欢迎加入中国茉莉花行动部落

我们来自同一个家园,那里毒草丛生。
我们来自同一个部落,那里毒蛇横行。
我们播种茉莉,为了呼吸自由的芳香。
我们移植鲜花,为了拥抱春天的曙光。

Wednesday, May 10, 2023

徐水良常年对我造谣诽谤,我将聘律师起诉徐水良


徐水良在过去的十多年里,几乎每天都要在各种社交媒体上张贴对我恶意造谣诽谤的文章,还通过电子邮件给我发送十分恶毒的造谣诽谤邮件。

我在过去的十多年里,从来就不回应徐水良的这类诽谤邮件。徐水良就不断宣称我不是他的对手,不敢正面迎战,还非要我承认他是我的猪队友。

近日,我正忙于发表10篇数学论文。如果我的学术论文被发表,我将聘请律师起诉徐水良对我恶意造谣诽谤,给我造成巨大的精神伤害。

徐水良常年对我公开造谣诽谤,我从来都不予反驳。可这里的很多人就信以为真。不妨举两个大家熟悉的例子。

一个例子是徐水良造谣说我曾经说过“徐水良是我的猪队友。”

我从来就没说过这句话。

我曾经说过:“宁要狼一样的对手,不要猪一样的队友。”这是我说过的。

大河就几次提及我的这句“语录”。

胡安宁就利用我的这个“语录”,将徐水良硬说成是我的猪队友。可谁都知道,我从来就不曾将徐水良当成是我的什么战友。

徐水良从此就反复发文,说我不敢承认他徐水良就是我的猪队友。我从来就不同徐水良纠缠这种问题。

最近的一个例子就是徐水良在独评几次发帖,说赛昆是共特,赛昆将我拉回独评。

我同赛昆没有任何私下联系,赛昆也从来不曾跟我有任何交往,不曾发过私信、电邮,更不曾打过电话。可徐水良就是四处造谣说赛昆将我拉进《独立评论》,说我被赛昆利用来《独立评论》对徐水良进行打压。《独立评论》的很多网友还真以为我同赛昆是上下线家生共呢。。

这就是徐水良的一贯造谣手段。

刘刚
2023年5月10日

附录:徐水良对我造谣诽谤文章链接和截屏,将保留这些截屏作为起诉徐水良的证据材料。

徐水良自2011年茉莉花革命后,几乎是每天都要群发大量邮件,以及在推特、脸书、博讯等媒体上发布大量对我造谣诽谤的文章。

仅仅是在2023年5月9日,徐水良在推特、脸书、博讯等社交媒体或徐水良个人的博客上就发布数十篇对我进行恶意造谣诽谤的文章,下面是其中的部分文章的链接和截屏。

日常评论(230509)继续评川粉、新自由主义、公有私有、逆种族主义、赛昆刘刚家生共、反人类圣经和神棍、郭台铭



笑笑高学历特线科盲刘刚
再笑造谣撒谎的刘刚附带蠢人顾晓军
刘刚改换主子







Tuesday, May 2, 2023

追忆当年教科书


首先说明,我的这篇短文仅仅是对当年大学生活的感慨,同徐水良没啥关系。

刚刚看到徐水良向大家推荐统计物理学教科书。

我无意继续同徐水良讨论这些问题。只是徐水良推荐的教科书不禁让我想起当年在大学时读过的几本《统计物理学》教科书,令我感慨万千。

相信在77年上大学的人都会有同样的经历。

我们的统计物理或称统计力学课用的是王竹溪编的一本薄薄的教科书,电动力学是曹昌琪的《电动力学》,还有一本很有名的教授的《电动力学》,电磁学是赵凯华(?)编辑的《电磁学》,褚圣麟的《原子物理学》 等等等等,相信当年各个大学都是通用的这几本教材,对这些教材,我们当年可都是能倒背如流了,现在全忘光了。如果曹昌琪不曾经给我当了三年的导师,我连他的名字也会忘得精光。

这些中文教科书都是让人兴味索然,我们就设法去买一些美国的通用教材。最著名的是伯克利的四大力学教材,有《波动学》(绿色封面)、《电动力学》、《统计物理》(蓝色封面)、《量子力学》(紫铜封面)等等。其中的《统计物理学》是我最喜欢读的,那简直就是一种享受,如同是看动画小人书或漫画一样。我至今还记得其中讲述不可逆过程的漫画:

一个酷似卓别林的小丑,提着拐杖,对着一个小木房丢了一颗原子弹。房子炸飞啦。小丑随后又对炸飞的房子废墟又丢了一个原子弹,结果是所有废墟碎片都按照被炸飞的轨迹逆向飞行,被炸飞的房子瞬间又被炸回来啦。

物理学和力学在当年好象区分不大。力学通常是指那些包含有微分方程式的物理学科就成为是力学。比如,电动力学,因为有麦克斯韦方程,就不叫做电动物理学了。电磁学的很多内容就是初级的电动力学。

电磁学的另一个升级版叫作电子线路,因为其中没有用到微分方程,就不敢自称为电磁力学了。

量子力学,因为有薛定谔方程,也就被称为是力学了。

统计物理学,在初级阶段叫做统计物理,高级的统计物理就称为是统计力学了。

光学,没什么微分方程,就一直没有升级为光力学。

热学,初级阶段叫做分子运动学或热物理,高级阶段就成为热力学了。

原子物理学一升级就成为量子力学了。

原子核物理学升级后就成为量子场论。

普通物理,一升级就成理论力学了。

还有,流体力学,弹性力学,材料力学,空气动力学,塑性力学, 等等等等,都要不叫作力学的初级课程。

但凡是没有用到微分方程的学科,都不敢称其为力学。

我这段话是经验之谈,开玩笑。

虽然是这样开玩笑,我们力学专业的人都认为力学没什么技术含量,认为力学系上古典学科,而物理系则是属于现代科学。更因为古典力学都让牛顿给糟蹋完了,剩下的任务就是完成牛顿还没来得及完成的修修补补工作,没什么机会得诺贝尔奖了,大家都争先恐后地转行学物理。

还记得有另一本英文版的著名的统计物理学,我忘记作者名字了,其中的题目特别好。我们科大图书馆只有一本。我们每个人都抢着去借那本书。我曾经借到过几次。就是现在,我还经常做梦我去图书馆去排队借那本书。

还有一本英文版统计物理学,好象是一位姓黄的美国华人编著的。也忘记名字了。后来到了单位工作后,我就将单位图书馆的那本黄某的统计物理学给包下来了。

前几天还记得有人在此提到铁摩欣科。我们在78年上《材料力学》是就是使用铁摩欣科的英文版《材料力学》,当时给我们上课的教授叫沈志荣,是美国的正宗博士,是当年同钱学森同一条轮船从美国回到祖国的,也是我毕业论文的导师。钱学森一直都是我们系的系主任。上沈教授的课,让我记住了一些相关的英文单词,我们学生都会牢记他经常使用的“homogeneous”一词。相信我们班上的学生都会不忘他传授的这个单词。我的毕业论文题目是“近海石油平台的震动模型分析”,其实就是帮助他查阅和归纳这方面的英文资料。沈志荣教授在那期间几乎每天下午都要去体育教研室,就是同宁柏下围棋。

当时科大的条件比较优越,我们所用的教科书基本上是免费发放,就连铁摩欣科的那本厚厚的英文硬皮书也是免费发放。如果不是免费发放的教科书,我们很多人就不买了。后来到了北大,才知道买书贵啊。

吉米多维奇的《高等数学解析题集》就更是我们那个年代尽人皆知几乎人手一本的参考书了。

最令人难忘的是上了方励之的《天体物理学和相对论》、阮图南的“量子场论”,我的物理学课都是到现代物理系或是物理系选修的。我有一个学期选了十几门课,既包括力学系的必修课,又同时去选修物理系和近代物理系的物理课程。我们那时是可以逃课的,参加期中和期末考试通过就给记学分。

凡是方励之授课,没上过的,我大多都会去听。因为那跟听评书一样令人开怀。方励之的那本科普读物《宇宙的創生》很有伯克利《统计物理学》的遗风,其中有大量的漫画。一方面在讲述宇宙的创生是始于和谐、美、平等。其中特别引用圆桌会议来比喻圆桌上平等和美的基本的要求,宇宙中就没有中心,处处是中心,圆形、椭圆形、球形是基本的存在形式,否则就会产生战争,产生灾难。而人类社会也是要求平等,最终的均衡是要逐渐地趋向于人人平等,没中心,没有国王,大家都是占有圆桌的平等的一个角,而不存在坐难坐北的席位,更不存在什么主席。

我们系的数学课要求比较高,各种数学课一直都是由徐澄波教授担任主教授,大概交了我们三年有余。徐澄波教授的授课令人难以忘怀,是北大数力系毕业,讲课的每一句话都配有十分有力度的动作。我们都不会忘记他讲授“Epsilon 非常小,小到不能再小”,那动作就如同是在反复地按死一个老共匪。徐澄波教授的夫人是浙大教授,后来,徐澄波教授也调到了浙大。我还同几位同学专程去他在浙大的家中拜访过他。

只有概率统计是同物理系一道上的大课。给我们上概率统计的老师也令人难以忘怀,最难忘记的是他几乎每堂课都会说几次“这个问题同学们并不百生”,引起一阵阵哄堂大笑,因为我们都知道他要说的是“陌生”,而他就一直都奇怪我们为何要笑。

刚刚想起那本蓝色硬皮的英文版《统计物理学》的作者叫黄克森,Kesen Huang.

