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Journal articles on the topic 'Exact polynomial'

Author: Grafiati

Published: 26 October 2022

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1

Guilloux, Antonin, and Julien Marché. "Volume function and Mahler measure of exact polynomials." Compositio Mathematica 157, no. 4 (April 2021): 809–34. http://dx.doi.org/10.1112/s0010437x21007016.

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We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.

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2

Achter, Jeffrey, and Cassandra Williams. "Local Heuristics and an Exact Formula for Abelian Surfaces Over Finite Fields." Canadian Mathematical Bulletin 58, no. 4 (December 1, 2015): 673–91. http://dx.doi.org/10.4153/cmb-2015-050-8.

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AbstractConsider a quartic q-Weil polynomial ƒ. Motivated by equidistribution considerations, we define, for each prime ℓ, a local factor that measures the relative frequency with which ƒ mod ℓ occurs as the characteristic polynomial of a symplectic similitude over 𝔽ℓ. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over 𝔽q with Weil polynomial ƒ.

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3

Borwein, Peter B. "Exact Inequalities for the Norms of Factors of Polynomials." Canadian Journal of Mathematics 46, no. 4 (August 1, 1994): 687–98. http://dx.doi.org/10.4153/cjm-1994-038-8.

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AbstractThis paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

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4

RÜHL, WERNER, and ALEXANDER TURBINER. "EXACT SOLVABILITY OF THE CALOGERO AND SUTHERLAND MODELS." Modern Physics Letters A 10, no. 29 (September 21, 1995): 2213–21. http://dx.doi.org/10.1142/s0217732395002374.

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Translationally invariant symmetric polynomials as coordinates for N-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland N-body Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra sl N in finitedimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.

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Chen, Yi-Chou. "Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results." Journal of Applied Mathematics 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/516159.

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LetFx, y=asxys+as-1xys-1+⋯+a0xbe a real-valued polynomial function in which the degreesofyinFx, yis greater than or equal to 1. For any polynomialyx, we assume thatT:Rx→Rxis a nonlinear operator withTyx=Fx, yx. In this paper, we will find an eigenfunctionyx∈Rxto satisfy the following equation:Fx, yx=ayxfor some eigenvaluea∈Rand we call the problemFx, yx=ayxa fixed point like problem. If the number of all eigenfunctions inFx, yx=ayxis infinitely many, we prove that (i) any coefficients ofFx, y, asx, as-1x,…, a0x, are all constants inRand (ii)yxis an eigenfunction inFx, yx=ayxif and only ifyx∈R.

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6

Paschos, Vangelis Th. "When polynomial approximation meets exact computation." Annals of Operations Research 271, no. 1 (July 30, 2018): 87–103. http://dx.doi.org/10.1007/s10479-018-2986-9.

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Bhattacharyya, Rajsekhar, Robert de Mello Koch, and Michael Stephanou. "Exact multi-restricted Schur polynomial correlators." Journal of High Energy Physics 2008, no. 06 (June 27, 2008): 101. http://dx.doi.org/10.1088/1126-6708/2008/06/101.

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8

Paschos, Vangelis Th. "When polynomial approximation meets exact computation." 4OR 13, no. 3 (July 8, 2015): 227–45. http://dx.doi.org/10.1007/s10288-015-0294-7.

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9

Zerz, Eva, Viktor Levandovskyy, and Kristina Schindelar. "Exact linear modeling with polynomial coefficients." Multidimensional Systems and Signal Processing 22, no. 1-3 (July 1, 2010): 55–65. http://dx.doi.org/10.1007/s11045-010-0125-0.

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10

Mueller, Matthias. "Polynomial Exact-3-SAT-Solving Algorithm." International Journal of Engineering & Technology 9, no. 3 (August 4, 2020): 670. http://dx.doi.org/10.14419/ijet.v9i3.30749.

