Abstract:
We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-N limit of the O(N) phi^4 model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.

Abstract:
We explore the quantum-classical crossover in the behaviour of a quantum field mode. The quantum behaviour of a two-state system - a qubit - coupled to the field is used as a probe. Collapse and revival of the qubit inversion form the signature for quantum behaviour of the field and continuous Rabi oscillations form the signature for classical behaviour of the field. We demonstrate both limits in a single model for the full coupled system, for states with the same average field strength, and so for qubits with the same Rabi frequency.

Abstract:
This paper is devoted to study of the classical-to-quantum crossover of the shot noise value in chaotic systems. This crossover is determined by the ratio of the particle dwell time in the system, $\tau_d$, to the characteristic time for diffraction $t_E \simeq \lambda^{-1} |\ln \hbar|$, where $\lambda$ is the Lyapunov exponent. The shot noise vanishes in the limit $t_E \gg \tau_d $, while reaches its universal quantum value in the opposite limit. Thus, the Lyapunov exponent of chaotic mesoscopic systems may be found by the shot noise measurements.

Abstract:
In this paper, we give Hurwitz zeta distributions with $0 < \sigma \ne 1$ by using the Gamma function. During the proof process, we show that the Hurwitz zeta function $\zeta (\sigma,a)$ does not vanish for all $0 <\sigma <1$ if and only if $a \ge 1/2$. Next we define Euler-Zagier-Hurwitz type of double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence. Moreover, we show that the Euler-Zagier-Hurwitz type of double zeta function $\zeta_2 (\sigma_1,\sigma_2\,;a)$ does not vanish when $0<\sigma_1<1$, $\sigma_2>1$ and $1<\sigma_1+\sigma_2<2$ if and only if $a \ge 1/2$.

Abstract:
We consider superfluid turbulence near absolute zero of temperature generated by classical means, e.g. towed grid or rotation but not by counterflow. We argue that such turbulence consists of a {\em polarized} tangle of mutually interacting vortex filaments with quantized vorticity. For this system we predict and describe a bottleneck accumulation of the energy spectrum at the classical-quantum crossover scale $\ell$. Demanding the same energy flux through scales, the value of the energy at the crossover scale should exceed the Kolmogorov-41 spectrum by a large factor $\ln^{10/3} (\ell/a_0)$ ($\ell$ is the mean intervortex distance and $a_0$ is the vortex core radius) for the classical and quantum spectra to be matched in value. One of the important consequences of the bottleneck is that it causes the mean vortex line density to be considerably higher that based on K41 alone, and this should be taken into account in (re)interpretation of new (and old) experiments as well as in further theoretical studies.

Abstract:
We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the first terms in the Laurent expansion at zero of the zeta function associated to a double sequence. We particularize this technique to the case of sums of sequences of spectral type, and we give two applications: the first concerning some special functions appearing in number theory, and the second the functional determinant of the Laplace operator on a product space.

Abstract:
We present an accurate numerical determination of the crossover from classical to Ising-like critical behavior upon approach of the critical point in three-dimensional systems. The possibility to vary the Ginzburg number in our simulations allows us to cover the entire crossover region. We employ these results to scrutinize several semi-phenomenological crossover scaling functions that are widely used for the analysis of experimental results. In addition we present strong evidence that the exponent relations do not hold between effective exponents.

Abstract:
The reduction of quantum scattering leads to the suppression of shot noise. In the present paper, we analyze the crossover from the quantum transport regime with universal shot noise, to the classical regime where noise vanishes. By making use of the stochastic path integral approach, we find the statistics of transport and the transmission properties of a chaotic cavity as a function of a system parameter controlling the crossover. We identify three different scenarios of the crossover.

Abstract:
We study the entire function zeta(n,s) which is the sum of l to the power -s, where l runs over the positive eigenvalues of the Laplacian of the circular graph C(n) with n vertices. We prove that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(n,s) is in K. To prove this, we look at the Dirac zeta function, which uses the positive eigenvalues of the Dirac operator D=d+d^* of the circular graph, the square root of the Laplacian. We extend a Newton-Coates-Rolle type analysis for Riemann sums and use a derivative which has similarities with the Schwarzian derivative. As the zeta functions zeta(n,s) of the circular graphs are entire functions, the result does not say anything about the roots of the classical Riemann zeta function zeta(s), which is also the Dirac zeta function for the circle. Only for Re(s)>1, the values of zeta(n,s) converge suitably scaled to zeta(s). We also give a new solution to the discrete Basel problem which is giving expressions like zeta_n(2) = (n^2-1)/12 or zeta_n(4) = (n^2-1)(n^2+11)/45 which allows to re-derive the values of the classical Basel problem zeta(2) = pi^2/6 or zeta(4)=pi^4/90 in the continuum limit.