Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

A sketch of Lie superalgebra theory

This article deals with the structure and representations of Lie superalgebras (₂-graded Lie algebras). The central result is a classification of simple Lie superalgebras over and .     

Free differential algebras: Their use in field theory and dual formulation

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p>1). We study here the algebra of FDA transformations. To every p-form in the FDA, we associate an extended Lie derivative l generating a corresponding gauge transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's.     

Supersymmetry and Kac-Moody algebras

The invariance algebra of the Majorana action contains a Kac-Moody algebra which, on shell, reduces to an Abelian algebra. In the absence of auxiliary fields in the Wess-Zumino model, supersymmetry transformations generate an infinite-dimensional Lie algebra, which is shown to be a Grassmannian extension of this Kac-Moody algebra. The corresponding Noether charges are discussed.     

An attempt to relate area-preserving diffeomorphisms to Kac-Moody algebras

In the same way as the Virasoro algebra can be connected with Kac-Moody algebras defined on the S ¹ circle, the area-preserving diffeomorphism algebra SDiff(), where is a two-dimensional surface, acts as a derivation algebra on super Kac-Moody algebras with one or two supersymmetries. Then a Sugawara-like construction with fermions of the nonextended SDiff() algebra is discussed.     

Superconformal current algebras and their unitary representations

A natural supersymmetric extension is defined of the current (= affine Kac-Moody Lie) algebra ; it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of are constructed. They extend to unitary representations of the semidirect sumS (G) of with the superconformal algebra of Neveu-Schwarz, for , or of Ramond, for =0.On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria     

From Dynkin diagram symmetries to fixed point structures

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra g induces an automorphism of g and a mapping between highest weight modules of g. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the orbit Lie algebra . In particular, the generating function for the trace of over weight spaces, which we call the twining character of g (with respect to the automorphism), is equal to a character of . The orbit Lie algebras of untwisted affine Lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.     

The motion of 90° wedge domains in BaTiO3 in an alternating electric field

The paper deals with the motion of 90° wedge domains in BaTiO₃ in an alternating field of 50 c/s. The critical field, the positional hysteresis loops with double asymmetry, the production of wedges with polarization perpendicular to the field and 180° substructure in the wedges were studied. The differences between the behaviour of the wedges and the individual 90° walls are pointed out which are caused by differences in the energy balance of these formations and by different interactions with 180° processes. The upper limit of contribution of the wedge motion to the initial permittivity is estimated. The results are discussed from the phenomenological point of view.

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90° BaTiO₃

90° BaTiO₃ , 50 Hz. , , , , , , 180° . 90° , 180° . , . .

     

The influence of oscillations on ionization in a discharge path

The paper solves the problem of gas ionization in a discharge path in a very dilute gas, where the free path of the electrons is much larger than the dimensions of the path and the transit time of the electrons between the electrodes is of the order of the period of the applied h-f voltage. It was found that for a certain ratio of the transit time of the electrons between the electrodes in the discharge path to the period of the h-f oscillation, resonance occurs when the wattless current component is zero. The electron density rises in the path and thus also the gas ionization.

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In conclusion, the author would like to thank F. Benda for preparing the equipment, M. Kivánek for preparing the equipment and some of the measurements, and A. Hrdá for the measurements and for working out the case with equally large a-c and d-c voltages within the framework of her thesis.     

О СУЩЕСТВОВАНИИ ОТРИ ЦАТЕЛЬНОГО ЯВЛЕНИЯ БАРКГАУЗЕНА

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Quantum Observables,Lie Algebra Homology and TQFT

Let us consider a Lie (super)algebra G spanned by T where T are quantum observables in BV formalism. It is proved that for every tensor c^... that determines a homology class of the Lie algebra G the expression c^...T...T is again a quantum observable. This theorem is used to construct quantum observables in the BV sigma model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.     

