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Статті в журналах з теми "Quadratic polynomial"

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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Sankari, Hasan, and Ahmad Abdo. "On Polynomial Solutions of Pell’s Equation." Journal of Mathematics 2021 (August 12, 2021): 1–4. http://dx.doi.org/10.1155/2021/5379284.

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Анотація:

Polynomial Pell’s equation is x 2 − D y 2 = ± 1 , where D is a quadratic polynomial with integer coefficients and the solutions X , Y must be quadratic polynomials with integer coefficients. Let D = a 2 x 2 + a 1 x + a 0 be a polynomial in Z x . In this paper, some quadratic polynomial solutions are given for the equation x 2 − D y 2 = ± 1 which are significant from computational point of view.

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2

AHMADI, OMRAN, FLORIAN LUCA, ALINA OSTAFE, and IGOR E. SHPARLINSKI. "ON STABLE QUADRATIC POLYNOMIALS." Glasgow Mathematical Journal 54, no. 2 (March 29, 2012): 359–69. http://dx.doi.org/10.1017/s001708951200002x.

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AbstractWe recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

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3

YU, PEI, and MAOAN HAN. "FOUR LIMIT CYCLES FROM PERTURBING QUADRATIC INTEGRABLE SYSTEMS BY QUADRATIC POLYNOMIALS." International Journal of Bifurcation and Chaos 22, no. 10 (October 2012): 1250254. http://dx.doi.org/10.1142/s0218127412502549.

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In this paper, we present four limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least four limit cycles with (3,1)-distribution. This result provides a positive answer to an open question in this research area.

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4

Coelho, Terence, and Bahman Kalantari. "How many real attractive fixed points can a polynomial have?" Mathematical Gazette 103, no. 556 (February 14, 2019): 65–76. http://dx.doi.org/10.1017/mag.2019.8.

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While the notion of roots of a quadratic polynomial is rudimentary in high school mathematics, that of its fixed points is uncommon. A real or complex number is a fixed point of a polynomial p (x) p (θ) = θ. The fact that the notion of fixed point of polynomials is not commonly covered in high school or undergraduate mathematics is surprising because the relevance of the fixed points of a quadratic can be demonstrated easily via iterative methods for the approximation of such numbers as , when the quadratic formula offers no remedy.

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5

Yuksel, Cem. "High-Performance Polynomial Root Finding for Graphics." Proceedings of the ACM on Computer Graphics and Interactive Techniques 5, no. 3 (July 25, 2022): 1–15. http://dx.doi.org/10.1145/3543865.

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Анотація:

We present a computationally-efficient and numerically-robust algorithm for finding real roots of polynomials. It begins with determining the intervals where the given polynomial is monotonic. Then, it performs a robust variant of Newton iterations to find the real root within each interval, providing fast and guaranteed convergence and satisfying the given error bound, as permitted by the numerical precision used. For cubic polynomials, the algorithm is more accurate and faster than both the analytical solution and directly applying Newton iterations. It trivially extends to polynomials with arbitrary degrees, but it is limited to finding the real roots only and has quadratic worst-case complexity in terms of the polynomial's degree. We show that our method outperforms alternative polynomial solutions we tested up to degree 20. We also present an example rendering application with a known efficient numerical solution and show that our method provides faster, more accurate, and more robust solutions by solving polynomials of degree 10.

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6

Rukhin, A. L. "Admissible polynomial estimators for quadratic polynomials of normal parameters." Journal of Mathematical Sciences 68, no. 4 (February 1994): 566–76. http://dx.doi.org/10.1007/bf01254283.

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7

Braś, M. "NORDSIECK METHODS WITH INHERENT QUADRATIC STABILITY." Mathematical Modelling and Analysis 16, no. 1 (April 8, 2011): 82–96. http://dx.doi.org/10.3846/13926292.2011.560617.

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Анотація:

We derive suffcient conditions which guarantee that the stability polynomial of Nordsieck method for ordinary differential equations has only two nonzero roots. Examples of such methods up to order four are presented which are A-and L-stable. These examples were obtained by computer search using the Schurcriterion applied to the quadratic factor of the resulting stability polynomials.

