In this paper, we investigate the representation of curves on the lightlike cone ℚ³₂ in Minkowski... more In this paper, we investigate the representation of curves on the lightlike cone ℚ³₂ in Minkowski space ℝ⁴₂ by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone ℚ³₂ in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function κ₂ = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions
In this paper we use MATLAB platform [7, 8] to calculate some quantities of 3-manifold sol3 like ... more In this paper we use MATLAB platform [7, 8] to calculate some quantities of 3-manifold sol3 like Christoffel symbols, curvatures, Einstein tensor and plot the geodesics of this space.
Abstract—In this work, we successfully extended one-dimensional differential transform method (DT... more Abstract—In this work, we successfully extended one-dimensional differential transform method (DTM), by presenting and proving some theorems, to solving nonlinear high-order multi-pantograph equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results. Keywords—Nonlinear multi-pantograph equation; Delay differential equation; ...
In this paper, we have introduced dual Lorentzian connection, bracket and curvature tensor on dua... more In this paper, we have introduced dual Lorentzian connection, bracket and curvature tensor on dual Lorentzian space D 3 1. We have studied a dual curve in different situations in dual Lorentzian space D 3 1 and have found Bishop Darboux vector and some relations according to this vector field, Bishop frame and focal curve of the present dual curve. It has been shown that Bishop Darboux vector has a similar amount in three different cases of a dual curve and the first dual focal curvature of the aforementioned curve is constant function.
Let ) , ( m R be a non-zero commutative Noetherian local ring , M be a non-zero finitely generate... more Let ) , ( m R be a non-zero commutative Noetherian local ring , M be a non-zero finitely generated R -module and 0 n be an integer. In this paper for any ) ( S R pec p we show that )) ( , ( 0 M H p n p is bounded from above by )). ( , ( ) / ( dim 0 M H m p R n m
One of the most basis important and basic problems in commutative algebra is to determine the set... more One of the most basis important and basic problems in commutative algebra is to determine the set of associated primes of a given module over a commutative non-Noetherian ring. Unfortunately, the study of usual set of associated primes of a module is not so helpful over the non-Noetherian rings. Cansidering this fact, there are several generalizations of this concept for these type of rings. For example, the set of Bourbaki ideals, the set of weak Bourbaki ideals, the set of Krull ideals, the set of Zariski-Samuel ideals are some family of generalizatios of the set of associated prime ideals for modules over non-Noetherian commutative rings. The main porpose of this thesis is to study these families of ideals.
In this paper we use MATLAB platform to calculate some quantities of 3-manifold sol3 like Christo... more In this paper we use MATLAB platform to calculate some quantities of 3-manifold sol3 like Christoffel symbols, curvatures, Einstein tensor and plot the geodesics of this space.
In this paper, we investigate the representation of curves on the lightlike cone$\mathbb {Q}^{3}_... more In this paper, we investigate the representation of curves on the lightlike cone$\mathbb {Q}^{3}_{2}$Q23in Minkowski space$\mathbb {R}^{4}_{2}$R24by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone$\mathbb {Q}^{3}_{2}$Q23in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function$\kappa _{2}=0$κ2=0, and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.
In this paper, we study curves in the lightlike cone. First, we show that any curves in the light... more In this paper, we study curves in the lightlike cone. First, we show that any curves in the lightlike cone are spacelike or lightlike, and then we characterize some curves with special cone curvature function in the 4, 5, and 6-dimensional lightlike cone. Finally, we consider the relationship between Frenet curvature functions and cone curvature functions for a spacelike curve on the lightlike cone.
In this paper, the stationary acceleration of the spherical general helix in a 3-dimensional Lie ... more In this paper, the stationary acceleration of the spherical general helix in a 3-dimensional Lie group is studied by using a bi-invariant metric. The relationship between the Frenet elements of the stationary acceleration curve in 4-dimensional Euclidean space and the intrinsic Frenet elements of the Lie group is outlined. As a consequence, the corresponding curvature and torsion of these curves are computed. In Minkowski space, for the curves on a timelike surface to have a stationary acceleration, a necessary and sufficient condition is refined.