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This article describes an algorithm which is supposed by the author to be capable of solving any instance of a 3-SAT CNF in maximal O(n^15), whereby n is the variable index range within the 3-SAT CNF to solve. The presented algorithm imitates the proceeding of an exponential, fail-safe solver. This exponential solver stores internal data in m-SAT clauses, with 3 <= m <= n. The polynomial solver works similarly, but uses 3-SAT clauses only to save the same data. The paper explains how, and proves why this can be achieved. On the supposition the algorithm is correct, the P-NP-Problem would be solved with the result that the complexity classes NP and P are equal.

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11

Molla, Hasib Uddin, and Goutam Saha. "Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods." GANIT: Journal of Bangladesh Mathematical Society 38 (January 14, 2019): 11–25. http://dx.doi.org/10.3329/ganit.v38i0.39782.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25

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12

Kotsireas, Ilias, Edmond Lau, and Richard Voino. "Exact implicitization of polynomial curves and surfaces." ACM SIGSAM Bulletin 37, no. 3 (September 2003): 78. http://dx.doi.org/10.1145/990353.990364.

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13

Luca, M. B., S. Azou, G. Burel, and A. Serbanescu. "On exact Kalman filtering of polynomial systems." IEEE Transactions on Circuits and Systems I: Regular Papers 53, no. 6 (June 2006): 1329–40. http://dx.doi.org/10.1109/tcsi.2006.870899.

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14

Samadi, S., M. O. Ahmad, and M. N. S. Swamy. "Exact Fractional-Order Differentiators for Polynomial Signals." IEEE Signal Processing Letters 11, no. 6 (June 2004): 529–32. http://dx.doi.org/10.1109/lsp.2004.827917.

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15

Feng, Yong, Xiaolin Qin, Jingzhong Zhang, and Xun Yuan. "Obtaining exact interpolation multivariate polynomial by approximation." Journal of Systems Science and Complexity 24, no. 4 (June 11, 2011): 803–15. http://dx.doi.org/10.1007/s11424-011-8312-0.

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16

Marshalko, Grigory, and Julia Trufanova. "Polynomial Approximations for Several Neural Network Activation Functions." Informatics and Automation 21, no. 1 (November 16, 2021): 161–80. http://dx.doi.org/10.15622/ia.2022.21.6.

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Active deployment of machine learning systems sets a task of their protection against different types of attacks that threaten confidentiality, integrity and accessibility of both processed data and trained models. One of the promising ways for such protection is the development of privacy-preserving machine learning systems, that use homomorphic encryption schemes to protect data and models. However, such schemes can only process polynomial functions, which means that we need to construct polynomial approximations for nonlinear functions used in neural models. The goal of this paper is the construction of precise approximations of several widely used neural network activation functions while limiting the degree of approximation polynomials as well as the evaluation of the impact of the approximation precision on the resulting value of the whole neural network. In contrast to the previous publications, in the current paper we study and compare different ways for polynomial approximation construction, introduce precision metrics, present exact formulas for approximation polynomials as well as exact values of corresponding precisions. We compare our results with the previously published ones. Finally, for a simple convolutional network we experimentally evaluate the impact of the approximation precision on the bias of the output neuron values of the network from the original ones. Our results show that the best approximation for ReLU could be obtained with the numeric method, and for the sigmoid and hyperbolic tangent – with Chebyshev polynomials. At the same time, the best approximation among the three functions could be obtained for ReLU. The results could be used for the construction of polynomial approximations of activation functions in privacy-preserving machine learning systems.

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17

Köbler, Johannes, and Wolfgang Lindner. "ORACLES IN $\Sigma^p_2$ ARE SUFFFICIENT FOR EXACT LEARNING." International Journal of Foundations of Computer Science 11, no. 04 (December 2000): 613–32. http://dx.doi.org/10.1142/s012905410000034x.

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We study the learnability of representation classes in Angluin's exact learning model. In particular, we consider the following three query types: equivalence queries, equivalence and membership queries, and membership queries only. We show in all three cases that polynomial query complexity implies already polynomial-time learnability, provided that the learner additionally has access to an oracle in [Formula: see text]. It follows that boolean circuits are polynomial-time learnable with equivalence queries and the help of an oracle in [Formula: see text].a

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18

Znojil, Miloslav. "Displaced Harmonic Oscillator V ∼ min [(x + d)2, (x − d)2] as a Benchmark Double-Well Quantum Model." Quantum Reports 4, no. 3 (August 24, 2022): 309–23. http://dx.doi.org/10.3390/quantum4030022.

