Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Mean-square exponential stability of stochastic Hopfield neural networks with time-varying discrete and distributed delays

In this Letter, the mean-square exponential stability problem for stochastic Hopfield neural networks with both discrete and distributed time-varying delays is investigated. By choosing a modified Lyapunov-Krasovskii functional, a delay-dependent criterion is established such that the stochastic neural network is mean-square exponentially stable. The derivative of discrete time-varying delay h(t) satisfies and the decay rate β can be any finite positive value without any other constraints. The assumptions given in this Letter are more general than the conventional assumptions (i.e., and β satisfies a transcendental equation or an inequality). Finally, numerical examples are provided to illustrate the effectiveness of the proposed sufficient conditions.     

The rule for a subdiffusive particle in an extremely diverse environment

We consider a subdiffusive continuous time random walker in an inhomogeneous environment. Each microscopic random time is drawn from a waiting time probability density function (WT-PDF) of the form: , 0<β?1. The parameter k is a random quantity also, and is drawn from a PDF, , 0?γ<1, for a cutoff parameter . We show that the effective WT-PDF, ψ(t), obtained by averaging φ(t;k) with p(k), exhibits a transition in the rule that governs the power of ψ(t). ψ(t) obeys, , and μ is given by two different formula. When, 1−γ>β, μ=β, but otherwise, μ=1−γ. The rule for the scaling of ψ(t) reflects the competition between two different mechanisms for subdiffusion: subdiffusion due to the heavily tailed φ(t;k) for individual jumps, and subdiffusion due to the collective effect of an environment made of many slow local regions. These two different mechanisms for subdiffusion are not additive, and compete each other. The reported transition is dimension independent, and disappears when the power β is also distributed, in the range, 0<β?1. Simulations exemplified the transition, and implications are discussed.     

Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance R_ee and the number of distinct visited nodes N_v of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈R_ee〉 shows the dynamical scaling behavior , where is the average minimum distance between all possible pairs of nodes in the network, N is the number of nodes, γ is the degree exponent of the SFN and t is the step number of RWs. Especially, in the limit t→∞ satisfies the relation , where d is the diameter of network with for γ≥3 or for γ<3. Based on the scaling relation 〈R_ee〉, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.     

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels and noise term for the anharmonic coordinates . For zero temperature, i.e. for , we prove that the support of the Fourier transform of and of the time averaged velocity-velocity correlation functions of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency we find that the energy E_T transferred to the harmonic system up to time T is proportional to T^α. If equals one of the phonon frequencies ω_ν, it is α=2. We prove that there is a zero measure set L such that for in its full measure complement R?L, it is α=0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L^′ such that for the dissipation rate is nonzero and may be subdissipative (0≤α<1) or superdissipative (1<α≤2), compared to ordinary dissipation (α=1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for , related to . For the small denominators do not lead to divergencies such that is a smooth and bounded function in t.     

A scale-free network with limiting on vertices

We propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity. The random graph process is defined as follows: at any time-step t, (i) with probability α(k) and independently of other time-step, each vertex is inactive which means it cannot be connected by more edges, where k is the degree of v_i at the time-step t; (ii) a new vertex v_t is added along with m edges incident with v_t at one time and its neighbors are chosen in the manner of preferential attachment. We prove that the degree distribution P(k) of this random graph process satisfies if α(⋅) is a constant α₀; and P(k)∝C₂k⁻³ if α(?)↓0 as ?↑∞, where C₁,C₂ are two positive constants. The analytical result is found to be in good agreement with that obtained by numerical simulations. Furthermore, we get the degree distributions in this model with m-varying functions by simulation.     

Domain and hull-enclosed area distributions in two-dimensional coarsening

We obtain the exact distribution of the areas enclosed by domain boundaries (“hulls”) during the coarsening dynamics of a two-dimensional nonconserved scalar field. We prove that the number of hulls per unit area that enclose an area greater than A has the form N_h(A,t)=2c/(A+λt), where . This result is also a demonstration of the validity of the dynamical scaling hypothesis in this system. We also show that domain areas (regions of aligned spins) have a similar distribution.     

Sn Mössbauer spectroscopy and specific heat studies of the stannides RETSn (RE=Gd-Er and T=Cu,Ag)

The stannides RETSn (RE=Gd-Er and T=Cu,Ag), NdPtSb type structure, space group P6₃mc, have been investigated by ¹¹⁹Sn Mössbauer spectroscopy and specific heat studies. Small transferred magnetic hyperfine fields are detected at the tin nuclei at 4.2 K in the ¹¹⁹Sn Mössbauer spectra of RECuSn (RE=Tb,Dy and Ho) which reveal that these compounds undergo magnetic transitions at low temperatures. Heat capacity (C) measurements show that the title compounds undergo antiferromagnetic ordering. In order to explore the magnetic behaviour below the Néel temperature (T_N), the magnetic part of heat capacity was obtained by subtracting the lattice part of heat capacity obtained from the isostructural non-magnetic stannides Y TSn (T=Cu,Ag). of GdCuSn exhibits an equal moment (EM) magnetic structure and also exhibits multiple transitions below T_N, revealing higher order exchange interactions. Among the REAgSn stannides, the magnetic part of heat capacity for RE=Dy and Er exhibits non-T³ behaviour at low temperatures.     

