Univariate Polynomial Of Degree Research Articles

Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

In the semialgebraic range searching problem, we are given a set of n points in ℝ d , and we want to preprocess the points such that for any query range belonging to a family of constant complexity semialgebraic sets (Tarski cells), all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem is well-understood: It can be solved using S(n) space and with Q(n) query time with \(S(n)Q(n)^d = \tilde{O}(n^d),\) where the \(\tilde{O}(\cdot)\) notation hides polylogarithmic factors and this trade-off is tight (up to n o (1) factors). In particular, there exist “low space” structures that use O(n) space with O ( n 1-1/ d } ) query time [ 8 , 25 ] and “fast query” structures that use O ( n d ) space with O (log n ) query time [ 9 ]. However, for general semialgebraic ranges, only “low space” solutions are known, but the best solutions [ 7 ] match the same trade-off curve as simplex queries, with O ( n ) space and \(\tilde{O}(n^{1-1/d})\) query time. It has been conjectured that the same could be done for the “fast query” case, but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semialgebraic range searching and other related problems. More precisely, we show that any data structure for reporting the points between two concentric circles, a problem that we call 2D annulus reporting, with Q ( n ) query time must use \(S(n)=\overset{\scriptscriptstyle o}{\Omega }(n^3/Q(n)^5)\) space, where the \(\overset{\scriptscriptstyle o}{\Omega }(\cdot)\) notation hides \(n^{o(1)}\) factors, meaning, for \(Q(n)=\log ^{O(1)}n\) , \(\overset{\scriptscriptstyle o}{\Omega }(n^3)\) space must be used. In addition, we study the problem of reporting the subset of input points in a polynomial slab defined by \(\lbrace (x,y)\in \mathbb {R}^2:P(x)\le y\le P(x)+w\rbrace\) , where \(P(x)=\sum _{i=0}^\Delta a_i x^i\) is a univariate polynomial of degree Δ and \(a_0, \ldots , a_\Delta , w\) are given at the query time, a problem that we call polynomial slab reporting. For this, we show a space lower bound of \(\overset{\scriptscriptstyle o}{\Omega }(n^{\Delta +1}/Q(n)^{(\Delta +3)\Delta /2})\) , which implies that for \(Q(n)=\log ^{O(1)}n\) , we must use \(\overset{\scriptscriptstyle o}{\Omega }(n^{\Delta +1})\) space. We also consider the dual semialgebraic stabbing problems of semialgebraic range searching and present lower bounds for them. In particular, we show that in linear space, any data structure that solves 2D annulus stabbing problems must use \(\Omega (n^{2/3})\) query time. Note that this almost matches the upper bound obtained by lifting 2D annuli to 3D. Like semialgebraic range searching, we also present lower bounds for general polynomial slab stabbing problems. Again, our lower bounds are almost tight for linear size data structures in this case.

THE NUMBER OF LIMIT CYCLES FROM ELLIPTIC HAMILTONIAN VECTOR FIELDS BY HIGHER ORDER MELNIKOV FUNCTIONS

In this paper, the perturbed Hamiltonian system $dH=\epsilon F_4+\epsilon^2F_3+\epsilon^3F_2+\epsilon^4F_1$, with $F_i$ the vector valued homogeneous polynomials of degree $i$. The Hamiltonian function is $H=y^2/2+U(X),$ where $U$ is a univariate polynomial of degree four without symmetry. By computing higher order Melnikov functions, the upper bounds for the number of limit cycles that bifurcate from $dH=0$ are deserved.

Note on an inequality of M.A. Malik

Let P(z):= ?nv=0 avzv be a univariate complex coefficient polynomial of degree n. It was shown by Malik [J London Math Soc, 1 (1969), 57-60] that if P(z) has all its zeros in |z| ? k, k ? 1, then max|z|=1 |P?(z)| ? n 1 + k max |z|=1 |P(z)|. In this paper, we prove an inequality for the polar derivative of a polynomial which besides give extensions and refinements of the above inequality also produce various inequalities that are sharper than the previous ones known in very rich literature on this subject.

