Algebraic Polynomials Research Articles

İZİ SIFIR OLAN JENERİK MATRİSLER CEBİRİ İÇİN NOWICKI SANISI

A locally nilpotent linear derivation δ of the commutative polynomial algebra K[Xd ]=K[x1,…,xd ] of rank d is called Weitzenböck. It is well known that the subalgebra K[Xd ]^δ of K[Xd ] consisting of polynomials which are sent to zero by δ is finitely generated. Let the Weitzenböck derivation δ act on K[Xd,Yd ] such that δ(yi )=xi, δ(xi )=0, i=1,…,d. The explicit form of generators of the algebra K[Xd,Yd ]^δ was conjectured by Nowicki in 1994. In this study, we consider the Nowicki conjecture in the algebra W generated by two traceless generic matrices with entries from commutative associative unitary polynomial algebra with six variables, and obtain the free generators of the algebra W^δ of constants in this algebra.

СИНТЕЗ ГИБРИДНОЙ СИСТЕМЫ УПРАВЛЕНИЯ НЕАФФИННЫМИ ОБЪЕКТАМИ

In the theory of automatic control, an urgent problem is the development of design methods bynonaffine control systems. In such systems, the control affects the input of the plant nonlinearly, so itaffects the state variables non-additively. The purpose of this article is to develop a design method thatensures the stability of the zero equilibrium position of a closed control system in a certain area.The object described by a nonlinear system of differential equations with one control is considered.A restriction is introduced, consisting in the differentiability of the right part of the differential equationsfor all state variables. The task of designing control in the form of a function of the reference signal, avector of state variables and control values at previous points in time is set. This problem is solved usinga quasilinear model of the control plant. This model of description allows you to preserve all the featuresof a nonlinear plant without simplifying them. In the quasilinear model, matrices and vectors arefunctions of the variables of the state of the control plant. The control is performed using an algebraicpolynomial matrix method. This method allows you to find control when the control condition of theplant are met in the form of inequalities. This article presents the expressions for calculating the controlaccording to the polynomial matrix method. Based on the given coefficients of the desired polynomial,as a result of solving an algebraic system of equations, coefficients are found that are a function of controland state variables. At the same time, the fulfillment of the controllability condition guarantees theexistence of a solution of the specified algebraic system. An expression has been found that allows calculatingthe control by the coefficients found. The article also finds a condition for the possibility ofproviding a non-zero value of the output controlled quantity of a nonlinear Hurwitz system in a steadystatemode. Under this condition, a zero value of the static error for the setting effect can also be provided.Further, the transformation of the obtained continuous control into a discrete one is proposed, whichis implemented in a digital computer. The article also provides a numerical example of the control designof a second-order nonlinear control and the results of modeling a closed nonaffine system.The given example confirms the theoretical results obtained. Thus, the proposed approach makes it possibleto design stable Hurwitz control systems for nonaffine objects using the algebraic polynomial matrixmethod with sufficiently small sampling periods of variables of the control object and small modules of theroots of the characteristic polynomial of the matrix of a closed system in its quasilinear model.

The squaring operation and the hit problem for the polynomial algebra in a type of generic degree

Let Pk be the graded polynomial algebra F2[x1,x2,…,xk] with the degree of each generator xi being 1, where F2 denote the prime field with two elements.The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra A. Equivalently, we want to find a vector space basis for F2⊗APk in each degree.In this paper, we study a generating set for the kernel of Kameko's squaring operation Sq˜⁎0:F2⊗APk⟶F2⊗APk in a so-called generic degree. By using this result, we explicitly compute the hit problem for k=5 in the respective generic degree.

ON $D_4$ INVARIANTS OF POLYNOMIAL ALGEBRAS

Let $D_4$ be the dihedral group of order $8$. In the present study, we give generators of the algebra of $D_4$ invariants in the polynomial algebra with four generators over a field of characteristic zero.