徐水良 牛津大学研究生教材《统计物理学中的蒙特卡罗方法》等大学教材 2023-04-26 12:18:58 [点击:257]
赛昆、刘刚等等对统计学几乎一窍不通,不仅他们的统计计算很可笑,听不懂樊教授关于蒙特卡罗方法的介绍,赛昆还坚持说胡话疯话,非常可笑地一再一再说蒙特卡罗方法不是统计方法,不属于统计学,完全不知道自己是在一次又一次出丑。

参见:https://twishort.com/7Dsoc

https://facebook.com/xushuiliang/posts/pfbid02QebkdJ72nGX38cTZBfiuwsz776UEuTGdqT39qCBoyWS8hSgNKeS9qzdP79woj3u2l

我的统计学是40年前在监狱自学的,忘记用的是哪个大学的教科书。在与赛昆等家生共辩论蒙特卡罗法是不是统计方法等时候,发觉不仅赛昆、刘刚等对统计学几乎一窍不通,而且其他很多朋友似乎也不懂统计学。于是上网搜索相关大学教科书,看看有什么教科书,可供我们学习。这是一部分结果:

比较权威的有:《牛津大学研究生教材:统计物理学中的蒙特卡罗方法》等教科书,据说是经典的。纸本书籍,亚马逊网站和书店有售:
https://www.amazon.cn/dp/B00CE3TLB0

刘刚 浙大力学属于数力系能说明什么? 2023-04-30 14:26:17  [点击:114]
中国的各个系所包括的分支专业并不是什么科学问题。

中国文革前各个大学大多都有数力系。这仅仅是为了师资资源管理的方便,并不能据此来论证力学就是数学的一个分支。

将力学划分到哪个系,不是科学问题,而仅仅是管理问题。在那个年代,力学教师都需要很强的数学,而物理系规模通常是过于庞大,为了均衡,就将力学划分到数学系。中国高中里将力学划分到物理课中。力学究竟应该是划分到物理中还是划分到数学中,中国是根据师资和教学管理的需要来划分的。

中国在文革前很少有计算机系。计算机专业大多都划分到无线电系或者是数学系。这也是根据师资管理的需要。只是后来需要招募大量的计算机专业的学生后,才划分出计算机系。

中国将计算机或力学划分到哪个系不能作为证据来证明计算机或力学是属于哪种学科,这就如同将马克思主义、习近平思想划分到哲学系,并不能证明习近平思想就成了哲学。中国的数学系学生都要学习党史,这并不能证明中共党史就是数学的一个分支!

徐水良的辩论术真是风××。也许徐水良想说的就是他曾经是浙大非数力系的某个系的肄业生。

另外,Monte Carlo 方法也不是什么神秘的方法。可以简单地概括为:对随机事件进行大量随机抽样再对某个随机变量求平均值。

在计算机领域,通常将这种方法也称为Randomization Method。有很多计算机领域的专家会将Randomization Method列为自己的研究方向,但很少有人会将Monte Carlo方法作为自己的研究方向,就如同不会将“投硬币”、“掷骰子”列为自己的主攻方向一样。如果有人自称是Monte Carlo Simulation方面的专家,那是指他会使用Distributed computation 或Cloud Computation进行计算机计算,应该是指如何进行大规模计算,是计算机领域的专家,而不是统计学领域的专家。

Monte Carlo Simulation方法是属于一种“傻瓜方法”,是找不到更好算法时的无奈选择。这个方法本身不是什么难题,难的是如何实现这种算法中所需要的大量计算,那就是属于计算机领域的问题了。

徐水良在讨论Monte Carlo方法是属于数学领域还是统计学领域,这就如同是在讨论“掷骰子”是属于数学领域还是统计学领域一样地无知。

有很多人会在自己的Resume上将Monte Carlo Simuluation列为一个skill。当面试的人看到这个Skill时,这仅仅是表明他的计算机skill比较强,会使用Monte Carlo方法进行模拟计算,其中包括的内容会是Distributed Computation, Grids Computation, 而丝毫不包括什么统计学和其它的什么数学。其数学知识很可能仅仅是求平均值而已。

Monday, May 1, 2023

三年前我对各主要国家新冠死亡人数的预估


三年前,也就是在2020年5月5日,我曾经发文估算美国的新冠预期死亡人数。见下述文章链接。

用数学模型估算各国预期新冠死亡数,美国将高达24万!

A Simple Model to Estimate the Expected Death Toll

按照我当时给出的估算,美国新冠死亡每年24万,三年后105万 

当时美国正在感染新冠的总人数是1,237,633人,当日死亡人数是2350。我根据当日的数据,美国当时感染新冠的人中将会有24万人死亡。从而预估新冠高峰期每年大约死亡24万。

很多人都说我的估算太夸张了,是大大高估了新冠死亡人数。

美国当时的总死亡人数是72271,按照我的估算,今日的死亡总人数应该是105万。

而现今美国的新冠死亡总人数是106万!同我的估算没什么太大差距嘛。

上图是我在2020年5月5日文章的部分截屏。那是预估半年后的死亡总数。如果将这些数字乘以2,可以看作是一年的新冠死亡人数,将上述数字乘以7,就可以作为到达2023年5月的新冠死亡人数。而按照我当时估算,全球新冠病人的死亡概率会高达21%,但美国这几个疫情泛滥大国的新冠死亡率是在10%左右。那么,这些数字还应该被除2。由此估算出的到达2023年5月的各国新冠死亡人数见下表。目前的新冠死亡人数同我的预估基本上相当。

使用当时估算的全球新冠病人死亡概率21%来估算,上图是到2023年5月各主要国家因新冠而死亡的预期人数。

当时美国及各个欧洲国家的新冠病人死亡率大约是10%,这些国家到达2023年的预期死亡人数应该是上表中的数据再除以2,如下表所示。


这个预估中对全球的新冠死亡人数严重偏低。那主要是因为很多国家尚未开始疫情,还有很多落后国家还来不及进行核算检测,也无法及时搜集数据。所以,当时的我对全球新冠病人的确诊数是严重偏低,我根据这种偏低的数目进行新冠死亡概率的预估值也就自然会严重偏低。

简述三年前我用Monte Carlo方法估算新冠病人的死亡概率

就因为我修改两个错别字,徐水良和杨巍就对我辱骂三年


翻看了一下我在3年前发的网文,以及徐水良、杨巍对我反复纠缠的旧帖,见下面的截屏和链接:





就从这些标题来看,徐水良和杨巍对我进行纠缠的就是我将原本打入的公式:

“确诊总人数 = 死亡总人数 + 康复总人数 + 死亡总人数”

又修改成为:

“确诊总人数 = 死亡总人数 + 康复总人数 + 病中总人数”

