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Journal articles on the topic 'M-Polynomial'

Author: Grafiati
Published: 3 June 2025
Last updated: 31 July 2025

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1

Blel, Mongi. "On some m-symmetric generalized hypergeometric d-orthogonal polynomials." Filomat 38, no. 4 (2024): 1279–89. http://dx.doi.org/10.2298/fil2404279b.

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In [9] I. Lamiri and M. Ouni state some characterization theorems for d-orthogonal polynomials of Hermite, Gould-Hopper and Charlier type polynomials. In [3] Y. Ben Cheikh I. Lamiri and M.Ouni give a characterization theorem for some classes of generalized hypergeometric polynomials containing for example, Gegenbauer polynomials, Gould-Hopper polynomials, Humbert polynomials, a generalization of Laguerre polynomials and a generalization of Charlier polynomials. In this work, we introduce a new class D of generalized hypergeometric m-symmetric polynomial sequence containing the Hermite polynomi
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2

Chami, Peter S., Bernd Sing, and Norris Sookoo. "Generalizing Krawtchouk Polynomials Using Hadamard Matrices." ISRN Applied Mathematics 2014 (March 4, 2014): 1–8. http://dx.doi.org/10.1155/2014/498135.

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We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged in a square matrix; in particular, the case where this matrix is a Hadamard matrix is considered. Orthogonality relations and recurrence relations are established, and coefficients for the expansion of any polynomial in terms of m-polynomials are obtained. We conclude this paper by an implementation of m-polynomials and some of the results obtained for them in Mathematica.
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3

Muhammad Hussian, Hasan Mahmood, Muhammad Abdullah, Sikandar Raza,. "Computing Topological Indices for Niobium Dioxide and Metal-Organic Frameworks via M-Polynomials." Power System Technology 48, no. 1 (2024): 211–27. http://dx.doi.org/10.52783/pst.268.

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This paper investigates the topological properties of metal-organic frameworks and niobium dioxide using M-polynomials. The stability and wide range of bonding that the molecule exhibits make it a promising candidate for a number of uses, such as energy storage, gas detection, and catalysis. We use M-polynomials to calculate several degree-based topological indices for metal-organic frameworks and niobium dioxide. The M-polynomial is one such fundamental polynomial that provides a way to derive a multitude of degree-based topological indices. These indices are crucial for research in chemistry
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4

Mufti, Zeeshan Saleem, Ali H. Tedjani, Rukhshanda Anjum, and Turki Alsuraiheed. "Entropy and Multi-Fractal Analysis in Complex Fractal Systems Using Graph Theory." Axioms 12, no. 12 (2023): 1126. http://dx.doi.org/10.3390/axioms12121126.

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In 1997, Sierpinski graphs, S(n,k), were obtained by Klavzar and Milutinovic. The graph S(1,k) represents the complete graph Kk and S(n,3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by Sie(Λ,t), already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidde
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5

QIN, XIAOER, and SHAOFANG HONG. "CONSTRUCTING PERMUTATION POLYNOMIALS OVER FINITE FIELDS." Bulletin of the Australian Mathematical Society 89, no. 3 (2013): 420–30. http://dx.doi.org/10.1017/s0004972713000646.

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AbstractIn this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form ${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$ over ${\mathbf{F} }_{{q}^{m} } $, where ${L}_{i} (x)$ and $B(x)$ are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms $xh({\lambda }_{j} (x))$ and $xh({\mu }_{j} (x))$, where ${\lambda }_{j} (x)$ i
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6

BERGELSON, VITALY, and DONALD ROBERTSON. "Polynomial multiple recurrence over rings of integers." Ergodic Theory and Dynamical Systems 36, no. 5 (2015): 1354–78. http://dx.doi.org/10.1017/etds.2014.138.

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We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^{m}$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne [Intersective polynomials and the polynomial Szemerédi theorem. Adv. Math.219(1) (2008), 369–388] on polynomials over the integers.
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7

Hegland, Markus. "Numerical methods for computing the greatest common divisor of univariate polynomials using floating point arithmetic." ANZIAM Journal 60 (August 15, 2019): C127—C139. http://dx.doi.org/10.21914/anziamj.v60i0.14059.