Bu calismada, kati hareketlerinin Lie grubu olan uzerindeki egrilerin sabit ivmeli olma sartlari... more Bu calismada, kati hareketlerinin Lie grubu olan uzerindeki egrilerin sabit ivmeli olma sartlari arastirilmistir. Bunun icin cesitli hipotezler verilmistir. Bu tez 7 bolum halinde duzenlenmistir. 1. Bolum girise ayrilmistir. 2. bolum temel kavramlara ayrilmistir. 3. bolumde, uzerindeki egrilerin sabit ivmeli olma sartlari arastirilmistir. 4. bolumde, yuzey uzerinde bir egri ve bu egrinin jeodezik catisi yardimiyla uzerinde bir egri tanimlanmis bu egrinin sabit ivmeli olma sartlari incelenmistir. 5. bolumde, egri bi-invaryant metrige sahip Lie gruplari uzerinde alinmis ve egrinin sabit ivmeli olma hipotezleri verilmistir. 6. bolumde, kompakt Riemann manifoldlarinda bir egrinin sabit ivmeli olmasi icin sartlar incelenmistir. Son bolumde, Lorentz uzayinda bir egri ele alinmistir. Bu egrinin elastik egri olmasi icin hipotezler verilmistir. Abstract In this thesis, conditions of the curves on the Lie group SE(3) of rigid body motions to be in the stationary acceleration state have been studied. For this reason, different kinds of hypotheses have been given. This thesis contains seven sections. The first section has been introduction. The second section has been composed of basic concepts. In the third section, conditions of curves on to be with stationary acceleration have been studied. In the forth section, a curve on the surface and an other curve on with the help of geodesic frame of the former curve have been described and stationary acceleration situations of this curve have been discussed. This curve has been considered in a Lie group with a bi-invariant metric in the fifth section and stationary acceleration hypotheses have been given for it. In the sixth section, stationary acceleration conditions of a curve in compact Riemannian manifolds have been investigated. In the last section, a curve in the Lorentz space have been considered. Hypotheses for this to be an elastic curve have been given.
In this paper, the tangent, binormal, normal and unit Darboux vectors of the dual curve on the on... more In this paper, the tangent, binormal, normal and unit Darboux vectors of the dual curve on the one-parameter dual spherical motion are obtained with respect to a unit dual orthogonal frame for dual 3-space D 3 .
ABSTRACT In this paper we shall prove the following result, which is a generalization of the Melk... more ABSTRACT In this paper we shall prove the following result, which is a generalization of the Melkersson's main result proved in [1616. Melkersson , L. ( 1999 ). Properties of cofinite modules and application to local cohomology . Math. Proc. Cambridge Philos. Soc. 125 : 417 – 423 . [CrossRef], [Web of Science ®]View all references]. Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I be a proper ideal of R and A be an Artinian R-module. Then A is I-cofinite if and only if Rad(I + AnnR (A)) = 𝔪. Also, we present an example to show that this result does not hold for an arbitrary local Noetherian ring in general. As an application of this result we prove the following generalization of the Lichtenbaum-Hartshorne Vanishing Theorem (see [55. Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press . [CrossRef]View all references, Theorem 8.2.1]). Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I an ideal of R and M be a nonzero finitely generated R-module of dimension n. Then the following conditions are equivalent: (i) . (ii) There exists a prime ideal 𝔭 in AsshR(M) such that Rad(𝔭 +I) = 𝔪.
The article illustrates the graphical study of geodesic motion on H3 , H2*R , Nil3 and sol3 using... more The article illustrates the graphical study of geodesic motion on H3 , H2*R , Nil3 and sol3 using the symbolic and graphical computation of MATLAB platform.
In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning pro... more In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.
In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning pro... more In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.
... Author(s): NEMAT ABAZARI Department of Mathematics, Ardabil Branch, Islamic Azad University, ... more ... Author(s): NEMAT ABAZARI Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran KAMAL BAHMANPOUR Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran. ... World Scientific is a Member of CrossRef. ...
