Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

We consider a family of random line tessellations of the Euclidean plane introduced in a more formal context by Hug and Schneider (Geom. Funct. Anal. 17:156, 2007) and described by a parameter α≥1. For α=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for α=2 it coincides with the typical Poisson-Voronoi cell. Let p _n (α) be the probability for the zero-cell to have n sides. We construct the asymptotic expansion of log p _n (α) up to terms that vanish as n→∞. Our methods are nonrigorous but of the kind commonly accepted in theoretical physics as leading to exact results. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when α>1, but gets delocalized for the Crofton cell, α=1, which is a singular point of the parameter range. The large-n expansion of log p _n (1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the typical n-sided cell of a Poisson line tessellation.     

Geometric structure and electronic properties of planar and nanotubular BN structures of the Haeckelite type

Planar and nanotubular structures that are based on boron and nitrogen and consist of tetragons, hexagons, and octagons are considered. By analogy with carbon nanoobjects of the same topology, these structures are referred to as Haeckelites. The geometric, electronic, and energy properties are thoroughly investigated for two variants of the regular mutual arrangement of the polygons. It is established that planar and nanotubular BN structures of the Haeckelite type are dielectrics with a band gap E _g ∼ 3.2–4.2 eV, which is less than the band gap E _g for BN nanotubes consisting only of hexagons. The cohesive energy of the BN nanotubes under investigation exceeds the cohesive energy of BN hexagonal nanotubes by 0.3 eV/atom.     

Structure of Shocks in Burgers Turbulence¶with Stable Noise Initial Data

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval. Received: 28 September 1998 / Accepted: 12 January 1999     

Neutrino radiation by an electon in quantizing nuclear and strong magnetic fields

The probability and intensity of neutrino radiation of a hydrogen-like atom in the strong magnetic field B >> Z ²α² B ₀, α = e ² = 1/137, B ₀ = m ²/e = 4.41⋅10¹³ G are determined. The temperature dependence of the intensity of an atom ensemble is analyzed.     

A precursor of market crashes: Empirical laws of Japan's internet bubble

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as X(t) = S(t)/S(0), where S(0), is the stock price for an onset time of the bubble. We selected approximately 3200 stocks traded on the Japanese Stock Exchange, and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which internet Bubble formed and crashed in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, P(S>x) ∼x^-α , with an exponent that moves over time. Furthermore we found that as the power-law exponents α approached two, the bubble burst. It is reasonable to suppose that it indicates that internet bubble is about to burst.     

On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data

Consider an inviscid Burgers equation whose initial data is a Lévy α-stable process Z with α>1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a partially negative answer to a Conjecture of Janicki and Woyczynski (J. Stat. Phys. 86(1–2):277–299, 1997). Along the way, we contradict a recent Conjecture of Z. Shi () about the lower tails of integrated stable processes.     

Gibbs Delaunay Tessellations with Geometric Hardcore Conditions

In this paper, we prove the existence of infinite Gibbs Delaunay tessellations on ℝ². The interaction depends on the local geometry of the tessellation. We introduce a geometric hardcore condition on small and large cells, consequently we can construct more regular infinite random Delaunay tessellations.     

The Discrete Spectrum in the Gaps of the Continuous One for Non-Signdefinite Perturbations with a Large Coupling Constant

Given two selfadjoint operators A and V=V ₊ -V _-, we study the motion of the eigenvalues of the operator A(t)=A-tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N ₊(λ,α), N _-(λ,α), N ₀(λ,α) defined as the number of the eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α>0. An abstract theorem on the asymptotics for these quantities is presented. Applications to Schr?dinger operators and its generalizations are given. Received: 9 April 1997 / Accepted: 26 August 1997     

Gravitational potential in fractional space

In this paper the gravitational potential with β-th order fractional mass distribution was obtained in α dimensionally fractional space. We show that the fractional gravitational universal constant G _α is given by , where G is the usual gravitational universal constant and the dimensionality of the space is α > 2.      

Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called “exactness of the mean-field theory”. It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the α-Potts model with annealed vacancies and the α-Potts model with invisible states.     

Three-dimensional array structures associated with Richtmyer-Meshkov and Rayleigh-Taylor instability

A boundary separating adjacent gas or liquid media is frequently unstable. Richtmyer-Meshkov and Rayleigh-Taylor instability cause the growth of intricate structures on such boundaries. All the lattice symmetries [rectangular (pmm2), square (p4mm), hexagonal (p6mm), and triangular (p3m1) lattices] which are of interest in connection with the instability of the surface of a fluid are studied for the first time. They are obtained from initial disturbances consisting of one (planar case, two-dimensional flow), two (rectangular cells), or three (hexagons and triangles) harmonic waves. It is shown that the dynamic system undergoes a transition during development from an initial, weakly disturbed state to a limiting or asymptotic stationary state (stationary point). The stability of these points (stationary states) is investigated. It is shown that the stationary states are stable toward large-scale disturbances both in the case of Richtmyer-Meshkov instability and in the case of Rayleigh-Taylor instability. It is discovered that the symmetry increases as the system evolves in certain cases. In one example the initial Richtmyer-Meshkov or Rayleigh-Taylor disturbance is a sum of two waves perpendicular to one another with equal wave numbers, but unequal amplitudes: a ₁(t=0)≠a ₂(t=0). Then, during evolution, the flow has p2 symmetry (rotation relative to the vertical axis by 180°), which goes over to p4 symmetry (rotation by 90°) at t→∞, since the amplitudes equalize in the stationary state: a ₁(t=∞)=a ₂(t=∞). It is shown that the hexagonal and triangular arrays are complementary. Upon time inversion (t→−t), “rephasing” occurs, and the bubbles of a hexagonal array transform into jets of a triangular array and vice versa. Zh. éksp. Teor. Fiz. 116, 908–939 (September 1999)     