Geometrical properties of local dynamics in Hamiltonian systems: The Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multi-dimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on N-dimensional tori. More specifically we introduce the Generalized Alignment Index of order k () as the volume of a generalized parallelepiped, whose edges are k initially linearly independent unit deviation vectors with respect to the orbit studied whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N-degree-of-freedom Hamiltonian systems that, for chaotic orbits, tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, fluctuates around non-zero values for 2≤k≤N and goes to zero for N<k≤2N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus: ∝t^{−2(k−N)+m} if 0≤m<k−N, and ∝t^−(k−N) if m≥k−N. The is a generalization of the Smaller Alignment Index (SALI) (). However, provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.     

Semi-ab initio calculations of superposition model and crystal field parameters for Co ions using the exchange charge model

This paper attempts for the first time to establish a reliable linkage between the two well-known and independent models of crystal field (CF), namely the exchange charge (ECM) and superposition models (SM). Our approach aims to show that the SM parameters can be reliably extracted from the distance dependence of the CF invariants for Co²⁺ as derived from the ECM through some semi-ab initio calculations which involved a single fitting parameter and a set of newly constructed procedures. Complete sets of the numerical values of SM parameters and t_k for Co²⁺ in its own host lattices of Li₂Co₃(SeO₃)₄, CoSO₄·H₂O, CoSeO₄·H₂O, and Co(OH)₂ are obtained and they are found to be around 13,000-16,000 cm⁻¹ for , 4100-5700 cm⁻¹ for , 4.1-5.0 for t₂ and 6.2-6.5 for t₄. The present results generally agree with but should be much better than those incomplete sets of results found by previous researchers using the conventional fitting approach. Plausible explanations for some noticeable discrepancies are also discussed together with the effects of different CF contributions on values of the SM parameters.     

Fluctuations of trading volume in a stock market

We consider the probability distribution function of the trading volume and the volume changes in the Korean stock market. The probability distribution function of the trading volume shows double peaks and follows a power law, P(V/〈V〉)∼(V/〈V〉)^−α at the tail part of the distribution with α=4.15(4) for the KOSPI (Korea composite Stock Price Index) and α=4.22(2) for the KOSDAQ (Korea Securities Dealers Automated Quotations), where V is the trading volume and 〈V〉 is the monthly average value of the trading volume. The second peaks originate from the increasing trends of the average volume. The probability distribution function of the volume changes also follows a power law, , where V_r=V(t)−V(t−T) and T is a time lag. The exponents β depend on the time lag T. We observe that the exponents β for the KOSDAQ are larger than those for the KOSPI.     

Finite and infinite dynamical systems of identical interacting particles

A dynamical system of infinite volume and of infinite number of identical interacting particles occupying energy levels has been constructed as the limit of an infinite sequence of finite, equivalent systems of increasing size and particle number. Systems both in equilibrium and in non-equilibrium state (designated S_∞=limS_k, , respectively, k=1,2,…) were investigated. The main results are:(i) The values in the T-limit (thermodynamic limit) of the physical quantities characterizing these systems are determined.(ii) The time evolution process both in and in systems is governed by the non-linear rate equations with common initial conditions p_i(t₀), where p_i(t)=n_i(t)/N are the occupation probabilities at time t. The time evolution process in the and systems is the same. The asymptotic approach to the equilibrium state is proved.(iii) For the case of the equilibrium state, the Boltzmann probability distribution p_i is given by the equation −lnp_i+a+e_ib=0 common to S_k and S_∞ systems with the same value of a and b. The term a=β⁻¹a_e, where a_e is the free energy per particle, and .(iv) The conditions for the equivalence of the systems being in equilibrium and also of the ones in non-equilibrium are stated.     

On the partial trace over collective spin degrees of freedom

We derive analytical properties for the degeneracy ν(N,j) occurring in the decomposition of the state space C2⊗N. We also investigate the dynamics of two qubits coupled via Ising interactions to separate spin baths, and we study the thermodynamic limit.     