ON AUTOMORPHISMS OF LIE ALGEBRA OF SYMMETRIC POLYNOMIALS

Let $L_{n}$ be the free Lie algebra of rank $n$ over a field $K$ of characteristic zero, $L_{n,c}=L_{n}/(L_{n}''+\gamma_{c+1}(L_{n}))$ be the free metabelian nilpotent of class $c$ Lie algebra, and $F_{n}=L_{n}/L_{n}''$ be the free metabelian Lie algebra generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ in these Lie algebras {\it symmetric} if $p(x_1,\ldots,x_n)=p(x_{\pi(1)},\ldots,x_{\pi(n)})$ for each element of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincides with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\text{\rm Inn}(L_{n,c}^{S_n})\cap \text{\rm Inn}(L_{n,c})$ and $\text{\rm Inn}(F_{n}^{S_n})\cap \text{\rm Inn}(F_{n})$ of inner automorphisms of the algebras $L_{n,c}^{S_n}$ and $F_{n}^{S_n}$ in the groups $\text{\rm Inn}(L_{n,c})$ and $\text{\rm Inn}(F_{n})$, respectively. In particular, we obtain the descriptions of the groups $\text{\rm Aut}(L_{2}^{S_2})\cap \text{\rm Aut}(L_{2})$ and $\text{\rm Aut}(F_{2}^{S_2})\cap \text{\rm Aut}(F_{2})$ of automorphisms of the algebras $L_{2}^{S_2}$ and $F_{2}^{S_2}$ in the groups $\text{\rm Aut}(L_{2})$ and $\text{\rm Aut}(F_{2})$, respectively.

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of , this provides an explicit connection with the generic superintegrable model on the -dimensional sphere and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of and , respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

The algebraic structure of the rank two Racah algebra is studied in detail. We provide an automorphism group of this algebra, which is isomorphic to the permutation group of five elements. This group can be geometrically interpreted as the symmetry of a folded icosidodecahedron. It allows us to study a class of equivalent irreducible representations of this Racah algebra. They can be chosen symmetric so that their transition matrices are orthogonal. We show that their entries can be expressed in terms of Racah polynomials. This construction gives an alternative proof of the recurrence, difference and orthogonal relations satisfied by the Tratnik polynomials, as well as their expressions as a product of two monovariate Racah polynomials. Our construction provides a generalization of these bivariate polynomials together with their properties.

Planar and spherical four-bar linkage [formula omitted] algebraic input–output equations

The algebraic polynomial input–output (IO) equations relating any two of the relative joint displacement parameters, called vi and vj, between any of the six distinct pairs of rigid links in arbitrary planar and spherical four-bar mechanisms are derived. First, the forward kinematics transformation matrices of the corresponding serial kinematic chains are computed in terms of their Denavit–Hartenberg parameters, but with all angles converted to tangent half-angle parameters. These matrices are mapped to their corresponding Study soma coordinates. The serial kinematic chain is closed by equating the soma coordinates to the identity array. Algebraic polynomial elimination methods are then used to obtain a single polynomial in terms of only the design and the selected IO joint displacement parameters. This yields six independent algebraic IO Equations for each of the planar and spherical 4R linkages; the same techniques are applied to derive six additional algebraic IO equations for each of the RRRP and PRRP planar linkages providing a catalogue of 24. The utility of these IO equation sets is demonstrated via discussion of the associated mobility and design parameter spaces.

The structure of matrix polynomial algebras

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and . For N = 2, being both different from zero, this reduces to the classical Zernike system. We prove that always provides a superintegrable system (for any value of γ n and N) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1 : 1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the symmetry algebra determined by the constants of the motion is also studied, giving rise to a th-order polynomial algebra. As a byproduct, the Hamiltonian is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that (and so the Zernike system as well) is endowed with a Poisson -coalgebra symmetry which would allow for further possible generalizations that are also discussed.

Levinson-type theorem and Dyn'kin problems

Questions relating to theorems of Levinson-Sjöberg-Wolf type in complex and harmonic analysis are explored. The well-known Dyn'kin problem of effective estimation of the growth majorant of an analytic function in a neighbourhood of its set of singularities is discussed, together with the problem, dual to it in certain sense, on the rate of convergence to zero of the extremal function in a nonquasianalytic Carleman class in a neighbourhood of a point at which all the derivatives of functions in this class vanish. The first problem was solved by Matsaev and Sodin. Here the second Dyn'kin problem, going back to Bang, is fully solved. As an application, a sharp asymptotic estimate is given for the distance between the imaginary exponentials and the algebraic polynomials in a weighted space of continuous functions on the real line. Bibliography: 24 titles.

