Möbius function for the set of acyclic directed backbone graphs

Consider a two-rooted graphG, the edges of which are directed in such a way that there are no cycles and every edge belongs to some self-avoiding walk from rootu to rootv which follows the direction of the edges. Au-v backbone ofG is a subgraph formed by taking the union of any subset of directed self-avoiding walks fromu tov. Let _uv be the set of all such backbones ofG partially ordered by set-inclusion. We prove the conjecture of Bhatti and Essam that the Möbius function of this set is given, for acyclicb,b _uv withbb, by(b,b)=(–1) ^c-c , wherec andc are the respective cycle ranks ofb andb. The significance of this result in percolation theory is reviewed together with previous results for other sets of subgraphs.     

Energy and momentum in general relativity

In general relativity, conservation of energy and momentum is expressed by an equation of the form /x= 0, where –gT represents the total energy, momentum, and stress. This equation arises from the divergence formula dV _v = (/x ^v )d ⁴ d. Here we show that this formula fails to account properly for the system of basis vectors e(x). We obtain the (invariant) divergence formula e dV _v = e (/x ^v + )d ⁴ d. Conservation of energy and momentum is therefore expressed by the covariant equation (/x ^v ) + = 0. We go on to calculate the variation of the action under uniform displacements in space-time. This calculation yields the covariant equation of conservation, as well as the fully symmetric energy tensor . Finally, we discuss the transfer of energy and momentum, within the context of Einstein's theory of gravitation.     

Light cone expansion and four point function

The behaviour of products of local fields for lightlike distances is investigated. If a light cone expansion ofA(x)A(y) exists, then already the four point function carries the singularity arising in the expansion for (x–y)²0. For a special class of field theories, discussed by S. Schlieder and E. Seiler, it is shown that the light cone expansion is possible. Notation. the Schwartz space of strongly decreasing testfunctions over ⁿ A=scalar field operator, which fulfils the Wightman axioms [we freely writeA(x),x ⁴ andA(g),g ]. =Hilbert space. =vacuum state. is the linear hull of the vectors (With respect to the definition of operators with complex argument cf.[6]!) By (x ²) (x ²) we denote a sequence of functions which converges to (x ²) as 0.     

Thes-f model with Coulomb repulsion: Thermodynamics of two atomic cluster. Spin 1/2

A cluster of two atoms described by thes-f model with Coulomb repulsion has been considered. The interaction between localized 4f electrons (S=1/2) is taken in the molecular field approximation. The thermodynamic quantities like magnetization, specific heat and correlation functions n , n , S ^z n , S ^z n , S ^z (n –n ), n n and S ⁺ a ⁺ a as functions of temperature are presented for different band fillingN=0, 0.5, 1, 1.5, 2. The dependence of Curie temperature onN is calculated. The phase diagram forN=1 (T=0K) shows the possibility of existence of two phases: paramagnetic and ferromagnetic.The Curie temperature and the specific heat as functions ofN exhibit similar trends as found in experiments on doped magnetic semiconductors.     

Higher spin BRS cohomology of supersymmetric chiral matter inD=4

We examine the BRS cohomology of chiral matter inN=1,D=4 supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators _{(a, b)} are products of the elementary chiral superfieldsS and and the derivative operatorsD , and . Such superfields _{(a, b)} can be chosen to have a symmetrized undotted indices _i and b symmetrized dotted indices . The result derived here is that each composite superfield _(a,b) is subject to potential supersymmetry anomalies ifa–b is an odd number, which means that _(a,b) is a fermionic superfield.     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.     

Kinetics and mechanism of copper fracture due to deformation in surface-active paths

The general kinetic characteristics of copper fracture in the presence of surface-active bismuth-lead baths during creep and elongation under tension are explained. It is shown that the subcritical stage of crack development controls the process, whereupon the effects of stresses , temperature, strain rate , surface energy at the copper-bath interface _SL, and surface energy at the grain boundaries _b on the rate of crack development l/ are analyzed. The basic conclusions are that: a)l/=(–) ( and being constants here); b) the crack development activating energy ) the reduction of energy _b, achieved by intergranular internal adsorption of 0.5% antimony, lowers the value of about 50 times; d) a 30% increase in surface energy _SL reduces the cracking rate 30 times, according to the relation (where A=6 · 10^–15 cm²); and e) .Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 7–15, May, 1976.     

Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, ) be an Abelian topological group, be a nonnegative finitely additive measure defined on L, {_n: n N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a L, {_n(a)}_{n N} is a -convergent sequence, for each nN, when { (a)} convergent to 0, {_n(a)} is -convergent, then when { (a)} convergent to 0, {_n(a)} are -convergent uniformly with respect to nN     

Inequalities for the Schattenp-norm. III

We present some inequalities for the Schattenp-norm of operators on a Hilbert space. It is shown, among other things, that ifA is an operator such that ReAa0, then for any operatorX, AX+XA* _p 2aX _p . Also, for any two operatorsA andB, A–B ₂ ² +A*B* ₂ ² 2A–B ₂ ² .     