The q-gamma and (q,q)-polygamma functions of Tsallis statistics

An axiomatic definition is given for the q-gamma function of Tsallis (non-extensive) statistical physics, the continuous analogue of the q-factorial of Suyari [H. Suyari, Physica A 368 (1) (2006) 63], and the q-analogue of the gamma function of Euler and Gauss. A working definition in closed form, based on the Hurwitz and Riemann zeta functions (including their analytic continuations), is shown to satisfy this definition. Several relations involving the q-gamma and other functions are obtained. The (q,q)-polygamma functions , defined by successive derivatives of , where ln_qa=(1−q)⁻¹(a^1−q−1),a>0 is the q-logarithmic function, are also reported. The new functions are used to calculate the inferred probabilities and multipliers for Tsallis systems with finite numbers of particles N?∞.     

Quantum mechanics of Klein-Gordon-type fields and quantum cosmology

With a view to address some of the basic problems of quantum cosmology, we formulate the quantum mechanics of the solutions of a Klein-Gordon-type field equation: (∂_t²+D)ψ(t)=0, where and D is a positive-definite operator acting in a Hilbert space . In particular, we determine all the positive-definite inner products on the space of the solutions of such an equation and establish their physical equivalence. This specifies the Hilbert space structure of uniquely. We use a simple realization of the latter to construct the observables of the theory explicitly. The field equation does not fix the choice of a Hamiltonian operator unless it is supplemented by an underlying classical system and a quantization scheme supported by a correspondence principle. In general, there are infinitely many choices for the Hamiltonian each leading to a different notion of time-evolution in . Among these is a particular choice that generates t-translations in and identifies t with time whenever D is t-independent. For a t-dependent D, we show that regardless of the choice of the inner product the t-translations do not correspond to unitary evolutions in , and t cannot be identified with time. We apply these ideas to develop a formulation of quantum cosmology based on the Wheeler-DeWitt equation for a Friedman-Robertson-Walker model coupled to a real scalar field with an arbitrary positive confining potential. In particular, we offer a complete solution of the Hilbert space problem, construct the observables, use a position-like observable to introduce the wave functions of the universe (which differ from the Wheeler-DeWitt fields), reformulate the corresponding quantum theory in terms of the latter, reduce the problem of the identification of time to the determination of a Hamiltonian operator acting in , show that the factor-ordering problem is irrelevant for the kinematics of the quantum theory, and propose a formulation of the dynamics. Our method is based on the central postulates of nonrelativistic quantum mechanics, especially the quest for a genuine probabilistic interpretation and a unitary Schrödinger time-evolution. It generalizes to arbitrary minisuperspace (spatially homogeneous) models and provides a way of unifying the two main approaches to the canonical quantum cosmology based on these models, namely quantization before and after imposing the Hamiltonian constraint.     

Equation of state of cubic hafnium(IV) nitride having Th3P4 -type structure

Pressure dependence of the specific volume, V(P), of the recently discovered high-pressure compound Hf₃N₄ having cubic Th₃P₄-type structure (c- Hf₃N₄) has been measured at room temperature up to 43.9 GPa in a diamond anvil cell using energy-dispersive X-ray powder diffraction combined with synchrotron radiation. A least-square fit of the Birch-Murnaghan equation of state to the experimental V(P)-data yielded for c- Hf₃N₄ the bulk modulus of and its first pressure derivative of . For fixed at 4 the bulk modulus of c- Hf₃N₄ was determined to be . The obtained B₀-value is only insignificantly below that estimated in preliminary measurements. Existing theoretical predictions for B₀ scatter around the present experimental data. The observation of a high bulk modulus of c- Hf₃N₄ supports the suggestion that this compound could have high hardness.     

Crystal structure and structural transition caused by charge-transfer phase transition for iron mixed-valence complex (n-C3H7)4N[FeFe(dto)3] (dto=C2O2S2)

(n-C₃H₇)₄N[Fe^IIFe^III(dto)₃] shows a new type of first order phase transition called charge-transfer phase transition around 120 K, where the charge transfer between Fe^II and Fe^III occurs reversibly. Recently, we have succeeded in obtaining single crystals of the title complex and determined the crystal structure at room temperature. Crystal data: space group P6₃, Z=2. Moreover, we have investigated the structural transition caused by the charge-transfer phase transition by means of powder X-ray diffraction measurement. When the temperature is decreased, the a-axis, which corresponds to the hexagonal ring size in two-dimensional honeycomb network structure of [Fe^IIFe^III(dto)₃]_∞, contracts by 0.1 Å at the charge-transfer transition temperature (T_CT), while the c-axis, perpendicular to the honeycomb network layer, elongates by 0.1 Å at T_CT. Consequently, when the temperature is decreased, the unit cell volume decreases without noticeable anomaly around T_CT, which is responsible for the quite small vibrational contribution to the entropy change, compared with usual spin crossover transition. Thus, the charge-transfer phase transition around 120 K for (n-C₃H₇)₄N[Fe^IIFe^III(dto)₃] is regarded as spin entropy driven phase transition.