Analytic in the unit polydisc functions of bounded L-index in direction

This paper contains an account of A.A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann-Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann-Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched.Bibliography: 69 titles.

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

The object of this research is fourth-order differential equations. The aim of the research is to study the analytical properties of the solutions of these differential equations. The general form of the considered equations is indicated, and also the choice of the research object is justified. Herein we studied fourth-order differential equations for which sets of resonances with all positive nontrivial resonances are absent. Besides, three of these equations satisfy the conditions of absence in the solutions of moving multivalued singular points. The solutions of the next three equations have movable special points of multivalued character. Moreover, we also investigated the analytical properties of one more fourth-order differential equation of another general form for which it is also possible to construct a two-parameter rational solution as there is a nontrivial negative resonance in the related set of resonances. The first integrals of the equations under study are found and their rational solutions are constructed from negative non-trivial resonances. The resonance method was used in this study. The obtained results can be used in the analytical theory of differential equations.

In this paper, the Jacobi-Poisson method of building first integrals by known integral manifolds for Hamiltonian differential systems is developed. The existence theorem of first integrals in the form of Poisson brackets from known integral manifolds for Hamiltonian systems is proved. The statement of finding additional first integrals by known first integrals and integral manifolds for Hamiltonian systems is given. In the case of polynomial Hamiltonian systems, we use the concept of partial integrals and concretize ours results. The Jacobi - Poisson method of construction first integrals in the form of Poisson brackets from known integral characteristics (integral manifolds, partial integrals, and first integrals) for general ordinary differential systems is proposed. Statements of finding first integrals for an ordinary differential system by integral characteristics of the auxiliary Hamiltonian system are obtained. The generalized Poisson theorem of construction first integrals is indicated. And, the question of the existence of partial integrals and first integrals in the form of Poisson brackets for ordinary polynomial differential systems is investigated. The results obtained in this paper can be used in the analytical theory of differential equations and in the analytical mechanics.

This annual conference was established at the National Nuclear Research University – Moscow Engineering Institute in the year 2012. The scope was initially on mathematical physics and mathematical modelling, while since this year theoretical physics was also included. The Conference was organized by three leading departments of the MEPhI in the field of theoretical and mathematical physics.Each year, more than 100 participants from different countries including Belorussia, Czech Republic, France, Great Britain, Japan, Moldova, Russia, Slovakia, Spain, Sweden and the USA take part in the Conference with oral talks.The aim of this conference is to bring together researchers with interests in: Analytical Theory of Differential Equations, Interaction of Radiation With Matter and Physics of Extreme Light, Nuclear Physics and Theory of Elementary Particles, Optimization Problems, Solid Mechanics, Soft Matter and Interdisciplinary Physics, Fluid Mechanics, Mathematical Modelling, Time Series.Nikolay A. KudryashovOleg V. NagornovSergey V. Popruzhenko

Making use of L. V. Kantorovich's variational approach and of the methods of the analytic theory of differential equations, one determines the stresses in a thin plate, bounded by two arbitrary curves and two parallel straight lines.

We prove that if an analytic function f with an isolated singular point at ∞ is a solution of the differential equation P(zlnz, f, f′) = 0, where P is a polynomial in all variables, then f has finite order. We study the asymptotic properties of a meromorphic solution with logarithmic singularity.

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).

Connections between conservative linear systems, Lax–Phillips scattering, and operator model theory are well known. A common thread in all the theories is a contractive, analytic, operator-valued function on the unit disc T(z) having a representation of the form T(z) = D + z C (I − zA)−1 B, known as the transfer or frequency-response function in the system-theory community, the scattering function in the mathematical physics community, and the characteristic operator function in the operator theory community. Here we consider analogues of this circle of ideas in the more general setting of multidimensional systems/multi-evolution scattering systems/multivariable function-theoretic operator theory. Three particular extensions are discussed; from the point of view of system theory, these involve (1) a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex Euclidean space, (2) a non-commutative multidimensional linear system with evolution along a free semigroup and with transfer function equal to a formal power series in non-commuting indeterminants, and (3) an overdetermined multidimensional linear system with transfer function identified as a bundle mapping between two Hermitian vector bundles over an algebraic curve embedded in complex projective space. This survey updates an earlier survey by the first author appearing in 2000.

Chapter 46 - Image generation by Blaschke products in the unit disk

The methods in the previous sections provide asymptotic approximations for functions defined by integrals. Many of these functions are solutions of specific differential equations: the Bessel function is an example. In the case of functions defined as the solutions of differential equations which cannot be solved explicitly, these integral methods described above naturally cannot be used. (We group in the class of explicit solutions those given in the form of an integral.) In these circumstances we must resort to differential equation methods to find asymptotic approximations for the functions. The subject of asymptotic methods for solving differential equations is large. In this and the following sections we give a brief introduction to the subject. In the case of ordinary differential equations the book by Wasow (1965) discusses several aspects with detail and rigour. A large part of each of the books by Erdelyi (1956) and Jeffreys (1966) is devoted to ordinary differential equations. In the partial differential equation area the books by Van Dyke (1964) and Cole (1968)† are of importance in the particular area of asymptotic analysis called singular perturbation theory.KeywordsAsymptotic ExpansionAsymptotic SolutionAsymptotic FormLiouville EquationDominant TermThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Within the theory of linear systems, the differential equation associated with sinusoidal waves has a central importance in signal analysis. Free and forced oscillations or transient and steady-state responses can be understood as general and particular solutions of linear differential equations. Resonances may be formulated as solutions of the differential equation for a system with a single degree of freedom. Given observation records of a system, estimating source information is a fundamental issue in machine diagnostics. Taking an observation signal as a solution of a differential equation, the waveform of an external source acting on a system of a single degree of freedom can be recovered using the differential equation because in principle the transient response is independent of the source effect. A mixture of transient and steady-state solutions creates beats if the frequency of the source is close to the frequency of the resonance. The differential equation can be extended to the wave equation that governs wave propagation in a medium. A typical phenomenon of wave propagation is the standing wave, which is a superposition of direct and reflection waves, subject to there being no magnitude differences between the two waves. If the coefficient of reflection is given by a complex number, the phase difference moves the nodes; however, the standing waves disappear as the magnitude difference increases.

This paper presents solutions of second-order ordinary differential equations on bond problems to estimate crack widths of reinforced concrete (RC) members subjected to repeated cyclic loads. Up to now, bond behaviors under repeated cyclic loads have usually been calculated by a difference scheme. In contrast, this research adopts analytical solutions of the differential equations to calculate bond behaviors in finite element models where a number of concrete cracks occur. The conventional differential equation for the bond problem has been available only under monotonic loading conditions. This paper extends the equation to cases where the reinforcement becomes plastic and the bond is subject to unloading/reloading conditions. The bond stress-slip relationship is modeled by a multilinear model that is composed of an envelope made of five lines, unloading/reloading paths, and a negative friction path. The solution of the differential equation is classified into 25 cases when the bond stress-slip relationship is on the envelope and seven cases when on the unloading/reloading paths or the negative friction path. The solutions are implemented into the smeared crack-based finite-element method (FEM) to evaluate the crack widths of a RC beam subject to repeated cyclic flexural/shear loads. The FEM program is incorporated with the discrete-like crack procedure developed by the authors and calculates stress redistributions due to cracking using the solutions of the differential equations. Typical distributions of reinforcement stress, bond stress, and slip along the reinforcement are presented.

SynopsisIn this paper entire solutions of differential equations with polynomial coefficients are considered and bounds on the maximum modulus and the index are obtained, when the equation is of second order and the coefficients are of second degree.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.