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 .     

The thermodynamic limit for a crystal

Consider a crystal with nucleii fixed at the lattice points in ³, interacting by Coulomb forces with quantized electrons in . We prove that the pressure tends to a limit as grows infinitely large.Supported by NSF Grant No. MCS80-03072     

Lorentzian Warped Products and Singularity

We show that to any convex function f: ⁿ there correspondinfinitely many geodesically complete metricsds² such that Ric() 0 for anynonspacelike vector . These metrics are constructedas the warped products of the natural metric in and the inner metric of a convexhyperface (the graph of f) in ^{n + 1}.     

Splitting the Cartesian point

It is argued that the point structure of space and time must be constructed from the primitive extensional character of space and time. A procedure for doing this is laid down and applied to one-dimensional and two-dimensional systems of abstract extensions. Topological and metrical properties of the constructed point systems, which differ nontrivially from the usual and ² models, are examined. Briefly, constructed points are associated with directions and the Cartesian point is split. In one-dimension each point splits into a point pair compatible with the linear ordering. An application to one-dimensional particle motion is given, with the result that natural topological assumptions force the number of left point, right point transitions to remain locally finite in a continuous motion. In general, Cartesian points are seen to correspond to certain filters on a suitable Boolean algebra. Constructed points correspond to ultrafilters. Thus, point construction gives a natural refinement of the Cartesian systems.     

Monopoles,braid groups,and the dirac operator

Using the relation between the space of rational functions on , the space ofSU(2)-monopoles on ³, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artin's braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field on ³ whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.The first author was supported by a grant from the NSF     

The group of automorphisms of the galilei group

The group of automorphisms of the Galilei groupG: Aut(G) is calculated. It is shown that Aut(G) has the structure of a semi-direct product byG of the group _m ^* ×_m where _m is the group of reals noted multiplicatively and _m ^* <_m is the subgroup of positive reals.     

Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ⁿ . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W _n of formal vector fields on ⁿ . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ⁿ , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff( ^m ) × FDiff ⁿ ) gives rise to classes of compatible Poisson structures on the space J ( ^m , ⁿ ) of infinite jets of smooth maps ^{m n} , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.     

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° . , . .

     

Bianchi identities,electromagnetic waves,and charge conservation in theP(4) theory of gravitation and electromagnetism

The Bianchi identities for theP(4)=O(1, 3) ^4* theory of gravitation and electromagnetism are decomposed into the standardO(1, 3) Riemannian Bianchi identity plus an additional ^4* component. When combined with the Einstein-Maxwell affine field equations the ^4* components of theP(4) Bianchi identities imply conservation of magnetic charge and the wave equation for the Maxwell field strength tensor. These results are analyzed in light of the special geometrical postulates of theP(4) theory. We show that our development is the analog of the manner in which the Riemannian Bianchi identities, when combined with Einstein's field equations, imply conservation of stress-energy-momentum and the wave equation for the LanczosH-tensor.     

Classical solutions of the chiral model,unitons, and holomorphic vector bundles

This paper deals with classical solutions of theSU(2) chiral model on ², and of a generalized chiral model on ²⁺¹. Such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space. With an appropriate boundary condition, the solutions (called 1-unitons in [9]) correspond to bundles over a compact 2-dimensional complex manifold, and the problem becomes one of algebraic geometry.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

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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, , , . , , . , .

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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

. . , . , . , . coo , 13. 5. 1954 . - .     

On the yellow colouring of lithium fluoride crystals

The paper deals with the causes of the yellow colouring of LiF crystals. In accordance with some other writers the authors arrived at the conclusion that colouring is caused by impurities. Selective absorption in the infra-red region at 2·8 is independent of this colouring. Analytical data were supplemented by crystal growing experiments in which defined admixtures of heavy metals, such as Co, Mn, Fe, Cr, Ni, Pt, Cu, were added to the melt. It is shown that, of the admixtures used, the most intense colouring is produced by manganese. Experiments on the effect of Cu and Pt were not entirely conclusive. In vacuum colour-producing impurities evaporate quite easily from the melt until their concentration drops below the critical limit required for colouring; in air this happens only if the charge is left in the melted state for a longer period (in our case a 1 kg charge was kept at 100°C above melting point for 36 hours). Results are improved if a dried gas, for instance nitrogen, is bubbled through the meit. A colourless crystal can be obtained in this manner even without using a vacuum; the starting material, however, must be sufficiently pure. A new method was worked out for preparing the salt by direct precipitation of LiCl and HF. Heavy metals are removed from the lithium component by means of cupral and dithizone. The construction of the apparatus used for the crystal growing experients in vacuum differs from that described in the appropriate literature.

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LiF

LiF. , . 2,8 . . , , , Mn, Fe, Cr, Ni, Pt, Cu. , . , . Pt . , , , ( : 1 kg-36 100° ). , , , . , , . LiCl HF. . , .

     

R-Operator, Co-Product and Haar-Measure for the Modular Double of

A certain class of unitary representations of U_q((2,)) has the property of being simultanenously a representation of for a particular choice of (q). Faddeev has proposed to unify the quantum groups U_q((2,)) and into some enlarged object for which he has coined the name ``modular double'. We study the R-operator, the co-product and the Haar-measure for the modular double of U_q((2,)) and establish their main properties. In particular it is shown that the Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator.     

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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. , , , . , [9] .

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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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. .

     

Double cascade transitions of gamma rays from radiative capture on Co59

Pairs of gamma quanta, giving a sum equal to the binding energy of the last neutron in a Co⁶⁰ isotope, were studied during radiation capture on a Co⁵⁹ nucleus. The energies of the gamma quanta giving such cascades were determined and an attempt was made to determine the relative intensities of the different cascades.

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- Co⁵⁹

Co⁵⁹ -, Co⁶⁰. -, , .

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In conclusion the authors thank J. Vávra for cooperation in the measurements.     

The propagation of excitation energy in perturbed crystals

The process of the propagation of excitation energy in perturbed dielectric crystals with a weak bond between the atoms having zero permanent dipole moment is studied. It is shown that on certain assumptions this process of propagation can be regarded as the motion of a Frenkel exciton in the electrostatic field of the defect. Frenkel's exciton can be characterized in this case as a neutral polarizable particle having induced dipole moment equal to the change in the induced dipole moment of the crystal during the excitation of one of its atoms.

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, . , . .

     

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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