黑人兄弟上台 - 平方数列求和 - 组合数学 - xarchiver

黑人兄弟奥巴马当选美国总统，算是今日滴最大新闻了。此人长着一副给人好印象的面孔，拥有算作离奇的身世（甚至自爆青少年时期迷茫滴吸毒历史），演说鼓噪人心。貌似对国人来说，奥巴马当选算是众望所归了。这里有來自Economist的一个flash实时演示，很有趣。

今天看题碰到了一个平方数列求和（1^2 + 2^2 + 3^2 + ...），我依然清楚的记得公式。可兴致大发的想要证明一把此公式滴俺却以失败告终。google到了答案，有多种方式。其中最典型的两种就是——三角形旋转叠加（这方法还真是挺形象 ;-) 不过解释中的“60度”应该改成“120度”）；公式演算：

(n+1)^3 = n^3 + 3n^2 + 3n + 1  → (n+1)^3 - n^3 = 3n^2 + 3n + 1 然后叠加就OK了

还有一种提到了组合数学，看了半天没看明白 囧，觉得可能是某些式子写错了。不了了之。查了一把，觉得Brualdi教授的《Introductory Combinatorics》是本好书。可惜rapidshare上没找到4th edition的链接，仅有的一个3rd editon还是个死链接。emule上倒是可以搜索到，可惜这种资料用emule下载真是比白云天上飘还慢，完全没法和某些热门电影的无敌水上飘相比。倒是找到这么一本《Foundations of Combinatorics with Applications》，preface中有这么一段——

Advice to Students

This book does not assume any previous knowledge of combinatorics or discrete mathematics. Except for a few items which can easily be skipped over and some of the material on "generating functions"
in Part IV, calculus is not required. What is required is a certain level of ability or "sophistication" in dealing with mathematical concepts. The level of mathematical sophistication that is needed is about the same as that required in a solid beginning calculus course.

You may have noticed similarities and differences in how you think about various fields of mathematics such as algebra and geometry. In fact, you may have found some areas more interesting or more difficult than others partially because of the different thought patterns required.

The field of combinatorics will also require you to develop some new thought patterns. This can sometimes be a difficult and frustrating process. Here is where patience, mathematical sophistication and a willingness to ask "stupid questions" can all be helpful. Combinatorics differs as much from mathematics you are likely to have studied previously as algebra differs from geometry. Some people find this disorienting and others find it fascinating. The introductions to the parts and to the chapters can help you orient yourself as you learn about combinatorics. Don't skip them.

Because of the newness of much of combinatorics, a significant portion of the material in this text was only discovered in this generation. Some of the material is closely related to current research. In contrast, the other mathematics courses you have had so far probably contained little if anything that was not known in the Nineteenth Century.（这句话俺太喜欢了！ :D 譬如Calculus就是19世纪滴东西，而现在国内大学里，哪怕是专业课程和Calculus几乎毫不沾边的学生都得学习之，的确比较无奈。对于我们这种数学功底还不错的学生来说，C不算困难。特别是（俺个人觉得）其可谓是高中数学内容的一种思考方式上的延伸，不需要太多思维转换即可轻松接受。可那些数学功底不好的学生咋办？君不见某些人连挂几次依旧不得不补考？真不知道那些白痴curriculum制定者是怎么想问题的。为啥不改成选修课程？） Welcome to the frontiers!

顺便推荐一个gnome下的好东西，xarchiver，比gnome自带的压缩包管理工具好用，界面也很整洁。其他linux桌面下应该也没问题，毕竟是x打头的软件嘛。

ps. 刚才在emule上已经下载完《Introductory Combinatorics》全书了（速度比俺想象的快不少啊），是机械工业的影印版，扫描得不清晰。不过看起来印刷质量还不错，字体也漂亮。还是买书看吧。不过近来开始准备考研，肯定是不能买得看了。也不知两个月时间可否顺利搞定考研和4门专业课以及若干门公选课。真是愤恨自己前三年都没啥明确的努力目标，不然保研出国起码也得搞定一个了！吾悲愤啊！ :( 如果搞不定，明年3月份就准备和众童鞋一起找工作去咯！ :D