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Journal articles on the topic 'Polynomial product'

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Published: 9 March 2023

Last updated: 11 March 2023

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1

Szilágyi, Zsolt. "On Chern classes of the tensor product of vector bundles." Acta Universitatis Sapientiae, Mathematica 14, no. 2 (December 1, 2022): 330–40. http://dx.doi.org/10.2478/ausm-2022-0022.

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Abstract We present two formulas for Chern classes (polynomial) of the tensor product of two vector bundles. In the first formula the Chern polynomial of the product is expressed as determinant of a polynomial in a matrix variable involving the Chern classes of the first bundle with Chern classes of the second bundle as coefficients. In the second formula the total Chern class of the tensor product is expressed as resultant of two explicit polynomials. Finally, formulas for the total Chern class of the second symmetric and the second alternating products are deduced.

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2

Sastre, Jorge, and Javier Ibáñez. "Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method." Mathematics 9, no. 14 (July 7, 2021): 1600. http://dx.doi.org/10.3390/math9141600.

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Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.

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3

GAN, C. S. "The complete product of annihilatingly unique digraphs." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1327–31. http://dx.doi.org/10.1155/ijmms.2005.1327.

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LetGbe a digraph withnvertices and letA(G)be its adjacency matrix. A monic polynomialf(x)of degree at mostnis called an annihilating polynomial ofGiff(A(G))=0.Gis said to be annihilatingly unique if it possesses a unique annihilating polynomial. Difans and diwheels are two classes of annihilatingly unique digraphs. In this paper, it is shown that the complete product of difan and diwheel is annihilatingly unique.

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4

Koelink, H. T. "Addition Formula For Big q-Legendre Polynomials From The Quantum Su(2) Group." Canadian Journal of Mathematics 47, no. 2 (April 1, 1995): 436–48. http://dx.doi.org/10.4153/cjm-1995-024-8.

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AbstractFrom Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.

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5

Knor, Martin, and Niko Tratnik. "A New Alternative to Szeged, Mostar, and PI Polynomials—The SMP Polynomials." Mathematics 11, no. 4 (February 13, 2023): 956. http://dx.doi.org/10.3390/math11040956.

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Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at x=1, we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors.

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6

DIAO, Y., G. HETYEI, and K. HINSON. "TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY." Journal of Knot Theory and Its Ramifications 18, no. 05 (May 2009): 561–89. http://dx.doi.org/10.1142/s0218216509007075.

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It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

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7

González, Manuel, and Joaquí M. Gutiérrez. "Polynomial Grothendieck properties." Glasgow Mathematical Journal 37, no. 2 (May 1995): 211–19. http://dx.doi.org/10.1017/s0017089500031116.

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AbstractA Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.

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8

Jiang, Xue, and Kai Cui. "The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors." Axioms 10, no. 3 (August 19, 2021): 193. http://dx.doi.org/10.3390/axioms10030193.

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Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

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9

Chen, Lin-An, Tzong Shi Lee, and Wenyaw Chan. "Tensor product polynomial splines." Communications in Statistics - Theory and Methods 26, no. 9 (January 1997): 2093–111. http://dx.doi.org/10.1080/03610929708832036.

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10

Hammerlindl, Andy. "Polynomial global product structure." Proceedings of the American Mathematical Society 142, no. 12 (August 15, 2014): 4297–303. http://dx.doi.org/10.1090/s0002-9939-2014-12255-6.

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11

DIAO, Y., and G. HETYEI. "Relative Tutte Polynomials of Tensor Products of Coloured Graphs." Combinatorics, Probability and Computing 22, no. 6 (September 4, 2013): 801–28. http://dx.doi.org/10.1017/s0963548313000370.

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The tensor product (G1,G2) of a graph G1 and a pointed graph G2 (containing one distinguished edge) is obtained by identifying each edge of G1 with the distinguished edge of a separate copy of G2, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to coloured graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the coloured tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion/contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot.