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8

Mollin, R. A. "A Completely General Rabinowi1sch Criterion for Complex Quadratic Fields." Canadian Mathematical Bulletin 39, no. 1 (March 1, 1996): 106–10. http://dx.doi.org/10.4153/cmb-1996-013-1.

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AbstractWe provide a criterion for the class group of a complex quadratic field to have exponent at most 2. This is given in terms of the factorization of a generalized Euler-Rabinowitsch polynomial and has consequences for consecutive distinct initial prime-producing quadratic polynomials which we cite as applications.

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9

Louboutin, Stéphane. "Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields." Canadian Journal of Mathematics 42, no. 2 (April 1, 1990): 315–41. http://dx.doi.org/10.4153/cjm-1990-018-3.

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Анотація:

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

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10

Llibre, Jaume, Bruno D. Lopes, and Paulo R. da Silva. "Bifurcations of the Riccati Quadratic Polynomial Differential Systems." International Journal of Bifurcation and Chaos 31, no. 06 (May 2021): 2150094. http://dx.doi.org/10.1142/s0218127421500942.

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In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

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11

Kwun, Young Chel, Mobeen Munir, Waqas Nazeer, and Shin Min Kang. "Some Fixed Points Results of Quadratic Functions in Split Quaternions." Journal of Function Spaces 2016 (2016): 1–5. http://dx.doi.org/10.1155/2016/3460257.

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Анотація:

We attempt to find fixed points of a general quadratic polynomial in the algebra of split quaternion. In some cases, we characterize fixed points in terms of the coefficients of these polynomials and also give the cardinality of these points. As a consequence, we give some simple examples to strengthen the infinitude of these points in these cases. We also find the roots of quadratic polynomials as simple consequences.

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12

NATHANSON, MELVYN B. "CANTOR POLYNOMIALS FOR SEMIGROUP SECTORS." Journal of Algebra and Its Applications 13, no. 05 (February 25, 2014): 1350165. http://dx.doi.org/10.1142/s021949881350165x.

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A packing function on a set Ω in Rn is a one-to-one correspondence between the set of lattice points in Ω and the set N0 of non-negative integers. It is proved that if r and s are relatively prime positive integers such that r divides s - 1, then there exist two distinct quadratic packing polynomials on the sector {(x, y) ∈ R2 : 0 ≤ y ≤ rx/s}. For the rational numbers 1/s, these are the unique quadratic packing polynomials. Moreover, quadratic quasi-polynomial packing functions are constructed for all rational sectors.

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13

Car, Mireille. "Quadratic forms with polynomial coefficients." Acta Arithmetica 113, no. 2 (2004): 131–55. http://dx.doi.org/10.4064/aa113-2-2.

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14

Silverman, Robert D. "The multiple polynomial quadratic sieve." Mathematics of Computation 48, no. 177 (January 1, 1987): 329. http://dx.doi.org/10.1090/s0025-5718-1987-0866119-8.

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15

Han, Xuli. "Piecewise quadratic trigonometric polynomial curves." Mathematics of Computation 72, no. 243 (March 26, 2003): 1369–78. http://dx.doi.org/10.1090/s0025-5718-03-01530-8.

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16

James, Donald G. "Quadratic forms over polynomial rings." Communications in Algebra 18, no. 1 (January 1990): 247–51. http://dx.doi.org/10.1080/00927879008823910.

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17

Zhang, Lina, and Xuesi Ma. "New Refinements and Improvements of Jordan’s Inequality." Mathematics 6, no. 12 (November 26, 2018): 284. http://dx.doi.org/10.3390/math6120284.

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Анотація:

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

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18

Mollin, R. A., and H. C. Williams. "Quadratic Non-Residues and Prime-Producing Polynomials." Canadian Mathematical Bulletin 32, no. 4 (December 1, 1989): 474–78. http://dx.doi.org/10.4153/cmb-1989-068-1.

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Анотація:

AbstractWe will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

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19

Foster, William H., and Ilia Krasikov. "Explicit Bounds for Hermite Polynomials in the Oscillatory Region." LMS Journal of Computation and Mathematics 3 (2000): 307–14. http://dx.doi.org/10.1112/s1461157000000310.