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For the displaced harmonic double-well oscillator, the existence of exact polynomial bound states at certain displacements d is revealed. The N-plets of these quasi-exactly solvable (QES) states are constructed in closed form. For non-QES states, the Schrödinger equation can still be considered “non-polynomially exactly solvable” (NES) because the exact left and right parts of the wave function (proportional to confluent hypergeometric function) just have to be matched in the origin.

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19

Yang, Yajun, and Sheldon P. Gordon. "Interpolation and Polynomial Curve Fitting." Mathematics Teacher 108, no. 2 (September 2014): 132–41. http://dx.doi.org/10.5951/mathteacher.108.2.0132.

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20

Franco-Medrano, Fermin, and Francisco J. Solis. "Stability of Real Parametric Polynomial Discrete Dynamical Systems." Discrete Dynamics in Nature and Society 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/680970.

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We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameterλand generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to realmth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept ofcanonical polynomial mapswhich are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termedProduct Position Functionfor a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termedstability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

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21

Zhang, Qiuyan, Lingling Liu, and Weinian Zhang. "Local Bifurcations of the Enzyme-Catalyzed Reaction Comprising a Branched Network." International Journal of Bifurcation and Chaos 25, no. 06 (June 15, 2015): 1550081. http://dx.doi.org/10.1142/s0218127415500819.

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An enzyme-catalyzed reaction system with four parameters a, b, c, κ is discussed. The system can be reduced to a quartic polynomial differential system with four parameters, which leads to difficulties in the computation of semi-algebraic systems of large degree polynomials. Those systems have to be discussed on subsets of special biological sense, none of which is closed under operations of the polynomial ring. In this paper, we overcome those difficulties to determine the exact number of equilibria and their qualitative properties. Moreover, we obtain parameter conditions for all codimension-1 bifurcations such as saddle-node, transcritical, pitchfork and Hopf bifurcations. We compute varieties of Lyapunov quantities under the limitations of biological requirements and prove that the weak focus is of at most order 2. We further obtain parameter conditions for exact number of limit cycles arising from Hopf bifurcations.

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22

Bhrawy, A. H., and A. S. Alofi. "An Accurate Spectral Galerkin Method for Solving Multiterm Fractional Differential Equations." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/728736.

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This paper reports a new formula expressing the Caputo fractional derivatives for any order of shifted generalized Jacobi polynomials of any degree in terms of shifted generalized Jacobi polynomials themselves. A direct solution technique is presented for solving multiterm fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using spectral shifted generalized Jacobi Galerkin method. The homogeneous initial conditions are satisfied exactly by using a class of shifted generalized Jacobi polynomials as a polynomial basis of the truncated expansion for the approximate solution. The approximation of the spatial Caputo fractional order derivatives is expanded in terms of a class of shifted generalized Jacobi polynomialsJnα,−β(x)withx∈(0,1), andnis the polynomial degree. Several numerical examples with comparisons with the exact solutions are given to confirm the reliability of the proposed method for multiterm FDEs.

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23

BROWN, RON, and JONATHAN L. MERZEL. "INVARIANTS OF DEFECTLESS IRREDUCIBLE POLYNOMIALS." Journal of Algebra and Its Applications 09, no. 04 (August 2010): 603–31. http://dx.doi.org/10.1142/s0219498810004130.

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Defectless irreducible polynomials over a Henselian valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete (also called "saturated") distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case in which a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f. In this case a (best possible) result is given showing that a sufficiently good approximation in an extension field K of F to a root of a defectless polynomial f over F guarantees the existence of an exact root of f in K. Also in the tame case a (best possible) result is given describing when a polynomial is sufficiently close to a defectless polynomial so as to guarantee that the roots of the two polynomials generate the same extension fields. Another application in the tame case gives a simple characterization of the minimal pairs (in the sense of N. Popescu et al.). A key technical result is a computation in the tame case of the Newton polygon of f(x+α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja's characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials. Much of the work described here will be seen not to require the hypothesis that (F, v) is Henselian.

