B. I. Zhilinskii

Université du Littoral Côte d'Opale (ULCO), Dunkirk, Nord-Pas-de-Calais, France

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Publications (86)139.18 Total impact

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    ABSTRACT: The present paper on the SO(3) invariants and covariants built from N vectors of the three--dimensional space is the follow-up of our previous article [1] dealing with planar vectors and SO(2) symmetry. The goal is to propose integrity basis for the set of SO(3) invariants and covariant free modules and easy-to-use generating families in the case of non-free covariants modules. The existence of such non-free modules is one of the noteworthy features unseen when dealing with finite point groups, that we want to point out. As in paper [1], the Molien function plays a central role in the conception of these bases. The Molien functions are computed and checked by the use of two independent paths. The first computation relies on the Molien integral [2] and requires the matrix representation of the group action on the N spatial vectors. The second path considers the Molien function for only one spatial vector as the elementary building material from which are worked out the other Molien functions.
    Full-text · Article · Nov 2015
  • Boris Zhilinskii
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    ABSTRACT: Dehn–Sommerville relations for simple (simplicial) polytopes are applied to primitive parallelohedra. New restrictions on numbers of k -faces of non-principal primitive parallelohedra are explicitly formulated for five-, six- and seven-dimensional parallelohedra.
    No preview · Article · Mar 2015 · Acta Crystallographica Section A: Foundations and Advances
  • Guillaume Dhont · Frédéric Patras · Boris I Zhilinskií
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    ABSTRACT: The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskií (2013 The action of the orthogonal group on planar vectors: invariants, covariants and syzygies J. Phys. A: Math. Theor. 46 455202) of the algebraic structures that emerge from the symmetry-adapted polynomials in the &$({{x}_{i}},{{y}_{i}})$; coordinates of n planar vectors under the action of the SO(2) group. The set of &$\left( m \right)$;-covariant polynomials contains all the polynomials that transform according to the weight &$m\in \mathbb{Z}$; of SO(2) and is a free module for &$|m|\leqslant n-1$; but a non-free module for &$|m|\geqslant n$;. The sum of the rational functions of the Molien function for &$\left( m \right)$;-covariants describes the decomposition of the ring of invariants or the module of &$\left( m \right)$;-covariants as a direct sum of submodules. A method for extracting the generating function for &$\left( m \right)$;-covariants from the comprehensive generating function for all polynomials is introduced. The approach allows the direct construction of the integrity basis for the module of &$\left( m \right)$;-covariants decomposed as a direct sum of submodules and gives insight into the expressions for the Molien functions found in our earlier paper. In particular, a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed, where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed.
    No preview · Article · Jan 2015 · Journal of Physics A Mathematical and Theoretical
  • Toshihiro Iwai · Boris Zhilinskii
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    ABSTRACT: Rearrangement of rotation-vibration energy bands in isolated molecules within semi-quantum approach is characterized by delta-Chern invariants, each of which is associated to a locally approximated semi-quantum Hamiltonian valid in a small neighborhood of a degeneracy point for the initial semi-quantum Hamiltonian and also valid in a small neighborhood of a critical point corresponding to the crossing of the boundary between iso-Chern domains in the control parameter space. For a full quantum model, a locally approximated Hamiltonian is assumed to take the form of a Dirac operator together with a specific boundary condition. It is demonstrated that the crossing of the boundary along a path with a delta-Chern invariant equal to ±1 corresponds to the transfer of one quantum level from a subspaces of quantum states to the other subspace associated with respective positive and negative energy eigenvalues of the local Dirac Hamiltonian.
    No preview · Article · Dec 2014 · Acta Applicandae Mathematicae
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    Toshihiro Iwai · Boris Zhilinskii
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    ABSTRACT: Rearrangement of rotation-vibration energy bands in isolated molecules within semi-quantum approach is characterized by delta-Chern invariants associated to a local semi-quantum Hamiltonian valid in a small neighborhood of a degeneracy point for the initial semi-quantum Hamiltonian and also valid in a small neighborhood of a critical point corresponding to the crossing of the boundary between iso-Chern domains in the control parameter space. For a full quantum model, a locally approximated Hamiltonian is assumed to take the form of a Dirac operator together with a specific boundary condition. It is demonstrated that the crossing of the boundary along a path with a delta-Chern invariant equal to $\pm1$ corresponds to the transfer of one quantum level from a subspaces of quantum states to the other subspace associated with respective positive and negative energy eigenvalues of the local Dirac Hamiltonian.