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AbstractWe apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

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20

Lu, Daogang, and Chao Guo. "Development and Validation of a Three-Dimensional Diffusion Code Based on a High Order Nodal Expansion Method for Hexagonal-zGeometry." Science and Technology of Nuclear Installations 2016 (2016): 1–21. http://dx.doi.org/10.1155/2016/6340652.

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A three-dimensional, multigroup, diffusion code based on a high order nodal expansion method for hexagonal-zgeometry (HNHEX) was developed to perform the neutronic analysis of hexagonal-zgeometry. In this method, one-dimensional radial and axial spatially flux of each node and energy group are defined as quadratic polynomial expansion and four-order polynomial expansion, respectively. The approximations for one-dimensional radial and axial spatially flux both have second-order accuracy. Moment weighting is used to obtain high order expansion coefficients of the polynomials of one-dimensional radial and axial spatially flux. The partially integrated radial and axial leakages are both approximated by the quadratic polynomial. The coarse-mesh rebalance method with the asymptotic source extrapolation is applied to accelerate the calculation. This code is used for calculation of effective multiplication factor, neutron flux distribution, and power distribution. The numerical calculation in this paper for three-dimensional SNR and VVER 440 benchmark problems demonstrates the accuracy of the code. In addition, the results show that the accuracy of the code is improved by applying quadratic approximation for partially integrated axial leakage and four-order approximation for one-dimensional axial spatially flux in comparison to flat approximation for partially integrated axial leakage and quadratic approximation for one-dimensional axial spatially flux.

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21

Kulkarni, Raghavendra G. "Intersect a quartic to extract its roots." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 16, no. 1 (December 1, 2017): 73–76. http://dx.doi.org/10.1515/aupcsm-2017-0006.

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AbstractIn this note we present a new method for determining the roots of a quartic polynomial, wherein the curve of the given quartic polynomial is intersected by the curve of a quadratic polynomial (which has two unknown coefficients) at its root point; so the root satisfies both the quartic and the quadratic equations. Elimination of the root term from the two equations leads to an expression in the two unknowns of quadratic polynomial. In addition, we introduce another expression in one unknown, which leads to determination of the two unknowns and subsequently the roots of quartic polynomial.

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22

Kulkarni, Raghavendra G. "Delving Deeper: Decomposing Quintics." Mathematics Teacher 102, no. 9 (May 2009): 710–13. http://dx.doi.org/10.5951/mt.102.9.0710.

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Анотація:

Polynomial decomposition, also referred to as polynomial factorization, is the process of splitting a given polynomial of degree n into its constituent factors—that is, polynomials of lower degree. A reducible polynomial over a given field—such as the real (ℝ), complex (ℂ), or rational numbers (ℚ)—is one that can be factored into polynomials of lower degree with coefficients in that field; otherwise, it is irreducible over the field (Thangadurai 2007). From the fundamental theorem of algebra, we know that a polynomial with integer coefficients is completely reducible into linear factors over the complex field, whereas it is reducible to linear and quadratic factors over ℝ; however, it may be irreducible over ℚ. If a polynomial (with rational coefficients) can be factored over ℚ, Gauss's lemma states, it can be factored over the integers as well. The fundamental theorem of algebra is only an existence proof and does not provide any procedure for factoring a polynomial.

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23

Kulkarni, Raghavendra G. "Delving Deeper: Decomposing Quintics." Mathematics Teacher 102, no. 9 (May 2009): 710–13. http://dx.doi.org/10.5951/mt.102.9.0710.

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Анотація:

Polynomial decomposition, also referred to as polynomial factorization, is the process of splitting a given polynomial of degree n into its constituent factors—that is, polynomials of lower degree. A reducible polynomial over a given field—such as the real (ℝ), complex (ℂ), or rational numbers (ℚ)—is one that can be factored into polynomials of lower degree with coefficients in that field; otherwise, it is irreducible over the field (Thangadurai 2007). From the fundamental theorem of algebra, we know that a polynomial with integer coefficients is completely reducible into linear factors over the complex field, whereas it is reducible to linear and quadratic factors over ℝ; however, it may be irreducible over ℚ. If a polynomial (with rational coefficients) can be factored over ℚ, Gauss's lemma states, it can be factored over the integers as well. The fundamental theorem of algebra is only an existence proof and does not provide any procedure for factoring a polynomial.