Inequalities for the polar derivative of a complex polynomial

Let P(z):= ?n v=0 avzv be a univariate complex coefficient polynomial of degree n. Then as a generalization of a well-known classical inequality of Tur?n [25], it was shown by Govil [7] that if P(z) has all its zeros in |z| ? k, k ? 1, then max |z|=1 |P?(z)| ? n/1 + kn max |z|=1 |P(z)|, whereas, if P(z) , 0 in |z| < k, k ? 1, it was again Govil [6] who gave an extension of the classical Erd?s-Lax inequality [13], by obtaining max |z|=1 |P?(z)| ? n/1 + kn max |z|=1 |P(z)|, provided |P?(z)| and |Q?(z)| attain maximum at the same point on |z| = 1, where Q(z) = ?znP (1/z). In this paper, we obtain several generalizations and refinements of the above inequalities and related results while taking into account the placement of the zeros and extremal coefficients of the underlying polynomial. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.

Braid monodromy of univariate fewnomials

Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Vi\`{e}te map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $\mathscr{V}_*$ at the level of fundamental groups realises an isomorphism between $\pi_1(\mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{C}$ of $\mathcal{C}_d$ with a collection of coordinate hyperplanes in $\mathbb{C}^{d+1}$, the image of the map $\mathscr{V} _* : \pi_1(\mathcal{C}) \rightarrow B_d$ is not known in general. In the present paper, we show that the map $\mathscr{V} _*$ is surjective provided that the support of the corresponding polynomials spans $\mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of $\mathscr{V} _*$ is the expected wreath product of $\mathbb{Z}/b\mathbb{Z}$ with $B_{d/b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.

On the probability that the maximum of a polynomial is at an endpoint of an interval

In this article we study the probability that the maximum over a symmetric interval [−a, a] of a univariate polynomial of degree at most n is attained at an endpoint. We give explicit formulas for the degree n = 1, 2,3 cases. The formula for the degree 3 case leads to a lower bound for the true probability. Numerical experiments indicate that this lower bound is actually quite a good approximation. We finish with some numerical examples indicating that these ideas may be of use in the problem of finding average Markov factors for the bounds of the uniform norm of the derivative a polynomial in terms of the norm of the polynomial itself.

Spectral norm of a symmetric tensor and its computation

We show that the spectral norm of a $d$-mode real or complex symmetric tensor in $n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For $n=2$ we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most $(d-1)^2+1$, except certain cases, which are completely analyzed. In particular, for $n=2$ the spectral norm of $d$-symmetric tensor is polynomially computable in $d$ with a given relative precision. For a fixed $n>2$ we show that the spectral norm of a $d$-mode symmetric tensor is polynomially computable in $d$ with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of $d$-mode symmetric qunits on $\mathbb{C}^n$ are polynomially computable for a fixed $n$.

The inverse kinematics of rolling contact of timelike curves lying on timelike surfaces

Rolling contact between two surfaces plays an important role in robotics and engineering such as spherical robots, single wheel robots, and multi-fingered robotic hands to drive a moving surface on a fixed surface. The rolling contact pairs have one, two, or three degrees of freedom (DOFs) consisting of angular velocity components. Rolling contact motion can be divided into two categories: spin-rolling motion and pure-rolling motion. Spin-rolling motion has three (DOFs), and pure-rolling motion has two (DOFs). Further, it is well known that the contact kinematics can be divided into two categories: forward kinematics and inverse kinematics. In this paper, we investigate the inverse kinematics of spin-rolling motion without sliding of one timelike surface on another timelike surface in the direction of timelike unit tangent vectors of their timelike trajectory curves by determining the desired motion and the coordinates of the contact point on each surface. We get three nonlinear algebraic equations as inputs by using curvature theory in Lorentzian geometry. These equations can be reduced as a univariate polynomial of degree six by applying the Darboux frame method. This polynomial enables us to obtain rapid and accurate numerical root approximations and to analyze the rolling rate as an output. Moreover, we obtain another outputs: the rolling direction and the compensatory spin rate.