    Preview · Article · Jul 2014
  • Guillaume Dhont · Boris I. Zhilinskií
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    ABSTRACT: The construction of invariant and covariant polynomials from the x, y components of n planar vectors under the SO(2) and O(2) orthogonal groups is addressed. Molien functions determined under the SO(2) symmetry group are used as a guide to propose integrity bases for the algebra of invariants and the modules of covariants. The Molien functions that describe the structure of the algebra of invariants and the free modules of (m)-covariants, m ⩽ n - 1, are written as a ratio of a numerator in λ with positive coefficients over a (1 - λ2)2n - 1 denominator. This form of single rational function is standard in invariant theory and has a clear symbolic interpretation. However, its usefulness is lost for the non-free modules of (m)-covariants, m ⩾ n, due to negative coefficients in the numerator. We propose for these non-free modules a new representation of the Molien function as a sum of n rational functions with positive coefficients in the numerators and different numbers of terms in the denominators. This non-standard form is symbolically interpreted in terms of a generalized integrity basis. Integrity bases are explicitly given for n = 2, 3, 4 planar vectors and m ranging from 0 to 5. The integrity bases obtained under the SO(2) symmetry group are subsequently extended to the O(2) group.
    No preview · Article · Nov 2013 · Journal of Physics A General Physics
  • Boris Zhilinskii · Toshihiro Iwai
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    ABSTRACT: Qualitatively different systems of molecular energy bands are studied on example of a parametric family of effective Hamiltonians describing rotational structure of triply degenerate vibrational state of a cubic symmetry molecule. The modification of band structure under variation of control parameters is associated with a topological invariant "delta-Chern". This invariant is evaluated by using a local Hamiltonian for the control parameter values assigned at the boundary between adjacent parameter domains which correspond to qualitatively different band structures.
    No preview · Article · Jul 2013 · Physics Letters A
  • B. I. Zhilinskii
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    ABSTRACT: Recent developments associated with old technique of generating functions and invariant theory which I have started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 years ago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decomposition of the number of state generating function into regular and oscillatory contributions and its relation with Todd polynomials and topological characterization of energy bands; qualitative aspects of resonant oscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.
    No preview · Article · Jan 2012 · Physics of Atomic Nuclei
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    Boris Zhilinskii
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    ABSTRACT: Hamiltonian monodromy is known to be the first obstruction to the existence of global action coordinates in integrable systems. Its manifestation in quantum systems can be seen as characteristic defects of the regular lattice formed by the joint eigenvalues of mutually commuting quantum operators. The relation between topology of singular fibers of classical integrable fibrations and patterns formed by joint spectrum of corresponding quantum systems is discussed. The notion of the sign of 'elementary monodromy defect' is introduced on the basis of 'cut and glue' construction of the lattice defects. Special attention is paid to non-elementary defects which generically appear in phyllotaxis patterns and can be associated with plant morphology.
    Preview · Article · Oct 2010 · Journal of Physics A Mathematical and Theoretical
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    Boris Zhilinskii
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    ABSTRACT: Hamiltonian monodromy is known to be the simplest obstruction to the existence of global action-angle variables in integrable models of classical dynamics. Recently, the corresponding quantum monodromy concept is introduced and shown to be an important qualitative feature of many different realistic models and concrete physical quantum systems. Vibrational structure of simple molecules, electronic states of hydrogen atom in external fields, coupling of angular momenta is discussed as basic physical examples. Starting from these examples new qualitative features of molecular systems leading naturally to generalized monodromy notions is introduced. Going finally to really complex systems the tentative relation between phyllotaxis and monodromy is suggested.
    Preview · Article · May 2010