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24

Tamang, Bal Bahadur, and Ajay Singh. "The Solvability of Polynomial Pell’s Equation." Journal of Institute of Science and Technology 25, no. 2 (December 25, 2020): 125–32. http://dx.doi.org/10.3126/jist.v25i2.33749.

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Анотація:

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

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25

BASU, SOHAM. "STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING." Bulletin of the Australian Mathematical Society 104, no. 2 (January 18, 2021): 249–55. http://dx.doi.org/10.1017/s0004972720001434.

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Анотація:

AbstractWithout resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

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26

Zakharov, Victor G. "Polynomial spaces reproduced by elliptic scaling functions." International Journal of Wavelets, Multiresolution and Information Processing 13, no. 06 (November 2015): 1550042. http://dx.doi.org/10.1142/s0219691315500423.

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In this paper, we consider the so-called elliptic scaling functions [V. G. Zakharov, Elliptic scaling functions as compactly supported multivariate analogs of the B-splines, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014) 1450018]. Any elliptic scaling function satisfies the refinement relation with a real isotropic dilation matrix; and, in the paper, we prove that any real isotropic matrix is similar to an orthogonal matrix and the similarity transformation matrix determines a positive-definite quadratic form. We prove that the polynomial space reproduced by integer shifts of a compactly supported function can be usually considered as a polynomial solution to a system of constant coefficient PDE’s. We show that the algebraic polynomials reproduced by a compactly supported elliptic scaling function belong to the kernel of a homogeneous elliptic differential operator that the differential operator corresponds to the quadratic form; and thus any elliptic scaling function reproduces only affinely-invariant polynomial spaces. However, in the paper, we present nonstationary elliptic scaling functions such that the scaling functions can reproduce no scale-invariant (only shift-invariant) polynomial spaces.

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27

MA, LIANGANG. "A continuity result on quadratic matings with respect to parameters of odd denominator rationals." Mathematical Proceedings of the Cambridge Philosophical Society 167, no. 02 (June 1, 2018): 369–88. http://dx.doi.org/10.1017/s0305004118000397.

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AbstractIn this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

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28

Horwitz, Alan. "Even compositions of entire functions and related matters." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 63, no. 2 (October 1997): 225–37. http://dx.doi.org/10.1017/s1446788700000665.

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Анотація:

AbstractWe examine when the composition of two entire functions f and g is even, and extend some of our results to cyclic compositions in general. If p is a polynomial, then we prove that f ^ p is even for a non-constant entire function f if and only if p is even, odd plus a constant, or a quadratic polynomial composed with an odd polynomial. Similar results are proven for odd compositions. We also show that p ^ f can be even when f and no derivative of f are even or odd, where p is a polynomial. We extend some results of an earlier paper to cyclic compositions of polynomials. We also show that our results do not extend in general to rational functions or polynomials in two variables.

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29

Berthomieu, Jérémy, Jean-Charles Faugère, and Ludovic Perret. "Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case." Journal of Complexity 31, no. 4 (August 2015): 590–616. http://dx.doi.org/10.1016/j.jco.2015.04.001.

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30

Zhang, Fan, Jinjiang Li, Peiqiang Liu, and Hui Fan. "Computing knots by quadratic and cubic polynomial curves." Computational Visual Media 6, no. 4 (October 17, 2020): 417–30. http://dx.doi.org/10.1007/s41095-020-0186-4.

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Анотація:

AbstractA new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.

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31

Borggaard, Jeff, and Lizette Zietsman. "On Approximating Polynomial-Quadratic Regulator Problems." IFAC-PapersOnLine 54, no. 9 (2021): 329–34. http://dx.doi.org/10.1016/j.ifacol.2021.06.090.

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32

Kim, Myung-Hwan, Yuanhua Wang, and Fei Xu. "UNIVERSAL QUADRATIC FORMS OVER POLYNOMIAL RINGS." Journal of the Korean Mathematical Society 45, no. 5 (September 30, 2008): 1311–22. http://dx.doi.org/10.4134/jkms.2008.45.5.1311.

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33

Llibre, Jaume, and Xiang Zhang. "Polynomial First Integrals of Quadratic Systems." Rocky Mountain Journal of Mathematics 31, no. 4 (December 2001): 1317–71. http://dx.doi.org/10.1216/rmjm/1021249443.

