e83052c2-73e2-4b32-947c-4322d05301f5 6 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Discrete differential "roots" at(3) . . . productive

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lo hi N f q algo u lo hi lo hi lo - N / + (range : j 1 u length range ρ j el at3 u j el f x - x roots u j el q x - x roots at : for u ρ augment 31 line :

x q x : x f x 3^: 12 mat

Δ 1.25 - 1.25 100 f q algo :

x q x f Δ 3 1 sys

x q x f Δ 31 sys

# # at

r 0.25 - : x y c x 2^y 2^+ r + : lo 0.0001 : hi 0.4999 : N 99 : 13 mat u lo hi lo hi lo - N / + (range : 41 sys

C j 1 u length range rho j el ρ u j el y c y roots at : 11 line for rho 21 line :

dummy

# # # at

u C augment u C - augment u - C augment u - C - augment 4 1 sys

r 0.5 : x f r x 2^- sqrt ± : 21 sys

x f r sqrt 0 + 75 black 1 5 mat 2 1 sys

Otherwise profitable Discrete-Difference "roots" at(3) . . .

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r 0.125 - :

x y u Clear 1

x y Γ x 2^y 2^+ r + :

x y Ω 1 x 4^(* 2 x 2^* y * + 3 x * y 2^* - 1 y 3^(* + :

Ω Γ 0.35355 0.04526 - 1 2 mat 3 1 sys

Intersection limit

t0 1 time :

lo 0.35 : hi 0.36 : N 1000 : 13 mat

find low limit u lo hi lo hi lo - N / + (range :

last value "on error" ... NO real root. data j 1 u length range rho j el at3 u j el y Ω y roots u j el y Γ y roots at : break try for rho 21 line :

data data eval :

1 time t0 - 45.9 s *

"real root" exists in the positive region left of 0.35355. On the other side, the differential region is negative ... thus => "NO real root".

k 1 data rows range : junction u k el data augment : junction junction eval : 31 sys

u 0.35355 : Yjunction u y Γ u y Ω ≡ y roots : 21 sys

1 time t0 - 46.2 s *

Yjunction 0.04526 -

data data rows row 0.0525 11 mat

junction 0.35355 0.5 0.35355 0.05 - 2 2 mat 2 1 sys

► NO real root at(3)

revisit Electron at(2)

Solve/evaluate contour

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x y Clear 1

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

# # at

x y axes x y / - :

εlectron 816 π * sqrt * : 351.67021

isolate εlectron 351.67021 :

L 1.78568578 : H 14.24552935 : N 200 : 13 mat x L H N xd : 21 line

x y Φ 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ εlectron / * - 0.005 + :

n 1 x length range Esol n el at2 x n el y Φ vectorize y solve at : for

plot branch_1 x Esol augment : branch_2 x Esol - augment : branch_3 x - Esol augment : branch_4 x - Esol - augment : 41 sys :

round limits:  more decimals solve complains 1.78568577383018 14.2455293828432 21 mat 6 round vectorize 1.785686 14.245529 21 mat

plot axes 2 1 sys

L 1.78568578 : H 14.24552935 : N 200 : 13 mat

branch_1 1.7857 2.8178 108 -^* 1.8483 0.2644 1.9109 0.378 1.9735 0.4678 2.0361 0.5457 2.0987 0.6161 2.1614 0.6815 2.224 0.743 2.2866 0.8016 2.3492 0.8578 2.4118 0.912 2.4744 0.9647 2.537 1.0159 2.5996 1.0659 2.6623 1.1148 2.7249 1.1628 2.7875 1.21 2.8501 1.2564 2.9127 1.3021 2.9753 1.3471 3.0379 1.3916 3.1005 1.4355 222 mat

plot