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1. I calculated this fractal with 4D complex numbers. (j and k axis fractal zoom)
2. Zooming Julia's crowd. I programmed this image sometime in the early millennium.
3. ~~~ least squares fitting example ~~~ For example: One phenomenon related to the time was that at time 1 the value 14 + 8j + 11l - 3n was measured. At time 2, a value of -3 + 10j + 15l - 7n was measured. It was found that the cause-effect of the phenomenon was approximately in accordance with the values: x0 = 1, y0 = 14 + 8j + 11l - 3n x1 = 2, y1 = -3 + 10j + 15l - 7n x2 = 3, y2 = -12 + 14j + 20l - 11n x3 = 4, y3 = -20 + 17j + 23l - 15n x4 = 5, y4 = -25 + 19j + 24l - 17n x5 = 6, y5 = -28 + 22j + 24l - 21n x6 = 7, y6 = -27 + 24j + 23l - 24n x7 = 8, y7 = -25 + 27j + 20l - 27n x8 = 9, y8 = -23 + 27j + 17l - 28n x9 = 10, y9 = -18 + 27j + 13l - 32n x10 = 11, y10 = -13 + 28j + 11l - 33n x11 = 12, y11 = -4 + 28j + 6l - 35n x12 = 13, y12 = 1 + 28j + 2l - 35n x13 = 14, y13 = 9 + 27j - 3l - 35n x14 = 15, y14 = 15 + 26j - 6l - 35n x15 = 16, y15 = 22 + 24j - 10l - 37n x16 = 17, y16 = 26 + 21j - 13l - 37n x17 = 18, y17 = 30 + 21j - 17l - 36n x18 = 19, y18 = 33 + 17j - 18l - 35n x19 = 20, y19 = 34 + 13j - 19l - 33n x20 = 21, y20 = 32 + 11j - 17l - 32n x21 = 22, y21 = 28 + 7j - 15l - 30n x22 = 23, y22 = 22 + 2j - 11l - 29n x23 = 24, y23 = 12 - 2j - 7l - 26n x24 = 25, y24 = -1 - 8j + 0l - 22n Based on the graphical analysis, it was decided to fit a cube parable to the phenomenon. Placing the measured response values in the least squares formula gives the cubic parabola coefficients: f(x) = k0 + k1x^1 + k2x^2 + k3x^3, where k0 = 31.479 + 2.882j + 2.508l + 1.642n k1 = -20.364 + 4.297j + 8.735l - 4.578n k2 = 2.068 - 0.179j - 1.029l + 0.124n k3 = -0.052 + 0.000j + 0.027l + 0.001n the recomplex cube parabola describes the phenomenon well. At initial values of x, the function values are: f (recomplex(1)) = 13.131 + 7.000j + 10.241l - 2.810n f (recomplex(2)) = -1.394 + 10.758j + 16.080l - 7.009n f (recomplex(3)) = -12.409 + 14.152j + 20.185l - 10.950n f (recomplex(4)) = -20.227 + 17.181j + 22.720l - 14.628n f (recomplex(5)) = -25.163 + 19.842j + 23.846l - 18.038n f (recomplex(6)) = -27.528 + 22.134j + 23.725l - 21.175n f (recomplex(7)) = -27.638 + 24.053j + 22.520l - 24.035n f (recomplex(8)) = -25.805 + 25.598j + 20.393l - 26.612n f (recomplex(9)) = -22.342 + 26.766j + 17.506l - 28.903n f (recomplex(10)) = -17.563 + 27.554j + 14.021l - 30.901n f (recomplex(11)) = -11.782 + 27.961j + 10.100l - 32.602n f (recomplex(12)) = -5.311 + 27.984j + 5.906l - 34.002n f (recomplex(13)) = 1.536 + 27.621j + 1.600l - 35.095n f (recomplex(14)) = 8.445 + 26.870j - 2.655l - 35.876n f (recomplex(15)) = 15.103 + 25.728j - 6.697l - 36.342n f (recomplex(16)) = 21.197 + 24.192j - 10.364l - 36.487n f (recomplex(17)) = 26.413 + 22.261j - 13.493l - 36.305n f (recomplex(18)) = 30.438 + 19.932j - 15.924l - 35.794n f (recomplex(19)) = 32.959 + 17.203j - 17.493l - 34.946n f (recomplex(20)) = 33.661 + 14.072j - 18.038l - 33.759n f (recomplex(21)) = 32.232 + 10.535j - 17.397l - 32.226n f (recomplex(22)) = 28.359 + 6.592j - 15.408l - 30.343n f (recomplex(23)) = 21.727 + 2.239j - 11.909l - 28.106n f (recomplex(24)) = 12.024 - 2.525j - 6.737l - 25.509n f (recomplex(25)) = -1.064 - 7.704j + 0.269l - 22.547n Algebra does not take a stand on the quality symbolically agreed for each axis. The unit vector l may represent, for example, snowfall as moles. The real axis, i and its companion features have remarkable automation where different qualities can communicate with each other. Multidimensional algebra is recursively generated from zero.
