Commutator Representations of Covariant Differential Calculi on Quantum Groups

Let (, d) be a first-order differential *-calculus on a *-algebra . We say that a pair (, F) of a *-representation of on a dense domain of a Hilbert space and a symmetric operator F on gives a commutator representation of if there exists a linear mapping : L( ) such that (adb) = (a)i[F, (b) ], a, b . Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra has a faithful commutator representation. For a class of bicovariant *-calculi on , there is a commutator representation such that F is the image of a central element of the quantum tangent space. If is the Hopf *-algebra of the compact form of one of the quantum groups SL _q (n+1), O _q (n), Sp _q (2n) with real trancendental q, then this commutator representation is faithful.     

Convergence of the quantum Boltzmann map

We consider a non-linear map on the space of density matrices, which we call the Boltzmann map . It is the composition of a doubly stochastic mapT on the space ofn-body states, and the conditional expectation onto the one-body space. WhenT is ergodic, then the iterates of take any initial state to the uniform distribution. If the energy levels are equally spaced, andT conserves energy and is ergodic on each energy shell, then iterates of take any initial state of finite energy to a canonical distribution.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

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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, . , . , , . , . .

     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

Exponential decay near resonance,without analyticity

We consider a quantum mechanical model which displays the behaviour associated with having a resonance or metastable state. The Hamiltonian depends on a parameter . When =0, there is an eigenstate ₀; when 0, ₀ dissolves into the continuous spectrum, showing approximate exponential decay. We prove this result without using dilatation analyticity. The model describes a two-state atom coupled to the quantized radiation field. The state space of the field is truncated, so that only the vacuum and one-photon states are included.This work was partially supported by NSF Grant DMS-8922941     

Magnetic relaxation spectrum of ferrites Mn x Fe3−x O4+γ

The initial permeability disaccommodation in ferritesMn _x Fe₃–xO₄₊ , 0·5x1, was studied in a temperature range around –200°C to +180°C. Four separate bands were found in the relaxation spectrum of these ferrites.

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Mn _x Fe_3–x O₄₊

Mn _x Fe_3–x O₄₊ , 0,5x1, –200°C +180°C. .

     

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.     

Photo-luminescence and growth defects in ZnS: Cu monocrystals

The paper describes microscopic investigation into the spectral distribution of photo-luminescence and its polarization on hexagonal ZnS monocrystals in relation to the structural defects of such crystals. These observations, as well as birefringent observations, show that the values of the birefringence and the degree of polarization in different stripes of the same crystal are the same within the limits of observational errors.

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ZnS: Cu

ZnS . , , , .

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In conclusion the author would like to thank K. Pátek, L. Souková and M. Mokonová for providing the monocrystals and for fruitful discussions.     

Über den Einfluss von Stickstopf auf die Exoelektronenemission von Germanium

Zusammenfassung Es wurde der Einfluß eines Bombardements mit Stickstoffionen vor der Erregung untersucht und gezeigt, daß die Stiekstoffatome eine große Rolle bei der Entstehung des Emissionsmaximums bei 150° spielen.

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, 150°C.

     

МОДЕЛИРОВАНИЕ ЭЛЕКТ РОННО-ОПТИЧЕСКИХ СИС ТЕМ С ОСЕВОЙ СИММЕТРИЕЙ П РИ ПОМОЩИ СЕТИ СОПРОТИВЛЕНИЙ

The paper explains the theory of modelling electrostatic fields by a resistance network. The conditions, which the resistance network must satisfy, are derived and the question of modelling electrodes of different shapes is solved. The finished network and the results obtained on it when modelling a jet for a linear h-f accelerator of electrons are described. Particular attention is paid to the influence of a space charge, the modelling of which is an advantage of this method.

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1- , 1964., , .

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. .     

Эффективностъ сцинт илляционных счетчик ов для быстрых нейтроно в

, ZnS(Ag) . ZnS(Ag) . Po-Be, Ra-Be Li(d, n), C(d, n), N(d, n), Be(d, n). -, Co⁶⁰, Cs¹³⁷ Ra, , -, Li(p, ). , .     