Elastic stress around linear dislocation in anisotropic medium

Relations are derived for the elastic stress field around a linear dislocation in an infinite medium with general anisotropy. The strongly deformed material around the core of the dislocation is cut out in the shape of an elliptic cylinder. The conditions of a free surface are used on the boundary thus formed. The calculation of the field around a crack in a crystal, the model of which was proposed by Fujita in [9], is given as an example.

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The author thanks Dr. F. Kroupa for valuable discussions and all-round help in solving the problem and Z. Hemanová for performing the numerical calculations.     

The application of the thermodynamics of irreversible processes to welding arcs

The self-regulation of an inert gas shielded metal welding arc is dealt with briefly. A thermodynamic equation is derived for the self-regulation of such an arc.

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Perturbation about the mean field critical point

We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well /4⁴ — /2² perturbation of a massless Gaussian lattice field in the weak coupling limit (0, proportional to ). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, , is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice ^d has dimensiond3.In both cases we derive an asymptotic estimate to first order (in or ²) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in or around the Gaussian or mean field critical points.The appendix, due to E. Speer, contains an extension of Weinberg's theorem on the divergence of Feynman graphs which is used in the proofs.Supported by NSF Grant # MCS 78-01885Supported by NSF Grant # PHY 78-15920     

The modular group and super-KMS functionals

Every normal, faithful, self-adjoint functional on a von Neumann algebraA canonically determines a one-parameter-weakly continuous *-automorphism group (the analog of the modular group) and a canonical ₂ grading onA, commuting with . We show that the functional satisfies the weak super-KMS property with respect to and Furthermore, we prove that and are the unique pair of a-weakly continuous one-parameter *-automorphism group and a grading of the algebra, commuting with each other, with respect to which is weakly super-KMS. The above results thus provide a complete extension of the theory of Tomita and Takesaki to the nonpositive case.Supported in part by the National Science Foundation under Grant DMS-8922002.     

A contribution to the theory of the triple crystal diffractometer

The first part of the paper gives a general equation for triple-crystal arrangement with perfect crystals on the assumption that the third crystal is rotated. It is shown that in the case of perfect crystals the shape of the reflection curve is practically independent of the vertical divergence. The case of mosaic crystals is also solved and the possibility of rotation by other than the third crystal is considered. A method is proposed for investigating the imperfection of a crystal which is different from methods used up to now. The paper is supplemented by some experimental results.

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тЕОРИь И стРУктУРА тЕ МпЕРАтУРНОгО ИжОБРА жЕНИь

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The theory and structure of the temperature image

The paper describes the theory of the formation of a temperature image when there are large temperature differences and introduces the concept of temperature distortion of an image. It also deals with the geometrical distortion of the temperature image by the longitudinal thermal conductivity of the receiver layer and gives a solution of the corresponding differential equation of the temperature image for the case of a general linear line test and general circular concentric test. Equations are derived for the temperature distributions of the temperature images of some optical lineated and circular test objects and their geometrical distortion due to the thermal conductivity of the receiver layer is investigated.

     

Сцинтилляционный сп ектрометр с сложение м импулъсов от двух кри сталлов

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Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

The classical theory of nuclear spin interaction in a ferromagnet

The Hamiltonian of nuclear spin interaction in a ferromagnet is derived by the classical method, which introduces the precession of nuclear spin into the equation of motion for magnetization. From this it is shown that the interaction Hamiltonian also depends on the magnitude and sign of nuclear precession frequency _N and the damping constant of ferromagnetic resonance A. The calculation of these parameters makes the Suhl [1] quantum mechanical derivation of the Hamiltonian of nuclear spin interaction in a ferromagnet more accurate. The influence of these parameters on the relaxation timeT ₂ is also discussed and is applied to the case of cubic cobalt.

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In conclusion the author thanks Dr. L. Valenta for significant remarks and valuable advice on this work. He also thanks J. Kvasnica and Z. roubek for suggestive discussions.