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Oscillation problems (investigation of zeros) were widely studied for solutions of ordinary differential equation (ODE). In this article, we transfer oscillation problems to the case of solutions of nonautonomous system of equations . By... more
Oscillation problems (investigation of zeros) were widely studied for solutions of ordinary differential equation (ODE). In this article, we transfer oscillation problems to the case of solutions of nonautonomous system of equations . By analogy with the theory of oscillation for one ODE, we consider number n(t1,t2,0) of zeros τi in (t1,t2) of the solutions, that is the number of those points τi, where x(τi) =0 and y(τi)=0. It turns out that the above bounds for n(t1,t2,0) can be given in terms of a( t) , b( t) , F1, F2, t1 and t2. Also by analogy with the concept of a-points in the complex analysis, we consider values a:=(a′,a″) in the (x, y)-plane and define a-points of the solutions as those points τi, where and . Denoting by n(t1,t2, a) the number of a-points in (t1,t2) of the solutions we give above bounds for the sum , where a1,a2,…,aq is a given totality of pairwise different points. Thus we obtain for the solutions of the above equation an analog of the second fundamental theorem in the Nevanlinna value distribution theory; the last one also considers a similar sum for the number of a-points of meromorphic functions. As an immediate application we obtain below bounds for the periods of periodic solutions.
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ABSTRACT A new, purely geometric inequality for analytic functions admitting no zeros is revealed.
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In this article we give a topological approach to the global behavior of arbitrary single-valued solutions in a simple connected domain of some general classes of complex differential equations with multi-valued coefficients. In... more
In this article we give a topological approach to the global behavior of arbitrary single-valued solutions in a simple connected domain of some general classes of complex differential equations with multi-valued coefficients. In particular, this permits us to describe certain globally multi-valued solutions as well as algebraic and algebroid solutions.
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Recently, a topological approach has been offered for studying the global behaviour of general classes of complex differential equations with multivalued coefficients. In this article, we use this approach to study some fairly general... more
Recently, a topological approach has been offered for studying the global behaviour of general classes of complex differential equations with multivalued coefficients. In this article, we use this approach to study some fairly general second-order algebraic differential equations with rational coefficients. We also apply this reasoning to some of the Painlevé equations as well as to a differential equation investigated
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ABSTRACT In this article we study a “generalized oscillation problem” for real smooth function f(x), x ∈ (a, b). Namely we obtain above bounds for the magnitude , where N(a, b, f − Aν) is the number of solutions f − Aν in (a, b) and , is... more
ABSTRACT In this article we study a “generalized oscillation problem” for real smooth function f(x), x ∈ (a, b). Namely we obtain above bounds for the magnitude , where N(a, b, f − Aν) is the number of solutions f − Aν in (a, b) and , is a totality of pairwise different real numbers. The obtained inequality is in some aspects similar to that of the second fundamental theorem in Nevanlinna value distribution theory.
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Man, Mathematician, Teacher, whom I have always admired Abstract. The concept of A-level sets of real functions u(x, y) (i.e., the solutions of u(x, y) = A = const) in a given domain admits numerous inter-pretations in applied sciences:... more
Man, Mathematician, Teacher, whom I have always admired Abstract. The concept of A-level sets of real functions u(x, y) (i.e., the solutions of u(x, y) = A = const) in a given domain admits numerous inter-pretations in applied sciences: level sets are potential lines, streamlines in hydrodynamics, meteorology and electromagnetics, isobars in gas-dynamics, isotherms in thermodynamics, etc. In fact, the level sets of u considered for all values A make the “map ” of this function and their interpretations in different sciences make the “maps ” of the corresponding processes. In this paper we study the geometry of these maps for broad classes of functions and arbitrary values A. In particular, we study how much twisted or, speaking in general, how turbulent these maps are. The concepts and results admit some immediate interpretations and can be stated in terms of flow rotation and turbulence. The study gives a new, in fact a geometric description of these applied phenomena.
Many corn concepts in pure mathematics and applied science are in fact level sets of real functions. For particular cases of functions they were studied in di erent branches of mathematics. In the present paper we start investigations of... more
Many corn concepts in pure mathematics and applied science are in fact level sets of real functions. For particular cases of functions they were studied in di erent branches of mathematics. In the present paper we start investigations of the geometry of level sets for large classes of real functions. Some methods are established that permit to estimate the length of level sets of arbitrary “smooth enough” functions.
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This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For... more
This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.
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This paper presents a new principle related to an arbitrary meromorphic function w in a given domain D. The main component of this principle gives (first time) lower bounds for {|w^{\prime}|} for a similar general class of functions. The... more
This paper presents a new principle related to an arbitrary meromorphic function w in a given domain D. The main component of this principle gives (first time) lower bounds for {|w^{\prime}|} for a similar general class of functions. The principle can qualitatively be stated as follows: any set of simple a-points of w contains a “large” subset of complex values, where we have lower bounds for {|w^{\prime}|} and upper bounds for {|w^{(h)}|} , {h>1} .