In this paper, we investigate the representation of curves on the lightlike cone ℚ³₂ in Minkowski... more In this paper, we investigate the representation of curves on the lightlike cone ℚ³₂ in Minkowski space ℝ⁴₂ by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone ℚ³₂ in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function κ₂ = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions
In this paper we use MATLAB platform [7, 8] to calculate some quantities of 3-manifold sol3 like ... more In this paper we use MATLAB platform [7, 8] to calculate some quantities of 3-manifold sol3 like Christoffel symbols, curvatures, Einstein tensor and plot the geodesics of this space.
Abstract—In this work, we successfully extended one-dimensional differential transform method (DT... more Abstract—In this work, we successfully extended one-dimensional differential transform method (DTM), by presenting and proving some theorems, to solving nonlinear high-order multi-pantograph equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results. Keywords—Nonlinear multi-pantograph equation; Delay differential equation; ...
In this paper, we have introduced dual Lorentzian connection, bracket and curvature tensor on dua... more In this paper, we have introduced dual Lorentzian connection, bracket and curvature tensor on dual Lorentzian space D 3 1. We have studied a dual curve in different situations in dual Lorentzian space D 3 1 and have found Bishop Darboux vector and some relations according to this vector field, Bishop frame and focal curve of the present dual curve. It has been shown that Bishop Darboux vector has a similar amount in three different cases of a dual curve and the first dual focal curvature of the aforementioned curve is constant function.
Let ) , ( m R be a non-zero commutative Noetherian local ring , M be a non-zero finitely generate... more Let ) , ( m R be a non-zero commutative Noetherian local ring , M be a non-zero finitely generated R -module and 0 n be an integer. In this paper for any ) ( S R pec p we show that )) ( , ( 0 M H p n p is bounded from above by )). ( , ( ) / ( dim 0 M H m p R n m
One of the most basis important and basic problems in commutative algebra is to determine the set... more One of the most basis important and basic problems in commutative algebra is to determine the set of associated primes of a given module over a commutative non-Noetherian ring. Unfortunately, the study of usual set of associated primes of a module is not so helpful over the non-Noetherian rings. Cansidering this fact, there are several generalizations of this concept for these type of rings. For example, the set of Bourbaki ideals, the set of weak Bourbaki ideals, the set of Krull ideals, the set of Zariski-Samuel ideals are some family of generalizatios of the set of associated prime ideals for modules over non-Noetherian commutative rings. The main porpose of this thesis is to study these families of ideals.
In this paper we use MATLAB platform to calculate some quantities of 3-manifold sol3 like Christo... more In this paper we use MATLAB platform to calculate some quantities of 3-manifold sol3 like Christoffel symbols, curvatures, Einstein tensor and plot the geodesics of this space.
In this paper, we investigate the representation of curves on the lightlike cone$\mathbb {Q}^{3}_... more In this paper, we investigate the representation of curves on the lightlike cone$\mathbb {Q}^{3}_{2}$Q23in Minkowski space$\mathbb {R}^{4}_{2}$R24by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone$\mathbb {Q}^{3}_{2}$Q23in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function$\kappa _{2}=0$κ2=0, and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.
In this paper, we study curves in the lightlike cone. First, we show that any curves in the light... more In this paper, we study curves in the lightlike cone. First, we show that any curves in the lightlike cone are spacelike or lightlike, and then we characterize some curves with special cone curvature function in the 4, 5, and 6-dimensional lightlike cone. Finally, we consider the relationship between Frenet curvature functions and cone curvature functions for a spacelike curve on the lightlike cone.
In this paper, the stationary acceleration of the spherical general helix in a 3-dimensional Lie ... more In this paper, the stationary acceleration of the spherical general helix in a 3-dimensional Lie group is studied by using a bi-invariant metric. The relationship between the Frenet elements of the stationary acceleration curve in 4-dimensional Euclidean space and the intrinsic Frenet elements of the Lie group is outlined. As a consequence, the corresponding curvature and torsion of these curves are computed. In Minkowski space, for the curves on a timelike surface to have a stationary acceleration, a necessary and sufficient condition is refined.
