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Let Φ n (k) (x) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values Φ n (k) (1) for fixed positive integers n. D. H. Lehmer proved that Φ n (k) (1) / Φ n (1) is a polynomial of the Euler totient function ϕ (n) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that Φ n (k) (1) / Φ n (1) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: 2 Φ n (3) (1) is divisible by ϕ (n) - 2 . Moreover, if k is greater than 1, then Φ n (2 k + 1) (1) is divisible by ϕ (n) - 2 k . The proof depends on a new combinatorial identity for general self-reciprocal polynomials over Z , which gives rise to a formula that expresses the value Φ n (k) (1) as a Z -linear combination of the coefficients in the minimal polynomial of 2 cos (2 π / n) - 2 . As a supplement, we show the monotonic increasing property of Φ n (x) on [ 1 , ∞) in two ways. [ABSTRACT FROM AUTHOR]

Differentiation matrices are in wide use in numerical algorithms, although usually studied in an ad hoc manner. We collect here in this review paper elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange and Hermite interpolational bases. We give new explicit formulations, and new explicit formulations for the pseudo-inverses which help to understand antidifferentiation, of many of these matrices. We also give the unique Jordan form for these (nilpotent) matrices and a new unified formula for the transformation matrix. [ABSTRACT FROM AUTHOR]

The algorithm for solving the matrix equation Xn = A is well-known, where A is an given matrix and A is the unknown square matrix. This paper is concerned with the general case of the polynomial matrix equation , where. The study of the general case help us to solve any polynomial matrix equation. The main difficulty to solve the polynomial matrix equation is that, in general, the function is not invertible. Even if the function h is invertible, it is difficult to find the type of the inverse function and its derivatives. We designed an algorithm, which enables us to bypass anything related with the inverse function of h. In our algorithm we just used the polynomial function h and its derivatives. As it is easily understood, this is a very effective procedure and our algorithm can be used for every polynomial function h and any square matrix A. All the possible cases concerning the Jordan canonical form of the matrix A are examined and a formula of the matrix hoq(A) is given where q(x) is the interpolating polynomial to the data (4.1.1, 4.1.2). Mathematical types to calculate the number of different roots of the polynomial matrix equation and their algebraic multiplicity are also presented (5). We want to point out that the general case helps us to solve any polynomial matrix equation, which play an important role in many applications. [ABSTRACT FROM AUTHOR]

For a positive integer m and a Hilbert space H an operator T in B (H) , the space of all bounded linear operators on H , is called m -selfadjoint if ∑ k = 0 m (− 1) k ( m k ) T ⁎ k T m − k = 0. In this paper, we show that if T ∈ B (H) and the spectrum of T consists of a finite number of points then it is m -selfadjoint if and only if it is an n -Jordan operator for some integer n. Moreover, we prove that if T is m -selfadjoint then T is nilpotent when it is quasinilpotent. Then we characterize m -selfadjoint weighted shift operators. Also, we show that if T is m -selfadjoint then so is p (T) when p (z) is a polynomial with real coefficients. After that, we investigate an elementary operator τ and a generalized derivation operator δ on the Hilbert-Schmidt class C 2 (H) which are m -selfadjoint. Finally, we prove that no m -selfadjoint operator on an infinite-dimensional Hilbert space, can be N -supercyclic, for any N ≥ 1. [ABSTRACT FROM AUTHOR]

Let A be a non-commutative prime ring with involution ∗ , of characteristic ≠ 2 (and 3) , with Z as the center of A and Π a mapping Π : A → A such that [ Π (x) , x ] ∈ Z for all (skew) symmetric elements x ∈ A. If Π is a non-zero CE-Jordan derivation of A , then A satisfies s 4 , the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A , then A satisfies s 4 or Π (y) = λ (y − y *) for all y ∈ A , and some λ ∈ C , the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results. [ABSTRACT FROM AUTHOR]

In this paper, new refinements and improvements of Jordan's and Kober's inequalities are presented. We give new polynomial bounds for the s i n c (x) and cos (x) functions based on the interpolation and approximation method. The results show that our bounds are tighter than the previous methods. [ABSTRACT FROM AUTHOR]

We define the notion of an almost polynomial identity of an associative algebra R, and show that its existence implies the existence of an actual polynomial identity of R. A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for the computation of the Jordan canonical form of regular matrix polynomials is proposed. The new method contains rank conditions of suitably defined block Toeplitz matrices and it does not require the computation of the Jordan chains or the Smith form. The Segré and Weyr characteristics are also considered. [Copyright &y& Elsevier]

In Part I we proposed a multi linear algebra framework to solve 0-dimensional systems of polynomial equations with simple roots. We extend this framework to incorporate multiple roots: a block term decomposition (BTD) of the null space of the Macaulay matrix reveals the dual (sub) space of a disjoint root in each term. The BTD is the joint triangularization of multiplication tables and a three-way generalization of the Jordan canonical form in the matrix case, intimately related to the border rank of a tensor. We hint at and illustrate flexible numerical optimization-based algorithms. [ABSTRACT FROM AUTHOR]

In the space of cubic polynomials, Milnor defined a notable curve S p , consisting of cubic polynomials with a periodic critical point, whose period is exactly p. In this paper, we show that for any integer p ≥ 1 , any bounded hyperbolic component on S p is a Jordan disk. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the eigenproblem on the large and low-rank matrix S = A B H , where A , B ∈ ℂ n × r are of full column rank and r ≪ n. To the best of our knowledge, there are no results on the relations between the Jordan decomposition and the Schur decomposition of B H A and those of A B H . Some known results are only on characteristic polynomials, elementary divisors, and Jordan blocks of A B H , and are purely theoretical and are not easy to use for computational purposes. Based on the Jordan decomposition and the Schur decomposition of the small matrix B H A ∈ ℂ r × r , we consider how to derive those of the large matrix A H B ∈ ℂ n × n in this work. The construction methods proposed are not only theoretical but also practical. Numerical experiments show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

We study the Schwarz problem for J-analytic vector-valued functions in an ellipse with a square matrix J admitting a nondiagonal Jordan form. We obtain conditions on the ellipse and matrix J necessary and sufficient for the existence and uniqueness of a solution to the Schwarz problem with an arbitrary boundary function of Hölder class. Under certain conditions on the matrix J, we show that the homogeneous Schwarz problem in an ellipse has a solution in the form of a vector polynomial of an arbitrary degree. [ABSTRACT FROM AUTHOR]

The problem of finding the distance from a given n×n matrix polynomial of degree k to the set of matrix polynomials having the elementary divisor (λ-λ0)j,j⩾r, for a fixed scalar λ0 and 2⩽r⩽kn is considered. It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor. For regular matrix polynomials the problem is shown to be equivalent to finding minimal structure preserving perturbations such that a certain block Toeplitz matrix becomes suitably rank deficient. This is then used to characterize the distance via two different optimizations. The first one shows that if λ0 is not already an eigenvalue of the matrix polynomial, then the problem is equivalent to computing a generalized notion of a structured singular value. The distance is computed via algorithms like BFGS and Matlab's globalsearch algorithm from the second optimization. Upper and lower bounds of the distance are also derived and numerical experiments are performed to compare them with the computed values of the distance. [ABSTRACT FROM AUTHOR]

The problem of the similarity of integer matrices with single eigenvalue over the ring of integers is considered. A criterion for a matrix to be similar to a Jordan block is obtained. In addition, a similarity criterion for matrices whose minimal polynomial has degree 2 is obtained. [ABSTRACT FROM AUTHOR]