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.     

Fractal chemical kinetics: Reacting random walkers

Computer simulations on binary reactions of random walkers (A + A A) on fractal spaces bear out a recent conjecture: ( ^–1 – ₀ ^–1) t ^f , where is the instantaneous walker density and ₀ the initial one, andf=d _s /2, whered _s is the spectral dimension. For the Sierpinski gaskets:d=2, 2f=1.38 (d _s =1.365);d=3, 2f=1.56 (d _s =1.547); biased initial random distributions are compared to unbiased ones. For site percolation:d= 2,p=0.60, 2f= 1.35 (d _s =1.35); d=3,p=0.32, 2f=1.37 (d _s =1.4); fractal-to-Euclidean crossovers are also observed. For energetically disordered lattices, the effective 2f (from reacting walkers) andd _s (from single walkers) are in good agreement, in both two and three dimensions.Supported by NSF Grant No. DMR 8303919.     

Direction-direction correlations of oriented polymers

We calculate the direction-direction correlations between the tangent vectors of an oriented self-avoiding walk (SAW). LetJ (x) andJ ^v (0) be components of unit-length tangent vectors of an oriented SAW, at the spatial pointsx and 0, respectively. Then for distances |x| much less than the average distance between the endpoints of the walk, the correlation function ofJ (x) withJ ^v (0) has, ind dimensions, the form . The dimensionless amplitudek(d) is universal, and can be calculated exactly in two dimensions by using Coulomb gas techniques, where it is found to bek(2)=12/25 ². In three dimensions, the -expansion to second order in together with the exact value ofk(2)in two dimensions allows the estimatek(3)=0.0178±0.0005. In dimensionsd4, the universal amplitudek(d) of the direction-direction correlation functions of an oriented SAW is the same as the universal amplitude of the direction-direction correlation functions of an oriented random walk, and is given byk(d)= ²(d/2)/(d–2) ^d .     

On the critical behavior of the magnetization in high-dimensional Ising models

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind4 dimensions the magnetization is continuous atT _c and its critical exponents take the classical values=3 and=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x–y| couplings, for which 2/(–1) replacesd in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation fellowship (M.A.).     

New observation of the , 1Δg, and 2Πg states and molecular constants with all Li2, Li2, and LiLi data

This paper reports our new observation of the , 1³Δ_g (v = 2–4), and 2³Π_g (v = 2–8) states of ⁶Li⁷Li by continuous wave perturbation facilitated optical–optical double resonance spectroscopy. Combining our new experimental term values of ⁶Li⁷Li with the available experimental data of ⁶Li₂ and ⁷Li₂, molecular constants and potential energy curves by Rydberg–Klein–Rees and direct-potential-fit techniques have been determined. Born-Oppenheimer breakdown parameters of the Li₂ 1³Δ_g and 2³Π_g states are calculated.     

Marriage of exact enumeration and 1/d expansion methods: Lattice model of dilute polymers

We consider the properties of a self-avoiding polymer chain with nearestneighbor contact energy on ad-dimensional hypercubic lattice. General theoretical arguments enable us to prescribe the exact analytic form of then-segment chain partition functionC _n ,and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind=2–6. This exact form provides the main ingredient to produce a large-n expansion ind ^–1of the chain free energy through fifth order with the full dependence on the contact energy retained. The -dependent chain connectivity constant and free energy amplitude are evaluated within thed ^–1expansion toO(d ^–5). Our general formulation includes for the first time self-avoiding walks, neighboravoiding walks, theta, and collapsed chains as particular limiting cases.     

Analyticity properties and a convergent expansion for the inverse correlation length of the low-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(z) of the truncated spin-spin correlation function of theZ ^d Ising model with + or — boundary conditions admits the representationm(z) = –(4d–4)ln z(1–_d1) + r(z) for smallz=e ^–, i.e., large inverse temperatures is ad-dependent analytic function atz = 0, already known in closed form ford = 1 and 2; ford = 3 b_n can be computed explicitly from a finite number of the Z^d limits of z = 0 Taylor series coefficients of the finite lattice correlation function at a finite number of points ofZ ^d.     

