A Generalization of the Collatz Problem. Building Cycles and a Stochastic Approach

The (3x+1)/2 problem is generalized into the n-furcation problem (l _i x+m _i )/n where i[0, 1, ..., n–1]. It is shown that, under some constraints on l _i and m _i , the main bijection property between the k less significant digits of the seed, written in base n, and the sequence of generalized parities of the k first iterates is preserved. This property is used to investigate a stochastic treatment of ensemble of large value seeds. The bijection property predicts stochasticity for a number of iterations equals to the number of significant digits of the seed. In fact, the stochastic approach is valid for much larger numbers, a property which is more easily shown by using increasing sequences than decreasing ones. Finally, we extend the stochastic approach to cases where the bijection theorem does not hold, introducing the matrix giving the probability that a j number (where j is the last significant digit of this number written in base n) gives a i number iterate.     

Far infrared sub-nanosecond pulse generation in GaP with a time-synchronized mode-locked double-frequency Nd: Glass laser system

Light pulses of 149 m wavelength and 700 ps duration are generated by non-collinear phase-matched difference frequency mixing of laser pulses at 1053.5 and 1061 nm in a (110) cut GaP crystal. The pump laser pulses are generated in a time-synchronized mode-locked double-frequency Nd:glass laser system consisting of a silicate glass branch and a phosphate glass branch. A photon conversion efficiency of 4 × 10^–6 is achieved. The non-linear susceptibility constant of GaP is determined to be d ₁₄ = (10 ± 1) pm V^–1.     

LD泵浦Nd:YAG 946 nm/1 064 nm双波长运转及腔内和频

用国产半导体激光二极管(LD)端面泵浦NdYAG晶体, 通过优化激光谐振腔反射膜系,调节1 064 nm谱线的线性损耗以达到与弱谱线946 nm的增益匹配,在室温下实现1 064 nm和946 nm双波长连续运转,并通过I类临界相位匹配LBO晶体腔内和频在国内首次实现500.8 nm青色激光连续输出.当泵浦注入功率为1.4W时和频青色激光最大输出达20 mW,光-光转换效率为1.4%,功率稳定性24 h内优于±3%.     

伪轨跟踪与混沌

本文对具有无限个点的紧致度量空间上的连续满射，研究了伪轨跟踪性质与Ｒｕｅｌｌｅ－Ｔａｋｅｎｓ意义下的混沌、拓扑混合及具有性质Ｐ的关系．     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

A Strong Law for Mixing Random Variables

A sharp almost sure bound is derived for limit points of average sum of weakly dependent random variables, which ensures strong laws of large numbers for and -mixing random variables, without assumptions on rate of tending to zero of and -mixing parameters _n and _n.     

A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().

     

Characterizing domain mixing effects in hydrogen bond-compatibilized polymer blends

The nature and extent of phase mixing in blends of hydroxyl-functionalized polystyrene and poly(ethyl acrylate) (PS/PEA), where the driving force for mixing is hydrogen bonding, are characterized by several techniques. Small-angle x-ray scattering (SAXS) shows a reduction in average domain size with increasing functionalization level, a result also evident from scanning electron microscopy (SEM). Together, the two techniques reveal a very broad distribution of domain sizes. At high functionalization levels, both SAXS and SEM indicate a high degree of “in-domain” mixing, with little or no pure PS or PEA remaining in the blends. Mathematical modeling of dynamic mechanical thermal analysis (DMTA) data is employed to quantify this progression. Initially, mixing is primarily interfacial, but as the functionalization level increases, the mixed interphase rapidly grows to occupy the entire material. In agreement with the SAXS and SEM results, DMTA modeling shows that further increases in the functionalization level suppress the amplitude of composition variations in the sample. The onset of extensive in-domain mixing coincides with the marked changes in stress-strain behavior observed previously in these materials. © 1994 John Wiley & Sons, Inc.     

Fermilab top mass and modified Fritzsch mass matrices

Fritzsch like mass matrices with non-zero 22-elements both in U sector and D sector have been investigated in the context of latest data regardingm _t ^phys , |V _ub|, |V _cb|, |V _td| and |V _ts|. Unlike several other phenomenological models, the present model not only accommodates the value ofm _t ^phys in the range 150–240 GeV, encompassing the CDF and D0 values, but is also able to reproduce |V _cb| ≊0.040 and |V _ub/V_cb| = 0.08±0.02 and |V _td| is predicted to lie in the range 0.005–0.014. Further, the angles of the unitarity triangle, related to the CP-violating asymmetries, are calculated to be in the ranges −1.0⩽sin2α⩽−0.1, 0.6 ⩽sin2α⩽1.0 and 0.48⩽sin2β⩽0.56, which are in agreement with other recent calculations.     

Comparison of calorimetric and IR-spectroscopic data for alcohols and alcoholic solutions

The heats of solution of alcohols in hexane can be considered as the energy necessary to break hydrogen bonds (H-bond). The amount of non H-bonded OH groups estimated from caloric data, are in good agreement with IR-spectroscopic data. Comparison of calorimetric and IR-spectroscopically determined H-bond energies permit the separation of intermolecular van der Waals effects from H-bond interactions. This separation shows that van der Waals interactions of alcohols or water should not be underestimated.