Smooth factorizations of matrix valued functions and their derivatives

A function f meromorphic on the complex plane c is called an α-Yosida function if \( {{f}^{\# }}(z) = O(|z{{|}^{\alpha }})\,as|z| \to \infty ,\alpha \geqslant - 1\;where\,{{f}^{\# }}(z) = |f'(z)|(/1 + |f(z){{|}^{2}}) \) We give a sufficient and necessary condition for a meromorphic function f to be not an α-Yosida function by proving a relationship between α-Yosida functions and normal families. We also give some integral characterizations of α-Yosida functions. Applying our results we know that the meromorphic solutions of some higher order algebraic differential equations are α-Yosida functions. Moreover, some estimates for the order of growth of these meromorphic solutions by studying α-Yosida functions are given.Keywordsα-Yosida functionsmeromorphic solutionsalgebraic differential equations

The general result of Gol’dberg’s theorem concerning the growth of meromorphic solutions of algebraic differential equations

In this paper, we treat the problem of determining the rate of growth of meromorphic functions on the plane, which are solutions of nth order algebraic differential equations whose coefficients are arbitrary entire functions (i.e., equations of the form, Ω(z,y,dy/dz, • • •, dny/dzn) — 0, where Ω is a polynomial in y, dy/dz, • • •, dny/dzn whose coefficients are arbitrary entire functions of z.)

We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no ‘sub-term’ occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows.

A method is proposed to transform a system of differential algebraic equations (D.A.E.) to a system of ordinary differential equations (O.D.E.) which can be solved relatively easily by standard numerical techniques.Two examples, including a model of an absorption tower, are given to illustrate the utility of the method. The example problems reveal that this easily implemented technique offers significant savings in CPU time compared to the numerical solution of the untransformed D.A.E.s particularly when the algebraic equations are nonlinear.Furthermore, it appears to be faster and/or more reliable than other numerical schemes which have been recently developed for equations of this type.

In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].

I. A. A. Gol'dberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions [in Russian], Nauka, Moscow (1970). 2. A. Z. Mokhon'ko, "On differential and functional equations with multipliers," Differents. Uravn., 15, No. 9, 1713-1715 (1979). 3. V. D. }~khon'ko, "On the meromorphic solutions of a class of functional equations," Ukr. Mat. Zh., 34, No. 2, 222-225 (1982). 4. A. Z. Mokhon'ko, "On a class of differential and functional equations," Differents. Uravn., 19, No. 3, 414-419 (1983). 5. J. Valiron, Analytic Functions [Russian translation], Gosudarstvennoe Izd. Fizikomatematicheskoi Literatury, Moscow (1957). 6. R. Goldstein, "On meromorphic solutions of certain functional equations," Aequat. Math., 1._8.8, I12-157 (1978). 7. J. M. Baker and F. Gross, "On factorizing entire functions," Proc. London Math. Sot., 3, No. I, 69-76 (1968). 8. B. L. Van der Waerden, Algebra [Russian translation], Nauka, ~scow (1976). 9. A. Z. Mokhon'ko, "The Nevanlinna characteristics of the composition of rational and algebroidal functions,".. Ukr. Mat. Zh., 34, No. 3, 388-396 (1982). 10. H. L. Selberg, "Uber die Wertverteilung der algebroiden Funktionen," Math. Z., 31, 709728 (1930). 11. A. Z. Hokhon'ko, "On the Nevanlinna characteristics for a class of meromorphic curves," Teor. Funktsii, Funkts. Anal. Prilozhen., 25, 95-105 (1976). 12. A. E. Eremenko, "Meromorphic solutions of algebraic differential equations," Usp. Hat. Nauk, 37, No. 4 (226), 53-82 (1982).

Hoping to find bounds on the approximation by rational functions of all members of certain classes of power series, it is necessary to restrict to “reasonable” classes of functions. Algebraic functions are one such class. Another class is the set of irrational power series solutions of linear homogeneous differential equations with polynomial coefficients. Tight bounds have been found in both cases. The jump to nonlinear differential equations is another matter. The present paper obtains tight bounds, if one restricts to certain classes of general solutions. The presence of enough algebraically independent parameters permits the desired type of bound. It is to be hoped that the more general case will ultimately be settled; however, other methods will certainly be needed.

A theorem is proved that enables one to replace a ${C^n}$ solution of a system of algebraic differential equations by analytic solutions nearby, such that each satisfies its own algebraic differential equation. As an application, we emend a proof of the Shannon-Pour-E1 thesis relating the outputs of analog computers to solutions of algebraic differential equations.

In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field.In the particular case where the coefficients are rational functions, our main result states that for any given sector S in the plane, there exists a positive real number N, depending only on the equation and the angle opening of S, such that any solution y(z), which is meromorphic in S and satisfies the condition z~"y -» co as z -> co in S, must be of the form exp J" czm(\ +o(l)) in subsectors, where c and m are constants.(From this, we easily obtain a similar representation for analytic solutions in S, which are not identically zero, and for which zKy -> 0 as z -» oo in S, where the positive real number K again depends only on the equation and the angle opening of S.)

We estimate the growth of the spherical derivative for meromorphic solutions of algebraic differential equations of arbitrary order under certain conditions on the growth of the coefficients.

Asymptotic properties of entire transcendental solutions of algebraic differential equations

On algebroid solutions of algebraic differential equations in the complex plane, II

On the Singular Solutions of Algebraic Differential Equations

For an analytic function f(z)=∑ k=0 ∞ a k z k on a neighbourhood of a closed disc D⊂ℂ, we give assumptions, in terms of the Taylor coefficients a k of f, under which the number of intersection points of the graph Γ f of f |D and algebraic curves of degree d is polynomially bounded in d. In particular, we show these assumptions are satisfied for random power series, for some explicit classes of lacunary series, and for solutions of algebraic differential equations with coefficients and initial conditions in ℚ. As a consequence, for any function f in these families, Γ f has less than βlog α T rational points of height at most T, for some α,β>0.