Loci of complex polynomials, part II: polar derivatives

2015 ◽  
Vol 159 (2) ◽  
pp. 253-273 ◽  
Author(s):  
BLAGOVEST SENDOV ◽  
HRISTO SENDOV
Keyword(s):  
Higher Order ◽  
Complex Polynomial ◽  
Complex Polynomials ◽  
Point Sets ◽  
Closed Set ◽  
Measurable Set ◽  
Closed Point ◽  
Polar Derivatives ◽  
Lebesgue Measurable Set ◽  
Derivatives Of

AbstractFor every complex polynomial p(z), closed point sets are defined, called loci of p(z). A closed set Ω ⊆ ${\mathbb C}$* is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such set with respect to inclusion. Using the notion locus, some classical theorems in the geometry of polynomials can be refined. We show that each locus is a Lebesgue measurable set and investigate its intriguing connections with the higher-order polar derivatives of p.

Duality between loci of complex polynomials and the zeros of polar derivatives

2018 ◽  
Vol 167 (01) ◽  
pp. 65-87 ◽  
Author(s):  
BLAGOVEST SENDOV ◽  
HRISTO SENDOV
Keyword(s):  
Complex Polynomial ◽  
Complex Polynomials ◽  
Polar Derivatives

AbstractThis work investigates the connections between the notion of a locus of a complex polynomial and the polar derivatives. Polar differentiation extends classical derivatives and provides additional flexibility. The notion of a locus was introduced in [8] and proved useful in providing sharp versions of several classical results in the area known as Geometry of Polynomials. The investigations culminated in the work [11]. A need was revealed for a unified treatment of bounded and unbounded loci of polynomials of degree at most n as well as a unified treatment of polar derivatives and ordinary derivatives. This work aims at providing such a framework.

scholarly journals Generalizations of some Zygmund-type integral inequalities for polar derivatives of a complex polynomial

2021 ◽  
Vol 110 (124) ◽  
pp. 57-69
Author(s):  
Abdullah Mir
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Integral Inequalities ◽  
Complex Polynomial ◽  
Algebraic Polynomials ◽  
Polar Derivative ◽  
Type Integral ◽  
Polar Derivatives ◽  
Derivatives Of

We prove some results for algebraic polynomials in the complex plane that relate the L-norm of the polar derivative of a complex polynomial and the polynomial under some conditions. The obtained results include several interesting generalizations of some Zygmund-type integral inequalities for polynomials and derive polar derivative analogues of some classical Bernsteintype inequalities for the sup-norms on the unit disk as well.

scholarly journals Growth of a polynomial not vanishing in a disk

2019 ◽  
Vol 73 (1) ◽  
pp. 41
Author(s):  
Abdullah Mir
Keyword(s):  
Upper Bound ◽  
Maximum Modulus ◽  
Higher Order ◽  
Complex Polynomial ◽  
Derivatives Of

This paper deals with the problem of finding some upper bound estimates for the maximum modulus of the derivative and higher order derivatives of a complex polynomial on a disk under the assumption that the polynomial has no zeros in another disk. The estimates obtained strengthen the well-known inequality of Ankeny and Rivlin about polynomials.

Poly(methyl-phenylsilylene) Derivatives as Photoconductors

1993 ◽  
Vol 58 (10) ◽  
pp. 2337-2348 ◽  
Author(s):  
Ivan Kmínek ◽  
Stanislav Nešpůrek ◽  
Eduard Brynda ◽  
Jiří Pfleger ◽  
Věra Cimrová ◽  
...  
Keyword(s):  
Spectral Properties ◽  
Long Wavelength ◽  
Langmuir Blodgett ◽  
Ultrathin Layers ◽  
Polar Derivatives ◽  
Derivatives Of

The attachment of long wavelength absorbing π-conjugated chromophores to poly(methyl-phenylsilylene) (PMPSi) via reactions of its formylated derivative is described. Some of the obtained polymers exhibit improved photostability in comparison with the parent polymer. Their spectral properties and photoconductivity are discussed. Ultrathin layers and multilayers were prepared from polar derivatives of PMPSi by the Langmuir-Blodgett technique and their photoconductive behaviour was studied.

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

Certain inequalities for the polar derivatives of polynomials

2009 ◽  
Vol 12 (1) ◽  
pp. 29-39 ◽  
Author(s):  
K. K. Dewan ◽  
Naresh Singh ◽  
Sunil Hans
Keyword(s):  
Polar Derivatives ◽  
Derivatives Of

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

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