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The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and... more
The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.
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Three results and a collection of problems for complex algebraic differential equations and their systems related with the so-called stability phenomenon are formulated. The latter, in particular, means that meromorphic (entire) solutions... more
Three results and a collection of problems for complex algebraic differential equations and their systems related with the so-called stability phenomenon are formulated. The latter, in particular, means that meromorphic (entire) solutions to algebraic differential equations P(z,f,f ' ,⋯,f (k) )=0, where P is a polynomial in all variables, preserve properties which they have on a “small” subset of ℂ.
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The concept of A-level sets of real functions u(x;y) (i.e., the solutions of u(x;y) = A = const) in a given domain admits numerous inter- pretations in applied sciences: level sets are potential lines, streamlines in hydrodynamics,... more
The concept of A-level sets of real functions u(x;y) (i.e., the solutions of u(x;y) = A = const) in a given domain admits numerous inter- pretations in applied sciences: level sets are potential lines, streamlines in hydrodynamics, meteorology and electromagnetics, isobars in gas-dynamics, isotherms in thermodynamics, etc. In fact, the level sets of u considered for all values A make the "map" of this function and their interpretations in dierent sciences make the "maps" of the corresponding processes. In this paper we study the geometry of these maps for broad classes of functions and arbitrary values A. In particular, we study how much twisted or, speaking in general, how turbulent these maps are. The concepts and results admit some immediate interpretations and can be stated in terms of flow rotation and turbulence. The study gives a new, in fact a geometric description of these applied phenomena.
Using a technique developed for meromorphic functions we give upper bounds for the length of ⌈-lines of monic polynomials for large classes of curves ⌈. When ⌈ is a circumference we deal with the Erdöos-Herzog-Piranian problem.
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ABSTRACT The authors introduce a new concept of jumping a-points for functions w(z) meromorphic in the complex plane. Let z 0 be an a-point of w(z), and let w(z)=a+c 1 (z-z 0 )+c 2 (z-z 0 ) 2 +⋯. The jumping a-point is described by some... more
ABSTRACT The authors introduce a new concept of jumping a-points for functions w(z) meromorphic in the complex plane. Let z 0 be an a-point of w(z), and let w(z)=a+c 1 (z-z 0 )+c 2 (z-z 0 ) 2 +⋯. The jumping a-point is described by some condition c 1 is very small compare to other coefficients in its Taylor series. They state generalizations of some main conclusions of the Nevanlinna value distribution theory related to multiple a-points. Let N(r,a,↑,w) be a counting function with respect to such jumping a-points. One of the main result is the following. For any distinct complex values ν=1,2,⋯,q, it holds ∑ ν=1 q lim inf r→∞ N(r,a,↑,w) T(r,w)≤4· The authors consider, as an application, the growth order of a meromorphic solution of certain algebraic complex differential equation.
ABSTRACT This paper presents some new identities and inequalities in integral and differential geometry, in real and complex analysis, in ODE. Ideas, methods, and problems come from some preceding studies related to Gamma lines. There are... more
ABSTRACT This paper presents some new identities and inequalities in integral and differential geometry, in real and complex analysis, in ODE. Ideas, methods, and problems come from some preceding studies related to Gamma lines. There are also certain bridges between the presented results and Nevanlinna theory of meromorphic functions.
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ABSTRACT A new program of mathematical studies primarily based on the theory of Gamma-lines and ideas of the Nevanlinna value distribution theory is presented. This program establishes new connections between a variety of mathematical... more
ABSTRACT A new program of mathematical studies primarily based on the theory of Gamma-lines and ideas of the Nevanlinna value distribution theory is presented. This program establishes new connections between a variety of mathematical fields: real and complex analysis, ordinary, partial and complex differential equations, differential geometry, real and complex algebraic geometry, and Hilbert’s topological problem 16. Preliminary results, related to some of the problems posed, are given. In addition, the usefulness of this program in applications will be discussed.
This paper has two novel components: we 1)investigate oscillation problems for solutions (x(t);y (t)), t 2 (t1;t2) of the autonomous system of equations y0 = F1 (x;y), x0 = F2 (x;y) and 2) transfer the spirit of classical Nevan- linna... more
This paper has two novel components: we 1)investigate oscillation problems for solutions (x(t);y (t)), t 2 (t1;t2) of the autonomous system of equations y0 = F1 (x;y), x0 = F2 (x;y) and 2) transfer the spirit of classical Nevan- linna value distribution theory (complex analysis) into the theory of ordinary difierential equations (ODE). By analogy with the theory of oscillation for one ODE we consider the number n(t1;t2;0) of zeros ¿i in (t1;t2) of the solutions, that is, the number of those points ¿i, where x(¿i) = 0 and y(¿i) = 0. It turns out that upper bounds for n(t1;t2;0) can be given in terms of F1, F2, t1 and t2. By analogy with the concept of a-points in complex analysis we consider values a := (a0;a00) in the (x;y)-plane and deflne a-points of the solutions as those points ¿i, where x(¿i) = a0 and y(¿i) = a00. Denoting by n(t1;t2;a) the number of a-points in (t1;t2) of the solutions, we give upper bounds for the sum Pq "=1 n(t1;t2;a"), where a1;a2;:::;aq is a give...
