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ε3f1(x0,y,t+t2)f1(x0,y,t)D0,0(f1)(x0,y,t+t1+t2)+ε3f1(x0,y,t)D0(f1)(x0,y,t+t1+t2)D0(f1)(x0,y,t+t2)+ε2f1(x0,y,t+t2)f1(x0,y,t)D0,0(f0)(x0,y,t+t1+t2)+ε2f1(x0,y,t)D0(f0)(x0,y,t+t2)D0(f1)(x0,y,t+t1+t2)+ε2f0(x0,y,t)f1(x0,y,t+t2)D0,0(f1)(x0,y,t+t1+t2)+ε2f0(x0,y,t+t2)f1(x0,y,t)D0,0(f1)(x0,y,t+t1+t2)+ε2f1(x0,y,t)g1(y,t+t2)D0,1(f1)(x0,y,t+t1+t2)+ε2f1(x0,y,t+t2)g1(y,t)D0,1(f1)(x0,y,t+t1+t2)+ε2f1(x0,y,t)D0(f0)(x0,y,t+t1+t2)D0(f1)(x0,y,t+t2)+ε2f0(x0,y,t)D0(f1)(x0,y,t+t1+t2)D0(f1)(x0,y,t+t2)+ε2g1(y,t)D0(f1)(x0,y,t+t1+t2)D1(f1)(x0,y,t+t2)+εf0(x0,y,t)f1(x0,y,t+t2)D0,0(f0)(x0,y,t+t1+t2)+εf0(x0,y,t+t2)f1(x0,y,t)D0,0(f0)(x0,y,t+t1+t2)+εf1(x0,y,t)g1(y,t+t2)D0,1(f0)(x0,y,t+t1+t2)+εf1(x0,y,t+t2)g1(y,t)D0,1(f0)(x0,y,t+t1+t2)+εf1(x0,y,t)D0(f0)(x0,y,t+t1+t2)D0(f0)(x0,y,t+t2)+εf0(x0,y,t)D0(f0)(x0,y,t+t2)D0(f1)(x0,y,t+t1+t2)+εg1(y,t)D1(f0)(x0,y,t+t2)D0(f1)(x0,y,t+t1+t2)+εf0(x0,y,t+t2)f0(x0,y,t)D0,0(f1)(x0,y,t+t1+t2)+εf0(x0,y,t)g1(y,t+t2)D0,1(f1)(x0,y,t+t1+t2)+εf0(x0,y,t+t2)g1(y,t)D0,1(f1)(x0,y,t+t1+t2)+εg1(y,t+t2)g1(y,t)D1,1(f1)(x0,y,t+t1+t2)+εf0(x0,y,t)D0(f0)(x0,y,t+t1+t2)D0(f1)(x0,y,t+t2)+εg1(y,t)D0(f0)(x0,y,t+t1+t2)D1(f1)(x0,y,t+t2)+εg1(y,t)D1(f1)(x0,y,t+t1+t2)D0(g1)(y,t+t2)+f0(x0,y,t+t2)f0(x0,y,t)D0,0(f0)(x0,y,t+t1+t2)+f0(x0,y,t)g1(y,t+t2)D0,1(f0)(x0,y,t+t1+t2)+f0(x0,y,t+t2)g1(y,t)D0,1(f0)(x0,y,t+t1+t2)+g1(y,t+t2)g1(y,t)D1,1(f0)(x0,y,t+t1+t2)+f0(x0,y,t)D0(f0)(x0,y,t+t1+t2)D0(f0)(x0,y,t+t2)+g1(y,t)D0(f0)(x0,y,t+t1+t2)D1(f0)(x0,y,t+t2)+g1(y,t)D1(f0)(x0,y,t+t1+t2)D0(g1)(y,t+t2)\displaystyle {{\varepsilon}}^{3} f_{1}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{3} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{0}\left(x_{0}, y, t\right) f_{1}\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{0}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}}^{2} f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}}^{2} g_{1}\left(y, t\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{1}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) f_{1}\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{1}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{1}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t\right) \mathrm{D}_{1}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t + t_{2}\right) f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{1, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{1}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t\right) \mathrm{D}_{1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(g_{1}\right)\left(y, t + t_{2}\right) + f_{0}\left(x_{0}, y, t + t_{2}\right) f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + f_{0}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + f_{0}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + g_{1}\left(y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{1, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) + g_{1}\left(y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{1}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) + g_{1}\left(y, t\right) \mathrm{D}_{1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(g_{1}\right)\left(y, t + t_{2}\right)

An elliptic curve is (in its simplest form) an equation of the form y2=x3+Ax+By^2 = x^3 + Ax + B for some values AA and BB. (Note that the plot of such a curve is not an ellipse! The name arises from the original study of these curves in connection with the computation of the arclength of a sector of an ellipse.)

\begin{CD} (0,0) @ >1/2>> (1,0) @>1/3>> (2,0) @. @. @. @. \ @V 1/2 VV @V2/3 VV @V1 VV @. @. @. @. \ (1,1) @ >2/3>> (2,1) @>1/2>> (3,1) @. @. @. @. \ @V 1/3 VV @V1/2 VV @V1 VV @. @. @. @. \ (2,2) @ > 1 >> (3,2) @>1/2>> (4,2) @>1/2>> (5,2)@>1/3>> (6,2)@. @. \ @. @. @V1/2VV @V2/3VV @V1 VV @. @. \ @. @. (5,3) @>2/3>> (6,3)@>1/2>> (7,3)@. @. \end{CD}

There is an n0>0n_0 > 0 such that an<2na_n < 2^n for all n>n0n > n_0.

broken ∫0∞x5 dx\int_0^\infty x^5\, \mathrm{d}x ?

abc text $ 123 abc

single $abc line

abc $a^b$ test


$a^b$

asdf $a^b$ asdf
asdf


$ cd ~
$ mkdir apps

$ nosetests pandas$
..........
OK (SKIP=2)