commit 7873ce23609d3ffc8f8f5b5d49f76072fe252411 Author: Adamo Date: Wed Dec 23 16:45:08 2020 -0500 Init and final. diff --git a/2D_oscillator/formalism.motes b/2D_oscillator/formalism.motes new file mode 100644 index 0000000..5856a33 --- /dev/null +++ b/2D_oscillator/formalism.motes @@ -0,0 +1,19 @@ +ℋ❙φ❭ = (ℋ𝓍𝓎 + ℋ𝓏)❙φ❭ = E❙φ❭. + +ℋ𝓍𝓎 = (P²𝓍 + P²𝓎)/(2μ) + ½ μω² (X² + Y²). + +ℋ𝓏 = P²𝓏/2μ. + +❙φ❭ = ❙φ𝓏𝓎❭ ⛒ ❙φ𝓏❭. + +ℋ𝓍𝓎❙φ❭ = E𝓍𝓎❙φ❭. + +E = E𝓍𝓎 + E𝓏. + +❬z❙φ𝓏❭ = (2πħ)⁻¹ exp(ιp𝓏z/ħ). + +E𝓏 = p𝓏²/2μ where p𝓏 is an arbitrary real constant. + +ℰ = ℰ𝓍𝓎 ⛒ ℰ𝓏. + + diff --git a/classbfs120904402944051.pdf b/classbfs120904402944051.pdf new file mode 100644 index 0000000..416d278 Binary files /dev/null and b/classbfs120904402944051.pdf differ diff --git a/griffiths/3.7 b/griffiths/3.7 new file mode 100644 index 0000000..b9903f8 --- /dev/null +++ b/griffiths/3.7 @@ -0,0 +1,30 @@ +3.7 + +Suppose f and g are eigenfunctions of Q, with eigenvalue q. Show any linear combination of f and g are eigenfunctions of Q with eigenvalue q. + +|Qf> = q|f> +|Qg> = w|g> + +A and B are (possibly complex) constants. + +|Q(A*f+B*g)> = A|Qf> + B|Qg> = Aq|f> + Bq|g> + = q(A|f> + B|g>). + +QED + +Check that f(x)=exp(x) and g(x)=exp(-x) are eigenfunctions of the operator d^2/dx^2, with the same eigenvalue. construct two linear combiations of f and g that are orthogonal eigenfunctions on the interval [-1,1] + + +if Q = d^2/dx^2 then we have the differential equation + +f'' = qf + +(e^x)'' = e^x, so this is an eigenfunction with q=1. + +(e^-x)'' = e^-x, so this is an eigenfunction with q=1. + +e^x + e^-x is also an eigenfunction of q, per earlier proof, and so is e^x - e^-x. These functions are just 2*sinh and 2*cosh, which are orthogonal functions because sinh is odd and cosh is even. + + + + diff --git a/griffiths/3.8 b/griffiths/3.8 new file mode 100644 index 0000000..9a1dae2 --- /dev/null +++ b/griffiths/3.8 @@ -0,0 +1,21 @@ +3.8 + +a) + +check that the eigenvalues of the hermitian operator in example 3.1 are real. show that the eigenfunctions are orthogonal. + +Qf = if' + +the eigenvalues are 0,+- 1, etc., which are obviously real. + + +pick two arbitrary eigenfunctions: + +f = A exp(-i q phi) +g = A exp(-i q' phi) + + = A*A int[exp(i q phi) exp(-i q' phi)] dphi[0,2pi] + = A*A int[exp(i (q - q') phi)] dphi[0,2pi] + = A*A [i(q-q')]^-1 [exp(i(q-q') phi)]|[0,2pi] + + diff --git a/hw/12/H12.pdf b/hw/12/H12.pdf new file mode 100644 index 0000000..b43f9a5 Binary files /dev/null and b/hw/12/H12.pdf differ diff --git a/hw/12/HW12.motes b/hw/12/HW12.motes new file mode 100644 index 0000000..ee6491e --- /dev/null +++ b/hw/12/HW12.motes @@ -0,0 +1,127 @@ +𝓗𝓍𝓎 = ½ (-∇² + ρ²). + +ρ² = x² + y² and ħ = m = ω = 1. + +Griffith's Eq. 2.71: + + Hₙ₊₁(ξ) = 2ξ Hₙ(ξ) - 2n Hₙ₋₁(ξ) + + + + +In 2D cartesian coordinates, the del operator is defined + + ∇f = [∂/∂x f, ∂/∂y f]. + + +∇²f = ∇⋅∇f = ∇⋅[∂/∂x f, ∂/∂y f] + = ∂²/∂x² f + ∂²/∂y² f. + +∴ ∇² = ∂²/∂²x + ∂²/∂²y. + +Then, + + 𝓗𝓍𝓎 = ½ (-(∂²/∂²x + ∂²/∂²y) + (x² + y²)). + +𝓗𝓍𝓎 = ½ (-(∂²/∂²x + ∂²/∂²y) + (x² + y²)) + = ½ (x² - ∂²/∂²x + y² - ∂²/∂²y) + = ½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y). + +𝓗𝓍 + 𝓗𝓎 = ½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y). + +∎ + +The Schrodinger Equation then reads, + + [½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y)] Ψ = (E𝓍 + E𝓎) Ψ. + +Assuming a separable solution Ψ(x,y) = X(x) Y(y), with E = E𝓍 + E𝓎. + +[½(x² - ∂²/∂²x) + ½(y² - ∂²/∂²y)] X(x) Y(y) = (E𝓍 + E𝓎) X(x) Y(y). + +½(x² - ∂²/∂²x) X(x) Y(y) + ½(y² - ∂²/∂²y) X(x) Y(y) + = (E𝓍 + E𝓎) X(x) Y(y). + +[1/X(x)] [½(x² - ∂²/∂²x) X(x)] + [1/Y(y)] [½(y² - ∂²/∂²y) Y(y)] + = (E𝓍 + E𝓎). + +So, I have two differential equations, + + ½(x² - ∂²/∂²x) X(x) = E𝓍 X(x), and + ½(y² - ∂²/∂²y) Y(y) = E𝓎 Y(y). + +The solutions to these differential equations are the same as for the 1D harmonic oscillator. They have eigenvalues (n + 1/2), where ħ = ω = 1, with n = 0,1,2,... . + +∴ the eigenvalues for the combined operator are n𝓍 + n𝓎 + 1. + +The degeneracy is pretty obvious, just from counting the possibilities: there is n+1 degeneracy for each value of n = n𝓍 + n𝓎. + +So, + n degeneracy + ───────────────── + 0 1 + 1 2 + 2 3 + 3 4 + 4 5 + 5 6 + + +The Hermite polynomials help to generate the eigenstates of this: + +Hₙ(x) = (-1)ⁿ exp(x²) d/dxⁿ exp(-x²/2) = (2x - d/dx)ⁿ * 1. + +The first six polynomials are + +H₀(x) = 1. +H₁(x) = 2x. +H₂(x) = 4x² - 2. +H₃(x) = 8x³ - 12x. +H₄(x) = 16x⁴ - 48x² + 12. +H₅(x) = 32x⁵ - 160x³ + 120x. +H₆(x) = 64x⁶ - 480x⁴ + 720x² - 120. + + +The wave functions involving these polynomials, with the unitizations given in the intro, are + + Ψₙₘ(x) = π^(-1/4) 1/√(2ⁿ n!) Hₙ(x) exp(-x²/2) + π^(-1/4) 1/√(2ᵐ m!) Hₘ(y) exp(-y²/2). + +Ψₙₘ(x) = 1/√π 1/√(2ⁿ n! 2ᵐ m!) Hₙ(x) Hₘ(y) exp(-(x²/2 + y²/2)). + +For n = {1,...,∞}, the lowest possible energy is 3, and this level has no degeneracy. + +For additional levels, + + + + +a𝓍 = 1/√2(x + ιp𝓍) +a𝓎 = 1/√2(y + ιp𝓎) +======= +The lowering operators are + + a𝓍 = 1/√2 (x + ιp𝓍) and + a𝓎 = 1/√2 (y + ιp𝓎). + +They are not hermitian, but x,y and p𝓍,p𝓎 are, so the raising operators are + + a𝓍᛭ = 1/√2 (x - ιp𝓍) and + a𝓎᛭ = 1/√2 (y - ιp𝓎). + +Applying these to the ground state ❙00❭, I can find the first six states, with normalization: + +a𝓍᛭❙00❭ = 1/√2 (x - ιp𝓍)❙00❭ + = 1/√2 (x❙00❭ - ιp𝓍❙00❭) + + + + + + +b) & c) + +I'm still working out the algebra, here. I will try to finish it as soon as I can, but I know I also have new work to do. + + +I finished much of this assignment, but need to get done faster in the future. diff --git a/hw/12/HW12.motes.ps b/hw/12/HW12.motes.ps new file mode 100644 index 0000000..80b6f57 --- /dev/null +++ b/hw/12/HW12.motes.ps @@ -0,0 +1,3883 @@ +%!PS-Adobe-3.0 +%%Title: Otho Ulrich - HW12 +%%Creator: paps version 0.6.7 by Dov Grobgeld +%%Pages: (atend) +%%BoundingBox: 0 0 595 842 +%%Orientation: Portrait +%%EndComments +%%BeginProlog +/papsdict 1 dict def +papsdict begin + +/inch {72 mul} bind def +/mm {1 inch 25.4 div mul} bind def + +% override setpagedevice if it is not defined +/setpagedevice where { + pop % get rid of its dictionary + /setpagesize { + 3 dict begin + /pageheight exch def + /pagewidth exch def + /orientation 0 def + % Exchange pagewidth and pageheight so that pagewidth is bigger + pagewidth pageheight gt { + pagewidth + /pagewidth pageheight def + /pageheight exch def + /orientation 3 def + } if + 2 dict + dup /PageSize [pagewidth pageheight] put + dup /Orientation orientation put + setpagedevice + end + } def +} +{ + /setpagesize { pop pop } def +} ifelse +/duplex { + statusdict /setduplexmode known + { statusdict begin setduplexmode end } {pop} ifelse +} def +/tumble { + statusdict /settumble known + { statusdict begin settumble end } {pop} ifelse +} def +% Turn the page around +/turnpage { + 90 rotate + 0 pageheight neg translate +} def +%%EndProlog +%%BeginSetup +% User settings +/pagewidth 595 def +/pageheight 842 def +pagewidth pageheight setpagesize +/column_width 523 def +/bodyheight 756 def +/lmarg 36 def +/ytop 792 def +/do_separation_line true def +/do_landscape false def +/do_tumble true def +/do_duplex true def +% Procedures to translate position to first and second column +/lw 20 def % whatever +/setnumcolumns { + /numcolumns exch def + /firstcolumn { /xpos lmarg def /ypos ytop def} def + /nextcolumn { + do_separation_line { + xpos column_width add gutter_width 2 div add % x start + ytop lw add moveto % y start + 0 bodyheight lw add neg rlineto 0 setlinewidth stroke + } if + /xpos xpos column_width add gutter_width add def + /ypos ytop def + } def +} def + +1 setnumcolumns +/showline { + /y exch def + /s exch def + xpos y moveto + column_width 0 rlineto stroke + xpos y moveto /Helvetica findfont 20 scalefont setfont s show +} def +/paps_bop { % Beginning of page definitions + papsdict begin + gsave + do_landscape {turnpage} if + % ps2pdf gets wrong orientation without this! + /Helvetica findfont setfont 100 100 moveto ( ) show + firstcolumn + end +} def + +/paps_eop { % End of page cleanups + grestore +} def +/papsdict 1 dict def +papsdict begin + +/conicto { + /to_y exch def + /to_x exch def + /conic_cntrl_y exch def + /conic_cntrl_x exch def + currentpoint + /p0_y exch def + /p0_x exch def + /p1_x p0_x conic_cntrl_x p0_x sub 2 3 div mul add def + /p1_y p0_y conic_cntrl_y p0_y sub 2 3 div mul add def + /p2_x p1_x to_x p0_x sub 1 3 div mul add def + /p2_y p1_y to_y p0_y sub 1 3 div mul add def + p1_x p1_y p2_x p2_y to_x to_y curveto +} bind def +/start_ol { gsave } bind def +/end_ol { closepath fill grestore } bind def +/draw_char { fontdict begin gsave 0.001000 dup scale last_x cvi last_y cvi translate load exec end grestore} def +/goto_xy { fontdict begin /last_y exch string_y cvs def /last_x exch string_x cvs def end } def +/goto_x { fontdict begin /last_x exch string_x cvs def end } def +/fwd_x { fontdict begin /last_x exch last_x cvi add string_x cvs def end } def +/c /curveto load def +/x /conicto load def +/l /lineto load def +/m /moveto load def +end +/paps_exec { + 1 dict begin + /ps exch def + /len ps length def + /pos 0 def + + % Loop over all the characters of the string + { + pos len eq {exit} if + + % Get character at pos + /ch ps pos 1 getinterval def + + % check for + + (+) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /yp ps pos 8 add 8 getinterval cvi def + /pos 16 pos add def + papsdict begin xp yp goto_xy end + } { + (*) ch eq { + /pos 1 pos add def + /xp ps pos 8 getinterval cvi def + /pos 8 pos add def + papsdict begin xp goto_x end + } { (>) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp 2 mul fwd_x end + } { (-) ch eq { + /pos 1 pos add def + /xp ps pos 4 getinterval cvi def + /pos 4 pos add def + papsdict begin xp neg 2 mul fwd_x end + } { + % Must be a 3 char sym. Load and exec + /name ps pos 3 getinterval def + papsdict begin name draw_char end + /pos 3 pos add def + } ifelse + } ifelse + } ifelse + } ifelse + } loop + end +} def +/fontdict 1 dict dup begin 16 string dup /string_x exch def /last_x exch def 16 string dup /string_y exch def /last_y exch def end def +papsdict begin fontdict begin +/AAA { start_ol +0 8784 m +1514 8784 l +2143 2326 l +2900 6480 l +4339 6480 l +5218 2326 l +5712 8784 l +7239 8784 l +6229 0 l +4614 0 l +3616 4593 l +2683 0 l +1080 0 l +0 8784 l +7272 fwd_x +end_ol + } def +/BAA { start_ol +6469 360 m +5871 108 5248 -18 x +4626 -144 3933 -144 x +2283 -144 1411 731 x +540 1606 540 3252 x +540 4846 1379 5771 x +2219 6696 3668 6696 x +5131 6696 5937 5823 x +6745 4950 6745 3364 x +6745 2664 l +2289 2664 l +2295 1938 2747 1580 x +3199 1224 4097 1224 x +4691 1224 5266 1396 x +5842 1569 6469 1944 x +6469 360 l +5014 3960 m +5002 4635 4664 4981 x +4326 5328 3669 5328 x +3076 5328 2723 4970 x +2371 4612 2307 3954 x +5014 3960 l +7272 fwd_x +end_ol + } def +/CAA { start_ol +4644 5598 m +4644 9144 l +6358 9144 l +6358 0 l +4644 0 l +4644 1019 l +4368 441 3930 148 x +3493 -144 2900 -144 x +1773 -144 1150 751 x +528 1647 528 3276 x +528 4927 1159 5811 x +1791 6696 2958 6696 x +3487 6696 