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12

Liu, Juan, and Laiyi Zhu. "Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points." Journal of Mathematics 2022 (January 4, 2022): 1–19. http://dx.doi.org/10.1155/2022/5649969.

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In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square − 1,1 2 . And, we prove that the growth order of the Lebesgue constant is O n + 2 2 . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O n . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square − 1,1 2 , the growth order of which is O ln n 2 .

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13

Glasby, S. P. "On the tensor product of polynomials over a ring." Journal of the Australian Mathematical Society 71, no. 3 (December 2001): 307–24. http://dx.doi.org/10.1017/s1446788700002950.

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AbstractGiven polynomials a and b over an integral domain R, their tensor product (denoted a ⊗ b) is a polynomial over R of degree deg(a) deg(b) whose roots comprise all products αβ, where α is a root of a, and β is a root of b. This paper considers basic properties of ⊗ including how to factor a ⊗ b into irreducibles factors, and the direct sum decomposition of the ⊗-product of fields.

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14

Aguirre-Hernández, Baltazar, Edgar-Cristian Díaz-González, Carlos-Arturo Loredo-Villalobos, and Faustino-Ricardo García-Sosa. "Properties of the Set of Hadamardized Hurwitz Polynomials." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/695279.

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We say that a Hurwitz polynomialptis a Hadamardized polynomial if there are two Hurwitz polynomialsftandgtsuch thatf∗g=p, wheref∗gis the Hadamard product offandg. In this paper, we prove that the set of all Hadamardized Hurwitz polynomials is an open, unbounded, nonconvex, and arc-connected set. Furthermore, we give a result so that a fourth-degree Hurwitz interval polynomial is a Hadamardized polynomial family and we discuss an approach of differential topology in the study of the set of Hadamardized Hurwitz polynomials.

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15

ELLIS-MONAGHAN, J., and I. MOFFATT. "Evaluations of Topological Tutte Polynomials." Combinatorics, Probability and Computing 24, no. 3 (October 10, 2014): 556–83. http://dx.doi.org/10.1017/s0963548314000571.

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We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

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16

Desmorat, R., N. Auffray, B. Desmorat, B. Kolev, and M. Olive. "Generic separating sets for three-dimensional elasticity tensors." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (June 2019): 20190056. http://dx.doi.org/10.1098/rspa.2019.0056.

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We define a generic separating set of invariant functions (a.k.a. a weak functional basis ) for tensors. We then produce two generic separating sets of polynomial invariants for three-dimensional elasticity tensors, one consisting of 19 polynomials and one consisting of 21 polynomials (but easier to compute), and a generic separating set of 18 rational invariants. As a by-product, a new integrity basis for the fourth-order harmonic tensor is provided.

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17

Białas, Stanisław, and Michał Góra. "On the Existence of Hurwitz Polynomials with no Hadamard Factorization." Electronic Journal of Linear Algebra 36, no. 36 (April 14, 2020): 210–13. http://dx.doi.org/10.13001/ela.2020.5097.

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A Hurwitz stable polynomial of degree $n\geq1$ has a Hadamard factorization if it is a Hadamard product (i.e., element-wise multiplication) of two Hurwitz stable polynomials of degree $n$. It is known that Hurwitz stable polynomials of degrees less than four have a Hadamard factorization. It is shown that, for arbitrary $n\geq4$, there exists a Hurwitz stable polynomial of degree $n$ which does not have a Hadamard factorization.

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18

Pin, J. E., and P. Weil. "Polynomial closure and unambiguous product." Theory of Computing Systems 30, no. 4 (August 1997): 383–422. http://dx.doi.org/10.1007/bf02679467.

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19

Dhanalakshmi K, Amalorpava Jerline J, and Benedict Michaelraj L. "Dichromatic Polynomial of Product Digraphs." Electronic Notes in Discrete Mathematics 53 (September 2016): 165–72. http://dx.doi.org/10.1016/j.endm.2016.05.015.

