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五个鲜为人知的数学定理,颠覆你的认知2019-07-18 15:11·深本思维官方号在数学里,有很多有趣而又深刻的数学定理,不但深受数学家们的喜爱,在数学迷的圈子里也广为流传,今天深本老师跟大家分享几个“反常识”的定理!一、喝醉的小鸟定理:喝醉的酒鬼总能找到回家的路,喝醉的小鸟则可能永远也回不了家。假设有一条水平直线,从某个位置出发,每次有 50% 的概率向左走1米,有50%的概率向右走1米。按照这种方式无限地随机游走下去,最终能回到出发点的概率是多少?答案是100% 。在一维随机游走过程中,只要时间足够长,我们最终总能回到出发点。现在考虑一个喝醉的酒鬼,他在街道上随机游走。假设整个城市的街道呈网格状分布,酒鬼每走到一个十字路口,都会概率均等地选择一条路(包括自己来时的那条路)继续走下去。那么他最终能够回到出发点的概率是多少呢?答案也还是 100% 。刚开始,这个醉鬼可能会越走越远,但最后他总能找到回家路。不过,醉酒的小鸟就没有这么幸运了。假如一只小鸟飞行时,每次都从上、下、左、右、前、后中概率均等地选择一个方向,那么它很有可能永远也回不到 出发点了。事实上,在三维网格中随机游走,最终能回到出发点的概率只有大约 34% 。这个定理是著名数学家波利亚(George Pólya)在 1921 年证明的。随着维度的增加,回到出发点的概率将变得越来越低。在四维网格中随机游走,最终能回到出发点的概率是 19.3% ,而在八维空间中,这个概率只有 7.3% 。二、“你在这里”定理:把一张当地的地图平铺在地上,则总能在地图上找到一点,这个点下面的地上的点正好就是它在地图上所表示的位置。也就是说,如果在商场的地板上画了一张整个商场的地图,那么你总能在地图上精确地作一个“你在这里”的标记。1912 年,荷兰数学家布劳威尔(Luitzen Brouwer)证明了这么一个定理:假设 D 是某个圆盘中的点集,f 是一个从 D 到它自身的连续函数,则一定有一个点 x ,使得 f(x) = x 。换句话说,让一个圆盘里的所有点做连续的运动,则总有一个点可以正好回到运动之前的位置。这个定理叫做布劳威尔不动点定理(Brouwer fixed point theorem)。除了上面的“地图定理”,布劳威尔不动点定理还有很多其他奇妙的推论。如果取两张大小相同的纸,把其中一张纸揉成一团之后放在另一张纸上,根据布劳威尔不动点定理,纸团上一定 存在一点,它正好位于下面那张纸的同一个点的正上方。这个定理也可以扩展到三维空间中去:当你搅拌完咖啡后,一定能在咖啡中找到一个点,它在搅拌前后的位置相同(虽然这个点在搅拌过程中可 能到过别的地方)。三、不能抚平的毛球定理:你永远不能理顺椰子上的毛。想象一个表面长满毛的球体,你能把所有的毛全部梳平,不留下任何像鸡冠一样的一撮毛或者像头发一样的旋吗?拓扑学告诉你,这是办不到的。这叫做毛球定理(hairy ball theorem),它也是由布劳威尔首先证明的。用数学语言来说就是,在一个球体表面,不可能存在连续的单位向量场。这个定理可以推广到更高维的空间:对于任意一个偶数维的球面,连续的单位向量场都是不存在的。毛球定理在气象学上有一个有趣的应用:由于地球表面的风速和风向都是连续的,因此由毛球定理,地球上总会有一个风速为 0 的地方,也就是说气旋和风眼是不可避免的。四、平分三明治定理:任意给定一个火腿三明治,总有一刀能把它切开,使得火腿、奶酪和面包片恰好都被分成两等份。而且更有趣的是,这个定理的名字真的就叫做“火腿三明治定理”(ham sandwich theorem)。它是由数学家亚瑟?斯通(Arthur Stone)和约翰?图基(John Tukey)在 1942 年证明的,在测度论中有着非常重要的意义。火腿三明治定理可以扩展到 n 维的情况:如果在 n 维空间中有 n 个物体,那么总存在一个 n - 1 维的超平面,它能把每个物体都分成“体积”相等的两份。这些物体可以是任何形状,还可以是不连通的(比如面包片),甚至可以是一些奇形怪状的点集,只要满足点集可测就行了。五、分球悖论巴拿赫-塔斯基悖论,又称分球悖论,是一条经过严格证明的数学定理。一个三维实心球,必定存在一种办法分成有限部分,然后仅仅通过旋转和平移,就可以组成两个和原来完全相同的球(半径相同,密度相同……所有性质都相同)。这是一条非常反常识的数学定理,基于“选择公理”严格地推导出来,而且不容置疑。这个定理还有更强的版本描述:一块石头经过分解,可以随意组合成任何东西,可以拼成一个星球,也可以拼成一个人,甚至藏进一个细胞之中!要理解其中的原理,需要对“无穷”这个概念有深刻的理解:这个比喻,是对“无穷”的一个通俗解释,分球悖论也可以通过这个比喻来解释。我们来类比,“球分成无限份”相当于“旅馆的无限个房间“,把这无限个房间分成偶数和奇数两类,我们再单独把这两类房间分开,分别称为“希尔伯特旅馆一”和“希尔伯特旅馆二”。如果我们不看序号,或者把两个旅馆的房间重新编号,请问:这两个新的旅馆,和原来的“希尔伯特旅馆”有区别吗?答案是:没有区别,两个新旅馆,和原来的旅馆一摸一样,房间数一样,每个房间的大小也一样。分球悖论指出:实心球也存在这样的分解办法,然后进行分类和重组,就能变“一”为“二”;两者本质上是一样的。有人可能会觉得,新的实心球,质量肯定变为原来的一半!其实不是的,因为在无穷面前,分球悖论并不满足质量守恒,比如我们假设每个单元的质量为Δm(无穷小),在我们分类的时候,Δm并没有被分解,我们分解的是“∞“。在数学中,“可数∞”的一半,还是“可数∞”,于是,我们确实得到了两个,和原来一模一样的实心球。或许,这正是数学和大自然,完美统一的表现。深本数学创始人邹老师已经开设了深本教育智慧分享群组,每天都会分享高效、深刻的数学解题方法以及先进的教学理念!各位家长、老师和孩子们如果有兴趣加入一同探讨更多对于数学学习与教育等问题,可以私信进群!他也不找我,我也不找人家,各有各的事,没有闲情逸致,坐在一起喷大江东了。