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24

CHENG, YAN-FU, and TONG-QING DAI. "EXACT SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR A NEW RING-SHAPED NONHARMONIC OSCILLATOR POTENTIAL." International Journal of Modern Physics A 23, no. 12 (May 10, 2008): 1919–27. http://dx.doi.org/10.1142/s0217751x08039621.

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The bound state solutions of the Schrödinger equation with a new ring-shaped nonharmonic potential are presented using exactly the Nikiforov–Uvarov method. It is found that the solutions of the angular wave function can be expressed by Jacobi polynomial and radial wave functions are given by the generalized Laguerre polynomials. We also discuss the special case for α = 0 and β = 0 respectively.

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25

Patnaik, L. M., and Satrajit Gupta. "Exact Output Response Computation of RC Interconnects Under General Polynomial Input Waveforms." VLSI Design 11, no. 2 (January 1, 2000): 75–84. http://dx.doi.org/10.1155/2000/48985.

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Accurate output response computation of RC interconnects under various input excitations is a key issue in deep submicron delay analysis. In this paper, we present an exact analysis of output response computation of a distributed RC interconnect under input signals that are polynomial in time (tn). A simple, recursive equation that helps us to calculate the interconnect response under higher order polynomial inputs in terms of the lower order polynomial responses is derived. To the best of our knowledge, this is the first exact output response analysis of RC interconnects under generalized polynomial inputs.

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26

Zhang, Xinjian, and Xiongwei Liu. "A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/823264.

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A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

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27

Roy, Pinaki, and Rajkumar Roychoudhury. "Exact Analytical Solutions of the Non Polynomial Oscilator." Zeitschrift für Naturforschung A 43, no. 4 (April 1, 1988): 360–62. http://dx.doi.org/10.1515/zna-1988-0411.

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Abstract We derive new exact (analytical) solutions of the non polynomial oscillator V(x) = x2 + λx2 / (1 + g x2) in one as well in three dimensions. The solutions are derived within the framework of supersymmetric quantum mechanics, and it is shown that exact solutions exist when the coupling constants satisfy a supersymmetric constraint.

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28

Aland, Sebastian, Dominic Dumrauf, Martin Gairing, Burkhard Monien, and Florian Schoppmann. "Exact Price of Anarchy for Polynomial Congestion Games." SIAM Journal on Computing 40, no. 5 (January 2011): 1211–33. http://dx.doi.org/10.1137/090748986.

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29

Weeks, Jeff. "Exact polynomial eigenmodes for homogeneous spherical 3-manifolds." Classical and Quantum Gravity 23, no. 23 (October 24, 2006): 6971–88. http://dx.doi.org/10.1088/0264-9381/23/23/023.

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30

Fu, Guosheng, Johnny Guzmán, and Michael Neilan. "Exact smooth piecewise polynomial sequences on Alfeld splits." Mathematics of Computation 89, no. 323 (January 13, 2020): 1059–91. http://dx.doi.org/10.1090/mcom/3520.

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31

Szanto, Agnes. "Certification of Approximate Roots of Exact Polynomial Systems." Notices of the American Mathematical Society 63, no. 10 (November 1, 2016): 1160–62. http://dx.doi.org/10.1090/noti1440.

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32

Gómez-Ullate, D., N. Kamran, and R. Milson. "Quasi-exact solvability in a general polynomial setting." Inverse Problems 23, no. 5 (August 24, 2007): 1915–42. http://dx.doi.org/10.1088/0266-5611/23/5/008.

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33

Nie, Jiawang. "An exact Jacobian SDP relaxation for polynomial optimization." Mathematical Programming 137, no. 1-2 (September 22, 2011): 225–55. http://dx.doi.org/10.1007/s10107-011-0489-4.