很明显,那是我偷懒使用Copy-Paste功能,而忘记了将“死亡”二字修正为“病中”二字。



徐水良分明知道我的字母公式是我要给出的公式,而文字公式则是错别字。

随后,徐水良和杨巍就连续不断地攻击我修改他们自己都能看出来的两个错别字(将“死亡”改正为“病中”)是蓄意对他们进行构陷。


并据此在过去的几年里连番不断地攻击我“人品恶劣”、“投共”、“共特”、“家生共”、“白痴”,等等等等。

从上面的截屏中可以看到,我根本就不曾对他们二人有什么诬陷伪造。我该文的

始发时间:2020-05-06 05:13:45
最后编辑时间: 2020-05-06 23:03:21

杨巍最早指出我这个错别字错误是在:2020-05-06 15:29:38

我修改过以后,就不曾针对这个问题回应他们二人。可他们就能凭空造谣说我诬陷他们伪造我的公式了。

就因为我在文章中曾经将“病中”二字错误地打成了“死亡”二字,徐水良和杨巍就连番累牍地对我攻击、批判、辱骂,这俩人是否是太闲了啊。

并且谁都可以看到,我对他们的胡搅蛮缠都是极力忍让,无意同他们深入讨论。

刘刚
2023年5月1日

徐水良说我不懂统计学,我可以承认徐水良最懂统计学


徐水良说我不懂统计学,我也承认我不懂统计学,我还可以承认徐水良最懂统计学。

我的那篇文章在一开头就声明那是任何初中生都能做出的算术应用题。我不过是用初中生都会的概率统计来估算美国的预期死亡率。

徐水良根本就没看懂我的那篇文章中的数字分析。我是根据已有的死亡人数和确诊人数来估算感染新冠的死亡概率,我也声明这个死亡概率也会随着时间发生变化,但变化不会太大。再根据这个死亡概率来乘上已经感染的总人数,以此来估算这些感染病人中将会死亡的人数。再考虑到感染新冠的人在半年期间里不是康复就会死亡,仅此就能预估美国的新冠死亡人数到年底至少是24万。

我没说疫情感染人数是线性分布。我强调新冠高峰期的年死亡人数至少是24万人。随后是假设24万是疫情期间平均年死亡人数,再假设新冠疫情从开始到结束的时间跨度是三年,再将24万乘以3,加上已经死亡的7万多人,就得到预估的总死亡人数将是105万。

我有说过疫情感染人数线性均匀分布吗?我有说过不是正态分布吗?难道正态分布就不能求平均值吗?就不能算出总死亡人数吗?

我仅仅是给出的大致的估算。

我是不懂统计学。可我列出了我曾经读过和学过的一系列教科书,包括:
概率统计分析
统计物理
热力学
流体力学
分子物理学
数值积分
Numerical Computation

等等,至少是翻烂了伯克利的《统计物理学》、Keshen Huang 的英文版《统计力学》,这些课程都是使用统计方法的科学,还指出我在

花旗银行
摩根斯坦利
Morgan Chase
Bear Stern

等等华尔街公司工作期间大量使用Monte Carlo方法计算IR Curve、Credit Curve、各种金融产品的Risks(Greeks),这些计算都要使用Distributed Computation、Grids Computation、Binomial Tree、Trinomial Tree进行Monte Carlo Simulation。

我在贝尔试验室工作期间,同几位同事一道开发了名为Spider的光纤通讯优化设计软件,其中的很多程序也要使用Monte Carlosimulation和Randomization方法进行选择最佳路径和进行Routing Optimization.我甚至暗示这些计算要使用Stochaotics Theory和Brownian Motion theory。我在NYU学习了二年的Finance Engineering高等金融班,其中大多是学习Randomization 、Numerical Computation, Monte Carlo Simulation的计算机计算方法求解Black-Scholes微分方程等等,我在哥大攻读计算机专业期间,就是每年给“Numerical Computation”的计算机课程担任助教。我列出了几篇我在AI和Optimization方面的学术论文,这当然都不属于是统计学,但都是用统计学作为重要工具之一。

我不愿向徐水良反复提及这些,因为徐水良根本就不懂其中的任何一门学科,甚至就没听说过上述学术方面的名词,没必要劳烦他上网去查阅这些名词,再让他宣称他是这些领域的头号专家。

徐水良宣称他最懂统计学,证据就是他曾经自学了概率论,还是在40年前,还是在中国的监狱里!还根本不能说出是哪所大学的教科书!

谁相信中国的70年代在监狱里能够搞到大学的教科书啊?在文革期间,市面上卖过哪所大学的教科书啊?行,你徐水良自学过概率论,我们都无人质疑你真的自学了概率论,承认你自学过,也认可你就是全球最顶级的概率统计专家。可面对我们这些在中国的名牌大学里以及在美国的常春藤大学里不是通过自学拿到学位的人,面对这些都反反复复地真正修练过概率统计、Stochastic Theory、Brownian Motion、热力学、统计物理学、统计力学等等大量使用统计方法的正牌大学生的人,徐水良为何就非要否认我们会对统计方法也略知一二呢?

徐水良还向大家推荐一部牛津的 《统计物理学中的蒙特卡罗方法》,还是上网搜索的!如果你徐水良真正阅读过这本书,我们还认可你的推荐。可徐水良这个自称在40年前在监狱里自学过概率统计的人,以及没有做过任何Monte Carlo数值计算的人,没有读过任何统计物理教科书的人,有什么资格向我们这些将伯克利《统计物理学》都翻烂的人推荐什么统计物理学啊?有什么资格向我们这些在美国华尔街进行过多年Monte Carlo 数值计算的人推荐什么“Monte Carlo方法”或统计方法啊?

徐水良就是不知道何为羞耻,就是不知道山外有山。

徐水良在过去的三年里,针对我的那一篇估算新冠死亡人数的文章反复追骂,我都不予理会,因为他根本就不理解这些方法,看不懂我的文章中的数字分析。我没必要同一个半文盲来讨论一些连初中生都能懂得的算术应用题问题,更没必要去同这些地地道道的科盲去讨论什么统计方法和Monte Carlo Simulation。

可是,这个徐水良就是得寸进尺,不依不饶。我不理会他,他居然据此将我反复说成是共匪、共特、自干五、投共、白痴。更是有人将我们说成是为了一些鸡毛蒜皮炒成一地鸡毛。

我不过是让大家看到,在过去的三年里是徐水良就我的一篇小学生算术题文章来反复追骂,而我就不理会徐水良和杨巍的这些反复追骂。是他们自己同他们自己在过去的三年里就一个鸡毛蒜皮炒成了一地鸡毛!

-----------
杨巍   当年“批判”你的是你一个非常明显的低级错误,然后你耍无赖 2023-05-01 14:22:09  [点击:28]
文章偷偷改了。

没料到你一直到今天还在想翻本,故未保留当时的文字。你要翻盘就该贴出当年的文字,让网友明白来龙去脉,而不是像今天那样再次耍赖,贴出一些与当年完全不同的东东。我没兴趣去弄懂你现在那些乱七八糟的玩意,只是对你那无赖人品有进一步的认识。
-----------
刘刚   你根本就没看懂我的文章中的公式,你批判我什么? 改贴 2023-05-01 15:17:08  [点击:8]
我将我在2020年5月5日的文章全文发出来了,徐水良也将你批判和辱骂我的文字全文发出来了,你还要什么原文?原文都在我的文集里,大家都会去阅读。

你辱骂我的原文之一是:

作者: 杨巍 "我一看到这个刘刚公式,就觉得好笑。" 2020-05-08 09:52:08 [点击:79]
我一看到这个刘刚公式:“确诊总人数 = 死亡总人数 + 康复总人数 + 死亡总人数”就觉得好笑,并断定“看得懂”的人都有病。

本来想等几天,看看除了老徐之外,还有谁会来说“看懂”,不料刘刚立即偷偷把刘刚公式改了,却又不说明,立即理直气壮地贴出来问老徐:

确诊总人数 = 死亡总人数 + 康复总人数 + 病中总人数
这个公式,完全错误。

-----

你是揪住我曾经写下:

“确诊总人数 = 死亡总人数 + 康复总人数 + 死亡总人数”

我不记得我是否写下那样的文字,如果真的写过,那也是人人都能看出来的Typo。

你宣称我将上述公式修改成了:
确诊总人数 = 死亡总人数 + 康复总人数 + 病中总人数

然后就宣称我的这个公式完全错误。

你不妨指出来这个公式错误在哪里?

你进而宣称我修改错别字是耍无赖,是构陷徐水良。有你这样耍无赖的么?

而徐水良根本就不是纠缠我的这个公式。

你和徐水良相互配合对我进行辱骂、批判,而你根本就不知道徐水良在批判我什么,你不仅不懂我的文章,甚至根本就不懂徐水良说的是什么。

你这不是胡搅蛮缠又是什么?