3907 6421 x +4326 6147 4644 5598 x +2248 3276 m +2248 2349 2566 1822 x +2883 1296 3440 1296 x +3998 1296 4321 1822 x +4644 2349 4644 3276 x +4644 4203 4321 4729 x +3998 5256 3440 5256 x +2883 5256 2566 4729 x +2248 4203 2248 3276 x +7272 fwd_x +end_ol + } def +/EAA { start_ol +1103 1512 m +3040 1512 l +3040 7200 l +1238 6768 l +1238 8352 l +3053 8784 l +4726 8784 l +4726 1512 l +6663 1512 l +6663 0 l +1103 0 l +1103 1512 l +7272 fwd_x +end_ol + } def +/FAA { start_ol +793 8784 m +6334 8784 l +6334 7554 l +3363 0 l +1597 0 l +4450 7272 l +793 7272 l +793 8784 l +7272 fwd_x +end_ol + } def +/GAA { start_ol +640 432 m +640 2448 l +1145 1955 1716 1697 x +2289 1440 2871 1440 x +3551 1440 3868 1769 x +4186 2099 4186 2820 x +4186 7272 l +2066 7272 l +2066 8784 l +5918 8784 l +5918 2820 l +5918 1227 5245 541 x +4574 -144 3029 -144 x +2471 -144 1864 1 x +1256 146 640 432 x +7272 fwd_x +end_ol + } def +/HAA { start_ol +4109 3096 m +3146 3096 2765 2855 x +2383 2614 2383 2030 x +2383 1595 2648 1337 x +2912 1080 3364 1080 x +4045 1080 4421 1580 x +4797 2082 4797 2981 x +4797 3096 l +4109 3096 l +6504 3751 m +6504 0 l +4797 0 l +4797 776 l +4485 334 3998 94 x +3511 -144 2923 -144 x +1803 -144 1177 444 x +552 1032 552 2088 x +552 3229 1297 3774 x +2043 4320 3604 4320 x +4797 4320 l +4797 4621 l +4797 5005 4494 5202 x +4191 5400 3604 5400 x +2988 5400 2409 5262 x +1831 5125 1203 4824 x +1203 6264 l +1773 6485 2360 6590 x +2947 6696 3604 6696 x +5207 6696 5855 6052 x +6504 5409 6504 3751 x +7272 fwd_x +end_ol + } def +/IAA { start_ol +6287 4261 m +6287 0 l +4579 0 l +4579 3997 l +4579 4712 4377 5019 x +4174 5328 3717 5328 x +3252 5328 2984 4911 x +2718 4495 2718 3757 x +2718 0 l +1010 0 l +1010 6552 l +2718 6552 l +2718 5577 l +2900 6111 3340 6403 x +3781 6696 4403 6696 x +5337 6696 5811 6084 x +6287 5473 6287 4261 x +7272 fwd_x +end_ol + } def +/JAA { start_ol +2548 1512 m +6241 1512 l +6241 0 l +675 0 l +675 1465 l +1614 2466 l +3288 4248 3646 4659 x +4086 5166 4279 5556 x +4474 5947 4474 6330 x +4474 6918 4118 7239 x +3763 7560 3111 7560 x +2648 7560 2068 7375 x +1490 7191 862 6840 x +862 8424 l +1490 8669 2092 8798 x +2694 8928 3246 8928 x +4638 8928 5433 8277 x +6229 7627 6229 6503 x +6229 5985 6055 5531 x +5882 5078 5459 4525 x +5149 4125 3751 2724 x +2994 1965 2548 1512 x +7272 fwd_x +end_ol + } def +/KAA { start_ol +2889 4388 m +2889 4695 3100 4903 x +3312 5112 3616 5112 x +3927 5112 4138 4903 x +4350 4695 4350 4388 x +4350 4088 4138 3880 x +3927 3672 3616 3672 x +3312 3672 3100 3876 x +2889 4082 2889 4388 x +3616 7488 m +3018 7488 2741 6750 x +2466 6012 2466 4388 x +2466 2772 2741 2034 x +3018 1296 3616 1296 x +4221 1296 4496 2034 x +4773 2772 4773 4388 x +4773 6012 4496 6750 x +4221 7488 3616 7488 x +722 4388 m +722 6661 1444 7794 x +2166 8928 3616 8928 x +5072 8928 5794 7797 x +6517 6666 6517 4388 x +6517 2116 5794 985 x +5072 -144 3616 -144 x +2166 -144 1444 988 x +722 2122 722 4388 x +7272 fwd_x +end_ol + } def +/LAA { start_ol +3615 3960 m +3035 3960 2676 3585 x +2318 3211 2318 2595 x +2318 1978 2676 1600 x +3035 1224 3615 1224 x +4197 1224 4558 1606 x +4919 1990 4919 2595 x +4919 3211 4560 3585 x +4203 3960 3615 3960 x +2331 4681 m +1667 4908 1320 5394 x +974 5881 974 6585 x +974 7669 1684 8298 x +2395 8928 3615 8928 x +4843 8928 5553 8298 x +6264 7669 6264 6585 x +6264 5887 5923 5400 x +5583 4914 4932 4681 x +5659 4457 6070 3880 x +6482 3304 6482 2497 x +6482 1233 5736 544 x +4990 -144 3616 -144 x +2248 -144 1503 544 x +757 1233 757 2497 x +757 3309 1173 3885 x +1590 4462 2331 4681 x +2513 6453 m +2513 5934 2809 5630 x +3106 5328 3616 5328 x +4133 5328 4429 5630 x +4726 5934 4726 6453 x +4726 6964 4429 7261 x +4133 7560 3616 7560 x +3111 7560 2812 7259 x +2513 6959 2513 6453 x +7272 fwd_x +end_ol + } def +/MAA { start_ol +2635 6264 m +4591 6264 l +4591 4104 l +2635 4104 l +2635 6264 l +2635 2160 m +4591 2160 l +4591 0 l +2635 0 l +2635 2160 l +7272 fwd_x +end_ol + } def +/NAA { start_ol +3223 3960 m +2296 3960 l +2296 5472 l +3223 5472 l +3868 5472 4224 5728 x +4579 5985 4579 6450 x +4579 6940 4224 7213 x +3868 7488 3223 7488 x +2730 7488 2166 7358 x +1603 7230 997 6984 x +997 8568 l +1603 8742 2189 8834 x +2776 8928 3317 8928 x +4702 8928 5477 8327 x +6252 7726 6252 6670 x +6252 5893 5806 5401 x +5360 4908 4532 4756 x +5472 4592 5964 4007 x +6458 3422 6458 2470 x +6458 1195 5641 525 x +4826 -144 3270 -144 x +2606 -144 1963 -32 x +1320 77 733 288 x +733 1872 l +1285 1625 1926 1496 x +2566 1368 3270 1368 x +3974 1368 4385 1690 x +4797 2014 4797 2561 x +4797 3232 4385 3595 x +3974 3960 3223 3960 x +7272 fwd_x +end_ol + } def +/OAA { start_ol +1191 216 m +1191 1800 l +1654 1505 2080 1364 x +2506 1224 2900 1224 x +3832 1224 4323 1847 x +4813 2470 4843 3710 x +4567 3331 4138 3141 x +3710 2952 3123 2952 x +1942 2952 1297 3703 x +651 4456 651 5836 x +651 7311 1373 8119 x +2095 8928 3416 8928 x +5019 8928 5744 7857 x +6469 6786 6469 4397 x +6469 2127 5583 991 x +4696 -144 2923 -144 x +2512 -144 2074 -51 x +1638 39 1191 216 x +3463 4392 m +4021 4392 4317 4808 x +4614 5224 4614 6015 x +4614 6799 4317 7215 x +4021 7632 3463 7632 x +2900 7632 2597 7215 x +2296 6799 2296 6015 x +2296 5230 2597 4810 x +2900 4392 3463 4392 x +7272 fwd_x +end_ol + } def +/PAA { start_ol +4074 6854 m +1860 3384 l +4074 3384 l +4074 6854 l +3921 8784 m +5736 8784 l +5736 3384 l +6699 3384 l +6699 1872 l +5736 1872 l +5736 0 l +4074 0 l +4074 1872 l +598 1872 l +598 3560 l +3921 8784 l +7272 fwd_x +end_ol + } def +/QAA { start_ol +2683 7344 m +2683 4752 l +3393 4752 l +4244 4752 4587 5042 x +4932 5332 4932 6048 x +4932 6763 4587 7053 x +4244 7344 3393 7344 x +2683 7344 l +950 8784 m +3328 8784 l +5143 8784 5938 8139 x +6734 7494 6734 6048 x +6734 4601 5938 3956 x +5143 3312 3328 3312 x +2683 3312 l +2683 0 l +950 0 l +950 8784 l +7272 fwd_x +end_ol + } def +/RAA { start_ol +505 8784 m +2571 8784 l +3616 4911 l +4655 8784 l +6734 8784 l +6734 0 l +5242 0 l +5242 7027 l +4315 3168 l +2935 3168 l +1995 7027 l +1995 0 l +505 0 l +505 8784 l +7272 fwd_x +end_ol + } def +/SAA { start_ol +6446 0 m +986 0 l +986 8784 l +6446 8784 l +6446 7272 l +2718 7272 l +2718 5328 l +6094 5328 l +6094 3816 l +2718 3816 l +2718 1512 l +6446 1512 l +6446 0 l +7272 fwd_x +end_ol + } def +/TAA { start_ol +2994 3891 m +1685 4386 1220 4930 x +757 5475 757 6395 x +757 7576 1519 8251 x +2283 8928 3615 8928 x +4221 8928 4825 8781 x +5430 8636 6023 8352 x +6023 6696 l +5465 7083 4889 7285 x +4314 7488 3750 7488 x +3123 7488 2787 7239 x +2453 6990 2453 6527 x +2453 6168 2696 5933 x +2940 5699 3715 5415 x +4461 5137 l +5519 4749 6017 4109 x +6517 3469 6517 2494 x +6517 1166 5733 510 x +4950 -144 3364 -144 x +2712 -144 2057 19 x +1404 183 793 504 x +793 2304 l +1486 1794 2134 1544 x +2783 1296 3416 1296 x +4056 1296 4408 1588 x +4761 1881 4761 2407 x +4761 2804 4525 3102 x +4291 3402 3845 3573 x +2994 3891 l +7272 fwd_x +end_ol + } def +/UAA { start_ol +4485 0 m +2754 0 l +2754 7272 l +528 7272 l +528 8784 l +6710 8784 l +6710 7272 l +4485 7272 l +4485 0 l +7272 fwd_x +end_ol + } def +/VAA { start_ol +3615 7344 m +2953 7344 2647 6645 x +2342 5947 2342 4388 x +2342 2836 2647 2137 x +2953 1440 3615 1440 x +4285 1440 4590 2137 x +4896 2836 4896 4388 x +4896 5947 4590 6645 x +4285 7344 3615 7344 x +540 4388 m +540 6631 1317 7779 x +2095 8928 3616 8928 x +5143 8928 5920 7779 x +6699 6631 6699 4388 x +6699 2152 5920 1003 x +5143 -144 3616 -144 x +2095 -144 1317 1003 x +540 2152 540 4388 x +7272 fwd_x +end_ol + } def +/WAA { start_ol +4051 8424 m +4051 6552 l +6300 6552 l +6300 5256 l +4051 5256 l +4051 2124 l +4051 1680 4264 1488 x +4479 1296 4979 1296 x +6300 1296 l +6300 0 l +4855 0 l +3376 0 2853 468 x +2331 937 2331 2206 x +2331 5256 l +651 5256 l +651 6552 l +2331 6552 l +2331 8424 l +4051 8424 l +7272 fwd_x +end_ol + } def +/XAA { start_ol +6287 4213 m +6287 0 l +4579 0 l +4579 3952 l +4579 4653 4377 4954 x +4174 5256 3717 5256 x +3246 5256 2982 4847 x +2718 4439 2718 3714 x +2718 0 l +1010 0 l +1010 9144 l +2718 9144 l +2718 5526 l +2900 6084 3340 6390 x +3781 6696 4403 6696 x +5337 6696 5811 6071 x +6287 5448 6287 4213 x +7272 fwd_x +end_ol + } def +/YAA { start_ol +3616 5328 m +3000 5328 2648 4785 x +2296 4243 2296 3276 x +2296 2308 2648 1766 x +3000 1224 3616 1224 x +4239 1224 4591 1766 x +4943 2308 4943 3276 x +4943 4243 4591 4785 x +4239 5328 3616 5328 x +574 3276 m +574 4852 1399 5773 x +2225 6696 3615 6696 x +5013 6696 5837 5773 x +6663 4852 6663 3276 x +6663 1699 5837 777 x +5013 -144 3615 -144 x +2225 -144 1399 777 x +574 1699 574 3276 x +7272 fwd_x +end_ol + } def +/ZAA { start_ol +622 3271 m +622 8784 l +2354 8784 l +2354 2838 l +2354 2183 2688 1811 x +3024 1440 3616 1440 x +4209 1440 4543 1811 x +4879 2183 4879 2838 x +4879 8784 l +6610 8784 l +6610 3271 l +6610 1481 5899 668 x +5189 -144 3616 -144 x +2048 -144 1335 668 x +622 1481 622 3271 x +7272 fwd_x +end_ol + } def +/aAA { start_ol +2272 2729 m +2272 7848 l +528 7848 l +528 9144 l +3992 9144 l +3992 2729 l +3992 1947 4233 1621 x +4474 1296 5049 1296 x +6422 1296 l +6422 0 l +4567 0 l +3341 0 2806 631 x +2272 1262 2272 2729 x +7272 fwd_x +end_ol + } def +/bAA { start_ol +6757 4680 m +6475 4968 6096 5112 x +5718 5256 5266 5256 x +4720 5256 4312 5062 x +3904 4869 3681 4501 x +3540 4275 3484 3953 x +3429 3632 3429 2977 x +3429 0 l +1708 0 l +1708 6552 l +3429 6552 l +3429 5521 l +3681 6082 4203 6388 x +4726 6696 5424 6696 x +5776 6696 6114 6603 x +6453 6512 6757 6336 x +6757 4680 l +7272 fwd_x +end_ol + } def +/cAA { start_ol +1297 6552 m +4702 6552 l +4702 1296 l +6840 1296 l +6840 0 l +839 0 l +839 1296 l +2982 1296 l +2982 5256 l +1297 5256 l +1297 6552 l +2982 9792 m +4702 9792 l +4702 7776 l +2982 7776 l +2982 9792 l +7272 fwd_x +end_ol + } def +/dAA { start_ol +6229 360 m +5794 108 5295 -18 x +4797 -144 4221 -144 x +2701 -144 1843 761 x +986 1667 986 3270 x +986 4879 1849 5787 x +2712 6696 4233 6696 x +4761 6696 5251 6572 x +5742 6449 6229 6192 x +6229 4608 l +5853 4920 5404 5088 x +4955 5256 4474 5256 x +3634 5256 3182 4739 x +2730 4223 2730 3270 x +2730 2317 3182 1806 x +3634 1296 4474 1296 x +4972 1296 5406 1455 x +5842 1616 6229 1944 x +6229 360 l +7272 fwd_x +end_ol + } def +/eAA { start_ol +1767 4320 m +5472 4320 l +5472 2592 l +1767 2592 l +1767 4320 l +7272 fwd_x +end_ol + } def +/fAA { start_ol +804 8784 m +2536 8784 l +2536 5400 l +4702 5400 l +4702 8784 l +6435 8784 l +6435 0 l +4702 0 l +4702 3888 l +2536 3888 l +2536 0 l +804 0 l +804 8784 l +7272 fwd_x +end_ol + } def +/gAA { start_ol +4714 3351 m +4714 4214 4379 4735 x +4045 5256 3499 5256 x +2959 5256 2628 4737 x +2296 4219 2296 3351 x +2296 2476 2628 1957 x +2959 1440 3499 1440 x +4045 1440 4379 1960 x +4714 2481 4714 3351 x +6435 484 m +6435 -1105 5721 -1812 x +5008 -2520 3405 -2520 x +2865 -2520 2336 -2449 x +1807 -2378 1262 -2232 x +1262 -648 l +1749 -905 2254 -1028 x +2759 -1152 3294 -1152 x +4021 -1152 4368 -802 x +4714 -452 4714 290 x +4714 1003 l +4462 532 4045 301 x +3628 72 3029 72 x +1902 72 1238 947 x +574 1822 574 3323 x +574 4874 1238 5784 x +1902 