A solution method for decomposing vector fields in Hamilton energy

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.

LAGRANGIAN VECTOR FIELD ON KOHLER MANIFOLD

In this paper.we discuss Lagrangian vector field on Kahler manifold and use it to describe and solve some problem in Newtonican and Lagrangian Mechanics on Kahler Manifold.

Projective vector fields on Finsler manifolds

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.

Conservative Vector Fields and the Intersect Rule

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.

Average vector field methods for the coupled Schrdinger KdV equations

The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.

Pressure and velocity cross-spectrum of normal modes in low-frequency acoustic vector field of shallow water and its application

The pressure and horizontal particle velocity combined descriptions in the very low frequency acoustic field of shallow wa- ter integrated with the concept of effective depth of Pekeris wave- guide is proposed, especially the active component of the pressure and horizontal particle velocity cross-spectrum, also called ho- rizontal complex cross acoustic intensity, when only two normal modes are trapped in the waveguide. Both the approximate theo- retic analysis and the numerical results show that the sign of the horizontal complex cross acoustic intensity active component is independent of the range when vertically deployed receiving dual sensors are placed in appropriate depths, the sum of which is equal to the waveguide effective depth, so it can be used to tell whether the sound source is near the surface or underwater; while the range rate is expected to be measured by utilizing the sign distribution characteristic of the reactive component. The further robustness analysis of the depth classification algorithm shows that the existence of shear waves in semi infinite basement and the change of acoustic velocity profiles have few effects on the application of this method, and the seabed attenuation will limit the detection range, but the algorithm still has a good robustness in the valid detection range.

Characteristics of electromagnetic vector field generated from rock fracturing

Rock fracturing is often accompanied by electromagnetic phenomenon.As a vector field,in addition to the intensity that is widely concerned,the generated electromagnetic field also has obvious direction-ality.To this end,a set of electromagnetic antennas capable of simultaneous three-axis measurement is used to monitor the electromagnetic vector field generated from rock fracturing based on Brazilian tests.The signal amplitude on each axis can represent the magnitude of actual magnetic flux density component on the three axes.The intensity and directional characteristics of electromagnetic signals received at different positions are studied using vector synthesis.The directionality of electromagnetic radiation measured using a three-axis electromagnetic antenna shows that the direction of the magnetic flux intensity generated by rock fracturing tends to be parallel to the crack surface,and the measured signal intensity is greater in a direction closer to the crack surface.

LIC color texture enhancement algorithm for ocean vector field data based on HSV color mapping and cumulative distribution function

Texture-based visualization method is a common method in the visualization of vector field data.Aiming at adding color mapping to the texture of ocean vector field and solving the ambiguity of vector direction in texture image,a new color texture enhancement algorithm based on the Line Integral Convolution(LIC)for the vector field data is proposed,which combines the HSV color mapping and cumulative distribution function calculation of vector field data.This algorithm can be summarized as follows:firstly,the vector field data is convoluted twice by line integration to get the gray texture image.Secondly,the method of mapping vector data to each component of the HSV color space is established.And then,the vector field data is mapped into HSV color space and converted from HSV to RGB values to get the color image.Thirdly,the cumulative distribution function of the RGB color components of the gray texture image and the color image is constructed to enhance the gray texture and RGB color values.Finally,both the gray texture image and the color image are fused to get the color texture.The experimental results show that the proposed LIC color texture enhancement algorithm is capable of generating a better display of vector field data.Furthermore,the ambiguity of vector direction in the texture images is solved and the direction information of the vector field is expressed more accurately.

THE INVARIANT CLOSED CONES OF HOMOGENEOUS VECTOR FIELDS OF DEGREE TWO IN R^(3)

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.

EMBEDDING THEOREMS ON CAMPANATO-MORREY SPACES FOR VECTOR FIELDS OF HoRMANDER TYPE

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.

THE VECTOR FIELDS ADMITTING ONE-PARAMETER SPATIAL SYMMETRY GROUP AND THEIR REDUCTION

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.

DIRECTIONAL DERIVATIVE OF VECTOR FIELD AND REGULAR CURVES ON TIME SCALES

The general idea in this paper is to study curves of the parametric equations where the parameter varies in a so-called time scale, which may be an arbitrary closed subset of the set of all real numbers. We introduce the directional derivative according to the vector fields.

AN ANALOGUE ROTATED VECTOR FIELD OF POLYNOMIAL SYSTEM

A class of polynomial system was structured, which depends on a parameter delta. When delta monotonous changes, more than one neighbouring limit cycles located in the vector field of this polynomial system can expand (or reduce) together with thee. But the expansion (or reduction) of these limit cycles is not surely monotonous. This vector field is like the rotated vector field. So these limit cycles of the polynomial system are called to constitute an 'analogue rotated vector field' with delta. They may become an effective tool to study the bifurcation of multiple limit cycle or fine separatrix cycle.

Some Properties of Homogeneous Vector Fields

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.

The Representation of a Broadband Vector Field

Compared to a scalar pressure sensor, a vector sensor can provide a higher signal-to-noise ratio （SNR） signal and more detailed intbrmation on the sound field. Study on vector sensors and their applications have become a hot topic. Research on the representation of a vector field is highly relevant for extending the scope of vector sensor technology. This paper discusses the range-frequency distribution of the vector field due to a broadband acoustic source moving in a shallow-water waveguide as the self noise of a surface ship, and the vector extension of the waveguide impulse response measured over a limited frequency range using an active source of known waveform. From theory analysis and numerical simulation, the range-frequency representation of a vector field exhibits an interference structure qualitatively similar to that of the corresponding pressure field but, being quantitatively different, provides additional information on the waveguide, especially through the vertical component. For the range-frequency representation, physical quantities that can better exhibit the interference characteristics of the wavegaide are the products of pressure and particle velocity and of the pressure and pressure gradient. An image processing method to effectively detect and isolate the individual striations from an interference structure was reviewed briefly. The representation of the vector impulse response was discussed according to two different measurement systems, also known as particle velocity and pressure gradient. The vector impulse response representation can not only provide additional information from pressure only but even more than that of the range-frequency representation.

Classification of Teleparallel Homothetic Vector Fields in Cylindrically Symmetric Static Space-Times in Teleparallel Theory of Gravitation

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.

A Note on Classification of Spatially Homogeneous Rotating Space-Times According to Their Teleparallel Killing Vector Fields in Teleparallel Theory of Gravitation

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.

Classification of Kantowski-Sachs and Bianchi Type Ⅲ Space-Times According to Their Killing Vector Fields in Teleparallel Theory of Gravitation

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.

Hybrid gradient vector fields for path-following guidance

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.

On a Hypothetical Vector Field Associated with the Classic Electromagnetic Wave in a Space of Four Dimensions

In two previous papers [1] and [2], a structure for vector products in n dimensions was presented, and at the same time it was possible to propose the existence of a vector analogous to the curl of a vector field, for a space of four dimensions. In continuation of these works, the objective is to develop, through dimensional analogy, the idea of a hypothetical vector field, associated with the classical electromagnetic wave. This hypothetical field has a possible mathematical existence only when considering a space of four dimensions. The properties of the electromagnetic wave are preserved and equations with mathematical forms analogous to those of Maxwell’s equations are presented.