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20

Pin, J. —E, and P. Weil. "Polynomial Closure and Unambiguous Product." Theory of Computing Systems 30, no. 4 (July 1, 1997): 383–422. http://dx.doi.org/10.1007/s002240000058.

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21

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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22

Qian, Jiang, Sujuan Zheng, Fan Wang, and Zhuojia Fu. "Bivariate Polynomial Interpolation over Nonrectangular Meshes." Numerical Mathematics: Theory, Methods and Applications 9, no. 4 (November 2016): 549–78. http://dx.doi.org/10.4208/nmtma.2016.y15027.

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AbstractIn this paper, bymeans of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

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23

JIN, XIAN'AN, and FUJI ZHANG. "ALEXANDER POLYNOMIAL FOR EVEN GRAPHS WITH REFLECTIVE SYMMETRY." Journal of Knot Theory and Its Ramifications 17, no. 10 (October 2008): 1241–56. http://dx.doi.org/10.1142/s0218216508006610.

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Based on the connection with Alexander polynomial of special alternating links, Murasugi and Stoimenow introduced the Alexander polynomial of even graphs. In this paper, we study the Alexander polynomial of spatial even graphs with reflective symmetry. Roughly speaking, we prove that the Alexander polynomial of one half of a spatial even graph with reflective symmetry is a divisor of that of the whole spatial even graph. Then, we apply the result to a family of special alternating links, expressing the Alexander polynomial of such a link as the product of Alexander polynomials of two smaller special alternating links derived from the two isotopic "halves" of the original link.

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24

Nonkané, Ibrahim, and Latevi Lawson. "Invariant differential operators and the generalized symmetric group." Gulf Journal of Mathematics 13, no. 2 (September 16, 2022): 19–32. http://dx.doi.org/10.56947/gjom.v13i2.738.

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In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as a D-module, under the map π: spec OX →spec OXG(r,n), where OXG(r,n) is the ring of invariant polynomial under the action of the wreath product G(r, p):= Z/rZ ~Sn. We first describe the generators of the simple components of π+(OX) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials

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25

DUNKL, CHARLES F. "A LAGUERRE POLYNOMIAL ORTHOGONALITY AND THE HYDROGEN ATOM." Analysis and Applications 01, no. 02 (April 2003): 177–88. http://dx.doi.org/10.1142/s0219530503000132.

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The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis, an analogous strange orthogonality is obtained for Meixner polynomials.

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26

XU, XUE-XIANG, LI-YUN HU, and HONG-YI FAN. "ON THE NORMALIZED TWO-MODE PHOTON-SUBTRACTED SQUEEZED VACUUM STATE." Modern Physics Letters A 24, no. 32 (October 20, 2009): 2623–30. http://dx.doi.org/10.1142/s0217732309031168.

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We show that the two-mode photon-subtracted squeezed state (TPSSS) is a squeezed two-variable Hermite polynomial excitation state, and we can therefore determine its normalization as a Jacobi polynomial of the squeezing parameter. Some new relations about the Jacobi polynomials are obtained by this analysis. We also show that the TPSSS can be treated as a two-variable Hermite-polynomial excitation on squeezed vacuum state. The technique of integration within an ordered product of operators brings convenience in our derivation.

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27

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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28

Saropah, Saropah. "Akar-akar Polinomial Separable sebagai Pembentuk Perluasan Normal pada Ring Modulo." CAUCHY 2, no. 3 (November 15, 2012): 148. http://dx.doi.org/10.18860/ca.v2i3.3124.