Bu calismada, kati hareketlerinin Lie grubu olan uzerindeki egrilerin sabit ivmeli olma sartlari... more Bu calismada, kati hareketlerinin Lie grubu olan uzerindeki egrilerin sabit ivmeli olma sartlari arastirilmistir. Bunun icin cesitli hipotezler verilmistir. Bu tez 7 bolum halinde duzenlenmistir. 1. Bolum girise ayrilmistir. 2. bolum temel kavramlara ayrilmistir. 3. bolumde, uzerindeki egrilerin sabit ivmeli olma sartlari arastirilmistir. 4. bolumde, yuzey uzerinde bir egri ve bu egrinin jeodezik catisi yardimiyla uzerinde bir egri tanimlanmis bu egrinin sabit ivmeli olma sartlari incelenmistir. 5. bolumde, egri bi-invaryant metrige sahip Lie gruplari uzerinde alinmis ve egrinin sabit ivmeli olma hipotezleri verilmistir. 6. bolumde, kompakt Riemann manifoldlarinda bir egrinin sabit ivmeli olmasi icin sartlar incelenmistir. Son bolumde, Lorentz uzayinda bir egri ele alinmistir. Bu egrinin elastik egri olmasi icin hipotezler verilmistir. Abstract In this thesis, conditions of the curves on the Lie group SE(3) of rigid body motions to be in the stationary acceleration state have been studied. For this reason, different kinds of hypotheses have been given. This thesis contains seven sections. The first section has been introduction. The second section has been composed of basic concepts. In the third section, conditions of curves on to be with stationary acceleration have been studied. In the forth section, a curve on the surface and an other curve on with the help of geodesic frame of the former curve have been described and stationary acceleration situations of this curve have been discussed. This curve has been considered in a Lie group with a bi-invariant metric in the fifth section and stationary acceleration hypotheses have been given for it. In the sixth section, stationary acceleration conditions of a curve in compact Riemannian manifolds have been investigated. In the last section, a curve in the Lorentz space have been considered. Hypotheses for this to be an elastic curve have been given.
In this paper, the tangent, binormal, normal and unit Darboux vectors of the dual curve on the on... more In this paper, the tangent, binormal, normal and unit Darboux vectors of the dual curve on the one-parameter dual spherical motion are obtained with respect to a unit dual orthogonal frame for dual 3-space D 3 .
ABSTRACT In this paper we shall prove the following result, which is a generalization of the Melk... more ABSTRACT In this paper we shall prove the following result, which is a generalization of the Melkersson's main result proved in [1616. Melkersson , L. ( 1999 ). Properties of cofinite modules and application to local cohomology . Math. Proc. Cambridge Philos. Soc. 125 : 417 – 423 . [CrossRef], [Web of Science ®]View all references]. Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I be a proper ideal of R and A be an Artinian R-module. Then A is I-cofinite if and only if Rad(I + AnnR (A)) = 𝔪. Also, we present an example to show that this result does not hold for an arbitrary local Noetherian ring in general. As an application of this result we prove the following generalization of the Lichtenbaum-Hartshorne Vanishing Theorem (see [55. Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press . [CrossRef]View all references, Theorem 8.2.1]). Let (R, 𝔪) be a Noetherian local ring such that is integral over R. Let I an ideal of R and M be a nonzero finitely generated R-module of dimension n. Then the following conditions are equivalent: (i) . (ii) There exists a prime ideal 𝔭 in AsshR(M) such that Rad(𝔭 +I) = 𝔪.
The article illustrates the graphical study of geodesic motion on H3 , H2*R , Nil3 and sol3 using... more The article illustrates the graphical study of geodesic motion on H3 , H2*R , Nil3 and sol3 using the symbolic and graphical computation of MATLAB platform.
In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning pro... more In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.
In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning pro... more In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.
... Author(s): NEMAT ABAZARI Department of Mathematics, Ardabil Branch, Islamic Azad University, ... more ... Author(s): NEMAT ABAZARI Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran KAMAL BAHMANPOUR Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran. ... World Scientific is a Member of CrossRef. ...
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