It's shown that on an n-dimensional LCS Lie group,there exist harmonic sections and left invariant vector fields defining harmonic maps only when it is endowed with a flat left invariant pseudo-Riemannian metric.M...It's shown that on an n-dimensional LCS Lie group,there exist harmonic sections and left invariant vector fields defining harmonic maps only when it is endowed with a flat left invariant pseudo-Riemannian metric.Moreover,we determine all the vector fields which are critical points of the energy functional restricted to vector fields of the same length on the n-dimensional pseudo-Riemannian LCS Lie group.展开更多
Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the...Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the existing systems.This derivation process consists of three steps:step 1,decomposing the vector field;step 2,solving the Hamilton energy function;and step 3,verifying uniqueness.In order to easily choose an appropriate decomposition method,we propose a classification criterion based on the form of system state variables,i.e.,type-I vector fields that can be directly decomposed and type-II vector fields decomposed via exterior differentiation.Moreover,exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition.Finally,we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior.This solution provides a new approach for deducing the Hamilton energy function,especially in high-dimensional systems.展开更多
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
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
In this paper we classify cylindrically symmetric static space-times according to their teleparallel homothetic vector fields using direct integration technique. It turns out that the dimensions of the teleparallel ho...In this paper we classify cylindrically symmetric static space-times according to their teleparallel homothetic vector fields using direct integration technique. It turns out that the dimensions of the teleparallel homothetic vector fields are 4, 5, 7 or 11, which are the same in numbers as in general relativity. In case of 4, 5 or 7 proper teleparallel homothetic vector fields exist for the special choice to the space-times. In the case of 11 teleparallel homothetic vector fields the space-time becomes Minkowski with all the zero torsion components. Teleparallel homothetic vector fields in this case are exactly the same as in general relativity. It is important to note that this classification also covers the plane symmetric static space-times.展开更多
In this paper we classify spatially homogeneous rotating space-times according to their teleparallel Killing vector fields using direct integration technique.It turns out that the dimension of the teleparallel Killing...In this paper we classify spatially homogeneous rotating space-times according to their teleparallel Killing vector fields using direct integration technique.It turns out that the dimension of the teleparallel Killing vector fields is 5 or 10.In the case of 10 teleparallel Killing vector fields the space-time becomes Minkowski and all the torsion components are zero.Teleparallel Killing vector fields in this case are exactly the same as in general relativity.In the cases of 5 teleparallel Killing vector fields we get two more conservation laws in the teleparallel theory of gravitation.Here we also discuss some well-known examples of spatially homogeneous rotating space-times according to their teleparallel Killing vector fields.展开更多
In this paper we classify Kantowski-Sachs and Bianchi type Ⅲ space-times according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the telepaxallel K...In this paper we classify Kantowski-Sachs and Bianchi type Ⅲ space-times according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the telepaxallel Killing vector fields are 4 or 6, which are the same in numbers as in general relativity. In case of 4 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 6 Killing vector fields the metric functions become constants and the Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case.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
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
In this paper we consider the most general form of non-static cylindrically symmetric space-times in order to study proper teleparallel homothetic vector fields using the direct integration technique and diagonal tetr...In this paper we consider the most general form of non-static cylindrically symmetric space-times in order to study proper teleparallel homothetic vector fields using the direct integration technique and diagonal tetrads. This study also covers static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times as well. Here, we will only discuss the cases which do not fall in the category of static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times. From the above study we show that very special classes of the above space-times yield 6, 7 and 8 teleparallel homothetic vector fields with non-zero torsion.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
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
We study the finite time domain dynamics of massive vector fields.We con-sider their standard Lagrangian and Hamiltonian densities and expand them into creation and annihilation operators.We integrate them via coheren...We study the finite time domain dynamics of massive vector fields.We con-sider their standard Lagrangian and Hamiltonian densities and expand them into creation and annihilation operators.We integrate them via coherent states path integral methods and extract the corresponding massive vector fields’Green functions in various dimensions.Then we consider the possible generation of massive vector fields from currents.展开更多
In this paper we present certain bilinear estimates for commutators on Besov spaces with variable smoothness and integrability,and under no vanishing assumptions on the divergence of vector fields.Such commutator esti...In this paper we present certain bilinear estimates for commutators on Besov spaces with variable smoothness and integrability,and under no vanishing assumptions on the divergence of vector fields.Such commutator estimates are motivated by the study of well-posedness results for some models in incompressible fuid mechanics.展开更多
Seed clearing is a critical stage during precision seed metering process to ensure high seed singulation.However,there is a lack of understanding of the dynamics in the seed clearing process.In this study,a model was ...Seed clearing is a critical stage during precision seed metering process to ensure high seed singulation.However,there is a lack of understanding of the dynamics in the seed clearing process.In this study,a model was developed to predict initial seed clearing angle,in the seed clearing process using vector fields.The model was applied to an existing high-speed metering device and soybean seeds,and the model was evaluated with bench testing results.Results showed that dynamic changes in forces and constraints of seeds during the seed clearing process could be abstracted as vectors,and the changes of vector directions could be described by their phase angles.The phase angles were functions of the rotational angle of the seed meter.The phase angle of the constraint boundary linearly increases with the increase of the rotational angle.The phase angle of the force fluctuates,as the rotational angle changes.Initial seed clearing angle obtained from the phase angles varies from 8°to 59°,depending on the seeder travel speed.When comparing the values of the initial seed clearing angles predicted by the model with those from the bench tests,the root mean square error(RMSE)were from 2.73 to 3.14,and the correlation(r)between predict and observer were all higher than 0.98,indicating that the model had reasonably good accuracy.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
We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution ...We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).展开更多
基金Supported by General Project for the Cultivation of Excellent Young Teachers of Anhui Province(YQYB2024018)。
文摘It's shown that on an n-dimensional LCS Lie group,there exist harmonic sections and left invariant vector fields defining harmonic maps only when it is endowed with a flat left invariant pseudo-Riemannian metric.Moreover,we determine all the vector fields which are critical points of the energy functional restricted to vector fields of the same length on the n-dimensional pseudo-Riemannian LCS Lie group.