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<div class="standard"><a id="magicparlabel-944">One of the most important uses of the ring and field theory is an extension of a broader field so that a polynomial can be found to have roots. In this study researchers took modulo a prima as follows indeterminate coeffcients to search for his roots extension the solutions of that it can seen normal. A field is subject to a polynomial form a set of polynomials , where is a coefficient field its terms modulo a prime number. Of the set of polynomial exists a polynomial is irreducible, it is necessary to extension the field to know the roots of the solution. Suppose to extension of the field is a field . Field is called extension the field over a field , if the field is subfield of the field and is irreducible polynomial in then can be factored as a product of linear factors in the splitting field. If the polynomial has different roots in the splitting field the polynomial is called polynomial separable. In this study polynomial separable is contained of odd degree in which the coefficients of the tribes polynomial is contained in the extension field. Polynomial is called a polynomial separable odd because it has different roots in the factors and there is one factor in a polynomial in the field. Splitting field that contains all the set of polynomials separable is called normal extension.</a></div>

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29

Rajkovic, Predrag, Sladjana Marinkovic, and Miomir Stankovic. "Orthogonal polynomials with varying weight of Laguerre type." Filomat 29, no. 5 (2015): 1053–62. http://dx.doi.org/10.2298/fil1505053r.

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In this paper, we define and examine a new functional product in the space of real polynomials. This product includes the weight function which depends on degrees of the participants. In spite of it does not have all properties of an inner product, we construct the sequence of orthogonal polynomials. These polynomials can be eigenfunctions of a differential equation what was used in some considerations in the theoretical physics. In special, we consider Laguerre type weight function and prove that the corresponding orthogonal polynomial sequence is connected with Laguerre polynomials. We study their differential properties and orthogonal properties of some related rational and exponential functions.

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30

Blais, Eric, Ryan O’Donnell, and Karl Wimmer. "Polynomial regression under arbitrary product distributions." Machine Learning 80, no. 2-3 (April 27, 2010): 273–94. http://dx.doi.org/10.1007/s10994-010-5179-6.

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31

Sohn, Moo Young, and Jaeun Lee. "Characteristic polynomials of some weighted graph bundles and its application to links." International Journal of Mathematics and Mathematical Sciences 17, no. 3 (1994): 503–10. http://dx.doi.org/10.1155/s0161171294000748.

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In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weightedK2(K¯2)-bundles over a weighted graphG?can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs areGAs an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

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32

MONKS, KEENAN, SARAH PELUSE, and LYNNELLE YE. "CONGRUENCE PROPERTIES OF BORCHERDS PRODUCT EXPONENTS." International Journal of Number Theory 09, no. 06 (September 2013): 1563–78. http://dx.doi.org/10.1142/s1793042113500437.

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In his striking 1995 paper, Borcherds [Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math.120 (1995) 161–213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

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33

Kashpur, O. F. "UNDAMENTAL POLYNOMIALS OF HERMITE’SINTERPOLATION FORMULA IN LINEAR NORMAL AND INEUCLIDEAN SPACES." Journal of Numerical and Applied Mathematics, no. 2 (2022): 50–58. http://dx.doi.org/10.17721/2706-9699.2022.2.06.

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In a linear infinite-dimensional space with a scalar product and in a finite-dimensional Euclidean space the interpolation Hermite polynomial with a minimal norm, generated by a Gaussian measure, contains fundamental polynomials are shown. The accuracy of Hermit’s interpolation formulas on polynomials of the appropriate degree are researched.

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34

Beletsky, Anatoly. "Factorization of the Degree of Semisimple Polynomials Over the Galois Fields of Arbitrary Characteristics." WSEAS TRANSACTIONS ON MATHEMATICS 21 (March 26, 2022): 160–72. http://dx.doi.org/10.37394/23206.2022.21.23.

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To semisimple polynomials over a Galois field of arbitrary characteristics we mean polynomials formed by the product of two coprime irreducible polynomials with a priori unknown degrees. The main task of this study is to develop an efficient algorithm for factorizing the degree of semisimple polynomials. The efficient factorization algorithms are those that provide a minimum of computational complexity. The proposed algorithm is reduced to solving a system of two equations for the unknown degrees of the factors of a semisimple polynomial. The right-hand sides of the system of equations are as follows: one of them is the degree n of a semisimple polynomial, known a priori, and the second, the cycle period C of the polynomial, is calculated using the so-called fiducial grid. At each rung of the ladder, the simplest recurrent modular computations are carried out, after which the cycle period C of the semisimple polynomial is determined, which is equal to the least common multiple of the degrees of the factors of the polynomial. Reducing the amount of calculations is achieved by switching from a linear scale when determining the cycle period C to a logarithmic one. The proposed factorization algorithms are invariant to the characteristic of the field generated by irreducible polynomials. Various options for the relationship between the parameters n and C are considered.