我只是大致看了一下他的标题,但还是看了下杨巍在三年前对我批判的文章,那还是近日才看了一下他究竟是在批判我什么。

我无须证明我会什么,没必要证明我懂什么。我懂不懂统计学、会不会Monte Carlo方法,都是无关紧要,即便是完全不懂,也没什么关系。我没必要向徐水良证明什么。

只有那些无知的科盲才是急急地向人们证明他不是科盲,不是文盲,向人们证明他在各行各业都是顶级专家,而给出来的证据不过就是他上网去搜索了一些相关名词,最多就是胡说他曾经在40年前的监狱里曾经自学过某个学科。另一个手段就是用胡搅蛮缠的方式证明其他人根本不懂他胡乱定义的某个概念、某个学术名词的归属问题。

---------
刘刚
2023年5月1日

下面是徐水良辱骂我的部分文章链接:


徐水良   是说你不懂统计学,用直线均匀分布统计死亡率。包括现在 2023-05-01 13:19:58  [点击:17]
这个帖子,表示你仍然完全不懂统计学,以为是死亡率是常数,一年24万,三年就乘3,24X3;五年就乘以5,一百年就是2400万。你那是均匀分布,永远不变,不适合流行病统计。流行病是不断变化的东西,连人类出生率都在不断变化,何况流行病。

实际上,任何流行病感染分布,都不是直线,而是正态分布,偏态分布或其他曲线分布。不是你到处套用直线线性均匀分布就能解决的。

你和赛昆不懂统计学,却非要在统计领域出丑。非常不智。赛昆和家生共们反正也不要脸皮,永远耍无赖胡搅蛮缠,不怕出丑。只要完成任务就行。难道你北大学生就不怕出丑?

徐水良 牛津大学研究生教材《统计物理学中的蒙特卡罗方法》等大学教材 2023-04-26 12:18:58 [点击:257]
赛昆、刘刚等等对统计学几乎一窍不通,不仅他们的统计计算很可笑,听不懂樊教授关于蒙特卡罗方法的介绍,赛昆还坚持说胡话疯话,非常可笑地一再一再说蒙特卡罗方法不是统计方法,不属于统计学,完全不知道自己是在一次又一次出丑。

参见:https://twishort.com/7Dsoc

https://facebook.com/xushuiliang/posts/pfbid02QebkdJ72nGX38cTZBfiuwsz776UEuTGdqT39qCBoyWS8hSgNKeS9qzdP79woj3u2l

我的统计学是40年前在监狱自学的,忘记用的是哪个大学的教科书。在与赛昆等家生共辩论蒙特卡罗法是不是统计方法等时候,发觉不仅赛昆、刘刚等对统计学几乎一窍不通,而且其他很多朋友似乎也不懂统计学。于是上网搜索相关大学教科书,看看有什么教科书,可供我们学习。这是一部分结果:

比较权威的有:《牛津大学研究生教材:统计物理学中的蒙特卡罗方法》等教科书,据说是经典的。纸本书籍,亚马逊网站和书店有售:
https://www.amazon.cn/dp/B00CE3TLB0

三年前我用数学模型预估美国新冠死亡每年24万,今日应该是105万!

本文网址:http://jasmine-action.blogspot.com/2023/05/24105.html

三年前,也就是在2020年5月5日,我曾经发文估算美国的新冠预期死亡人数。见下述文章链接。

用数学模型估算各国预期新冠死亡数,美国将高达24万!

A Simple Model to Estimate the Expected Death Toll

按照我当时给出的估算,美国新冠死亡每年24万,三年后105万 

当时美国正在感染新冠的总人数是1,237,633人,当日死亡人数是2350。我根据当日的数据,美国当时感染新冠的人中将会有24万人死亡。从而预估新冠高峰期每年大约死亡24万。

很多人都说我的估算太夸张了,是大大高估了新冠死亡人数。

美国当时的总死亡人数是72271,按照我的估算,今日的死亡总人数应该是105万。

而现今美国的新冠死亡总人数是106万!同我的估算没什么太大差距嘛。



上图是我在2020年5月5日文章的部分截屏。那是预估半年后的死亡总数。如果将这些数字当作是一年的新冠死亡人数,将上述数字乘以4,就可以作为到达2023年底的新冠死亡人数,见下表。目前的新冠死亡人数同我的预估基本上相当。




简述三年前我用Monte Carlo方法估算新冠病人的死亡概率

http://duping.net/XHC/show.php?bbs=11&post=1462551


下面我给出的这个小学算术公式,应该是属于小学生的应用算术题。可千万不要将这个问题上升为统计学疑难问题。

大约是在三年前,我根据上述网页发布的一些统计数据,给出用Monte Carlo方法计算新冠死亡概率的公式,我当时没有使用Monte Carlo这个名称,就是防范徐水良这种人说我卖弄学问。

当时杨巍和徐水良对我进行大批判,我认为这都是些鸡毛蒜皮之事,没必要跟他们讨论这种问题。现在才发现,他们没有理解我的原文中所要表达的意思。现在我不妨再补充解释一下我当时所要表达的意思。

首先,我发现这个网页的统计数据存在一些问题。按理来说,表中列出的总确诊人数应该有如下关系:

总确诊人数 = 新冠死亡总人数 + 正在康复中的总人数 + 已经康复的总人数

但是,很多国家给出的这些数据并不满足上述条件。

上面的截图是昨天的统计数据,美国的统计数据满足上述条件,但全球的总确诊人数却仍然有6百万左右的误差。这是现在的统计数据,在三年前,这个数据的差别还是很大的。随着时间的推移,这个误差在逐渐减小。

所以,我只是指出这个统计中应该有这样的关系成立:

总确诊人数 = 新冠死亡总人数 + 正在康复中的总人数 + 已经康复的总人数

这并不是我提出的数学模型。这个公式只是表明这其中的一个数据不是独立的,是可以根据另外的三个数据推导出来的。

徐水良和杨巍等人就开始对我指出的这个等式进行全面地大批判,直到今天又是掀起一个高潮。我都是不予理会。

另一个问题是,我提出可以用如下方法估算出新冠病毒感染病毒后的死亡概率:

先估算出当日的样本数:

昨天的当日检测人数 = 昨天的报告的总检测人数 - 前天报告的总检测人数

再估算出总人口中的感染病毒的总人数应该为:(假设检测到提交报告结果的时差为一天)

(今天的确诊总人数/昨天的检测总人数)× 总人口数

那么,感染新冠病毒的人的死亡率就可以估算为:

感染新冠死亡概率 = 新冠总死亡人数 /((今天的当日确诊人数/昨天的当日检测人数)× 总人口数)

感染新冠死亡概率 = 新冠总死亡人数 × 昨天的当日检测人数 / 今天的当日确诊人数 / 总人口数

感染新冠死亡概率 = 新冠总死亡人数 × (昨天的报告的总检测人数 - 前天报告的总检测人数) / 今天的当日确诊人数 / 总人口数

这个方法实际上也就是Monte Carlo 算法的数学部分。

在那个网页中,确实也列出了各国的总检测人数,但那其中有很多重复检测,作为随机样本的误差率就比较大。

按照上述公式,输入上述网页给出的数据,就能大致估算各个国家的感染新冠病人的死亡概率。

当然,一定会有更好的方法估算出新冠感染后的死亡概率。但是,就上述网页给出的一些统计数据,大概也只能这样估算死亡概率了。

我所要估算的是感染新冠后的死亡概率,是一种条件概率,不同于新冠病毒造成的总人口的死亡率!我说的是感染新冠的人的死亡概率,而后者说的是不管什么人在新冠大流行期间会通过感染新冠而死亡的概率。

我上述文章发布后,就遭到了徐水良、杨巍等人的连番累牍的攻击、辱骂,直到今天,又发起了一次大批判高潮。

徐水良据此就批判了我三年,说我不懂装懂,不懂得什么是概率统计。

相信杨巍和徐水良还是无法理解上述问题。


---------

刘刚   
2020年5月1日

关于新冠问题的统计数据  2023-04-30 23:27:48 



当年新冠隔离期间,就每天都查看下述网页发布的各国的各种新冠数据,这个网页给出了各国新冠的当日确诊数、死亡数等等统计数据,我就是将这个网页上的数据截屏一部分,也没作什么太多的计算,不过就是一些加减乘除而已,随便发表了一些评论。我那不过是因为疫情隔离闲得无聊,就随便转发一下疫情统计数据,解闷而已。

