Actions of finite groups on Amitsur rings

Action of finite groups on Amitsur rings is considered. We prove that for an Amitsur ring without |G|-torsion, the ring of invariants is approximated by finitely many Amitsur rings. It is also shown that the class of Amitsur rings is invariant under Montgomery equivalence.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 514–522, September-October, 1995.Supported by the Russian Foundation for Fundamental Research (RFFR), grant No. 93-01-16171.     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula. First two authors were partially supported by the grant MEC, PN I + D + i MTM2004-00958. Partially supported by the grants RFBR-04-01-00762, NSh-4719.2006.1 The author is thankful to the University of Valladolid for hospitality.     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of (\Bbb C²,0)({\Bbb C}^2,0) by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula.     

Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincaré map P^T is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincaré map has in the cylindrical phase space an invariant circle, on which P^T is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of P^T on this curve. Supported by National Natural Science Foundation of China (10771155, 10571131) and Natural Science Foundation of Jiangsu Province (BK 2006046).     

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras. Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998. The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work. In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376.     

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given. Project supported by the National Natural Science Foundation of China (No: 10171044), the Natural Science Foundation of Jiangsu Province (No: BK2001024), the Foundation for University Key Teachers of the Ministry of Education of China     

Denumerable controlled Markov chains with average reward criterion: Sample path optimality

We consider discrete-time nonlinear controlled stochastic systems, modeled by controlled Makov chains with denumerable state space and compact action space. The corresponding stochastic control problem of maximizing average rewards in the long-run is studied. Departing from the most common position which usesexpected values of rewards, we focus on a sample path analysis of the stream of states/rewards. Under a Lyapunov function condition, we show that stationary policies obtained from the average reward optimality equation are not only average reward optimal, but indeed sample path average reward optimal, for almost all sample paths.Research supported by a U.S.-México Collaborative Research Program funded by the National Science Foundation under grant NSF-INT 9201430, and by CONACyT-MEXICO.Partially supported by the MAXTOR Foundation for applied Probability and Statistics, under grant No. 01-01-56/04-93.Research partially supported by the Engineering Foundation under grant RI-A-93-10, and by a grant from the AT&T Foundation.     

The algebraic structure ofH-dissipative operators in a finite-dimensional space

We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998. The authors axe grateful to Professor A. A. Shkalikov for useful discussions. The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.     

Parabolic stable Higgs bundles over complete noncompact Riemann surfaces

LetM be an open Riemann surface with a finite set of punctures, a complete Poincaré-like metric is introduced near the punctures and the equivalence between the stability of an indecomposable parabolic Higgs bundle, and the existence of a Hermitian-Einstein metric on the bundle is established. Project surpported partially by the National Natural Science Foundation of China (Grant No. 19701034).     

Poincaré–Treshchev Mechanism in Multi-scale,Nearly Integrable Hamiltonian Systems

This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53–83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419–1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré–Treshchev mechanism.     

Analytic continuation and superconvergence of series of homogeneous polynomials

LetD be a domain in ⁿ (n1) andx ⁰ D. We prove that a necessary and sufficient condition for the existence of a semicontinuous regular methodA such that the series expansion of any real-analytic functionf inD in homogeneous polynomials aroundx ⁰ is uniformly summed by this method tof(x) on compact subsets ofD is thatD be rectilinearly star-shaped with respect tox ⁰.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 708–714, November, 1996.In conclusion, I wish to my sincere gratitude to my scientific supervisor Prof. Dolzhenko for setting the problem and his attention to my work.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Separately continuous selectors

The existence of separately continuous selectors of a separately lower and upper semicontinuous many-valued multivariate map is proved, provided its domain is a compact set and its range is a convex closed subset of a metrizable convex compact set. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 209–216, February, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00264.     

Use of the Lagrange method in nonlinear analysis

The problem of the spectrum of a nonlinear system and its relation to ellipsoid transform widths are studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 334–342, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Gas-dynamic helical motions with pressure and density depending on time alone

For the submodel of helical motions invariant with respect to the sum of rotation and translation, we consider solutions with pressure and density depending on time alone. Consistency of the system is studied by proceeding to Lagrangian variables. Equivalence of solutions is determined in terms of the five-dimensional admissible group. All solutions of the form described are calculated to within equivalence.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 133–141, January, 1996.The work was financially supported by the Russian Foundation for Basic Research under grant No. 93-013-17326.     

The number of periodic solutions of polynomial differential equations

We estimate the number of periodic solutions for special classes ofnth-order ordinary differential equations with variable coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 720–727, November, 1998. The author thanks Yu. S. Il'yashenko for setting the problems, permanent advice, and overall support. The author is also thankful to D. A. Panov for numerous discussions. This research was supported by the CRDF Foundation under grant MR1-220, by the INTAS Foundation under grant No. 93-05-07, and by the Russian Foundation for Basic Research under grant No. 95-01-01258.     

Singular integral operators on complicated contours and pseudodifferential operators

We apply the pseudodifferential operator technique to the study of algebras of singular operators on complicated contours. This technique is used to construct a symbolic calculus for theC ^*-algebra generated by singular integral operators whose coefficients may have singularities of the second kind on complicated contours; the curves forming a node are not required to have a tangent at the node. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 67–85, July, 1995. This work was partially supported by the Russian Foundation for Basic Research under grant No. 93-04-691.     

Denominators for the Poincaré series of invariants of matrices with involution

The goal of this paper is to study some Poincaré series associated to the invariants of the symplectic and odd orthogonal groups. These series turn out to be rational functions and our main results will describe the denominators. This work will generalize some known results on the invariants of the general linear groups. In addition to whatever intrinsic interest we hope our results may have, the subject involves an interesting interplay of invariant theory and complex variables. The first author gratefully acknowledges Support from DePaul University Research Council. The second author was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by an Internal Research Grant from Bar-Ilan University.     

Periodic orbits of competitive and cooperative systems

In this paper we consider the existence, location and stability type of periodic orbits of competitive and cooperative systems of autonomous ordinary differential equations. Particular attention is given to the existence of invariant manifolds related to periodic orbits and these results are used to improve a result of Hirsch for three dimensional irreducible competitive and cooperative systems. In particular, the Poincaré-Bendixson theorem holds for such three dimensional systems.     

On action-angle coordinates and the Poincaré coordinates

This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants “∮_γ p dq”. The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did.     

Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $$\mathbb{R}^{12} $$

A quadratic polynomial differential system can be identified with a single point of through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems of having a rational first integral of degree 2. We show that there are only 24 topologically different phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide a unique representative of every class modulo an affine change of variables and a rescalling of the time variable. Moreover, each one of these 24 representatives is determined by a set of invariant conditions and each respective first integral is given in invariant form directly in The authors are partially supported by a MEC/FEDER grant MTM2005-06098-C02-01, and a CONACIT grant number 2005SGR-00550. Partially supported by CRDF-MRDA CERIM-1006-06