The infinite number of generalized dimensions of fractals and strange attractors

We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions D_q, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities D_q. We prove that lim _q→0D_q = fractal dimension (D), lim_q→1D_q = information dimension (σ) and D_q=2 = correlation exponent (v). D_q with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > D_q for any q′ > q. For homogeneous fractals D_q = D_q. A particularly interesting dimension is D_q=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D_∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D_∞.     

Critical Slowing Down in One-Dimensional Maps and Beyond

This is a brief review on critical slowing down near the Feigenbaum period-doubling bifurcation points and its consequences. The slowing down of numerical convergence leads to an “operational” fractal dimension D=2/3 at a finite order bifurcation point. There is a cross-over to D ₀=0.538... when the order goes to infinity, i.e., to the Feigenbaum accumulation point. The problem of whether there exists a “super-scaling” for the dimension spectrum D _q ^W that does not depend on the primitive word W underlying the period-n-tupling sequence seems to remain open     

Diffusion on two-dimensional percolation clusters with multifractal jump probabilities

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension D_F and a multifractal distribution of probabilities of dimensionsD _q (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD' _q =D _q +D _F -d, for a finite interval ofq.     

Early stages of generation of two-dimensional structures by the Hastings-Levitov method of conformal mapping dynamics

Two-dimensional structures obtained by the Hastings-Levitov conformal mapping were studied for a relatively small number of mappings n. The fractal dimension D of these structures is computed by the recent Davidovitch-Procaccia technique [6] as a function of n. For small n < n₀ (where n₀ is the number of particles at the first layer), D exponentially decreases, which should have supported the conclusion made in [6] about the possibility of determining the fractal dimension with an arbitrary accuracy using a relatively small number of mappings n≈ n₀. On the other hand, it turned out that D irregularly deviates from a certain quantity D₀ depending on the initial size of the bump \(\sqrt {\lambda _0 } \), which contradicts the main assertion of     

Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.     

Generalized inverse participation numbers in metallic-mean quasiperiodic systems

From the quantum mechanical point of view, the electronic characteristics of quasicrystals are determined by the nature of their eigenstates. A practicable way to obtain information about the properties of these wave functions is studying the scaling behavior of the generalized inverse participation numbers Z_q ~ N - ^{D_q (q - 1)}Z_q \sim N - ^{D_q (q - 1)} with the system size N. In particular, we investigate d-dimensional quasiperiodic models based on different metallic-mean quasiperiodic sequences. We obtain the eigenstates of the one-dimensional metallic-mean chains by numerical calculations for a tight-binding model. Higher dimensional solutions of the associated generalized labyrinth tiling are then constructed by a product approach from the one-dimensional solutions. Numerical results suggest that the relation D _q ^dd = dD _q ^1d holds for these models. Using the product structure of the labyrinth tiling we prove that this relation is always satisfied for the silver-mean model and that the scaling exponents approach this relation for large system sizes also for the other metallic-mean systems.     

Pore size distribution in porous glass: fractal dimension obtained by calorimetry

By differential Scanning Calorimetry (DSC), at low heating rate and using a technique of fractionation, we have measured the equilibrium DSC signal (heat flow) J _q ⁰ of two families of porous glass saturated with water. The shape of the DSC peak obtained by these techniques is dependent on the sizes distribution of the pores. For porous glass with large pore size distribution, obtained by sol-gel technology, we show that in the domain of ice melting, the heat flow J_q is related to the melting temperature depression of the solvent, ΔT _m , by the scaling law: J _q ⁰∼ΔT _m ^{- (1 + D)}. We suggest that the exponent D is of the order of the fractal dimension of the backbone of the pore network and we discuss the influence of the variation of the melting enthalpy with the temperature on the value of this exponent. Similar D values were obtained from small angle neutron scattering and electronic energy transfer measurements on similar porous glass. The proposed scaling law is explained if one assumes that the pore size distribution is self similar. In porous glass obtained from mesomorphic copolymers, the pore size distribution is very sharp and therefore this law is not observed. One concludes that DSC, at low heating rate ( q? 2^°C/min) is the most rapid and less expensive method for determining the pore distribution and the fractal exponent of a porous material. Received 23 July 1999 and Received in final form 16 February 2001     

Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter

Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξ〉_q ≡ D(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e.,

$ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $

and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξ〉_q is stronger for odd q than for even q.

