Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. III. Triangular-Lattice Chromatic Polynomial

We study the chromatic polynomial P _G (q) for m×n triangular-lattice strips of widths m12_P,9_F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin–Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m,n and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P _G (q) for m×n rectangular subsets of the square lattice, with m8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin–Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B ₂,B ₃,B ₄,B ₅ are limiting points of partition-function zeros as n whenever the strip width m is 7 (periodic transverse b.c.) or 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B ₁₀) cannot be a chromatic root of any graph.     

Structural stability properties of friedman cosmology

A dynamical system with Robertson-Walker symmetries and the equation of the statep=, 01, considered both as a conservative and nonconservative system, is studied with respect to its structural stability properties. Different cases are shown and analyzed on the phase space (x=R ^D , y=x).     

Norm inequalities for fractional powers of positive operators

It is shown that ifA, B andX are operators on a Hilbert space such thatA andB are positive andX belongs to a norm ideal associated with some unitarily invariant norm |·|, then for 0 r 1 we have |A ^r XB ^r | |X|^1-r |AXB| ^r . This is an extension of the classical Heinz-Kato inequality which was originally proved for the usual operator norm. Other related inequalities are also discussed.     

A Monomial Basis for the Virasoro Minimal Series M(p,p′) : The Case 1<p′/p<2

Quadratic relations are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The first case is the (2,1) primary fields for the (p,p)-minimal series M_r,s (1rp–1,1sp–1) for the Virasoro algebra where 1<p/p<2. We restrict ourselves to the case p3, for which the (2,1) primary field exists. The second case is the intertwiners corresponding to the two-dimensional representation for the level k integrable highest weight modules V() (0k) for the affine Lie algebra      

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 .     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Quantitative method for determination of fibre texture in wires I. Principle of method and analytical expression of absorption factor

An X-ray diffraction method is described for the quantitative determination of the fibre texture in cylindrical specimens (wires), which does not require specimen preparation. The integrated intensity of the diffracted radiation from a certain atomic plane is measured in the direction parallel to the plane determined by the axis of the wire and by the direct beam for different orientations of the axis of the wire. These measurements can be carried out in practice using an X-ray counter diffractometer or Weissenberg goniometer. The pole figure is determined from the dependence of the diffracted intensity on the orientation of the wire after correcting the intensities for the absorption of radiation in the specimen. The geometric arrangement enables the absorption factor to be calculated analytically. Its form was verified experimentally by measuring the dependence of the intensity of the 220 reflection on the orientation of the axis of a cylindrical specimen of Al containing randomly oriented crystals.

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( ), . , , , . . , . . 220 .

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The author wishes to thank Z. iký for help in carrying out the measurements and calculations.     

Electron-electron interaction and magnon scattering in the d-electron dominated resistivity of Ni80−xFe x Si8B12 glasses

Our investigation of the electrical resistivity (T) betweenT=0.1 K and 295 K revealed for all Fe-concentrations, 2.4x16, a minimum of betweenT _min=7 K and 15 K. Above the minimum, scattering from structural disorder prevails in the paramagnetic alloy (x=2.4), while in the ferromagnetic glasses (x5.6) coherent magnon scattering appears to dominate. With one exception, (TT _min) increases as down to the lowest temperatures for all investigated Fe-contents (x10.4) in zero magnetic field as well as in applied 6 T. The value of the slopes, 6.3(6) ( cm)^–1 K^–1/2, agrees with the universal scaling behaviour reported recently by Cochrane and Strom-Olsen for systems with strong electron-electron interaction. A large, linear rise of (T<T _min,B=0) observed in the most diluted sample (x=2.4) is tentatively attributed to quantum localization in the non-reentrant spin glass phase. The magnitude of the temperature effects on (T) above as well as belowT _min indicates significantd-band contributions to the conductivity.     

Zeros of the finite-size scaling region partition function for a model with a wetting transition

We derive a finite-size scaling representation for the partition function for an Onsager-Temperley string model with a wetting transition, and analyze the zeros of this partition function in the complex scaled coupling parameter of relevance. The system models the one-dimensional interface between two phases in a rectangular two-dimensional region (x, y) ²,–L yL,oxN. The two phases are at coexistence. The string or interface has a surface tension 2KkT per unit length and an extra Boltzmann weighta per unit length if it touches the surfaces aty=±L. There is a critical valuea _c=1/2K and fora>a _c the string is confined to one of the surfaces, while fora a _c the string moves roughly in the rectangular region. The finite-size scaling parameters are =a _c ² N/L ² and =L(a–a _c)/a _c ² . We find that for || large, the zeros of the scaled partition function lie close to the lines arg()=±/4 with re()>0. We discuss the motion of all the zeros as changes by both analytic and numerical arguments.     

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.     