居然就被徐水良、杨巍等人给上升到“刘刚模型”、“刘刚公式”等等大帽子。我当时就觉得好笑得很,还真帮我解闷了,没跟他们就这些无聊问题进行什么深入讨论。

现在将这些问题又翻出来,好象是红卫兵找到了我在30年代当过叛徒的证据一样来对我进行大批判。我将这些讨论就当成是帮我解闷啦。

再次感谢老徐、杨巍在纽约大隔离期间伴我度过无聊时光。其实,他们的长篇大论我都没有细看,至今都懒得看。

刚才才看到杨巍的这个帖文,说我要老美照抄老共的隔离方式。

我记得我当年只是发文想说明美国的这种软隔离实际上是等于没隔离。如果认定这种病毒是通过空气传染的,是必须要隔离,那就需要采取严格隔离,而不是象当时的纽约那样,即强迫人在家隔离,在同时又允许人们上街买菜遛狗,那种隔离就不如不隔离,应该允许人们继续上班工作,或者就是长痛不如短痛,采取更严格的隔离,要给被隔离的人派饭、送日用品。


------------------

在2020年5月,还没人敢说美国最终会死亡超过24万人,我是说即便美国不再感染新冠,也会在2020年年底因新冠死亡至少24万人。这就引起徐水良等人的大肆批判和辱骂。


徐水良   在武肺问题上批评刘刚投共的几篇文章(可惜没有把统计问题汇编进去)

必须高度警惕土共及其情报机构一种可能的阴谋

徐水良 2020-05-06日

徐水良等人至今都没有去核实我的预期死亡数字,就是跟我胡搅蛮缠3年。

徐水良何必不去反驳一下我的预期死亡数字啊?

刘刚
2023年5月1日

Friday, March 10, 2023

诚请数学高手对我的这篇学术论文批评指正



A Mathematic Model for Supply-Demand Equilibrium and the Optimal Solution for Labor Assignment

Gang Liu


  • Gang Liu

Abstract

Supply function and Production Possibility Frontier (PPF) are basic concepts in Economics. We present a model that can give mathematic formula for PPF and the supply as a function of the price vector and capability parameters. With this Supply function, the generic Supply-Demand equilibrium problems can be solved numerically. We apply the supply-demand equilibrium to give optimal solutions for team work management problems or labor assignment problems. Concrete examples are given for managing an engineer team in Boeing Corporation.

Keywords: Resource Allocation, Labor Assignment, Teamwork Management, Supply Function, Production Possibility Frontier, Supply-Demand equilibrium

Cite this paper: Gang Liu , A Mathematic Model for Supply-Demand Equilibrium and the Optimal Solution for Labor Assignment, American Journal of Economics, Vol. 4 No. 2, 2014, pp. 83-98. doi: 10.5923/j.economics.20140402.01.

1. Introduction

Production Possibility Frontier (PPF) is one of the basic concepts in Economics. PPF can be simply defined as the boundary of frontier of the economy’s production capabilities. Most of Economic books normally have a whole chapter to discuss PPF and its applications [1, 4, 10, 11]. Most of these text books discuss the PPF and supply function for two products and in qualitative format, as shown in Figure 1. It is well-known that the PPF is convex and the supply-price curve is an upward-sloping curve. However, no one has given the PPF nor supply function in mathematic formats. The law of supply and the law of supply-demand equilibrium are basic laws in modern economics. However, without knowing the supply function in mathematic format, these laws can be only discussed quantitatively, and the general supply-demand equilibrium cannot be solved numerically.
In early 2002, we have given the PPF and supply function in mathematic format for any number of products [6]. In this paper, we first introduce the model that can formulate the PPF in mathematic format, and then derive the supply as a function of the prices of all products. Some of the basic economic theorems can be derived from the PPF formula, such as the convexity of the PPF curve, the Law of Supply, as well as supply elasticity. We also proposed methods for solving the general supply-demand equilibrium numerically.
There are wide applications of the proposed PPF functions. As an example, we apply the PPF function and the Supply-Demand equilibrium equation to give optimal solution for team work management and labor assignment problems. Our analysis and test results show that the proposed model can improve production efficiency by 40% for most of the team work management problems.
The rest of the paper is organized as follows. Section 2 introduces some useful notations and definitions. Section 3 formulates the PPF and the supply function for a micro system. Section 4 further formulates the PPF and the supply function for a macro system. Section 5 discusses the lower Production Possibility Frontier. Section 6 discusses the Law of Supply and the Elasticity of Supply. Section 7 introduces two methods for solving the supply-demand equilibrium. Section 8 introduces two special production possibility curves that can be used to show how efficiency of the PPF. Section 9 applies our supply function to a dummy system to show concrete PPF curves and supply curves. Section 10 discusses the best scenario and the worst scenario. Section 11 applies our model to solve the labor assignment problem for a labor oriented team. Section 12 shows a concrete example of labor assignment problem for project oriented team. In Section 13 and 14, we analyze the efficiency improvements by the optimal solution. Section 15 introduces two management methods that can reach the optimal solution for team work management. Conclusions are presented in Section 16.

2. Notations and Definitions

In order to describe our model and the PPF curve, let us first introduce some notations and definitions.

2.1. Notations

: An economic system that contains some sub systems. It can be as small as a family, a firm, and can be as large as a country or the whole world.  is also called a macro system compared with its sub systems.
: An integer represents the total number of the sub systems in .
: An integer represents the total number of the products that are produced in .
: The ith unit or sub system in system . It could be an individual member, or a group of people, such as a firm or another economic system in . However, there are no overlaps between different Units. For example, if an individual has been included in one unit, the same individual can’t be included in another unit.  is also called a micro system compared with its parent system.
: The kth product that can be produced in system . Actually, the product here means any labor activities that can be done by any unit in . It can be a real product, a project, a job, a task, and anything that need to be done or produced by any unit in .
: A time span that the above mentioned products are produced in .
, which is the set of non-negative M-dimensional real vectors and with the original point  excluded.
: The same as . However, we use it to particularly represent the Production Space with its kth Cartesian coordinate representing the amount of a parameter related to  product.
: The demand amount of  product that requested by .
: The maximum amount of  product that can be produced by  within time . Normally  can be produced by  when it uses all of its resources to work on product . All the  make up an  matrix, which is called the micro capability matrix of system .
: The amount of  product that are actually produced by . All the  make up a vector in , and is called micro supply vector.
: The amount of  product produced by the macro system . It is also called the macro supply of system . All the  make up a vector in , and is called a macro supply vector of system .
: The price of the  product in system .

2.2. Definitions

DefinitionMicro System and Macro System: An economic system is called a macro system if it contains some sub-systems, such as the system  introduced above. An economic system is called a micro system if it is contained in a macro system, such as the sub-system  described above.
DefinitionSystem Parameter: An economic parameter is called a system parameter if it is associated with an economic system. For example, the supply amount, demand amount, and price are all examples of system parameters.
DefinitionMicro Parameter and Macro Parameter: A system parameter is called a micro parameter if it is associated with a micro system, and is called a macro parameter if it is associated with a macro system. Some parameters may be associated with both micro system and macro system. As long as there is no confusion, we use the same alphabet character to represent a system parameter: if  denotes a macro parameter associated with a macro system , then  will be used to denote the same micro parameter associated with micro system .
For example,  and  represent the macro and micro supply amounts of product  and  denote the macro PPF of system  and the micro PPF of the micro system  correspondingly.
DefinitionVector Parameter and Scalar Parameter: a system parameter may also be associated with each of the production, for example, supply amount normally means the amount of a product that can be provided by an economic system. In a system with  products, a system parameter could be -dimensional with each component representing the amount associated with one product. These -dimensional parameters make up a vector in the production space , thus is called a Vector Parameter. A vector parameter can be represented by a vector in . Examples of vector parameters are: Supply, Demand, and Price.
A system parameter is called a scalar parameter if it is one dimensional. Examples of scalar parameters are: income, GDP, revenue, etc.
Additive Parameter: A system parameter is called additive parameter if the corresponding macro parameter can be derived by aggregating the same parameter over all of its micro systems. Or precisely, system parameter X is additive if
(1)
Where  is a parameter reachable and meaningful to micro system , while  is the same parameter reachable and meaningful to macro system .
In our model, we tried to decompose the macro system  into some micro systems , and divide the products into some products or tasks in such a way that some of the system parameters are additive, particularly, the supply amount, the demand amount, revenue, and income are all additive. However, the price parameter is not an additive parameter.
Based on the above notations and definitions, we can say that the supply and demand are vector parameters and are additive, whereas the price is also a vector parameter, but not additive. The income, revenue, GDP, profit and loss are all additive scalar parameters. We are only interested in those parameters in this paper.

