TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Zeta-function of a nonlinear system N2 - Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn. T3 - Preprint - (2004) 19 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26795 ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - An index formula for Toeplitz operators N2 - We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first order partial differential equations in a bounded domain of Euclidean space with infinitely differentiable boundary. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)12 KW - Toeplitz operators KW - Fredholm property KW - index Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72499 SN - 2193-6943 VL - 3 IS - 12 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - The method of Fischer-Riesz equations for elliptic boundary value problems N2 - We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)24 KW - Boundary value problems for first order systems KW - Green formula KW - Fischer-Riesz equations KW - regularisation Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61792 ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Class of Toeplitz Operators in Several Variables N2 - We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)17 KW - Cauchy data spaces KW - Laplace-Beltrami operator KW - Toeplitz operators KW - Fredholm property Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68932 SN - 2193-6943 ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - Weak boundary values of solutions of Lagrangian problems N2 - We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 2 KW - nonlinear equations KW - Lagrangian system KW - weak boundary values KW - quasilinear Fredholm operator KW - mapping degree Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72617 SN - 2193-6943 VL - 4 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - A spectral theorem for deformation quantisation N2 - We present a construction of the eigenstate at a noncritical level of the Hamiltonian function. Moreover, we evaluate the contributions of Morse critical points to the spectral decomposition. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 4 KW - star product KW - WKB method KW - spectral theorem Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-72425 SN - 2193-6943 VL - 4 IS - 4 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Makhmudov, K. O. A1 - Makhmudov, O. I. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A nonstandard Cauchy problem for the heat equation JF - Mathematical Notes N2 - We consider the Cauchy problem for the heat equation in a cylinder C (T) = X x (0, T) over a domain X in R (n) , with data on a strip lying on the lateral surface. The strip is of the form S x (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem. Y1 - 2017 U6 - https://doi.org/10.1134/S0001434617070264 SN - 0001-4346 SN - 1573-8876 VL - 102 SP - 250 EP - 260 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Elin, Mark A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - Analytic Semigroups of Holomorphic Mappings and Composition Operators JF - Computational Methods and Function Theory N2 - In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. KW - Non-linear semigroups KW - Composition operators KW - Analytic extension KW - Holomorphic mappings Y1 - 2017 U6 - https://doi.org/10.1007/s40315-017-0227-x SN - 1617-9447 SN - 2195-3724 VL - 18 IS - 2 SP - 269 EP - 294 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Mera, Azal A1 - Stepanenko, Vitaly A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Successive approximation for the inhomogeneous burgers equation JF - Journal of Siberian Federal University : Mathematics & Physics N2 - The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term. KW - Navier-Stokes equations KW - classical solution Y1 - 2018 U6 - https://doi.org/10.17516/1997-1397-2018-11-4-519-531 SN - 1997-1397 SN - 2313-6022 VL - 11 IS - 4 SP - 519 EP - 531 PB - Siberian Federal University CY - Krasnoyarsk ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Golusin-Krylov formulas in complex analysis JF - Complex variables and elliptic equations N2 - This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces. KW - Analytic continuation KW - inegral formulas KW - Cauchy problem Y1 - 2017 U6 - https://doi.org/10.1080/17476933.2017.1395872 SN - 1747-6933 SN - 1747-6941 VL - 63 IS - 7-8 SP - 1142 EP - 1167 PB - Routledge CY - Abingdon ER - TY - JOUR A1 - Mera, Azal A1 - Shlapunov, Alexander A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Navier-Stokes Equations for Elliptic Complexes JF - Journal of Siberian Federal University. Mathematics & Physics N2 - We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations. KW - Navier-Stokes equations KW - classical solution Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-1-3-27 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 1 SP - 3 EP - 27 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolai Nikolaevich T1 - The de Rham Cohomology through Hilbert Space Methods JF - Journal of Siberian Federal University. Mathematics & physics N2 - We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer. KW - De Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-4-455-465 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 4 SP - 455 EP - 465 PB - Sibirskij Federalʹnyj Universitet CY - Krasnoyarsk ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Removable singularities of CR functions on singular boundaries N2 - The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. T3 - Preprint - (2000) 18 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25836 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Anisotropic edge problems N2 - We investigate elliptic pseudodifferential operators which degenerate in an anisotropic way on a submanifold of arbitrary codimension. To find Fredholm problems for such operators we adjoint to them boundary and coboundary conditions on the submanifold.The algebra obtained this way is a far reaching generalisation of Boutet de Monvel's