Spectral Statistics in the Quantized Cardioid Billiard

The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the first 11000 quantal energy levels for odd and even symmetry respectively. It is found that the level-spacing distribution is in good agreement with the GOE distribution of random-matrix theory. In case of the number variance and rigidity we observe agreement with the random-matrix model for short-range correlations only, whereas for long-range correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosine-modulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits instead of exclusively Gaussian peaks.Comment: 20 pages, uu-encoded ps.Z-fil

Level spacings and periodic orbits

Starting from a semiclassical quantization condition based on the trace formula, we derive a periodic-orbit formula for the distribution of spacings of eigenvalues with k intermediate levels. Numerical tests verify the validity of this representation for the nearest-neighbor level spacing (k=0). In a second part, we present an asymptotic evaluation for large spacings, where consistency with random matrix theory is achieved for large k. We also discuss the relation with the method of Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472] for two-point correlations.Comment: 4 pages, 2 figures; major revisions in the second part, range of validity of asymptotic evaluation clarifie

CMB Anisotropy of Spherical Spaces

The first-year WMAP data taken at their face value hint that the Universe might be slightly positively curved and therefore necessarily finite, since all spherical (Clifford-Klein) space forms M^3 = S^3/Gamma, given by the quotient of S^3 by a group Gamma of covering transformations, possess this property. We examine the anisotropy of the cosmic microwave background (CMB) for all typical groups Gamma corresponding to homogeneous universes. The CMB angular power spectrum and the temperature correlation function are computed for the homogeneous spaces as a function of the total energy density parameter Omega_tot in the large range [1.01, 1.20] and are compared with the WMAP data. We find that out of the infinitely many homogeneous spaces only the three corresponding to the binary dihedral group T*, the binary octahedral group O*, and the binary icosahedral group I* are in agreement with the WMAP observations. Furthermore, if Omega_tot is restricted to the interval [1.00, 1.04], the space described by T* is excluded since it requires a value of Omega_tot which is probably too large being in the range [1.06, 1.07]. We thus conclude that there remain only the two homogeneous spherical spaces S^3/O* and S^3/I* with Omega_tot of about 1.038 and 1.018, respectively, as possible topologies for our Universe.Comment: A version with high resolution sky maps can be obtained at http://www.physik.uni-ulm.de/theo/qc

Human osteochondritis dissecans fragment-derived chondrocyte characteristics ex vivo, after monolayer expansion-induced de-differentiation, and after re-differentiation in alginate bead culture

Background Autologous chondrocyte implantation (ACI) is a therapy for articular cartilage and osteochondral lesions that relies on notch- or trochlea-derived primary chondrocytes. An alternative cell source for ACI could be osteochondritis dissecans (OCD) fragment-derived chondrocytes. Assessing the potential of these cells, we investigated their characteristics ex vivo and after monolayer expansion, as monolayer expansion is an integral step of ACI. However, as monolayer expansion can induce de-differentiation, we asked whether monolayer-induced de-differentiation can be reverted through successive alginate bead culture. Methods Chondrocytes were isolated from the OCD fragments of 15 patient knees with ICRS grades 3–4 lesions for ex vivo analyses, primary alginate bead culture, monolayer expansion, and alginate bead culture following monolayer expansion for attempting re-differentiation. We determined yield, viability, and the mRNA expression of aggrecan and type I, II, and X collagen. Results OCD fragment-derived chondrocyte isolation yielded high numbers of viable cells with a low type I:II collagen expression ratio ( 1. Conclusion OCD fragment derived human chondrocytes may hold not yet utilized clinical potential for cartilage repair. Keywords: Chondrocyte; Articular cartilage; De-differentiation Re-differentiation; Monolayer expansion; Alginate bead cultur

Distribution of Eigenvalues for the Modular Group

The two-point correlation function of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that ion the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital, but can be obtained upon request from the first Autho

Zeta Function Zeros, Powers of Primes, and Quantum Chaos

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed

Periodic orbit spectrum in terms of Ruelle--Pollicott resonances

Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory ``p'' returns to its initial conditions after some fixed time tau_p. Our aim is to investigate the spectrum tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle--Pollicott resonances). For large periods, corrections to the well--known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random matrix theory and discrete maps are also considered. In particular, a random matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards

Osteoarthritis-Induced Metabolic Alterations of Human Hip Chondrocytes

Osteoarthritis (OA) alters chondrocyte metabolism and mitochondrial biology. We explored whether OA and non-OA chondrocytes show persistent differences in metabolism and mitochondrial function and different responsiveness to cytokines and cAMP modulators. Hip chondrocytes from patients with OA or femoral neck fracture (non-OA) were stimulated with IL-1β, TNF, forskolin and opioid peptides. Mediators released from chondrocytes were measured, and mitochondrial functions and glycolysis were determined (Seahorse Analyzer). Unstimulated OA chondrocytes exhibited significantly higher release of IL-6, PGE 2 and MMP1 and lower production of glycosaminoglycan than non-OA chondrocytes. Oxygen consumption rates (OCR) and mitochondrial ATP production were comparable in unstimulated non-OA and OA chondrocytes, although the non-mitochondrial OCR was higher in OA chondrocytes. Compared to OA chondrocytes, non-OA chondrocytes showed stronger responses to IL-1β/TNF stimulation, consisting of a larger decrease in mitochondrial ATP production and larger increases in non-mitochondrial OCR and NO production. Enhancement of cAMP by forskolin prevented IL-1β-induced mitochondrial dysfunction in OA chondrocytes but not in non-OA chondrocytes. Endogenous opioids, present in OA joints, influenced neither cytokine-induced mitochondrial dysfunction nor NO upregulation. Glycolysis was not different in non-OA and OA chondrocytes, independent of stimulation. OA induces persistent metabolic alterations, but the results suggest upregulation of cellular mechanisms protecting mitochondrial function in OA

Intermediate statistics in quantum maps

We present a one-parameter family of quantum maps whose spectral statistics are of the same intermediate type as observed in polygonal quantum billiards. Our central result is the evaluation of the spectral two-point correlation form factor at small argument, which in turn yields the asymptotic level compressibility for macroscopic correlation lengths