2.3. Known Parameters and Unknowns

Using the above notations, we can list the input data in table format. Basically, we assume the following data are known input data:
, the production capability matrix for each micro system and for each product.
: The macro demand of  product that needs to be accomplished by system .
We assume the following parameters are unknowns and need to be resolved: Price, micro supply, macro supply, micro income, macro revenue.

3. Micro PPF and Micro Supply Function of a Micro System

For any micro system , let us assume that it can arrange its resources to make any requested products. However, its production amount for product  should be limited to or bounded by  due to its limited resources and capabilities. Any of its feasible production state will be a point or a vector in the production space . All of its feasible production vectors should be a bounded range in the production space . We call this bounded range as its micro Production Possibility Range (PPR) and denoted as . The upper boundary of  is called the micro PPF, denoted as . Normally, as long as the micro system  is small enough compared with macro system , each of the micro PPF can be approximated by the linear plane curve that passing the  axis at  in production space .
Given the micro capability matrix, the micro PPF  can be formulated as:
(2)
 can also be expressed in the following format:
(3)
Where
(4)
 is called the intrinsic price vector of the micro system.
Actually,  is a linear plane curve with vector  as its normal vector. It passes the  axis at the points:
(5)
Where  is the unit vector on the  axis.  and is on the  axis of the production space. We call these  points as the micro vertex points.
Given the price vector as  and the micro supply vector of the micro system , the total income or revenue of the micro system  can be expressed as:
(6)
As all the supply vectors  is feasible to micro system , there should have some possible micro supply vectors that can maximize the income or revenue of the micro system . Such feasible micro supply vectors can be formulated as:
(7)
As the objective function  to be maximized is a linear function, according to Lagrange’s theorem, the max-min can only be found on the boundary of the region . Then the above equation is equivalent to:
(8)
By applying Lagrange’s theorem again, the Max-Min can be found at some of the  vertex points listed in Equation (5). Then, the above equation can be further simplified as:
(9)
Or more explicitly,
(10)
Where  is an  function defined as:
(11)
And  is the total number of vertex points that have  maximized for . Number  can be formulated as:
(12)
By substituting the above Equation into Equation (10), we have:
(13)
It can be easily checked that Equation (13) gives the mean vector of all the vertex points that can have  maximized. It gives at least one of the micro supply vectors  that can have the micro income  maximized. It is an explicit mathematical format for the micro supply as a function of the capability matrix and the price vector. However, it is a discrete function and thus is difficult to deal with. The key step and major contribution of our model is to formulate a continuous and smooth function to approximate the Equation (13). Here are the key steps. Equation (13) is equivalent to the following function:
(14)
Where  is called the macro capability parameter defined as:
(15)
It can be easily checked that Equation (14) is exactly the same as Equation (13) once .
In practical applications, we can simply drop the limited function by assigning  with a large number, thus the Equation (14) can be simply expressed as:
(16)
It can be shown that the above micro supply function does depend on the direction of the price vector, but doesn’t depend on the length of the price vector. So, we can normalize the price vector to 1 without any impacts on the supply function. Then, we always have:
 for ;
Normally, it will be good enough if we assign  as:
Equation (16) will be a good approximation for the micro supply function, and it gives the Micro PPF  once the price vector goes through all possible directions in the production space  will be good enough for most of the practical applications. We use  to get most of the results listed in this paper.

4. Macro PPF and Macro Supply Function of a Macro System

We have formulated the micro PPF  and the micro supply as a function of the price vector and the capability parameters, as shown in Equation (16). As each of the micro system  has a maximum production frontier and a limited range as its PPR, the macro system should also have a macro PPR in the production space , and must have a macro PPF. As the supply parameter is an additive parameter, we can get the macro supply vector by aggregating all the micro supply vectors, that is:
(17)
Once the price vector goes through all possible directions, the above Equation will give all the points on the macro PPF , as shown in Figure 3. So, the above Equation is not only a supply function, but also a function that gives the macro PPF .
Figure 4.shows the Supply-Price curves for various  values. Once  is big enough, the Supply-Price curves become step curves. Keeping in mind that each micro system is targeting at maximizing its income, so, when the price  rises, a micro system  may want to switch all of its resources to work on product , and thus to have the macro supply  be suddenly increased by an amount of . At the same time, because  switched from working on  to work on some other products, such as  (), it will have the supply  decreased suddenly by an amount of . This explains why the Supply-Price curve shows as a step curve, also explained the Law of Supply as discussed in later sections.
The supply functions and the PPF given in Equation (17) are the key contributions of this paper. The rest of the paper discusses applications of this supply function, or compare our PPF with some other production possibility frontiers and see how much efficiency can be improved by using our proposed model.

5. Macro Worst Production Possibility Frontier of the Macro System

Similarly, we can formulate the Worst Production Possibility Frontier (WPPF) for the micro as well as the macro system. Let  denote the micro WPPF of micro system , and  denote the macro WPPF of the macro system  is defined as the lower boundary under the conditions that all of its micro systems have reached their micro PPF . The WPPF  is defined as the set of micro supply vectors that are on the  and results a macro supply vector on the macro WPPF  if aggregated for all micro systems.
 and  can be formulated using the same Equations as shown in Equation (16) and (17), as long as we set . We will not rewrite these Equations here.
Let  be the region bounded by  and , and  be the region bounded by  and  is called the micro Maximum Production Possibility Range (MPPR), and  is called the macro MPPR.
Given a price vector , through Equations (16) and (17),we can get a macro supply vector  on , a macro supply vector  on , a micro supply vector  on , and a micro supply vector  on . It can be shown that the price vector  is the normal vector of those curves at the corresponding points as indicated by these supply vectors [6].

6. The Supply Elasticity and the Law of Supply

The Supply Elasticity is defined as the absolute value of the ratio of the percentage change in quantity supplied to the percentage change in price, which brings about the change in supply. Let  denote the supply elasticity for the  product. By applying partial differentiation to Equation (16) and (17), we can easily have:
(18)
(19)
The Law of Supply is one of cornerstones in Economics theory. It states that as the price of a commodity rises, producers supply more. So far, the law of supply has been widely accepted as an empirical law. Using the analytical formats of the supply function, as shown in Equation (16) and (17), we can give a more precise and extended format for the Law of Supply as follows:
The Law of Supply: Given an ideal economic system[6] and assuming that all other things remain unchanged (e.g., prices of other products and all production capability parameters remain unchanged), the supply of a commodity rises as the price of that commodity rises, and decreases as the price of any other commodity rises. Or in mathematical formats:
(20)
(21)
Equation (20) is equivalent to say that the elasticity  given in Equation (19) is non-negative. Noting that  and , thus all items in Equation (19) is non-negative, then the elasticity  which is a sum of some non-negative numbers is also non-negative. Thus Equation (20) is proven.
By applying partial differentiation to Equation (16), we have:
,
,
for  and 
Equation (21) follows the above two equations immediately, and thus the Law of Supply is proven.

7. Supply-Demand Equilibrium

Supply-Demand Equilibrium is one of the most important theorems in Economics. It states that Equilibrium is defined to be the price-quantity pair where the quantity demanded is equal to the quantity supplied, represented by the intersection of the demand and supply curves. The general format for the Supply-Demand Equilibrium can be expressed as:
(22)
Where both  and  are functions of . Thus, the above general equilibrium equation gives  equations with  prices as unknowns. Generally, once we know the function format for  and , the equilibrium price vector can be solved from Equation(22).
We do not try to give a generic format for demand functions in this paper, instead, we just simply assume the demand is given as a vector with fixed direction, thus Equation (22) can be expressed as:
(23)
We propose two methods to solve the General Equilibrium Equation (23).