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Computing the greatest common divisor (GCD) for two polynomials in floating point arithmetic is computationally challenging and even standard library software might return the result GCD=1 even when the polynomials have a nontrivial GCD. Here we review Euclid's algorithm and test a variant for a class of random polynomials. We find that our variant of Euclid's method often produces an acceptable result. However, close monitoring of the norm of the vector of coefficients of the intermediate polynomials is required.
 
 References R. M. Corless, P. M. Gianni, B. M. Trager, and S. M. Wat
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8

Heintze, Sebastian. "Pillai's conjecture for polynomials." Glasnik Matematicki 58, no. 1 (2023): 67–74. http://dx.doi.org/10.3336/gm.58.1.05.

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In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation -17ex p^n - q^m = f. We prove that for any non-constant polynomial \( f \) there are only finitely many quadruples \( (n,m,\deg p,\deg q) \) consisting of integers \( n,m \geq 2 \) and non-constant polynomials \( p,q \) such that Pillai's equation holds. Moreover, we will give some examples that there can still be infinitely many possibilities for the polynomials \( p,q \).
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9

Stankov, Dragan. "On linear combinations of Chebyshev polynomials." Publications de l'Institut Math?matique (Belgrade) 97, no. 111 (2015): 57–67. http://dx.doi.org/10.2298/pim150220001s.

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We investigate an infinite sequence of polynomials of the form: a0Tn(x) + a1Tn?1(x) + ... + amTn?m(x) where (a0, a1,..., am) is a fixed m-tuple of real numbers, a0, am ? 0, Ti(x) are Chebyshev polynomials of the first kind, n = m, m+ 1, m+ 2,.... Here we analyze the structure of the set of zeros of such polynomial, depending on A and its limit points when n tends to infinity. Also the expression of envelope of the polynomial is given. An application in number theory, more precise, in the theory of Pisot and Salem numbers, is presented.
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10

Almira, J. M. "Aichinger equation on commutative semigroups." Filomat 37, no. 16 (2023): 5301–11. http://dx.doi.org/10.2298/fil2316301a.

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We consider Aichinger?s equation f (x1 +... + xm+1) = Xm+1 i=1 1i(x1, x2, ...,bxi,..., xm+1) for functions defined on commutative semigroups which take values on commutative groups. The solutions of this equation are, under very mild hypotheses, generalized polynomials. We use the canonical form of generalized polynomials to prove that compositions and products of generalized polynomials are again generalized polynomials and that the bounds for the degrees are, in this new context, the natural ones. In some cases, we also show that a polynomial function defined on a semigroup can uniquely be e
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11

Yasin H, Mohammed, M. Suresh, Zebene G. Tefera, and Samuel A. Fufa. "M-Polynomial and NM-Polynomial Methods for Topological Indices of Polymers." International Journal of Mathematics and Mathematical Sciences 2024 (February 2, 2024): 1–16. http://dx.doi.org/10.1155/2024/1084450.

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Topological indices (TIs) are numerical tools widely applied in chemometrics, biomedicine, and bioinformatics for predicting diverse physicochemical attributes and biological activities within molecular structures. Despite their significance, the challenges in deriving TIs necessitate novel approaches. This study addresses the limitations of conventional methods in dealing with dynamic molecular structures, focusing on the neighborhood M-polynomial (NM-polynomial), a pivotal polynomial for calculating degree-based TIs. Current literature acknowledges these polynomials but overlooks their limit
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12

FAN, HONG-YI, LI-YUN HU, and XUE-XIANG XU. "LEGENDRE POLYNOMIALS AS THE NORMALIZATION OF PHOTON-SUBTRACTED SQUEEZED STATES." Modern Physics Letters A 24, no. 20 (2009): 1597–603. http://dx.doi.org/10.1142/s021773230902996x.

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By converting the photon-subtracted squeezed state (PSSS) to a squeezed Hermite-polynomial excitation state we find that the normalization factor of PSSS is an m-order Legendre polynomial of the squeezing parameter, where m is the number of subtracted photons. Some new relations about the Legendre polynomials are obtained by this analysis. We also show that the PSSS can also be treated as a Hermite-polynomial excitation on squeezed vacuum state.
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13

Virk, Abaid ur Rehman, and Muhammad Quraish. "Some Invariants of Flower Graph." Applied Mathematics and Nonlinear Sciences 3, no. 2 (2018): 427–32. http://dx.doi.org/10.21042/amns.2018.2.00033.