Effect of hydrogen doping on the magnetic properties of Gd2CuO4

We report the results of ac-susceptibility and dc-magnetization measurements for H_yGd₂CuO₄ (0y0.54). It is shown thatH doping lowers the weak ferromagnetic component in the material. The distinct hysteresis loops observed atT=77 K for both non- and hydrogenated samples change its shape withy. The magnetic ordering temperatures T _N ^Cu and T _N ^Gd , as determined from the temperature dependencies of ac-susceptibility, remain unchanged with sample's hydrogenation. This result seems to indicate that extra electrons are not doped onto the Cu-O planes of Gd₂CuO₄. The frequency dependencies ofx(, T) andx(, T) for bothy=0 andy=0.15 samples are analysed., The maximums ofx andx found at about 200K are considered in terms of susceptibility dependence on the spin-lattice relaxation time (). The anomalies in ac-susceptibility found recently in Gd₂CuO₄ atT _a=8 K andT _b=9.5 K decrease significantly withy. Results are discussed in the context of available data on 214T-type compounds.     

A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions

We rigorously derive a linear kinetic equation of Fokker–Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obstacle of the Lorentz gas generates a potential V( ), where V is a smooth radially symmetric function with compact support, and >0. The density of obstacles diverges as ^– , where >0. We prove that when 0< <1/8 and =2+1, the probability density of a test particle converges as 0 to a solution of our kinetic equation.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Microscopic selection principle for a diffusion-reaction equation

We consider a model of stochastically interacting particles on , where each site is assumed to be empty or occupied by at most one particle. Particles jump to each empty neighboring site with rate/2 and also create new particles with rate 1/2 at these sites. We show that as seen from the rightmost particle, this process has precisely one invariant distribution. The average velocity of this particle V() then satisfies ^–1/2V() as. This limit corresponds to that of the macroscopic density obtained by rescaling lengths by a factor ^1/2 and letting. This density solves the reaction-diffusion equation , and under Heaviside initial data converges to a traveling wave moving at the same rate .     

Ultrasound in spin glasses

The change of the sound velocity v(,T) and the damping of sound waves (,T) in spin glasses are calculated in the frame-work of an Ising model with a random distribution of exchange interactions. The calculation is based on linearized equations of motion for the spins and on an improved mean field approximation which includes the Onsager reaction field. Near to the freezing temperatureT _f and at high temperatures v(,T) and (,T) turn out to be approximately proportional to the real and the imaginary parts of the dynamical susceptibility. For the special case of infinite range interactions atT=T _f one has v(, T_f) ( )^1/2 and (, T_f) (/)^1/2 where is the relaxation time of independent spins. However, already slightly aboveT _f the frequency dependence of both quantities becomes rather small for RKKY spin glasses. At high temperatures both, v(,T) and (,T) vary asT ^–1.SFB 125 Aachen-Jülich-Köln     

Properties of theL p -solutions of linear impulsive equations in a banach space

Conditions for the existence ofL _p-solutions (1p) of linear impulsive equations in a Banach space are found.     

An Equation for the Dissipation Rate Correlation and Its Implications for the Intermittency Exponent μ in Turbulence

We derive an equation satisfied by the dissipation rate correlation function, for the homogeneous, isotropic state of fully-developed turbulence from the the Navier–Stokes equation. In the equal time limit we show that the equation leads directly to two intermittency exponents ₁=2– ₆ and ₂=z₄– ₄, where the 's are exponents of velocity structure functions and z₄ is a dynamical exponent characterizing the fourth order structure function. We discuss the contributions of the pressure terms to the equation and the consequences of hyperscaling.     

μSR studies in the heavy fermion systems UNi2Al3, UPd2Al3 and U2PtC2

We have performed Muon Spin Relaxation (SR) measurements of several heavy fermion systems: UNi₂Al₃ (single crystal), UPd₂Al₃ (single crystal) and U₂PtC₂ (polycrystal). ZF -SR measurements of UNi₂Al₃ show magnetic order below the Néel temperatureT _N=4.6K, with the muon precession frequency(T 0)=2.5MHz at low temperatures. The local field distribution obtained at low temperatures in UNi₂Al₃ resembles that of the incommensurate spin density wave (SDW) system (TMTSF)₂PF₆, suggesting an incommensurate spin arrangement in UNi₂Al₃. TF -SR measurements give the penetration depth(T 0)=6100Å for UPd₂Al₃ and(T 0)=7000Å for U₂PtC₂ respectively. Since UPd₂Al₃ and U₂PtC₂ are clean superconductors (/l-0), we can combine n _s/m^* with the Sommerfeld constant n _s ^1/3 m^* to calculate the Fermi temperatureT _F ^3/4–1/4, which represents the characteristic energy scale of the superconducting carriers. We obtainT _C/T_F 0.01 for both UPd₂Al₃ and U₂PtC₂, comparable to other exotic superconductors, but quite different from simple metal BCS superconductors, whereT _C/T_F 0.01.