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35

Hansen, Jennie C. "Factorization in Fq[x] and Brownian Motion." Combinatorics, Probability and Computing 2, no. 3 (September 1993): 285–99. http://dx.doi.org/10.1017/s0963548300000687.

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We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.

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Baker, B. M., and D. E. Handelman. "Positive Polynomials and Time Dependent Integer-Valued Random Variables." Canadian Journal of Mathematics 44, no. 1 (December 1, 1991): 3–41. http://dx.doi.org/10.4153/cjm-1992-001-6.

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Let {Pi} be a sequence of real (Laurent) polynomials each of which has no negative coefficients, and suppose that f is a real polynomial. Consider the problem of deciding whetherfor all integers k, there exists Nsuch that the product of polynomials(*) Pk+1. Pk+2.....Pk+N·ƒ has no negative coefficients.

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37

Gopalan, M. A., K. Geetha, and Manju Somanath. "Special Dio 3 - Tuples." Bulletin of Society for Mathematical Services and Standards 10 (June 2014): 22–25. http://dx.doi.org/10.18052/www.scipress.com/bsmass.10.22.

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We search for three distinct polynomials with integer coefficient such that the product of anytwo members of the set minus their sum and increased by a non-zero integer (or polynomial withinteger coefficient) is a perfect square.

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38

Zaheer, Neyamat. "A generalization of Lucas' theorem to vector spaces." International Journal of Mathematics and Mathematical Sciences 16, no. 2 (1993): 267–76. http://dx.doi.org/10.1155/s0161171293000316.

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The classical Lucas' theorem on critical points of complex-valued polynomials has been generalized (cf. [1]) to vector-valued polynomials defined onK-inner product spaces. In the present paper, we obtain a generalization of Lucas' theorem to vector-valued abstract polynomials defined on vector spaces, in general, which includes the above result of the author [1] inK-inner product spaces. Our main theorem also deduces a well-known result due to Marden on linear combinations of polynomial and its derivative. At the end, we discuss some examples in support of certain claims.

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Loh, Jian Rong, Chang Phang, and Abdulnasir Isah. "New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations." Advances in Mathematical Physics 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/3821870.

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It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.

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Gopalan, M. A., V. Geetha, and S. Vidhyalakshmi. "DIO 3 - Tuples for Special Numbers - I." Bulletin of Society for Mathematical Services and Standards 10 (June 2014): 1–6. http://dx.doi.org/10.18052/www.scipress.com/bsmass.10.1.

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We search for three distinct polynomials with integer coefficients such that the product of any two members of the set added with their sum and increased by a non-zero integer (or polynomial with integer coefficients) is a perfect square.

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Chbili, Nafaa, Shamma Al Dhaheri, Mei Tahnon, and Amna Abunamous. "The Characteristic Polynomials of Symmetric Graphs." Symmetry 10, no. 11 (November 2, 2018): 582. http://dx.doi.org/10.3390/sym10110582.

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In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ , with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ [ F ] by one of Γ \ F . A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ .

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Dowlin, Nathan. "The knot Floer cube of resolutions and the composition product." Journal of Knot Theory and Its Ramifications 29, no. 03 (March 2020): 2050006. http://dx.doi.org/10.1142/s0218216520500066.

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We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot [Formula: see text], we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of [Formula: see text].

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WAGNER, E. "KHOVANOV–ROZANSKY GRAPH HOMOLOGY AND COMPOSITION PRODUCT." Journal of Knot Theory and Its Ramifications 17, no. 12 (December 2008): 1549–59. http://dx.doi.org/10.1142/s0218216508006750.

