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1

On the derivative of a polynomial

Jain Vinay Kumar

Journal of Mathematics and Applications

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2023

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Vol. 46

5--13

EN

For an arbitrary polynomial P(z), let M(P,r) =max|z|=r|P(z)| and m(P,r)=min|z|=r|P(z)|, (r >0). For a polynomial p(z)=Pn [wzór] of degreen, having all its zeros in|z|≤k, (k≥1), with a zero of orders, (s≥0), at 0 and F0, F1, F2, Gn-s, F3, F4, Hn-s, Fn-s, B0, B1, En-1, B2, B3, Dn-1 and Bn-1, as in Theorem, we have obtained a refinement [wzór] of our old result (1997), there by obtaining a new refinement of known results [wzór].

2

On rational functions related to algorithms for a computation of roots. I

Baran Mirosław

Science, Technology and Innovation

|

2019

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Vol. 7, no. 4

17--25

EN

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k – th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

3

On the maximum modulus of a polynomial

Jain Vinay Kumar

Journal of Mathematics and Applications

|

2019

|

Vol. 42

109--116

EN

For a polynomial p(z) of degree n, having no zeros in |z| < 1 Ankeny and Rivlin had shown that for R ≥ 1 [wzór]. Using Govil, Rahman and Schmeisser’s refinement of the generalization of Schwarz’s lemma we have obtained a refinement of Ankeny and Rivlin’s result. Our refinement is also a refinement of Dewan and Pukhta’s refinement of Ankeny and Rivlin’s result.

4

Number of zeros of a polynomial in a specific region with restricted coefficients

Mir Abdullah, Ahmad Abrar, Malik Adil Hussain

Journal of Mathematics and Applications

|

2019

|

Vol. 42

135--146

EN

This paper focuses on the problem concerning the location and the number of zeros of polynomials in a specific region when their coefficients are restricted with special conditions. We obtain extensions of some classical results concerning the number of zeros of polynomials in a prescribed region by imposing the restrictions on the moduli of the coefficients, the real parts(only) of the coefficients, and the real and imaginary parts of the coefficients.

5

Integration of polynomials over n-dimensional simplices

Rouigueb Abdenebi, Maiza Mohamed, Tkourt Abderahmane, Cherchour Imed

Annals of Computer Science and Information Systems

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2019

|

Vol. 18

157--163

EN

Integrating an arbitrary polynomial function f of degree D over a general simplex in dimension n is well-known in the state of the art to be NP-hard when D and n are allowed to vary, but it is time-polynomial when D or n are fixed. This paper presents an efficient algorithm to compute the exact value of this integral. The proposed algorithm has a time-polynomial complexity when D or n are fixed, and it requires a reasonable time when the values of D and n are less than 10 using widely available standard calculators such as desktops.

6

Accelerating multivariate cryptography with constructive affine stream transformations

Carenzo Michael, Polak Monika

Annals of Computer Science and Information Systems

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2019

|

Vol. 18

221--225

EN

On December 20th, 2016, the National Institute of Standards and Technology (NIST) formally initiated a competition to solicit, evaluate, and standardize one or more quantum-resistant cryptographic algorithms. Among the current candidates is a cryptographic primitive which has shown much promise in the post-quantum age, Multivariate Cryptography. These schemes compose two affine bijections S and T with a system of multivariate polynomials. However, this composition of S and T becomes costly as the data encrypted grows in size. Here we present Constructive Affine Stream (CAS) Transformations, a set of algorithms which enable specialized, large-scale, affine transformations in O(n) space and O(n log n) time, without compromising security. The goal of this paper is to address the practical problems related to affine transformations common among almost all multivariate cryptographic schemes.

7

Number of zeros of a polynomial (Lacunary-type) in a disk

Qasim I., Rasool T., Liman A.

Journal of Mathematics and Applications

|

2018

|

Vol. 41

181--194

EN

The problem of finding out the region which contains all or a prescribed number of zeros of a polynomial [wzór] has a long history and dates back to the earliest days when the geometrical representation of complex numbers was introduced. In this paper, we present certain results concerning the location of the zeros of Lacunary-type polynomials [wzór] in a disc centered at the origin.

8

Digraph-building method for finding a set of minimal realizations of 2-D dynamic systems

Markowski K. A., Hryniów K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2018

|

Vol. 66, nr 5

589--597

EN

This paper presents a digraph-building method designed to find the determination of realization of two-dimensional dynamic system. The main differences between the method proposed and other state-of-the-art solutions used include finding a set of realizations (belonging to a defined class) instead of only one realization, and the fact that obtained realizations have minimal size of state matrices. In the article, the proposed method is described, compared to state-of-the-art methods and illustrated with numerical examples. To the best of authors’ knowledge, the method shown in the paper is superior to all other state-of-the-art solutions both in terms of number of solutions and their matrix size. Additionally, MATLAB function for determination of realization based on the set of state matrices is included.