6696 3017 6696 x +3582 6696 4021 6428 x +4462 6160 4714 5673 x +4714 6552 l +6435 6552 l +6435 484 l +7272 fwd_x +end_ol + } def +/hAA { start_ol +3174 6764 m +3174 7126 2849 7248 x +2754 7272 2680 7272 x +1683 7259 1370 6620 x +1251 6139 1033 5970 x +949 5909 829 5837 x +517 5656 348 5656 x +264 5656 240 5728 x +252 5764 300 5873 x +649 6789 1454 7323 x +1611 7434 1779 7518 x +2585 7920 3451 7920 x +4147 7896 4160 7267 x +4160 6421 3703 4548 x +3944 4680 4100 4680 x +7263 4680 l +7935 6517 8489 7423 x +8633 7678 9066 7859 x +9234 7920 9343 7920 x +9451 7920 9451 7846 x +9451 7834 8958 6789 x +7394 3485 7394 992 x +7407 84 7912 72 x +8068 72 8188 108 x +8440 763 9066 837 x +9114 837 9138 837 x +9246 824 9258 763 x +9054 130 8212 -295 x +7659 -576 7153 -576 x +6589 -576 6468 -52 x +6409 201 6409 687 x +6409 2037 7081 4163 x +6841 4032 6697 4032 x +3570 4032 l +3018 2009 2308 298 x +2188 34 1731 -144 x +1551 -216 1443 -216 x +1346 -205 1334 -165 x +1334 -145 1443 96 x +2200 2027 2777 4347 x +3174 5931 3174 6764 x +9288 fwd_x +end_ol + } def +/iAA { start_ol +4965 4176 m +5447 4176 5567 3784 x +5591 3688 5591 3594 x +5567 3108 5073 3096 x +4798 3096 4677 3258 x +4557 3420 4539 3582 x +4521 3744 4485 3744 x +4293 3744 3703 2714 x +3570 2472 3498 2315 x +3498 1918 3245 724 x +3186 475 3186 381 x +3210 156 3390 144 x +3883 144 4821 1293 x +4906 1389 5338 1951 x +5494 1855 l +4701 838 4629 753 x +3883 -59 3246 -72 x +2344 -48 2067 1491 x +1213 -59 528 -72 x +96 -72 -84 284 x +-144 415 -144 545 x +-120 1067 420 1080 x +769 1080 961 700 x +1058 547 1105 535 x +1346 559 2044 1855 x +2067 2145 2320 3586 x +2344 3753 2344 3811 x +2320 3918 2212 3930 x +1743 3907 156 1782 x +0 1878 l +1094 3322 1599 3755 x +2103 4163 2477 4176 x +3259 4152 3475 2750 x +4280 4163 4965 4176 x +5256 fwd_x +end_ol + } def +/jAA { start_ol +3150 3272 m +3150 2489 2092 1031 x +1551 279 1551 168 x +1575 84 1683 72 x +2248 72 4052 2446 x +4268 2749 4496 3053 x +5014 4046 l +6517 4151 l +4352 72 l +4942 279 5218 472 x +5843 922 6625 1942 x +6781 1833 l +5650 374 4629 -11 x +4496 -59 4244 -144 x +2934 -2651 1262 -2664 x +697 -2664 492 -2299 x +420 -2176 420 -2055 x +420 -1350 2056 -632 x +2236 -559 2921 -279 x +4232 2441 l +2885 567 2260 113 x +1899 -144 1611 -144 x +1081 -144 888 418 x +829 614 829 834 x +829 1849 2031 3316 x +2428 3802 2441 3863 x +2416 3947 2320 3960 x +1767 3935 168 1760 x +0 1869 l +1755 4176 2416 4176 x +2873 4176 3065 3694 x +3150 3489 3150 3272 x +793 -2175 m +793 -2421 1069 -2507 x +1141 -2520 1202 -2520 x +1947 -2495 2826 -513 x +805 -1473 793 -2175 x +6552 fwd_x +end_ol + } def +/lAA { start_ol +6276 4032 m +938 4032 l +613 4032 613 4289 x +636 4523 938 4536 x +6276 4536 l +6589 4523 6601 4289 x +6601 4032 6276 4032 x +6276 2304 m +938 2304 l +613 2304 613 2561 x +636 2795 938 2808 x +6276 2808 l +6589 2795 6601 2561 x +6601 2304 6276 2304 x +7200 fwd_x +end_ol + } def +/nAA { start_ol +5832 5726 m +1611 763 l +1490 619 1395 619 x +1202 619 1166 799 x +1154 835 1154 859 x +1166 943 1262 1063 x +5470 6025 l +5614 6169 5686 6169 x +5879 6169 5915 5989 x +5927 5953 5927 5929 x +5915 5834 5832 5726 x +1815 7350 m +1815 3528 l +2668 3528 l +2896 3504 2909 3312 x +2885 3108 2668 3096 x +528 3096 l +289 3119 276 3312 x +300 3528 649 3540 x +1118 3540 1382 3528 x +1382 6768 l +600 6516 l +505 6480 492 6480 x +348 6480 301 6657 x +289 6683 289 6708 x +289 6849 433 6912 x +625 7004 1815 7350 x +4257 3184 m +4257 3539 4677 3906 x +5086 4248 5567 4248 x +6229 4248 6637 3705 x +6865 3393 6865 3031 x +6865 2647 6529 2250 x +6156 1816 4653 432 x +6456 432 l +6456 469 l +6480 707 6672 720 x +6877 695 6889 475 x +6889 0 l +4124 0 l +4124 519 l +5711 1999 l +6348 2637 6409 2877 x +6432 2949 6432 3034 x +6432 3467 5999 3708 x +5796 3816 5567 3816 x +5026 3816 4762 3367 x +4701 3268 4677 3168 x +4605 2989 4461 2976 x +4293 3013 4257 3184 x +7200 fwd_x +end_ol + } def +/oAA { start_ol +3534 2921 m +3534 4401 4437 6144 x +4953 7142 5230 7250 x +5254 7263 5266 7263 x +5458 7263 5494 7081 x +5506 7045 5506 7022 x +5506 6889 5158 6264 x +4291 4617 4268 2885 x +4280 1033 5470 -1118 x +5506 -1213 5506 -1251 x +5506 -1443 5314 -1479 x +5278 -1490 5254 -1490 x +5037 -1490 4557 -636 x +3703 889 3558 2452 x +3534 2693 3534 2921 x +7200 fwd_x +end_ol + } def +/pAA { start_ol +6024 3096 m +1190 3096 l +866 3096 866 3353 x +889 3587 1190 3600 x +6024 3600 l +6337 3587 6348 3353 x +6348 3096 6024 3096 x +7200 fwd_x +end_ol + } def +/qAA { start_ol +6682 6768 m +3857 0 l +3327 0 l +502 6768 l +6682 6768 l +5961 6264 m +1224 6264 l +3593 598 l +5961 6264 l +7200 fwd_x +end_ol + } def +/rAA { start_ol +2236 7740 m +2236 8088 2657 8447 x +3065 8784 3547 8784 x +4208 8784 4617 8241 x +4845 7929 4845 7567 x +4845 7183 4509 6786 x +4136 6352 2633 4968 x +4437 4968 l +4437 5005 l +4460 5243 4653 5256 x +4857 5231 4870 5011 x +4870 4536 l +2103 4536 l +2103 5055 l +3691 6535 l +4327 7173 4388 7413 x +4412 7485 4412 7570 x +4412 8003 3979 8244 x +3775 8352 3547 8352 x +3006 8352 2741 7920 x +2680 7824 2657 7728 x +2585 7548 2441 7536 x +2272 7572 2236 7740 x +7200 fwd_x +end_ol + } def +/sAA { start_ol +3847 3168 m +3847 754 l +3835 444 3606 432 x +3354 432 3354 754 x +3354 3168 l +1190 3168 l +866 3168 866 3425 x +889 3659 1190 3672 x +3354 3672 l +3354 6021 l +3354 6336 3606 6336 x +3835 6312 3847 6021 x +3847 3672 l +6012 3672 l +6337 3659 6348 3425 x +6324 3180 6012 3168 x +3847 3168 l +7200 fwd_x +end_ol + } def +/tAA { start_ol +3811 5184 m +4965 5184 5722 4334 x +6384 3582 6384 2550 x +6384 1287 5482 523 x +4773 -72 3811 -72 x +2488 -59 1719 1030 x +1719 -1872 l +2898 -1872 l +3198 -1896 3210 -2111 x +3187 -2363 2898 -2376 x +577 -2376 l +264 -2351 252 -2111 x +276 -1872 577 -1872 x +1226 -1872 l +1226 2287 l +1202 2503 l +1202 3693 2128 4534 x +2368 4763 2632 4907 x +3173 5184 3811 5184 x +3811 4680 m +2668 4680 2080 3776 x +1719 3227 1719 2556 x +1719 1506 2524 871 x +3090 432 3811 432 x +4821 432 5447 1237 x +5891 1823 5891 2556 x +5891 3666 5098 4277 x +4557 4680 3811 4680 x +7200 fwd_x +end_ol + } def +/uAA { start_ol +1767 -1251 m +1767 -1118 2116 -492 x +2982 1154 3006 2885 x +3006 4689 1887 6733 x +1839 6817 1803 6889 x +1767 6986 1767 7022 x +1767 7214 1959 7250 x +1995 7263 2019 7263 x +2236 7263 2716 6409 x +3570 4881 3715 3318 x +3739 3078 3739 2849 x +3739 1370 2837 -372 x +2320 -1370 2044 -1479 x +2019 -1490 2008 -1490 x +1815 -1490 1779 -1310 x +1767 -1274 1767 -1251 x +7200 fwd_x +end_ol + } def +/vAA { start_ol +3547 1368 m +3666 1368 l +4207 1368 4424 920 x +4496 751 4496 581 x +4496 74 4039 -131 x +3858 -216 3666 -216 x +3547 -216 l +3029 -216 2801 219 x +2716 400 2716 581 x +2716 1077 3173 1283 x +3354 1368 3547 1368 x +7200 fwd_x +end_ol + } def +/wAA { start_ol +3955 2646 m +6168 504 l +6276 504 l +6589 491 6601 257 x +6601 0 6276 0 x +4689 0 l +4365 0 4365 257 x +4388 491 4689 504 x +5470 504 l +3606 2297 l +1719 504 l +2536 504 l +2849 491 2862 257 x +2862 0 2536 0 x +938 0 l +613 0 613 257 x +636 491 938 504 x +1046 504 l +3258 2646 l +1298 4536 l +1213 4536 l +889 4536 889 4793 x +913 5027 1213 5040 x +2549 5040 l +2862 5015 2873 4793 x +2873 4536 2549 4536 x +1995 4536 l +3606 2971 l +5254 4536 l +4677 4536 l +4352 4536 4352 4793 x +4376 5027 4677 5040 x +5999 5040 l +6312 5027 6324 4793 x +6324 4536 5999 4536 x +5915 4536 l +3955 2646 l +7200 fwd_x +end_ol + } def +/xAA { start_ol +3390 15 m +1130 4536 l +938 4536 l +613 4536 613 4793 x +636 5027 938 5040 x +2320 5040 l +2633 5027 2644 4793 x +2644 4536 2320 4536 x +1695 4536 l +3667 556 l +5603 4536 l +4953 4536 l +4629 4536 4629 4793 x +4653 5027 4953 5040 x +6276 5040 l +6589 5027 6601 4793 x +6601 4536 6276 4536 x +6108 4536 l +3029 -1728 l +3811 -1728 l +4124 -1740 4136 -1974 x +4136 -2232 3811 -2232 x +1010 -2232 l +685 -2232 685 -1974 x +709 -1740 1010 -1728 x +2536 -1728 l +3390 15 l +7200 fwd_x +end_ol + } def +/yAA { start_ol +1503 4556 m +1503 4900 2946 5113 x +3379 5184 3595 5184 x +4701 5184 5242 4512 x +5530 4152 5530 3708 x +5530 504 l +6179 504 l +6492 491 6504 257 x +6504 0 6179 0 x +5037 0 l +5037 812 l +3967 -216 2741 -216 x +1599 -216 1107 520 x +866 884 866 1330 x +866 2286 1851 2769 x +2513 3096 3403 3096 x +4124 3096 5037 2838 x +5037 3693 l +5037 4306 4316 4559 x +3980 4680 3558 4680 x +2970 4680 2128 4410 x +1851 4320 1731 4320 x +1551 4320 1515 4497 x +1503 4532 1503 4556 x +5037 1346 m +5037 2414 l +4316 2592 3498 2592 x +2128 2592 1587 1931 x +1359 1656 1359 1319 x +1359 648 2067 396 x +2369 288 2729 288 x +3787 288 4701 1054 x +4869 1188 5037 1346 x +7200 fwd_x +end_ol + } def +/zAA { start_ol +3836 4680 m +2885 4680 2164 3763 x +2140 3728 2092 3657 x +2031 3585 2008 3549 x +2008 504 l +2549 504 l +2860 491 2873 257 x +2873 0 2549 0 x +974 0 l +649 12 636 257 x +661 491 974 504 x +1515 504 l +1515 4536 l +1107 4536 l +781 4536 781 4793 x +805 5027 1107 5040 x +2008 5040 l +2008 4198 l +2718 4982 3318 5124 x +3570 5184 3883 5184 x +4893 5184 5446 4487 x +5771 4068 5771 3551 x +5771 504 l +6180 504 l +6493 491 6504 257 x +6504 0 6180 0 x +4870 0 l +4545 0 4545 257 x +4568 491 4870 504 x +5278 504 l +5278 3459 l +5278 4153 4641 4500 x +4293 4680 3836 4680 x +7200 fwd_x +end_ol + } def +/ABA { start_ol +6035 7272 m +6035 504 l +6684 504 l +6997 491 7009 257 x +7009 0 6684 0 x +5542 0 l +5542 1078 l +4689 -203 3354 -216 x +2200 -216 1418 689 x +757 1463 757 2489 x +757 3710 1647 4519 x +2380 5184 3354 5184 x +4676 5171 5542 3926 x +5542 6768 l +4893 6768 l +4568 6768 4568 7025 x +4592 7259 4893 7272 x +6035 7272 l +3390 4680 m +2369 4680 1731 3873 x +1251 3272 1251 2489 x +1251 1418 2056 769 x +2646 288 3403 288 x +4413 288 5050 1082 x +5542 1695 5542 2466 x +5542 3573 4725 4223 x +4136 4680 3390 4680 x +7200 fwd_x +end_ol + } def +/BBA { start_ol +5254 3459 m +5254 4201 4557 4524 x +4208 4680 3775 4680 x +3126 4680 2658 4285 x +2428 4081 2117 3710 x +1984 3555 l +1984 504 l +2525 504 l +2849 491 2862 257 x +2838 12 2525 0 x +938 0 l +625 24 613 257 x +636 491 938 504 x +1490 504 l +1490 5616 l +841 5616 l +528 5628 517 5868 x +541 6107 841 6120 x +1490 6120 l +1490 6768 l +841 6768 l +517 6768 517 7025 x +541 7259 842 7272 x +1984 7272 l +1984 6120 l +3679 6120 l +3991 6107 4003 5868 x +3980 5628 3679 5616 x +1984 5616 l +1984 4165 l +2705 5038 3451 5159 x +3643 5184 3836 5184 x +4918 5184 5459 4440 x +5747 4019 5747 3504 x +5747 504 l +6288 504 l +6612 491 6625 257 x +6601 12 6288 0 x +4713 0 l +4388 0 4388 257 x +4412 491 4713 504 x +5254 504 l +5254 3459 l +7200 fwd_x +end_ol + } def +/CBA { start_ol +1346 5040 m +1346 4403 l +1851 5100 2428 5171 x +2524 5184 2608 5184 x +3378 5171 3775 4343 x +4473 5160 5134 5184 x +5807 5184 6180 4595 x +6396 4271 6396 3888 x +6396 504 l +6805 504 l +7117 491 7130 257 x +7130 0 6805 0 x +5915 0 l +5915 3828 l +5915 4307 5506 4572 x +5314 4680 5122 4680 x +4557 4680 3980 3871 x +3920 3789 3872 3718 x +3872 504 l +4280 504 l +4593 491 4605 257 x +4605 0 4280 0 x +3390 0 l +3390 3792 l +3390 4284 2982 4559 x +2801 4680 2608 4680 x +1995 4656 