Die Integralbeziehungen bei der stationären laminaren Strömung mit geschlossenen Stromlinien in annähernd kreisförmigen Gebieten

Zusammenfassung Für die statioäre lineare ebene Strömung mit geschlossenen Stromlinien bei einer großen Reynoldsschen Zahl in annähernd kreisförmigen Gebieten werden zwei Integralbeziehungen (5) und (9) abgeleitet. Nach der ersten von diesen ist das Integral der Reibungskraft entlang der Grenze des Strömungsgebiets direkt proportional der Länge dieser Grenze und der Größe der Rotation der Geschwindigkeit im Kern der Strömung. Aus der zweiten Beziehung geht hervor: Das Volumen der in der Zeiteinheit den Schnitt der Rezirkulationsgrenzschicht durchfließenden Flüssigkeit ist direkt proportional dem Integral entlang der Gebietsgrenze aus der Differenz der Quadrate der Strömungsgeschwindigkeit an der Grenze des Kerns und der Geschwindigkeit an der Gebietsgrenze und umgekehrt proportional der Größe der Geschwindigkeitsrotation im Kern und der Länge der Gebietsgrenze.

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(5) (9). . : , , .

     

The basic equations of the dynamics of the continuous distribution of dislocations II. Interpretation of general theory

The basic equations of the dynamics of the continuous distribution of dislocations analogical to Maxwell equations are derived in a series of papers [I, II, III]. The analogy of the elastic and electromagnetic fields is analyzed. In part [II] a physical interpretation is given of the tensors of stress and distortion and expressions are considered for the energy density of the elastic field of the dislocations. The basic equations are generalized for the case of a non-zero Poisson ratio and a finite body. A detailed interpretation is given of the tensor components of dislocation current density.

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II.

, . . , . . .

     

The crystal structure of kettnerite,CaBi[OF¦CO3]

Macroscopic, microscopic, goniometric, qualitative spectrographic and quantitative chemical data of kettnerite [2] with some new observations are given. The unit cell dimensions determined from the powder and rotating crystal photographs area ₀=b ₀=5.36±0.02 Å,c ₀=13.59±0.03 Å. There are four molecules in the unit cell. The crystal structure of kettnerite was studied from the ordinary and generalized projections of the Patterson function along the [010] axis. Direct evidence of tetragonal layers (Ca. 2 F, Ca) and (Bi, 2 O, Bi) parallel to the basal face was found. The spatial arrangement of these layers corresponds to the symmetry of the space groupP 4/nmm. These layers alternate in the [001] direction being interleaved by single CO₃ layers. Both the biaxial character of the mineral and the uncertainty concerning the rotation of the CO₃ groups indicate a lower, most probably an orthorhombic symmetry. The highest possible symetry is that ofCmma. The structure is related to that of the typeX ₁ found by Sillén et al. for several bismuth oxyhalides and especially to that of bismutite.

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CaBi[OF¦ CO₃]

, , , [2] . , :a ₀=b ₀=5,36±0,02 Å, ₀=13,59±0,03 Å;Z=4. [010]. (Ca, 2 F, ) (Bi, 2 O, Bi), (010). 4/. , ₃. ₃ . C. , , ₁, .

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We wish to express gratitude to our colleagues Dr. A. Línek and Ing. C. Novák (Institute of Technical Physics, Czech. Acad. Sci.) for the use of the M1 equipment [11]. We are also very grateful to Mr. Z. iký for carrying out most of the numerical calculations.     

ПОЛЕ ПЕРЕМЕННОГО ДИП ОЛЯ В НЕОДНОРОДНОМ ПОЛУПРОСТРАНСТВЕ

, . , ₁, ₂. - .     

Electron microscope study of dislocations in slip bands of bent Fe-3·2% Si alloy single crystals

Dislocations in Fe-3·2% Si alloy single crystals deformed by bending are studied in this paper. The specimens were cut parallel to the slip plane into platelets, polished, and studied in places at various distances from the neutral plane with an electron microscope. A large number of jogs of various sizes was found on screw dislocations. Many loops are formed by the motion of dislocations with jogs and by the stopping of moving dislocations by obstacles. Some of them grow as a result of stress. The activity of sources on large jogs is stopped at an early stage by a cross slip. The formation and growth of loops increases the dislocation density and causes the growth of slip bands.

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Fe-3,2% Si

Fe-3,2% Si, , , . . , . . . .