Three-variable Polynomials Research Articles

Enhancing High-Alloy Steel Cutting with Abrasive Water Injection Jet (AWIJ) Technology: An Approach Using the Response Surface Methodology (RSM).

The common machining technologies for difficult-to-machine materials do not remarkably ensure acceptable efficiency and precision in bulk materials cutting. High-energy abrasive water injection jet (AWIJ) treatment can cut diverse materials, even multi-layer composites characterized by divergent properties, accurately cutting complex profiles and carrying them out in special circumstances, such as underwater locations or explosion hazard areas. This work reports research on the AWIJ machining quality performance of X22CrMoV12-1 high-alloy steel. The response surface method (RSM) was utilized in modeling. The most influencing process control parameters on cut kerf surface roughness-abrasive flow rate, pressure, and traverse speed-were tested. The result is a mathematical model of the process in the form of a three-variable polynomial. The key control parameter affecting the cut slot roughness turned out to be the traverse speed. In contrast, pressure has a less significant effect, and the abrasive mass flow rate has the slightest impact on the cut slot roughness. Under the optimal conditions determined as a result of the tests, the roughness of the intersection surface Sq does not exceed 2.3 μm. Based on the ANOVA, we confirmed that the model fits over 96% appropriately with the research outcomes. This method reduces the computations and sharply determines the optimum set of control parameters.

$SO(3)$ квазі-мономіальні сім'ї многочленів

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

A polynomial invariant of Kauffman type for knotoids

We construct a three-variable polynomial invariant for unoriented kd([Formula: see text])-linkoid diagrams including knotoid ones as a certain weighted sum of polynomials on oriented kd([Formula: see text])-linkoid diagrams associated with a given kd([Formula: see text])-linkoid diagram.

Cyclotomic expansion of generalized Jones polynomials

In (Compos. Math. 152(7): 1333–1384, 2016), Berest and Samuelson proposed a conjecture that the Kauffman bracket skein module of any knot in \(S^3\) carries a natural action of a rank 1 double-affine Hecke algebra \(SH_{q,t_1, t_2}\) depending on 3 parameters \(q, t_1, t_2\). As a consequence, for a knot K satisfying this conjecture, we defined a three-variable polynomial invariant \(J^K_n(q,t_1,t_2)\) generalizing the classical coloured Jones polynomials \(J^K_n(q)\). In this paper, we give explicit formulas and provide a quantum group interpretation for the polynomials \(J^K_n(q,t_1,t_2)\). Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by Habiro (Invent. Math. 171(1): 1–81, 2008) : as in the classical case, they imply the integrality of \(J^K_n(q,t_1,t_2)\) and, in fact, make sense for an arbitrary knot K independent of whether or not it satisfies the conjecture of Berest and Samuelson (Compos. Math. 152(7): 1333–1384, 2016). When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of \(J^K_n(q,t_1)\) are expressed in terms of Macdonald orthogonal polynomials.

A new family of orthogonal polynomials in three variables

In this paper we introduce a six-parameter generalization of the four-parameter three-variable polynomials on the simplex and we investigate the properties of these polynomials. Sparse recurrence relations are derived by using ladder relations for shifted univariate Jacobi polynomials and bivariate polynomials on the triangle. Via these sparse recurrence relations, second order partial differential equations are presented. Some connection relations are obtained between these polynomials. Also, new results for the four-parameter three-variable polynomials on the simplex are given. Finally, some generating functions are derived.

New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found.

Mahler measures of elliptic modular surfaces

In this article we develop a new method for relating Mahler measures of three-variable polynomials that define elliptic modular surfaces to L-values of modular forms. Using an idea of Deninger, we express the Mahler measure as a Deligne period of the surface and then apply the first author's extension of the Rogers-Zudilin method to Kuga-Sato varieties to arrive at an L-value.

Double affine Hecke algebras and generalized Jones polynomials

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

The elliptic trilogarithm and Mahler measures of K3 surfaces

Abstract The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of certain families of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves.

Mahler measure and elliptic curve L-functions at s = 3

Abstract We study the Mahler measure of some three-variable polynomials that are conjectured to be related to L-functions of elliptic curves at s = 3 by Boyd. The connection with L-functions can be explained with the use of a regulator and a result of Goncharov. Finally, we prove a relationship between two formulas.

Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial

Bollobás and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are disks. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing Tutte's concept of activity to quasi-trees, we prove a quasi-tree expansion of the Bollobás–Riordan–Tutte polynomial.

New families of generating functions for certain class of three-variable polynomials

Abstract Recently, Srivastava, Ozarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Ozarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1–12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them.