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AbstractLet G be a graph and let mij(G), i, j ≥ 1, represents the number of edge of G, where i and j are the degrees of vertices u and v respectively. In this article, we will compute different polynomials of flower graph f(n×m), namely M polynomial and Forgotten polynomial. These polynomials will help us to find many degree based topological indices, included different Zagreb indices, harmonic indices and forgotten index.
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14

Grytsiuk, Pavlo. "Features of generation sequences of refined Fibonacci polynomial matrices." Vìsnik Nacìonalʹnogo unìversitetu "Lʹvìvsʹka polìtehnìka". Serìâ Ìnformacìjnì sistemi ta merežì 17 (June 2025): 343–65. https://doi.org/10.23939/sisn2025.17.343.

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The features of generating the n th sequence of refined Fibonacci polynomial matrices of the m th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2)th number, are considered. The obtained Fibonacci matrices allow finding both their determinants and inverse matrices suitable for matrix encryption of block data. It was found that even over the last decade a significant number of publications have been published, each of which substantiates different approaches to generating sequences of Fibonacci polynomial matrices and proves the feasibility of their use for encryp
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15

Paluga, Rolando N. "Monophonic Polynomial of the Join of Graphs." Journal of the Indonesian Mathematical Society 31, no. 1 (2025): 1686. https://doi.org/10.22342/jims.v31i1.1686.

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The monophonic polynomial of a graph $G$, denoted by $M(G,x)$, is the polynomial $M(G,x) = \sum_{k=m(G)}^{|G|}M(G, k)x^k $, where $|G|$ is the order of $G$ and $M(G, k)$ is the number of monophonic sets in $G$ with cardinality $k$. In this paper, we delve into some characterizations of monophonic sets in the join of two graphs and use it to determine its corresponding monophonic polynomial. Moreover, we also present the monophonic polynomials of the complete graph $K_n$ $(n \geq 1)$, the path $P_n$ $(n \geq 3)$, the cycle $C_n$ $(n \geq 4)$, the fan $F_n$ $(n \geq 3)$, the wheel $W_n$ $(n \geq
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16

K, Mageshwaran, Gopinath S, and Siluvaidasan R. "UNVEILING THE RELATIONSHIP BETWEEN M-POLYNOMIAL BASED TOPOLOGICAL INDICES AND INVERSE GRAPHS OF FINITE CYCLIC GROUPS: A COMPREHENSIVE STUDY." Journal of the Indonesian Mathematical Society 30, no. 3 (2024): 447–67. https://doi.org/10.22342/jims.30.3.1565.447-467.

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In the discipline of graph theory, topological indices are extremely important. The M-polynomial is a powerful tool for determining a graph's topological indices. The use of M-polynomials to describe macro-molecules and biochemical networking is a novel concept. Also, the M-polynomial of various micro-structural allows us to calculate a variety of topological indices. The chemical substances and biochemical networks are correlated with their chemical characteristics and bio-active compounds using these findings. In this research, we use the M-polynomial to create special essential topological
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17

ZHANG, XU, YUMING SHI, and GUANRONG CHEN. "CONSTRUCTING CHAOTIC POLYNOMIAL MAPS." International Journal of Bifurcation and Chaos 19, no. 02 (2009): 531–43. http://dx.doi.org/10.1142/s0218127409023172.

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This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different n
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18

Yasmeen, Farhana, Muhammad Imran, Shehnaz Akhter, Yasir Ali, and Kashif Ali. "On topological polynomials and indices for metal-organic and cuboctahedral bimetallic networks." Main Group Metal Chemistry 45, no. 1 (2022): 136–51. http://dx.doi.org/10.1515/mgmc-2022-0012.

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Abstract A molecular graph consists of bonds and atoms, where atoms are present as vertices and bonds are present as edges. We can look at topological invariants and topological polynomials that furnish bioactivity and physio-chemical features for such molecular graphs. These topological invariants, which are usually known as graph invariants, are numerical quantities that relate to the topology of a molecular graph. Let m pq (X) be the number of edges in X such that (ζ a , ζ b ) = (p, q), where ζ a (or ζ b ) present the degree of a (or b). The M-polynomial for X can be determined with the hel
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19

Rasool, Kavi B., Payman A. Rashed, and Ahmed M. Ali. "Relations Between Vertex–Edge Degree Based Topological Indices and Mve-Polynomial of r−Regular Simple Graph." European Journal of Pure and Applied Mathematics 16, no. 2 (2023): 773–83. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4698.

