▽α·(‖z‖^-pγ|▽αu|^p-2▽αu)+V(z, t)u^p-1, uq...In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials: uq/ t=▽α·(‖z‖^-pγ|▽αu|^p-2▽αu)+V(z, t)u^p-1, uq/ t=▽α·(‖z‖^-2γ▽αu^m)+V(z, t)u^m, uq/ t=u^μ▽α·(u^τ|▽αu|^p-2▽αu)+V(z, t)u^p-1+μ+τin a cylinder Ω×(0, T) with initial condition u(z, 0)=u0(z) ≥ 0 and vanishing on the boundary Ω×(0, T), where Ω is a Carnot-Carathéodory metric ball in Rd+k and the time-dependent singular potential function is V(z, t) ∈ L^1loc (Ω×(0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.展开更多
The purpose of this paper is to investigate the nonexistence of positive so lutions of the following doubly nonlinear degenerate parabolic equations:{∂u=▽k·(u^(m-1)|▽ku|^(p-2)▽ku)+(w)u^(m+p-2),u(w,0)=u0(w)≥0,...The purpose of this paper is to investigate the nonexistence of positive so lutions of the following doubly nonlinear degenerate parabolic equations:{∂u=▽k·(u^(m-1)|▽ku|^(p-2)▽ku)+(w)u^(m+p-2),u(w,0)=u0(w)≥0,u(w,t)=0,inΩ×(0,T),inΩ,on∂Ω×(0,T),where Q is a Carnot-Carathéodory metric bal in IR^(2n+1)generated by Greiner vector fields,V∈L_(loc)(Ω),k∈N,m∈R,1<p<2n+2k and m+p-2>0.The better lower bound p*for m+p is found and the nonexistence results are proved for p*≤m+p<3.展开更多
In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity...In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.展开更多
In this paper,we classify static spherically symmetric(SS)perfect fluid space-times via conformal vector fields(CVFs)in f(T)gravity.For this analysis,we first explore static SS solutions by solving the Einstein field ...In this paper,we classify static spherically symmetric(SS)perfect fluid space-times via conformal vector fields(CVFs)in f(T)gravity.For this analysis,we first explore static SS solutions by solving the Einstein field equations in f(T)gravity.Secondly,we implement a direct integration technique to classify the resulting solutions.During the classification,there arose 20 cases.Studying each case thoroughly,we came to know that in three cases the space-times under consideration admit proper CVFs in f(T)gravity.In one case,the space-time admits proper homothetic vector fields,whereas in the remaining 16 cases either the space-times become conformally flat or they admit Killing vector fields.展开更多
In this work we study Lie symmetries of planar quasihomogeneous polynomialvector fields from different points of view, showing its integrability. Additionally, we show thatcertain perturbations of such vector fields w...In this work we study Lie symmetries of planar quasihomogeneous polynomialvector fields from different points of view, showing its integrability. Additionally, we show thatcertain perturbations of such vector fields which generalize the so–called degenerate infinityvector fields are also integrable.展开更多
This paper studies the nonzero normal vector fields of immersions homotopic to a map g: Mn→N2n-1. In the case of the stable normal bundle of g being orientable, rather complete results are obtained.
文摘This paper covers the concept of a conservative vector field, and its application in vector physics and Newtonian mechanics. Conservative vector fields are defined as the gradient of a scalar-valued potential function. Gradient fields are irrotational, as in the curl in all conservative vector fields is zero, by Clairaut’s Theorem. Additionally, line integrals in conservative vector fields are path-independent, and line integrals over closed paths are always equal to zero, properties proved by the Gradient Theorem of multivariable calculus. Gradient fields represent conservative forces, and the associated potential function is analogous to potential energy associated with said conservative forces. The Intersect Rule provides a new, unique shortcut for determining if a vector field is conservative and deriving potential functions, by treating the indefinite integral as a set of infinitely many functions which satisfy the integral.