7.1. Solving the General Equilibrium as an Equation Problem

By combining Equation (23) and Equation (17), the General Equilibrium Equation can be formulated as the following equation problem:
(24)
The above formula gives  equations with  for as the  variables. Then the price victor  can be solved by solving the above equation through many mature methods, such as Newton’s method [5, 9], Brent’s method [2, 13]. Thus all the micro supplies  can be given through Equation (16), and all the macro supplies can be given through Equation (23) or Equation (17).

7.2. Solving the General Equilibrium as an Linear Programming Problem

The General Equilibrium Equation (23) can also be formulated as the following optimization problem:
(25)
This is a linear programming problem with  and  as variables. It can be easily solved through the T-forward method [7, 8, 3]. Equation (24) and (25) should give the same solution.

8. Some Special Production Possibility Curves

In this section, we introduce two special scenarios with special production possibility curves. These curves can be used to compare with the PPF and can show how efficiency of the production states on the PPF.

8.1. Linear Production Possibility Frontier

We have introduced the micro vertex points as shown in Equation (5). A plane curve can be formulated by passing through these vertex points as following:
(26)
Let us call this plane curve as the micro Linear Production Possibility Frontier (LPPF), and denoted as . Similarly, we can construct a macro LPPF curve  for the macro system , which can be formulated as:
(27)
Using the price vector as a reference vector, the above equation can be expressed as:
(28)
Note that the above Equation cannot be treated as the supply function, although it is expressed as functions of the price vector. It can be shown that:
(29)

8.2. Self Sufficient Scenario

Another special scenario is the so called Self Sufficient case, in which every micro system just simply works on itself to provide all products with the amounts to be proportional to the requested demands. Within this scenario, there is no cooperation among micro systems. Let us call the PPF Curve for this scenario as SPPF. Let  and  be the micro SPPF and macro SPPF for the Self Sufficient scenario.  is the same as the plane curve  as shown in Equation (26). However, the macro SPPF  is different from the Macro LPPF .
By definition, given the requested demand as , the micro SPPF  can be expressed as:
(30)
And the macro SPPF  can be formulated as:
(31)

9. A Dummy System with Dummy Data

In order to give concrete examples to demonstrate our model and methodology, let us create a dummy system with dummy data. The dummy system  is called the carpenter system, which contains  units or members, and need to make  products. The carpenter system needs to produce 4 table legs, 1 table top, and 6 chairs, which make up the 3 components of the demand vector. The micro capability matrix and demand vector are listed in Table 1. The requested amounts or demands are listed in row 3, and the production capabilities are listed in row 5 to 14.
Table 1. Input data for the carpenter system with 10 members and 3 products
     
Given the input data as listed in Table 1, by applying Equation (16), (26), and (30), the micro curves , and  are derived and are drawn in Figure 2. These four curves are all converged to the same plane curve.
Figure 2The micro PPF, the micro WPPF , the micro LPPF , and the Micro SPPF  are all shown as the same plane curve in the production space
According to Equation (16), curves  and  could be different from the plane curve for some small . However, the difference can be only shown up for 3 or more dimensional case. If we set all other prices to 0 except two of them, then the above four curves always converge to the plane curve.
Figure 3. shows the macro PPF , along with some other macro curves, including WPPF , LPPF , and SPPF . Figure 4 draws the supply-price curves for various  value. By comparing with Figure 1, our model gives the PPF and supply functions in more detailed formats and make the general supply-demand equilibrium solvable numerically.
Figure 3The macro PPF  is a convex curve, the macro WPPF  and the macro SPPF  are concave curves, whereas the macro LPPF  is a linear plane curve in the production space. The maximum PPR  is the region bounded by  and . The extension line of the demand vector  intersects with these 4 curves at the points , and  respectively. These curves and points are used for improved efficiency analysis

10. The Best and the Worst Scenarios

Suppose each of the micro system  has reached its micro PPF, then the aggregated macro state of the macro system must be in the macro MPPR. Also, suppose the macro system  is requested to produce the demand vector. The demand vector  is as shown as the line  in Figure 3. Figure 3. also shows the four points, and, which are the intersection points of the line  with the curves, and.
For all possible production states in, the Line  gives all possible states that are proportional to the requested demand. Any other states in  may have some products wasted or not needed compared with the requested demand. So, for all possible states in, only the states on line  can best meet the requested demand. We are only interested in the states that are on the line.
For all states on the line, the point  is the best scenario, because it gives the maximum possible amount for all supplies, is the worst scenario, whereas  and  are somewhere in the middle. Point  can be treated as the average state for all points in the macro MPPR.
Let us assume that point  is the average state that the macro system reached without using any optimal management method. Now, let  denote the improved production efficiency by comparing the best scenario  with the average scenario, then  can be formulated as:
(32)
Figure 4Supply Curves. When price  rises and all other prices remain unchanged, the supply  rises, whereas the supply for other products decreases, for example, Supply  decreases. When  is large, the supply curves  and  become step curves

11. Application 1: Labor Assignment for Labor Oriented Team

The mathematic formula for the Macro PPF and the supply function may have lot of applications and can improve the production efficiency significantly. Here we just introduce one of the applications of the macro PPF and the supply function: finding the optimal solution for team work management problems.
We have developed a software tool that can find optimal solution for labor assignment problems by solving equation problem listed in Equation (24), or by solving linear programming problem as listed in Equation (25).By applying this software tool to the Carpenter team with the input data as listed in Table 1, we find the optimal solution as shown in Table 2.
The solution given in Table 2 tells us at least the following information:
Table 2. The optimal solution for the Carpenter team management problem
     
It tells all the information about the production state, including the amount of a production to be made by each team member. Each micro production state is a point on the corresponding micro PPF.
Most of the team members should work on only one product. Some of the team members may work on multiple products. For example, team member David should work only on “Table Legs” and make 8 “Table Legs” within requested time T, while Richard should work on all 3 products, and make 6.1 “Table legs”, 0.59 “Table Tops”, and 0.06 “Chairs”.
The final macro Supply is 28 “Table legs”, 7 “Table Tops”, and 42 “Chairs”. This macro Supply state is on the macro PPF  and proportional to the requested production demand vector .
The last column gives the Earnings or Incomes for each team member.
The row with header “Supply/Demand” gives the number defined in Equation (23). It is the ratio between the optimal solution and the requested demand amount.
The row with header “Optimal Intrinsic Price” gives the intrinsic prices which can be used to calculate the earnings for each team member. The intrinsic price vector for these 3 products should be (0.51, 0.68, 0.52). Once this intrinsic price vector is known, each team member will reach his maximum income by making the products with the amounts as requested in the optimal solution. In other words, the whole team will realize the optimal solution automatically as long as each sub unit has maximized its income.
The row with header “Self Sufficient Supply” lists the supply state  on the SPPF .
The row with header “Linear Macro Supply” lists the supply state  on the LPPF .
The row with header “Improved Efficiency” lists the improved efficiency ratio by the optimal solution compared with the Linear Macro Scenario.
The row with header “Loop Count” lists the number of loops for our numerical calculation to converge to the results with required precision.
The optimal solution not only depends on the capability matrix, but also depends on the direction of the demand vector. Given a team member and his capability matrix, we cannot tell which product is the most specialized product for him. The optimal solution may request him to work on one product. However, once the direction of the demand vector changes, the same team member might be requested to work on some other products to have the whole team to reach the optimal solution.

