A Novel INDI based Guidance Law for Fixed Wing Aircrafts: Derivation and Application

In this paper, we comprehensively present and derive two INDI principle based guidance laws for fixed wing aircrafts. More specifically, two control methods are mathematically derived in detail, where the first decouples the lateral and the longitudinal channels while the second takes the interactions into account. The cumbersome mathematical operations involved in the derivation process aim at reaching a more concise control method and also at providing the community with clearer physical concepts behind this formula. The reason for manipulating transformation matrices is to find a univariate function and to isolate the variable as a virtual input. Efficient and modular guidance control law is then permitted. Lastly, the proposed guidance methods are applied to a 6 dof nonlinear platform under various flight modes to demonstrate the feasibility and advantages.

A Novel INDI based Guidance Law for Fixed Wing Aircrafts: Derivation and Application

In this paper, we comprehensively present and derive two INDI principle based guidance laws for fixed wing aircrafts. More specifically, two control methods are mathematically derived in detail, where the first decouples the lateral and the longitudinal channels while the second takes the interactions into account. The cumbersome mathematical operations involved in the derivation process aim at reaching a more concise control method and also at providing the community with clearer physical concepts behind this formula. The reason for manipulating transformation matrices is to find a univariate function and to isolate the variable as a virtual input. Efficient and modular guidance control law is then permitted. Lastly, the proposed guidance methods are applied to a 6 dof nonlinear platform under various flight modes to demonstrate the feasibility and advantages.

On PDE Characterization of Smooth Hierarchical Functions Computed by Neural Networks

Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that are computable by a given network. We study real, infinitely differentiable (smooth) hierarchical functions implemented by feedforward neural networks via composing simpler functions in two cases: (1) each constituent function of the composition has fewer in puts than the resulting function and (2) constituent functions are in the more specific yet prevalent form of a nonlinear univariate function (e.g., tanh) applied to a linear multivariate function. We establish that in each of these regimes, there exist nontrivial algebraic partial differential equations (PDEs) that are satisfied by the computed functions. These PDEs are purely in terms of the partial derivatives and are dependent only on the topology of the network. Conversely, we conjecture that such PDE constraints, once accompanied by appropriate nonsingularity conditions and perhaps certain inequalities involving partial derivatives, guarantee that the smooth function under consideration can be represented by the network. The conjecture is verified in numerous examples, including the case of tree architectures, which are of neuroscientific interest. Our approach is a step toward formulating an algebraic description of functional spaces associated with specific neural networks, and may provide useful new tools for constructing neural networks.

A Study of Modified Newton-Raphson Method

A basic alteration of the standard Newton technique is investigated and described for the approximation of the roots of a univariate function. For a similar number of functions and evaluation of the derivative, an altered strategy combines quicker, with the convergence of the modified NR’s method being 2.4 as compared with the regular NR method which is 2. Some of the example shows the faster convergence accomplished with the modified NR method. This modification of Newton’s technique is generally basic and strong. It is bound to converge to the solution rather than the higher order or Newton-Raphson method itself. In this paper, the modification of NR strategy introduced which offers expanded rate of convergence over NR standard method.

On the fast approximation of point clouds using Chebyshev polynomials

Suppose a large and dense point cloud of an object with complex geometry is available that can be approximated by a smooth univariate function. In general, for such point clouds the “best” approximation using the method of least squares is usually hard or sometimes even impossible to compute. In most cases, however, a “near-best” approximation is just as good as the “best”, but usually much easier and faster to calculate. Therefore, a fast approach for the approximation of point clouds using Chebyshev polynomials is described, which is based on an interpolation in the Chebyshev points of the second kind. This allows to calculate the unknown coefficients of the polynomial by means of the Fast Fourier transform (FFT), which can be extremely efficient, especially for high-order polynomials. Thus, the focus of the presented approach is not on sparse point clouds or point clouds which can be approximated by functions with few parameters, but rather on large dense point clouds for whose approximation perhaps even millions of unknown coefficients have to be determined.

Logarithmic concavity of the inverse incomplete beta function with respect to the first parameter

The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as a univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.