     

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.     

Correlation between the Soret coefficient and the static structure factor in a polymer blend

Mutual mass diffusion and thermal diffusion has been investigated in poly(dimethylsiloxane)/ poly(ethylmethylsiloxane) (PDMS/PEMS) polymer blends of equal weight fractions. Molar masses ranged from below 1 to over 20 kg/mol. Both the mutual mass (D) and the thermal diffusion (D_T) coefficient contain a thermally activated factor with an activation temperature of 1415 K. The molar mass dependence of D_T is due to an end-group effect of the local friction coefficient. The thermal diffusion coefficient in the limit of long chains and infinite temperature is D_T^0, = - 1.69×10^-7cm²(sK)^-1. The Soret coefficient S_T of blends far enough away from a critical point is proportional to the static structure factor S(q = 0).     

Roughness and fractality of fracture surfaces as indicators of mechanical quantities of porous solids

The 3D profile surface parameter H _q and fractal dimension D were tested as indicators of mechanical properties inferred from fracture surfaces of porous solids. High porous hydrated cement pastes were used as prototypes of porous materials. Both the profile parameter H _q and the fractal dimension D showed capability to assess compressive strength from the fracture surfaces of hydrated pastes. From a practical point of view the 3D profile parameter H _q seems to be more convenient as an indicator of mechanical properties, as its values suffer much less from statistical scatter than those of fractal dimensions.     

Representations of the coordinate ring of GL q (3)

It is shown that finite-dimensional irreducible representations of the quantum matrix algebraM _q (3) (the coordinate ring of GL _q (3)) exist only whenq is a root of unity (q ^p = 1). The dimensions of these representations can only be one of the following values:p ³,p ³/2,p ³/4, orp ³/8. The topology of the space of states ranges between two extremes, from a three-dimensional torusS ¹ ×S ¹ ×S ¹ (which may be thought of as a generalization of the cyclic representation) to a three-dimensional cube [0, 1] × [0, 1] × [0, 1].     

On the Buchdahl Inequality for Spherically Symmetric Static Shells

A classical result by Buchdahl [6] shows that for static solutions of the spherically symmetric Einstein equations, the ADM mass M and the area radius R of the boundary of the body, obey the inequality 2M/R ≤ 8/9. The proof of this inequality rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl’s hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in [R ₀, R ₁], R ₀ > 0, of matter models for which the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1. We show a Buchdahl type inequality for shells which are thin; given an there is a κ > 0 such that 2M/R ₁ ≤ 1 − κ when . It is also shown that for a sequence of solutions such that R ₁/R ₀ → 1, the limit supremum of 2M/R ₁ of the sequence is bounded by ((2Ω + 1)² − 1)/(2Ω + 1)². In particular if Ω = 1, which is the case for Vlasov matter, the bound is 8/9. The latter result is motivated by numerical simulations [3] which indicate that for non-isotropic shells of Vlasov matter 2M/R ₁ ≤ 8/9, and moreover, that the value 8/9 is approached for shells with R ₁/R ₀ → 1. In [1] a sequence of shells of Vlasov matter is constructed with the properties that R ₁/R ₀ → 1, and that 2M/R ₁ equals 8/9 in the limit. We emphasize that in the present paper no field equations for the matter are used, whereas in [1] the Vlasov equation is important.     