9

On the derivative of a polynomial with prescribed zeros

Jain V. K.

Journal of Mathematics and Applications

|

2017

|

Vol. 40

97--120

EN

For a polynomial [wzór] of degree n having all its zeros in │z│ ≤ K, K ≥1 it is known that max [wzór]. By assuming a possible zero of order m, 0 ≤ m ≤ n - 4, at z = 0, of p(z) for n ≥ k + m + 1 with integer k ≥ 3 we have obtained a new refinement of the known result.

10

Zastosowanie aproksymacji wielomianowej do wyznaczania załomków EKG

Maciejewski M.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

|

2017

|

T. 7, nr 4

92--95

PL

W celu wdrożenia elementów systemu telemedycznego związanego z diagnostyką [3], konieczne jest wcześniejsze zweryfikowanie wartości diagnostycznej algorytmów decyzyjnych odpowiedzialnych za wykrywanie stanów zagrożenia życia. Analiza przebiegu EKG jest w stanie dać informację o wielu takich stanach związanych z zaburzeniami układu krążenia [8]. W tym celu konieczne jest podjęcie szeregu działań prowadzących do budowy odpowiednich modeli. Pierwszym krokiem jest filtracja i przygotowanie danych [1], następnie ekstrakcja parametrów z przebiegów EKG, analiza wyników, porównanie ich z posiadanymi modelami oraz postawienie diagnozy. Każdy z tych kroków wymaga zastosowania odpowiedniego podejścia w celu zminimalizowania popełnianego błędu [4], wynikającego z niekiedy znacznie zniekształconego sygnału [7]. W celu ekstrakcji parametrów czasowych z odfiltrowanego i przygotowanego sygnału EKG konieczne jest najpierw wykrycie załomka R w zespole QRS [6], następnie wyznaczenie załomków P, Q, S, T i znalezienie ich początku i końca oraz określenie interesujących nas interwałów [2]. Zaproponowana tutaj metoda bazuje na aproksymacji przebiegu w oknie czasowym zawierającym dany załomek wielomianem określonego rzędu. Takie podejście pozwala następnie na wyznaczenie punktów przegięcia i, co za tym idzie, granic załomka. Metoda została zastosowana do przetwarzania przebiegów zarejestrowanych w warunkach laboratoryjnych w spoczynku i w trakcie kontrolowanego wysiłku, wyniki zostały porównane i przedstawione w niniejszej pracy.

EN

To succesfully implement a telemedical system for diagnostic purposes it is necessary to verify the diagnostic value of the decision algorithms used to detect life threatening situations. ECG analysis is a useful tool for obtaining information about the overall patient condition, especially for the circulatory system. Proper recognition cannot be performed without creation of proper models, The first step is signal filtration and data preparation, followed by parameter extraction, comparison with the model and diagnosis presentation. Each of these steps reqires a certain approach to minimize the error. Proper filtration needs to be performed. Then, the QRS complex is detected and rythm is calculated. Afterwards, the remaining waves are detected. To be able to perform valuable time dependencies it is necessary to exactly mark the beginnings and ends of intervals. The proposed method is based on opproximating the signal around the wave with a polynomial of a certain degree. This allows detection of inflection points corresponding to the borders of the wave. The method was applied to a set of ECG signals recorced during rest and activity, the results are presented and discussed.

11

A refinement of Schwarz's lemma and its applications

Jain V. K.

Journal of Mathematics and Applications

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2016

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Vol. 39

69--80

EN

By using the value of the second derivative of the function at 0, along with the values of the function and its first derivative at 0, we have obtained a refinement of well known Schwarz’s lemma and have used this refinement to obtain refinements, of Aziz and Rather’s inequalities [2004] for a polynomial of degree n having no zeros in |z| < k, (k ≥ 1).

12

The average time complexity of probabilistic algorithms for finding generators in finite cyclic groups

Adamski T., Nowakowski W.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2015

|

Vol. 63, nr 4

989--996

EN

Generators of finite cyclic groups play important role in many cryptographic algorithms like public key ciphers, digital signatures, entity identification and key agreement algorithms. The above kinds of cryptographic algorithms are crucial for all secure communication in computer networks and secure information processing (in particular in mobile services, banking and electronic administration). In the paper, proofs of correctness of two probabilistic algorithms (for finding generators of finite cyclic groups and primitive roots) are given along with assessment of their average time computational complexity.