henkuailundaohujinlinliao。zaizhaodaisuoli,yigehenyouzilideguanyuanzhiwenta[17]:“laoziliangwanwuqianlichangzhengguolai,yecai381kuaiqianyigeyue。niyinianzhuanjishiwan,bushitoujidaobashishime?”xingnengzuiqianganzhuoshouji meiyouzhiyi!zhengjiuzheY90dianjingshoujishoufapingce2022-02-28 00:06·kuaikejiyi、qianyan:shuangrenMshuangfengshansanrexitongjiachi dazaozuiqianganzhuoshoujiyikuaidingjideRTX 3090xianka,ruguozhigei100Wgonghao,nametakenenglianyibandexingnengduwufashifangchulai。tongyangde,yikeAMDruilong7 6800Uchuliqi,zaibutongdeqingbaobenzhonggonghaoshedinggebutong,15W、25W、35Wdengdengduyou。haowuyiwen,35Wxiadexingnengyuanqiangyu15W。shoujishichangyiran!duiyuxinyidaizuolong8 Gen 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730 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60FPS,:junhengmoshixia,mandiandezhengjiuzheY90dianjingshoujikeyiyunxingjiangjin3gexiaoshi。ruguogaiweiyeshoumoshi,youxixuhangshijiandagaishi2.2xiaoshi。qi、zongjie:wukezhengyidezuiqianganzhuoshouji、meiyouzhiyiciciceshigeibizheyizhongniudaoshajideganjue,dingjidezuolong8 Gen 1chuliqipeiheshoujishangzuiqiangdesanrexitong,zhujiuliaodangqianzuiqiangdeanzhuoshouji。ceshixiaojieruxia:1、duxiexingnengzhengjiuzheY90shidangjinshimianshangwukezhengyidezuiqianganzhuoshouji,bujinjinzhishiCPUheGPUxingnengzuiqiang,tadeshancunxingnengtongyangyeshiaoshiqunxiong,UFS 3.1+SSD RAID 0cipanzhenliejiejin3000MB/sshunxuduqusudujihushiqitaqijianshoujide2bei。lingwai18GB RAM + 640GB ROMsuiranyouzuoyaodexianyi,danyefuhezhengjiuzheY90zuiqianganzhuoshoujideshenfen。2、wendumuqianmeiyounayikuanshoujiyouxinenggouzhaganzhengjiuzheY90dianjingshoujidexingneng,jibianqiangru《yuanshen》,zhengjiuzheY90zhixuyaoshichu6chengdexingnengjiukeyipaomanjigaohuazhi60FPS,bingqiewenduyejinjinzhiyou40du。dangqianyezhiyouzhengjiuzheY90nengzuodaoruci,ruguohuanchengqitashouji,yunxingyuanshenshiwenduzhishaoyaogao20du,ranhoukaishijiangpinjianggonghao。ranercishidezhengjiuzheY90quehuanyouyuliyunxingdaogenggaozhenglv,zhejiushiqiangdasanrexitongdailaidexingnengzhibian。3、manxuezuolong8 Gen1zhiyouGFXbenchnengzhenzhengfahuichuzhengjiuzheY90dianjingshoujidequanbuxingneng。zhekuan3Dceshichengxuyouyumeiyouxianzhizhenglv,yejiushishuoshoujixingnengyouduoqiang,tajiunengpaochuduogaodezhenglv。biru“1080Pbawanglong”lipingdezhenglvchaoguoliao400FPS。zaijiashangGFXbenchceshihaoshijiaochang,ciqiandeqijianshoujiyunxingdaoliaohouqijibenshangduhuijiangpin。erwomenrangzhengjiuzheY90dianjingshoujipaowansuoyouceshixiangmu,haoshi1geduoxiaoshi,zhenggeguochengzhongzuolong8 Gen 1chuliqijihuduzaimanxueyunxing。4、danhexingnengweishimezhengjiuzheY90dianjingshoujidanhexingnengbingmeiyoubiqitaqijianshoujigaoduoshao?leisiGeekBenchzheyangdeceshiruanjian,ceshidanhexingnengzhiyouyigeCortex-X2hexinyoufuzai,qita7gehexinheAdreno 730 GPUdushikongzaiyunxing。