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ABSTRACT By a recent method to estimate the derivatives $|w^{(k)}(z_{i})|,$ $k>1$ , at certain $a$ -points of a meromorphic function $w(z)$ in terms of the Ahlfors-Shimizu characteristic and of $|w^{\prime}(z_{i})|$ , we improve... more
ABSTRACT By a recent method to estimate the derivatives $|w^{(k)}(z_{i})|,$ $k>1$ , at certain $a$ -points of a meromorphic function $w(z)$ in terms of the Ahlfors-Shimizu characteristic and of $|w^{\prime}(z_{i})|$ , we improve some classical results on the growth of meromorphic solutions of certain algebraic differential equations. Moreover, we offer similar results for equations involving inverse derivatives and derivatives of a power $w^{t}$ of a meromorphic function $w$ .
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ABSTRACT The paper announces some new inequalities that refer to broken lines, curves and real and complex functions. Their derivation is based on a new principle of angles and lengths for curves. The inequality in the complex analysis,... more
ABSTRACT The paper announces some new inequalities that refer to broken lines, curves and real and complex functions. Their derivation is based on a new principle of angles and lengths for curves. The inequality in the complex analysis, called the principle of derivatives, is valid for analytic functions in arbitrary domains and extends to a broad class of sufficiently smooth complex functions. A new inequality follows for real functions of two variables concerning their level sets. For all cases mentioned above, both for curves and functions, we obtain some analogues of the second fundamental theorem in Nevanlinna theory of meromorphic functions. At the end we discuss a new point-domain inequality dealing with finite point sets in an arbitrary domain.
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... 25 D. Singerman, Superficies de Riemann y cristalografía. 26 Pilar Onús Báguena, Tratamiento de Datos, Grafos y Didáctica de las Matemáticas. ... Un punto de vista didáctico-matemático. 29 Fernando Etayo Gordejuela, Subvariedades... more
... 25 D. Singerman, Superficies de Riemann y cristalografía. 26 Pilar Onús Báguena, Tratamiento de Datos, Grafos y Didáctica de las Matemáticas. ... Un punto de vista didáctico-matemático. 29 Fernando Etayo Gordejuela, Subvariedades reales de variedades complejas. ...
ABSTRACT Meromorphic solutions of algebraic differential equations has been intensively investigated during the last three decades. These studies were mainly concerned with the growth of solutions and numbers of their a-points. In the... more
ABSTRACT Meromorphic solutions of algebraic differential equations has been intensively investigated during the last three decades. These studies were mainly concerned with the growth of solutions and numbers of their a-points. In the present article we transfer to study location of a-points of meromorphic solutions for some important classes of algebraic differential equations. To this end, we apply the properties of proximitly and comparability of a-points. These properties offer additional information to the classical value distribution theory by describing locations of a-points of arbitrary meromorphic functions. In particular, we show for first-order algebraic differential equations that mutual locations of different a-points of the solutions are completely determined in terms of the equation.
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ABSTRACT Let w(z) denote a function meromorphic in the complex plane . In the present article we extend the Proximity Property of a-points to study some other phenomena related to geometry of neighborhoods of the a-points and estimations... more
ABSTRACT Let w(z) denote a function meromorphic in the complex plane . In the present article we extend the Proximity Property of a-points to study some other phenomena related to geometry of neighborhoods of the a-points and estimations for derivatives of w(z) and its inverse functions F(z). Particularly we establish that (a) for “good” a-points z i(a, w) of w the magnitudes of are estimated in terms of where u is a given integer >1. Also we introduce so called “inverse meromorphic derivatives” and give above bounds for in terms of .
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ABSTRACT This article has two novel aspects in complex differential equations: it starts the studies of the global behaviour of solutions in arbitrary domains and also the studies of gamma-lines of the solutions. This is done for... more
ABSTRACT This article has two novel aspects in complex differential equations: it starts the studies of the global behaviour of solutions in arbitrary domains and also the studies of gamma-lines of the solutions. This is done for solutions of some classes of equations involving the basic Schrödinger type equations, also those with entire coefficients, which we consider in ‘almost’ arbitrary domains. In particular case, when the domain is the complex plane, we study deficiency of gamma-lines of solutions of algebraic Schrödinger equations and prove that the deficiency is equal to zero for any curve which does not pass through zero.