dr Agnieszka Martens

Zakład Teorii Ośrodków Ciągłych i Nanostruktur (ZTOCiN)
Pracownia Mechaniki Analitycznej i Teorii Pola (PMAiTP)
stanowisko: główny specjalista
telefon: (+48) 22 826 12 81 wew.: 430
pokój: 117
e-mail: amartens

Doktorat
2006-06-29Hamiltonowskie i kwantowe układy z symetriami i więzami. Modele nieliniowe i ich zastosowania fizyczne 
promotor -- prof. dr hab. Jan Sławianowski, IPPT PAN
596
 
Ostatnie publikacje
1.Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Mechanics of affine bodies. Towards affine dynamical symmetry, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ISSN: 0022-247X, DOI: 10.1016/j.jmaa.2016.08.042, Vol.446, pp.493-520, 2017

Streszczenie:

In this paper we discuss certain dynamical models of affine bodies, including problems of partial separability and integrability. There are some reasons to expect that the suggested models are dynamically viable and that on the fundamental level of physical phenomena the “large” affine symmetry of dynamical laws is more justified and desirable than the restricted invariance under isometries.

Słowa kluczowe:

Homogeneous deformation, structured media, affinely-invariant dynamics, elastic vibrations encoded in kinetic energy, Calogero-Moser and Sutherland integrable lattices

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
35p.
2.Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Quantized mechanics of affinely-rigid bodies, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.4501, Vol.40, No.18, pp.6900-6918, 2017

Streszczenie:

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configurationmanifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group GL(n,R) and the space of translations R^n, where n equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.

Słowa kluczowe:

Homogeneously deformable body, Peter-Weyl analysis, Schrödinger quantization.

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
25p.
3.Martens A., Test affinely-rigid bodies in Riemannian spaces and their quantization, ACTA PHYSICA POLONICA B, ISSN: 0587-4254, DOI: 10.5506/APhysPolB.46.843, Vol.46, No.4, pp.843-862, 2015

Streszczenie:

Discussed are some classical and quantization problems of test affinely-rigid bodies moving in Riemannian spaces. We investigate the systems with potential energies for which the variables can be separated. The special case of constant curvature two-dimensional spaces is discussed. Some explicit solutions are found using the Sommerfeld polynomial method.

Słowa kluczowe:

test affinely-rigid body, Riemannian manifolds, Sommerfeld polynomial method

Afiliacje autorów:

Martens A.-IPPT PAN
15p.
4.Martens A., Test rigid bodies in Riemannian spaces and their quantization, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(13)60038-5, Vol.71, No.3, pp.381-398, 2013

Streszczenie:

Discussed are some classical and quantization problems of rigid bodies of infinitesimal size moving in Riemannian spaces. The rigorous meaning of “infinitesimal size” consists in replacing an extended body by the structured material point with internal degrees of freedom (co-moving orthonormal frame). The special case of constant curvature two-dimensional spaces is discussed. The Sommerfeld polynomial method is used to perform the quantization of such problems.

Słowa kluczowe:

test rigid body, Riemannian manifolds, Sommerfeld polynomial method

Afiliacje autorów:

Martens A.-IPPT PAN
15p.
5.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, ISSN: 1531-3492, DOI: 10.3934/dcdsb.2012.17.699, Vol.17, No.2, pp.699-733, 2012

Streszczenie:

The main leitmotivs of this paper are the essential nonlinearities, symmetries and the mutual relationships between them. By essential nonlinearities we mean ones which are not interpretable as some extra perturbations imposed on a linear background deciding about the most important qualitative features of discussed phenomena. We also investigate some discrete and continuous systems, roughly speaking with large symmetry groups. And some remarks about the link between two concepts are reviewed. Namely, we advocate the thesis that the most important non-perturbative nonlinearities are those implied by the assumed “large” symmetry groups. It is clear that such a relationship does exist, although there is no complete theory. We compare the mechanism of inducing nonlinearity by symmetry groups of discrete and continuous systems. Many striking and instructive analogies are found, e.g., analogy between analytical mechanics of systems of affine bodies and general relativity, tetrad models of gravitation, and Born-Infeld nonlinearity. Some interesting, a bit surprising problems concerning Noether theorem are discussed, in particular in the context of large symmetry groups.

Słowa kluczowe:

Essential nonlinearity, dynamical symmetries, affine bodies, general relativity, tetrad models and micromorphic continua

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
30p.
6.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Generalized Weyl–Wigner–Moyal–Ville formalism and topological groups, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.1531, Vol.35, pp.17-42, 2012

Streszczenie:

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’.

