格子乘法、Napier grid 与熙圆指算 摘要：格子乘法原名叫 lattice multiplication、 是印度人十二世纪发明、阿拉伯首先加以传播的,十三世纪初由意大利数学家斐波那契将它与阿拉伯数字一起传入欧洲但未得到普及，十七世纪才被爱丁堡数学家以纳皮尔以Napier grid 发表并在西方被普遍接受。在美国因熙圆指型研究而遇到 Napier grid 这个名字被我们关注。同意百度将其叫做格子乘法的名字。 先摘一段维基上的陈述： Historically the grid calculation tweaked slightly was the basis of a method called lattice multiplication,which was the standard method of multiple-digit multiplication developed inmedieval Arabic and Hindu mathematics. Lattice multiplication was introducedinto Europe by Fibonacci at the start of thethirteenth century along with the so-called Arabic numerals themselves;although, like the numerals also, the ways he suggested to calculate with themwere initially slow to catch on. Napier's bones were a calculating helpintroduced by the Scot John Napier in 1617 to assist lattice methodcalculations. Napier grid 其实就是 lattice multiplication。该 方法应该是意大利数学家 斐波那契 十三世纪初 (1202) 引进阿拉伯 - 印度发明的数字符号时同时引进欧州的，但均未被欧洲普遍认知和接受。十七世界初，苏格兰愛丁堡数学家约翰 纳皮尔 (1550~1617) 臨死前 介绍的这个方法如今还是英国和苏格兰学校的 （照片引自维基） 标准方法 ( 可搜索百度 Napier grid)。 以下叙述，是我自己的理解和学习它的总结。我的做法省地方，全部例子和照片都出自本人。 优点 ：整齐，简便，只要会一位数乘法（九九表），行列对齐了，把结果写进去，再会 加法，按斜线把和数写下来。 它的格式以行与列为准，条理分明，便于捡查；错在那位数，一看便知，所以不容易出错。尤其在位数较大时，它反而省地方：全部计算都在以被乘数 数位+2 为宽度、乘数位数+2为高度所决定的那个方块里面。 具体方法与步骤 ： 1 ，被乘数由左向右写在顶部，每数占一格，乘数写在右侧，从上往下每数占一格，小数点不占格子，仅在两格之间的线上点一下。 2 ，每个格子右上角与左下角之间划一条对角线，各个格子的对角线互相连接着。 3 ，被乘数与乘数的各位之数相乘。不必考虑什么位，只管每个位上的一位数。也可不限先后，乘积 肯定 不会超过 81， 写进它们所在行与列对应的格子里，乘积若有十位数则写在格子斜线上方，乘积的个位数写在斜线下方 , 注意每个格子都必须都填上，没有填上肯定是遗漏了！写完后，仔细检查一遍。 4 ，从右下方开始，自右向左、自下而上地将同一条斜线下 / 上的数加起来。注意，都是一位数相加。不会心算的请用一双熙圆手计算或者帮助记住临时结果，将最后结果写在格子的下方或左侧，和数大于十的进位给左侧或上方，可以在左上那个位值上用点或数记一下。 5 ，最终答案是从左侧最上开始往下读到底部，然后续往右读到最右边。若有小数点，将被乘数小数点与乘数的小数点按行列的线到相交处，確定交叉点后再后沿着斜线走到底，那里就是答案的小数点。 先举个二位数乘法的例来说明： 74*69=5106 再举两个六位数的乘积，其结果是十二位数： 789654*425815=406246518010 这在计算器上，只能给出这样的结果： 计算器显然已经不管用了，我们再举一个可以用计算器验证的例子： 389*976548=379877172 再举个带小数点的题目： 98765.4*32.1=3170369.34 这个计算器也可以算出来。 只要结果超过十位，手机里的计算器就不管用了。请看下面这个带小数点的，真不如自巳算： 9 876.5*43210.6=426769490.90 而计算器则把 .90 给进位了。不如手算可靠。 我感觉，这个格子乘法（可参见百度），我们也别把纳皮尔这个名字加像美国人那样加在在前面了，因为发明人不是他。但是这个方法确实很好玩。 它把任何位数的乘法全部化简为一位数的乘法和加法。什么叫简约？这就是简约。 事实上，这比计算器可靠。而一位数的计算则最最重要。把这个搞熟练了，根本不要惧怕它。这个格子乘法对于克服对多位数的畏惧很有效。 由于熙圆手(Xi-Yuan hand)对于一位数计算全面有效，所以与它配套教孩子挺进多位数计算毫无困难。 ￣ LYT 20171130 于哈德森 li-yt@126.com
新发现，大脑的网格细胞，负责GPS - 全球定位系统。 Note that this information is purely for education purpose. as the link may be shifted later, the full article, well written, is supplemented for my own convenience for study, for the completion of my own record of literature library. Some interesting stuff: He was researching physiology and they were interested in the intersection of behavior and physiology. But, she said, they wouldn’t take no for an answer. “We sat there for hours. He really couldn’t get us out of his office,” Dr. May-Britt Moser said. “Both of us come from nonacademic families and nonacademic places,” Edvard said. “The places where we grew up, there was no one with any university education, no one to ask. There was no recipe on how to do these things.” “And how to act politely,” May-Britt interjected. “It was just a way to get to the point where we wanted to be. But seen now, when I know the way people normally do it,” he said, smiling at the memory of his younger self, “I’m quite impressed.” PROFILES IN SCIENCEA Sense of Where You Are Brian Cliff Olguin for The New York Times May-Britt Moser with Thelma, one of the lab rats used in the research she and her husband, Edvard I. Moser, are conducting. By JAMES GORMAN Published: April 29, 2013 FACEBOOK TWITTER GOOGLE+ SAVE E-MAIL SHARE PRINT SINGLE PAGE REPRINTS TRONDHEIM, Norway — In 1988, two determined psychology students sat in the office of an internationally renowned neuroscientist in Oslo and explained to him why they had to study with him. Science Times