Lifted projective Reed–Solomon codes

Lifted Reed–Solomon codes, introduced by Guo et al. (in: Kleinberg (ed) Proceedings of the 4th conference on innovations in theoretical computer science, ITCS’13, ACM, New York, 2013), are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials whose restriction along any affine line can be interpolated as a low degree univariate polynomial. In this work, we give a formal definition of their analogues over projective spaces, and we study some of their parameters and features. Local correcting algorithms are first derived from the very nature of these codes, generalizing the well-known local correcting algorithms for Reed–Muller codes. We also prove that the lifting of both Reed–Solomon and projective Reed–Solomon codes are deeply linked through shortening and puncturing operations. It leads to recursive formulae on their dimension and their monomial bases. We finally emphasize the practicality of lifted projective Reed–Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms.

Higher Order Melnikov Functions for Studying Limit Cycles of Some Perturbed Elliptic Hamiltonian Vector Fields

In this paper, we study the number of limit cycles in the perturbed Hamiltonian system $$dH=\varepsilon F_1+\varepsilon ^2 F_2+\varepsilon ^3 F_3$$ with $$F_i$$ , the vector valued homogeneous polynomials of degree i and $$4-i$$ for $$i=1,2,3$$ , and small positive parameter $$\varepsilon $$ . The Hamiltonian function has the form $$H=y^2/2+U(x)$$ , where U is a univariate polynomial of degree four without symmetry. We compute higher order Melnikov functions until we obtain reversible perturbations. Then we find the upper bounds for the number of limit cycles that can bifurcate from the periodic orbits of $$dH=0$$ .

A Geometric Approach to Obtain the Closed-Form Forward Kinematics of H4 Parallel Robot

AbstractTo obtain the closed-form forward kinematics of parallel robots, researchers use algebra-based method to transform and simplify the constraint equations. However, this method requires a complicated derivation that leads to high-order univariate variable equations. In fact, some particular mechanisms, such as Delta, or H4 possess many invariant geometric properties during movement. This suggests that one might be able to transform and reduce the problem using geometric approaches. Therefore, a simpler and more efficient solution might be found. Based on this idea, we developed a new geometric approach called geometric forward kinematics (GFK) to obtain the closed-form solutions of H4 forward kinematics in this paper. The result shows that the forward kinematics of H4 yields an eighth degree univariate polynomial, compared with earlier reported 16th degree. Thanks to its clear physical meaning, an intensive discussion about the solutions is presented. Results indicate that a general H4 robot can have up to eight nonrepeated real solutions for its forward kinematics. For a specific configuration of H4, the nonrepeated number of real roots could be restricted to only two, four, or six. Two traveling plate configurations are discussed in this paper as two typical categories of H4. A numerical analysis was also performed for this new method.

Cubic perturbations of elliptic Hamiltonian vector fields of degree three

The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field XεXε:{x˙=Hy+εf(x,y)y˙=−Hx+εg(x,y),H =12y2 +U(x) which bifurcate from the period annuli of X0 for sufficiently small ε. Here U is a univariate polynomial of degree four without symmetry, and f,g are arbitrary cubic polynomials in two variables.We take a period annulus and parameterize the related displacement map d(h,ε) by the Hamiltonian value h and by the small parameter ε. Let Mk(h) be the k-th coefficient in its expansion with respect to ε. We establish the general form of Mk and study its zeroes. We deduce that the period annuli of X0 can produce for sufficiently small ε, at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of X0 of higher degrees are also studied.

Duality of multiple root loci

The multiple root loci among univariate polynomials of degree n are indexed by partitions of n. We study these loci and their conormal varieties. The projectively dual varieties are joins of such loci where the partitions are hooks. Our emphasis lies on equations and parametrizations that are useful for Euclidean distance optimization. We compute the ED degrees for hooks. Among the dual hypersurfaces are those that demarcate the set of binary forms whose real rank equals the generic complex rank.