基金the National Natural Science Foundation of China(Grant Nos.12305054,12172340,and 12371506)。
文摘Hamilton energy,which reflects the energy variation of systems,is one of the crucial instruments used to analyze the characteristics of dynamical systems.Here we propose a method to deduce Hamilton energy based on the existing systems.This derivation process consists of three steps:step 1,decomposing the vector field;step 2,solving the Hamilton energy function;and step 3,verifying uniqueness.In order to easily choose an appropriate decomposition method,we propose a classification criterion based on the form of system state variables,i.e.,type-I vector fields that can be directly decomposed and type-II vector fields decomposed via exterior differentiation.Moreover,exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition.Finally,we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior.This solution provides a new approach for deducing the Hamilton energy function,especially in high-dimensional systems.
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In this paper we classify cylindrically symmetric static space-times according to their teleparallel homothetic vector fields using direct integration technique. It turns out that the dimensions of the teleparallel homothetic vector fields are 4, 5, 7 or 11, which are the same in numbers as in general relativity. In case of 4, 5 or 7 proper teleparallel homothetic vector fields exist for the special choice to the space-times. In the case of 11 teleparallel homothetic vector fields the space-time becomes Minkowski with all the zero torsion components. Teleparallel homothetic vector fields in this case are exactly the same as in general relativity. It is important to note that this classification also covers the plane symmetric static space-times.
文摘In this paper we classify spatially homogeneous rotating space-times according to their teleparallel Killing vector fields using direct integration technique.It turns out that the dimension of the teleparallel Killing vector fields is 5 or 10.In the case of 10 teleparallel Killing vector fields the space-time becomes Minkowski and all the torsion components are zero.Teleparallel Killing vector fields in this case are exactly the same as in general relativity.In the cases of 5 teleparallel Killing vector fields we get two more conservation laws in the teleparallel theory of gravitation.Here we also discuss some well-known examples of spatially homogeneous rotating space-times according to their teleparallel Killing vector fields.
文摘In this paper we classify Kantowski-Sachs and Bianchi type Ⅲ space-times according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the telepaxallel Killing vector fields are 4 or 6, which are the same in numbers as in general relativity. In case of 4 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 6 Killing vector fields the metric functions become constants and the Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case.
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.
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.
基金the National Research FoundationNRF,of South Africa for research funding through two grants
文摘In this paper we consider the most general form of non-static cylindrically symmetric space-times in order to study proper teleparallel homothetic vector fields using the direct integration technique and diagonal tetrads. This study also covers static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times as well. Here, we will only discuss the cases which do not fall in the category of static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times. From the above study we show that very special classes of the above space-times yield 6, 7 and 8 teleparallel homothetic vector fields with non-zero torsion.
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.
文摘We study the finite time domain dynamics of massive vector fields.We con-sider their standard Lagrangian and Hamiltonian densities and expand them into creation and annihilation operators.We integrate them via coherent states path integral methods and extract the corresponding massive vector fields’Green functions in various dimensions.Then we consider the possible generation of massive vector fields from currents.
文摘In this paper we present certain bilinear estimates for commutators on Besov spaces with variable smoothness and integrability,and under no vanishing assumptions on the divergence of vector fields.Such commutator estimates are motivated by the study of well-posedness results for some models in incompressible fuid mechanics.
基金supported by the National Natural Science Foundation of China(Grant No.32372009)the National Natural Science Foundation of China(Grant No.31971802).
文摘Seed clearing is a critical stage during precision seed metering process to ensure high seed singulation.However,there is a lack of understanding of the dynamics in the seed clearing process.In this study,a model was developed to predict initial seed clearing angle,in the seed clearing process using vector fields.The model was applied to an existing high-speed metering device and soybean seeds,and the model was evaluated with bench testing results.Results showed that dynamic changes in forces and constraints of seeds during the seed clearing process could be abstracted as vectors,and the changes of vector directions could be described by their phase angles.The phase angles were functions of the rotational angle of the seed meter.The phase angle of the constraint boundary linearly increases with the increase of the rotational angle.The phase angle of the force fluctuates,as the rotational angle changes.Initial seed clearing angle obtained from the phase angles varies from 8°to 59°,depending on the seeder travel speed.When comparing the values of the initial seed clearing angles predicted by the model with those from the bench tests,the root mean square error(RMSE)were from 2.73 to 3.14,and the correlation(r)between predict and observer were all higher than 0.98,indicating that the model had reasonably good accuracy.
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supported by National Natural Science Foundation of China(Grant No.11401374)Shanghai YangFan Project(Grant No.14YF1401400)
文摘We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).
