Solution spectrum of nonlinear diffusion equations

The stationary version of the nonlinear diffusion equation–c/t +D c = ₁c – ₂c² can be solved with the ansatz c = _p ^{= 1/} A_p(coshkx) ^–p , inducing a band structure with regard to the ratio ₁/₂. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c = skp = 1/ A _pq (coshkx)^–p [(cosht) ^–q–1 sinht + b(cosht)^–q] represents a solution spectrum of the time-dependent nonlinear equation, and the stationary version can be found from the asymptotic behavior of the expansion. The solutions can be associated with reactive processes propagating along molecular chains, and their applicability to biophysical processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheroids.     

Quasi-free “second quantization”

Araki and Wyss considered in 1964 a mapAQ(A) of one-particle trace-class observables on a complex Hilbert-space into the fermionC*-algebraU() over . In particular they considered this mapping in a quasi-free representation.We extend the mapAQ(A) in a quasi-free representation labelled byT, 0TI, to allAB()_sa such that tr(T A(1–T)A)< withQ(A) now affiliated with the algebra. This generalizes some well-known results of Cook on the Fock-representationT=0.     

A generalization of Dirac nonlinear electrodynamics,and spinning charged particles

In this paper—dedicated to Prof. Asim O. Barut—we generalize the Diracnon-linear electrodynamics by introducing two potentials(namely, the vector potential A and the pseudo-vector potential ⁵B of the electromagnetic theorywith charges and magnetic monopoles) and by imposing the pseudoscalar part of the product * to be zero, with A+⁵B. We show that the field equations of such a theory possess a soliton-like solution which can representa priori a charged particle, since it is endowed with a Coulomb field plus the field of a magneticdipole. The rest energy of the soliton is finite, and the angular momentum stored in its electromagnetic field can be identified—for suitable choices of the parameters—with the spin of the charged particle. Thus, this approach seems to yield a classical model for the charged (spinning) particle which does not encounter the problems met by earlier attempts in the same direction.Work partially supported by INFN, CNR, MURST and by FUNREI and CNPq.     

Asymptotic Distribution of Eigenvalues for a Class of Second-Order Elliptic Operators with Irregular Coefficients in R d

Let A=A ₀+v(x) where A ₀ is a second-order uniformly elliptic self-adjoint operator in R ^d and v is a real valued polynomially growing potential. Assuming that v and the coefficients of A ₀ are Hölder continuous, we study the asymptotic behaviour of the counting function N(A,) () with the remainder estimates depending on the regularity hypotheses. Our strongest regularity hypotheses involve Lipschitz continuity and give the remainder estimate N(A,)O({}^–), where may take an arbitrary value strictly smaller than the best possible value known in the smooth case. In particular, our results are obtained without any hypothesis on critical points of the potential.     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.     

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     

Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

Spectral properties of certain composition operators arising in statistical mechanics

By applying the theory of linear positive operators in a Banach space we derive spectral properties of certain composition operators in the Banach spaceA () of holomorphic functions over some domain . Examples of such operators are provided by the so called generalized transfer matrices of classical one-dimensional lattice systems.     

Dynamics of vector spin glasses: The Gabay-Toulouse line

The dynamics of ann-component vector spin glass with infinite range interactions are investigated near and above the Gabay-Toulouse (GT) line. The local transverse susceptibility _T for 0 varies along the whole GT lineT _c1 (H) as ^v , with a field and temperature independent critical exponentv=1/2. The longitudinal susceptibility _L () remains analytic for all (T, H)T _c1 (H), except for a cross-over fromv=1 tov=1/2 forH0 at the freezing temperatureT=T _f . The dynamic susceptibilities _T () and _L () are already coupled above the GT line via self-energy terms. BelowT _c1 (H), this coupling is strongly enhanced by other mechanisms.     

Scaling function for energy-density correlations: Dispersion theory andε-expansion forT>T c

A dispersion representation for the static energy-density correlation function ² (q) ²(–q) _c =C(q,T)=A+Bt ^–h(z ²), wherez=q , t=(T—T)_c/T _c and is the correlation length, is discussed.h(z ²) is calculated to order ² in the zero-field critical region (T>T _c) for the standard isotropicn-component ⁴Ginzburg-Landau-Wilson model. Utilizing a procedure similar to that introduced by Bray for the two-point correlation function, the-expansion results are used in conjunction with an approximant for the spectral functionF(z/2) Imh(—z ²) based on the asymptotically exact short-distance expansion resulth ^–1(z ²)z ^/v[D ₀+D ₁ z ^{–(1 —)/v} +D ₂ z ^–1/v ] to predict quantitatively the full momentum dependence ofC(q,T) forT>T _c. In contrast to the two-point correlation function,C(q,T) is found to be a monotonic function as the critical temperature is approached at fixedq (forT>T _c).     

Pattern formation in systems with nonlocal interactions

One-dimensional lattices with harmonic coupling between neighboring lattice sites and an on-site anharmonic potential V()=A²ⁿ⁺² + ⁿ⁺² + C² + D are examined in the displacive limit. Kink solutions, interpolating between the coexistent phases =0 and =±(C/A)^1/2n at theT=0 first-order phase transition pointB ²=4AC,A, C>0,B<0,D=0 are found in simple analytic form and their dependence on the degree of anharmonicity (n=2, 4, 6, ...) is discussed. It is shown that, at the phase transition point, the kinks are accompanied by a continuous spectrum of periodic nonlinear excitations (periodons) having finite energy density.Work supported by the Swiss National Science Foundation     

ОПРЕДЕЛЕНИЕ ВРЕМЕНИ ЖИЗНИ μ-МЕЗОНОВ ПО АНОМАЛЬНОМУ ПОГЛ0Щ ЕНИЮ МЕТОДОМ ЯДЕРНЫХ ЭМУЛЬСИЙ

- ;- . ( ) . , - .     

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.     

Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L ^m/2( ^m ) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|^– , (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.     

BRST quantization of a scalar particle in a curved background

The BRST formalism is employed to quantize a scalar particle and interactions with an external scalar field (x ) and vector gauge fieldA (x ) in the background of an arbitrary gravitational field. The second-quantized actions are obtained.     

On Shirokov's “One New Effect of the Einsteinian theory of Gravitation”

In this note we study M.F. Shirokov's One New Effect of Einstein's theory of Gravitation in the Kerr field. We confine ourselves to a circular trajectory (r=const) in the plane=/2. In the limit that terms of the second order and higher ina/r (a is the angular momentum per unit mass) are negligible we find an additional shift betweenT andT _r (orT ) of amount –(8a/r)(m/r) ^3/2. Thus vibrations lie further behindr(or) vibrations in this case than in the Schwarzschild field.     

A reply to “a comment on generalized electromagnetism and Dirac algebra”

Issues raised by W. A. Rodrigues, Jr. are discussed.1. This is not a new result; see,e.g., Rohrlich.⁽³⁾ 2. A typographical error in Eq. (77) is corrected here: The productj A in the right-hand parentheses was erroneously transcribed in Ref. (2) as A.3. I define electromagnetic fieldF = A to be that generated by electric charges and the magnetoelectric fieldG = M to be that generated by magnetic monopoles:F F +₅ G. 4. Rodrigues, on the other hand, takes the position that the importance of the Lagrangian formulation should be downgraded if not discarded altogether: ... it is redundant to look for Lagrangians.⁽¹⁾ 5. In fact, he reformulates it using the language of differential forms.6. It is interesting to observe that this bilinear form has the additional virtue of being appropriate for dealing with the monopolecharge parity question, which was pointed out long ago.⁽¹⁴⁾ 7. In fact, even mathematics looks to Nature for its authority.⁽¹⁶⁾ There is evidence that Rodrigues does not understand this concept.⁽¹⁷⁾     

First experiment to test whether space-time is flat. II. Effects of orbit

The gyroscope in an orbiting satellite will be acted on by additional gravitational fields due to the rotation of the earth and due to the orbital velocity of the satellite. According to special relativistic gravitational theory, we deduce _L ^(S) —the gyroscope's precession rate due to the orbital velocity—and _S ^(S) —the gyroscope's precession rate due to the earth's rotation in the polar orbit case. The results are _L ^(S) = (2/3) _L ^(G) , _S ^(S) = (3/2) cos (1 - sin² cos²)^1/2 _S ^(G) , where and are the gyroscope's polar angles, and _L ^(G) and _S ^(G) are the geodetic and frame-dragging precession rates predicted by general relativity, respectively.     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR     

Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study

We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang–Swendsen–Kotecký (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length 5000; the data are consistent with ()=Ae ^{2 p} (1+a ₁ e ^– + ...) as , with p1. The staggered susceptibility behaves as _stagg ^5/3. For q=4 the model is disordered (2) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.