Class Of Rational Functions Research Articles

Необходимое и достаточное условия существования рациональных решений однородных разностных уравнений с постоянными коэффициентами

A necessary and a sufficient condition for solvability of homogeneous difference equations with constant coefficients in the class of rational functions are obtained. The necessary condition is a restriction on the Newton polyhedron of the characteristic polynomial. In the two-dimensional case, this condition is the existence of parallel sides on the polygon. The sufficient condition is the equality to zero of certain sums of the coefficients of the equation. If the sufficient condition is satisfied, the solution is the class of rational functions whose denominators form a subring in the ring of polynomials. This subring can be associated with an edge of the Newton polyhedron of the characteristic polynomial of the equation.

A Parametric Family of Triangular Norms and Conorms with an Additive Generator in the Form of an Arctangent of a Linear Fractional Function

At present, fuzzy modeling has established itself as an effective tool for designing and developing systems for various purposes that are used to solve problems of control, diagnostics, forecasting, and decision making. One of the most important problems is the choice and justification of an appropriate functional representation of the main fuzzy operations. It is known that, in the class of rational functions, such operations can be represented by additive generators in the form of a linear fractional function, a logarithm of a linear fractional function, and an arctangent of a linear fractional function. The paper is devoted to the latter case. Restrictions on the parameters, under which the arctangent of a linear fractional function is an increasing or decreasing generator, are defined. For each case, a corresponding fuzzy operation (a triangular norm or a conorm) is constructed. The theoretical significance of the research results lies in the fact that the obtained parametric families enrich the theory of Archimedean triangular norms and conorms and provide additional opportunities for the functional representation of fuzzy operations in the framework of fuzzy modeling. In addition, in fact, we formed a scheme for study functions that can be considered additive generators and constructed the corresponding fuzzy operations.

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

In this paper the complex dynamics of a smallest biochemical system model in three-dimensional systems with the reaction scheme. This model is described by a system of three nonlinear ordinary differential equations with five positive real parameters, are analyzed and studied. We present a thorough analysis of their invariant algebraic surfaces and exponential factors and investigate the integrability and nonintegrabilty of this model. Particularly, we show the non-existence of polynomial, rational, Darboux and local analytic first integrals in a neighborhood of the equilibrium. Moreover, we prove that, the model is not integrable in the sense of Bogoyavlensky in the class of rational functions.

The depth of continued fraction expansion for some classes of rational functions

For nonzero polynomials [Formula: see text] and [Formula: see text] over a field [Formula: see text], let [Formula: see text] be the depth (length) of the continued fraction expansion for [Formula: see text]. An upper bound on [Formula: see text], for nonzero polynomial [Formula: see text] and rational function [Formula: see text] is obtained. Applying this result, an upper bound on the depth of a linear fractional transformation is also established.

Inequalities for complex rational functions

In this paper, we consider a class of rational functions $r(s(z))$ of degree at most $mn$, where $s(z)$ is a polynomial of degree $m$ and obtain a certain sharp compact generalization of well-known inequalities for rational functions.

A new method for finding cycles by semilinear control

In the paper two new results that prove typically real and local univalent properties of certain class of rational functions in the unit disc D are presented. They provide a theoretical justification of a new method for finding periodic solutions of the discrete vector dynamical systems. The main goal of the paper is to propose and investigate the two-delays generalization of the scheme introduced by Ö. Morgül. The analysis shows that the coefficients in the proposed method can be chosen in a way that improves the admissible range of parameters. The method is elementary and easy to implement. Furthermore, numerical experiments with several well known maps show effectiveness of the control scheme.

Coefficient Estimates of some Classes of Rational Functions

Coefficient Estimates of some Classes of Rational Functions

Bernstein-type inequalities for certain class of meromorphic functions

In this paper, we shall obtain some Bernstein-type inequalities for certain class of meromorphic functions with restricted zeros and prescribed poles. Our results extend some well known polynomial inequalities to a class of rational functions.

Orthogonal rational functions on the extended real line and analytic on the upper half plane

Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.