1346 3718 x +1346 504 l +1755 504 l +2067 491 2080 257 x +2056 12 1755 0 x +445 0 l +145 24 132 257 x +156 491 457 504 x +866 504 l +866 4536 l +457 4536 l +132 4536 132 4793 x +156 5027 456 5040 x +1346 5040 l +7200 fwd_x +end_ol + } def +/DBA { start_ol +2405 -216 m +1359 -216 830 911 x +517 1602 517 2462 x +529 3541 1177 4536 x +949 4536 l +625 4548 613 4788 x +636 5027 949 5040 x +2260 5040 l +2561 4989 2585 4788 x +2561 4536 2260 4536 x +1743 4536 l +1010 3510 1010 2521 x +1010 915 1936 420 x +2188 288 2380 288 x +2777 288 3078 625 x +3342 928 3342 1410 x +3342 3075 l +3414 3256 3630 3292 x +3786 3292 3822 3124 x +3835 3099 3835 3075 x +3835 1409 l +3835 601 4532 372 x +4676 324 4785 324 x +5326 324 5711 855 x +6144 1470 6144 2556 x +6144 3485 5591 4317 x +5519 4426 5422 4559 x +4893 4559 l +4604 4597 4568 4812 x +4592 5052 4893 5064 x +6215 5064 l +6540 5040 6553 4812 x +6529 4572 6216 4559 x +5988 4559 l +6637 3553 6637 2475 x +6637 922 5819 195 x +5362 -192 4762 -192 x +3955 -192 3631 430 x +3595 480 3583 517 x +3258 -29 2837 -154 x +2644 -216 2405 -216 x +7200 fwd_x +end_ol + } def +/EBA { start_ol +3859 7359 m +3859 504 l +5530 504 l +5843 491 5855 257 x +5855 0 5530 0 x +1695 0 l +1370 12 1359 257 x +1382 491 1695 504 x +3367 504 l +3367 6710 l +2008 5351 l +1900 5256 1755 5256 x +1562 5315 1526 5531 x +1539 5652 1659 5772 x +3246 7359 l +3859 7359 l +7200 fwd_x +end_ol + } def +/FBA { start_ol +3787 6408 m +2176 6408 1527 4932 x +1251 4307 1251 3792 x +1251 2916 l +1251 1319 2453 648 x +3126 288 4041 288 x +4857 288 5700 663 x +5736 675 5760 687 x +5760 2448 l +4088 2448 l +3763 2448 3763 2705 x +3786 2939 4088 2952 x +6421 2952 l +6745 2939 6757 2705 x +6734 2460 6421 2448 x +6252 2448 l +6252 417 l +5158 -216 4003 -216 x +2224 -216 1346 976 x +757 1783 757 2914 x +757 3805 l +757 5130 1659 6044 x +2524 6912 3763 6912 x +4954 6899 5760 6256 x +5760 6430 l +5760 6768 6012 6768 x +6240 6744 6252 6446 x +6252 5361 l +6240 5052 6012 5040 x +5794 5063 5771 5332 x +5722 5870 4965 6186 x +4437 6408 3787 6408 x +7200 fwd_x +end_ol + } def +/GBA { start_ol +6252 4248 m +6180 4248 5904 4379 x +5567 4680 5266 4680 x +4773 4680 4111 4165 x +3715 3848 2982 3163 x +2982 504 l +5134 504 l +5447 491 5458 246 x +5434 12 5134 0 x +1334 0 l +1021 12 1010 257 x +1033 491 1334 504 x +2488 504 l +2488 4536 l +1587 4536 l +1274 4548 1262 4793 x +1287 5027 1587 5040 x +2982 5040 l +2982 3801 l +4257 4986 4953 5146 x +5134 5184 5302 5184 x +5952 5184 6373 4710 x +6493 4564 6504 4485 x +6468 4281 6252 4248 x +7200 fwd_x +end_ol + } def +/HBA { start_ol +3847 5040 m +3847 504 l +5771 504 l +6096 491 6108 257 x +6084 12 5771 0 x +1431 0 l +1105 0 1105 257 x +1130 491 1431 504 x +3354 504 l +3354 4536 l +1936 4536 l +1611 4536 1611 4793 x +1634 5027 1936 5040 x +3847 5040 l +3823 7488 m +3823 6264 l +3114 6264 l +3114 7488 l +3823 7488 l +7200 fwd_x +end_ol + } def +/IBA { start_ol +6252 6624 m +6216 6624 5999 6647 x +5086 6768 4593 6768 x +3703 6768 3378 6212 x +3258 5998 3258 5762 x +3258 5040 l +5530 5040 l +5843 5027 5855 4793 x +5855 4536 5530 4536 x +3258 4536 l +3258 504 l +5398 504 l +5711 491 5722 257 x +5699 12 5398 0 x +1587 0 l +1262 0 1262 257 x +1287 491 1587 504 x +2765 504 l +2765 4536 l +1707 4536 l +1382 4536 1382 4793 x +1406 5027 1707 5040 x +2765 5040 l +2765 5767 l +2765 6591 3534 7021 x +4016 7272 4617 7272 x +5447 7272 6288 7128 x +6493 7052 6504 6876 x +6468 6648 6252 6624 x +7200 fwd_x +end_ol + } def +/JBA { start_ol +2236 5040 m +4881 5040 l +5194 5027 5206 4793 x +5206 4536 4881 4536 x +2236 4536 l +2236 1305 l +2236 614 2970 372 x +3246 288 3583 288 x +4581 288 5411 687 x +5506 735 5578 771 x +5699 844 5771 844 x +5940 844 5988 650 x +5999 625 5999 600 x +5999 320 5098 28 x +4316 -216 3606 -216 x +2392 -216 1936 547 x +1743 875 1743 1274 x +1743 4536 l +853 4536 l +529 4548 517 4793 x +541 5027 853 5040 x +1743 5040 l +1743 6448 l +1743 6768 1995 6768 x +2224 6744 2236 6448 x +2236 5040 l +7200 fwd_x +end_ol + } def +/KBA { start_ol +5254 3459 m +5254 4201 4557 4524 x +4208 4680 3775 4680 x +3126 4680 2658 4285 x +2428 4081 2117 3710 x +1984 3555 l +1984 504 l +2525 504 l +2849 491 2862 257 x +2838 12 2525 0 x +938 0 l +625 24 613 257 x +636 491 938 504 x +1490 504 l +1490 6768 l +841 6768 l +517 6768 517 7025 x +541 7259 842 7272 x +1984 7272 l +1984 4165 l +2705 5038 3451 5159 x +3643 5184 3836 5184 x +4918 5184 5459 4440 x +5747 4019 5747 3504 x +5747 504 l +6288 504 l +6612 491 6625 257 x +6601 12 6288 0 x +4713 0 l +4388 0 4388 257 x +4412 491 4713 504 x +5254 504 l +5254 3459 l +7200 fwd_x +end_ol + } def +/LBA { start_ol +2837 7272 m +4376 7272 l +3967 4246 l +3895 3828 3607 3816 x +3306 3839 3246 4246 x +2837 7272 l +7200 fwd_x +end_ol + } def +/MBA { start_ol +5434 1402 m +5434 2045 4557 2299 x +4401 2335 3198 2549 x +1983 2754 1659 3235 x +1490 3487 1490 3825 x +1490 4534 2308 4930 x +2849 5184 3547 5184 x +4568 5171 5194 4620 x +5194 4673 l +5206 5026 5446 5040 x +5674 5015 5686 4694 x +5686 3845 l +5674 3524 5446 3512 x +5242 3512 5194 3765 x +5194 3776 5194 3789 x +5134 4367 4329 4583 x +3991 4680 3595 4680 x +2608 4680 2188 4188 x +2031 3996 2031 3780 x +2031 3288 2873 3072 x +3114 3012 3667 2929 x +4965 2723 5422 2421 x +5963 2046 5976 1381 x +5976 570 5086 110 x +4460 -216 3619 -216 x +2464 -203 1731 444 x +1731 316 l +1719 11 1490 0 x +1238 0 1238 322 x +1238 1312 l +1238 1635 1490 1635 x +1707 1599 1731 1371 x +1731 1287 l +1731 793 2452 492 x +2957 288 3583 288 x +4701 288 5206 844 x +5434 1111 5434 1402 x +7200 fwd_x +end_ol + } def +/NBA { start_ol +1984 3240 m +1984 504 l +5760 504 l +5760 1913 l +5760 2233 6012 2233 x +6240 2209 6252 1910 x +6252 0 l +842 0 l +517 0 517 257 x +541 491 841 504 x +1490 504 l +1490 6264 l +841 6264 l +517 6264 517 6521 x +541 6755 842 6768 x +5999 6768 l +5999 5140 l +5999 4824 5747 4824 x +5519 4846 5506 5134 x +5506 6264 l +1984 6264 l +1984 3744 l +3727 3744 l +3727 4239 l +3727 4536 3979 4536 x +4207 4512 4219 4222 x +4219 2701 l +4207 2399 3979 2448 x +3727 2448 3727 2745 x +3727 3240 l +1984 3240 l +7200 fwd_x +end_ol + } def +/OBA { start_ol +5542 4045 m +5542 5040 l +6684 5040 l +6997 5027 7009 4793 x +7009 4536 6684 4536 x +6035 4536 l +6035 -1728 l +6685 -1728 l +6997 -1740 7009 -1974 x +7009 -2232 6685 -2232 x +4365 -2232 l +4039 -2232 4039 -1974 x +4063 -1740 4365 -1728 x +5542 -1728 l +5542 1306 l +4725 144 3378 144 x +2127 144 1357 1041 x +757 1748 757 2669 x +757 3866 1683 4609 x +2416 5184 3390 5184 x +4689 5184 5494 4105 x +5519 4069 5542 4045 x +3390 4680 m +2320 4680 1684 3907 x +1251 3359 1251 2669 x +1251 1683 2044 1087 x +2633 648 3390 648 x +4437 648 5073 1409 x +5542 1956 5542 2646 x +5542 3668 4725 4263 x +4136 4680 3390 4680 x +7200 fwd_x +end_ol + } def +/PBA { start_ol +1251 5626 m +1251 6134 1803 6690 x +2524 7403 3487 7416 x +4473 7416 5158 6661 x +5699 6073 5699 5355 x +5699 4755 5290 4216 x +4917 3737 3967 2850 x +1479 537 l +1479 504 l +5254 504 l +5254 915 l +5254 1224 5506 1224 x +5735 1200 5747 906 x +5747 0 l +1010 0 l +1010 721 l +3847 3379 l +4917 4421 5109 4852 x +5206 5091 5206 5355 x +5206 6073 4545 6564 x +4064 6912 3487 6912 x +2621 6912 2067 6262 x +1828 5973 1731 5649 x +1647 5421 1479 5409 x +1310 5409 1262 5577 x +1251 5602 1251 5626 x +7200 fwd_x +end_ol + } def +/QBA { start_ol +5254 6552 m +5254 6768 l +1755 6768 l +1755 6348 l +1742 6024 1514 6012 x +1274 6036 1262 6351 x +1262 7272 l +5747 7272 l +5747 6480 l +3787 252 l +3691 12 3547 0 x +3354 0 3318 180 x +3306 216 3306 239 x +3306 300 3330 396 x +5254 6552 l +7200 fwd_x +end_ol + } def +/RBA { start_ol +3547 1368 m +3666 1368 l +4207 1368 4424 920 x +4496 751 4496 581 x +4496 74 4039 -131 x +3858 -216 3666 -216 x +3547 -216 l +3029 -216 2801 219 x +2716 400 2716 581 x +2716 1077 3173 1283 x +3354 1368 3547 1368 x +3547 5040 m +3666 5040 l +4183 5040 4412 4608 x +4496 4428 4496 4248 x +4496 3756 4039 3540 x +3858 3456 3666 3456 x +3547 3456 l +3029 3456 2801 3888 x +2716 4068 2716 4248 x +2716 4739 3173 4955 x +3354 5040 3547 5040 x +7200 fwd_x +end_ol + } def +/SBA { start_ol +5254 3240 m +2008 3240 l +2008 504 l +2657 504 l +2970 491 2982 257 x +2982 0 2657 0 x +974 0 l +649 12 636 257 x +661 491 974 504 x +1515 504 l +1515 6264 l +1215 6264 l +889 6264 889 6521 x +913 6755 1215 6768 x +2657 6768 l +2970 6755 2982 6521 x +2982 6264 2657 6264 x +2008 6264 l +2008 3744 l +5254 3744 l +5254 6264 l +4605 6264 l +4280 6264 4280 6521 x +4304 6755 4605 6768 x +6048 6768 l +6360 6755 6373 6521 x +6373 6264 6048 6264 x +5747 6264 l +5747 504 l +6301 504 l +6612 491 6625 257 x +6601 12 6301 0 x +4605 0 l +4280 0 4280 257 x +4304 491 4605 504 x +5254 504 l +5254 3240 l +7200 fwd_x +end_ol + } def +/TBA { start_ol +5355 2210 m +5355 0 l +4673 0 l +4673 2197 l +4673 2718 4443 2979 x +4215 3240 3757 3240 x +3206 3240 2889 2926 x +2571 2613 2571 2074 x +2571 0 l +1884 0 l +1884 3672 l +2571 3672 l +2571 3102 l +2812 3431 3144 3587 x +3476 3744 3910 3744 x +4621 3744 4988 3352 x +5355 2962 5355 2210 x +7272 fwd_x +end_ol + } def +/UBA { start_ol +3811 -648 m +3811 -1803 l +3811 -1827 3811 -1827 x +3775 -2019 3606 -2055 x +3583 -2055 l +3414 -2019 3390 -1803 x +3390 -648 l +2392 -648 l +2176 -610 2139 -426 x +2164 -228 2392 -216 x +3390 -216 l +3390 817 l +3426 1033 3606 1069 x +3799 1046 3811 817 x +3811 -216 l +4834 -216 l +5037 -253 5073 -426 x +5037 -623 4809 -648 x +3811 -648 l +7200 fwd_x +end_ol + } def +/VBA { start_ol +3871 2089 m +3871 -1753 l +4725 -1753 l +4953 -1777 4965 -1969 x +4942 -2173 4725 -2185 x +2585 -2185 l +2344 -2162 2332 -1969 x +2356 -1753 2705 -1742 x +3174 -1742 3439 -1753 x +3439 1489 l +2657 1237 l +2560 1198 2549 1198 x +2404 1198 2356 1375 x +2344 1401 2344 1426 x +2344 1567 2488 1630 x +2680 1726 3871 2089 x +7200 fwd_x +end_ol + } def +/WBA { start_ol +3511 5342 m +3511 4795 4028 4486 x +4316 4320 4617 4320 x +5242 4320 l +5434 4320 5470 4131 x +5483 4092 5483 4068 x +5483 3866 5303 3828 x +5266 3816 5242 3816 x +4617 3816 l +2176 3767 1984 2003 x +1972 1848 1972 1704 x +1972 912 2910 624 x +3306 504 3775 504 x +5724 480 5747 -768 x +5735 -1344 5290 -1728 x +5470 -1728 l +5783 -1752 5796 -1967 x +5796 -2144 5567 -2219 x +5519 -2232 5470 -2232 x +3955 -2232 l +3642 -2207 3631 -1967 x +3655 -1728 3955 -1728 x +4521 -1728 l +5037 -1301 l +5230 -1100 5266 -780 x +5266 -757 l +5242 0 3775 0 x +2549 0 1911 642 x +1490 1078 1490 1697 x +1490 3431 3018 4048 x +3282 4158 3559 4218 x +3042 4669 3029 5333 x +3042 6282 3823 6626 x +4160 6768 l +1984 6768 l +1984 6140 l +1972 5843 1743 5832 x +1490 5832 1490 6148 x +1490 7272 l +5699 7272 l +5699 6912 l +4016 6186 l +3619 5996 3547 5770 x +3511 5616 3511 5342 x +7200 fwd_x +end_ol + } def +/XBA { start_ol +2391 -648 m +2175 -610 2139 -426 x +2163 -228 2391 -216 x +4833 -216 l +5037 -253 5073 -426 x +5037 -623 4809 -648 x +2391 -648 l +7200 fwd_x +end_ol + } def +/YBA { start_ol +3847 6264 m +3847 504 l +5519 504 l +5843 491 5855 257 x +5832 12 5519 0 x +1683 0 l +1359 0 1359 257 x +1382 491 1683 504 x +3354 504 l +3354 6264 l +1684 6264 l +1359 6264 1359 6521 x +1382 6755 1684 6768 x +5519 6768 l +5843 6755 5855 6521 x +5832 6276 5519 6264 x +3847 6264 l +7200 fwd_x +end_ol + } def +/ZBA { start_ol +1251 504 m +1251 6264 l +842 6264 l +517 6264 517 6521 x +541 6755 842 6768 x +3487 6768 l +4726 6768 5542 5782 x +6252 4916 6252 3727 x +6252 3053 l +6252 1647 5314 745 x +4533 0 3487 0 x +842 0 l +517 0 517 257 x +541 491 842 504 x +1251 504 l +5760 3821 m +5760 4407 5375 5101 x +4713 6251 3462 6264 x +1743 6264 l +1743 504 l +3547 504 l +4461 504 5146 1318 x +5760 2048 5760 2946 x +5760 3821 l +7200 fwd_x +end_ol + } def +/aBA { start_ol +6432 1001 m +6432 802 5927 451 x +4965 -216 3715 -216 x +2344 -216 1575 750 x +1010 1463 1010 2441 x +1010 3819 2008 4604 x +2754 5184 3775 5184 x +4905 5171 5650 4511 x +5650 4678 l +5663 5027 5904 5040 x +6132 5015 6144 4696 x +6144 3579 l +6132 3260 5904 3247 x +5686 3247 5650 3512 x +5650 3524 5650 3537 x +5602 4114 4857 4451 x +4340 4680 3739 4680 x +2500 4680 1887 3777 x +1503 3211 1503 2453 x +1503 1310 2380 697 x +2957 288 3739 288 x +5050 299 5988 1110 x +6120 1224 6204 1224 x +6373 1224 6420 1059 x +6432 1024 6432 1001 x +7200 fwd_x +end_ol + } def +/bBA { start_ol +6252 2376 m +1251 2376 l +1418 1152 2428 588 x +2993 288 3679 288 x +4713 288 5542 760 x +5686 842 5783 925 x +5904 1008 5976 1008 x +6156 1008 6192 832 x +6204 797 6204 774 x +6204 471 5338 144 x +4473 -216 3667 -216 x +2344 -216 1479 726 x +757 1535 757 2598 x +757 3807 1695 4567 x +2464 5184 3498 5184 x +4929 5184 5699 4167 x +6252 3416 6252 2376 x +1251 2880 m +5747 2880 l +5555 3972 4581 4440 x +4088 4680 3498 4680 x +2236 4680 1599 3744 x +1346 3360 1251 2880 x +7200 fwd_x +end_ol + } def +/cBA { start_ol +3606 5184 m +4881 5184 5686 4241 x +6348 3480 6348 2453 x +6348 1257 5434 472 x +4653 -216 3606 -216 x +2356 -216 1539 702 x +866 1474 866 2489 x +866 3697 1779 4507 x +2572 5184 3606 5184 x +3606 4680 m +2536 4680 1864 3873 x +1359 3260 1359 2489 x +1359 1431 2188 781 x +2813 288 3606 288 x +4665 288 5338 1069 x +5855 1684 5855 2453 x +5855 3549 5026 4198 x +4401 4680 3606 4680 x +7200 fwd_x +end_ol + } def +/dBA { start_ol +2489 1728 m +4088 1728 l +2344 -1429 l +2176 -1715 1972 -1728 x +1707 -1728 1635 -1478 x +1623 -1429 1623 -1382 x +1623 -1334 1648 -1251 x +2489 1728 l +7200 fwd_x +end_ol + } def +/eBA { start_ol +3847 7272 m +3847 504 l +5771 504 l +6096 491 6108 257 x +6084 12 5771 0 x +1431 0 l +1105 0 1105 257 x +1130 491 1431 504 x +3354 504 l +3354 6768 l +1948 6768 l +1623 6768 1623 7025 x +1647 7259 1948 7272 x +3847 7272 l +7200 fwd_x +end_ol + } def +/fBA { start_ol +1731 5040 m +1731 4045 l +2464 4982 3354 5136 x +3606 5184 3883 5184 x +5158 5184 5927 4273 x +6504 3592 6504 2669 x +6504 1472 5591 730 x +4857 144 3883 144 x +2536 156 1731 1306 x +1731 -1728 l +2909 -1728 l +3222 -1740 3234 -1974 x +3234 -2232 2909 -2232 x +589 -2232 l +264 -2232 264 -1974 x +288 -1740 589 -1728 x +1238 -1728 l +1238 4536 l +589 4536 l +264 4536 264 4793 x +288 5027 589 5040 x +1731 5040 l +3871 4680 m +2826 4680 2188 3918 x +1731 3359 1731 2669 x +1731 1683 2536 1087 x +3126 648 3871 648 x +4906 648 5542 1397 x +6012 1956 6012 2646 x +6012 3668 5194 4263 x +4617 4680 3871 4680 x +7200 fwd_x +end_ol + } def +/gBA { start_ol +3859 6768 m +3859 -1008 l +5025 -1008 l +5338 -1020 5350 -1254 x +5350 -1512 5026 -1512 x +3367 -1512 l +3367 7272 l +5026 7272 l +5338 7259 5350 7025 x +5350 6768 5025 6768 x +3859 6768 l +7200 fwd_x +end_ol + } def +/hBA { start_ol +2621 7416 m +4147 7416 5146 6258 x +6132 5100 6132 3364 x +6132 1460 5194 446 x +4557 -216 3715 -216 x +2718 -216 2104 635 x +1635 1291 1635 2130 x +1635 3200 2392 3893 x +2946 4392 3643 4392 x +4906 4379 5639 2905 x +5663 3360 5663 3420 x +5663 5572 4039 6493 x +3306 6912 2669 6912 x +2320 6912 2020 6788 x +1948 6768 1900 6768 x +1695 6806 1671 7022 x +1671 7288 2188 7416 x +2392 7416 2621 7416 x +5567 2251 m +5002 3294 4437 3657 x +4052 3888 3655 3888 x +2910 3888 2441 3197 x +2128 2712 2128 2130 x +2128 1148 2802 603 x +3198 288 3703 288 x +4942 288 5459 1875 x +5519 2057 5567 2251 x +7200 fwd_x +end_ol + } def +/iBA { start_ol +5796 7611 m +1864 -757 l +1743 -974 1599 -974 x +1418 -974 1370 -793 x +1359 -769 1359 -745 x +1370 -673 1418 -553 x +5350 7815 l +5470 8020 5614 8032 x +5796 8032 5843 7851 x +5855 7827 5855 7804 x +5843 7719 5796 7611 x +7200 fwd_x +end_ol + } def +/jBA { start_ol +3354 -1008 m +3354 6768 l +2189 6768 l +1876 6780 1864 7014 x +1864 7272 2188 7272 x +3847 7272 l +3847 -1512 l +2188 -1512 l +1876 -1499 1864 -1265 x +1864 -1008 2189 -1008 x +3354 -1008 l +7200 fwd_x +end_ol + } def +/kBA { start_ol +3606 3384 m +4003 3384 4160 3004 x +4208 2882 4208 2759 x +4208 2392 3859 2220 x +3739 2160 3607 2160 x +3234 2160 3065 2515 x +3006 2637 3006 2772 x +3006 3151 3354 3323 x +3475 3384 3606 3384 x +7200 fwd_x +end_ol + } def +/lBA { start_ol +5879 1244 m +6276 1244 6432 865 x +6481 742 6481 619 x +6481 253 6132 81 x +6012 20 5880 20 x +5506 20 5338 375 x +5278 497 5278 632 x +5278 1011 5627 1183 x +5747 1244 5879 1244 x +1310 1244 m +1707 1244 1864 865 x +1912 742 1912 619 x +1912 253 1563 81 x +1443 20 1311 20 x +938 20 769 375 x +709 497 709 632 x +709 1011 1058 1183 x +1179 1244 1310 1244 x +3606 5379 m +4003 5379 4160 5000 x +4208 4878 4208 4755 x +4208 4388 3859 4216 x +3739 4155 3607 4155 x +3234 4155 3065 4511 x +3006 4632 3006 4767 x +3006 5146 3354 5319 x +3475 5379 3606 5379 x +7200 fwd_x +end_ol + } def +/mBA { start_ol +3859 504 m +5122 504 l +5434 491 5447 257 x +5447 0 5122 0 x +2104 0 l +1779 0 1779 257 x +1803 491 2104 504 x +3367 504 l +3367 6264 l +1359 6264 l +1359 5370 l +1359 5040 1107 5040 x +878 5064 866 5370 x +866 6768 l +6348 6768 l +6348 5370 l +6324 5052 6108 5040 x +5904 5040 5868 5285 x +5855 5333 5855 5370 x +5855 6264 l +3859 6264 l +3859 504 l +7200 fwd_x +end_ol + } def +/nBA { start_ol +1803 0 m +1803 6012 l +5411 6012 l +5411 0 l +1803 0 l +7200 fwd_x +end_ol + } def +/oBA { start_ol +5578 1823 m +5578 2664 4653 3000 x +4424 3084 3270 3324 x +2080 3576 1683 4082 x +1382 4467 1382 5045 x +1382 5985 2224 6526 x +2813 6912 3595 6912 x +4629 6899 5350 6192 x +5350 6429 l +5350 6768 5602 6768 x +5830 6743 5843 6439 x +5843 5188 l +5830 4872 5602 4860 x +5362 4860 5350 5148 x +5302 5832 4581 6192 x +4147 6408 3631 6408 x +2644 6408 2163 5772 x +1911 5436 1911 5027 x +1911 4259 2813 3935 x +3006 3863 4160 3611 x +5386 3335 5796 2818 x +6108 2408 6108 1794 x +6108 759 5182 181 x +4521 -216 3631 -216 x +2356 -203 1599 660 x +1599 318 l +1599 0 1346 0 x +1118 23 1105 321 x +1105 1658 l +1105 1980 1359 1980 x +1576 1956 1599 1692 x +1635 984 2417 563 x +2957 288 3619 288 x +4713 288 5266 959 x +5578 1344 5578 1823 x +7200 fwd_x +end_ol + } def +/pBA { start_ol +5290 4033 m +5290 5040 l +6432 5040 l +6745 5027 6757 4793 x +6757 4536 6432 4536 x +5783 4536 l +5783 -329 l +5783 -1220 5050 -1798 x +4496 -2232 3799 -2232 x +2428 -2232 l +2103 -2232 2103 -1974 x +2127 -1740 2428 -1728 x +3823 -1728 l +4593 -1728 5037 -1087 x +5290 -700 5290 -241 x +5290 1317 l +4485 156 3234 144 x +2116 144 1370 1018 x +757 1735 757 2669 x +757 3819 1623 4573 x +2320 5184 3234 5184 x +4485 5184 5267 4069 x +5278 4045 5290 4033 x +3271 4680 m +2272 4680 1671 3907 x +1251 3359 1251 2669 x +1251 1683 2020 1076 x +2574 648 3271 648 x +4257 648 4857 1409 x +5290 1956 5290 2646 x +5290 3681 4497 4276 x +3956 4680 3271 4680 x +7200 fwd_x +end_ol + } def +/qBA { start_ol +5278 0 m +5278 804 l +4268 -216 3090 -216 x +2091 -216 1683 560 x +1490 923 1490 1372 x +1490 4536 l +841 4536 l +517 4536 517 4793 x +541 5027 842 5040 x +1984 5040 l +1984 1379 l +1984 760 2513 445 x +2777 288 3078 288 x +4280 288 5278 1395 x +5278 4536 l +4388 4536 l +4064 4548 4052 4793 x +4075 5027 4388 5040 x +5771 5040 l +5771 504 l +6179 504 l +6492 491 6504 257 x +6504 0 6179 0 x +5278 0 l +7200 fwd_x +end_ol + } def +/rBA { start_ol +4160 2160 m +4749 2160 5278 2775 x +5819 3402 5819 4198 x +5819 6768 l +6721 6768 l +7033 6744 7045 6504 x +7009 6300 6721 6276 x +6312 6276 l +6312 4190 l +6312 2650 5073 1959 x +4557 1668 4160 1668 x +3859 1644 l +3859 504 l +4268 504 l +4581 480 4593 257 x +4569 11 4268 0 x +3018 0 l +2693 23 2680 257 x +2704 492 3017 504 x +3414 504 l +3414 1644 l +3150 1656 l +2621 1656 2008 2067 x +986 2795 974 4165 x +974 6264 l +564 6264 l +300 6264 240 6444 x +228 6480 228 6503 x +252 6755 564 6768 x +1454 6768 l +1454 4174 l +1454 3184 2200 2545 x +2668 2160 3137 2160 x +3414 2147 l +3414 6264 l +3137 6264 l +2909 6264 2813 6431 x +2801 6480 2801 6503 x +2824 6755 3137 6768 x +4147 6768 l +4424 6732 4460 6600 x +4473 6552 4473 6503 x +4460 6276 4147 6264 x +3859 6264 l +3859 2147 l +4160 2160 l +7200 fwd_x +end_ol + } def +/sBA { start_ol +5145 2232 m +1947 2232 l +1298 504 l +2248 504 l +2561 491 2572 257 x +2572 0 2248 0 x +433 0 l +108 0 108 257 x +132 491 433 504 x +805 504 l +2957 6264 l +1515 6264 l +1190 6264 1190 6521 x +1213 6755 1515 6768 x +3967 6768 l +6337 504 l +6781 504 l +7094 491 7106 257 x +7106 0 6781 0 x +4906 0 l +4581 0 4581 257 x +4605 491 4906 504 x +5819 504 l +5145 2232 l +4965 2736 m +3619 6264 l +3439 6264 l +2139 2736 l +4965 2736 l +7200 fwd_x +end_ol + } def +/tBA { start_ol +1731 7272 m +1731 3938 l +2621 5171 3895 5184 x +5122 5184 5891 4253 x +6504 3504 6504 2514 x +6504 1270 5614 460 x +4869 -216 3895 -216 x +2572 -203 1731 1066 x +1731 0 l +589 0 l +264 0 264 257 x +288 491 589 504 x +1238 504 l +1238 6768 l +589 6768 l +264 6768 264 7025 x +288 7259 589 7272 x +1731 7272 l +3871 4680 m +2849 4680 2212 3873 x +1731 3272 1731 2489 x +1731 1443 2524 781 x +3114 288 3871 288 x +4870 288 5519 1082 x +6012 1695 6012 2453 x +6012 3549 5206 4210 x +4617 4680 3871 4680 x +7200 fwd_x +end_ol + } def +/uBA { start_ol +3908 3461 m +6204 504 l +6409 504 l +6721 491 6733 257 x +6733 0 6409 0 x +4821 0 l +4496 0 4496 257 x +4521 491 4821 504 x +5567 504 l +3583 3066 l +1611 504 l +2380 504 l +2693 491 2705 257 x +2705 0 2380 0 x +805 0 l +481 0 481 257 x +505 491 805 504 x +1010 504 l +3270 3461 l +1118 6264 l +938 6264 l +613 6264 613 6521 x +636 6755 938 6768 x +2260 6768 l +2572 6755 2585 6521 x +2585 6264 2260 6264 x +1743 6264 l +3606 3856 l +5447 6264 l +4906 6264 l +4581 6264 4581 6521 x +4605 6755 4906 6768 x +6240 6768 l +6553 6743 6565 6521 x +6565 6264 6240 6264 x +6060 6264 l +3908 3461 l +7200 fwd_x +end_ol + } def +/vBA { start_ol +3872 3054 m +3872 504 l +5134 504 l +5447 491 5458 257 x +5458 0 5134 0 x +2116 0 l +1792 0 1792 257 x +1815 491 2116 504 x +3378 504 l +3378 3054 l +1226 6264 l +938 6264 l +613 6264 613 6521 x +636 6755 938 6768 x +2272 6768 l +2585 6755 2597 6521 x +2597 6264 2272 6264 x +1815 6264 l +3642 3546 l +5434 6264 l +4953 6264 l +4629 6276 4617 6521 x +4641 6755 4953 6768 x +6276 6768 l +6589 6755 6601 6521 x +6576 6276 6276 6264 x +5988 6264 l +3872 3054 l +7200 fwd_x +end_ol + } def +/wBA { start_ol +5302 0 m +4701 0 l +3606 3124 l +2536 0 l +1923 0 l +913 4536 l +685 4536 l +361 4536 361 