Properties of the skeleton of aggregates grown on a Cayley tree

We discuss and analyze a family of trees grown on a Cayley tree, that allows for a variable exponent in the expression for the mass as a function of chemical distance, M(l)l ^dl . For the suggested model, the corresponding exponent for the mass of the skeleton,d _l ^s , can be expressed in terms ofd _l asd _l ^s = 1,d _l d _l ^c = 2;d _l ^s = d _l –1,d ₁ d _l ^c = 2, which implies that the tree is finitely ramified ford _l 2 and infinitely ramified whend _l 2. Our results are derived using a recursion relation that takes advantage of the one-dimensional nature of the problem. We also present results for the diffusion exponents and probability of return to the origin of a random walk on these trees.     

On the correlation for Kac-like models in the convex case

The aim of this paper is to stu the behavior asm tends to of a family of measures exp[- ^(m)(x)]dx ^(m) on ^m , where ^(m) is a potential on ^m which is a perturbation in a suitable sense of the harmonic potential _j x _j ² .     

Computer simulation of some dynamical properties of the Lorentz gas

We carried out molecular dynamics simulations of a Lorentz gas, consisting of a lone hydrogen molecule moving in a sea of stationary argon atoms. A Lennard-Jones form was assumed for the H₂-Ar potential. The calculations were performed at a reduced temperatureK ^* =kT/_{H 2–Ar} = 4.64 and at reduced densities ^*= _{Ar Ar} ³ in the range 0.074–0.414. The placement of Ar atoms was assumed to be random rather than dictated by equilibrium considerations. We followed the trajectories of many H₂ molecules, each of which is assigned in turn a velocity given by the Maxwell-Boltzmann distribution at the temperature of the simulation. Solving the equations of motion classically, we obtained the translational part of the incoherent dynamic structure factor for the H₂ molecule,S _tr(q, ). This was convoluted with the rotational structure factorS _rot(q, ) calculated assuming unhindered rotation to obtain the total structure factorS(q, ). Our results agree well with experimental data on this function obtained by Egelstaffet al. At the highest density ( ^*=0.414) we studied the dependence ofS(q, ) on system size (number of Ar atoms), number of H₂ molecules for which trajectories are generated, and the length of time over which these trajectories are followed.     

Consequences of the complex character of the internal symmetry in supersymmetric theories

The consequences of the invariance of the superpotential under the complexificationG ^c of the internal symmetry group on the determination of the possible patterns of symmetry and supersymmetry breaking are established in a globally supersymmetric theory. In particular, in the case of global internal symmetry we show that a vacuum associaated to a pointz, whereG _z ^c G _z ^c is always degenerate with a vacuum associated to a pointz, whereG _z ^c =G _z ^c ; all the other degeneracies of the minimum of the potential on an orbit ofG ^c are also determined and shown to be completely removed when the internal symmetry is gauged. The zeroes of theD-term of a supersymmetric gauge theory are characterized as the points of the closed orbits ofG ^c which are at minimum distance from the origin; at these pointsG _z ^c =G _z ^c . It is rigorously proved that the minimum of the potential is zero if the gradient of the superpotential vanishes somewhere. It is also shown that theD-term necessarily vanishes at the minimum of the potential if the direction of spontaneous supersymmetry breaking is invariant byG.Partially supported by the Swiss National Science Foundation and INFN, Sezione di PadovaOn leave of absence from the Department of Physics of the University of Padova, Italy     

Charge compensation and the luminescence of Cr3+ in KMgF3

Studies of the photoluminescence spectra of Cr³⁺ ions in KMgF₃ crystals co-doped with Cr³⁺ and Ni²⁺ ions are reported. Several crystal field sites are identified by the different R-line spectra due to the ² E⁴ A ₂ transition and broadband luminescences associated with the ⁴ T ₂⁴ A ₂ transitions. Cr³⁺ ions substituting without local charge compensation on the octahedral Mg²⁺ site give rise to a low temperature R line in photoluminescence at =702.3 nm with a radiative decaytime of 3 ms at T=14 K. At T=300 K this isotropic centre gives rise to an unpolarized broadband ⁴ T ₂⁴ A ₂ emission, which results from the thermal occupancy of an excited ⁴ T ₂ state just above the ² E level which, at lower temperature, gives rise to emission in the R-line. Other crystal field sites are due to some Cr³⁺ ions having Mg²⁺ or K⁺ vacancies in nearest-neighbour positions, these vacancies being required to maintain charge neutrality in doped fluoride perovskites. The Cr³⁺–K⁺ vacancy complex results in the centre having trigonal symmetry, and low temperature, photoluminescence via R ₁ and R ₂ lines at 716.8 nm and 716.0 nm, respectively. Finally, Cr³⁺ ions having a nearest neighbour Mg²⁺ vacancy have tetragonal symmetry, experiencing weak crystal fields. In consequence, the ⁴ T ₂ level lies below ² E and the photoluminescence spectrum at low temperature takes the form of a polarized broad ⁴ T ₂⁴ A ₂ band with peak at 760 nm and radiative decaytime of 54 s.     

