A new class of Ornstein transformations with singular spectrum

It is shown that for any family of probability measures in Ornstein type constructions, the corresponding transformation has almost surely a singular spectrum. This is a new generalization of Bourgain's theorem [J. Bourgain, On the spectral type of Ornstein class one transformations, Israel J. Math. 84 (1993) 53–63], same result is proved for Rudolph's construction [D. Rudolph, An example of a measure-preserving map with minimal self-joining and applications, J. Anal. Math. 35 (1979) 97–122].     

Decay rates of energy of the 1D damped original nonlinear wave equation

We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity.     

Boundedness of solutions of nonlinear systems of differential equations

We first give an example to illustrate that the results in [12] concerning the boundedness of solutions of nonlinear oscillatory equations are not true. And then we obtain sufficient or necessary conditions for the boundedness of solutions of the nonlinear system of differential equations

Generalized evaluation subgroups of product spaces relative to a factor

For any -complexes and , we show that . We use this fact to compute generalized evaluation subgroups of generalized tori relative to a sphere.

     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Nonstationary Markov chains and convergence of the annealing algorithm

We study the asymptotic behavior as timet + of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt + , a nonstationary Markov chain may exhibit phase transitions. Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC/log(t/t ₀) ast+, with a computable constantC andt ₀ the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t^–1+) for some 0<1, with =0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal annealing schedule. This optimal choice reflects the competition between two physical effects: (a) The adiabatic effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the super-cooling effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly.     

Effect of hydrocolloids on starch thermal transitions,as measured by DSC

Differential scanning calorimetry (DSC) was used to analyze the influence of different hydrocolloids (xanthan, guar, and locust bean gums, carboxymethylcellulose and sodium alginate) on the gelatinization of corn starch in systems with starch concentration ranging between 0.1 and 0.7 g starch/g mixture. The reduction of available water produced a shift in gelatinization temperature, especially of the conclusion temperature. The effect was more marked for ionic hydrocolloids. The influence of hydrocolloids on glass transition temperature (T _g) of gelatinized starch suspensions and on the glass transition temperature of the maximally freeze-concentrated solute/unfrozen water matrix (T _g) was also studied.T _g onset values ranged between –4.5 and –5.5C for corn starch pastes with and without hydrocolloids. Those hydrocolloids that increased the viscosity of the unfrozen matrix inhibited additional ice formation during thawing (devitrification).Starch concentration and final heating temperature were found to be relevant factors affecting the kinetics of amylopectin retrogradation during frozen storage at –4C. Xanthan gum failed to prevent amylopectin retrogradation; this observation could be attributed to the fact that gums act outside the starch granule, while amylopectin retrogradation takes place within the granule.     

Cathodoluminescence Depth Profiling in SiO2:Ge Layers

 For investigation of the luminescent center profile cathodoluminescence measurements are used under variation of the primary electron energy E ₀ = 2…30 keV. Applying a constant incident power regime (E ₀·I ₀ = const), the depth profiles of luminescent centers are deduced from the range of the electron energy transfer profiles dE/dx. Thermally grown SiO₂ layers of thickness d = 500 nm have been implanted by Ge⁺-ions of energy 350 keV and doses (0.5–5)10¹⁶ ions/cm². Thus Ge profiles with a concentration maximum of (0.4 – 4) at% at the depth of d_m≅240 nm are expected. Afterwards the layers have been partially annealed up to T _a = 1100 °C for one hour in dry nitrogen. After thermal annealing, not only the typical violet luminescence (λ = 400 nm) of the Ge centers is strongly increased but also the luminescent center profiles are shifted from about 250 nm to 170 nm depth towards the surface. This process should be described by Ge diffusion processes, precipitation and finally Ge nanocluster formation. Additionally, a Ge surface layer is piled-up extending to a depth of roughly 25 nm.     

A modified steepest descent method with applications to maximizing likelihood functions

In maximizing a non-linear function G(), it is well known that the steepest descent method has a slow convergence rate. Here we propose a systematic procedure to obtain a 1–1 transformation on the variables , so that in the space of the transformed variables, the steepest descent method produces the solution faster. The final solution in the original space is obtained by taking the inverse transformation. We apply the procedure in maximizing the likelihood functions of some generalized distributions which are widely used in modeling count data. It was shown that for these distributions, the steepest descent method via transformations produced the solutions very fast. It is also observed that the proposed procedure can be used to expedite the convergence rate of the first derivative based algorithms, such as Polak-Ribiere, Fletcher and Reeves conjugate gradient methods as well.     

Singular perturbed Markov chains and exact behaviors of simulated annealing processes

We study asymptotic properties of discrete and continuous time generalized simulated annealing processesX(·) by considering a class of singular perturbed Markov chains which are closely related to the large deviation of perturbed diffusion processes. Convergence ofX(t) in probability to a setS ₀ of desired states, e.g., the set of global minima, and in distribution to a probability concentrated onS ₀ are studied. The corresponding two critical constants denoted byd and withd are given explicitly. When the cooling schedule is of the formc/logt, X(t) converges weakly forc>0. Whether the weak limit depends onX(0) or concentrates onS ₀ is determined by the relation betweenc, d, and . Whenc>, the expression for the rate of convergence for each state is also derived.