12. Application 2: Labor Assignment for Project Oriented Team

Application 1 gives an example of a team working on products, which is more applicable to a labor oriented team.
A product oriented team is basically a team working on products and the demands can be simply formulated as the amounts of the products. A project oriented team is a team working on projects, while each project is required to be delivered in a given deadline. For a project oriented team, the capability parameter  can be explained as the delivery ratio (or amount) of the kth project  by team member . The demand for each project is 1 within required delivery time.
Let  denote the required delivery time for the kth project , and  denote the fastest delivery time of team member  to deliver the kth project  under the condition that  works only on project . Then the equivalent demand vector can be expressed as:
(33)
And the equivalent production capability parameters can be expressed as:
(34)
Then, our proposed model for labor assignment is applicable to project oriented teams.
We have applied our proposed model to manage an engineer team in Boeing Corporation. Let us call this team as Boeing team. The Boeing team includes 10 Engineers to work on 6 projects that need to be delivered in requested deadlines. The total budget for these projects is 100,000 USD.
Table 3 lists the fastest delivery time for each team member to deliver each project. If we inverse each number in this table, it will be the capability parameter matrix. Please note that we have given dummy names for each team member. The Boeing Team was given a strict deadline to finish these projects. The manager had been worried about whether they could deliver these projects and struggled for labor assignment among the team members.
By solving Equilibrium Equation (24), or solving the linear programming problem as listed in Equation (25), we can give the optimal labor assignment solution for the Boeing team. Table 4 lists the amount to be delivered in unit time for each team member and each project. Table 5 lists the optimal assignment in terms of time allocation for each team member. Table 6 lists the income, total work days, and the pay-rate for each team member.
With our optimal solution for labor assignment, the Boeing Team can deliver all of their 6 projects on time and all the actual delivery time are reduced by half compared with the original requested delivery time.
Table 3. The delivery capability matrix and requested delivery time for Boeing team
     
Table 4. The optimal solution for a project oriented team in Boeing Corporation
     
Table 5. Optimal Time allocation for a project oriented team in Boeing Corporation
     
Table 6. Income and pay rate at the Equilibrium state
     

13. Production Efficiency Improvement for the Carpenter Team

Note that, even with the worst scenario case, all team members have worked hard to reach their micro PPF states. In other words, although all team members have exhausted all of their resources to work on the requested products, the production output are very different for the macro system. The inefficiency for the worst scenario is completely due to improper management. With proper arrangement, we can have the team to improve production efficiency by 148% compared with the worst scenario.
Table 7 lists the macro supplies for the optimal solution, the worst scenario solution, the Linear Macro Supply scenario, as well as the Self-Sufficient scenario. It also lists the improved production efficiency ratios by comparing with the Self-Sufficient scenario as well as the Linear Macro Supply scenario.
Table 7. Improved Production Efficiency Ratios for the Carpenter Team
     
By comparing with the Self-Sufficient Supply scenario, we can find out how much efficiency improved due to the division of labors and corporations among team members. The Linear Macro Supply scenario can improve the production efficiency by 25%, whereas our optimal solution can improve the production efficiency by 79%. These improved efficiencies come from the division of labor and cooperation among team members. Some managements with good quality are required to realize such kind efficiency improvements. Our proposed method gives the optimal solution for team work management.
By comparing with the supply state , we can find the improved efficiency ratio by the optimal solution. The improved efficiency for the carpenter team is:
(35)

14. Estimate the Improved Production Efficiency for Generic Cases

As discussed in the previous sections, by applying our optimal management method, the Carpenter team can improve its production efficiency by 43%. That is amazing! However, people might argue that the Carpenter team is just a special case. In this section, we estimate the production efficiency improvement ratio that can be brought by the optimal solution for generic cases. Our analysis shows that the optimal solution can really improve production efficiency by 40% for most of the team work management problems.
To simplify our analysis, we assume the demand vector as in this section. Let  be the intersection point of the demand vector  (or its extension) with the curve , and  denote the length of the line . Then  can be easily found as:
(36)
As the macro PPF  is a convex curve, we can use  to approximate the PPF , with  to be a value that can make the  most close to . Normally, once we find one point on , we can calculate the value of which can make the  most close to the . Further, we assume the average production state is roughly at the point  on the curve  (or denoted as). If  is used to approximate , the improved efficiency by reaching the macro PPF can be estimated as:
(37)
Actually, the more team members the team has, the more convex the macro PPF tends to be. Based on our test results, we found that  is a good approximation for  when; and even  will be a good approximation for  when  and.
To make it more conservative, we use  to approximate the macro PPF. Then the production efficiency ratio that can be improved by the optimal solution can be estimated as:
(38)
Table 8 lists the possible production efficiency improved by the optimal solution for a team working on  products.
Based on our test results and the above analysis, our optimal solution for team work management normally can improve the production efficiency by 40% for most of team work problems!
Figure 5 can be used to approximate the PPF  and LPPF . Here all the curves are shown in 2-dimensional. You need to imagine it in M-dimensional
Table 8. Production Efficiency Improved by the Optimal Solution for a Team
     

15. Micro and Macro Management

By applying the proposed method and the software tool, we can find the optimal solution for the team work management, or labor assignment problems. Suppose you are the team manager, how are you going to manage your team to reach the optimal management? Here we propose two management methods: Micro Management and Macro Management.

15.1. Micro Management

Our optimal solution for team work management problems, as shown in Table 2 and Table 4, lists all the micro information for each team member, including micro supply amount , and the micro income amount . Using these micro information, a team manager can tell each team member  to work on the  with the amount  as indicated in the optimal solution. Note that most of the team members may work on only one product. The team manager can also tell each team member  about how much he should be paid, which would be . So, with micro management, a team manager manages each of the team members in micro details, including what and how much need to be done, as well as how much should be paid as return. That is why we call this management method as Micro Management.

15.2. Macro Management

Our optimal solution for team work management problems, as shown in Table 2 and Table 4, also gives the intrinsic prices . Given this intrinsic price vector, each team member will reach his maximum income by working on the product with the amount as requested by the optimal solution. Suppose each team member is targeting at maximizing his income, the team manager needs only declare the intrinsic prices, then, every team member will automatically to work on the product with the amount such that the whole team will reach its optimal solution as indicated in the optimal solution.
The team manager doesn’t manage the team members in micro details, but manage it in macro level, which simply tells how much will be paid for product or project to be delivered. That is why we call this management method as macro management.

16. Conclusions

The main contributions of this paper include a new model that can formulate the Supply function and the Production Possibility Frontier as a function of the capability parameters and the price vector. The Law of Supply, the supply elasticity can be easily derived from the proposed model. Another contribution is the methods for solving the Supply-Demand equilibrium. Lastly, we present a model and method that can give optimal solution for labor assignment and team work management problems based on the Supply-Demand equilibrium. Our test results and generic analysis show that the proposed management method can improve the team work efficiency significantly. For most cases, it can improve the production efficiency by 40%!


下面是该论文中的几个图表:


Figure 1. This is a snapshot from Samuelson’s book “Economics” [11]. Without knowing the mathematic formats of the PPF and supply function, most of the Economics books can only discuss PPF, Supply, and supply-demand equilibrium quantitatively. We proposed a model that can give PPF and Supply as a mathematic function of the price as shown in Equation (17)















References

[1]  [Bar10] R. R. Barthwal, Industrial Economics: An Introductory Textbook. P. 31, 2010.
[2]  [Bre73] R. P. Brent. Algorithms for Minimization without Derivatives. Chapter 4. Englewood Cliffs, NJ: Prentice-Hall, 1973. ISBN 0-13-022335-2.
[3]  [DS97] Robert Dorfman; Paul A. Samuelson; Robert M. Solow. Linear Programming and Economic Analysis. McGraw-Hill College Division, 1997. ISBN: 0486654915.
[4]  [Gil07] Andrew Gillespie, Foundations of Economics. Ch. 2. Oxford University Press, 2007.
[5]  [Kel03] C. T. Kelley, Solving Nonlinear Equations with Newton's Method, no 1 in Fundamentals of Algorithms, SIAM, 2003. ISBN 0-89871-546-6.
[6]  [Liu02] G. Liu, Analytic Economics, P. 39-90, 2002. ISBN 10: 0971893403 / ISBN 13: 9780971893405.
[7]  [Liu14] G. Liu, “T-Forward Method: A Closed-Form Solution and Polynomial Time Approach for Convex Nonlinear Programming”. Algorithms Research, Vol.3, No.1: 1-23. 2014.
[8]  [Liu14] G. Liu, “T-Forward Method: A Closed-Form Solution and Polynomial Time Approach for Convex Quadratic Programming”. Applied Mathematics, 2014.
[9]  [OR00] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Classics in Applied Mathematics, SIAM, 2000. ISBN 0-89871-461-3.
[10]  [PR09] Robert Pindyck, Daniel Rubinfeld, Microeconomics, The Pearson Series in Economics, 2012. ISBN-10: 013285712X / ISBN-13: 978-0132857123.
[11]  [Sam04] Paul A. Samuelson and William D. Nordhaus, Economics. Chapter 1-5.McGraw-Hill; 16th Edition, 1998. ISBN 7-111-06448-8.
[12]  [Sam67] Paul A. Samuelson. “Summary on Factor-Price Equalization”. International Economics Review, Vol. 8, No. 3, 8(3): 286-295, 1967.
[13]  [Sta13] Steven A. Stage. “Comments on An Improvement to the Brent's Method”, International Journal of Experimental Algorithms 4 (1): 1–16.2013.