Diffusion Coefficient of a Brownian Particle with a Friction Function Given by a Power Law

Nonequilibrium biological systems like moving cells or bacteria have been phenomenologically described by Langevin equations of Brownian motion in which the friction function depends on the particle’s velocity in a nonlinear way. An important subclass of such friction functions is given by power laws, i.e., instead of the Stokes friction constant γ ₀ one includes a function γ(v)∼v ^2α . Here I show using a recent analytical result as well as a dimension analysis that the diffusion coefficient is proportional to a simple power of the noise intensity D like D ^{(1−α)/(1+α)} (independent of spatial dimension). In particular the diffusion coefficient does not depend on the noise intensity at all, if α=1, i.e., for a cubic friction F _fric=−γ(v)v∼v ³. The exact prefactor is given in the one-dimensional case and a fit formula is proposed for the multi-dimensional problem. All results are confirmed by stochastic simulations of the system for α=1, 2, and 3 and spatial dimension d=1, 2, and 3. Conclusions are drawn about the strong noise behavior of certain models of self-propelled motion in biology.     

An all-electron LCGTO study of square and hexagonal plutonium monolayers

The linear combinations of Gaussian type orbitals fitting function (LCGTO-FF) method is used to study the electronic and geometrical properties of plutonium monolayers with square and hexagonal symmetry. The effects of several common approximations are examined: (1) scalar-relativity vs. full-relativity (i.e., with spin-orbit coupling included); (2) paramagnetic vs. spin-polarized; and (3) local-density approximation (LDA) vs. generalized- gradient approximation (GGA). The results indicate that spin-orbit coupling has a much stronger effect on the monolayer properties compared to the effects of spin-polarization. In general, the GGA is found to predict a larger lattice constant and a smaller cohesive energy compared to LDA predictions. We also find a significant compression of the monolayers compared to the bulk, contradicting the only other published result on a Pu monolayer. The current result supports the existence of a δ-like surface on α-Pu. Received 17 October 2001 Published online 6 June 2002     

Conserved Noise Restricted Curvature Model

A restricted curvature model with conservation of total number of particles is introduced. The surface width W of the model grows as t ^β at the beginning with β≈0.25 and becomes saturated at L ^α for t≫L ^z with α≈1.5, where L is the system size. The conservation law leads to a new universality class following sixth-order linear equation with conservative noise. The relation between our model and the equation is discussed.     

Universality in diffusion front growth dynamics

We have studied the scaling properties of diffusion fronts by numerical calculations based on the mean field approach in the context of a lattice gas model, performed in a triangular lattice. We find that the height-height correlation function scales with time t and length l as C(l, t) ≈l ^α f (t/l ^α/β) with α = 0.62±0.01 and β = 0.39±0.02. These exponent values are identical to those characterising the roughness of the diffusion fronts evolving through a square lattice [1,2], thus confirming their universality. Received 14 November 2001 / Received in final form 20 April 2002 Published online 31 July 2002     

On the Uniqueness of Gravitational Centre

The dual volume of order α of a convex body A in R ⁿ is a function which assigns to every a ∈ A the mean value of α-power of distances of a from the boundary of A with respect to all directions. We prove that this function is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n (for α = 0 and for α = n the function is constant). It implies that the dual volume of a convex body has the unique minimizer for α > n or α < 0 and has the unique maximizer for 0 < α < n. The gravitational centre of a convex body in R³ coincides with the maximizer of dual volume of order 2, thus it is unique.      

A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes

Anomalous diffusion processes are often classified by their mean square displacement. If the mean square displacement grows linearly in time, the process is considered classical. If it grows like t ^β with β<1 or β>1, the process is considered subdiffusive or superdiffusive, respectively. Processes with infinite mean square displacement are considered superdiffusive. We begin by examining the ways in which power-law mean square displacements can arise; namely via non-zero drift, nonstationary increments, and correlated increments. Subsequently, we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present. Finally, we introduce an alternative classification scheme based on renormalization groups. This scheme classifies processes with stationary increments such as Brownian motion and fractional Brownian motion in the same groups as the mean square displacement scheme, but does a better job of classifying processes with nonstationary increments and/or processes with infinite second moments such as α-stable Lévy motion. A numerical approach to analyzing data based on the renormalization group classification is also presented.     

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order <Emphasis Type="Italic">α</Emphasis>

In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E _α(h^αD _x ^α)f(x) where E _α() denotes the Mittag-Leffler function, and D _x ^α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.      

Generalized two-mode coherent-entangled state with real parameters

The coherent-entangled state |α, x; λ> with real parameters λ is proposed in the two-mode Fock space, which exhibits the properties of both the coherent and entangled states. The completeness relation of |α, x; λ> is proved by virtue of the technique of integral within an ordered product of operators. The corresponding squeezing operator is derived, with its own squeezing properties. Furthermore, generalized P-representation in the coherent-entangled state is constructed. Finally, it is revealed that superp...