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Упит dcterms_subject_txt:unimodular roots

30 items

  • Approximation of the number of roots that do not lie on the unit circle of a self-reciprocal polynomial

    Dragan Stankov (2024)
    We introduce the ratio of the number of roots not equal to 1 in modulus of a reciprocal polynomial Rd(x) to its degree d. For some sequences of reciprocal polynomials we show that the ratio has a limit L when d tends to infinity. Each of these sequences is defined using a two variable polynomial P(x,y) so that Rd(x) = P(x,xn). For P(x,y) we present the theorem for the limit ratio which is analogous to the Boyd-Lawton limit formula ...
    reciprocal polynomial, envelope, unimodular roots

  • Mean-based iterative methods for finding multiple roots in nonlinear chemistry problems

    Dejan Ćebić, Nebojša M. Ralević (2021)
    нелинеарне једначине, поступци засновани на срединама, вишеструки корени,
    ... fourth order only for simple roots, while they lose the optimal properties when roots are multiple. In order to overcome this drawback, we construct a generalized version of Cordero et al. mean-based methods, capable to efficiently find multiple roots as well as sim- ple roots. Firstly, in the following ...
    ... s with multiple roots from [9] are used to check the dynamical behavior of the methods, 1. p 1 (z) = (z2 − 1)5 with roots ±1 of multiplicity m = 5, 2. p 2 (z) = (x3 + 4x 2 − 10)3 with roots −2.6826150 ± 0.3582593i and 1.365230,  where m = 3, 3. p 3 (z) = (z3 − z)4 with roots ±1 and 0,  where ...
    ... Mean-based iterative methods for finding multiple roots in nonlinear chemistry problems Dejan Ćebić, Nebojša M. Ralević Дигитални репозиторијум Рударско-геолошког факултета Универзитета у Београду [ДР РГФ] Mean-based iterative methods for finding multiple roots in nonlinear chemistry problems | Dejan Ćebić ...

  • The number of nonunimodular roots of a reciprocal polynomial

    Dragan Stankov (2023)
    We introduce a sequence Pd of monic reciprocal polynomials with integer coefficients having the central coefficients fixed as well as the peripheral coefficients. We prove that the ratio of the number of nonunimodular roots of Pd to its degree d has a limit L when d tends to infinity. We show that if the coefficients of a polynomial can be arbitrarily large in modulus then L can be arbitrarily close to 0. It seems reasonable to believe that if ...
    Algebraic integer, the house of algebraic integer, maximal modulus, reciprocal polynomial, primitive polynomial, Schinzel-Zassenhaus conjecture, Mahler measure, method of least squares, cyclotomic polynomials

  • Roots of trinomials of bounded height

    Stankov Dragan (2014)

  • On the optimality of some multi-point methods for finding multiple roots of nonlinear equation

    Ralević Nebojša, Ćebić Dejan (2016)

  • The third order mean-based Jarratt-type method for finding simple roots of nonlinear equation

    Ralević Nebojša, Ćebić Dejan, Pavkov Ivan (2015)

  • A variant of McDougall-Wotherspoon method for finding simple roots of nonlinear equations

    Glišović Nataša, Ralević Nebojša, Ćebić Dejan (2018)

  • A necessary and sufficient condition for an algebraic integer to be a Salem number