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One of the more exciting polynomials among the newly presented graph algebraic polynomials is the M−Polynomial, which is a standard method for calculating degree−based topological indices. In this paper, we define the Mve−polynomials based on vertex edge degree and derive various vertex–edge degree based topological indices from them. Thus, for any graph, we provide some relationships between vertex–edge degree topological indices. Also, we discuss the general Mve−polynomial of r−regular simple graph. Finally, we computed the Mve−polynomial of the 2−ary tree graph.
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20

Monga, Zahid Bashir, and Wali Mohammad Shah. "ON CAUCHY-TYPE BOUNDS FOR THE EIGENVALUES OF A SPECIAL CLASS OF MATRIX POLYNOMIALS." Ural Mathematical Journal 9, no. 1 (2023): 113. http://dx.doi.org/10.15826/umj.2023.1.009.

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Let \(\mathbb{C}^{m\times m}\) be the set of all \(m\times m\) matrices whose entries are in \(\mathbb{C},\) the set of complex numbers. Then \(P(z):=\sum\limits_{j=0}^nA_jz^j,\) \(A_j\in \mathbb{C}^{m\times m},\) \(0\leq j\leq n\) is called a matrix polynomial. If \(A_{n}\neq 0\), then \(P(z)\) is said to be a matrix polynomial of degree \(n\). In this paper we prove some results for the bound estimates of the eigenvalues of some lacunary type of matrix polynomials.
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21

Angelis, Valerio De. "Polynomial beta functions." Ergodic Theory and Dynamical Systems 14, no. 3 (1994): 453–74. http://dx.doi.org/10.1017/s0143385700007975.

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AbstractThe pointwise spectral radii of irreducible matrices whose entries are polynomials with positive, integral coefficients are studied in this paper. Most results are derived in the case that the resulting algebraic function, the beta function of S. Tuncel, is in fact a polynomial. We show that the set of beta functions forms a semiring, and the spectral radius of a matrix of beta functions is again a beta function. We also show that the coefficients of a polynomial beta function p must be real algebraic integers, and p satisfies (after a change of variables if necessary) the inequality f
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22

Raza, Zahid, and Mark Essa K. Sukaiti. "M-Polynomial and Degree Based Topological Indices of Some Nanostructures." Symmetry 12, no. 5 (2020): 831. http://dx.doi.org/10.3390/sym12050831.

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The association of M-polynomial to chemical compounds and chemical networks is a relatively new idea, and it gives good results about the topological indices. These results are then used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this paper, an effort is made to compute the general form of the M-polynomials for two classes of dendrimer nanostars and four types of nanotubes. These nanotubes have very nice symmetries in their structural representations, which have been used to determine the corresponding M-polynomials. Furthermo
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23

Urieles, Alejandro, María José Ortega, William Ramírez, and Samuel Vega. "New results on the q-generalized Bernoulli polynomials of level m." Demonstratio Mathematica 52, no. 1 (2019): 511–22. http://dx.doi.org/10.1515/dema-2019-0039.

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AbstractThis paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B_n^{[m - 1]}(x;q) of level m, as well as some others identities which connect this polynomial class with the q-generalized Bernoulli polynomials of level m, as well as the q-gamma function, and the q-Stirling numbers of the second kind and the q-Bernstein polynomials.
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24

Delladio, Silvano. "m-approximate Taylor polynomial." manuscripta mathematica 163, no. 3-4 (2019): 481–505. http://dx.doi.org/10.1007/s00229-019-01167-0.

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25

Kim, Dae, and Taekyun Kim. "A generalization of power and alternating power sums to any Appell polynomials." Filomat 31, no. 1 (2017): 141–57. http://dx.doi.org/10.2298/fil1701141k.

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The classical power sum and alternating power sum identities can be stated as ?m,i=0 sn(i)= 1/n+1 (Bn+1(m+1)- Bn+1), ?m,i=0(-1)i sn (i) = 1/2 ((-1)m En(m+1) + En), where sn(x)=xn is the simplest possible Appell polynomial for the Sheffer pair (1,t). The impetus for this research starts from the question that what if we replace sn(x)=xn by any Appell polynomial. In this paper, we give a generalization of power and alternating power sums to any Appell polynomials.
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26

Fan, Austen Z., Paraschos Koutris, and Sudeepa Roy. "Circuits and Formulas for Datalog over Semirings." Proceedings of the ACM on Management of Data 3, no. 2 (2025): 1–22. https://doi.org/10.1145/3725230.