jinjinshiyigehexinmanzaiduiyudangqiandeqijianshoujilaishuo,dubuhuizaochengsanreshangdemafan,yinerdanhexingnengdunengpaodaohengao。danshiruguo8gehexinyijiAdreno 730dugaofuzaiyunxing,zhegeshihoushoujizhijiandexingnengchayicaihuitixianchulai。5、xuhangrichangshiyong,zhengjiuzheY90dianjingshoujidexuhangheqitaqijianshoujichuyutongyishuizhun,birushiyong“dianchigougou”nengcechu9xiaoshidexuhangshijian。zaizuikekedeyouxihuanjingxia(yuanshenmanzheng+zuigaohuazhi),zhengjiuzheY90dianjingshoujikeyizhicheng2~3xiaoshideyouxishijian。zhengjiuzheY90dianjingshoujibiaopeiliao68Wkuaichong,0~100%chongmanzhixuyao38fenzhong,lingwaizaishoujicemianyeshejiliaoyigeType-Cchongdiankou,chongdianshihengpingwanyouxidetiyanbiqitashoujihaohenduo。6、paizhaosuiranzhengjiuzheY90shiyikuandianjingshouji,zaipaizhaofangmianbijiaobaoshou。shoujihouzhilianggejingtou,fenbieshi6400wanzhushe(CMOSxinghaokenengshisuoniIMX 686)+ 1300wanxiangsu120°chaoguangjiaojingtou,manzurichangjilushenghuobingmeiyouwenti。zhengjiuzheY90dianjingshoujidedansheng,youruyiwangqingquan,rangyuanbensiqichenchendegaoduananzhuoshoujishichangkandaoliaoxindechuangxinfangxiang!yuanlaigaoduanshoujihuannengzuochengzheyang!yuanlaizuolong8 Gen 1chuliqixingnengjingranzhemeqiang!
最(窜耻颈)近(闯颈苍)我(奥辞)常(颁丑补苍驳)常(颁丑补苍驳)思(厂颈)索(厂耻辞)一(驰颈)个(骋别)问(奥别苍)题(罢颈):儿(贰谤)子(窜颈)对(顿耻颈)父(贵耻)母(惭耻)来(尝补颈)说(厂丑耻辞)是(厂丑颈)什(厂丑颈)么(惭别)?
dajiangshoukuanyunzaiwurenjiDJI FlyCart 30fabu,125000yuanqi2023-08-16 21:13·ITzhijiaITzhijia 8 yue 16 rixiaoxi,dajiangjinrizhengshifabushoukuanminyongyunzaiwurenji DJI FlyCart 30,jidazaizhong、changhangcheng、qiangxinhao、gaozhinengyuyishen,shiyongyushandi、anji、xiangcunyunshuchangjingjigeleiyingjichangjingxiadewuziyunshu。IT zhijiafudajiangyunzaiwurenji DJI FlyCart 30 guanfangjieshaoruxia:FC30 caiyong 4 zhou 8 jiangduoxuanyigouxing,shuangdianmoshixiazuidazaizhong 30 gongjin、manzaizuidahangcheng 16 gongli。zhengjijubei IP55 fanghudengji,keshiying-20°C zhi 45°C [3] degongzuohuanjingwendu,zuidafeixinghaibagaodu 6000 mi,yongyouchaoqianghuanjingshiyingxing,keshiyingquantianhou、kuanwenyu、kuahaibadezuoyechangjing。jubeihuoxiangjikongdiaoliangzhongfuzaimoshi,huoxiangzhichikuaichaihezidongchengzhong,kongdiaozhichizhinengxiaobaihejinjirongduandenggongneng,weiyonghudailailinghuoduoyang、wendingkekaodeshiyongxuanze。quanxinshuangdianchixitong、20 gongli O3 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首(Shou)发(Fa)2024-05-31 12:12·一(Yi)个(Ge)80后(Hou)的(De)社(She)会(Hui)视(Shi)野(Ye)
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我的父母还健在都已经80多岁了身体一天不如一天经常需要吃药做检查我的父母也是地地道道的农民没有退休金的我有一个弟弟和一个妹妹妹妹远嫁外地照顾父母指望不上她而且在农村有种说法女儿是嫁出去的不继承父母的财产也不给父母养老给父母多少钱全凭女儿的孝心
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