    Dragan Stankov (2019)
    We present a necessary and sufficient condition for a root greater than unity of a monic reciprocal polynomial of an even degree at least four, with integer coefficients, to be a Salem number. This condition requires that the minimal polynomial of some power of the algebraic integer has a linear coefficient that is relatively large. We also determine the probability that an arbitrary power of a Salem number, of certain small degrees, satisfies this condition.
    Algebraic integer, the house of algebraic integer, maximal modulus, reciprocal polynomial, primitive polynomial, Schinzel-Zassenhaus conjecture, Mahler measure, method of least squares, cyclotomic polynomials
    ... set of roots of these polynomials is also finite, and all the powers α, α2, α3, . . . of a root α of P (x) are in this set. Therefore for some p, q, αp = αq, p 6= q. Since α 6= 0 it follows that αp−q = 1. Vice versa, if P (x) is the product of cyclotomic polynomials then all its roots are roots of 1 so ...
    ... fixed number of roots on the complex unit circle U = {z ∈ C : |z| = 1}. Let p(z) = adzd +ad−1zd−1 + · · ·+a1z +a0 be a d-th degree self-reciprocal polynomial. If the inequality (1.1) |ad−l| > 1 2 ( d d − 2l ) d ∑ k=0,k 6=l,d−l |ak|, l < d/2 holds, then p(z) has exactly d − 2l roots on U and these ...
    ... that there is n such that each of d − 2 unimodal roots of Pn(x) could be arbitrarily close to exactly one root of xd−2 + 1 (see [10, Lemma 2]) and to show that then the coefficients of Pn(x) will satisfy the condition (1.2). It is obvious that roots of xd−2 + 1 are exp(±π+2jπ d−2 i), j = 0, 1, . . ...

  • The number of unimodular roots of some reciprocal polynomials

    Dragan Stankov (2020)
    We introduce a sequence P2n of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of P2n and its degree d has a limit when d tends to infinity. We present an algorithm for calculation the limit and a numerical method for its approximation. If P2n is the sum of a fixed number of monomials we determine the central coefficients such that the ratio has the minimal limit. ...
    Algebraic integer, the house of algebraic integer, maximal modulus, reciprocal polynomial, primitive polynomial, Schinzel-Zassenhaus conjecture, Mahler measure, method of least squares, cyclotomic polynomials
    ... Измењено: 2024-12-25 10:25:20 The number of unimodular roots of some reciprocal polynomials Dragan Stankov Дигитални репозиторијум Рударско-геолошког факултета Универзитета у Београду [ДР РГФ] The number of unimodular roots of some reciprocal polynomials | Dragan Stankov | Cmptes rendus ...
    ... corresponds to an unimodular root of x120 +x63 +x61 +x59 +x57 +1. Figure 3. Unit circle sectors where unimodular roots of x120 + x63 + x61 + x59 + x57 +1 are located. Each intersection point of graph of f1 = cos60t and f2 = −cos t − cos2t (see Figure 2) corre- sponds to an unimodular root of the reciprocal ...
    ... 3.1. Small limit points of C (P2n) In the case of trinomials i.e. if k = 0, |a0| ≤ 2 then all roots of P2n(x) = x2n +a0xn +1 obviously are unimodular. If |a0| > 2 then P2n does not have any unimodular root so that C (P ) tends either to zero or to one as n approaches infinity. In the case of qu ...

  • An alternative to Mahler Measure of polynomials

    Dragan Stankov (2024)
    We introduce the ratio of the number of roots of a polynomial Pd, greater than one in modulus, to its degree d as an alternative to Mahler measure. We investigate some properties of the limit ratio. We generalise this definition for a two variable polynomial P(x,y) using the Cauchy’s argument principle. We present an algorithm for calculating the limit ratio and a numerical method for its approximation. We estimated the limit ratio for some families of polynomials. Some examples ...
    mahler measure, argument principle, limit ratio

  • The alternative to Mahler measure of polynomials in several variables

    Dragan Stankov (2024)
    We introduce the ratio of the number of roots of a polynomial Pd, greater than one in modulus, to its degree d as an alternative to Mahler measure. We investigate some properties of the alternative. We generalise this definition for a polynomial in several variables using Cauchy’s argument principle. If a polynomial in two variables do not vanish on the torus we prove the theorem for the alternative which is analogous to the Boyd-Lawton limit formula for Mahler measure. ...
    mahler measure, argument principle, Boyd-Lawton limit formula

  • The external aggregation Newton's method for solving nonlinear equations and applications

    Marija Paunović, Dejan Ćebić, Nebojša Ralević (2020)
    Њутнов метод, агрегациона функција, ред конвергенције
    ... modification of Newton’s method based on aggregation function applied on finding multiple roots of nonlinear equations, and numerically verified theoretical results on the examples with simple and multiple roots. Numerical analysis of the proposed approach with obtained results and the related discussion ...
    ... properties of the proposed method are discussed and it is shown that the order of convergence for multiple roots is one. It is proved that the iterative sequence defined by this method converges to roots, for any initial iteration sufficiently close to root. The rest of the paper is as follows. In Section ...
    ... parallel plates (see Maroju et al. (2017)). Here we are interested in finding simple roots 4.965114..., 0.541919808... and −0.3090932... respectively. In the fourth and the fifth example, we are searching for multiple roots −2.85 and 2 respectively. The fourth test function arised from the beam positioning ...