4793 x +384 5027 685 5040 x +2020 5040 l +2332 5027 2344 4793 x +2344 4536 2020 4536 x +1382 4536 l +2248 673 l +3282 3751 l +3895 3751 l +4965 673 l +5796 4536 l +5194 4536 l +4870 4536 4870 4793 x +4893 5027 5194 5040 x +6529 5040 l +6841 5027 6853 4793 x +6853 4536 6529 4536 x +6301 4536 l +5302 0 l +7200 fwd_x +end_ol + } def +/xBA { start_ol +4039 0 m +3198 0 l +1177 4536 l +685 4536 l +361 4536 361 4793 x +384 5027 685 5040 x +2500 5040 l +2826 5027 2837 4793 x +2813 4548 2500 4536 x +1719 4536 l +3523 504 l +3751 504 l +5519 4536 l +4701 4536 l +4376 4536 4376 4793 x +4401 5027 4701 5040 x +6529 5040 l +6841 5027 6853 4793 x +6853 4536 6529 4536 x +6035 4536 l +4039 0 l +7200 fwd_x +end_ol + } def +/zBA { start_ol +5855 4197 m +5855 2991 l +5855 1218 4942 326 x +4365 -216 3606 -216 x +2464 -216 1828 916 x +1359 1773 1359 2991 x +1359 4197 l +1359 5992 2284 6873 x +2849 7416 3606 7416 x +4761 7416 5386 6283 x +5855 5414 5855 4197 x +3606 6912 m +2644 6912 2152 5757 x +1851 5036 1851 4135 x +1851 3064 l +1851 1550 2572 768 x +3029 288 3606 288 x +4569 288 5062 1442 x +5362 2163 5362 3064 x +5362 4135 l +5362 5649 4641 6431 x +4185 6912 3606 6912 x +7200 fwd_x +end_ol + } def +/ACA { start_ol +4653 4536 m +1756 4536 l +1431 4536 1431 4793 x +1454 5027 1756 5040 x +5145 5040 l +5145 -326 l +5145 -1399 4352 -1930 x +3883 -2232 3281 -2232 x +1731 -2232 l +1406 -2232 1406 -1974 x +1429 -1740 1731 -1728 x +3270 -1728 l +4063 -1728 4448 -1078 x +4653 -742 4653 -321 x +4653 4536 l +4617 7488 m +4617 6264 l +3908 6264 l +3908 7488 l +4617 7488 l +7200 fwd_x +end_ol + } def +/BCA { start_ol +0 3888 m +7214 3888 l +7214 3384 l +0 3384 l +0 3888 l +7200 fwd_x +end_ol + } def +/CCA { start_ol +1503 6331 m +1503 6525 1923 6850 x +2644 7416 3619 7416 x +4725 7416 5338 6661 x +5735 6181 5735 5571 x +5735 4540 4857 4084 x +4677 3989 4496 3940 x +5603 3480 5915 2559 x +5999 2280 5999 2001 x +5999 995 5134 329 x +4412 -216 3487 -216 x +2633 -216 1731 246 x +1166 562 1154 806 x +1154 976 1323 1024 x +1359 1037 1382 1037 x +1454 1037 1839 783 x +2608 288 3498 288 x +4473 288 5086 965 x +5506 1435 5506 2003 x +5506 2801 4725 3333 x +4111 3744 3354 3744 x +3029 3744 3029 4001 x +3054 4235 3354 4248 x +5109 4248 5230 5366 x +5242 5460 5242 5556 x +5242 6282 4545 6674 x +4124 6912 3606 6912 x +2633 6912 2103 6365 x +1828 6115 1743 6103 x +1551 6103 1515 6284 x +1503 6307 1503 6331 x +7200 fwd_x +end_ol + } def +/DCA { start_ol +4521 2016 m +1262 2016 l +1262 2592 l +4003 7272 l +5014 7272 l +5014 2520 l +5422 2520 l +5747 2520 5747 2262 x +5724 2028 5422 2016 x +5014 2016 l +5014 504 l +5422 504 l +5735 491 5747 257 x +5747 0 5422 0 x +3606 0 l +3282 0 3282 257 x +3306 491 3606 504 x +4521 504 l +4521 2016 l +4521 2520 m +4521 6768 l +4232 6768 l +1731 2520 l +4521 2520 l +7200 fwd_x +end_ol + } def +/ECA { start_ol +3799 4248 m +3101 4248 2380 3931 x +2128 3816 2020 3816 x +1815 3852 1792 4077 x +1792 7272 l +5181 7272 l +5506 7259 5519 7025 x +5495 6780 5182 6768 x +2284 6768 l +2284 4383 l +3162 4752 3872 4752 x +4953 4752 5566 3867 x +5999 3237 5999 2400 x +5999 1044 5073 304 x +4424 -216 3547 -216 x +2452 -216 1551 477 x +1166 793 1154 963 x +1154 1157 1334 1193 x +1370 1206 1395 1206 x +1467 1206 1815 928 x +2597 288 3523 288 x +4641 288 5181 1181 x +5506 1724 5506 2425 x +5506 3487 4749 3982 x +4329 4248 3799 4248 x +7200 fwd_x +end_ol + } def +/FCA { start_ol +5868 6768 m +5843 6768 5614 6850 x +5410 6912 5097 6912 x +4111 6912 3258 6158 x +2103 5142 2103 3420 x +2103 3336 2128 2905 x +2874 4379 4124 4392 x +5037 4392 5650 3589 x +6132 2945 6132 2130 x +6132 975 5339 270 x +4773 -216 4052 -216 x +2885 -216 2200 965 x +1635 1941 1635 3364 x +1635 5305 2826 6463 x +3800 7416 5146 7416 x +5808 7416 6036 7144 x +6096 7062 6096 7003 x +6060 6802 5868 6768 x +2200 2251 m +2657 493 3764 312 x +3920 288 4064 288 x +4882 288 5339 1027 x +5639 1512 5639 2130 x +5639 3027 5001 3560 x +4593 3888 4113 3888 x +3078 3888 2260 2360 x +2248 2349 2224 2311 x +2212 2263 2200 2251 x +7200 fwd_x +end_ol + } def +/GCA { start_ol +4136 4806 m +4136 4842 l +4160 5040 4509 5040 x +4568 5040 l +4568 6747 l +4568 7224 4051 7368 x +3907 7416 3763 7416 x +3270 7416 2909 7009 x +2837 6928 2729 6789 x +2729 5040 l +2993 5040 l +3234 5015 3246 4829 x +3246 4806 l +3223 4620 2994 4608 x +2020 4608 l +1828 4645 1792 4818 x +1828 5040 2188 5040 x +2308 5040 l +2308 7344 l +1935 7344 1887 7471 x +1875 7521 1875 7572 x +1899 7763 2127 7776 x +2729 7776 l +2729 7353 l +2765 7353 2921 7512 x +3246 7848 3800 7848 x +4473 7848 4821 7367 x +4990 7104 4990 6804 x +4990 5040 l +5170 5040 l +5410 5015 5422 4829 x +5422 4806 l +5398 4620 5170 4608 x +4363 4608 l +4147 4632 4136 4806 x +7200 fwd_x +end_ol + } def +/HCA { start_ol +3354 5266 m +3354 6937 l +3354 7263 3606 7263 x +3835 7238 3847 6937 x +3847 5266 l +5447 5783 l +5531 5819 5627 5819 x +5796 5819 5843 5620 x +5855 5595 5855 5571 x +5832 5383 5603 5295 x +4003 4789 l +4978 3442 l +5062 3298 5062 3226 x +5062 3034 4881 2998 x +4845 2985 4821 2985 x +4689 3009 4581 3154 x +3595 4501 l +2621 3154 l +2488 2985 2369 2985 x +2176 2985 2139 3178 x +2128 3214 2128 3237 x +2139 3322 2224 3442 x +3198 4789 l +1599 5296 l +1370 5384 1359 5558 x +1359 5758 1539 5796 x +1575 5807 1599 5807 x +1647 5807 1755 5771 x +3354 5266 l +7200 fwd_x +end_ol + } def +/ICA { start_ol +4942 266 m +4942 -437 l +4942 -1626 4280 -2099 x +3992 -2293 3643 -2293 x +2873 -2293 2524 -1469 x +2344 -1008 2344 -437 x +2344 266 l +2344 1528 3101 1904 x +3354 2026 3643 2026 x +4485 2026 4809 1152 x +4942 763 4942 266 x +3643 1594 m +3018 1594 2849 937 x +2777 644 2777 231 x +2777 -389 l +2790 -1849 3643 -1861 x +4496 -1825 4509 -389 x +4509 231 l +4388 1533 3643 1594 x +7200 fwd_x +end_ol + } def +/JCA { start_ol +2272 982 m +2272 1330 2693 1689 x +3101 2026 3583 2026 x +4244 2026 4653 1483 x +4881 1171 4881 810 x +4881 425 4545 28 x +4172 -405 2669 -1789 x +4473 -1789 l +4473 -1752 l +4496 -1514 4689 -1501 x +4893 -1526 4906 -1746 x +4906 -2221 l +2139 -2221 l +2139 -1702 l +3727 -222 l +4363 415 4424 655 x +4448 727 4448 812 x +4448 1245 4015 1486 x +3811 1594 3583 1594 x +3042 1594 2777 1162 x +2716 1066 2693 970 x +2621 790 2477 778 x +2308 814 2272 982 x +7200 fwd_x +end_ol + } def +/KCA { start_ol +3642 1642 m +3150 1642 2813 1324 x +2693 1210 2621 1210 x +2441 1246 2405 1411 x +2405 1671 2909 1896 x +3282 2074 3642 2074 x +4388 2074 4737 1527 x +4893 1278 4893 979 x +4881 456 4340 100 x +5025 -345 5037 -958 x +5037 -1620 4412 -2004 x +4027 -2245 3583 -2245 x +2945 -2245 2441 -1909 x +2224 -1741 2212 -1608 x +2248 -1441 2416 -1405 x +2464 -1405 2765 -1585 x +3138 -1813 3583 -1813 x +4160 -1813 4460 -1381 x +4604 -1177 4604 -961 x +4604 -457 4063 -205 x +3799 -97 3498 -85 x +3258 -48 3246 136 x +3246 334 3511 346 x +3595 357 3691 357 x +4219 357 4401 735 x +4460 852 4460 983 x +4460 1407 4003 1584 x +3823 1642 3642 1642 x +7200 fwd_x +end_ol + } def +/LCA { start_ol +4509 3780 m +5843 3146 5855 1928 x +5855 916 5037 271 x +4413 -216 3606 -216 x +2536 -216 1864 576 x +1359 1173 1359 1928 x +1371 3146 2705 3780 x +1526 4483 1479 5449 x +1479 6355 2248 6951 x +2862 7416 3607 7416 x +4604 7416 5254 6676 x +5735 6129 5735 5449 x +5735 4459 4665 3863 x +4581 3816 4509 3780 x +3607 6912 m +2741 6912 2261 6274 x +1972 5884 1972 5424 x +1972 4693 2657 4279 x +3090 4032 3607 4032 x +4473 4032 4965 4633 x +5242 4999 5242 5413 x +5242 6192 4557 6640 x +4124 6912 3607 6912 x +3606 3528 m +2680 3528 2164 2848 x +1851 2423 1851 1926 x +1851 1101 2561 603 x +3029 288 3606 288 x +4496 288 5026 955 x +5362 1392 5362 1913 x +5362 2775 4641 3236 x +4172 3528 3606 3528 x +7200 fwd_x +end_ol + } def +/MCA { start_ol +3606 8352 m +3114 8352 2777 8033 x +2657 7920 2585 7920 x +2405 7956 2369 8121 x +2369 8381 2873 8606 x +3246 8784 3606 8784 x +4352 8784 4701 8237 x +4857 7987 4857 7689 x +4845 7166 4304 6809 x +4989 6364 5001 5751 x +5001 5089 4376 4704 x +3991 4464 3547 4464 x +2909 4464 2405 4800 x +2188 4968 2176 5100 x +2212 5268 2380 5304 x +2428 5304 2729 5124 x +3102 4896 3547 4896 x +4124 4896 4424 5328 x +4568 5531 4568 5747 x +4568 6251 4027 6503 x +3763 6611 3462 6624 x +3222 6661 3210 6845 x +3210 7043 3475 7056 x +3559 7067 3655 7067 x +4183 7067 4365 7445 x +4424 7562 4424 7692 x +4424 8116 3967 8293 x +3787 8352 3606 8352 x +7200 fwd_x +end_ol + } def +/NCA { start_ol +4063 1514 m +2790 -637 l +4063 -637 l +4063 1514 l +2236 -733 m +3835 1954 l +4496 1954 l +4496 -637 l +4749 -637 l +4918 -673 4942 -853 x +4942 -1009 4809 -1057 x +4773 -1068 4749 -1068 x +4496 -1068 l +4496 -1789 l +4749 -1789 l +4917 -1825 4942 -2005 x +4906 -2185 4749 -2221 x +3595 -2221 l +3426 -2209 3414 -2005 x +3439 -1813 3595 -1789 x +4063 -1789 l +4063 -1069 l +2236 -1069 l +2236 -733 l +7200 fwd_x +end_ol + } def +/OCA { start_ol +4027 8344 m +2754 6192 l +4027 6192 l +4027 8344 l +2200 6096 m +3799 8784 l +4460 8784 l +4460 6192 l +4713 6192 l +4882 6156 4906 5976 x +4906 5820 4773 5772 x +4737 5761 4713 5761 x +4460 5761 l +4460 5040 l +4713 5040 l +4881 5004 4906 4824 x +4870 4644 4713 4608 x +3559 4608 l +3390 4620 3378 4824 x +3403 5016 3559 5040 x +4027 5040 l +4027 5760 l +2200 5760 l +2200 6096 l +7200 fwd_x +end_ol + } def +/PCA { start_ol +3739 226 m +3414 226 2921 -39 x +2777 -123 2705 -123 x +2608 -86 2572 171 x +2572 2098 l +4545 2098 l +4725 2085 4737 1882 x +4713 1678 4545 1666 x +3018 1666 l +3018 442 l +3511 645 3655 658 x +3728 658 3787 658 x +4461 658 4846 31 x +5062 -342 5062 -776 x +5062 -1643 4388 -2065 x +4027 -2293 3595 -2293 x +2885 -2293 2368 -1824 x +2224 -1679 2212 -1607 x +2212 -1438 2320 -1366 x +2332 -1354 2344 -1354 x +2405 -1354 2680 -1560 x +3101 -1861 3583 -1861 x +4329 -1861 4569 -1221 x +4641 -1004 4641 -763 x +4641 141 3859 226 x +3800 226 3739 226 x +7200 fwd_x +end_ol + } def +/QCA { start_ol +3703 7056 m +3378 7056 2885 6790 x +2741 6706 2669 6706 x +2572 6743 2536 7000 x +2536 8928 l +4509 8928 l +4689 8915 4701 8712 x +4677 8508 4509 8496 x +2982 8496 l +2982 7272 l +3475 7475 3619 7488 x +3692 7488 3751 7488 x +4425 7488 4810 6861 x +5026 6487 5026 6053 x +5026 5186 4352 4764 x +3991 4536 3559 4536 x +2849 4536 2332 5005 x +2188 5150 2176 5222 x +2176 5391 2284 5463 x +2296 5475 2308 5475 x +2369 5475 2644 5269 x +3065 4968 3547 4968 x +4293 4968 4533 5608 x +4605 5825 4605 6066 x +4605 6971 3823 7056 x +3764 7056 3703 7056 x +7200 fwd_x +end_ol + } def +/RCA { start_ol +4208 2266 m +4833 2266 4917 2026 x +4929 1992 4929 1968 x +4893 1797 4713 1762 x +4677 1762 4593 1780 x +4377 1834 4196 1834 x +3390 1834 2910 1100 x +2585 582 2585 -55 x +2585 -127 l +3126 526 3727 538 x +4581 538 4845 -291 x +4942 -583 4942 -913 x +4942 -999 l +4942 -1561 4521 -1964 x +4185 -2256 3739 -2269 x +2644 -2269 2284 -1080 x +2152 -632 2152 -122 x +2152 -49 l +2152 1041 2898 1733 x +3462 2266 4208 2266 x +2669 -803 m +2862 -1726 3595 -1824 x +3667 -1837 3739 -1837 x +3763 -1837 l +4221 -1837 4437 -1320 x +4509 -1136 4509 -1000 x +4413 69 3691 106 x +3367 106 2946 -324 x +2693 -594 2669 -803 x +7200 fwd_x +end_ol + } def +/SCA { start_ol +4172 9000 m +4797 9000 4881 8759 x +4893 8725 4893 8701 x +4857 8530 4677 8496 x +4641 8496 4557 8514 x +4341 8568 4160 8568 x +3354 8568 2874 7833 x +2549 7315 2549 6678 x +2549 6606 l +3090 7259 3691 7272 x +4545 7272 4809 6441 x +4906 6149 4906 5819 x +4906 5734 l +4906 5171 4485 4768 x +4149 4476 3703 4464 x +2608 4464 2248 5653 x +2116 6100 2116 6610 x +2116 6683 l +2116 7774 2862 8466 x +3426 9000 4172 9000 x +2633 5929 m +2826 5006 3559 4908 x +3631 4896 3703 4896 x +3727 4896 l +4185 4896 4401 5412 x +4473 5596 4473 5733 x +4377 6802 3655 6840 x +3331 6840 2910 6409 x +2657 6139 2633 5929 x +7200 fwd_x +end_ol + } def +/TCA { start_ol +5699 5040 m +5699 4597 l +2056 504 l +5386 504 l +5386 1177 l +5386 1503 5638 1503 x +5866 1478 5879 1175 x +5879 0 l +1382 0 l +1382 442 l +5001 4536 l +1984 4536 l +1984 3908 l +1972 3611 1743 3600 x +1490 3600 1490 3916 x +1490 5040 l +5699 5040 l +7200 fwd_x +end_ol + } def +/UCA { start_ol +3821 3339 m +3945 3579 4138 3691 x +4332 3803 4609 3803 x +5107 3803 5313 3459 x +5519 3116 5519 2165 x +5519 33 l +4896 33 l +4896 2142 l +4896 2922 4795 3110 x +4696 3298 4438 3298 x +4144 3298 4035 3096 x +3927 2893 3927 2142 x +3927 33 l +3305 33 l +3305 2142 l +3305 2928 3199 3112 x +3093 3298 2818 3298 x +2548 3298 2444 3096 x +2342 2893 2342 2142 x +2342 33 l +1720 33 l +1720 3714 l +2342 3714 l +2342 3397 l +2460 3597 2644 3700 x +2829 3803 3058 3803 x +3340 3803 3528 3688 x +3715 3574 3821 3339 x +7272 fwd_x +end_ol + } def +/VCA { start_ol +1310 5040 m +5902 5040 l +6215 5027 6228 4793 x +6228 4536 5902 4536 x +5494 4536 l +5494 504 l +5904 504 l +6216 491 6228 257 x +6228 0 5904 0 x +4460 0 l +4136 0 4136 257 x +4160 491 4460 504 x +5001 504 l +5001 4536 l +2212 4536 l +2212 504 l +2754 504 l +3065 491 3078 257 x +3078 0 2752 0 x +1323 0 l +997 12 985 257 x +1010 491 1323 504 x +1719 504 l +1719 4536 l +1310 4536 l +985 4536 985 4793 x +1009 5027 1310 5040 x +7200 fwd_x +end_ol + } def +/WCA { start_ol +3606 7416 m +5747 4708 l +5855 4586 5855 4489 x +5855 4295 5675 4259 x +5639 4248 5614 4248 x +5483 4259 5375 4405 x +3606 6651 l +1839 4405 l +1719 4248 1599 4248 x +1406 4248 1370 4430 x +1359 4466 1359 4491 x +1370 4600 1467 4709 x +3606 7416 l +7200 fwd_x +end_ol + } def +/XCA { start_ol +1526 3744 m +2188 1179 l +4593 8424 l +6950 8424 l +7202 8400 7214 8208 x +7189 8004 6950 7992 x +4942 7992 l +2152 0 l +1141 3240 l +384 3240 l +132 3253 120 3498 x +144 3730 384 3744 x +1526 3744 l +7200 fwd_x +end_ol + } def +/YCA { start_ol +4063 6769 m +3847 2849 l +3811 2524 3595 2513 x +3378 2536 3354 2849 x +3138 6769 l +3138 6805 3126 6865 x +3126 6937 3126 6961 x +3126 7310 3450 7407 x +3534 7430 3606 7430 x +3943 7430 4052 7117 x +4075 7033 4075 6961 x +4075 6950 4075 6889 x +4063 6805 4063 6769 x +3522 1008 m +3678 1008 l +4099 1008 4268 641 x +4329 518 4329 396 x +4329 -7 3944 -166 x +3811 -216 3679 -216 x +3523 -216 l +3090 -216 2934 163 x +2885 273 2885 396 x +2885 799 3258 958 x +3378 1008 3522 1008 x +7200 fwd_x +end_ol + } def +/ZCA { start_ol +3821 7227 m +3945 7467 4138 7579 x +4332 7691 4609 7691 x +5107 7691 5313 7347 x +5519 7004 5519 6053 x +5519 3921 l +4896 3921 l +4896 6030 l +4896 6810 4795 6998 x +4696 7186 4438 7186 x +4144 7186 4035 6984 x +3927 6781 3927 6030 x +3927 3921 l +3305 3921 l +3305 6030 l +3305 6816 3199 7000 x +3093 7186 2818 7186 x +2548 7186 2444 6984 x +2342 6781 2342 6030 x +2342 3921 l +1720 3921 l +1720 7602 l +2342 7602 l +2342 7285 l +2460 7485 2644 7588 x +2829 7691 3058 7691 x +3340 7691 3528 7576 x +3715 7462 3821 7227 x +7272 fwd_x +end_ol + } def +/bCA { start_ol +6035 696 m +6035 516 5674 216 x +5134 -216 4449 -216 x +3619 -216 3282 555 x +3138 905 3138 1327 x +3138 4536 l +2405 4536 l +2092 4548 2080 4788 x +2103 5040 2405 5040 x +3631 5040 l +3631 1339 l +3643 300 4449 288 x +5037 288 5519 747 x +5711 936 5794 936 x +5976 936 6023 768 x +6035 732 6035 696 x +7200 fwd_x +end_ol + } def +/cCA { start_ol +2838 6192 m +4376 6192 l +3967 4694 l +5506 5112 l +5506 3541 l +3967 3946 l +4376 2376 l +2838 2376 l +3246 3946 l +1707 3541 l +1707 5112 l +3246 4694 l +2838 6192 l +7200 fwd_x +end_ol + } def +/dCA { start_ol +4527 8412 m +4527 -1438 l +2712 -1438 l +2712 8412 l +4527 8412 l +7272 fwd_x +end_ol + } def +/eCA { start_ol +1902 -762 m +4091 3622 l +1908 8014 l +3152 8014 l +5337 3622 l +3152 -762 l +1902 -762 l +7272 fwd_x +end_ol + } def +/fCA { start_ol +4725 0 m +4352 588 l +3811 -203 3006 -216 x +2188 -216 1659 515 x +1262 1050 1262 1707 x +1262 2731 2116 3254 x +2344 3388 2597 3474 x +2056 4363 2008 4628 x +1995 4712 1995 4833 x +1995 5566 2621 5999 x +3018 6264 3475 6264 x +3883 6250 4268 5996 x +4496 6120 4581 6120 x +4749 6120 4798 5940 x +4809 5904 4809 5868 x +4785 5724 4617 5616 x +4232 5427 l +3859 5760 3475 5760 x +2957 5760 2644 5303 x +2488 5051 2488 4798 x +2500 4413 3018 3596 x +4352 1469 l +4725 2185 4906 3024 x +5422 3024 l +5735 3011 5747 2777 x +5747 2520 5422 2520 x +5278 2520 l +5001 1517 4641 999 x +4965 504 l +5422 504 l +5735 491 5747 257 x +5747 0 5422 0 x +4725 0 l +4064 1047 m +2801 3047 l +2044 2876 1828 2169 x +1755 1948 1755 1716 x +1755 972 2308 544 x +2633 288 2993 288 x +3655 300 4064 1047 x +7200 fwd_x +end_ol + } def +/gCA { start_ol +2224 2176 m +2224 0 l +1082 0 l +757 0 757 257 x +780 491 1082 504 x +1731 504 l +1731 6768 l +1082 6768 l +757 6768 757 7025 x +780 7259 1082 7272 x +2224 7272 l +2224 2766 l +4304 4536 l +4039 4536 l +3715 4536 3715 4793 x +3739 5027 4039 5040 x +5603 5040 l +5915 5027 5927 4793 x +5927 4536 5603 4536 x +5037 4536 l +3150 2958 l +5639 504 l +6180 504 l +6493 491 6504 257 x +6504 0 6180 0 x +4605 0 l +4280 0 4280 257 x +4304 491 4605 504 x +4942 504 l +2777 2646 l +2224 2176 l +7200 fwd_x +end_ol + } def +end end +%%EndSetup +%%Page: 1 1 +paps_bop +(+ 36000 808000AAABAACAA+ 65088 808000EAAFAA+ 86904 808000GAAHAAIAA+ 115992 808000JAAKAAEAALAA+ 152352 808000EAAEAAMAANAAOAAMAAPAAEAA+ 217800 808000QAARAA+ 239616 808000SAATAAUAA)paps_exec +(* 232052VAAWAAXAAYAA+ 268412 808000ZAAaAAbAAcAAdAAXAA+ 319316 808000eAA+ 333860 808000fAAAAAEAAJAA)paps_exec +(* 515368QAAHAAgAABAA+ 551728 808000EAA)paps_exec +36 798.000000 moveto 559 798.000000 lineto 0 setlinewidth stroke +(+ 36000 778824hAAiAAjAA+ 60120 778824lAA+ 74520 778824nAA+ 88920 778824oAApAAqAArAA+ 124920 778824sAA+ 139320 778824tAArAAuAAvAA)paps_exec +()paps_exec +(+ 36000 754776tAArAA+ 57600 754776lAA+ 72000 754776wAArAA+ 93600 754776sAA+ 108000 754776xAArAA+ 129600 754776yAAzAAABA+ 158400 754776BBA+ 172800 754776lAA+ 187200 754776CBA+ 201600 754776lAA+ 216000 754776DBA+ 230400 754776lAA+ 244800 754776EBAvAA)paps_exec +()paps_exec +(+ 36000 730728FBAGBAHBAIBAIBAHBAJBAKBALBAMBA+ 115200 730728NBAOBAvAA+ 144000 730728PBAvAAQBAEBARBA)paps_exec +()paps_exec +(+ 64800 704736SBATBAUBAVBAoAAWBAuAA+ 122472 704736lAA+ 136872 704736PBAWBA+ 158472 704736SBATBAoAAWBAuAA+ 201744 704736pAA+ 216144 704736PBAzAA+ 237744 704736SBATBAXBAVBAoAAWBAuAA)paps_exec +()paps_exec +()paps_exec +()paps_exec +()paps_exec +(+ 36000 644616YBAzAA+ 57600 644616PBAZBA+ 79200 644616aBAyAAGBAJBAbBAMBAHBAyAAzAA+ 151200 644616aBAcBAcBAGBAABAHBAzAAyAAJBAbBAMBAdBA+ 244800 644616JBAKBAbBA+ 273600 644616ABAbBAeBA+ 302400 644616cBAfBAbBAGBAyAAJBAcBAGBA+ 367200 644616HBAMBA+ 388800 644616ABAbBAIBAHBAzAAbBAABA)paps_exec +()paps_exec +(+ 64800 620568qAAIBA+ 86400 620568lAA+ 100800 620568gBAhBAiBAhBAwAA+ 144000 620568IBAdBA+ 165600 620568hBAiBAhBAxAA+ 201600 620568IBAjBAvAA)paps_exec +()paps_exec +()paps_exec +(+ 36000 584496qAArAAIBA+ 64800 584496lAA+ 79200 584496qAAkBAqAAIBA+ 115200 584496lAA+ 129600 584496qAAkBAgBAhBAiBAhBAwAA+ 187200 584496IBAdBA+ 208800 584496hBAiBAhBAxAA+ 244800 584496IBAjBA)paps_exec +(+ 64800 572472lAA+ 79200 572472hBArAAiBAhBAwAArAA+ 129600 572472IBA+ 144000 572472sAA+ 158400 572472hBArAAiBAhBAxAArAA+ 208800 572472IBAvAA)paps_exec +()paps_exec +(+ 36000 548424lBA+ 50400 548424qAArAA+ 72000 548424lAA+ 86400 548424hBArAAiBAhBArAAwAA+ 136800 548424sAA+ 151200 548424hBArAAiBAhBArAAxAAvAA)paps_exec +()paps_exec +(+ 36000 524376mBAKBAbBAzAAdBA)paps_exec +()paps_exec +(+ 48096 499176hAAiAAjAA+ 72216 499176lAA+ 86616 499176nAA+ 101016 499176oAApAAoAAhBArAAiBAhBArAAwAA+ 173016 499176sAA+ 187416 499176hBArAAiBAhBArAAxAAuAA+ 245016 499176sAA+ 259416 499176oAAwAArAA+ 288216 499176sAA+ 302616 499176xAArAAuAAuAAvAA)paps_exec +()paps_exec +(+ 36000 473976hAAiAAjAA+ 60120 473976lAA+ 74520 473976nAA+ 88920 473976oAApAAoAAhBArAAiBAhBArAAwAA+ 160920 473976sAA+ 175320 473976hBArAAiBAhBArAAxAAuAA+ 232920 473976sAA+ 247320 473976oAAwAArAA+ 276120 473976sAA+ 290520 473976xAArAAuAAuAA)paps_exec +(+ 64800 461952lAA+ 79200 461952nAA+ 93600 461952oAAwAArAA+ 122400 461952pAA+ 136800 461952hBArAAiBAhBArAAwAA+ 187200 461952sAA+ 201600 461952xAArAA+ 223200 461952pAA+ 237600 461952hBArAAiBAhBArAAxAAuAA)paps_exec +(+ 64800 449928lAA+ 79200 449928nAAoAAwAArAA+ 115200 449928pAA+ 129600 449928hBArAAiBAhBArAAwAAuAA+ 187200 449928sAA+ 201600 449928nAAoAAxAArAA+ 237600 449928pAA+ 252000 449928hBArAAiBAhBArAAxAAuAAvAA)paps_exec +()paps_exec +(+ 36000 424728hAAiAA+ 53568 424728sAA+ 67968 424728hAAjAA+ 86832 424728lAA+ 101232 424728nAAoAAwAArAA+ 137232 424728pAA+ 151632 424728hBArAAiBAhBArAAwAAuAA+ 209232 424728sAA+ 223632 424728nAAoAAxAArAA+ 259632 424728pAA+ 274032 424728hBArAAiBAhBArAAxAAuAAvAA)paps_exec +()paps_exec +(+ 36000 400680nBA)paps_exec +()paps_exec +(+ 36000 376632mBAKBAbBA+ 64800 376632oBAaBAKBAGBAcBAABAHBAzAApBAbBAGBA+ 151200 376632NBAOBAqBAyAAJBAHBAcBAzAA+ 216000 376632JBAKBAbBAzAA+ 252000 376632GBAbBAyAAABAMBAdBA)paps_exec +()paps_exec +(+ 64800 351432gBAnAAoAAwAArAA+ 108000 351432pAA+ 122400 351432hBArAAiBAhBArAAwAAuAA+ 180000 351432sAA+ 194400 351432nAAoAAxAArAA+ 230400 351432pAA+ 244800 351432hBArAAiBAhBArAAxAAuAAjBA+ 309600 351432rBA+ 324000 351432lAA+ 338400 351432oAANBAiAA+ 361080 351432sAA+ 375480 351432NBAjAAuAA+ 403632 351432rBAvAA)paps_exec +()paps_exec +(+ 36000 326232sBAMBAMBAqBACBAHBAzAApBA+ 100800 326232yAA+ 115200 326232MBAbBAfBAyAAGBAyAAtBAeBAbBA+ 187200 326232MBAcBAeBAqBAJBAHBAcBAzAA+ 252000 326232rBAoAAwAAdBAxAAuAA+ 302400 326232lAA+ 316800 326232uBAoAAwAAuAA+ 352800 326232vBAoAAxAAuAAdBA+ 396000 326232wBAHBAJBAKBA+ 432000 326232NBA+ 446400 326232lAA+ 460800 326232NBAiAA+ 476280 326232sAA+ 490680 326232NBAjAAvAA)paps_exec +()paps_exec +(+ 36000 301032gBAnAAoAAwAArAA+ 79200 301032pAA+ 93600 301032hBArAAiBAhBArAAwAAuAA+ 151200 301032sAA+ 165600 301032nAAoAAxAArAA+ 201600 301032pAA+ 216000 301032hBArAAiBAhBArAAxAAuAAjBA+ 280800 301032uBAoAAwAAuAA+ 316800 301032vBAoAAxAAuAA+ 352800 301032lAA+ 367200 301032oAANBAiAA+ 389880 301032sAA+ 404280 301032NBAjAAuAA+ 432432 301032uBAoAAwAAuAA+ 468432 301032vBAoAAxAAuAAvAA)paps_exec +()paps_exec +(+ 36000 276984nAAoAAwAArAA+ 72000 276984pAA+ 86400 276984hBArAAiBAhBArAAwAAuAA+ 144000 276984uBAoAAwAAuAA+ 180000 276984vBAoAAxAAuAA+ 216000 276984sAA+ 230400 276984nAAoAAxAArAA+ 266400 276984pAA+ 280800 276984hBArAAiBAhBArAAxAAuAA+ 338400 276984uBAoAAwAAuAA+ 374400 