The Numerical Inversion of the Laplace Transform in a Multi-Precision Environment

This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.

A Synthesis of Multi-Precision Multiplication and Squaring Techniques for 8-Bit Sensor Nodes＂ State-of-the-Art Research and Future Challenges

Multi-precision multiplication and squaring are the performance-critical operations for the implementation of public-key cryptography, such as exponentiation in RSA, and scalar multiplication in elliptic curve cryptography （ECC）. In this paper, we provide a survey on the multi-precision multiplication and squaring techniques, and make special focus on the comparison of their performance and memory footprint on sensor nodes using 8-bit processors, Different from the previous work, our advantages are in at least three aspects. Firstly, this survey includes the existing techniques for multi- precision multiplication and squaring on sensor nodes over prime fields. Secondly, we analyze and evaluate each method in a systematic and objective way. Thirdly, this survey also provides suggestions for selecting appropriate multiplication and squaring techniques for concrete implementation of public-key cryptography. At the end of this survey, we propose the research challenges on efficient implementation of the multiplication and the squaring operations based on our observation.

Rapid Determination of Complete Distribution of Pore and Throat in Tight Oil Sandstone of Triassic Yanchang Formation in Ordos Basin, China

This study aimed to investigate the complete distribution of reservoir space in tight oil sandstone combining casting slices, field emission scanning electron microscopy(FE-SEM), the pore-throat theory model, high-resolution image processing, mathematical statistics, and other technical means. Results of reservoir samples from the Xin’anbian area of Ordos Basin showed that the total pore radius curve of the tight oil sandstone reservoir exhibited a multi-peak distribution, and the peaks appeared to be more focused on the ends of the range. This proved that pores with a radius of 1–50,000 nm provided the most significant storage space for tight oil, indicating that special attention should be paid to this range of the pore size distribution. Meanwhile, the complete throat radius curve of the tight oil sandstone reservoir exhibited a multipeak distribution. However, the peak values were distributed throughout the scales. This confirmed that the throat radius in the tight oil sandstone reservoir was not only in the range of hundreds of nanometers but was also widely distributed in the scale approximately equal to the pore size. The new rapid determination method could provide a precise theoretical basis for the comprehensive evaluation, exploration, and development of a tight oil sandstone reservoir.