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34

Marszałek, Roman, and Władyslaw Narkiewicz. "Finite polynomial orbits in quadratic rings." Ramanujan Journal 12, no. 1 (August 2006): 91–130. http://dx.doi.org/10.1007/s11139-006-9578-z.

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35

Llibre, Jaume, and Claudia Valls. "Algebraic Limit Cycles on Quadratic Polynomial Differential Systems." Proceedings of the Edinburgh Mathematical Society 61, no. 2 (February 27, 2018): 499–512. http://dx.doi.org/10.1017/s0013091517000244.

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Анотація:

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

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36

OH, SUNG-KWUN, DONG-WON KIM, and WITOLD PEDRYCZ. "HYBRID FUZZY POLYNOMIAL NEURAL NETWORKS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10, no. 03 (June 2002): 257–80. http://dx.doi.org/10.1142/s0218488502001478.

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Анотація:

We propose a hybrid architecture based on a combination of fuzzy systems and polynomial neural networks. The resulting Hybrid Fuzzy Polynomial Neural Networks (HFPNN) dwells on the ideas of fuzzy rule-based computing and polynomial neural networks. The structure of the network comprises of fuzzy polynomial neurons (FPNs) forming the nodes of the first (input) layer of the HFPNN and polynomial neurons (PNs) that are located in the consecutive layers of the network. In the FPN (that forms a fuzzy inference system), the generic rules assume the form "if A then y = P(x) " where A is fuzzy relation in the condition space while P(x) is a polynomial standing in the conclusion part of the rule. The conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as constant, linear, quadratic, and modified quadratic. As the premise part of the rules, both triangular and Gaussian-like membership functions are considered. Each PN of the network realizes a polynomial type of partial description (PD) of the mapping between input and out variables. HFPNN is a flexible neural architecture whose structure is based on the Group Method of Data Handling (GMDH) and developed through learning. In particular, the number of layers of the PNN is not fixed in advance but is generated in a dynamic way. The experimental part of the study involves two representative numerical examples such as chaotic time series and Box-Jenkins gas furnace data.

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37

Yu, Pei, and Yanni Zeng. "Visualization of Four Limit Cycles in Near-Integrable Quadratic Polynomial Systems." International Journal of Bifurcation and Chaos 30, no. 15 (December 9, 2020): 2050236. http://dx.doi.org/10.1142/s0218127420502363.

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Анотація:

It has been known for almost 40 years that general planar quadratic polynomial systems can have four limit cycles. Recently, four limit cycles were also found in near-integrable quadratic polynomial systems. To help more people to understand limit cycles theory, the visualization of such four numerically simulated limit cycles in quadratic systems has attracted researchers’ attention. However, for near-integral systems, such visualization becomes much more difficult due to limitation on choosing parameter values. In this paper, we start from the simulation of the well-known quadratic systems constructed around the end of 1979, then reconsider the simulation of a recently published quadratic system which exhibits four big size limit cycles, and finally provide a concrete near-integral quadratic polynomial system to show four normal size limit cycles.

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38

K, Vedavathi, and A. Swapna. "Deep Learning based Handwritten Polynomial Equation Solver." International Journal for Research in Applied Science and Engineering Technology 10, no. 6 (June 30, 2022): 1880–83. http://dx.doi.org/10.22214/ijraset.2022.44171.

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Abstract: Polynomials are algebraic expressions involving a sum of powers in one or more variables multiplied by its coefficients. If x is a variable, a0x n +a1x n-1 +a2x n-2 +....+an then it's a (n) powers polynomial. Human is capable to solve this type of mathematical problems. In this work, we propose a system in which machines can achieve the cognitional skills that can understand the problem by visual context. By taking an input image of Handwritten polynomial equations and simplifies the problem by generating the answer as an output. Here machine can able to solve quadratic, cubic, quartic, quantic, sextic as well as (n) powers polynomials. This proposed work can be workable in an embedded system as well as a mobile application. In this scope for recognition purposes, we use a CNN model. Robust handwritten character recognition is a tricky job in the area of image processing. Among all the problem handwritten mathematical expression recognition is one of the complicated issue in the area of computer vision research. Segmentation and classification of specific character makes the task more difficult. In this paper a group of handwritten quadratic equation as well as a single quadratic equation are considered to recognize and make a solution for those equations. Horizontal compact projection analysis and combined connected component analysis methods are used for segmentation. For classification of specific character we apply Convolutional Neural Network. Each of the correct detection, character string operation is used for the solution of the equation. The proposed workflow system automatically simplifies the Handwritten polynomial equation and has been done a really good performance. Developing an automatic equation recognizer and solver has been a desire of the researchers who worked in the field of NLP for many years.