Podcast Two Norwegian scientists on what a rat brain knows about location; new research into how some traveling primates take digestive cues from their hosts; demystifying talk of a cure for H.I.V./AIDS. 9:42 Your Brain’s GPS play max volume 9:41 6:31 Eating Like a Local (Monkey) play max volume 6:30 9:09 Weakness in AIDS’s Armor play max volume 9:08 Related Grid Cells: ‘Crystals of the Brain’ (April 30, 2013) Connect With Us on Social Media @nytimesscience on Twitter. Science Reporters and Editors on Twitter Like the science desk on Facebook. Enlarge This Image Unfortunately, the researcher, Per Oskar Andersen, was hesitant, May-Britt Moser said as she and her husband, Edvard I. Moser, now themselves internationally recognized neuroscientists , recalled the conversation recently. He was researching physiology and they were interested in the intersection of behavior and physiology. But, she said, they wouldn’t take no for an answer. “We sat there for hours. He really couldn’t get us out of his office,” Dr. May-Britt Moser said. “Both of us come from nonacademic families and nonacademic places,” Edvard said. “The places where we grew up, there was no one with any university education, no one to ask. There was no recipe on how to do these things.” “And how to act politely,” May-Britt interjected. “It was just a way to get to the point where we wanted to be. But seen now, when I know the way people normally do it,” he said, smiling at the memory of his younger self, “I’m quite impressed.” So, apparently, was Dr. Andersen. In the end, he yielded to the Mosers’ combination of furious curiosity and unwavering determination and took them on as graduate students. They have impressed more than a few people since. In 2005, they and their colleagues reported the discovery of cells in rats’ brains that function as a kind of built-in navigation system that is at the very heart of how animals know where they are, where they are going and where they have been. They called them grid cells. “I admire their work tremendously,” said Eric Kandel , the Nobel laureate neuroscientist who heads the Kavli Institute for Brain Science at Columbia and who has followed the Mosers’ careers since they were graduate students. John O’Keefe of University College London, whose discovery in the 1970s of so-called place cells in the brain that register specific places, like the corner deli or grandma’s house, and who was one of the Mosers’ mentors, said that the discovery of the grid cells was “incredibly significant.” The workings of the grid cells show that in the brain “you are constantly creating a map of the outside world,” said Cori Bargmann , of Rockefeller University, who is one of the two leaders of a committee set up to plan the National Institutes of Health’s contribution to President Obama’s recently announced neuroscience initiative. Often, the workings of billions of neurons that produce our thoughts are opaque. But electrical recordings of signals emitted by grid cells show a map “with a framework and coordinates that are completely intuitive,” Dr. Bargmann said. And to find such a straightforward system is, in its own way, “just mind-boggling.” What is the brain doing being so mysteriously unmysterious? The implications of the discovery are both practical and profound. The cells have been proved to exist in primates, and scientists think they will be found in all mammals, including humans. The area in the brain that contains the grid cell navigation system is often damaged early in Alzheimer’s disease , and one of the frequent early symptoms of Alzheimer’s patients is that they get lost. The Mosers do not work on humans, but any clues to understanding how memory and cognitive ability are lost are important. On the