数学启蒙教育从罗马数字累数算到笔算再到珠算

从数学教育发展的历史来看,在数学启蒙教育中,历经罗马数字累数算,进而发展创新到笔算;笔算的应用,不仅促进数学教育的大发展,同时对数学科学的研究也起到了巨大的推动作用。目前,笔算仍广泛应用于数学启蒙教育之中,珠算珠心算的计算功能比笔算强大,更适合于数学启蒙教育。

Spectrum Allocation for Cognitive Radio Networks with Non-Deterministic Bandwidth of Spectrum Hole

The spectrum allocation for cognitive radio networks(CRNs) has received considerable studies under the assumption that the bandwidth of spectrum holes is static. However, in practice, the bandwidth of spectrum holes is time-varied due to primary user/secondary user(PU/SU) activity and mobility, which result in non-determinacy. This paper studies the spectrum allocation for CRNs with non-deterministic bandwidth of spectrum holes. We present a novel probability density function(PDF) through order statistics as well as its simplified form to describe the statistical properties of spectrum holes, with which a statistical spectrum allocation model based on stochastic multiple knapsack problem(MKP) is formulated for spectrum allocation with non-deterministic bandwidth of spectrum holes. To reduce the computational complexity, we transform this stochastic programming problem into a constant MKP through exploiting the properties of cumulative distribution function(CDF), which can be solved via MTHG algorithm by using auxiliary variables. Simulation results illustrate that the proposed statistical spectrum allocation algorithm can achieve better performance compared with the existing algorithms when the bandwidth of spectrum holes is time-varied.

Robust user equilibrium model based on cumulative prospect theory under distribution-free travel time

The assumption widely used in the user equilibrium model for stochastic network was that the probability distributions of the travel time were known explicitly by travelers. However, this distribution may be unavailable in reality. By relaxing the restrictive assumption, a robust user equilibrium model based on cumulative prospect theory under distribution-free travel time was presented. In the absence of the cumulative distribution function of the travel time, the exact cumulative prospect value(CPV) for each route cannot be obtained. However, the upper and lower bounds on the CPV can be calculated by probability inequalities.Travelers were assumed to choose the routes with the best worst-case CPVs. The proposed model was formulated as a variational inequality problem and solved via a heuristic solution algorithm. A numerical example was also provided to illustrate the application of the proposed model and the efficiency of the solution algorithm.

On the Probability Density Function of the Real and Imaginary Parts in WFRFT Signals

Recently a Hybrid Carrier （HC） scheme based on Weighted-type Fractional Fourier Transform （WFRFT） was proposed and developed, which contains Single Carrier （SC） and Multi-Carrier （MC） synergetie transmission. The wide interest is primarily due to its appealing characteristics, such as the robust performances in different types of selective fading channels and a great deal of potential for secure communications. According to the literatures, the HC signal and SC or MC signal probability distributions are different. In particular, some benefits of this HC scheme are brought by the quasi-Gaussian distribution of WFRFT signals. However, until now researchers have only presented statistic properties through computer simulations, and the accurate expressions of signals are not derived yet. In this paper, we derive the accu- rate and rigorously established closed-form expressions of Probability Density Function （PDF） of WFRFT signal real and imaginary parts with a large number of QPSK subcarriers, and this PDF can describe the behavior of data modulated by WFRFT, avoiding the complex computation for extensive computer simulations. Furthermore, the components of PDF expression are described and analyzed, and it is revealed that the tendency of signal quasi-Gaussian changes with the increasing of the parameter a （a in （0,1]）. To validate the analytical results, extensive simulations have been conducted, showing a very good match between the analytical results and the real situations. The contribution of this paper may be useful to deduce the closed form expressions of Bit Error Ratio （BER）, the Complementary Cumulative Distribution Function （CCDF） of Peak to Average Power Ratio （PAPR）, and other analytical studies which adopt the PDF.