Zeros Of Derivatives Research Articles

On the Koebe Quarter Theorem for Certain Polynomials of Even Degree

AbstractWe continue research on problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in $$\mathbb T:=\{z\in \mathbb C:|z|=1\}$$ T : = { z ∈ C : | z | = 1 } . We found the minimal disc containing all images of $$\mathbb D:=\{z\in \mathbb C: |z|<1\}$$ D : = { z ∈ C : | z | < 1 } and the maximal disc contained in all images of $$\mathbb D$$ D through polynomials of degree 6. Moreover, we determine the extremal functions for both problems. Furthermore, we state the conjecture concerning polynomials of higher even degrees.

ZEROES: Robust Derivative-Based Demodulation Method for Optical Camera Communication

Most of Optical Camera Communication (OCC) systems benefit from the rolling shutter mechanism of Complementary Metal-Oxide Semiconductor (CMOS) cameras to record the brightness evolution of the Light-Emitting Diode (LED) through dark and bright strips within images. While this technique enhances the maximum achievable data rate, the main difficulty lies in the demodulation of the signal extracted from images, subject to blooming effect. Thus, two main approaches were proposed to deal with this issue, using adaptive thresholds whose value evolves according to amplitude changes or detecting signal variations with the first-order derivative. As the second method is more robust, a new demodulation method based on the detection of the zeros of the first-order derivative of the extracted signal was proposed in this paper. Obtained results clearly show an improvement in the extracted signal demodulation compared to other methods, achieving a raw Bit Error Rate (BER) of 10−3 around 50 cm in a Line-Of-Sight scenario, and increasing the maximum communication distance by 43.5%, reaching 330 cm in the case of a Non-Line-Of-Sight transmission.

Monotonicity of zeros of derivatives of Bessel functions

Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the nth derivative of the Bessel function of the first kind Jν(x) are all real when ν>n−1. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every n∈N, the positive zeros of Jν(n)(x) are increasing functions of the parameter ν, for ν∈(n−1,∞). We provide two apparently distinct proofs of the conjecture.

Zeros of Fractional Derivatives of Polynomials

We investigate the behavior of fractional derivatives of polynomials. In particular, we consider the locations and the asymptotic behavior of their zeros and give bounds for their Mahler measure.

Zeros of higher derivatives of Riemann zeta function

In this article, we extend the result of Conrey [4, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of ζ(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's ηk-function cluster near the critical line.

Transfer Matrix Method for the Analysis of Multiple Natural Frequencies

Multiple natural frequencies may be encountered when analyzing the essential natural vibration of a symmetric mechanical system or sub-structure system or a system with special parameters. The transfer matrix method (TMM) is a useful tool for analyzing the natural vibration characteristics of mechanical or structural systems. It derives a nonlinear eigen-problem (NEP) in general, even a transcendental eigen-problem. This investigation addresses the NEP in TMM and proposes a novel method, called the determinant-differentiation-based method, for calculating multiple natural frequencies and determining their multiplicities. Firstly, the characteristic determinant is differentiated with respect to frequency, transforming the even multiple natural frequencies into the odd multiple zeros of the differentiation of the characteristic determinant. The odd multiple zeros of the first derivative of the characteristic determinant and the odd multiple natural frequencies can be obtained using the bisection method. Among the odd multiple zeros, the even multiple natural frequencies are picked out by the proposed judgment criteria. Then, the natural frequency multiplicities are determined by the higher-order derivatives of the characteristic determinant. Finally, several numerical simulations including the multiple natural frequencies show that the proposed method can effectively calculate the multiple natural frequencies and determine their multiplicities.

On the zeros of derivatives of Bessel functions

The positive zeros of J ν ‴ ( x ) and J ν ( n + 1 ) ( x ) are studied by using classical analysis and the properties of J ν ( x ) . It is proved that J ν ‴ ( x ) has a unique zero in specific intervals. Regarding J ν ( n + 1 ) ( x ) , it is proved that its positive zero j ν , m ( n + 1 ) is an increasing function with respect to ν, for ν > n . Moreover, the first two Rayleigh sums for j ν , m ( n ) are calculated. The obtained results extend and complement previously known results and also answer an open problem regarding the monotonicity of j ν , m ( n ) . As a consequence of these results, a lower bound for j ν , 1 ( n + 1 ) is deduced, as well as an inequality between j ν , 1 ( n + 1 ) and j ν , 1 ( n ) .

On a Blaschke-type condition for the zeros of derivatives of R. Nevanlinna class functions in the disk

A necessary and sufficient Blaschke-type condition is obtained for the zeros of derivatoves of Nevanlinna class functions.

Equidistribution of the Zeros of Higher Order Derivatives in Polynomial Dynamics

Equidistribution of the Zeros of Higher Order Derivatives in Polynomial Dynamics

Non-real Zeros of Derivatives

A number of results are proved concerning non-real zeros of derivatives of real meromorphic functions.

Zeros of derivatives of 𝐿-functions in the Selberg class on ℜ(𝑠)<1/2

In this article, we show that the Riemann hypothesis for an L L -function F F belonging to the Selberg class implies that all the derivatives of F F can have at most finitely many zeros on the left of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974 [Acta Math. 133 (1974), pp. 49–65].

Zeros of random polynomials and their higher derivatives

We consider zeros of higher derivatives of various random polynomials and show that their limiting empirical measures agree with those of roots of corresponding random polynomials. Examples of random polynomials include those whose roots are given by i.i.d. random variables and those whose zeros are nearly all deterministic except for a fixed number of random roots. As an application, we show that such a phenomenon holds for the random polynomials whose roots follow the distribution of the 2D Coulomb gas ensemble.

VALIDATED UV- SPECTROPHOTOMETRIC METHODS FOR THE SIMULTANEOUS ESTIMATION OF DEXTROMETHORPHAN AND CETRIZINE

Using UV Spectrophotometric methods, a simple, precise and accurate approach is devised and verified for the simultaneous assessment of Dextromethorphan (DEX) and Cetrizine (CET) in combined dosage form. The approach 1 is the Dual wavelength method, which was developed, evaluates the difference in the absorbance of mixtures at wavelengths where single drugs have the same absorbance and vice versa. Approach 2 is a mean-centered ratio method that relies on the production of calibration by dividing the spectra of one substance by the spectra of the other. Approach 3 is a derivative spectrum method that relies on the derivative spectrum's zero crossing points to create calibration for two medications in the presence of the second. Simultaneous estimation of DEX and CET using the Q-absorption method and the simultaneous equation method has already been published. The two techniques Simultaneous equation method (Approach 4) and Q-Absorption method (Approach 5) are repeated to preserve the same circumstances throughout the experiment. To assess the accuracy and precision of all five approaches, six replication tests and recovery investigations utilizing recognized synthetic blends are used. The calibrations are used to analyze two pharmaceuticals contained in the tablet. The procedures have been validated in accordance with ICH recommendations.

Devising an approach to the geometric modeling of railroad tracks along curvilinear sections

This paper proposes an approach to arranging curvilinear sections of the railroad track, which involves the replacement of two transitional ones and a circular curve with one curvilinear section. Modeling this section implies the construction of a curve in the plan and profile, the joint consideration of which forms the spatial appearance of the curve, which ensures a smooth transition between the rectilinear rails. The need for such transitions is due to the terrain, the need to bypass settlements, and the presence of geological, geographical, and other obstacles that occur when laying railroads. A curvilinear section is modeled in the plan using a curve that is represented in natural parameterization under the law of curvature distribution in the form of a fifth-power polynomial. At the same time, at the start and end points of the section, the curvature and its derivative accept zero values. The outer rail elevation retraction (modeling in the profile) is performed using the curve built, whose sections are also represented in natural parameterization with the dependences of the curvature distribution on the arc length in the form of fourth-power polynomials. At the docking point of the sections, the third order of smoothness is ensured, which implies the equality of values of the functions, their derivatives, the curvature, and a derivative of the curvature from the length of its natural arc. Measures have been proposed to ensure the predefined track gauge retraction. The application of the proposed approach to model a railroad track along the curvilinear section with a variable-radius curve could make it possible to achieve a favorable curvature distribution, a smooth elevation retraction of the outer rail and the track gauge. That would consequently improve the safety of rolling stock running along the curvilinear section of the track, reduce the lateral and vertical efforts that predetermine the wear of rails and wheelsets.

Location of zeros of the polar derivative of a polynomial

Location of zeros of the polar derivative of a polynomial

Simultaneous Estimation of Protein and Nucleic Acid Using Derivative Spectrophotometric Method

Purpose: Simple and precise first derivative zero crossing spectrophotometric method has been developed for the simultaneous estimation of protein and nucleic acid without prior separation. Approach: The concentration ranges for protein and nucleic acid were taken in the range of 60-140 mg mL-1 and 20-100 mg mL-1 respectively in 0.1 M NaOH. Absorption spectra of the samples were recorded between 200 nm to 400 nm against a reagent blank. Zero-order spectra of protein and nucleic acid were stored individually within the above concentration ranges and were derivatized in first order using scaling factor 5 for both the substances. First derivative amplitudes were recorded at 261 nm and 289 nm for estimation of protein and RNA respectively for pure substances and binary mixture. Finding: Limit of detection was found to be 2.90 mg mL-1 and 0.36 mg mL-1 and limit of quantitation was 8.80 mg mL-1 and 1.02 mg mL-1 respectively for protein and nucleic acid respectively. Precision was found to be 1.38 % for protein and 1.25 % for RNA. Reproducibility was found to be 1.54 % for protein and 1.69 % for RNA. Conclusions: Thus, proposed method can be adapted for simultaneous determination of protein and nucleic acid and better alternate technique for immunoassays and electrophoretic methods. Keywords: First derivative zero crossing spectrophotometry; Protein; Nucleic acid; Ribonucleic acid; Analytical method validation

On the zeros of polar derivative of polynomials with restricted coefficients

On the zeros of polar derivative of polynomials with restricted coefficients

On the zeros of the polar derivative of a polynomial

On the zeros of the polar derivative of a polynomial

Zeros of derivatives of Gaussian random polynomials on plane domains

In this paper, we study the expected density of zeros of derivatives of Gaussian random polynomials on plane domains of C. The Gaussian random polynomials are of the form RN=∑n=0Nanpn where an are i.i.d. complex Gaussian random variables and {pn} are orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain relative to a weight. The main result is that the expected density of zeros of any k-th derivatives of RN tends to the equilibrium measure of the domain.

On the Koebe Quarter Theorem for certain polynomials

We study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in {mathbb {T}}:={zin {mathbb {C}}:|z|=1}. We found minimal disc containing all images of {mathbb {D}}:={zin {mathbb {C}}: |z|<1} and maximal disc contained in all images of {mathbb {D}} through polynomials of degree 3 and 4. Moreover we determine the extremal functions for both problems.