The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions

Existence and uniqueness results are established for solutions to the Becker-Döring cluster equations. The density is shown to be a conserved quantity. Under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined. Denoting the set of equilibrium solutions byc ⁽⁾, 0 _s , the principal result is that if the initial density _{0 s} then the solution converges strongly toc ^(o), while if ₀ > _s the solution converges weak* toc ^(s). In the latter case the excess density ₀– _s corresponds to the formation of larger and larger clusters, i.e. condensation. The main tools for studying the asymptotic behaviour are the use of a Lyapunov function with desirable continuity properties, obtained from a known Lyapunov function by the addition of a special multiple of the density, and a maximum principle for solutions.     

Solutions of the braid equation related to a Hopf algebra

Two solutionsT andT of the braid equation acting onA A (whereA is a Hopf algebra) are described. IfA is a cocommutative, thenT=. IfA is commutative, thenT= ( denotes the flip: (a b) =b a for anya,b A).Supported by a grant of the Ministry of Education of Poland.     

Investigation of nematic order by coherent neutron scattering

The mathematical relationship between the orientational order parameters and the coherent neutron scattering cross section for a nematic liquid crystal is given. For deuterated para-azoxyanisole the single-molecule part of the cross section is evaluated within the meanfield approximation and combined with experimental results to give information about molecular orientational order in terms of P ₂, P ₄ and P ₆. Both P ₂ and P ₄ are found necessary for describing the molecular order. Discrepancies between experimental and theoretical results are interpreted as possibly reflecting the inadequacy of the meanfield theory of Maier and Saupe.     

The spectrum of a quasiperiodic Schrödinger operator

The spectrum (H) of the tight binding Fibonacci Hamiltonian (H _mn= _m,n+1+ _m+1,n + _m,n v(n),v(n)= ((n–1)), 1/ is the golden number) is shown to coincide with the dynamical spectrum, the set on which an infinite subsequence of traces of transfer matrices is bounded. The point spectrum is absent for any , and (H) is a Cantor set for 4. Combining this with Casdagli's earlier result, one finds that the spectrum is singular continuous for 16.On leave from the Central Research Institute for Physics, Budapest, Hungary     

Self-consistent calculation of statics and dynamics of the roughening transition

Statics and dynamics of the modified kinetic discrete Gaussian model are treated selfconsistently using a Gaussian probability assumption. A non-trivial roughening temperatureT _R is found in exactly two dimensions only. The free energyF, the correlation length and the interface roughness h ² are found to behave—lnFlnh ²(T _R –T)^–1 for temperaturesT approachingT _R from below. The linear relaxation rate of the order parameter is found to be proportional to ^–2. As a model for crystal growth, the growth rate depends linearly upon the chemical potential difference aboveT _R , shows a metastable regime belowT _R with a spinodal limit of metastability _c , beyond which oscillatory growth starts. The critical behavior of _c is found to be ln _c –(T _R –T)^–1+O(ln (T _R –T)).     

Monte Carlo study of the ising model phase transition in terms of the percolation transition of “physical clusters”

Finite squareL×L Ising lattices with ferromagnetic nearest neighbor interaction are simulated using the Swendsen-Wang cluster algorithm. Both thermal properties (internal energyU, specific heatC, magnetization |M|, susceptibility) and percolation cluster properties relating to the physical clusters, namely the Fortuin-Kasteleyn clusters (percolation probability P , percolation susceptibility _p, cluster size distributionn _l) are evaluated, paying particular attention to finite-size effects. It is shown that thermal properties can be expressed entirely in terms of cluster properties, P being identical to |M| in the thermodynamic limit, while finite-size corrections differ. In contrast, _p differs from even in the thermodynamic limit, since a fluctuation in the size of the percolating net contributes to, but not to _p. NearT _c the cluster size distribution has the scaling properties as hypothesized by earlier phenomenological theories. We also present a generalization of the Swendsen-Wang algorithm allowing one to cross over continuously to the Glauber dynamics.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

On the stability of spin waves at finite temperatures

The temperature dependent exchange stiffnessD, defined for long wavelength spin waves by=Dq ², is calculated for very weak itinerant ferromagnets. It is found that for a general single band of d-electrons,D reduces to a very simple form givingD(T)=D(0)/ ₀ in the limit ₀0. The stability condition for spin waves at finite temperatures is thenD(0) > 0 and/ ₀ > 0, where is the relative occupation ± spin sub-bands and ₀ its value at 0°K.The authors are grateful to E. P. Wohlfarth for helpful discussions.     

Three-Mode Entangled State Representation of Continuum Variables and Optical Four-Wave Mixing

We establish a new three-mode entangled state representation , of continuum variables, which make up a complete set. Using optical four-wave mixing and a beam splitter transform we can prepare , . Based on , a new number-difference--operational-phase uncertainty relation is established and the corresponding squeezing dynamics is discussed.     

Amplitude-distortion factors for the inelastic scattering of an electron at a neon atom in an electric field

Inelastic electron scattering at a neon atom in the presence of an external electric field is considered. The distortion factors of the cross section _M (F) are calculated for a series of 4d and 5d levels. It is shown that the distortion factor in strong fields is a nonlinear and nonmonotonic function of the field. The distortion factors and transition probabilitiesA , are compared. The correlation between the behavior ofA , and _M (F) is explained by the dependence of the mean dipole moment of the Stark state on the field strength.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 97–110, January, 1996.