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In this paper, we study circuits and formulas for provenance polynomials of Datalog programs. We ask the following question: given an absorptive semiring and a fact of a Datalog program, what is the optimal depth and size of a circuit/formula that computes its provenance polynomial? We focus on absorptive semirings as these guarantee the existence of a polynomial-size circuit. Our main result is a dichotomy for several classes of Datalog programs on whether they admit a formula of polynomial size or not. We achieve this result by showing that for these Datalog programs the optimal circuit dept
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27

Ebadian, A., M. Bidkhamandm, and Eshaghi Gordji. "Number of zeros of a polynomial in a given domain." Tamkang Journal of Mathematics 42, no. 4 (2011): 531–36. http://dx.doi.org/10.5556/j.tkjm.42.2011.1037.

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In this paper, we obtain results concerning the bound for the number of zeros for the polynomial $p(z)$ which generalize earlier well-known result due to Bidkham and Dewan [{\it On the zeros of a polynomial}, Numerical Methods and Approximation Theory III, Ni\v{s} (1987), 121--128] and Mohammad [{\it On the zeros of polynomials}, Amer. Math. Monthly, 72 (1965), 631-633]. We also obtain result for location of zeros of polynomial $p(z)=\sum\limits_{i=0}^m \frac{a_i}{(i!)^\lambda} z^i+a_nz^n$, $a_n\neq 0$, $0\leq m\leq n-1$, $\lambda\geq 0$.\
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28

Yang, Hong, A. Q. Baig, W. Khalid, Mohammad Reza Farahani, and Xiujun Zhang. "M-Polynomial and Topological Indices of Benzene Ring Embedded in P-Type Surface Network." Journal of Chemistry 2019 (November 3, 2019): 1–9. http://dx.doi.org/10.1155/2019/7297253.

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The representation of chemical compounds and chemical networks with the M-polynomials is a new idea, and it gives nice and good results of the topological indices. These results are used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this article, particular attention will be put on the derivation of M-polynomial for the benzene ring embedded in the P-type surface network in 2D. Furthermore, the topological indices based on the degrees are also derived by using the general form of M-polynomial of the benzene ring embedded in the P
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29

Ali, Ashaq, Waqas Nazeer, Mobeen Munir, and Shin Min Kang. "M-Polynomials And Topological Indices Of Zigzag And Rhombic Benzenoid Systems." Open Chemistry 16, no. 1 (2018): 73–78. http://dx.doi.org/10.1515/chem-2018-0010.

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AbstractM-polynomial of different molecular structures helps to calculate many topological indices. This polynomial is a new idea and its beauty is the wealth of information it contains about the closed forms of degree-based topological indices of molecular graph G of the structure. It is a well-known fact that topological indices play significant role in determining properties of the chemical compound [1, 2, 3, 4]. In this article, we computed the closed form of M-polynomial of zigzag and rhombic benzenoid systemsbecause of their extensive usages in industry. Moreover we give graphs of M-poly
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30

Samaruk, N. M. "$SO(3)$ квазі-мономіальні сім'ї многочленів". Carpathian Mathematical Publications 16, № 1 (2024): 40–52. http://dx.doi.org/10.15330/cmp.16.1.40-52.

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Let $H$ be a subgroup of the affine space group ${\rm Aff} 3)$, considered with its natural action on the vector space of three-variable polynomials. The polynomial family $\{ B_{m,n,k}(x,y,z) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $\{ x^m y^n z^k \}$ and $\{ B_{m,n,k}(x,y,z) \}$ have identical atrices. We derive a criterion for quasi-monomiality when the group $H$ is the special orthogonal group $SO(3)$. This criterion is expressed through the exponential generating function of the polynomial family $\{B_{m,n,k}(x,y,z)\}$. It has been pr
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31

Samaruk, N. M. "Quasi-monomials with respect to subgroups of the plane affine group." Matematychni Studii 59, no. 1 (2023): 3–11. http://dx.doi.org/10.30970/ms.59.1.3-11.

