Polynomials of Least Deviation from Zero in Sobolev p-Norm

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm ($$1<p<\infty $$ 1 < p < ∞ ) for the case $$p=1$$ p = 1 . Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least $$n-\mathbf {d}^*$$ n - d ∗ zeros on the convex hull of the support of the measure, where $$\mathbf {d}^*$$ d ∗ denotes the number of terms in the discrete part.

Distributions of zeros and poles of -point Padé approximants to complex-symmetric functions defined at complex points

UDC 517.5 The knowledge of the location of zeros and poles Padé and -point Padé approximations to a given function provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of complex-symmetric functions.The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of All considered cases are graphically illustrated.Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them.

Lee-Yang zeros of the antiferromagnetic Ising model

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

On the regions containing all the zeros of polynomials and related analytic functions

In this paper, by using standard techniques we shall obtain results with relaxed hypothesis which give zero bounds for the larger class of polynomials. Our results not only generalizes several well-known results but also provide better information about the location of zeros. We also obtain a similar result for analytic functions. In addition to this, we show by examples that our result gives better information on the zero bounds of polynomials than some known results.

SURVEY AND ADJUSTMENT OF CONTROLLER OF A GRAPH-ANALYTICAL METHOD TO A SECOND ORDER OBJECT

This paper proposes to adjust the controller using a graph-analytical method in the complex plane. The various configurations regarding the location of zeros and poles to those of the object are also considered. Adjusted controllers are surveyed, such as they are integrated into control systems, and some of the quality indicators of an automatic control system are analyzed.

The location of critical points of polynomials

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials.

Speiser’s Theorem on the Road

In this note we discuss the Gauss-Lucas theorem (for the zeros of the derivative of a polynomial) and Speiser’s equivalent for the Riemann hypothesis (about the location of zeros of the Riemann zeta-function). We indicate similarities between these results and present there analogues in the context of elliptic curves, regular graphs, and finite Euler products.