Three- and five-dimensional considerations of the geometry of Einstein's* g-unified field theory

We study the algebraic and differential geometric structures of three- and five-dimensional^* g-unified field theory, with emphasis on the five-dimensional^* g-unified field theory, in which we derive a new set of powerful recurrence relations which hold in a five-dimensional generalized Riemannian manifoldX ₅ , prove a necessary and sufficient condition for the existence and uniqueness of the solution of the system of the Einstein equations in the first two classes, and find a precise tensorial representation of the Einstein connection in terms of^*g .     

On the Existence of a Maximizer for the Strichartz Inequality

It is shown that a maximizing function u _*L ² does exist for the Strichartz inequality e ^it _x ² u _L ⁶ _t (L ⁶ _x )Su _L ², with S>0 being the sharp constant.     

On the directional dependence of composite field operators

The Wilson expansion of the field operator productA ₁(x ₁)A ₂(x ₂) may be used to define composite operators which are local with respect to 1/2(x ₁+x ₂) and depend in addition on a vector proportional to the distancex ₁–x ₂. It is proved that the composite operators are polynomials in , for fixed ² 0, and that their dependence on ² only involves powers of ² and lg².This work was supported in part by the National Science Foundation Grant No. GP-25609.     

The effect of the crystal field strength on the optical spectra of Cr3+ in gallium garnet laser crystals

We have investigated the basic spectroscopic properties of Cr³⁺-doped gallium garnets. Weak crystal fields (Dq/B=2.39–2.55) lead to broad⁴ T ₂⁴ A ₂ fluorescence in the 700–950 nm range depending on the chemical composition of the crystals. At room temperature the⁴ T ₂⁴ A ₂ transition is homogeneously broadened, whereas the 4 K spectra of the² E⁴ A ₂ transition show inequivalent Cr³⁺ sites and inhomogeneous linewidths of 8 cm^–1 in these laser crystals. From 300 to 4 K, the oscillator strength of the⁴ T ₂⁴ A ₂ transition is reduced by a factor of about 0.6 due to the temperature dependent electron phonon coupling. The transition probability of the² E⁴ A ₂ transition correlates with the energetic position of the⁴ T ₂ level, which is admixed into the² E level via spin-orbit coupling. The comparison of the² E⁴ A ₂ phonon sideband structure in garnets of different chemical compositions allows the assignment of some peaks to GaO₄-tetrahedron modes of the garnet structure.     

Method of action-angle variables and the classical dynamics of a nonlinear lagrangian

The method of action-angle variables is used to obtain the complete periodic solutions of a nonlinear chiral Lagrangian system with the Lagrangian of the formL=1/2 {q² [(q · q)²/(1 - q²)] - [k ₀q²/(1 - q²)]q=(q ₁,q ₂,q ₃) by making suitable canonical transformations. Usual semiclassical quantization procedure may then be applied to obtain the energy levels, which is shown to be in good agreement with exact results.     

The density of states at the fermi level of the non-crystalline semiconductors

Conclusion On the basis of our experimental results on glassy As₂Te₃ we have shown that the results on the films of As₂Te₃, concerning the conductivity _F (approximately linearly rising with the frequency) and the following density of states at the Fermi level are affected mainly by the thin film nature of samples.The estimated value of the density of statesG(E _F ) at the Fermi level for glassy As₂Te₃ isG(E _F ) 10¹⁸ eV^–1 cm^–3 which is no longer in contradiction with the other results.     

On the number of invariant measures for higher-dimensional chaotic transformations

LetS be a bounded region inR ^N and letP={S _l } _i=1 ^m be a partition ofS into a finite number of closed subsets having piecewiseC ² boundaries of finite (N–1)-dimensional measure. Let :SS be piecewiseC ² onP and expanding in the sense that there exists 0<<1 such that for anyi=1,2,...,m, DT _i ^–1<, whereDT _i ^–1 is the derivative matrix ofT _i ^–1 and · is the Euclidean matrix norm. We prove that for some classes of such mappings, for example, Jabtonski transformations or convexity-preserving transformations, the number of crossing points constitutes a bound for the number of ergodic absolutely continuous -invariant measures. We give examples showing that in general the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we show that it is possible to construct piecewise expandingC ² transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic, absolutely continuous invariant measures.     

The cohomology of the space of magnetic monopoles

Denote byX _q the reduced space ofSU ₂ monopoles of chargeq in ³. In this paper the cohomology ofX _q , the cohomology with compact supports ofX _q , and the image of the latter in the former are all calculated as representations of /q which acts onX ₂. This provides a non-trivial lower bound for theL ² cohomology ofX _q which is compatible with some conjectures of Sen. It is also shown that, granted some assumptions about the metric onX _q , itsL ² cohomology does not exceed this bound in the situation referred to in the paper as the coprime case.The work described here was carried out partly at the University of Texas at Austin.     