Słowa kluczowe:

Weyl–Wigner–Moyal–Ville formalism, topological groups, classical momentum mapping, unitary projective representations, Pontryagin duality, ‘Umklapp–Prozessen’

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
25p.
7.Gołubowska B., Kovalchuk V., Martens A., Rożko E.E., Sławianowski J.J., Some strange features of the Galilei group, Journal of Geometry and Symmetry in Physics, ISSN: 1312-5192, DOI: 10.7546/jgsp-26-2012-33-59, Vol.26, pp.33-59, 2012

Streszczenie:

Discussed are certain strange properties of the Galilei group, connected first of all with the property of mechanical energy-momentum covector to be an affine object, rather than the linear one. Its affine transformation rule is interesting in itself and dependent on the particle mass. On the quantum level this means obviously that we deal with the projective unitary representation of the group rather than with the usual representation. The status of mass is completely different than in relativistic theory, where it is a continuous eigenvalue of the Casimir invariant. In Galilei framework it is a parameter characterizing the factor of the projective representation, in the sense of V. Bargmann. This “pathology” from the relativistic point of view is nevertheless very interesting and it underlies the Weyl-Wigner-Moyal-Ville approach to quantum mechanics.

Słowa kluczowe:

Galilei group, affine transformation, particle mass, projective unitary representation, Weyl-Wigner-Moyal-Ville formalism

Afiliacje autorów:

Gołubowska B.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Sławianowski J.J.-IPPT PAN
8.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.1462, Vol.34, pp.1512-1540, 2011

Streszczenie:

Discussed are geometric structures underlying analytical mechanics of systems of affine bodies. Presented is detailed algebraic and geometric analysis of concepts like mutual deformation tensors and their invariants. Problems of affine invariance and of its interplay with the usual Euclidean invariance are reviewed. This analysis was motivated by mechanics of affine (homogeneously deformable) bodies, nevertheless, it is also relevant for the theory of unconstrained continua and discrete media. Postulated are some models where the dynamics of elastic vibrations is encoded not only in potential energy (sometimes even not at all) but also (sometimes first of all) in appropriately chosen models of kinetic energy (metric tensor on the configuration space), like in Maupertuis principle. Physically, the models may be applied in structured discrete media, molecular crystals, fullerens, and even in description of astrophysical objects. Continuous limit of our affine-multibody theory is expected to provide a new class of micromorphic media.

Słowa kluczowe:

Systems of affine bodies, mutual deformation tensors, affine and Euclidean invariance, structured media, elastic vibrations, geometric structures

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
25p.
9.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part I. Group Algebras as a Framework for Quantum-Mechanical Models with Symmetries, Journal of Geometry and Symmetry in Physics, DOI: 10.7546/jgsp-21-2011-61-94, Vol.21, pp.61-94, 2011

Streszczenie:

We use the mathematical structure of group algebras and $H^+$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2) and its quotient SO(3, R). The proposed scheme is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. Secondly, the other promising area of applications is Schroedinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schroedinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of SU(2).

Słowa kluczowe:

Systems of angular momenta, models with symmetries, quantum dynamics, spin systems, Schroedinger quantum mechanics, quasiclassical limit

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
10.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part II. Quantum Mechanics on Lie Groups and Methods of Group Algebras, Journal of Geometry and Symmetry in Physics, DOI: 10.7546/jgsp-22-2011-67-94, Vol.22, pp.67-94, 2011

Streszczenie:

In Part I of this series we have presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment there was very “ascetic” in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Relying on differential geometry one is able to introduce explicitly representation of important physical quantities and to formulate the general ideas of quasiclassical representation and classical analogy.

Słowa kluczowe:

Systems of angular momenta, models with symmetries, quantum dynamics, Lie group structures, quasiclassical representation, classical analogy

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
11.Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Quasiclassical and Quantum Systems of Angular Momentum. Part III. Group Algebra su(2), Quantum Angular Momentum and Quasiclassical Asymptotics, Journal of Geometry and Symmetry in Physics, DOI: 10.7546/jgsp-23-2011-59-95, Vol.23, pp.59-95, 2011

Streszczenie:

This is the third part of our series “Quasiclassical and Quantum Systems of Angular Momentum”. In two previous parts we have discussed the methods of group algebras in formulation of quantum mechanics and certain quasiclassical problems. Below we specify to the special case of the group SU(2) and its quotient SO(3,R), and discuss just our main subject in this series, i.e., angular momentum problems. To be more precise, this is the purely SU(2)-treatment, so formally this might also apply to isospin. However. it is rather hard to imagine realistic quasiclassical isospin problems.