13

Optimisation of polynomial railway transition curves of even degrees

Zboiński K., Woźnica P.

Archives of Transport

|

2015

|

Vol. 35, iss. 3

71--86

EN

This paper represents new results obtained by its authors while searching for the proper shape of polynomial railway transition curves (TCs). The search for the proper shape means the evaluation of the curve properties based on chosen dynamical quantities and generation of such shape with use of mathematically understood optimisation methods. The studies presented now and in the past always had got a character of the numerical tests. For needs of this work advanced vehicle model, dynamical track-vehicle and vehicle-passenger interactions, and optimisation methods were exploited. In this software complete rail vehicle model of 2-axle freight car, the track discrete model, and non-linear description on wheel-rail contact are used. That part of the software, being vehicle simulation software, is combined with library optimisation procedures into the final computer programme. The main difference between this and previous papers by the authors are the degrees of examinated polynomials. Previously they tested polynomial curves of odd degrees, now they focus on TCs of 6th, 8th and 10th degrees with and without curvature and superelevation ramp tangence in the TC’s terminal points. Possibility to take account of fundamental demands (corresponding values of curvature in terminal points) concerning TC should be preserved. Results of optimisation are compared both among themselves and with 3rd degree parabola. The aim of present article is to find the polynomial TCs’ optimum shapes which are determined by the possible polynomial configurations. Only one dynamical quantities being the results of simulation of railway vehicle advanced model is exploited in the determination of quality function (QF1). This is: minimum of integral of vehicle body lateral acceleration.

14

Kryptograficzna ochrona danych na podstawie minimalnie nadmiernych wielomianowo-skalarnych modularnych systemów liczbowych

Selianinau M.

Prace Naukowe Akademii im. Jana Długosza w Częstochowie. Technika, Informatyka, Inżynieria Bezpieczeństwa

|

2014

|

T. 2

371-382

PL

Obecnie jakość realizacji procedur przetwarzania informacji zależy w dużym stopniu od wybranego modelu matematycznego organizacji tego procesu i zrealizowanej na jego podstawie technologii informacyjnej. Istnieje szereg naukowych i praktycznych rodzajów działalności, gdzie występuje potrzeba przetwarzania informacji przedstawionej w formie wielomianów. Działania na wielomianach odgrywają istotną rolę we współczesnej algebrze komputerowej, cyfrowym przetwarzaniu sygnałów, teorii kodowania, kryptografii itd. W związku z tym duże zainteresowanie z znajduje modularna technika równoległych struktur obliczeniowych zdefiniowanych w zakresach wielomianów. W artykule opisano algorytm kryptograficzny, który pozwala wykonywać operacje blokowego szyfrowania informacji przy użyciu minimalnie nadmiernych wielomianowo skalarnych modularnych systemów liczbowych. W takich systemach na górnym poziomie jako podstawy wybierane są nierozkładalne wielomiany, a na dolnym m poziomie wykorzystuje się minimalnie nadmierne modularne kodowanie elementów w z zakresu skalarów. W tym przypadku wielomian z pozycyjnym przedstawieniem współczynników może być jednoznacznie definiowany jako zbiór reszt według wybranych modułów systemu liczbowego. Użycie minimalnie nadmiernego kodowania pozwala istotnie zwiększyć efektywność metod i algorytmów kryptograficznych kosztem optymalizacji procedur niemodułowych.

EN

At the present time, quality of the execution of information processing procedures is largely determined by the selected mathematical model of the organization of information processing and the information technology implemented on this basis. There are many scientific and applied researches which demand processing the information presented in the form of polynomials. Operations over the polynomials are very important in modern computer algebra, digital signal processing, coding theory, cryptography, etc. At the same time, modular technology of parallel computing structures defined on polynomial ranges is of great interest. This article describes the cryptographic algorithm that allows us to perform the block encryption of information using minimal redundant polynomial-scalar modular number systems. In these systems, at the upper level the normalized polynomials of the first degree are used as a basis, whereas at the lower level the elements of scalar range are represented in minimal redundant modular code. In this case a polynomial with the positional representation of the coefficients can be uniquely defined as a sequence of residues with respect to selected bases. The efficiency of the cryptographic methods and algorithms is significantly increased due to the optimization of the non-modular procedures when using the minimal redundant coding.