4. That is, you cannot talk about any technology at code level. A new way to generate pseudorandom numbers was in the next- and rnd-functions. But it doesn't hurt if the forum is forbidden to display the code. At least I enjoy reading other people's codes, and at best I try the other person's code myself.
5. // 64-bit random number generator. // CODE DELETED }
6. Scale over 20 years ago. I program now the fractal carbon colors. Each pixel contains 256 * 256 mosaic information, resulting in an image resolution of 216 (65 536).
8. By eliminating symbolic signs for multiplication, complex spaces are possible. I studied this problem for over a decade until it was resolved. Complex spaces fulfill all municipal conditions. Please find the code below to test the municipal rules. ! Moderator Note No one should have to reverse engineer your code to understand what you are talking about. Explain what you want to say.
9. #include <math.h> #include <stdio.h> typedef unsigned __int64 uint64; #define TRUE 1 #define FALSE 0 #define INC 3 #define NEW 7 #define SEQUEL 9 class rnd64 { public: rnd64(void); ~rnd64(void); uint64 rnd(void); private: int prime_number(int); uint64 next(void); int status; uint64 dx; uint64 dy; uint64 end; uint64 x, y; uint64 R16A; uint64 R16B; uint64 Z; uint64 increaseA; uint64 increaseB; }; rnd64::rnd64(void) { dx=0xfedca201L; dy=0x012357bfL; R16A=(uint64)1; R16B=(uint64)2; Z=Z=0x00; x=5; y=3; end=2; increaseA=0x012357bfL; increaseB=0xfedca201L; } rnd64::~rnd64(void) { } int rnd64::prime_number(int st) { if (st == INC) { x+=2; y=3; if (x>=(1<<16)) x=5; end=unsigned(sqrt(double(x)+0.25)); return SEQUEL; } else { if (!(x%y)) return FALSE; if ((y+=2)>end) return TRUE; return SEQUEL; } } inline uint64 uabs(uint64 &a, uint64 &b) { return a>b? a-b: b-a; } inline void swap(uint64 &a, uint64 &b) { uint64 c=a; a=b; b=c; } uint64 rnd64::next(void) { status=prime_number(NEW); if (status == TRUE) { increaseB-=increaseA; increaseA+=x; prime_number(INC); } else if (status == FALSE) { prime_number(INC); } R16A -= increaseA; increaseA-=dx; R16B += increaseB; increaseB+=dy; if (status==TRUE) swap(R16A, R16B); R16A += (R16A>>32)^(R16B<<32); R16B -= (R16A<<32)^(R16B>>32); return R16A^R16B; } uint64 rnd64::rnd(void) { uint64 p1=next(); if (status==TRUE) { uint64 a, b; uint64 p2=next(); a=b = Z; a=b = Z; ++a[unsigned(p1%2)]; ++b[unsigned(p2%2)]; uint64 A=uabs(a, a); uint64 B=uabs(b, b); if (Z == Z) { if (unsigned(next()%2)) { A=1; B=0; } else { A=0; B=1; } } if (A < B) { ++Z[unsigned(p1%2)]; return p1; } else { ++Z[unsigned(p2%2)]; return p2; } } else { ++Z[unsigned(p1%2)]; return p1; } } void main(void) { rnd64 R; int st=0; uint64 bit; bit=bit=0x00; for (;;) { uint64 P=R.rnd(); ++bit[unsigned(P%2)]; if (st==0 && bit>bit) { printf("%d", bit>bit? 1: 0); st=1; } else if (st==1 && bit>bit) { printf("%d", bit>bit? 1: 0); st=0; } } }
10. k0∑xj0 + k1∑xj1 + k2∑xj2 +,…,+ kn∑xjn+0 = ∑yjxj0 k0∑xj1 + k1∑xj2 + k2∑xj3 +,…,+ kn∑xjn+1 = ∑yjxj1 k0∑xj2 + k1∑xj3 + k2∑xj4 +,…,+ kn∑xjn+2 = ∑yjxj2 . . . k0∑xjn + k1∑xjn+1 + k2∑xjn+2 +,…,+ kn∑xjn+n = ∑yjxjn One of the most beautiful algebra formulas is the least squares polynomial formula.
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