Some families of generating functions for a certain class of three-variable polynomials

The present investigation is a continuation of the works initiated by Srivastava [A contour integral involving Fox's H-function, Indian J. Math. 14 (1972), pp. 1–6] and followed by several papers such as (among others) [A. Altın, E. Erkuş, and M.A. Özarslan, Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct. 17 (2006), pp. 315–320; B. González, J. Matera, and H.M. Srivastava, Some q-generating functions and associated generalized hypergeometric polynomials, Math. Comput. Modelling 34 (2001), pp. 133–175; S.-D. Lin, Y.-S. Chao, and H.M. Srivastava, Some families of hypergeometric polynomials and associated integral representations, J. Math. Anal. Appl. 294 (2004), pp. 399–411; S.-D. Lin, H.M. Srivastava, and P.-Y. Wang, Some families of hypergeometric transformations and generating relations, Math. Comput. Modelling 36 (2002), pp. 445–459; E. Özergin, M.A. Özarslan, and H.M. Srivastava, Some families of generating functions for a class of bivariate polynomials, Math. Comput. Modelling 50 (2009), pp. 1113–1120]. In this study, we introduce a family of three-variable polynomials and derive a number of two-sided linear generating functions between these polynomials and another family of two-variable polynomials. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials. Several other results of the above-mentioned types are also considered.

A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, \( \bar z \) and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.

FORCE-UNCONSTRAINED POSES OF THE 3-PRR AND 4-PRR PLANAR PARALLEL MANIPULATORS

Force-unconstrained (singular) poses of the 3-PRR planar parallel manipulator (PPM), where the underscore indicates the actuated joint, and the 4-PRR, a redundant PPM with an additional actuated branch, are presented. The solution of these problems is based upon concepts of reciprocal screw quantities and kinematic analysis. In general, non-redundant PPMs such as the 3-PRR are known to have two orders of infinity of force-unconstrained poses, i.e., a three-variable polynomial in terms of the task-space variables (position and orientation of the mobile platform). The inclusion of redundant branches eliminates one order of infinity of force-unconstrained configurations for every actuated branch beyond three. The geometric identification of force-unconstrained poses is carried out by assuming one variable for each order of infinity. In order to simplify the algebraic procedure of these problems, the assumed or “free” variables are considered to be joint displacements. For both manipulators, an effective elimination technique is adopted. For the 3-PRR, the roots of a 6th-order polynomial determine the force-unconstrained poses, i.e., surfaces in a three dimensional space defined by the task-space variables. For the 4-PRR, a 64th-order polynomial determines curves of force-unconstrained poses in the same dimensional space.

Scalings of mixed-mode regimes in a simple polynomial three-variable model of nonlinear dynamical systems.

We describe scaling laws for a control parameter for various sequences of bifurcations of the LSn mixed-mode regimes consisting of single large amplitude maximum followed by n small amplitude peaks. These regimes are obtained in a normalized version of a simple three-variable polynomial model that contains only one nonlinear cubic term. The period adding bifurcations for LSn patterns scales as 1/n at low n and as 1/n2 at sufficiently large values of n. Similar scaling laws 1/k at low k and 1/k2 at sufficiently high values of k describe the period adding bifurcations for complex k(LSn)(LS(n + 1)) patterns. A finite number of basic LSn patterns and infinite sequences of complex k(LSn)(LS(n + 1)) patterns exist in the model. Each periodic pattern loses its stability by the period doubling bifurcations scaled by the Feigenbaum law. Also an infinite number of the broken Farey trees exists between complex periodic orbits. A family of 1D return maps constructed from appropriate Poincaré sections is a very fruitful tool in studies of the dynamical system. Analysis of this family of maps supports the scaling laws found using the numerical integration of the model.

Resonance frequency of a rectangular microstrip patch

The microstrip-patch resonant frequency problem is formulated in terms of an integral equation using vector Fourier transforms. Using Galerkins's method in solving the integral equation, the resonant frequency of the microstrip patch is studied with both Chebyshev polynomials and sinusoidal functions as basis functions. In the case of the Chebyshev polynomials, the edge singularity is included, but it is not important for convergence. Furthermore, the resonant frequency of the microstrip patch is ascertained with a perturbation calculation. The results for Galerkin's method and experiments are in good agreement. The perturbation calculation agrees asymptotically with Galerkin's method. With the aim of developing a computer-aided design formula, the solutions obtained by Galerkin's method are interpolated with a three-variable polynomial.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Three-way companion matrices of three-variable polynomials

The existence of the companion matrices of three-variable polynomials is investigated. A theorem giving necessary and sufficient conditions for the existence of a three-way companion matrix is stated and proved. The construction of such a matrix is given too. The problem of the factorization of three-variable polynomials is also solved.