Qabo rahyo0518

Raha ayaa ku boodaya khadka lambarka waxaadna isku dayeysaa inaad qabsato. Boodka iyo qabashada marwalba way isbedelayaan. Raha wuxuu ka bilaabmaa meesha $$s \in \mathbb{Z}$$ dhaqaaq kasta oo ay sameysana wuxuu ku boodaa masaafo $$z \in \mathbb{Z}$$ (haddii $$z>0$$ , wuu boodayaa midigta, haddii kale bidix). $$z$$ waa la mid boodbooyin kasta. Snapping wuxuu ka kooban yahay qeexida booska integer. Midna ma garanayo $$z$$ iyo $$s$$ midna. Waxaan muujineynaa inay jirto wado markasta lagu soo qabto raha.

Marka hore, $$a_1 = s$$ iyo $$a_{n+1} = a_n + z = s + n \cdot z$$ wata $$s,z \in \mathbb{Z}$$ .

$$h:\mathbb{N} \to \mathbb{Z}^2: h(2^k r) = \left ( (-1)^{k+1} \left \lfloor \frac{k+1}{2} \right \rfloor, (-1)^{\frac{r+1}{2}} \left \lfloor \frac{r+1}{4} \right \rfloor \right )$$

sida shaqada u xilsaaraysa (si sax ah) tiro lambar ah tirooyin dhan lambar kasta oo dabiici ah. Xulashada shaqadan waa iyada oo loo marayo shaqooyinka $$f(n) = (-1)^n \left \lfloor \frac{n}{2} \right \rfloor$$ , the $$\mathbb{N}$$ saaran $$\mathbb{Z}$$ iyo $$g(2^kr) = (k+1, \frac{r+1}{2})$$ , oo $$\mathbb{N}$$ saaran $$\mathbb{N}^2$$ khariidad si asluubeysan, dhiirigalin leh.

Waxaan hadda muujineynaa qiyaasta $$h$$ ( $$h$$ sidoo kale waa cirbad, laakiin uma baahnin hantidan).

Ha $$(x,y) = (2^{k_1} r_1, 2^{k_2} r_2) \in\mathbb{Z}^2$$ . Laakiin markaa

$$h \left ( 2^{2 \cdot 2^{k_1} r_1 - 1} \cdot (4 \cdot 2^{k_2} r_2 - 1) \right ) = (2^{k_1} r_1, 2^{k_2} r_2) = (x,y).$$

Sidaa awgeed: $$\forall (s,z) \in \mathbb{Z}^2 \, \exists \, m \in \mathbb{N}$$ leh $$h(m) = (x_m,y_m) = (s, z)$$ .

Tusaale ahaan, haddii ay tahay waqtigeenna inaan u dhaqaaqno $$n = 88$$ , waxaan xisaabineynaa $$h(88)=(2,3)$$ waxaan doorannaa $$2 + 88 \cdot 3 = 266$$ oo ah booska.

Ka dib marka sida saxda ah $$m$$ ula dhaqaaqdo $$x_m + m \cdot y_m = s + m \cdot z = a_m$$ doorashadu waxay ku $$x_m + m \cdot y_m = s + m \cdot z = a_m$$ raha.

Marka lagu daro $$h$$ , shaqooyin kale oo badan sida shaqada isku dheelitirka ee Cantor ama wareegga biiyaha ayaa suuragal ah.

Halkan waa hirgelinta fudud ee JavaScript:

See the Pen catch the frog by David Vielhuber (@vielhuber) on CodePen.

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