[2.2]Paracyclophane: theoretical study of its lower excited states and of the zero-field splitting parameters D

The lower excited singlet and triplet states and the zero-field splitting parameters D of [2.2]paracyclophane are studied within a semiempirical π theory which takes into account overlap effects between the two benzene rings, transanular and through-bond interaction via the methylene bridges. Whereas the singlet energies depend strongly on the through-bond interaction and the mutual polarization of σ core and π system this is not the case for the energies and zero-field splitting parameters D of the two lowest triplet states. The deformations of the benzene rings in [2.2]paracyclophane lead only to a small decrease of the excitation energies of about 0·2 eV. The D parameter can be written as a sum D = D_A + D_B + D_AB with the intrasubunit contributions D_A and D_B of the conjugated subunits A and B of the phane and an intersubunit term D_AB . We demonstrate that the deformations reduce the intrasubunit terms D_A and D_B and that they are crucial for the decrease of the D values of [2.2]paracyclophane with respect to p-xylene. The difference between the D values of the first and second triplet states is governed by the intersubunit term D_AB which has a different sign in the two states. However, this difference does not depend markedly on the transanular interaction. A further reduction of D_A and D_B in the first triplet state only is caused by transanular interaction by means of symmetrical charge-transfer terms in the wavefunction.     

Analysis of R. P. Smith's and D. C. Parris-R. B. Mclellan's carbon diffusivity data

A phenomenological model is proposed to account for the variation of carbon diffusivityD(N₁) with composition in austenite. This model is based on Parris-McLellan's statement that the increase ofD(N₁) with carbon concentrationN ₁ is due to the increase in chemical driving force. The gradient of the carbon activity in Wagner's and in Zupp-Stevenson's interpretation is presumed to be the intrinsic driving force. The predictions of the model are compared with carbonD(N₁) values both with [1] and without [5] a chemical concentration gradient. TheD(0) values and the carbon-carbon interaction coefficients ₁ are calculated by the use of the least squares method from the experimental data. The application of Zupp-Stevenson's definition of the activity coefficient ₁ results in a better fit of the theory to the experimental data.     

Response of fcc metals and L12 and D022 type trialuminides to uniaxial loading along [100] and [001]: ab initio DFT calculations

Ab initio density-functional calculations have been used to investigate the response of the face-centred cubic (fcc) metals Al and Cu, and of the L1₂- and D0₂₂-type trialuminides Al₃(Sc,Ti,V) to uniaxial loading along the [100] and [001] directions. The results obtained under uniaxial strains are compared to the response to biaxial (epitaxial) strains. The ideal tensile and compressive strengths and their limitation by shear instabilities along these deformation paths have been calculated. Although the response of both pure fcc metals could be expected to be very similar, our results show a fundamental difference: whereas for Cu a special invariant state with C ₂₂?=?C ₂₃, leading to a bifurcation from the tetragonal to an orthorhombic deformation path, is reached at a strain of 10%, for Al this state is reached only at a strain of 33% close to the critical strain defining the ideal tensile strength. The reaction of the L1₂-type trialuminides is comparable to the response of Al; no bifurcation to an orthorhombic deformation path is predicted. The response of the D0₂₂-type trialuminides is different from that of the L1₂-type phases because of the difference in the stacking of the atomic planes along the [001] direction. For D0₂₂-type trialuminides, the uniaxial compression along this direction or epitaxial tension in the (001) plane leads to the formation of a stress-free D0₃ structure, in complete analogy to the fcc???bcc transformations observed for the pure metals. Under uniaxial [100] loading the guiding symmetry along the deformation path is orthorhombic and leads to the formation of special structures under both tension and compression parts, which are related to the D0₃ structure in the same way as the parent D0₂₂-lattice is related to the L1₂ structure.     

