Polynomial ring representations of endomorphisms of exterior powers

An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$ , and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$ , see Definition 1.1, where $f_c^n$ is the n-st iteration of f c . In this paper, we prove that if u is a rational critical point of f, then there exists an M f > 0 such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$ .

Dirichlet characters of the rational polynomials

<abstract><p>Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $\end{document} </tex-math></disp-formula></p> <p>where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity ^{[<xref ref-type="bibr" rid="b6">6</xref>]} under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.</p></abstract>

Symbolic Computation of Inverse Kinematics for General 6R Manipulators Based on Raghavan and Roth’s Solution

We discuss the symbolic computation of inverse kinematics for serial 6R manipulators with arbitrary geometries (general 6R manipulators) based on Raghavan and Roth’s solution. The elements of the matrices required in the solution were symbolically calculated. In the symbolic computation, an algorithm for simplifying polynomials upon considering the symbolic constraints (constraints of the trigonometric functions and those of the rotation matrix), a method for symbolic elimination of the joint variables, and an efficient computation of the rational polynomials are presented. The elements of the matrix whose determinant produces a 16th-order single variable polynomial (characteristic polynomial) were symbolically calculated by using structural parameters (parameters that define the geometry of the manipulator) and hand configuration parameters (parameters that define the hand configuration). The symbolic determinant of the matrix consists of huge number of terms even when each element is replaced by a single symbol. Instead of expressing the coefficients in a characteristic polynomial by structural parameters and hand configuration parameters, we substituted appropriate rational numbers that strictly satisfy the constraints of the symbols for the elements of the matrix and calculated the determinant (numerical error free calculation). By numerically calculating the real roots of the rational characteristic polynomial and the joint angles for each root, we verified the formulation for the symbolic computation.

A Note on the Classical Gauss Sums

The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums mod p ( a prime with p ≡ 1 mod 12 ) to give an exact evaluation formula for it.

On Classical Gauss Sums and Some of Their Properties

The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it.