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Let $H$ be a subgroup of the plane affine group ${\rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family $\{ B_{m,n}(x,y) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ \{ x^m y^n \} $ and $\{ B_{m,n}(x,y) \}$ have \textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family $\{ B_{m,n}(x,y) \}$.
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32

Gould, M. D. "Characteristic identities for semi-simple Lie algebras." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 26, no. 3 (1985): 257–83. http://dx.doi.org/10.1017/s0334270000004501.

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AbstractWe present a new derivation of the polynomial identities satisfied by certain matrices A with entries Aij (i, j = 1,…, n) from the universal enveloping algebra of a semi-simple Lie algebra. These polynomial identities are exhibited in a representation-independent way as p(A) = 0 where p(x) (herein called the characteristic polynomial of A) is a polynomial with coefficients from the centre Z of the universal enveloping algebra. The minimum polynomial identity m(A) = 0 of the matrix A over Z is also obtained and it is shown that p(x) and m(x) possess properties analogous to the character
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33

Dassani*, Mamta, and Mukesh kushwaha. "Some Generalized Results to unify Classical Polynomials." International Journal of Basic Sciences and Applied Computing 3, no. 3 (2021): 1–5. http://dx.doi.org/10.35940/ijbsac.b0201.033321.

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Present work of this paper deals with the unification of classical polynomials in which we have defined a generalized polynomial set analogous to that of associated Legendre polynomial P (x) m n by taking 5the use of Operator. Also we have derived explicit form, Operational Formulae generating functions for this function.
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MamtaDassani and kushwaha Mukesh. "Some Generalized Results to unify Classical Polynomials." International Journal of Basic Sciences and Applied Computing (IJBSAC) 3, no. 3 (2021): 1–5. https://doi.org/10.35940/ijbsac.B0201.033321.

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Present work of this paper deals with the unification of classical polynomials in which we have defined a generalized polynomial set analogous to that of associated Legendre polynomial P (x) m n by taking 5the use of Operator. Also we have derived explicit form, OperationalFormulae generating functions for this function.
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35

Li, Hai-Xia, Sarfaraz Ahmad, and Iftikhar Ahmad. "Topology-Based Analysis of OTIS (Swapped) Networks OKn and OPn." Journal of Chemistry 2019 (November 7, 2019): 1–11. http://dx.doi.org/10.1155/2019/4291943.

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In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In this paper, M-polynomial OKn and OPn networks are computed. The M-polynomial is rich in information about degree-based topological indices. By applying the basic rules of calculus on M-polynomials, the first and second Zagreb indices, modified second Zagreb index, general Randić index, inverse Randić index, symmetric division index, harmonic index, inverse sum index, and augmented Zagreb index are recovered.
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36

T. Anitha Baby and Y. A. Shiny. "Connected 2-domination polynomials of some graph operations." Malaya Journal of Matematik 8, no. 04 (2020): 2329–32. http://dx.doi.org/10.26637/mjm0804/0176.

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In this paper, we derive the connected 2-domination polynomials of some graph operations. The connected 2-domination polynomial of a graph \(\mathrm{G}\) of order \(m\) is the polynomial \(D_{c 2}(G, x)=\sum_{j=\gamma_{c_2}(G)}^m d_{c 2}(G, j) x^j\), where \(d_{c 2}(G, j)\) is the number of connected 2-dominating sets of \(G\) of size \(j\) and \(\gamma_{c 2}(G)\) is the connected 2-domination number of \(G\).
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37

Duda, Dominik. "The maximal coefficient of ternary cyclotomic polynomials with one free prime." International Journal of Number Theory 10, no. 04 (2014): 1067–80. http://dx.doi.org/10.1142/s1793042114500158.

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A cyclotomic polynomial Φn(x) is said to be ternary if n = pqr, with p, q and r distinct odd primes. Let M(p, q) be the maximum (in absolute value) coefficient appearing in the polynomial family Φpqr(x) with p < q < r, p and q fixed. Here a stronger version of the main conjecture of Gallot, Moree and Wilms regarding M(p, q) is established. Furthermore it is shown that there is an algorithm to compute M(p): = max {M(p, q): q > p}. Our methods are the most geometric used so far in the study of ternary cyclotomic polynomials.
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38

Fujimoto, Hirotaka. "On uniqueness polynomials for meromorphic functions." Nagoya Mathematical Journal 170 (2003): 33–46. http://dx.doi.org/10.1017/s0027763000008527.