基金the support of the National Natural Science Foundation of China under Grant No.62076204 and Grant No.62006193in part by the Postdoctoral Science Foundation of China under Grants No.2021M700337in part by the Fundamental Research Funds for the Central Universities under Grant No.3102019ZX016。
文摘Guidance path-planning and following are two core technologies used for controlling un-manned aerial vehicles(UAVs)in both military and civilian applications.However,only a few approaches treat both the technologies simultaneously.In this study,an innovative hybrid gradient vector fields for path-following guidance(HGVFs-PFG)algorithm is proposed to control fixed-wing UAVs to follow a generated guidance path and oriented target curves in three-dimensional space,which can be any combination of straight lines,arcs,and helixes as motion primitives.The algorithm aids the creation of vector fields(VFs)for these motion primitives as well as the design of an effective switching strategy to ensure that only one VF is activated at any time to ensure that the complex paths are followed completely.The strategies designed in earlier studies have flaws that prevent the UAV from following arcs that make its turning angle too large.The proposed switching strategy solves this problem by introducing the concept of the virtual way-points.Finally,the performance of the HGVFs-PFG algorithm is verified using a reducedorder autopilot and four representative simulation scenarios.The simulation considers the constraints of the aircraft,and its results indicate that the algorithm performs well in following both lateral and longitudinal control,particularly for curved paths.In general,the proposed technical method is practical and competitive.
基金Research supported in part by he National Sience Foundation Grant # DMS93-15963
文摘This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from the known Poincare estimates. The main ingredients of the proofs rely on the fractional maximal functions. These results evidently have applications to the regularity of subelliptic PDE.
文摘1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). The isotropy subgroup of a point
文摘In this paper, the authors investigate the invariant cones of quadratic homoge- neous polynomial vector fields in three variables. Necessary and sufficient conditions for the existence of non-isolated invariant closed cones are obtained by the algebraic expressions in terms of the coefficients of certain quadratic homogeneous polynomials.
文摘For a n-dimensional vector fields preserving some n-form, the following conclusion is reached by the method of Lie group. That is, if it admits an one-parameter, n-form preserving symmetry group, a transformation independent of the vector field is constructed explicitly, which can reduce not only dimesion of the vector field by one, but also make the reduced vector field preserve the corresponding ( n - 1)-form. In partic ular, while n = 3, an important result can be directly got which is given by Me,ie and Wiggins in 1994.
文摘In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.
文摘For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit sets for any m>1. Similar results arc extended to highel-dimensional polynomial homogeneous vector fields under certain conditions.
基金supported by the National Natural Science Foundation of China (Nos. 12001007,12201358)the Natural Science Foundation of Shandong Province (No. ZR2021QA051)+1 种基金the Natural Science Foundation of Anhui Province (No. 1908085QA03)Starting Research Funds of Shandong University of Science and Technology (Nos. 0104060511817, 0104060540626)
文摘Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable.In this article,the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn,and give the relationships between the minimal unit vector fields and the geodesic vector fields,the strongly normal unit vectors respectively.
基金This work is sponsored by the China Scholarship Council with Grant Number 20200636-0116.
文摘In this paper,we consider the divergence degenerate elliptic equation with bounded coefficients constructed by Hörmander's vector fields.We prove a De Giorgi type result,i.e,the local Holder continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here.As a consequence,the Harnack inequality of weak solutions is also given.
基金Supported by National Natural Science Foundation of China(Grant Nos.11961061,11461064,11761069)Natural Science Foundation of Xinjiang Uygur Autonomous Region,China(Grant No.2015211C277)。
文摘In this paper,we study the conformal vector fields on Finsler warped product manifolds.We obtain a system of equivalent equations that the conformal vector fields on Finsler warped product manifolds satisfy and completely characterize conformal vector fields on such manifolds.Further,by solving the equation,we give the classification.And we also give some examples.
