算法化

The algorithmization of counterfactuals now brings us a giant step closer to understanding why evolution has endowed humans with the illusion of free will, and how it manages to keep that illusion so vivid in our brain.

算法化反事实如今让我们距离了解为什么进化给与了人类自由意志错觉及它是如何让这种错觉在我们的头脑中栩栩如生迈出了巨大的一步。

However, taking into account the 16th and 17th centuries when European mathematicians still held the view that negative is "ridiculous number", Fibonacci's awareness of the negative undoubtedly is of great significance.4、Fibonacci's understanding of the irrational depends on the geometry, and the arithmetic of Fibonacci for square root was the tendency of algorithm.

虽他忽视二次方程之负数解，但考虑到16和17世纪，欧洲数学家仍认为负数是"荒谬的数"，斐波那契对负数的认识无疑具有重要的意义。4、斐波那契对无理数的认识有赖于几何图形，"开方法"趋于算法化。

This paper shows the methods of finding the sum of the Geometric Progression for the first n terms and the rule for extracting square root in Llvat , which is a famous mathematical work in the 12~ century Hindu, and compare them with the corresponding traditional Chinese algorithmic methods, in order to show the characteristics of mathematics in the two countries.

以12世纪印度著名数学著作《丽罗娃底》中的等比数列求和与开方为例，与中国传统数学中的相应算法进行比较，以展现中、印数学的算法化特色。

Firstly, Wavelet-Galerkin algorithm for solving the first kind of singular integral equation with the Hilbert kernel is proposed, we use the characteristic of periodic wavelet on L~2([0,1]) and Hilbert kernel to solve and make stiff matrix lower dimensions and become sparser through thresholding,thus the cost of computation is reduced. Because of the singularity of Hilbert kernel we use Tikhonov regularization method to solve the system of stiff equation. At last the convergence and numerical result of approximate solution are given. Secondly, an approach of regularization based on Fourier is presented for sideways heat equation; we give the theory proof and error estimate.

首先，提出了含Hilbert核的第一类奇异积分方程的小波伽辽金(Wavelet-Galerkin)数值算法，该算法中利用了L~2([0，1])上的周期小波和Hilbert核的特点进行处理，使得刚性矩阵维数降低并且通过阈值使得它更加稀疏，减少了计算量；由于Hilbert核的奇异性，通过Tikhonov正则化方法求解所得到的刚性方程组，给出了收敛性和数值结果；其次，对标准的一维逆热传导方程给出了一种基于Fourier正则化方法，给出了理论证明及其误差估计，解决了文献中算法与理论误差估计的不相匹配的现象，该正则化方法不仅保留了测量数据的部分高频成份，且与文献中的算法具有同样的计算量和误差估计。

Based on the summarizing of the exiting algorithms of the texture mapping and antialiasing, this paper goes deeply into the basic theory and processing of texture mapping. The texture compress algorithm which can satisfy the need of the memory and time is given for the stochastic sampling in the inverse texture mapping; the geometry transformation in the texture mapping are discussed and the concepts of the unitary parametrization and the patch parametrization is proposed, some kinds of the unitary parametrization methods of parametric surfaces and the geometry transformation in the patch parametrization are also presented; in order to solve the aliasing in the parametrization of intermediate surface-sphere in two-part texture mapping, an area-preserving transformation is presented also; to improve the quality of the graph the aliasing in the texture mapping is analyzed from the aspect of signal process, and the modified summed-area tables algorithms is proposed; the distortion of the scanline conversion algorithms under perspective transformation is analyzed and the proper interpolation algorithm is presented so the aliasing problem of the polyhedral object after the unitary parametrization is efficiently solved.

本文在总结现有的纹理映射和反走样算法的基础上，深入研究了纹理映射的基本原理与过程，根据反向纹理映射对纹理随机采样的特殊要求，提出了一种可满足存储空间和寻址时间的纹理压缩存储算法；通过对纹理映射中几何变换理论的进一步深入探讨，提出了整体参数化和面片参数化的概念，并给出了参数化曲面的各种整体参数化方法和面片参数化的几何变换形式；针对现有的两步映射算法所存在的中间曲面—球面的参数化问题，提出一种等积映射方法，较好地实现了由非参数化面片拼接而成的多面体表面的纹理映射；从信号采样的角度深入分析了纹理映射中走样产生的原因，提出了改进的区域求和表算法，使图形生成的效果有所提高；还分析了一般的扫描线算法在透视变换中所存在的问题，提出了一种正确的插值算法，从而解决了整体参数化后多面体表面纹理映射的几何变形问题。

Because this arithmetic changes megillah into simpleness , it becomes possible to consider some fiber breaks at the same time in the crack core evolvement process.

该算法化繁为简，便于考虑裂纹扩展过程中多根纤维同时断裂这一因素。

In the former representation, if no restrictions are imposed on formulas, then there is no algorithm to realize the reasoning in the resulted knowledge base; in the latter representation, the reasoning in the ontology represented by the description logic is decidable, while in general, for mathematical knowledge described by the first order logic which contains the ontology represented by the description logic, there is no algorithm to realize its reasoning.

在前者的表示中，如果对公式不作任何限制，那么得到的知识库中的推理不是可算法化的；在后者的表示中，以描述逻辑描述的本体中的推理是可算法化的，而以本体为逻辑语言的一部分的一阶逻辑所表示的数学知识中的推理一般是不可算法化的。

Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality and mechanization, in Chapter 3, we firstly present the new generalized tanh function method, and apply it to construct the exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation.

第三章基于非线性发展方程求解代数化、算法化、机械化的指导思想，以吴方法和符号计算为工具，首先提出一种新的广义tanh函数方法，并将其应用于(2+1)-维Kadomtsev-Petviashvili方程的精确解构造。

From the 13th to the 16th century, Liber Abaci affected the development of arithmetic mathematical, the revolution in business Mathematics, and the mathematics education in Europe.

十三世纪意大利数学家斐波那契的《计算之书》(Liber Abaci，1202初版、1228再版)是中世纪晚期欧洲重要的数学著作，在13到16世纪对欧洲算法化数学发展、商业数学革命、以及数学教育变革等方面产生了广泛的影响。

This doing might be restructuring or reformatting it, extracting smaller bits of information from it, algorithmically modifying the content of the information, or performing calculations that depend on the textual information.

这个&做一些事情&可以包括重组文本结构，重排格式，提取更小的文本信息，算法化地自动处理文本的内容，或者根据文本信息做一些计算。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。