Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

2009 ◽  
Vol 61 (5) ◽  
pp. 847-853
Author(s):  
A. I. Podvysotskaya
Keyword(s):  
Lower Bound ◽  
Bernstein Inequality ◽  
Algebraic Polynomials ◽  
First Derivative

scholarly journals Point-wise estimates for the derivative of algebraic polynomials

2021 ◽  
Vol 56 (2) ◽  
pp. 208-211
Author(s):  
A. V. Savchuk
Keyword(s):  
Algebraic Polynomial ◽  
Bernstein Inequality ◽  
Sufficient Condition ◽  
Algebraic Polynomials

We give a sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum\limits_{k=0}^{n}a_kz^k$, $a_n\not=0,$ such that the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ is true for all $z,\ |z|\le 1$.

On the Laplacian characteristic polynomials of mixed graphs

2015 ◽  
Vol 30 ◽  
Author(s):  
Dariush Kiani ◽  
Maryam Mirzakhah
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Characteristic Polynomials ◽  
Laplacian Spectral Radius ◽  
Mixed Graph ◽  
First Derivative ◽  
Signless Laplacian ◽  
Mixed Graphs

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial of a mixed graph is reconstructible from the collection of the Laplacian characteristic polynomials of its edge deleted subgraphs. Then, it is investigated how graph modifications affect the mixed Laplacian characteristic polynomial. Also, a connection between the Laplacian characteristic polynomial of a non-singular connected mixed graph and the signless Laplacian characteristic polynomial is provided, and it is used to establish a lower bound for the spectral radius of L(G). Finally, using Coates digraphs, the perturbation of the mixed Laplacian spectral radius under some graph transformations is discussed.

scholarly journals Strong result for real zeros of random algebraic polynomials

2001 ◽  
Vol 14 (4) ◽  
pp. 351-359 ◽  
Author(s):  
T. Uno
Keyword(s):  
Lower Bound ◽  
Random Variables ◽  
Strong Result ◽  
Algebraic Polynomials ◽  
Exceptional Set ◽  
Gaussian Random Variables ◽  
Real Zeros ◽  
Random Algebraic Polynomials

An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

Novel UV Spectrophotometric Method for the Quantitative Analysis of CefpodoximeProxetil in Pharmaceutical Formulations by First Derivative Technique

2017 ◽  
Vol 8 (03) ◽  
Author(s):  
Potdar S. S. ◽  
Karajgi S. R. ◽  
Simpi C. C. ◽  
Kalyane N. V.
Keyword(s):  
Quantitative Analysis ◽  
Spectrophotometric Method ◽  
Dosage Form ◽  
Pharmaceutical Formulations ◽  
First Derivative ◽  
Good Linearity ◽  
Beer's Law ◽  
Beer’S Law ◽  
Tablet Dosage ◽  
Ich Guideline

The spectrophotometric method for estimation of CefpodoximeProxetil employed first derivative amplitude UV spectrophotometric method for analysis using methanol as solvent for the drug. CefpodoximeProxetil has absorbance maxima at 235nm and obeys Beer’s law in concentration range 10-50µg/ml with good linearity i.e. r2 about 0.999. The recovery studies established accuracy of the proposed method; result validated according to ICH guideline. Results were found satisfactory and reproducible. The method was successfully for evaluation of CefpodoximeProxetil in tablet dosage form without interference of common excipients.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

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