Generalized susceptibilities and phonon anomalies in Pd and Pt metals

The generalized susceptibility, χ(q), in Pd and Pt for q along the [100], [110], [111], and [120] directions was determined from their APW and RAPW energy band structures, respectively, using the analytic tetrahedron linear energy scheme of Rath and Freeman. The band structures were previously found to yield Fermi surface radii, temperature dependencies of the static magnetic susceptibility, χ(T), resistivity, and a spin lattice relaxation, T₁T, in very good agreement with experiment. In the χ(q) calculations, we used 2048 tetrahedra in 1/48th irreducible BZ and the energy eigenvalues for bands 4, 5, and 6 which cross the Fermi energy as fitted to a Fourier series representation. The intraband parts of χ(q) at q = 0 for both metals are found to agree with the density of states at the Fermi energy to without 0.5%. Our results show that the dominant contribution to χ_intra arises from the dominant band 5 whose “jungle-gym” FS has strong nesting features; the main peak for Pd occurs at the same q value (= 0.65π/a) for q along the [0q0], [q, q, 0], and [q, q, q] directions. The locus of this main peak is a square in the (0, 0, 1) plane. The maximum of χ_intra for q along the [110] and [111] directions are 23% and 13%, respectively, higher than the value of χ(q) at q = 0. For q along the [010] and [120] directions, the peak is, however, lower than the value of χ_intra at q = 0. Hence, while phonon anomalies are predicted for the [110] and [111] directions, no anomaly is predicted for either the [100] or [120] direction. The predicted q value for the [110] anomaly, $\text{q}\text{= 0.65π/a}$ is close to the experimental value of ～0.7 π/a. Although there may be a hint of an anomaly at 0.56 [111] in the measurements, a more detailed investigation of this region is called for. For platinum, χ_intra for q along the [010], [110] and [111] directions has main peaks which occur at q = 0.68 π/a, 0.75 π/a, and 0.85 π/a, respectively. Here too, this main peak comes from the nesting of the jungle-gym Fermi surface which is not, however, as flat as that of palladium. Anomalies are predicted (although weaker in Pt than in Pd) along [110] and [111] but not along [100] and [120]. The [110] anomaly is close to the measured q value (～0.7–0.8 π/a). Also in agreement with experiment, we predict a weaker [110] anomaly for Pt than for Pd. In both Pd and Pt, weaker anomalies are predicted for the [111] direction than for the [110] direction.     

Local Index Formula on the Equatorial Podleś Sphere

We discuss spectral properties of the equatorial Podleś sphere S _q ². As a preparation we also study the ‘degenerate’ (i.e. q=0) case (related to the quantum disk). Over S _q ² we consider two different spectral triples:one related to the Fock representation of the Toeplitz algebra and the isopectral one given in [7]. After the identification of the smooth pre-C ^*-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the C ^*-algebra (the nontrivial generator of the K ₀-group) as well as the pairing with the q-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.     

各向异性扩散DLA的标度性质

本文利用坐标变换的方法讨论了各向异性扩散对DLA集团标度行为的影响。对二维正方形点阵,解析计算了各向异性扩散DLA集团的Hausdorff维数。结果表明,此时生长集困的Hausdorff维数D随各向异性扩散概率p连续变化,其最大值为5/3,最小值为3/2,还将这里所得到的理论结果与Jullien等人的数值结果进行了比较,发现它们符合得很好。最后,还解析讨论并计算了各向异性扩散DLA的广义维数D_q(p)。     

Exploring CP violation and penguin effects through B0d→D+D- and B0s→D+sD-s

The decay B⁰ _d→D⁺D^- offers an interesting probe of CP violation, but it requires control of penguin effects, which can be done through B⁰ _s→D⁺ _sD^- _s by means of the U-spin flavour symmetry of strong interactions. Recently, the Belle collaboration reported indications of large CP violation in the B⁰ _d decay, which were, however, not confirmed by BaBar, and first signals of the B⁰ _s channel were observed at the Tevatron. In view of these developments and the quickly approaching start of the LHC, we explore the allowed region in observable space for CP violation in B⁰ _d→D⁺D^-, perform theoretical estimates of the relevant hadronic penguin parameters and observables, and we address questions both about the most promising strategies for the extraction of CP-violating phases and about the interplay with other measurements of CP violation and the search for new physics. As far as the latter aspect is concerned, we point out that the B⁰ _q→D_q ⁺D^- _q system provides a setting for the determination of the B⁰ _q–B̄⁰ _q mixing phases (q∈{d,s}) that is complementary to the conventional B⁰ _d→J/ψK_S and B⁰ _s→J/ψφ modes with respect to possible new-physics effects in the electroweak penguin sector.