Słowa kluczowe:

Systems of angular momenta, models with symmetries, quantum dynamics, quasiclassical isospin problems

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Martens A.-IPPT PAN
Gołubowska B.-IPPT PAN
Rożko E.E.-IPPT PAN
12.Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Quantized excitations of internal affine modes and their influence on Raman spectra, ACTA PHYSICA POLONICA B, ISSN: 0587-4254, DOI: 10.5506/APhysPolB.41.165, Vol.41, No.1, pp.165-218, 2010

Streszczenie:

Discussed is the structure of classical and quantum excitations of internal degrees of freedom of multiparticle objects like molecules, fullerens, atomic nuclei, etc. Basing on some invariance properties under the action of isometric and affine transformations we reviewed some new models of the mutual interaction between rotational and deformative degrees of freedom. Our methodology and some results may be useful in the theory of Raman scattering and nuclear radiation.

Słowa kluczowe:

Internal degrees of freedom, isometric and affine invariance, classical and quantum excitations, multiparticle objects, Raman scattering, nuclear radiation

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
20p.
13.Martens A., Sławianowski J.J., Affinely-rigid body and oscillatory dynamical models on GL(2,R), ACTA PHYSICA POLONICA B, ISSN: 0587-4254, Vol.41, No.8, pp.1847-1880, 2010

Streszczenie:

Discussed is a model of the two-dimensional affinely-rigid body with the double dynamical isotropy. We investigate the systems with potential energies for which the variables can be separated. The special stress is laid on the model of the harmonic oscillator potential and certain anharmonic alternatives. Some explicit solutions are found on the classical, quasiclassical (Bohr–Sommerfeld) and quantum levels.

Słowa kluczowe:

affinely-rigid body, harmonic oscillator potential, Sommerfeld polynomial method

Afiliacje autorów:

Martens A.-IPPT PAN
Sławianowski J.J.-IPPT PAN
20p.
14.Martens A., Quantization of two-dimensional affine bodies with stabilized dilatations, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, Vol.62, No.2, pp.145-155, 2008

Streszczenie:

Discussed are some quantization problems of two-dimensional affine bodies. Quantum dilatational motion is stabilized by some appropriately chosen model potentials. lsochoric part of the dynamics is geodetic, i.e. potential-free. Surprisingly enough, this is compatible with the existence of discrete spectrum (bounded quantum motion). The Sommerfeld polynomial method is used to perform the quantization of such problems.

Słowa kluczowe:

affinely-rigid body, two-polar decomposition, Sommerfeld polynomial method

Afiliacje autorów:

Martens A.-IPPT PAN
15.Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(05)80002-3, Vol.55, No.1, pp.1-46, 2005

Streszczenie:

Discussed is the quantized version of the classical description of collective and internal affine modes as developed in Part I. We perform the Schroedinger quantization and reduce effectively the quantized problem from $n^2$ to $n$ degrees of freedom. Some possible applications in nuclear physics and other quantum many-body problems are suggested. Discussed is also the possibility of half-integer angular momentum in composed systems of spinless particles.

Słowa kluczowe:

Collective modes, affine invariance, Schroedinger quantization, quantum many-body problem

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Kovalchuk V.-IPPT PAN
Sławianowska A.-other affiliation
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Zawistowski Z.J.-IPPT PAN
16.Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Affine symmetry in mechanics of collective and internal modes. Part I. Classical models, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, DOI: 10.1016/S0034-4877(04)80026-0, Vol.54, No.3, pp.373-427, 2004

Streszczenie:

Discussed is a model of collective and internal degrees of freedom with kinematics based on affine group and its subgroups. The main novelty in comparison with the previous attempts of this kind is that it is not only kinematics but also dynamics that is affinely-invariant. The relationship with the dynamics of integrable one-dimensional lattices is discussed. It is shown that affinely-invariant geodetic models may encode the dynamics of something like elastic vibrations.

Słowa kluczowe:

Collective modes, affine invariance, integrable lattices, nonlinear elasticity

Afiliacje autorów:

Sławianowski J.J.-other affiliation
Kovalchuk V.-IPPT PAN
Sławianowska A.-other affiliation
Gołubowska B.-IPPT PAN
Martens A.-IPPT PAN
Rożko E.E.-IPPT PAN
Zawistowski Z.J.-IPPT PAN