most profound level, Dr. O’Keefe, the Mosers and others speculate that the way the brain records and remembers movement in space may be the basis of all memory. This idea resonates with the memory palaces of the Renaissance, imagined buildings that used spatial cues as memory aids. The technique dates to the ancient Greeks . In this regard, neuroscience may be catching up with intuition. A Welcome Ambush Edvard, 51, and May-Britt Moser, 50, now direct the Kavli Institute for Systems Neuroscience and the Centre for the Biology of Memory at the Norwegian University of Science and Technology here in Trondheim. They have a steady stream of findings coming from their lab, and a slew of awards, the latest of which, the Perl-U.N.C. Neuroscience Prize , they received April 16 at the University of North Carolina. But they did not grow up in a center of academic ferment or intellectual competition. They were born and raised on islands off the coast of Norway a couple of hundred miles north of Bergen, part of an area known as Norway’s Bible Belt. They went to the same high school, but didn’t really get to know each other until they met again at the University of Oslo in the 1980s. May-Britt, who grew up on a farm, remembers an environment in which drinking, card playing and dancing were all frowned upon. When she called home from Oslo announcing that she had been to a bar and had her first beer, her mother said, “And what’s next?” The Mosers married in 1985 while still undergraduates. By the time they had finished their doctorates, in 1995, they had two daughters, but they were ready to see the world, to train in laboratories outside Norway. And they did spend time in England, with Dr. O’Keefe, and in Scotland, with Richard Morris at the University of Edinburgh. But the Mosers’ travels were cut short when they were ambushed by a job offer too good to refuse, from the university in Trondheim, where they have been ever since. http://www.nytimes.com/2013/04/30/science/may-britt-and-edvard-moser-explore-the-brains-gps.html?hpw_r=0 1 2 NEXT PAGE » A version of this article appeared in print on April 30, 2013, on page D 1 of the New York edition with the headline: A Sense of Where You Are .
Wang, S., M. K. Cowles, et al. (2008). Grid computing of spatial statistics: using the TeraGrid forGi* analysis. Concurrency and Computation: Practice and Experience 20(14): 1697-1720. The massive quantities of geographic information that are collected by modern sensing technologies are difficult to use and understand without data reduction methods that summarize distributions and report salient trends. Statistical analyses, therefore, are increasingly being used to analyze large geographic data sets over a broad spectrum of spatial and temporal scales. Computational Grids coordinate the use of distributed computational resources to form a large virtual supercomputer that can be applied to solve computationally intensive problems in science, engineering, and commerce. This paper presents a solution to computing a spatial statistic, Gi*(d) using Grids. Our approach is based on a quadtree-based domain decomposition that uses task-scheduling algorithms based on GridShell and Condor. Computational experiments carried out on the TeraGrid were designed to evaluate the performance of solution processes. The Grid-based approach to computing values for Gi*(d) shows improved performance over the sequential algorithm while also solving larger problem sizes. The solution demonstrated not only advances knowledge about the application of the Grid in spatial statistics applications but also provides insights into the design of Grid middleware for other computationally intensive applications. Copyright 2008 John Wiley Sons, Ltd.
标签: Grid