15

Some inequalities for the polar derivative of a polynomial with restricted zeros

Zireh A., Hosseini S. A.

Journal of Mathematics and Applications

|

2014

|

Vol. 37

119--129

EN

Let p(z) be a polynomial of degree n and for any complex number (alfa), D(alfa)p(z) = np(z)+(alfa-z)p′(z) denote the polar derivative of the polynomial p(z) with respect to. In this paper, we obtain new results concerning maximum modulus of the polar derivative of a polynomial with restricted zeros. Our result generalize certain well-known polynomial inequalities.

16

On the zeros of an analytic function

Jain V.

Journal of Mathematics and Applications

|

2014

|

Vol. 37

51--58

17

Wybrane aspekty metodyczne wykorzystania aparatu Proctora w badaniach zagęszczalności gleb

Błażejczak D., Dawidowski J. B., Malinowski M., Śnieg K.

Inżynieria Rolnicza

|

2013

|

R. 17, nr 4, t. 1

27--34

PL

Celem pracy było poszukiwanie funkcji opisującej przebieg zależności gęstości objętościowej szkieletu gleby od jej wilgotności na podstawie aproksymacji punktów Proctora. Znajomość tej zależności umożliwia przewidywanie skutków oddziaływania na glebę kół pojazdów rolniczych. Poszukiwania ograniczono do czterech wielomianów – od drugiego do piątego stopnia. Materiał badawczy pochodził z warstwy podornej pięciu profili glebowych. W materiale oznaczono: skład granulometryczny, gęstość fazy stałej, zawartość próchnicy, odczyn gleby oraz granice plastyczności i płynności gleby. Zagęszczanie próbek metodą Proctora wykonano zgodnie z PN-88/B-04481. Wartości gęstości maksymalnej i wilgotności optymalnej obliczano w miejscu, gdzie uzyskane wielomiany opisujące dane doświadczalne przyjmowały wartość maksymalną. Stwierdzono, że do wyznaczania wartości wilgotności optymalnej i gęstości maksymalnej, najbardziej nadaje się wielomian stopnia czwartego.

EN

The objective of the paper was to search for a function which describes relations of the bulk intensity of the soil skeleton to its moisture based on approximation of the Proctor points. Knowledge on this relation enables prediction of the influence of agricultural vehicles wheels on soil. Searching was limited to four polynomials – from the second to the fifth degree. Material for research originated from the subsoil of five soil profiles. The following were determined in material: grain size distribution, density of a solid phase, humus content, soil reaction and limits of elasticity and soil liquidity. Concentration of samples with the Proctor method was carried out according to PN-88/B-04481. Values of the maximum density and optimal moisture were calculated in the place, where the obtained polynomials describing experimental data accepted the maximum value. It was found out that for determination of the value of optimal moisture and the maximum density, polynomial of the fourth degree is the most suitable.

18

The Darboux property for polynomials in Golomb's and Kirch's topologies

Szczuka P.

Demonstratio Mathematica

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2013

|

Vol. 46, nr 2

429--435

EN

In this paper, we present conditions which are equivalent to the Darboux property for non-constant polynomials in Golomb's and Kirch's topologies on the set of positive integers.

19

Application of the polynomial chaos approximation to a stochastic parametric vibrations problem

Brząkała A.

Archives of Civil and Mechanical Engineering

|

2013

|

Vol. 13, no. 2

220--228

EN

In the paper the application of the polynomial chaos expansion in case of parametric vibrations problem is presented. Hitherto this innovative approach has not been applied to such a stochastic problem. The phenomenon is described by a nonlinear ordinary differential equation with periodic coefficients. It can be observed among others in cable-stayed bridges due to periodic excitation caused by a deck or a pylon. The analysis is focused on a real situation for which the problem of parametric resonance was observed (a cable of the Ben–Ahin bridge). The characteristic of the viscous damper is considered as a log-normal random variable. The results obtained by the use of the polynomial chaos approximations are compared with the ones based on the Monte Carlo simulation. The convergence of both methods is discussed. It is found that the polynomial chaos yields a better convergence then the Monte Carlo simulation, if resonant vibrations appear.

20

Location of zeros of polynomials

Zireh A.

Journal of Applied Analysis

|

2012

|

Vol. 18, nr 1

159-166

EN

In this paper, we obtain a result concerning the location of zeros of a polynomial p(z)= αo+a1z+···+αnzn, where αi are complex coefficients and z is a complex variable. We obtain a ring shaped region containing all the zeros of a polynomial involving binomial coefficients and t,z-Fibonacci numbers. This result generalizes some well-known inequalities.