Bernstein-Nikolskii-Markov-type inequalities for algebraic polynomials in a weighted Lebesgue space

In this paper, we study Bernstein, Markov and Nikol?skii type inequalities for arbitrary algebraic polynomials with respect to a weighted Lebesgue space, where the contour and weight functions have some singularities on a given contour.

A Tannaka–Krein extension theorem for topological semigroups

We prove a Tannaka–Krein extension theorem for linear mappings taking values in $C^*$-algebras, that are defined on algebras of trigonometric polynomials on topological semigroups. We discuss an application of this result to an approximation problem on th

Наилучшее приближение и значения поперечников некоторых классов аналитических функций в весовом пространстве Бергмана B_(2,γ)

In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function f in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the m th order of the r th order derivative f^((r)) in the weighted Bergman space B_(2,γ). Also using the modulus of continuity of the m-th order of the derivative f(r), we introduce a class of functions W_m^((r) ) (h,Φ) analytic in the unit circle and defined by a given majorant Φ, h∈(0,π⁄n], n>r, monotonically increasing on the positive semiaxis. Under certain conditions on the majorant Φ, for the introduced class of functions, the exact values of some known n-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the n-widths of functional classes in various Banach spaces developed by V.M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space B_(2,γ).

The de Rham cohomology of the algebra of polynomial functions on a simplicial complex

The de Rham cohomology of the algebra of polynomial functions on a simplicial complex

Graded Automorphisms of the Algebra of Polynomials in Three Variables

Graded Automorphisms of the Algebra of Polynomials in Three Variables

The Non-Commutative Korteweg—De Vries Hierarchy and Combinatorial Poppe Algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show that within the class of polynomial partial differential fields, at every order, each member of the non-commutative potential Korteweg--de Vries hierarchy is unique. Indeed, we prove to all orders, that each such member matches the non-commutative Lax hierarchy field, which is therefore a polynomial partial differential field. We achieve this by constructing the abstract combinatorial algebra that underlies the non-commutative potential Korteweg-de Vries hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by the set of all compositions endowed with the Poppe product. This product is the abstract representation of the product rule for Hankel operators pioneered by Ch. Poppe for integrable equations such as the Sine-Gordon and Korteweg-de Vries equations. Integrability of the hierarchy members translates, in the combinatorial algebra, to proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions'. Proving the existence of such Poppe polynomial expansions boils down to solving a linear algebraic problem for the expansion coefficients, which we solve constructively to all orders.

Уточнение результатов численных расчетов частот поперечных колебаний стержня переменного сечения с упругим закреплением экстраполяционным методом

Thearticle describes a numerical experiment with the results of calculating the natural frequencies of transverse vibrations of a rod of variable cross-section with elastic fixation at one of the ends. The possibility of refining by extrapolation the calculated values of natural frequencies obtained by the method of algebraic polynomials of the fifth degree is analyzed. It is shown that ex-trapolation made it possible to increase the accuracy of calculations by the numerical method by 2-3 orders of magnitude

Linear combinations of two Bernstein polynomials

In the present note we construct a linear combination $ L_{kn} $ of two Bernstein polynomials $ B_n $ and $ B_{kn} $, for natural number $ k\geq 2 $. The new operator $ L_{kn} $ reproduces all algebraic polynomials of degree $ \leq 2 $. A direct pointwise estimate is proved which shows that the rate of approximation of $ f\in C[0, 1] $ such that $ \|\varphi f^{(3)}\|_{C[0, 1]}<\infty $ by $ L_{kn} $ is $ O\left(\frac{1}{n^{\frac{5}{2}}} \right) $, when $ x $ is close to the ends of the interval $ [0, 1] $.

On the growth of the modulus of the derivative of algebraic polynomials in bounded and unbounded domains with cusps

In this present work, we study the behavior of the derivatives of algebraic polynomials in the bounded and unbounded regions bounded by piecewise smooth curve with interior and exterior cusps.