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34

Camerini, P. M., G. Galbiati, and F. Maffioli. "Random pseudo-polynomial algorithms for exact matroid problems." Journal of Algorithms 13, no. 2 (June 1992): 258–73. http://dx.doi.org/10.1016/0196-6774(92)90018-8.

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35

Constantine, Kenneth B., Yong B. Lim, and W. J. Studden. "Admissible and optimal exact designs for polynomial regression." Journal of Statistical Planning and Inference 16 (January 1987): 15–32. http://dx.doi.org/10.1016/0378-3758(87)90052-8.

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36

King, J. R. "Exact polynomial solutions to some nonlinear diffusion equations." Physica D: Nonlinear Phenomena 64, no. 1-3 (April 1993): 35–65. http://dx.doi.org/10.1016/0167-2789(93)90248-y.

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37

Simeone, Bruno. "An asymptotically exact polynomial algorithm for equipartition problems." Discrete Applied Mathematics 14, no. 3 (July 1986): 283–93. http://dx.doi.org/10.1016/0166-218x(86)90032-6.

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38

Islam, Md Shafiqul, and Md Bellal Hossain. "On the Use of Piecewise Standard Polynomials in the Numerical Solutions of Fourth Order Boundary Value Problems." GANIT: Journal of Bangladesh Mathematical Society 33 (January 13, 2014): 53–64. http://dx.doi.org/10.3329/ganit.v33i0.17659.

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This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the polynomials satisfy the given boundary conditions. The linear combinations of each polynomial are exploited in the Galerkin weighted residual approximation. The derived formulation is illustrated through various numerical examples. Our approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. The approximate solutions converge to the exact solutions monotonically even with desired large significant digits. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 53-64 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17659

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GE, Q., and D. ŠTEFANKOVIČ. "A Graph Polynomial for Independent Sets of Bipartite Graphs." Combinatorics, Probability and Computing 21, no. 5 (July 10, 2012): 695–714. http://dx.doi.org/10.1017/s0963548312000296.

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We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.

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40

Kostić, Dragutin, and Vladimir Šinik. "Approximate Algorithm for Determining Pulse Edges of a PWM Inverter Based on Natural Sampling." Mathematical Problems in Engineering 2009 (2009): 1–23. http://dx.doi.org/10.1155/2009/495360.

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The paper presents a new algorithm for determination of pulse edges of a modulated wave of a PWM voltage inverter which offers a possibility that natural sampling is realized with an arbitrary accuracy without applying an iterative procedure. The basic idea is to express the angles which determine pulse edges of the modulated signal as polynomials of amplitude modulation index. Geometric interpretation of sampling of the polynomial algorithm is identical with the geometric interpretation of natural algorithm, but the transcendental equation whose solution defines pulse edges of the modulated signal is replaced by a simple procedure of finding values of a polynomial whose coefficients are determined in advance by an exact procedure. This approach gives the possibility of digital implementation of polynomial sampling method using the low-cost microprocessor platforms.

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41

Oboyi, J., S. E. Ekoro, and P. T. Bukie. "Numerical solution of initial value problems by rational interpolation method using Chebyshev polynomials." Global Journal of Pure and Applied Sciences 25, no. 2 (September 6, 2019): 185–94. http://dx.doi.org/10.4314/gjpas.v25i2.8.

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In this research, a modified rational interpolation method for the numerical solution of initial value problem is presented. The proposed method is obtained by fitting the classical rational interpolation formula in Chebyshev polynomials leading to a new stability function and new scheme. Three numerical test problems are presented in other to test the efficiency of the proposed method. The numerical result for each test problem is compared with the exact solution. The approximate solutions are show competitiveness with the exact solutions of the ODEs throughout the solution interval.Keywords and Phrases: Chebyshev polynomial, Rational Interpolation, Minimaxpolynomial, Initial Value Problems and Ordinary Differential Equations (ODEs)

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Gallas, Jason A. C. "Preperiodicity and systematic extraction of periodic orbits of the quadratic map." International Journal of Modern Physics C 31, no. 12 (October 19, 2020): 2050174. http://dx.doi.org/10.1142/s0129183120501740.