algebra of boundary value problems with transmission property. We construct left and right regularisers and prove theorems on hypoellipticity and local solvability. T3 - Preprint - (2002) 09 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26280 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Harmonic integrals on domains with edges N2 - We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary. T3 - Preprint - (2004) 20 KW - domains with singularities KW - de Rham complex KW - Neumann problem KW - Hodge theory Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26800 ER - TY - INPR A1 - Gauthier, Paul M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A covering property of the Riemann zeta-function N2 - For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails. T3 - Preprint - (2004) 03 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26683 ER - TY - INPR A1 - Kytmanov, Alexander A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Holomorphic Lefschetz formula for manifolds with boundary N2 - The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f : M -> M in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschtz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschtz formula on a compact complex manifold with boundary T3 - Preprint - (2002) 17 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26354 ER - TY - INPR A1 - Sultanov, Oskar A1 - Kalyakin, Leonid A1 - Tarkhanov, Nikolai Nikolaevich T1 - Elliptic perturbations of dynamical systems with a proper node N2 - The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 4 KW - dynamical system KW - singular perturbation KW - asymptotic methods Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70460 SN - 2193-6943 VL - 3 IS - 4 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Duality by reproducing kernels N2 - Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A. T3 - Preprint - (2001) 26 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26095 ER - TY - INPR A1 - Aizenberg, Lev A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Bohr phenomenon for elliptic equations N2 - In 1914 Bohr proved that there is an r ∈ (0, 1) such that if a power series converges in the unit disk and its sum has modulus less than 1 then, for |z| < r, the sum of absolute values of its terms is again less than 1. Recently analogous results were obtained for functions of several variables. The aim of this paper is to comprehend the theorem of Bohr in the context of solutions to second order elliptic equations meeting the maximum principle. T3 - Preprint - (1999) 18 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25547 ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Formal Poincaré lemma N2 - We show how the multiple application of the formal Cauchy-Kovalevskaya theorem leads to the main result of the formal theory of overdetermined systems of partial differential equations. Namely, any sufficiently regular system Au = f with smooth coefficients on an open set U ⊂ Rn admits a solution in smooth sections of a bundle of formal power series, provided that f satisfies a compatibility condition in U. T3 - Preprint - (2007) 02 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30231 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Euler characteristic of Fredholm quasicomplexes N2 - By quasicomplexes are usually meant perturbations of complexes small in some sense. Of interest are not only perturbations within the category of complexes but also those going beyond this category. A sequence perturbed in this way is no longer a complex, and so it bears no cohomology. We show how to introduce Euler characteristic for small perturbations of Fredholm complexes. The paper is to appear in Funct. Anal. and its Appl., 2006. T3 - Preprint - (2006) 11 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30117 ER - TY - INPR A1 - Krupchyk, K. A1 - Tarkhanov, Nikolai Nikolaevich A1 - Tuomela, J. T1 - Generalised elliptic boundary problems N2 - For elliptic systems of differential equations on a manifold with boundary, we prove the Fredholm property of a class of boundary problems which do not satisfy the Shapiro-Lopatinskii property. We name these boundary problems generalised elliptic, for they preserve the main properties of elliptic boundary problems. Moreover, they reduce to systems of pseudodifferential operators on the boundary which are generalised elliptic in the sense of Saks (1997). T3 - Preprint - (2005) 23 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29994 ER - TY - INPR A1 - Aizenberg, Lev A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Stable expansions in homogeneous polynomials N2 - An expansion for a class of functions is called stable if the partial sums are bounded uniformly in the class. Stable expansions are of key importance in numerical analysis where functions are given up to certain error. We show that expansions in homogeneous functions are always stable on a small ball around the origin, and evaluate the radius of the largest ball with this property. T3 - Preprint - (2005) 16 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29925 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Root functions of elliptic boundary problems in domains with conic points of the boundary N2 - We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain whose boundary is a smooth surface away from a finite number of points, each of them possesses a neighbourhood where the boundary is a conical surface. T3 - Preprint - (2005) 07 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29812 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Operator algebras related to the Bochner-Martinelli Integral N2 - We describe a general method of computing the square of the singular integral of Bochner-Martinelli. Any explicit formula for the square applies in a familiar way to describe the C*-algebra generated by this integral. T3 - Preprint - (2005) 04 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29789 ER - TY - JOUR A1 - Stepanenko, Victor A1 - Tarkhanov, Nikolai