ReLU Networks Are Universal Approximators via Piecewise Linear or Constant Functions

This letter proves that a ReLU network can approximate any continuous function with arbitrary precision by means of piecewise linear or constant approximations. For univariate function [Formula: see text], we use the composite of ReLUs to produce a line segment; all of the subnetworks of line segments comprise a ReLU network, which is a piecewise linear approximation to [Formula: see text]. For multivariate function [Formula: see text], ReLU networks are constructed to approximate a piecewise linear function derived from triangulation methods approximating [Formula: see text]. A neural unit called TRLU is designed by a ReLU network; the piecewise constant approximation, such as Haar wavelets, is implemented by rectifying the linear output of a ReLU network via TRLUs. New interpretations of deep layers, as well as some other results, are also presented.

SPATIAL DISTRIBUTION AND STRUCTURAL VARIATION OF Copaifera arenicola IN A CAATINGA FRAGMENT

The study of spatial distribution of tree populations has proven to be important for revealing how individuals are horizontally organized in the environment, facilitating the structural understanding and forms of colonization and dispersion of propagules. The present work aimed at studying the pattern of spatial distribution of tree species Copaiferaarenicola [(Ducke) J. Costa e L.P.Queiroz] and its structural relation with the altimetric profile in a Caatinga fragment in Ribeira do Pombal municipality, Bahia. Census of all individuals in the area with circumference at breast height (CBH) ≥ 6 cm was performed. The spatial distribution analysis was conducted for the whole population using Ripley K univariate function, with maximum search radius (h) of 128 m. 409 individuals were found, corresponding to absolute density of 89.49 ind. ha-1 and 0.681 m². ha-1 of basal area. The group of C. arenicola individuals corresponds to a stable population in expansion phase, presenting higher number of young and medium individuals. The pattern of spatial distribution of individuals in the area under study was the uniform arrangement. None of the altimetric classes of the area had a different influence on the structure and distribution of arboreal individuals.

Adaptive anisotropic response surface method based on univariate dimension-reduction model and its high-order revision

Purpose Based on univariate dimension-reduction model, this study aims to propose an adaptive anisotropic response surface method (ARSM) and its high-order revision (HARSM) to balance the accuracy and efficiency for response surface method (RSM). Design/methodology/approach First, judgment criteria for the constitution of a univariate function are derived mathematically, together with the practical implementation. Second, by combining separate polynomial approximation of each component function of univariate dimension-reduction model with its constitution analysis, the anisotropic ARSM is proposed. Third, the high-order revision for component functions is introduced to improve the accuracy of ARSM, namely, HARSM, in which the revision is also anisotropic. Finally, several examples are investigated to verify the accuracy, efficiency and convergence of the proposed methods, and the influence of parameters on the proposed methods is also performed. Findings The criteria for constitution analysis are appropriate and practical. Obtaining the undetermined coefficients for every component functions is easier than the existing RSMs. The existence of special component functions is useful to improve the efficiency of the ARSM. HARSM is helpful for improving accuracy significantly and it is more robust than ARSM and the existing quadratic polynomial RSMs and linear RSM. ARSM and HARSM can achieve appropriate balance between precision and efficiency. Originality/value The constitution of univariate function can be determined adaptively and the nonlinearity of different variables in the response surface can be treated in an anisotropic way.

Research on Minimum Time Interception Problem with a Tangent Impulse under Relative Motion Models

The minimum time interception problem with a tangent impulse whose direction is the same as the satellite’s velocity direction is studied based on the relative motion equations of elliptical orbits by the combination of analytical, numerical, and optimization methods. Firstly, the feasible domain of the true anomaly of the target under the fixed impulse point is given, and the interception solution is transformed into a univariate function only with respect to the target true anomaly by using the relative motion equation. On the basis of the above, the numerical solution of the function is obtained by the combination of incremental search and the false position method. Secondly, considering the initial drift when the impulse point is freely selected, the genetic algorithm-sequential quadratic programming (GA-SQP) combination optimization method is used to obtain the minimum time interception solution under the tangent impulse in a target motion cycle. Thirdly, under the high-precision orbit prediction (HPOP) model, the Nelder-Mead simplex method is used to optimize the impulse velocity and transfer time to obtain the accurate interception solution. Lastly, the effectiveness of the proposed method is verified by simulation examples.