Absence of continuous symmetry breaking in a one-dimensional n−2 model

For a one-dimensional array ofS ^N–1 spins (N 2) with isotropic pair interactions (and more general systems) with J(j–i) obeying sup_n[n^–1 ₁ ⁿ j ²|J(j)|]<, we prove that every equilibrium state is invariant under the natural action ofSO(N). In particular, there is no long-range order of the conventional type. Included is the caseJ(n)=n ^–2.Research partially supported by U.S.N.S.F. Grant No. MCS-78-01885.S. Fairchild Scholar at Caltech. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, New Jersey 08544.     

Electro-Strong Interaction and High Q 2 Events at HERA

A previously proposed unified field theory of electro-strong interactions requires two scales of length within hadrons, 10^–15 and 10^–18 m respectively, and the onset of new phenomena at the shorter scale. Studies at the HERA electron–proton collider at the shorter scale have revealed a possible excess of high-transverse momentum events, as expected, with Q ² 30,000 (GeV/c²). The collider is currently being upgraded. This will permit a clearer test to be carried out.     

Critical behaviour of directed branched polymers and the dynamics at the Yang-Lee edge singularity

Relying on a field theoretic model due to Day and Lubensky we establish the one-to-one correspondence of the directed branched polymer problem ind dimensions to (relaxational) critical dynamics at the Yang-Lee edge ind–1 spatial dimensions; like their isotropic counterparts the directed polymer exponents andv are uniquely determined by the static Yang-Lee exponent whereasv requires in addition the dynamic Yang-Lee exponentz. JoiningO(²)-expansions about the upper critical dimensiond _c =7 to exact results atd=1 and 2 by Padé-interpolations we obtain good agreement with series expansion data for low dimensions.     

Anomalous fluctuations in random walk dynamics

The anomalous dispersion of noninteracting particles randomly walking in a network is considered. It is shown that the existence of large dangling branches attached to a backbone induces a l/f-like behavior in the current autocorrelation function at low frequencies. The waiting times associated with dangling loops scale liket ^–3/2. The size of the dangling branches provides a lower cutoff to the power law behavior. When the side branches are infinite, self-similar structures, the power law behavior persists up to a zero frequency. The currents we consider are created either by a bias on the random walk or by a current source. We consider both the total current, which is often referred to in the literature, and the current measured at endpoints of a specimen attached to a (model) battery. The differences and similarities between the two corresponding correlations are analyzed. In particular, we find that in the second case l/f noise exists only for large bias. When a statistical distribution of dangling branches is considered, we find that the largest power of frequency in the spectrum is 1.13. Much of our results are true when the dangling branches are replaced by traps having waiting time distributions that equal those of the branches. The waiting time associated with a power law distribution of dangling loops (m ^–x:m is the length of the loop) scales liket ^–1 –(x/2). However, it is shown that geometry alone can be responsible for the appearance of power laws in the spectra. Random geometry can be regarded as a model (or source) of random hopping times.     

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L¹L estimates for the time evolution of Hamiltonians H=–+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is (1+|x|)|V(x)|dx<, whereas in d=3 it is |V(x)|C(1+|x|)^–3–.Supported by the NSF grant DMS-0070538 and a Sloan fellowship.     

Anderson localization in different one-dimensional systems with off-diagonal disorder and spin-dependence

The localization properties of certain spin-dependent, one-dimensional electronic systems with only off-diagonal disorder are studied. In higher dimensions (d=2,3) the models considered would correspond to different universality classes, whereas ford=1 no qualitative difference is found: ForE=0, all eigenstates are exponentially localized, whereas forE0 the localization length diverges logarithmically, such that exactly atE=0 the geometric average of the transmission coefficient would decay with increasing chain lengthL as exp (-const. ·L ^1/2), instead of the usual, exponential decay.ForE=0, in the interior of the band, the localization lengthr ₀ diverges W ₂ ^–2 in the limit of weak disorder (W ₂0), whereas just at the band edge one has roughlyr ₀W ₂ ^–2/3. A universal recursion relation, depending only on the energy and on certain randomly distributed determinants, determines the localization length and the density of states for all systems considered.     