DIRECT KINEMATIC ANALYSIS OF A FAMILY OF 4-DOF PARALLEL MANIPULATORS WITH A PASSIVE CONSTRAINING LEG

This paper presents direct kinematic analysis of a family of 3R1T parallel manipulators, while R and T denote the rotational and translational degrees of freedom respectively. The manipulators consist of two rigid bodies, a movable platform and a fixed (base) connected to each other by four active legs and one constraining passive leg. First, the direct position kinematics of the manipulators is analyzed. For a general manipulator of this class, this analysis results in a univariate polynomial of degree 30 along with a set of other univariate polynomials of degree 16 and 4 respectively. However, for a special architecture of the manipulators, it is shown that the direct position kinematics leads to a minimal univariate polynomial of degree 12. A numerical example is also included to confirm the results. Moreover, direct velocity and direct kinematic singularities of the manipulators are analyzed using Jacobian matrices.

The Position Analysis of Spherical Seven-Bar Mechanism

In this paper, we turn plane seven-bar mechanism into spherical seven-bar mechanism, using quaternion to construct mathematical model for spherical seven-bar mechanism. Three constraint equations are obtained according to the angles constraint. Using Sylvester resultant elimination by two steps, a 32 degree univariate polynomial equation can be obtained. A numerical example confirms that analytical solutions of spherical seven-bar mechanism are 32 and with the help of Mathematic software to solve the location parameters.

Kinematics Analysis of a Parallel Robotic Manipulator

The parallel robotic manipulator has attracted many researchers’ attention and it also has growing applications to different areas. In this paper an algebraic method for solving the direct kinematics analysis problem for a parallel robotic manipulator. Based on the presented algebraic method, the problem is derived into a 40th degree univariate polynomial. All complete sets of 40 solutions to the problem are obtained. The proposed method is exemplified by a numerical example.

A New Algorithm for a Spatial Serial Robot

In this paper a new algorithm to compute all the closed-form inverse kinematics solutions of a spatial serial robot. Based on the method, A 16th degree univariate polynomial of the spatial serial robot is obtained without factoring out or deriving the greatest common divisor. We also obtain all the closed-form solutions for the inverse kinematics of the robot. Finally a numerical example is given to demonstrate the algorithm process.

Closed-form forward kinematics for a symmetrical 6-6 Stewart platform using algebraic elimination

Abstract This paper studies the forward kinematics of a symmetrical 6-6 Stewart platform, in which both the base and the mobile platform are hexagons and the joint centers satisfy some conditions. A concise algebraic elimination algorithm to solve the closed-form forward kinematics of the Stewart platform is presented. Based on the presented algebraic method, an interesting result that comes out of our study is that the forward kinematics problem is reduced to solve a univariate polynomial equation of degree at most 14. The 14th degree univariate polynomial is derived from the determinant of the 15 × 15 Sylvester’s matrix, which is relatively small in size, without factoring out or deriving the greatest common divisor. The algorithm is comparatively concise and requires fairly less computation time. The result is verified by a numerical example.

Inverse kinematic analysis of the general 6R serial manipulators based on double quaternions

In this paper, double quaternions and Dixon resultant are introduced to solve inverse kinematics analysis of the general 6R robot. First, we show that homogeneous transforms matrix in terms of double quaternions lead to double kinematics equations of 6R robots; Second, a resultant procedure was constructed via linear algebra and Dixon resultant formulation; A 16th degree univariate polynomial is yielded from the resultant matrix. Finally a numerical example is used to demonstrate the effectiveness of the solution process.

An Algebraic Method for Direct Position Analysis of the General Stewart Platform Manipulator Robot

Stewart platform manipulator robot is a six degree of freedom, parallel manipulator, which consists of a base platform, a mobile platform and six limbs connected at six distinct points on the base platform and the mobile platform respectively. The direct position analysis problem of Stewart platform relates to the determination of the mobile platform pose for a given set of the lengths of the limbs. In this paper, we present a concise algebraic method for solving the direct position analysis problem for the fully parallel manipulator with general geometry, often referred to as General Stewart platform manipulator. Based on the presented algebraic method, we derive a 40th degree univariate polynomial from a determinant of 20×20 Sylvester’s matrix, which is relatively small in size. We also obtain a complete set of 40 solutions to the most general Stewart platform. The proposed method is comparatively concise and reduces the computational burden. Finally the method is demonstrated by a numerical example.