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In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov–Rozansky graph homology.

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Pechenik, Oliver, and Dominic Searles. "Decompositions of Grothendieck Polynomials." International Mathematics Research Notices 2019, no. 10 (September 16, 2017): 3214–41. http://dx.doi.org/10.1093/imrn/rnx207.

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AbstractWe investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $K$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $n$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $K$-theory, and we state our results in this more general context.

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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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BIRMAN, JOAN, PETER BRINKMANN, and KEIKO KAWAMURO. "POLYNOMIAL INVARIANTS OF PSEUDO-ANOSOV MAPS." Journal of Topology and Analysis 04, no. 01 (March 2012): 13–47. http://dx.doi.org/10.1142/s1793525312500033.

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We investigate the structure of the characteristic polynomial det (xI - T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det (xI - T). The degrees of the new polynomials are invariants of [F] and we give simple formulas for computing them by a counting argument from an invariant train-track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.

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Vladeva, Dimitrinka. "Derivations of polynomial semirings." International Journal of Algebra and Computation 30, no. 01 (September 12, 2019): 1–12. http://dx.doi.org/10.1142/s0218196719500620.

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The aim of this paper is the investigation of derivations in semiring of polynomials over idempotent semiring. For semiring [Formula: see text], where [Formula: see text] is a commutative idempotent semiring we construct derivations corresponding to the polynomials from the principal ideal [Formula: see text] and prove that the set of these derivations is a non-commutative idempotent semiring closed under the Jordan product of derivations — Theorem 3.3. We introduce generalized inner derivations defined as derivations acting only over the coefficients of the polynomial and consider [Formula: see text]-derivations in classical sense of Jacobson. In the main result, Theorem 5.3, we show that any derivation in [Formula: see text] can be represented as a sum of a generalized inner derivation and an [Formula: see text]-derivation.

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Guo, Nan, and Yongbin Li. "The Accuracy of Low-Altitude Photogrammetry of Drones." International Journal of Pattern Recognition and Artificial Intelligence 34, no. 08 (November 7, 2019): 2059029. http://dx.doi.org/10.1142/s0218001420590296.

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This paper uses the Haida iFLY-U3 fixed-wing UAV for image data acquisition. Based on UAV low-altitude photogrammetry technology, field control measurement method, such as Agisoft PhotoScan and Pix4D Mapper software, can process and produce DEM and DOM. Practical research on the processes and key technologies of digital products is under process. The field checkpoint measurement for the accuracy of DOM digital products made by low-altitude digital photogrammetry technology creates a basis for the practical application and development of digital products. The main contents of this paper are as follows: (a) The basic principles and processes of digital products for drone remote sensing image production, such as control point layout and measurement methods. (b) Based on the UAV photography technology, the digital product DOM and the field measurement control point accuracy evaluation are generated. (c). The polynomial curve digital model method is used to solve the elevation correction value, and the quadratic polynomial fitting model of the elevation point of the internal digital product is established, and the precision analysis is carried out.

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Benedetto, Robert L. "An Elementary Product Identity in Polynomial Dynamics." American Mathematical Monthly 108, no. 9 (November 2001): 860. http://dx.doi.org/10.2307/2695559.

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Aleshina, Sofia, and Ilya Vyugin. "POLYNOMIAL VERSION ON THE SUM-PRODUCT PROBLEM." Automation and modeling in design and management 2020, no. 2 (June 23, 2020): 4–10. http://dx.doi.org/10.30987/2658-6436-2020-2-4-10.

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This work is about the generalization of sum-product problem. The general principle of it was formulated in the Erdos-Szemeredi’s hypothesis. Instead of the Minkowski sum in this hypothesis, the set of values f(x,y) of a homogeneous polynomial f lin two variables, where x and y belong to subgroup G of is considered. The lower bound on the cardinality of such set is obtained. This topic has an applied value in the theory of information and dynamics in calculating the probabilities of events, as well as in various methods of encoding and decoding information.

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