Lista ostatnich monografii
1.
473
Sławianowski J.J., Schroeck Jr. F.E., Martens A., Why must we work in the phase space?, IPPT Reports on Fundamental Technological Research, 1, pp.1-162, 2016
2.
67
Sławianowski J.J., Kovalchuk V., Gołubowska B., Martens A., Rożko E.E., Dynamical systems with internal degrees of freedom in non-Euclidean spaces, IPPT Reports on Fundamental Technological Research, 8, pp.1-129, 2006
3.
484
Sławianowski J.J., Kovalchuk V., Sławianowska A., Gołubowska B., Martens A., Rożko E.E., Zawistowski Z.J., Invariant geodetic systems on Lie groups and affine models of internal and collective degrees of freedom, Prace IPPT - IFTR Reports, Warszawa, 7, pp.1-164, 2004
Lista rozdziałów w ostatnich monografiach
1.
388
Sławianowski J.J., Martens A., Geometry, Integrability, Mechanics and Quantization, rozdział: Affinely-rigid body and oscillatory two-dimensional models, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.89-105, 2015
2.
392
Martens A., Geometry, Integrability, Mechanics and Quantization, rozdział: Affine models of internal degrees of freedom and their quantization, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.306-317, 2015
3.
353
Kielanowski P., Martens A., Fizycy wspominają, rozdział: Rozmowa z Andrzejem Trautmanem – Moje pierwsze 50 lat na Hożej, red. Andrzej Michał Kobos, Polska Akademia Umiejętności, Kraków, pp.489-523, 2014
4.
252
Gołubowska B., Kovalchuk V., Martens A., Rożko E.E., Sławianowski J.J., Geometry, Integrability and Quantization XIII, rozdział: Some strange features of the Galilei group, Editors: Ivailo M. Mladenov, Andrei Ludu and Akira Yoshioka, Avangard Prima, Sofia, pp.150-175, 2012
5.
35
Sławianowski J.J., Kovalchuk V., Martens A., Gołubowska B., Rożko E.E., Geometry, Integrability and Quantization XII, rozdział: Quasiclassical and Quantum Dynamics of Systems of Angular Momenta, Editors: Ivailo M. Mladenov, Gaetano Vilasi and Akira Yoshioka, Avangard Prima, Sofia, pp.70-155, 2011

Prace konferencyjne
1.Sławianowski J.J., Martens A., Affinely-rigid body and oscillatory two-dimensional models, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-94-109, Vol.XVI, pp.94-109, 2015

Streszczenie:

Discussed are some classical and quantization problems of the affinely-rigid body in two dimensions. Strictly speaking, we consider the model of the harmonic oscillator potential and then discuss some natural anharmonic modifications. It is interesting that the considered doubly-isotropic models admit coordinate systems in which the classical and Schrödinger equations are separable and in principle solvable in terms of special functions on groups.

Słowa kluczowe:

affinely-rigid body, quantization problems, two-dimensional models, harmonic oscillator potential, anharmonic modifications, doubly-isotropic models, Schrödinger equation, special functions, separability problem

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Martens A.-IPPT PAN
2.Martens A., Affine models of internal degrees of freedom and their quantization, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/giq-16-2015-207-218, Vol.XVI, pp.207-218, 2015

Streszczenie:

We discuss some classical and quantization problems of infinitesimal affinely-rigid bodies moving in two-dimensional manifolds. Considered are highly symmetric models for which the variables can be separated. We follow the standard procedure of quatization in Riemannian manifolds, i.e., we use the Hilbert space of wave functions in the sense of the usual Riemannian measure (volume element).

Słowa kluczowe:

affine models, internal degrees of freedom, quantization problems, highly symmetric models, separability problem, Riemannian manifolds, Hilbert space

Afiliacje autorów:

Martens A.-IPPT PAN
3.Sławianowski J.J., Martens A., The dynamics of the field of linear frames and gauge gravitation, Geometry, Integrability and Quantization, ISSN: 1314-3247, DOI: 10.7546/gip-14-2013-201-214, Vol.XIV, pp.201-214, 2013

Streszczenie:

The paper is motivated by gauge theories of gravitation and condensed matter, tetrad models of gravitation and generalized Born-Infeld type nonlinearity. The main idea is that any generally-covariant and GL(n,R)-invariant theory of the n-leg field (tetrad field when n=4) must have the Born-Infeld structure. This means that Lagrangian is given by the square root of the determinant of some second-order twice covariant tensor built in a quadratic way of the field derivatives. It is shown taht there exist interesting solutions of the group-theoretical structure. Some models of the interaction between gravitation and matter are suggested. It turns out that in a sense the space-time dimension n=4, the normal-hyperbolic signature and velocity of light are integration constants of our differential equations.

Słowa kluczowe:

gauge gravitation, linear frames, condensed matter

Afiliacje autorów:

Sławianowski J.J.-IPPT PAN
Martens A.-IPPT PAN