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Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree [Formula: see text], while for the [Formula: see text] period-20 orbits the degree rises already to [Formula: see text]. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map.

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MATTAREI, SANDRO. "ROOT MULTIPLICITIES AND NUMBER OF NONZERO COEFFICIENTS OF A POLYNOMIAL." Journal of Algebra and Its Applications 06, no. 03 (June 2007): 469–75. http://dx.doi.org/10.1142/s0219498807002338.

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It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero coefficients, over a finite field of characteristic larger than the degree.

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Townsend, M. A., and S. Gupta. "Automated Modeling and Rapid Solution of Robot Dynamics Using the Symbolic Polynomial Technique." Journal of Mechanisms, Transmissions, and Automation in Design 111, no. 4 (December 1, 1989): 537–44. http://dx.doi.org/10.1115/1.3259035.

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Fast and accurate solutions of the dynamic equations of a robot arm are required for real time on-line control. In this paper we present a new method for rapidly evaluating the exact dynamic state of a robot. This method uses a combination of symbolic and numerical computations on the equations of motion, which are developed in the form of polynomials—hence the name, the symbolic polynomial technique.

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Ito, Noboru, and Jun Yoshida. "Crossing change on Khovanov homology and a categorified Vassiliev skein relation." Journal of Knot Theory and Its Ramifications 29, no. 07 (June 2020): 2050051. http://dx.doi.org/10.1142/s0218216520500510.

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Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

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Burmasheva, N. V., and E. Yu Prosviryakov. "Exact solutions of the Navier–Stokes equations for describing an isobaric one-directional vertical vortex flow of a fluid." Diagnostics, Resource and Mechanics of materials and structures, no. 2 (April 2021): 30–51. http://dx.doi.org/10.17804/2410-9908.2021.2.030-051.

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The article proposes a family of exact solutions to the Navier–Stokes equations for describing isobaric inhomogeneous unidirectional fluid motions. Due to the incompressibility equation, the velocity of the inhomogeneous Couette flow depends on two coordinates and time. The expression for the velocity field has a wide functional arbitrariness. This exact solution is obtained by the method of separation of variables, and both algebraic operations (additivity and multiplicativity) are used to substantiate the importance of modifying the classical Couette flow. The article contains extensive bibliographic information that makes it possible to trace a change in the exact Couette solution for various areas of the hydrodynamics of a Newtonian incompressible fluid. The fluid flow is described by a polynomial depending on one variable (horizontal coordinate). The coefficients of the polynomial functionally depend on the second (vertical) coordinate and time; they are determined by a chain of the simplest homogeneous and inhomogeneous partial differential parabolic-type equations. The chain of equations is obtained by the method of undetermined coefficients after substituting the exact solution into the Navier–Stokes equation. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining velocity are polynomials. It is shown that the topology of the vorticity vector and shear stresses has a complex structure even without convective mixing (creeping flow).

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Algazin, O. D. "Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip." Mathematics and Mathematical Modeling, no. 3 (August 3, 2018): 1–12. http://dx.doi.org/10.24108/mathm.0318.0000120.

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In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

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NISHIMURA, HARUMICHI. "QUANTUM COMPUTATION WITH RESTRICTED AMPLITUDES." International Journal of Foundations of Computer Science 14, no. 05 (October 2003): 853–70. http://dx.doi.org/10.1142/s0129054103002059.

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In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.

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Chen, Ray-Bing, and Mong-Na Lo Huang. "Exact D-optimal designs for weighted polynomial regression model." Computational Statistics & Data Analysis 33, no. 2 (April 2000): 137–49. http://dx.doi.org/10.1016/s0167-9473(99)00054-7.

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Znojil, Miloslav. "Symmetrized quartic polynomial oscillators and their partial exact solvability." Physics Letters A 380, no. 16 (April 2016): 1414–18. http://dx.doi.org/10.1016/j.physleta.2016.02.035.

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