Group-Theoretic analysis of the mixing angle in the electroweak gauge group

In this paper we provide strong mathematical support for the idea that the experimentally measured magnitude 1 –M _W ² /M _Z ² associated with sin² _W in the standard model of electroweak interactions cannot be simultaneously identified with the squared quotient of the electric charge by theSU(2) charge,e ²/g ². In fact, the natural, mathematical requirement that the Weinberg rotation between the gauge fields associated with the third component of the weak isospin (T ₃) and the hypercharge (Y) proceeds from aglobal Lie-group homomorphism of theSU(2) U(l) _Y gauge group in some locally isomorphic Lie group [which then proves to beU(2)], and not from alocal (Lie algebra)isomorphism, imposes strong restrictions so as to fix the single valuee ²/g ²=1/2. The two definitions of sin² _W can only be identified in the asymptotic limit corresponding to an earlier stage of the universe before the spontaneous symmetry breaking had taken place.     

Note on the interfacial tension of phase-separated polymer solutions

Earlier theoretical calculations of the interfacial tension of phase-separated polymer solutions as a function of the degree of polymerizationN and the temperatureT, based partly on the mean-field approximation, had led toN ^–1/4(1–T/T _c )^3/2 for fixedN1 andT approaching the critical solution temperatureT _c It is here remarked that the scaling procedure of de Gennes then modifies this toN ^–0.37(1–T/T _c )^1.26, which is in close accord with the experimentalN ^–0.44(1–T/T _c )^1.26. The simplest mean-field picture yieldsN ^–1/2(1–T/T _c )^3/2.     

Gauge invariance of the Euler-Lagrange expressions

We prove, for a Lagrangian density L(g_ij;A _i ⁱ ;A _i ^j ), that the gauge invariance of the Euler-Lagrange expressionsE ⁱ (L) implies the existence of a gauge-invariant scalar densityL ₁, such thatE ⁱ (L) =E ⁱ (L₁). We then prove the uniqueness of the Yang-Mills field equations.     

Proton-14N double resonance study of the structural phase transitions in the perovskite type layer compound (CH3NH3)2CdCl4

Using a proton-nitrogen double resonance technique we have determined the quadrupole coupling of¹⁴N in the room temperature orthorhombic (Cmca), the low temperature tetragonal (P4₂/ncm), and the monoclinic low temperature (P2₁/c) phases of (CH₃NH₃)₂CdCl₄. In all these phases all nitrogens are chemically equivalent demonstrating that the disorder in the orientations and H-bonding arrangements of the CH₃-NH₃ groups in theC m c a andP4₂/ncm phases is indeed dynamic and not static. In the monoclinic phase the¹⁴N quadrupole coupling constant equalse ² qQ/h=880 kHz and the asymmetry parameter is=0.20, wherease ² qQ/h=790 kHz,=0.1 in the tetragonal low temperature phase ande ² qQ/h=726 kHz,=0.21 in the room temperature orthorhombic phase. The observed increase in the¹⁴N quadrupole coupling constant on going from the orthorhombic phase to the tetragonal low temperature phase which is coupled with a simultaneous decrease in the asymmetry parameter can be understood in terms of a partial freezing in of the dynamic disorder in the C-N bond directions whereas the¹⁴N quadrupole coupling tensor in the monoclinic phase is characteristic of a frozen in C-N bond in a deformed lattice, where the N-H — Cl bonds are of different length.     

Interchangeability and bounds on the effective conductivity of the square lattice

The effective conductivity ^* of an infinitely interchangeable two-component random medium is considered. This class of media includes cell materials in the continuum and the bond lattice on ^d , where the cells or bonds are randomly assigned the conductivities ₁ and ₂ ( ₁, ₂ne0) with probabilitiesp ₁ andp ₂=1–p ₁. A rigorous basis for the very old and widely used low volume fraction expansion of ^* is established, by proving that ^* is an analytic function ofp ₂ in a suitable domain containing [0, 1]. In the case of the bond lattice ind=2, rigorous fourth-order upper and lower bounds on ^* valid for allp ₂, ₁, and ₂ are derived. The four perturbation coefficients entering into the bounds are obtained from the first-order volume fraction coefficient using the method of infinite interchangeability.     

The homogeneity of the ball inR 3 and special relativity

Einstein's velocity addition formula ofspecial relativity (SR) defines a transformation _v of the ballB _c of radiusc inR ³, representing all possible velocities in an inertial systemK, onto identical ballB _c , which represents the velocities in another systemK, moving with velocity v relative toK. Since _v maps the zero velocity ofB _c into arbitrary vector v ofB _c ,B _c is homogeneous under all possible _v.A similar homogeneity of the unit ballB inL(G, H) under a set of maps _a, a B, arises also in theLine Transmission Theory (TLT) for a lossless line. HereL(G, H) is the space of all linear operators between Hilbert spacesG,H, representing the signals on the line in the two directions. The explicit form of _a is obtained naturally in TLT.     

Note on the quantum displacement of the critical point

The quantum corrections to the law of corresponding states are studied by calculating the critical pressure, temperature, and density to first order in Planck's constanth on an exactly soluble model. The ratio of the critical parameters to the corresponding classical values are found to be (p _c/p _c ⁰)^1/2=_c/_c ⁰ = T_c/T_c ⁰ = 1–0.67, with=h _c ^1/3(mkT _c)^–1/2. The critical ratio is independent ofh to first order. The results are compared with critical data for noble gases and hydrogen isotopes.