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AbstractA polynomial P(w) is called a uniqueness polynomial (or a uiqueness polynomial in a broad sense) if P(f) = cP(g) (or P(f) = P(g)) implies f = g for any nonzero constant c and nonconstant meromorphic functions f and g on C. We consider a monic polynomial P(w) without multiple zero whose derivative has mutually distinct k zeros ej with multiplicities qj. Under the assumption that P(el) ≠ P(em) for all distinct l and m, we prove that P(w) is a uniqueness polynomial in a broad sense if and only if ∑l<mqlqm > ∑l ql. We also give some sufficient conditions for uniqueness polynomials.
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39

Prasad, Bandhu. "Coding theory on h(x) extension of m sequences for Fibonacci numbers." Discrete Mathematics, Algorithms and Applications 07, no. 02 (2015): 1550008. http://dx.doi.org/10.1142/s1793830915500081.

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In this paper, we introduced h(x) extension of m sequences of Fibonacci numbers polynomials of order m and a new Sn matrix of order m where h(x)(> 0) is a polynomial with real coefficients. Thereby, we discuss various properties of Sn matrix and the Fibonacci coding theory followed from this matrix. We established the relations among the code matrix elements, error deduction and correction for this coding theory. In general, correct ability potential of this method increases and it is independent of h(x). But h(x) being a polynomial, improves the cryptography protection.
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40

HABIBI, MOHAMMAD, and RAOUFEH MANAVIYAT. "A GENERALIZATION OF NIL-ARMENDARIZ RINGS." Journal of Algebra and Its Applications 12, no. 06 (2013): 1350001. http://dx.doi.org/10.1142/s0219498813500011.

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Let R be a ring, M a monoid and ω : M → End (R) a monoid homomorphism. The skew monoid ring R * M is a common generalization of polynomial rings, skew polynomial rings, (skew) Laurent polynomial rings and monoid rings. In the current work, we study the nil skew M-Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials to skew monoid rings. We resolve the structure of nil skew M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil skew M-Armendariz, unifying and generalizing a number of known nil Armendariz-lik
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41

Bedoya, D., M. Ortega, W. Ramírez, and A. Urieles. "New Biparametric Families of Apostol-Frobenius-Euler Polynomials level-m." Matematychni Studii 55, no. 1 (2021): 10–23. http://dx.doi.org/10.30970/ms.55.1.10-23.

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We introduce two biparametric families of Apostol-Frobenius-Euler polynomials of level-$m$. We give some algebraic properties, as well as some other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers of the second kind, the generalized Apostol--Bernoulli polynomials, the generalized Apostol--Genocchi polynomials, the generalized Apostol--Euler polynomials and Jacobi polynomials. Finally, we will show the differential properties of this new family of polynomials.
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42

Li, Yingfang, Li Yan, Mohammad R. Farahani, Muhammad Imran та Muhammad K. Jamil. "Computing the Theta Polynomial Θ(G, x) and the Theta Index Θ(G) of Titania Nanotubes TiO2(m, n)". Journal of Computational and Theoretical Nanoscience 14, № 1 (2017): 715–17. http://dx.doi.org/10.1166/jctn.2017.6262.

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Let G = (V,E) be a simple connected molecular graph in chemical graph theory, where the vertex/atom set and edge/bond set of G denoted by V(G) and E(G), respectively and its vertices correspond to the atoms and the edges correspond to the bonds. Two counting polynomials the “Omega Ω(G,x) and Theta Θ(G,x)” polynomials of a molecular graph G were defined by Diudea as Ω(G,x) = ΣeE(G) xn(E) and Θ(G,x) = ΣeE(G) xn(E), where n(E) denotes the number of edges co-distant with the edge E. From definition of these counting polynomials, we can obtain the Theta polynomial by inserting the coefficient n(E)
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43

Zhao, Shuang. "Matching Forcing Polynomials of Constructable Hexagonal Systems." Match Communications in Mathematical and in Computer Chemistry 88, no. 3 (2022): 767–97. http://dx.doi.org/10.46793/match.88-3.767z.