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39

Han, Maoan, Tatjana Petek, and Valery G. Romanovski. "On Some Symmetries of Quadratic Systems." Symmetry 12, no. 8 (August 4, 2020): 1300. http://dx.doi.org/10.3390/sym12081300.

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Анотація:

We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented.

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40

Li, Zong Huai, Jiang Hong Gong, Zhi Jian Peng, and He Zhuo Miao. "Description of the Nanoindentation Unloading Behavior of Brittle Ceramics with a Modified Surface." Key Engineering Materials 336-338 (April 2007): 2422–25. http://dx.doi.org/10.4028/www.scientific.net/kem.336-338.2422.

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Анотація:

The nanoindentation unloading behavior of some brittle ceramics with modified surfaces was analyzed. It was found that the unloading data may be described well with a quadratic polynomial. The physical meaning of the quadratic polynomial in describing the nanoindentation unloading behavior was then discussed by considering the effect of residual contact stress on the force-displacement relationship. It was suggested that the quadratic polynomial may be considered as a modified form of the basic forcedisplacement relationship for the contact of an isotropic elastic half-space by a rigid conical punch.

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41

BIRMAJER, DANIEL, JUAN B. GIL та MICHAEL D. WEINER. "FACTORIZATION OF QUADRATIC POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ". Journal of Algebra and Its Applications 06, № 06 (грудень 2007): 1027–37. http://dx.doi.org/10.1142/s021949880700265x.

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We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring ℤ[[x]] of formal power series over the integers. In particular, for polynomials of the form pn + pm βx + αx2 with n,m ≥ 1 and p prime, we show that reducibility in ℤ[[x]] is equivalent to reducibility in ℤp[x], the ring of polynomials over the p-adic integers.

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42

Toral, Fábio Luiz Buranelo, Roberto Augusto de Almeida Torres Júnior, Paulo Sávio Lopes, Luiz Otávio Campos da Silva, and João Cruz Reis Filho. "Modeling the effect of the age of dam at calving on the weaning weight of Charolais-Zebu crossbred calves." Revista Brasileira de Zootecnia 38, no. 7 (July 2009): 1229–37. http://dx.doi.org/10.1590/s1516-35982009000700011.

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The objective of this study was to evaluate alternatives for modeling the age of dam at calving (AOD) effect on the weaning weight of Charolais-Zebu crossbred calves. Data from 56,965 calves were analyzed, using statistical models considering the fixed effects of the contemporary groups, sire and dam genetic groups, and AOD. The AOD effect was fitted to models using annual age classes, and ordinary quadratic to quintic-ordered polynomials (OP) or segmented polynomials (SP) with two, three, four, six and twelve evenly spaced intervals. In the case of segmented polynomials, general linear and quadratic effects and only one quadratic additional term from each knot were considered. The AOD effects were nested within sex of calf in all cases. According to the fitting criteria, the F-test for the reduction of residual sum of squares, coefficient of determination, residual sum of squares and mean of squared residuals, the three interval segmented polynomial (two knots) fitted to the data as well as the more complex polynomials.

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43

Friedlander, J. B., and H. Iwaniec. "Square-free values of quadratic polynomials." Proceedings of the Edinburgh Mathematical Society 53, no. 2 (April 30, 2010): 385–92. http://dx.doi.org/10.1017/s0013091508000989.

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AbstractWe consider the quadratic polynomial m2 + D and study the asymptotic formula for the number of integers m, 1≤ m ≤ M, for which the values of the polynomial are square-free. We are interested in particular in the question of how small we can take m in relation to D and still have the asymptotic hold.

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44

Journal, Baghdad Science. "An Approximate Solution of some Variational Problems Using Boubaker Polynomials." Baghdad Science Journal 15, no. 1 (March 4, 2018): 106–9. http://dx.doi.org/10.21123/bsj.15.1.106-109.

