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  • 取整函数的性质

    我们通常将(y=[x])(y=lfloor x floor)记作关于(x)取整函数,也称为高斯函数,其意义是不超过x的最大整数

    ( ext{Lemma 0:})

    [lfloor b floor le b<lfloor b floor+1 ]

    ( ext{Lemma 0':})

    [forall a,b,cin N_+ ,lfloorlfloorfrac{a}{b} floor/c floor=lfloorfrac{a}{bc} floor ]

    ( exttt{Proof:})

    [a=lfloorfrac{a}{b} floor imes b+r_1=lfloorfrac{a}{bc} floor imes bc+r_2,r_1in[0, b),r_2in[0, bc),r_2-r_1in(-bc,bc) ]

    [lfloorlfloorfrac{a}{b} floor/c floor=lfloor frac{a-r_1}{bc} floor=lfloor lfloorfrac{a}{bc} floor + frac{r_2-r_1}{bc} floor=lfloorfrac{a}{bc} floor ]

    ( ext{Lemma 1:})

    [ain Z,bin R ]

    [alelfloor b floor Leftrightarrow ale b ]

    ( exttt{Proof:})

    [alelfloor b floor,lfloor b floorle b Rightarrow ale b ]

    [ale b Rightarrow a<lfloor b floor+1 Leftrightarrow ale lfloor b floor ]

    (整数的离散性:(x,yin Z,x<yLeftrightarrow xle y-1))

    ( ext{Lemma 2:})

    [x,yin Z ]

    [xle lfloor frac{n}{y} floorLeftrightarrow ylelfloor frac{n}{x} floor ]

    ( exttt{Proof:})

    [ ext{By lemma1:}xlelfloor frac{n}{y} floorLeftrightarrow xle frac{n}{y} Leftrightarrow ylefrac{n}{x}Leftrightarrow yle lfloor frac{n}{x} floor ]

    ( ext{Proposition 3:})

    [x,nin Z ]

    [xlelfloorfrac{n}{lfloorfrac{n}{x} floor} floor ]

    ( exttt{Proof:})

    [ ext{By lemma2: }xlelfloorfrac{n}{lfloorfrac{n}{x} floor} floorLeftrightarrowlfloorfrac{n}{x} floorlelfloorfrac{n}{x} floor ]

    ( ext{Theorem 4:})

    [xin Z,lfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x} floor} floor} floor=lfloorfrac{n}{x} floor ]

    ( exttt{Proof:})

    [ ext{By prosition3: }lfloorfrac{n}{x} floorlelfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x} floor} floor} floor--(1),xlelfloorfrac{n}{lfloorfrac{n}{x} floor} floor ]

    [Rightarrowfrac{n}{x}gefrac{n}{lfloorfrac{n}{lfloorfrac{n}{x} floor} floor}gelfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x} floor} floor} floor ]

    [ ext{By lamma1: }lfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x} floor} floor} floorlelfloorfrac{n}{x} floor--(2) ]

    [(1) ext{ and }(2)Rightarrowlfloorfrac{n}{lfloorfrac{n}{lfloorfrac{n}{x} floor} floor} floor=lfloorfrac{n}{x} floor ]

    ( ext{Corollary 5:})

    [yin Z_+,maxleft{xin Z_+|lfloorfrac{n}{x} floor=lfloorfrac{n}{y} floor ight}=lfloorfrac{n}{lfloorfrac{n}{y} floor} floor ]

    ( exttt{Proof:})

    [forall xin Z_+ , ext{that } lfloorfrac{n}{x} floor=lfloorfrac{n}{y} floor ]

    [ ext{By proposition3: }x le lfloorfrac{n}{lfloorfrac{n}{x} floor} floor=lfloorfrac{n}{lfloorfrac{n}{y} floor} floor ]

    [ exttt{original author: 11Dimensions} ]

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  • 原文地址:https://www.cnblogs.com/hkr04/p/Integer-valued-function.html
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