  • An Efficient Class of Iterative Root-Finding Methods with Applications

    Dejan Ćebić, Nebojša Ralević (2021)
    ... N Determining the roots of the nonlinear equation f(x) = 0 is an important task in many applications and engineering problems. In practice, very often nonlinear equations cannot be solved analytically and therefore iterative methods are being used to find the approximate roots. Probably the most ...
    ... University of Novi Sad Novi Sad, Serbia nralevic@uns.ac.rs Abstract—This paper presents a new class of four-point itera- tive methods for finding simple roots of nonlinear equations. The class is constructed by adding the simple structured fourth step to the previously developed three-point eighth-order methods ...
    ... (B is the Boltzmann constant) in order to find wavelength δ, one has to find the maximum energy density of Ω(δ). The maximum is provided by the roots of function f(x5) = x 5 − 1 + e−x. The desired root is α ≈ 4.9651... and the initial value is x0 = 3. Tables I-V display the numerical calculations ...

  • The Reciprocal Algebraic Integers Having Small House

    Dragan Stankov (2021)
    Algebraic integer, the house of algebraic integer, maximal modulus, reciprocal polynomial, primitive polynomial, Schinzel-Zassenhaus conjecture, Mahler measure, method of least squares, cyclotomic polynomials
    ... minimum of α over α of degree d which are not roots of unity. Let an α attaining m(d) be called extremal. We say that α is reciprocal if α−1 is a conjugate of α, i.e., XdP(1/X) = P(X). Let mr(d) denote the minimum of α over reciprocal α of degree d which are not roots of unity. Let an α attaining mr(d) be called ...
    ... of [a over a of degree d which are not roots of unity. Let an @ attaining m(d) be called extremal. We say that « is reciprocal if a~! is a conjugate of q, i.e., X@P(1/X) = P(X). Let mr(d) denote the minimum of [@ over reciprocal aw of degree d which are not roots of unity. Let an @ attaining mr(d) be ...
    ... easy to verify that P(xk) = k √ P(x) . (2.1) Let mrp(d) denote the minimum of α over reciprocal algebraic integer α of even degree d which are not roots of unity and which have a primitive minimal polynomial. Let mrp(d) is attained for αd with minimal reciprocal primitive polynomial Rd(x). Let αd be ...

  • The Newton method for solving nonlinear equations based on aggregation operators

    Nebojša M. Ralević, Dejan Ćebić (2019)
    ... method. The convergence properties of the proposed method are discussed and it is shown that the order of convergence for simple roots is three. by this method converges to roots, for any initial iteration sufficiently close to root. Theoretical results are verified on some relevant nonlinear equations ...
    ... (3) where x0 is an initial approximation sufficiently close to α. The convergence order of the classical Newton’s method is quadratical for simple roots with the error equation en+1 = f ′′(α) 2 f ′(α) · e2 n +O(e3 n). Let α be a root of the function f and suppose that sequence {xn} converges to α ...
    ... logic, theory and applications, Prentice Hall, New Jersey (1995) [8] T. Lukić, N. M. Ralević, Geometric Mean Newton’s Method for Simple and Multiple Roots, Applied Mathematics Letters 21 (2008), 30–36. [9] T. Lukić, N. M. Ralević, A. Lukity, Application of Aggregation Operators in Solution of Nonlinear ...