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In this paper, an approximate solution of nonlinear two points boundary variational problem is presented. Boubaker polynomials have been utilized to reduce these problems into quadratic programming problem. The convergence of this polynomial has been verified; also different numerical examples were given to show the applicability and validity of this method.

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45

Krumm, David. "A local–global principle in the dynamics of quadratic polynomials." International Journal of Number Theory 12, no. 08 (October 19, 2016): 2265–97. http://dx.doi.org/10.1142/s1793042116501360.

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Анотація:

Let [Formula: see text] be a number field, [Formula: see text] a quadratic polynomial, and [Formula: see text]. We show that if [Formula: see text] has a point of period [Formula: see text] in every non-archimedean completion of [Formula: see text], then [Formula: see text] has a point of period [Formula: see text] in [Formula: see text]. For [Formula: see text] we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over [Formula: see text] for which this local–global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn–Poonen–Schaefer in the case [Formula: see text]. More precisely, we show that for every quadratic polynomial [Formula: see text] there exist infinitely many primes [Formula: see text] such that [Formula: see text] does not have a point of period four in the [Formula: see text]-adic field [Formula: see text]. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period five.

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46

ZHANG, XU, YUMING SHI, and GUANRONG CHEN. "CONSTRUCTING CHAOTIC POLYNOMIAL MAPS." International Journal of Bifurcation and Chaos 19, no. 02 (February 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 non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

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47

Le, Thi, Mohand Ouanes, and Ahmed Zidna. "Computing real zeros of a polynomial by branch and bound and branch and reduce algorithms." Yugoslav Journal of Operations Research 24, no. 1 (2014): 53–69. http://dx.doi.org/10.2298/yjor120620004l.

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Анотація:

In this paper we propose two algorithms based on branch and bound method and reduced interval techniques to compute all real zeros of a polynomial. Quadratic bounding functions are proposed which are better than the well known linear underestimator. Experimental result shows the efficiency of the two algorithms when facing ill-conditioned polynomials.

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48

Wang, Chuanfu, and Qun Ding. "A Class of Quadratic Polynomial Chaotic Maps and Their Fixed Points Analysis." Entropy 21, no. 7 (July 4, 2019): 658. http://dx.doi.org/10.3390/e21070658.

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When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors.

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49

Anwar, Y. R., H. Tasman, and N. Hariadi. "Determining implicit equation of conic section from quadratic rational Bézier curve using Gröbner basis." Journal of Physics: Conference Series 2106, no. 1 (November 1, 2021): 012017. http://dx.doi.org/10.1088/1742-6596/2106/1/012017.

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Abstract The Gröbner Basis is a subset of finite generating polynomials in the ideal of the polynomial ring k[x 1,…,xn ]. The Gröbner basis has a wide range of applications in various areas of mathematics, including determining implicit polynomial equations. The quadratic rational Bézier curve is a rational parametric curve that is generated by three control points P 0(x 0,y 0), P 1(xi ,yi ), P 2(x 2,y 2) in ℝ2 and weights ω 0, ω 1, ω 2, where the weights ω i are corresponding to control points Pi (xi, yi ), for i = 0,1, 2. According to Cox et al (2007), the quadratic rational Bézier curve can represent conic sections, such as parabola, hyperbola, ellipse, and circle, by defining the weights ω 0 = ω 2 = 1 and ω 1 = ω for any control points P 0(x 0, y 0), P 1(x 1, y 1), and P 2(x 2, y 2). This research is aimed to obtain an implicit polynomial equation of the quadratic rational Bézier curve using the Gröbner basis. The polynomial coefficients of the conic section can be expressed in the term of control points P 0(x 0, y 0), P 1(x 1, y 1), P 2(x 2, y 2) and weight ω, using Wolfram Mathematica. This research also analyzes the effect of changes in weight ω on the shape of the conic section. It shows that parabola, hyperbola, and ellipse can be formed by defining ω = 1, ω > 1, and 0 < ω < 1, respectively.

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50

Chen, Jie. "Algebraic Gordian distance." Journal of Knot Theory and Its Ramifications 28, no. 04 (April 2019): 1950024. http://dx.doi.org/10.1142/s021821651950024x.

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Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with algebraic Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.

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