Self-avoiding walk in five or more dimensions I. The critical behaviour

We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford5. We prove that the numberc _n ofn-step self-avoiding walks satisfiesc _n ~A ⁿ , where is the connective constant (i.e. =1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forc _n(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (^––Z)^1/2. The critical two-point function is shown to decay at least as fast as x^–2, and its Fourier transform is shown to be asymptotic to a multiple ofk ^–2 ask0 (i.e. =0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.Supported by the Nishina Memorial Foundation and NSF grant PHY-8896163.Supported by NSERC grant A9351     

O(N) vector model with twisted boundary conditions

Prompted by a recent article of Chakravarty, we reexamine theO(N) vector model with twisted boundary conditions ind dimensions in the various frameworks of the =d–2 expansion, the =4–d expansion, and the large-N expansion. These continuum models describe the physics below the critical temperatureT _c and nearT _c of a latticeO(N) spin model. We determine the effect of the twisting on finite-size scaling functions, for various geometries.On leave from G. Nadjakov Institute of Solid State Physics, 1784 Sofia, Bulgaria.     

One- and two-dimensional C–H/N–H dipolar correlation experiments under fast magic-angle spinning for determining the peptide dihedral angle φ

A recently proposed ¹³C–¹H recoupling sequence operative under fast magic-angle spinning (MAS) [K. Takegoshi, T. Terao, Solid State Nucl. Magn. Reson. 13 (1999) 203–212.] is applied to observe ¹³C–¹H and ¹⁵N–¹H dipolar powder patterns in the ¹H–¹⁵N–¹³C–¹H system of a peptide bond. Both patterns are correlated by ¹⁵N-to-¹³C cross polarization to observe one- or two-dimensional (1D or 2D) correlation spectra, which can be simulated by using a simple analytical expression to determine the H–N–C–H dihedral angle. The 1D and 2D experiments were applied to N-acetyl[1,2-¹³C,¹⁵N] -valine, and the peptide φ angle was determined with high precision by the 2D experiment to be ±155.0°±1.2°. The positive one is in good agreement with the X-ray value of 154°±5°. The 1D experiment provided the value of φ=±156.0°±0.8°.     

Critical behavior of short range Potts glasses

We study by means of Monte Carlo simulations and the numerical transfer matrix technique the critical behavior of the short rangep=3 state Potts glass model in dimensionsd=2,3,4 with both Gaussian and bimodal (±J) nearest neighbor interactions on hypercubic lattices employing finite size scaling ideas. Ind=2 in addition the degeneracy of the glass ground state is computed as a function of the number of Potts states forp=3, 4, 5 and compared to that of the antiferromagnetic ground state. Our data indicate a transition into a glass phase atT=0 ind=2 with an algebraic singularity, aT=0 transition ind=3 with an essential singularity of the form exp(const.T ^–2), and an algebraic singularity atT0.25 ind=4. We conclude that the lower critical dimension of the present model isd _c =3 or very close to it. Some of the critical exponents are estimated and their respective values discussed.     

Annealing behavior ot trap-centers in silicon containing A-swirl defects

The annealing behavior of trap-centers was studied in float-zone silicon wafers containing A-swirl defects. Samples from areas of high and low A-swirl density were annealed in nitrogen ambient between 100° and 900 °C, and analysed using the Deep Level Transient Spectroscopy. The results indicate, that two levels atE _c }-0.07 eV, _n=4.6×10^–16 cm², andE _c–0.49eV, _n=6.6×10^–16cm² are caused by one defect, for which the silicon di-selfinterstitial is a likely interpretation. A level atE _c }-0.11 eV was assigned to interstitial carbon. Both defects annealed out at about 170 °C. After 600 °C annealing an additional level atE _c–0.2 eV was detected, which was attributed to an interstitial silicon carbon complex. Heat treatment at 800 °C generated a new level atE _c–0.49 eV, _n=2.9×10^–16cm² only in the area of high A-swirl defect density. This level was also observed after oxidation and subsequent annealing of silicon.     

A tunable frequency stabilized Ar+ laser

The frequency of a single-mode Ar⁺ laser atv=582 THz (=515 nm) is stabilized by means of an external iodine cell to a Doppler broadened absorption profile. The —unmodulated — stabilized laser frequency is reproducible to ±1 MHz or ±1.5×10^–9 v and can be shifted within a 180 MHz wide range.