  • An optimal sixteenth order family of methods for solving nonlinear equations and their basins of attraction

    Dejan Ćebić, Nebojša Ralević, Marina Marčeta (2020)
    нелинеарна једначина, 16.ред конвергенције, оптималне методе, подељене разлике
    ... test examples have been employed to analyze the dynamical behavior: • p1(z) = z2 + 1 with roots ±i, [29]; • p2(z) = z5 + z with roots 0,±0.70710678± 0.70710678i, [32]; • p3(z) = (ez+1 − 1)(z − 1) with roots ±1, [7]; (a) MKT Method (b) MSGG Method (c) MSSSL Method (d) MMBMMethod (e) MBAMMMethod ...
    ... Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots, Math. Comput. Simul. 109(2015), 74–91. [7] C.Chun, B.Neta, Comparative study of eighth-order methods for finding simple roots of nonlinear equations, Numer. Algorithms 74(2017), 1169–1201. An optimal sixteenth ...
    ... 000 Novi Sad, Serbia Received March 23, 2020; accepted August 24, 2020 Abstract. We propose a new family of iterative methods for finding simple roots of non- linear equations. The proposed method is the four-point method with convergence order 16, which consists of four steps: the Newton step, an ...

  • Pleistocene rhinoceros from Bogovina Cave: the first report of Stephanorhinus hundsheimensis Toula, 1902 (Mammalia, Rhinocerotidae) from Serbia

    Predrag Radović, Miloš Radonjić, Emmanuel Billia (2020)
    Finds of Pleistocene rhinoceros are rare in Serbia, and only one species (the woolly rhinoceros Coelodonta antiquitatis Blumenbach, 1799) has been reported so far. The current paper presents the dental material of an extinct so-called Hundsheim rhinoceros, Stephanorhinus hundsheimensis Toula, 1902 from Bogovina Cave (East ern Serbia). Both the morphological and metric characteristics of the teeth are consis tent with the attribution to S. hundsheimensis. Unfortunately, the rhinoceros material originated from an uncertain geological context, so there is no ...
    rhinoceros, Stephanorhinus hundsheimensis, Serbia

  • Multiword Expressions between the Corpus and the Lexicon: Universality, Idiosyncrasy and the Lexicon-Corpus Interface

    Verginica Barbu Mititelu, Voula Giouli, Kilian Evang, Daniel Zeman, Petya Osenova, Carole Tiberius, Simon Krek, Stella Markantonatou, Ivelina Stoyanova, Ranka Stankovic, Christian Chiarcos (2024)
    Predstavljamo trenutne aktivnosti na definisanju interfejsa leksikona i korpusa koji će služiti kao referenca u prikazu polileksemskih jedinica - višečlanih izraza - (različitih tipova - imenskih, glagolskih, itd.) u specijalizovanim leksikonima i povezivanju ovih unosa sa njihovim pojavljivanjima u korpusima. Konačni cilj je korišćenje ovakvih resursa za automatsko identifikovanje višečlanih izraza u tekstu. Uključivanje nekoliko prirodnih jezika ima za cilj univerzalnost rešenja koje nije usredsređeno na određeni jezik, kao i prilagođavanje idiosinkrazijama. Raspravljaju se izazovi u leksikografskom opisu višerečnih ...
    multiword expression lexicon, corpus, proof-of-concept lexicon encoding

  • Mineral Resources of Serbia: Environmental, Societal and Economic Challenges

    Vladimir Simić, Rade Jelenković, Dragana Životić (2019)

  • The Use of the Omeka Semantic Platform for the Development of the University of Belgrade, Faculty of Mining and Geology Digital Repository

    Petar Popović, Mihailo Škorić, Biljana Rujević (2021)
    Under the regulations of the Ministry of Education, Science and technological Development, a digital repository based on the Omeka S data storage platform has been developed for the Faculty of Mining and Geology. The platform has been upgraded with the required modular extensions, Solr index and automatic OCR. Furthermore, document indexing and search have been fine-tuned with the aid of e-dictionaries of the Serbian language, which has brought about outstanding results in terms of usage facilitation and overall ...
    Дигитални репозиторијум, Омека, претрега дигиталних библиотека