Nikolaevich T1 - The Cauchy problem for Chaplygin's system N2 - We discuss the Cauchy problem for the so-called Chaplygin system which often appears in gas, aero- and hydrodynamics. This system can be thought of as a nonlinear analogue of the Cauchy-Riemann system in the plane. We pose Cauchy data on a part of the boundary and apply variational approach to construct a solution to this ill-posed problem. The problem actually gives insight to fundamental questions related to instable problems for nonlinear equations. Y1 - 2010 UR - http://www.informaworld.com/openurl?genre=journal&issn=1747-6933 U6 - https://doi.org/10.1080/17476930903394978 SN - 1747-6933 ER - TY - JOUR A1 - Garifullinevich, Rustem Nail A1 - Suleimanov, Bulat Irekovich A1 - Tarkhanov, Nikolai Nikolaevich T1 - Phase shift in the Whitham zone for the Gurevich-Pitaevskii special solution of the Korteweg-de Vries equation N2 - We get the leading term of the Gurevich-Pitaevskii special solution of the KdV equation in the oscillation zone without using averaging methods. Y1 - 2010 UR - http://www.sciencedirect.com/science/journal/03759601 U6 - https://doi.org/10.1016/j.physleta.2010.01.057 SN - 0375-9601 ER - TY - JOUR A1 - Glebov, Sergei A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Autoresonance in a dissipative system N2 - We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations. Y1 - 2010 UR - http://iopscience.iop.org/1751-8121/ U6 - https://doi.org/10.1088/1751-8113/43/21/215203 SN - 1751-8113 ER - TY - JOUR A1 - Glebov, Sergei A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Weakly nonlinear dispersive waves under parametric resonance perturbation N2 - We consider a solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance. Y1 - 2010 UR - http://www3.interscience.wiley.com/cgi-bin/issn?DESCRIPTOR=PRINTISSN&VALUE=0022-2526 U6 - https://doi.org/10.1111/j.1467-9590.2009.00460.x SN - 0022-2526 ER - TY - JOUR A1 - Palamodov, Victor A1 - Tarkhanov, Nikolai Nikolaevich T1 - Nonregular boundary problems for elliptic systems N2 - We discuss explicit boundary value problems for solutions of the Fueter equation in R-4 which are normally solvable. The results extend to nonlinear first order elliptic systems. Y1 - 2009 UR - http://www.springerlink.com/content/120002 U6 - https://doi.org/10.1007/s00006-009-0159-2 SN - 0188-7009 ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators JF - Journal of differential equations N2 - We consider a Sturm-Liouville boundary value problem in a bounded domain D of R-n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on partial derivative D. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact selfadjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types. (C) 2013 Elsevier Inc. All rights reserved. KW - Sturm-Liouville problem KW - Discontinuous Robin condition KW - Root function KW - Lipschitz domain KW - Non-coercive problem Y1 - 2013 U6 - https://doi.org/10.1016/j.jde.2013.07.029 SN - 0022-0396 SN - 1090-2732 VL - 255 IS - 10 SP - 3305 EP - 3337 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Glebov, Sergei A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Forced nonlinear resonance in a system of coupled oscillators JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time t similar to s(-2) one component of the system is described for the most part by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes for the most part the behavior of the envelope in a special case. The analytic results agree with numerical simulations. Y1 - 2011 U6 - https://doi.org/10.1063/1.3578047 SN - 1054-1500 VL - 21 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Gauthier, P. M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - On the instability of the Riemann hypothesis for curves over finite fields JF - Journal of approximation theory N2 - We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) an analog of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. No number theory is required to understand the theorems and their proofs, for it is known that the zeta-functions of curves over finite fields are very explicit meromorphic functions. We study the approximation properties of these meromorphic functions. KW - Zeta-function Y1 - 2012 U6 - https://doi.org/10.1016/j.jat.2011.12.002 SN - 0021-9045 VL - 164 IS - 4 SP - 504 EP - 515 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Elin, Mark A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - Separation of boundary singularities for holomorphic generators JF - Annali di matematica pura ed applicata N2 - We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates rather strikingly the particular role of the binary operation au broken vertical bar given by 1/ f au broken vertical bar g = 1/f + 1/g on generators. KW - Semigroup KW - Holomorphic map KW - Unit disk KW - Angular derivatives Y1 - 2011 U6 - https://doi.org/10.1007/s10231-010-0165-y SN - 0373-3114 VL - 190 IS - 4 SP - 595 EP - 618 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Tarkhanov, Nikolai Nikolaevich T1 - The dirichlet to Neumann operator for elliptic complexes JF - Transactions of the American Mathematical Society N2 - We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary. KW - Elliptic complexes KW - Dirichlet to Neumann operator KW - inverse problems Y1 - 2011 SN - 0002-9947 VL - 363 IS - 12 SP - 6421 EP - 6437 PB - American Mathematical Soc. CY - Providence ER - TY - JOUR A1 - Bagderina, Yulia Yu. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Differential invariants of a class of Lagrangian systems with two degrees of freedom JF - Journal of mathematical analysis and applications KW - Equivalence KW - Differential invariant KW - Euler-Lagrange equations Y1 - 2014 U6 - https://doi.org/10.1016/j.jmaa.2013.08.015 SN - 0022-247X SN - 1096-0813 VL - 410 IS - 2 SP - 733 EP - 749 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Kiselev, Oleg A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of trajectories at a separatrix under autoresonance JF - Journal of mathematical physics N2 - The subject of this paper is solutions of an autoresonance equation. We look for a connection between the parameters of the solution bounded as t -> -infinity, and the parameters of two two-parameter families of solutions as t -> infinity. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In this way we describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. (C) 2014 AIP Publishing LLC. Y1 - 2014 U6 - https://doi.org/10.1063/1.4875105 SN - 0022-2488 SN - 1089-7658 VL - 55 IS - 6 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Class of Toeplitz Operators in Several Variables JF - Advances in applied Clifford algebras N2 - We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory. KW - Cauchy data spaces KW - Laplace-Beltrami operator KW - Toeplitz operators KW - Fredholm property Y1 - 2015 U6 - https://doi.org/10.1007/s00006-015-0546-9 SN - 0188-7009 SN - 1661-4909 VL - 25 IS - 4 SP - 811 EP - 828 PB - Springer CY - Basel ER - TY - JOUR A1 - Fedchenko, Dmitri A1 - Tarkhanov, Nikolai Nikolaevich T1 - An index formula for Toeplitz operators JF - Complex variables and elliptic equations N2 - We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first-order partial differential equations in a bounded domain in R-n with smooth boundary. KW - Toeplitz operators KW - Fredholm property KW - index KW - Primary: 47B35 KW - Secondary: 47L80 Y1 - 2015 U6 - https://doi.org/10.1080/17476933.2015.1050007 SN - 1747-6933 SN - 1747-6941 VL - 60 IS - 12 SP - 1764 EP - 1787 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - Normally solvable nonlinear boundary value problems JF - Nonlinear analysis : theory, methods & applications ; an international multidisciplinary journal N2 - We investigate nonlinear problems which appear as Euler-Lagrange equations for a variational problem. They include in particular variational boundary value problems for nonlinear elliptic equations studied by F. Browder in the 1960s. We establish a solvability criterion of such problems and elaborate an efficient orthogonal projection method for constructing approximate solutions. KW - Nonlinear Laplace operator KW - Boundary value problem KW - Dirichlet to Neumann operator Y1 - 2014 U6 - https://doi.org/10.1016/j.na.2013.09.024 SN - 0362-546X SN - 1873-5215 VL - 95 SP - 468 EP - 482 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - The capture of a particle into resonance at potential hole with dissipative perturbation JF - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science N2 - We study the capture of a particle into resonance at a potential hole with dissipative perturbation and external periodic excitation. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance. Y1 - 2014 U6 - https://doi.org/10.1016/j.chaos.2013.11.003 SN - 0960-0779 SN - 1873-2887 VL - 58 SP - 27 EP - 39 PB - Elsevier CY - Oxford ER - TY - INPR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of autoresonance trajectories upon a separatrix N2 - We study asymptotic properties of solutions to the primary resonance equation with large amplitude on a long time interval. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 2 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56880 ER - TY - INPR A1 - Dyachenko, Evgueniya A1 - Tarkhanov, Nikolai Nikolaevich T1 - Degeneration of boundary layer at singular points N2 - We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)23 KW - Heat equation KW - Dirichlet problem KW - characteristic points KW - boundary layer Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60135 ER - TY - INPR A1 - Makhmudov, Olimdjan A1 - Tarkhanov, Nikolai Nikolaevich T1 - An extremal problem related to analytic continuation N2 - We show that the usual variational formulation of the problem of analytic continuation from an arc on the boundary of a plane domain does not lead to a relaxation of this overdetermined problem. To attain such a relaxation, we bound the domain of the functional, thus changing the Euler equations. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 4 KW - Extremal problem KW - Euler equations KW - p-Laplace operator KW - mixed problems Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63634 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - A simple numerical approach to the Riemann hypothesis N2 - The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 9 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57645 SN - 2193-6943 ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - Spectral projection for the dbar-Neumann problem N2 - We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)12 KW - dbar-Neumann problem KW - strongly pseudoconvex domains KW - spectral kernel function Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-58616 SN - 2193-6943 ER - TY - JOUR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Deformation quantization and boundary value problems JF - International journal of geometric methods in modern physics : differential geometery, algebraic geometery, global analysis & topology N2 - We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator. KW - Symplectic manifold KW - star product KW - trace KW - index Y1 - 2016 U6 - https://doi.org/10.1142/S0219887816500079 SN - 0219-8878 SN - 1793-6977 VL - 13 SP - 176 EP - 195 PB - World Scientific CY - Singapore ER - TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - Boundary value problems for elliptic complexes N2 - The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 3 KW - elliptic complexes KW - Fredholm property KW - index Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86705 SN - 2193-6943 VL - 5 IS - 3 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Hilbert boundary value problem for generalised Cauchy-Riemann equations N2 - We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 1 KW - Dirac operator KW - Clifford algebra KW - Riemann-Hilbert problem KW - Fredholm operator Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86109 SN - 2193-6943 VL - 5 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER -