Bifurcations of circle maps: Arnol'd tongues,bistability and rotation intervals

We study the bifurcations of two parameter families of circle maps that are similar tof _b,w (x)=x+w+(b/2) sin (2x) (mod1). The bifurcation diagram is constructed in terms of setsT _r , whereT _r is the set of parameter values (b, w) for whichf _{b, w} has an orbit with rotation numberr. We show that the known structure whenb<1 (forr rational,T _r is an Arnol'd tongue and forr irrational, it is an arc) extends nicely into the regionb>1, wheref _{b, w} is no longer injective and can have an interval of rotation numbers. Specifically, the tongues overlap in a uniform, monotonic manner and forr irrational,T _r opens into a tongue. Our other theorems give information about the dynamics off _{b, w} (e.g., bistability or aperiodicity) for (b, w) in various subsets of parameter space.     

On trace representation of linear functionals on unbounded operator algebras

In this paper we consider the following problem: Given a *-algebraA of unbounded operators, under what conditions is every strongly positive linear functionalf onA a trace functional, i.e. of the formf(a)=Trtta,aA, wheret is an appropriate positive nuclear operator. Further, the linear functionalsf onA which can be represented asf(a)=Trta (f andt not necessarily positive) are characterized by their continuity in a certain topology. Some applications (canonical commutation relations on the Schwartz space, integrable representations of enveloping algebras) are discussed.     

Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

General two-mode squeezed states

The states |A ₁ A ₂ are considered, where the operators are associated with a unitary representation of the groupSp(4,), and the two-mode Glauber coherent states |A ₁ A ₂> are joint eigenstates of the destruction operatorsa ₁ anda ₂ for the two independent oscillator modes. We show that they are ordinary coherent states with respect to new operatorsb ₁ andb ₂, which are themselves general linear (Bogolibov) transformations of the original operatorsa ₁,a ₂ and their hermitian conjugatesa ¹ ,a ² . We further show how they may be regarded as the most general two-mode squeezed states. Most previous work on two-mode squeezed states appears to be based on more restrictive definitions than our own, and thereby reduces to special cases which are unified within our treatment.     

On the statistics of quantised dissipative systems

Magnetic circular dichroism (MCD) of theA- andB-absorption band region has been obtained at 4.2 K and 50 kG in KI:Ga⁺, KI:In⁺, and KI:Sn²⁺. The MCD spectra indicate the complex nature of these bands more clearly than the absorption spectra themselves do. TheA-band MCD consists in all cases of a positive and a negative part reflecting the structure of the absorption band. TheB-band MCD shows three peaks, two positive peaks at 4.34 and 4.41₅ eV (4.09 and 4.17₅ eV) and a negative peak at 4.38 eV (4.12₅ eV) in KI:Ga⁺ (KI:In⁺). TheB-band in KI:Sn²⁺ consists of a shoulder (b ₀) at 3.76 eV and a main band which has at least 5 sub-peaks (b ₁~b ₅) at 3.82₁, 3.84₁, 3.86₁, 3.88₀, and 3.89₅ eV; each of the subpeaks (b ₁~b ₅) gives a derivative-like MCD.The MCD shape functionf() for the transitiona _1g ² a _1g t _1u has been obtained for one set of parameter values by using the classical Franck-Condon approximation and the Monte Carlo integration method. The result can explain the observed salient features of theB- as well asA-band MCD's, indicating the validity of the Franck-Condon approximation and the interaction mode coordinates.     

Magnetic circular dichroism of theA andB bands in KI:Ga+, KI:In+, and KI:Sn2+

Magnetic circular dichroism (MCD) of theA- andB-absorption band region has been obtained at 4.2 K and 50 kG in KI:Ga⁺, KI:In⁺, and KI:Sn²⁺. The MCD spectra indicate the complex nature of these bands more clearly than the absorption spectra themselves do. TheA-band MCD consists in all cases of a positive and a negative part reflecting the structure of the absorption band. TheB-band MCD shows three peaks, two positive peaks at 4.34 and 4.41₅ eV (4.09 and 4.17₅ eV) and a negative peak at 4.38 eV (4.12₅ eV) in KI:Ga⁺ (KI:In⁺). TheB-band in KI:Sn²⁺ consists of a shoulder (b ₀) at 3.76 eV and a main band which has at least 5 sub-peaks (b ₁~b ₅) at 3.82₁, 3.84₁, 3.86₁, 3.88₀, and 3.89₅ eV; each of the subpeaks (b ₁~b ₅) gives a derivative-like MCD.The MCD shape functionf() for the transitiona _1g ² a _1g t _1u has been obtained for one set of parameter values by using the classical Franck-Condon approximation and the Monte Carlo integration method. The result can explain the observed salient features of theB- as well asA-band MCD's, indicating the validity of the Franck-Condon approximation and the interaction mode coordinates.     

Phase transitions on fractal lattices with long-range interactions

A fractal latticeF is defined here to comprise all points of the forma + ma+ m² a+ ... +m^qa^(q), whereq is a nonnegative integer anda, a,..., a^(q)A, whereA is a finite set of points in some Euclidean space. Providedm is not too small (in particular,m must be at least 2), the dimension ofF is shown to beD = log n/logm, wheren is the number of points inA. It is shown further that an Ising model onF, with a ferromagnetic pair interaction r^– between spins separated by a distancer, has a phase transition ifD < < 2D. On the other hand, for > 2D, provided a certain condition which rules out periodic lattices is satisfied, there can be no finite-temperature transition leading to spontaneous magnetization.     

On rational functions of first-class complexity

It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.     

Instability of layered quantum liquids: 4. Intraplane and interplane correlations

We calculate the local-field correctionsG(q, q _z) f electron superlattices with interlayer distanced in order to describe many-body effects. A generalized Hubbard expression for the local-field correction is derived. The sum-rule version of the self-consistent approach for the local-field correction is developed and used to discuss the effects of exchange and correlation. The RPA-parameterr _s and the ratiod/a ^* determine the many-body effects.a ^* is the effective Bohr radius. The stability region for the Fermi liquid behavior of the layered electron gas is discussed: (i) forda ^* the critical RPA-parameterr _sc1 is determined by exchange andintraplane correlation effects, (ii) forda ^* withr _sc1 correlation effects are small and the instability point is mainly determined by exchange effects, (iii) forda ^* withr _sc<1 exchange andinterplane correlation effects are both important for the instability point. Our results are important for high-T _c superconductors, organic superconductors and semiconductor superlattices.     

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.     

Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Ifq is ap ^th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU _q (sl ₂), whose coproduct yields the truncated tensor product of physical representations ofU _q (sl ₂), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (quasi-quantum group algebras). We describe a space of functions on the quasi quantum plane, i.e. of polynomials in noncommuting complex coordinate functionsz _a , on which multiplication operatorsZ _a and the elements of can act, so thatz _a will transform according to some representation ^f of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU _q (sl ₂) algebra we show that the subspaces of monomials inz _a ofn ^th degree vanish fornp–1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p–2). Assuming that the representation ^f of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.     

Sigma models withA k singularities in Euclidean spacetime of dimension 0 ≤D < 4 and in the limitN → ∞

For the case of the single-O(N)-vector linear sigma models the critical behaviour following from anyA _k singularity in the action is worked out in the double scaling limitN→∞,f _r →f _r ^c , 2≤r≤k. After an exact elimination of Gaussian degrees of freedom, the critical objects such as coupling constants, indices and susceptibility matrix are derived for allA _k and spacetime dimensions 0≤D<4. There appear exceptional spacetime dimensions where the degreek of the singularityA _k is more strongly constrained than by the renormalizability requirement.     

The hard-sphere fluid: New exact results with applications

A theorem for convolution integrals is proved and then applied to extend the second zero-separation theorem to the bridge functionb(r) and direct-correlation tail functionsd(r). This theorem allows us to exactly relateb(r)/r andd(r)/ratr=0 for the hard-sphere fluid to the contact value of the radial distribution functiong(r) atr= ⁺. From this we obtain immediately the exact values of b(r)/r and d(r)/r atr=0 through second order in number density . Using our results to compare the exact and Percus-Yevick (PY) bridge function, we find that they differ significantly. After obtaining the bridge function and tail function and their derivatives atr=0 andr= through, we suggest new approximations forb(0) andd(0) as well as an analytical integral-equation theory to improve the PY approximation in the pure hard-sphere fluid. The major deficiency of that approximation has been its poor assessment of the cavity function inside the hard-core region. Our theory remedies this defect in a way that yields ay(r) that is self-consistent with respct to the virial and compressibility relations and also the two zero-separation relations involvingy(r) and its spatial derivative atr=0.     

Moments of the Percus-Yevick hard-sphere correlation function

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ . We compare exact moments with shell approximationsM _n [h ^s ] corresponding to integration fromr=0 tos+1 fors=3–8, and with hybrid approximationsM _n [h ^s +h ^a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to . For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a ] is even better thanM _n [h ⁸]     

The orthogonality postulate in axiomatic quantum mechanics

Letp(A,,E) be the probability that a measurement of an observableA for the system in a state will lead to a value in a Borel setE. An experimental function is a function f from the set of all statesI into [0,1] for which there are an observableA and a Borel setE such thatf()=p(A, , E) for all I. A sequencef ₁,f ₂,... of experimental functions is said to be orthogonal if there is an experimental functiong such thatg+f ₁+f ₂+...=1, and it is said to be pairwise orthogonal iff _i+f _j 1 forij. It is shown that if we assume both notions to be equivalent then the setL of all experimental functions is an orthocomplemented partially ordered set with respect to the natural order of real functions with the complementationf=1–f, each observableA can be identified with anL-valued measure _A, each state can be identified with a probability measurem onL and we havep(A,,E)=m oA(E). Thus we obtain the abstract setting of axiomatic quantum mechanics as a consequence of a single postulate.     

A particle model for spinodal decomposition

We study a one-dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a small drift toward regions of higher concentration. We prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift. There is a critical valuea _c ofa. Fora<a _c, the density values are unrestricted, while foraa _c, they should all be to the right or to the left of a given interval (a). The diffusion coefficient of the limiting equation can be continued analytically to (a), and, in the interior of (a), it has negative values which should correspond to particle aggregation phenomena. We also show that the dynamics can be obtained as a limit of a Kawasaki evolution associated to a Kac potential. The coefficienta plays the role of the inverse temperature. The critical value ofa coincides with the critical inverse temperature in the van der Waals limit and (a) with the spinodal region. It is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic, the system evolves according to an integrodifferential equation. The instanton solutions of this equation, as studied by Dal Passo and De Mottoni, are then related to the phase transition region in the thermodynamic phase diagram; analogies with the Cahn-Hilliard equations are also discussed.This paper is dedicated to Jerry Percus with great affection on the occasion of his 65th birthday.     

Self-gravitating radiation in anti-de sitter space

Equilibrium configurations of self-gravitating massless thermal radiation inside spherical boxes of radiusR in asymptotically anti-de Sitter space (A = -3/b ₂) are constructed numerically for a range of central densities. For each box radius considered (R/b = 0, 1/2, 1, 2, 4, ), there is a unique configuration with maximal total mass and entropy, and another (at a lower central density) with maximum asymptotic red-shifted temperature. With the box removed toR=, the maximum total mass and entropy of self-gravitating thermal radiation areM _max 0.4598b0.7964(–A)^–1/2 andS _max1.3560a ^1/4 b ^3/2 3.0910a ^1/4(–A)^–3/4, and the maximum red-shifted temperature is     

Generalized q-Modified Laguerre Functions

In this paper we define a new q-special function A _n (x, b, c; q). The new function is a generalization of the q-Laguerre function and the Stieltjes–Wigert function. We deduced all the properties of the function A _n (x, b, c; q). Finally, lim _q1 A _n ((1 – q)x, –, 1;q) gives L _n ^(,)(x,q), which is a -modification of the ordinary Laguerre function.     

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.