基金supported by NSFC(Grant No.11371056)supported by a US NSF grant
文摘In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander's condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander's condition, but they also hold for Grushin vector fields as well with obvious modifications.
基金partially supported by the MINECO/FEDER(Grant No.MTM2008–03437)AGAUR(Grant No.2009SGR-410)+1 种基金ICREA Academia and FP7-PEOPLE-2012-IRSES 316338 and 318999supported by Portuguese National Funds through FCT-Fundao para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD
文摘We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R2 having an invariant ellipse.More precisely,a quadratic system having an invariant ellipse can be written into the form x=x2+y2-1+y(ax+by+c),y=x(ax+by+c),and the ellipse becomes x2+y2=1.We prove that(i) this quadratic system is analytic integrable if and only if a=0;(ii) if x2+y2=1 is a periodic orbit,then this quadratic system is Liouvillian integrable if and only if x2+y2=1 is not a limit cycle;and(iii) if x2+y2=1 is not a periodic orbit,then this quadratic system is Liouvilian integrable if and only if a=0.
文摘In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.
基金Supported by Nature Science Fund of Shaanxi Province(Grant No.2012JM1014)
文摘In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials: uq/ t=▽α·(‖z‖^-pγ|▽αu|^p-2▽αu)+V(z, t)u^p-1, uq/ t=▽α·(‖z‖^-2γ▽αu^m)+V(z, t)u^m, uq/ t=u^μ▽α·(u^τ|▽αu|^p-2▽αu)+V(z, t)u^p-1+μ+τin a cylinder Ω×(0, T) with initial condition u(z, 0)=u0(z) ≥ 0 and vanishing on the boundary Ω×(0, T), where Ω is a Carnot-Carathéodory metric ball in Rd+k and the time-dependent singular potential function is V(z, t) ∈ L^1loc (Ω×(0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.
基金support from Nature Science Fund of China(No.11771354).
文摘The purpose of this paper is to investigate the nonexistence of positive so lutions of the following doubly nonlinear degenerate parabolic equations:{∂u=▽k·(u^(m-1)|▽ku|^(p-2)▽ku)+(w)u^(m+p-2),u(w,0)=u0(w)≥0,u(w,t)=0,inΩ×(0,T),inΩ,on∂Ω×(0,T),where Q is a Carnot-Carathéodory metric bal in IR^(2n+1)generated by Greiner vector fields,V∈L_(loc)(Ω),k∈N,m∈R,1<p<2n+2k and m+p-2>0.The better lower bound p*for m+p is found and the nonexistence results are proved for p*≤m+p<3.
基金This work is supported by the National Natural Science Foundation of China(No.1177-1354).
文摘In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result,i.e.,the local Hölder regularity for weak solutions.After deriving the parabolic Sobolev inequality,(1,1)type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma,the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Hölder space.As a consequence,we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.
文摘In this paper,we classify static spherically symmetric(SS)perfect fluid space-times via conformal vector fields(CVFs)in f(T)gravity.For this analysis,we first explore static SS solutions by solving the Einstein field equations in f(T)gravity.Secondly,we implement a direct integration technique to classify the resulting solutions.During the classification,there arose 20 cases.Studying each case thoroughly,we came to know that in three cases the space-times under consideration admit proper CVFs in f(T)gravity.In one case,the space-time admits proper homothetic vector fields,whereas in the remaining 16 cases either the space-times become conformally flat or they admit Killing vector fields.
基金supported by the Spanish MCYT,Grant Number BFM 2002-04236-C01-01
文摘In this work we study Lie symmetries of planar quasihomogeneous polynomialvector fields from different points of view, showing its integrability. Additionally, we show thatcertain perturbations of such vector fields which generalize the so–called degenerate infinityvector fields are also integrable.
文摘This paper studies the nonzero normal vector fields of immersions homotopic to a map g: Mn→N2n-1. In the case of the stable normal bundle of g being orientable, rather complete results are obtained.