Feasible Trajectories Generation for Autonomous Driving Vehicles

This study presents smooth and fast feasible trajectory generation for autonomous driving vehicles subject to the vehicle physical constraints on the vehicle power, speed, acceleration as well as the hard limitations of the vehicle steering angle and the steering angular speed. This is due to the fact the vehicle speed and the vehicle steering angle are always in a strict relationship for safety purposes, depending on the real vehicle driving constraints, the environmental conditions, and the surrounding obstacles. Three different methods of the position quintic polynomial, speed quartic polynomial, and symmetric polynomial function for generating the vehicle trajectories are presented and illustrated with simulations. The optimal trajectory is selected according to three criteria: Smoother curve, smaller tracking error, and shorter distance. The outcomes of this paper can be used for generating online trajectories for autonomous driving vehicles and auto-parking systems.

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Classical n-body system in geometrical and volume variables: I. Three-body case

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

Nonsymmetric Macdonald Superpolynomials

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

A Novel Algorithm for Path Planning of the Mobile Robot in Obstacle Environment

—In this paper, a design method of smoothing the path generated by a novel algorithm is proposed, which makes the mobile robot can more rapidly and smoothly follow the path and reach the target point. No matter the attitude vector angle is an acute angle or obtuse angle, there is no doubt that we can find the right curve, including polar polynomial curves and piecewise polynomial functions, which makes the path length and the circular arc tend to be similar and guarantees the shorter path length. In the condition of meeting the dynamic characteristics of the mobile robot, the tracking speed and quality are improved. Therefore, the symmetric polynomial curve and the piecewise polynomial function curve are used to generate a smooth path. This novel algorithm improves the path tracking accuracy and the flexibility of the mobile robot. At the same time, it expands the application range of mobile robot in structured environment.

Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables $z_j = x_j + x_j^{-1}$

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that\[P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1})=Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}).\] We denote this polynomial $Q$ by $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomials of the form $\Phi_n(\operatorname{s}_{\lambda/\mu}^{(2n)})$, where $\operatorname{s}_{\lambda/\mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,\ldots,x_n,x^{-1}_1,\ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.

Spectrum of the q-Schrödinger equation by means of the variational method based on the discrete q-Hermite I polynomials

In this work, the [Formula: see text]-Schrödinger equations with symmetric polynomial potentials are considered. The spectrum of the model is obtained for several values of [Formula: see text], and the limiting case as [Formula: see text] is considered. The Rayleigh–Ritz variational method is adopted to the system. The discrete [Formula: see text]-Hermite I polynomials are handled as basis in this method. Furthermore, the following potentials with numerous results are presented as applications: [Formula: see text]-harmonic, purely [Formula: see text]-quartic and [Formula: see text]-quartic oscillators. It is also shown that the obtained results confirm the ones that exist in the literature for the continuous case.

A New Approach to Solve Perturbed Symmetric Eigenvalue Problems

The objective of this study is to ef-ciently resolve a perturbed symmetric eigen-value problem, without resolving a completelynew eigenvalue problem. When the size of aninitial eigenvalue problem is large, its multipletimes solving for each set of perturbations can becomputationally expensive and undesired. Thistype of problems is frequently encountered inthe dynamic analysis of mechanical structures.This study deals with a perturbed symmetriceigenvalue problem. It propose to develop atechnique that transforms the perturbed sym-metric eigenvalue problem, of a large size, toa symmetric polynomial eigenvalue problem ofa much reduced size. To accomplish this, weonly need the introduced perturbations, the sym-metric positive-de nite matrices representing theunperturbed system and its rst eigensolutions.The originality lies in the structure of the ob-tained formulation, where the contribution of theunknown eignsolutions of the unperturbed sys-tem is included. The e ectiveness of the pro-posed method is illustrated with numerical tests.High quality results, compared to other existingmethods that use exact reanalysis, can be ob-tained in a reduced calculation time, even if theintroduced perturbations are very signi cant.

Arithmetic Table as an Integral Part of all Computational Mathematics

The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.