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Harary et al. put forward the concept on the minimum cardinality over all subsets of perfect matching M that are not included by any other ones, to be the forcing number for M. A counting polynomial for perfect matchings possessing the same forcing number was introduced by Zhang et al., using the name `forcing polynomial'. This research deduces recurrence formulas of forcing polynomials for monotonic constructable hexagonal systems and constructable hexagonal systems with one turning. From them, a characterization of continuity of forcing spectrum for hexagonal systems with forcing edges can b
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44

B, Basavanagoud, Goutam Veerapur, and Pooja B. "M-Polynomial of Windmill Graphs." International Journal of Mathematics Trends and Technology 68, no. 7 (2022): 66–74. http://dx.doi.org/10.14445/22315373/ijmtt-v68i7p510.

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45

Mulay, S. B. "Polynomial-mappings and M-equivalence." Journal of Algebra 302, no. 2 (2006): 862–80. http://dx.doi.org/10.1016/j.jalgebra.2006.04.027.

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46

Nakajima, Y., Y. Kawano, and H. Sekigawa. "Efficient quantum circuits for approximating the Jones polynomial." Quantum Information and Computation 8, no. 5 (2008): 489–500. http://dx.doi.org/10.26421/qic8.5-8.

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Freedman, Kitaev, and Wang proved the equivalence between quantum field theory and quantum computation, and consequently showed that the problem of approximating the Jones polynomial (a knot invariant) at the fifth root of unity is BQP-complete. Recently, Aharonov, Jones, and Landau proposed a concrete quantum algorithm, called the AJL algorithm, that approximates the Jones polynomial at the $k$th root of unity in polynomial time. In this paper, we propose a new method for implementing the AJL algorithm, which improves the performance from $O(mn\log^2k)$ to $O(mn)$. Here, $n$ is the number of
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47

S. Garcia, Elino, and EMIL C. ALCANTARA. "An Exploration of Synthetic Division for Non-Linear Polynomial Divisors." International Multidisciplinary Research Journal 4, no. 1 (2022): 10–21. http://dx.doi.org/10.54476/5292497.

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This paper is an exploration of the synthetic division in compact form. The main goal was to develop an alternative algorithm on division of polynomials whose dividend is P(x) = a1xn + a2xn−1 + a3xn−2 + ⋯ + anx + an+1 and the divisor is D(x) = b1xm + b2xm−1 + b3xm−2 … + bmx + bm+1 , where n > m, a1 ≠ 0, b1 ≠ 0, and ai′s and bi′s including an+1 and bm+1 are constant. The quotient is in the form Q(x) = q1xn−m + q2xn−m−1 + q3xn−m−2 + ⋯ + qn−mx + qn−m+1 while the remainder is R(x) = r1xm−1 + r2xm−2 + ⋯ + rm−1 + rm. Specifically, this aimed to develop an algorithm using synthetic division in com
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48

Athanasiadis, Christos A. "A Combinatorial Reciprocity Theorem for Hyperplane Arrangements." Canadian Mathematical Bulletin 53, no. 1 (2010): 3–10. http://dx.doi.org/10.4153/cmb-2010-004-7.

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AbstractGiven a nonnegative integer m and a finite collection of linear forms on ℚd, the arrangement of affine hyperplanes in ℚd defined by the equations α(x) = k for α ∈ and integers k ∈ [–m,m] is denoted by . It is proved that the coefficients of the characteristic polynomial of are quasi-polynomials in m and that they satisfy a simple combinatorial reciprocity law.
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49

Cheng, Zhiyun, Hongzhu Gao, and Mengjian Xu. "Transcendental linking polynomials of virtual string links." Journal of Knot Theory and Its Ramifications 28, no. 03 (2019): 1950022. http://dx.doi.org/10.1142/s0218216519500226.

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In this paper, we define transcendental polynomial invariants for two-component virtual string links. One of these invariants is a strictly refinement of the linking polynomial in [M. Xu and H. Gao, Linking polynomials of virtual string links, Sci. China Math. 61(7) (2018) 1287–1302]. It is a homotopy invariant and can distinguish some virtual string links from their mirror images. We also define a transcendental polynomial invariant for two-component flat virtual string links. These invariants can be used to study the periodicity and linking crossing number of virtual string links.
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50

LLIBRE, JAUME, ANA CRISTINA MEREU, and MARCO ANTONIO TEIXEIRA. "Limit cycles of the generalized polynomial Liénard differential equations." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 2 (2009): 363–83. http://dx.doi.org/10.1017/s0305004109990193.

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AbstractWe apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.
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