  • No preview · Article · Jun 2009 · Russian Mathematical Surveys
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    J.B. Delos · G Dhont · D.A. Sadovskií · B.I. Zhilinskií
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    ABSTRACT: a b s t r a c t Monodromy is the simplest obstruction to the existence of global action–angle variables in integrable Hamiltonian dynamical sys-tems. We consider one of the simplest possible systems with monodromy: a particle in a circular box containing a cylindrically symmetric potential-energy barrier. Systems with monodromy have nontrivial smooth connections between their regular Liouville tori. We consider a dynamical connection produced by an appro-priate time-dependent perturbation of our system. This turns studying monodromy into studying a physical process. We explain what aspects of this process are to be looked upon in order to uncover the interesting and somewhat unexpected dynamical behavior resulting from the nontrivial properties of the connection. We compute and analyze this behavior.
    Preview · Article · Apr 2009 · Annals of Physics

  • No preview · Article · Jan 2009
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    B. I. Zhilinskií
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    ABSTRACT: Presence of energy bands in quantum energy spectra of molecules reflects the existence of “slow” and “fast” motions in corresponding classical problem. Generic qualitative modifications of energy bands under the variation of some strict or approximate integrals or motion considered as control parameters are analyzed within purely quantum description, within semi-quantum one (slow dynamical variables are classical; fast variables are quantum) and within purely classical one. In quantum approach the reorganization of bands is seen from the redistribution of energy levels between bands. In semi-quantum approach the system of bands is represented by a complex vector bundle with the base space being the classical phase space for slow variables. The topological invariants (Chern classes) of the bundle are related to the number of states in bands through Fedosov deformation quantization. In purely classical description the reorganization of energy bands is manifested through the presence of Hamiltonian monodromy.
    Preview · Article · Nov 2008 · Journal of Mathematical Chemistry
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    Boris Zhilinskii
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    ABSTRACT: We present a method which enables one to calculate generating functions counting the number of linearly independent tensor operators of different degrees which should be included in phenomenological effective Hamiltonians constructed from boson creation and annihilation operators for several degrees of freedom in the presence of resonances and symmetry. The method is based on the application of the Molien generating function technique and the Hadamard product of rational functions. The latter leads to the representation of the answer in a form of a rational function. The technique is illustrated by the example of effective Hamiltonians for vibrational polyads in a methane-type molecule, which is a dynamical system with nine degrees of freedom formed by one non-degenerate, one doubly degenerate and two triply degenerate modes in resonance 2:1:1:2:2:2:1:1:1.
    Preview · Article · Sep 2008 · Journal of Physics A Mathematical and Theoretical
  • J. B. Delos · G. Dhont · D. A. Sadovskií · B. I. Zhilinskií
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    ABSTRACT: Hamiltonian monodromy —a topological property of the bundle of regular tori of a static Hamiltonian system which obstructs the existence of global action-angle variables— occurs in a number of integrable dynamical systems. Using as an example a simple integrable system of a particle in a circular box with quadratic potential barrier, we describe a time-dependent process which shows that monodromy in the static system leads to interesting dynamical effects.
    No preview · Article · Jul 2008 · EPL (Europhysics Letters)
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    M S Hansen · F Faure · B I Zhilinskií
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    ABSTRACT: We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2). In a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra.
    Preview · Article · Oct 2007 · Journal of Physics A Mathematical and Theoretical
  • G. Pierre · D. A. Sadovskii · B. I. Zhilinskii
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    ABSTRACT: Qualitative changes in the rotational structure of a finite particle quantum system are studied. The crossover phenomenon is explained from the point of view of consecutive quantum bifurcations. The generic organization of bifurcations is related to the stratification of the space of dynamical variables imposed by the invariance group of the Hamiltonian.
    No preview · Article · Jul 2007 · EPL (Europhysics Letters)
  • V. B. Pavlov-Verevkin · D. A. Sadovskii · B. I. Zhilinskii
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    ABSTRACT: Using a simple exactly soluble quantum model, it is shown that the diabolic points may be associated with the qualitative phenomenon of the redistribution of the energy levels between different branches in the energy spectra.
    No preview · Article · Jul 2007 · EPL (Europhysics Letters)
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    K. Efstathiou · D. A. Sadovskií · B. I. Zhilinskií
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    ABSTRACT: We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of al possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture tha the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonanc corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000 Physica 142, 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and conside briefly the 1 : 2 zone.
    Preview · Article · Jul 2007 · Proceedings of The Royal Society A

Publication Stats

1k Citations
139.18 Total Impact Points

Institutions

  • 2008-2015
    • Université du Littoral Côte d'Opale (ULCO)
      • Département de Physique
      Dunkirk, Nord-Pas-de-Calais, France
  • 2004-2008
    • French National Centre for Scientific Research
      Lutetia Parisorum, Île-de-France, France
  • 2007
    • University of Burgundy
      Dijon, Bourgogne, France
  • 1984-2007
    • Moscow State Forest University
      Mytishi, Moskovskaya, Russia
  • 2001
    • Pohang University of Science and Technology
      • Department of Physics
      Geijitsu, Gyeongsangbuk-do, South Korea
  • 1994-1995
    • Aarhus University
      • Department of Chemistry
      Århus, Central Jutland, Denmark
  • 1982
    • Lomonosov Moscow State University
      • Division of Chemistry
      Moskva, Moscow, Russia