Abstract:
The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating distribution by minimizing the Kullback-Leibler information (relative entropy) of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints. We study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases. We prove that (i) the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with, and (ii) the approximate discrete distribution weakly converges to the given continuous distribution. Moreover, we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem.

Abstract:
A technique for discretizing efficiently the solution of a Linear descriptor (regular) differential input system with consistent initial conditions, and Time-Invariant coefficients (LTI) is introduced and fully discussed. Additionally, an upper bound for the error that derives from the procedure of discretization is also provided. Practically speaking, we are interested in such kind of systems, since they are inherent in many physical, economical and engineering phenomena. 1. Introduction: Preliminary Results During the discretization (or sampling) process, we should replace the original continuous-time systems with finite sequences of values at specified discrete-time points. This important process is commonly used whenever the differential systems involve digital inputs, and by having numerical data, the sampling operation and the quantization are necessary. Additionally, the discretization (or sampling) process is occurred whenever significant measurements for the system are obtained in an intermittent fashion. For instance, we can consider a radar tracking system, where there is information about the azimuth and the elevation, which is obtained as the antenna of the radar rotates. Consequently, the scanning operation of the radar produces many important sampled data. In our approach, we consider the LTI descriptor (or generalized) differential input systems of type where matrices (i.e., is the algebra of square matrices with elements in the field ) and are constants; the state has consistent initial conditions. We shall call ？？a consistent initial condition for (1.1) at if there is a solution for (1.1), which is defined on some interval , such that , the input , and the are related to the matrix pencil theory, since its algebraic, geometric, and dynamic properties stem from the structure of the associated pencil, that is, . Moreover, for the sake of simplicity, we set in the sequel and . Now, in what it follows, the pencil is regular, that is, . Practically speaking, descriptor (or generalized) regular (or singular) differential systems constitute a more general class than linear state space systems do. Considering applications, these kinds of systems appear in the modelling procedure of many physical, engineering, mechanical, actuarial, and financial problems. For instance, in engineering, in electrical networks, and in constrained mechanics, the reader may consult [1–6], and so forth. In Economics, the famous Leontief input-output singular dynamic model is well known; see for instance some of the numerous references [2, 3, 7–12], and so forth. In

Abstract:
利用美国冰雪资料中心(National Snow and Ice Data Center)提供的近40年逐周的卫星反演雪盖资料,定义了各季节新增(融化)雪盖而积指数(fresh snow extent),即增/融雪覆盖率P_(FSE)、增/融雪面积A_(FSE)、欧亚大陆北部增/融雪面积之和T_(FSE),针对欧亚大陆各季节平均的雪盖面积本身(snow extent,P_(SE)、A_(SE)、T_(SE)和其增(融)雪盖面积,分析比较二者的变化特征.结果表明,欧亚大陆各季节平均的雪盖面积和相应增(融)雪盖面积不论是气候态分布还是其年际、十年际变化均有明显不同,其中以冬、春季差别更为明显;夏、秋季二者虽有较好的一致性,但增(融)雪盖面积的变率明显强于雪盖而积本身;另外,冬季欧洲新增雪盖对欧业北部冬季雪盖面积以及其后的春季雪盖都有较显著的影响,而春季欧洲和中纬度亚洲地区的融雪则受到冬、春两季雪盖情况的影响.进一步分析欧亚大陆冬、春两季增(融)雪盖与ENSO关系显示,二者除在个别地区(两伯利业北部、欧洲中东部以及青藏高原)存在较明显关系外,整体上,欧亚大陆北部雪盖变化既不受控于ENSO,也不会显著影响ENSO.

Abstract:
This paper presents a systematic way to examine the origin of variety in falling snow. First, we define shape diversity as the logarithm of the number of possible distinguishable crystal forms for a given resolution and set of conditions, and then we examine three sources of diversity. Two sources are the range of initial-crystal sizes and variations in the trajectory variables. For a given set of variables, diversity is estimated using a model of a crystal falling in an updraft. The third source is temperature-updraft heterogeneities along each trajectory. To examine this source, centimeter-scale data on cloud temperature and updraft speed are used to estimate the spatial frequency (m 1) of crystal feature changes. For air-temperature heterogeneity, this frequency decays as p 0.66, where p is a measure of the temperature-deviation size. For updraft-speed heterogeneity, the decay is p 0.50. By using these frequencies, the fallpath needed per feature change is found to range from ~0.8 m, for crystals near 15°C, to ~8 m near 19°C – lengths much less than total fallpath lengths. As a result, the third source dominates the diversity, with updraft heterogeneity contributing more than temperature heterogeneity. Plotted against the crystal's initial temperature ( 11 to 19°C), the diversity curve is "mitten shaped", having a broad peak near 15.4°C and a sharp subpeak at 14.4°C, both peaks arising from peaks in growth-rate sensitivity. The diversity is much less than previous estimates, yet large enough to explain observations. For example, of all snow crystals ever formed, those that began near 15°C are predicted to all appear unique to 1 μm resolution, but those that began near 11°C are not.