Analysis of additive generators of fuzzy operations represented by rational functions

This article presents an approach for determining additive generators of commutative and associative operations. Its applicability for finding generators of triangular norms and conorms is shown. Conditions for the parameters of increasing generators that generate triangular conorms in the class of rational functions are determined.

Quadrature formulas with variable nodes and Jackson–Nikolskii inequalities for rational functions

We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive (q,p)-inequalities of Jackson–Nikolskii type for various classes of rational functions, complex polynomials and their logarithmic derivatives (simple partial fractions). It is shown that our (∞,2)- and (∞,4)-inequalities are sharp in a number of main theorems. Our inequalities extend and refine several results obtained earlier by other authors.

Hyperbolic dimension and Poincaré critical exponent of rational maps

We study the Poincare series of rational maps. By investigating the property of conical Julia set and dissipative measure, we prove that the Poincare critical exponents are equal to the hyperbolic dimensions for a large class of rational maps.

Some classes of rational functions for pictures

With the aid of homogeneous morphisms, we turn the deterministic two-dimensional two-way ordered restarting automaton and its extended variant into devices that compute transductions of pictures, and we study the resulting classes of transductions in detail.

Bernstein type inequalities for rational functions

In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions.

Uniform boundedness ofS-units in arithmetic dynamics

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set f(K) contains only finitely many S-units. We conjecture that the number of such S-units is bounded by a function of |S| and d (independently of K and f). We prove this conjecture for several classes of rational functions, and show that the full conjecture follows from the Bombieri--Lang conjecture.

Galois representations from pre-image trees: an arboreal survey

— Given a global fieldK and a rational function φ ∈ K(x), one may take pre-images of 0 under successive iterates of φ, and thus obtain an infinite rooted tree T∞ by assigning edges according to the action of φ. The absolute Galois group of K acts on T∞ by tree automorphisms, giving a subgroup G∞(φ) of the group Aut(T∞) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of G∞(φ) allows one to prove density results on the set of primes of K that divide at least one element of a given orbit of φ. Following an overview of the history of the subject and two of its fundamental questions, we survey in Section 2 cases where G∞(φ) is known to have finite index in Aut(T∞). While it is tempting to conjecture that such behavior should hold in general, we exhibit in Section 3 four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about G∞(φ), thanks in part to a surprising application of probability theory, as we discuss in Section 4. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of φ, which we survey briefly in Section 5. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of φ proves decisive, just as the topology of these orbits is decisive in complex dynamics. 2010 Mathematics Subject Classification. — 11R32, 37P15, 11F80.

The best uniform polynomial approximation of two classes of rational functions

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the theoretical results.

A rigidity theorem for special families of rational functions

We study the question of whether for a given nonconstant holomorphic function f there is a pair of domains U;V such that f is the only nonconstant holomorphic function with f(U) µ V. We show existence of such a pair for several classes of rational functions, namely maps of degree 1 and 2 as well as arbitrary degree Blaschke products. We give explicit constructions of U and V , where possible. Consequences for the generalized Kobayashi and Caratheodory metrics are also presented.

Nonlinear Regularization Path for Quadratic Loss Support Vector Machines

Regularization path algorithms have been proposed to deal with model selection problem in several machine learning approaches. These algorithms allow computation of the entire path of solutions for every value of regularization parameter using the fact that their solution paths have piecewise linear form. In this paper, we extend the applicability of regularization path algorithm to a class of learning machines that have quadratic loss and quadratic penalty term. This class contains several important learning machines such as squared hinge loss support vector machine (SVM) and modified Huber loss SVM. We first show that the solution paths of this class of learning machines have piecewise nonlinear form, and piecewise segments between two breakpoints are characterized by a class of rational functions. Then we develop an algorithm that can efficiently follow the piecewise nonlinear path by solving these rational equations. To solve these rational equations, we use rational approximation technique with quadratic convergence rate, and thus, our algorithm can follow the nonlinear path much more precisely than existing approaches such as predictor-corrector type nonlinear-path approximation. We show the algorithm performance on some artificial and real data sets.

Error bounds for uniform asymptotic expansions—modified bessel function of purely imaginary order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.