276984vBAoAAxAAuAA)paps_exec +(+ 64800 263808lAA+ 79200 263808oAANBAiAA+ 101880 263808sAA+ 116280 263808NBAjAAuAA+ 144432 263808uBAoAAwAAuAA+ 180432 263808vBAoAAxAAuAAvAA)paps_exec +()paps_exec +(+ 36000 239760gBAEBAiBAuBAoAAwAAuAAjBA+ 100800 239760gBAnAAoAAwAArAA+ 144000 239760pAA+ 158400 239760hBArAAiBAhBArAAwAAuAA+ 216000 239760uBAoAAwAAuAAjBA+ 259200 239760sAA+ 273600 239760gBAEBAiBAvBAoAAxAAuAAjBA+ 338400 239760gBAnAAoAAxAArAA+ 381600 239760pAA+ 396000 239760hBArAAiBAhBArAAxAAuAA+ 453600 239760vBAoAAxAAuAAjBA)paps_exec +(+ 64800 226584lAA+ 79200 226584oAANBAiAA+ 101880 226584sAA+ 116280 226584NBAjAAuAAvAA)paps_exec +()paps_exec +(+ 36000 202536oBAcBAdBA+ 64800 202536YBA+ 79200 202536KBAyAAxBAbBA+ 115200 202536JBAwBAcBA+ 144000 202536ABAHBAIBAIBAbBAGBAbBAzAAJBAHBAyAAeBA+ 237600 202536bBAOBAqBAyAAJBAHBAcBAzAAMBAdBA)paps_exec +()paps_exec +(+ 64800 177336nAAoAAwAArAA+ 100800 177336pAA+ 115200 177336hBArAAiBAhBArAAwAAuAA+ 172800 177336uBAoAAwAAuAA+ 208800 177336lAA+ 223200 177336NBAiAA+ 238680 177336uBAoAAwAAuAAdBA+ 281880 177336yAAzAAABA)paps_exec +(+ 64800 164160nAAoAAxAArAA+ 100800 164160pAA+ 115200 164160hBArAAiBAhBArAAxAAuAA+ 172800 164160vBAoAAxAAuAA+ 208800 164160lAA+ 223200 164160NBAjAA+ 239976 164160vBAoAAxAAuAAvAA)paps_exec +()paps_exec +(+ 36000 140112mBAKBAbBA+ 64800 140112MBAcBAeBAqBAJBAHBAcBAzAAMBA+ 136800 140112JBAcBA+ 158400 140112JBAKBAbBAMBAbBA+ 201600 140112ABAHBAIBAIBAbBAGBAbBAzAAJBAHBAyAAeBA+ 295200 140112bBAOBAqBAyAAJBAHBAcBAzAAMBA+ 367200 140112yAAGBAbBA+ 396000 140112JBAKBAbBA+ 424800 140112MBAyAACBAbBA+ 460800 140112yAAMBA+ 482400 140112IBAcBAGBA+ 511200 140112JBAKBAbBA)paps_exec +(+ 36000 128088EBAZBA+ 57600 128088KBAyAAGBACBAcBAzAAHBAaBA+ 122400 128088cBAMBAaBAHBAeBAeBAyAAJBAcBAGBAvAA+ 208800 128088mBAKBAbBAxAA+ 244800 128088KBAyAAxBAbBA+ 280800 128088bBAHBApBAbBAzAAxBAyAAeBAqBAbBAMBA+ 367200 128088oAAzAA+ 388800 128088sAA+ 403200 128088EBAiBAPBAuAAdBA+ 446400 128088wBAKBAbBAGBAbBA+ 489600 128088BBA+ 504000 128088lAA+ 518400 128088DBA+ 532800 128088lAA)paps_exec +(+ 36000 116064EBAdBA+ 57600 116064wBAHBAJBAKBA+ 93600 116064zAA+ 108000 116064lAA+ 122400 116064zBAdBAEBAdBAPBAdBAvAAvAAvAA+ 194400 116064vAA)paps_exec +()paps_exec +(+ 36000 90864lBA+ 50400 90864JBAKBAbBA+ 79200 90864bBAHBApBAbBAzAAxBAyAAeBAqBAbBAMBA+ 165600 90864IBAcBAGBA+ 194400 90864JBAKBAbBA+ 223200 90864aBAcBACBAtBAHBAzAAbBAABA+ 288000 90864cBAfBAbBAGBAyAAJBAcBAGBA+ 352800 90864yAAGBAbBA+ 381600 90864zAAiAA+ 397080 90864sAA+ 411480 90864zAAjAA+ 428256 90864sAA+ 442656 90864EBAvAA)paps_exec +()paps_exec +(+ 36000 66816mBAKBAbBA+ 64800 66816ABAbBApBAbBAzAAbBAGBAyAAaBAxAA+ 144000 66816HBAMBA+ 165600 66816fBAGBAbBAJBAJBAxAA+ 216000 66816cBAtBAxBAHBAcBAqBAMBAdBA+ 280800 66816ACAqBAMBAJBA+ 316800 66816IBAGBAcBACBA+ 352800 66816aBAcBAqBAzAAJBAHBAzAApBA+ 417600 66816JBAKBAbBA+ 446400 66816fBAcBAMBAMBAHBAtBAHBAeBAHBAJBAHBAbBAMBARBA)paps_exec +(+ 36000 53640JBAKBAbBAGBAbBA+ 79200 53640HBAMBA+ 100800 53640zAAsAAEBA+ 129600 53640ABAbBApBAbBAzAAbBAGBAyAAaBAxAA+ 208800 53640IBAcBAGBA+ 237600 53640bBAyAAaBAKBA+ 273600 53640xBAyAAeBAqBAbBA+ 316800 53640cBAIBA+ 338400 53640zAA+ 352800 53640lAA+ 367200 53640zAAiAA+ 382680 53640sAA+ 397080 53640zAAjAAvAA)paps_exec +()paps_exec +paps_eop +showpage +%%Page: 2 2 +paps_bop +(+ 36000 808000AAABAACAA+ 65088 808000EAAFAA+ 86904 808000GAAHAAIAA+ 115992 808000JAAKAAEAALAA+ 152352 808000EAAEAAMAANAAOAAMAAPAAEAA+ 217800 808000QAARAA+ 239616 808000SAATAAUAA)paps_exec +(* 232052VAAWAAXAAYAA+ 268412 808000ZAAaAAbAAcAAdAAXAA+ 319316 808000eAA+ 333860 808000fAAAAAEAAJAA)paps_exec +(* 515368QAAHAAgAABAA+ 551728 808000JAA)paps_exec +36 798.000000 moveto 559 798.000000 lineto 0 setlinewidth stroke +(+ 36000 779976oBAcBAdBA)paps_exec +(+ 64800 767952zAA+ 108000 767952ABAbBApBAbBAzAAbBAGBAyAAaBAxAA)paps_exec +(+ 64800 755928BCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCABCA)paps_exec +(+ 64800 743904zBA+ 144000 743904EBA)paps_exec +(+ 64800 731880EBA+ 144000 731880PBA)paps_exec +(+ 64800 719856PBA+ 144000 719856CCA)paps_exec +(+ 64800 707832CCA+ 144000 707832DCA)paps_exec +(+ 64800 695808DCA+ 144000 695808ECA)paps_exec +(+ 64800 683784ECA+ 144000 683784FCA)paps_exec +()paps_exec +()paps_exec +(+ 36000 647712mBAKBAbBA+ 64800 647712SBAbBAGBACBAHBAJBAbBA+ 122400 647712fBAcBAeBAxAAzAAcBACBAHBAyAAeBAMBA+ 208800 647712KBAbBAeBAfBA+ 244800 647712JBAcBA+ 266400 647712pBAbBAzAAbBAGBAyAAJBAbBA+ 331200 647712JBAKBAbBA+ 360000 647712bBAHBApBAbBAzAAMBAJBAyAAJBAbBAMBA+ 446400 647712cBAIBA+ 468000 647712JBAKBAHBAMBARBA)paps_exec +()paps_exec +(+ 36000 621720SBATBAoAAwAAuAA+ 79272 621720lAA+ 93672 621720oAApAAEBAuAAGCA+ 136872 621720bBAwAAfBAoAAwAArAAuAA+ 194472 621720ABAiBAABAwAAGCA+ 237672 621720bBAwAAfBAoAApAAwAArAAiBAPBAuAA+ 316872 621720lAA+ 331272 621720oAAPBAwAA+ 360072 621720pAA+ 374472 621720ABAiBAABAwAAuAAGCA+ 424872 621720HCA+ 439272 621720EBAvAA)paps_exec +()paps_exec +(+ 36000 597672mBAKBAbBA+ 64800 597672IBAHBAGBAMBAJBA+ 108000 597672MBAHBAwAA+ 136800 597672fBAcBAeBAxAAzAAcBACBAHBAyAAeBAMBA+ 223200 597672yAAGBAbBA)paps_exec +()paps_exec +(+ 36000 573624SBAICAoAAwAAuAA+ 79200 573624lAA+ 93600 573624EBAvAA)paps_exec +(+ 36000 561600SBAVBAoAAwAAuAA+ 79200 561600lAA+ 93600 561600PBAwAAvAA)paps_exec +(+ 36000 549576SBAJCAoAAwAAuAA+ 79200 549576lAA+ 93600 549576DCAwAArAA+ 122400 549576pAA+ 136800 549576PBAvAA)paps_exec +(+ 36000 537552SBAKCAoAAwAAuAA+ 79200 537552lAA+ 93600 537552LCAwAAMCA+ 122400 537552pAA+ 136800 537552EBAPBAwAAvAA)paps_exec +(+ 36000 525528SBANCAoAAwAAuAA+ 79200 525528lAA+ 93600 525528EBAFCAwAAOCA+ 129600 525528pAA+ 144000 525528DCALCAwAArAA+ 180000 525528sAA+ 194400 525528EBAPBAvAA)paps_exec +(+ 36000 513504SBAPCAoAAwAAuAA+ 79200 513504lAA+ 93600 513504CCAPBAwAAQCA+ 129600 513504pAA+ 144000 513504EBAFCAzBAwAAMCA+ 187200 513504sAA+ 201600 513504EBAPBAzBAwAAvAA)paps_exec +(+ 36000 501480SBARCAoAAwAAuAA+ 79200 501480lAA+ 93600 501480FCADCAwAASCA+ 129600 501480pAA+ 144000 501480DCALCAzBAwAAOCA+ 187200 501480sAA+ 201600 501480QBAPBAzBAwAArAA+ 244800 501480pAA+ 259200 501480EBAPBAzBAvAA)paps_exec +()paps_exec +()paps_exec +(+ 36000 465408mBAKBAbBA+ 64800 465408wBAyAAxBAbBA+ 100800 465408IBAqBAzAAaBAJBAHBAcBAzAAMBA+ 172800 465408HBAzAAxBAcBAeBAxBAHBAzAApBA+ 244800 465408JBAKBAbBAMBAbBA+ 288000 465408fBAcBAeBAxAAzAAcBACBAHBAyAAeBAMBAdBA+ 381600 465408wBAHBAJBAKBA+ 417600 465408JBAKBAbBA+ 446400 465408qBAzAAHBAJBAHBATCAyAAJBAHBAcBAzAAMBA)paps_exec +(+ 36000 453384pBAHBAxBAbBAzAA+ 79200 453384HBAzAA+ 100800 453384JBAKBAbBA+ 129600 453384HBAzAAJBAGBAcBAdBA+ 180000 453384yAAGBAbBA)paps_exec +()paps_exec +(+ 64800 427392rBATBAUCAoAAwAAuAA+ 115344 427392lAA+ 129744 427392VCAWCAoAApAAEBAiBADCAuAA+ 194544 427392EBAiBAXCAoAAPBAGCA+ 244944 427392zAAYCAuAA+ 273744 427392SBATBAoAAwAAuAA+ 317016 427392bBAwAAfBAoAApAAwAArAAiBAPBAuAA)paps_exec +(+ 151200 413424VCAWCAoAApAAEBAiBADCAuAA+ 216000 413424EBAiBAXCAoAAPBAZCA+ 266544 413424CBAYCAuAA+ 295344 413424SBAUCAoAAxAAuAA+ 338616 413424bBAwAAfBAoAApAAxAArAAiBAPBAuAAvAA)paps_exec +()paps_exec +(+ 36000 387432rBATBAUCAoAAwAAuAA+ 86544 387432lAA+ 100944 387432EBAiBAXCAVCA+ 136944 387432EBAiBAXCAoAAPBAGCA+ 187344 387432zAAYCA+ 208944 387432PBAZCA+ 230688 387432CBAYCAuAA+ 259488 387432SBATBAoAAwAAuAA+ 302760 387432SBAUCAoAAxAAuAA+ 346032 387432bBAwAAfBAoAApAAoAAwAArAAiBAPBA+ 425232 387432sAA+ 439632 387432xAArAAiBAPBAuAAuAAvAA)paps_exec +()paps_exec +()paps_exec +(+ 36000 351360mBAKBAbBA+ 64800 351360eBAcBAwBAbBAGBAHBAzAApBA+ 129600 351360cBAfBAbBAGBAyAAJBAcBAGBAMBA+ 201600 351360yAAGBAbBA)paps_exec +()paps_exec +(+ 64800 326160yAAiAA+ 80280 326160lAA+ 94680 326160EBAiBAXCAPBA+ 130680 326160oAAwAA+ 152280 326160sAA+ 166680 326160bCAfBAiAAuAA+ 200736 326160yAAzAAABA)paps_exec +(+ 64800 312984yAAjAA+ 81576 312984lAA+ 95976 312984EBAiBAXCAPBA+ 131976 312984oAAxAA+ 153576 312984sAA+ 167976 312984bCAfBAjAAuAAvAA)paps_exec +()paps_exec +(+ 36000 287784mBAKBAbBAxAA+ 72000 287784yAAGBAbBA+ 100800 287784zAAcBAJBA+ 129600 287784KBAbBAGBACBAHBAJBAHBAyAAzAAdBA+ 208800 287784tBAqBAJBA+ 237600 287784wAAdBAxAA+ 266400 287784yAAzAAABA+ 295200 287784fBAiAAdBAfBAjAA+ 331632 287784yAAGBAbBAdBA+ 367632 287784MBAcBA+ 389232 287784JBAKBAbBA+ 418032 287784GBAyAAHBAMBAHBAzAApBA+ 475632 287784cBAfBAbBAGBAyAAJBAcBAGBAMBA)paps_exec +(+ 36000 275760yAAGBAbBA)paps_exec +()paps_exec +(+ 64800 250560yAAiAAcCA+ 91656 250560lAA+ 106056 250560EBAiBAXCAPBA+ 142056 250560oAAwAA+ 163656 250560pAA+ 178056 250560bCAfBAiAAuAA+ 212112 250560yAAzAAABA)paps_exec +(+ 64800 237384yAAjAAcCA+ 92952 237384lAA+ 107352 237384EBAiBAXCAPBA+ 143352 237384oAAxAA+ 164952 237384pAA+ 179352 237384bCAfBAjAAuAAvAA)paps_exec +()paps_exec +(+ 36000 211392sBAfBAfBAeBAxAAHBAzAApBA+ 100800 211392JBAKBAbBAMBAbBA+ 144000 211392JBAcBA+ 165600 211392JBAKBAbBA+ 194400 211392pBAGBAcBAqBAzAAABA+ 244800 211392MBAJBAyAAJBAbBA+ 288000 211392dCAzBAzBAeCAdBA+ 331344 211392YBA+ 345744 211392aBAyAAzAA+ 374544 211392IBAHBAzAAABA+ 410544 211392JBAKBAbBA+ 439344 211392IBAHBAGBAMBAJBA+ 482544 211392MBAHBAwAA)paps_exec +(+ 36000 199368MBAJBAyAAJBAbBAMBAdBA+ 93600 199368wBAHBAJBAKBA+ 129600 199368zAAcBAGBACBAyAAeBAHBATCAyAAJBAHBAcBAzAARBA)paps_exec +()paps_exec +(+ 36000 173376yAAiAAcCAdCAzBAzBAeCA+ 91872 173376lAA+ 106272 173376EBAiBAXCAPBA+ 142272 173376oAAwAA+ 163872 173376pAA+ 178272 173376bCAfBAiAAuAAdCAzBAzBAeCA)paps_exec +(+ 93600 159408lAA+ 108000 159408EBAiBAXCAPBA+ 144000 159408oAAwAAdCAzBAzBAeCA+ 194616 159408pAA+ 209016 159408bCAfBAiAAdCAzBAzBAeCAuAA)paps_exec +()paps_exec +()paps_exec +()paps_exec +()paps_exec +()paps_exec +()paps_exec +(+ 36000 75240tBAuAA+ 57600 75240fCA+ 72000 75240aBAuAA)paps_exec +()paps_exec +(+ 36000 51192YBALBACBA+ 64800 51192MBAJBAHBAeBAeBA+ 108000 51192wBAcBAGBAgCAHBAzAApBA+ 165600 51192cBAqBAJBA+ 194400 51192JBAKBAbBA+ 223200 51192yAAeBApBAbBAtBAGBAyAAdBA+ 288000 51192KBAbBAGBAbBAvAA+ 331200 51192YBA+ 345600 51192wBAHBAeBAeBA+ 381600 51192JBAGBAxAA+ 410400 51192JBAcBA+ 432000 51192IBAHBAzAAHBAMBAKBA+ 482400 51192HBAJBA+ 504000 51192yAAMBA)paps_exec +(+ 36000 39168MBAcBAcBAzAA+ 72000 39168yAAMBA+ 93600 39168YBA+ 108000 39168aBAyAAzAAdBA+ 144000 39168tBAqBAJBA+ 172800 39168YBA+ 187200 39168gCAzAAcBAwBA+ 223200 39168YBA+ 237600 39168yAAeBAMBAcBA+ 273600 39168KBAyAAxBAbBA+ 309600 39168zAAbBAwBA+ 338400 39168wBAcBAGBAgCA+ 374400 39168JBAcBA+ 396000 39168ABAcBAvAA)paps_exec +paps_eop +showpage +%%Page: 3 3 +paps_bop +(+ 36000 808000AAABAACAA+ 65088 808000EAAFAA+ 86904 808000GAAHAAIAA+ 115992 808000JAAKAAEAALAA+ 152352 808000EAAEAAMAANAAOAAMAAPAAEAA+ 217800 808000QAARAA+ 239616 808000SAATAAUAA)paps_exec +(* 232052VAAWAAXAAYAA+ 268412 808000ZAAaAAbAAcAAdAAXAA+ 319316 808000eAA+ 333860 808000fAAAAAEAAJAA)paps_exec +(* 515368QAAHAAgAABAA+ 551728 808000NAA)paps_exec +36 798.000000 moveto 559 798.000000 lineto 0 setlinewidth stroke +()paps_exec +()paps_exec +(+ 36000 755928YBA+ 50400 755928IBAHBAzAAHBAMBAKBAbBAABA+ 115200 755928CBAqBAaBAKBA+ 151200 755928cBAIBA+ 172800 755928JBAKBAHBAMBA+ 208800 755928yAAMBAMBAHBApBAzAACBAbBAzAAJBAdBA+ 295200 755928tBAqBAJBA+ 324000 755928zAAbBAbBAABA+ 360000 755928JBAcBA+ 381600 755928pBAbBAJBA+ 410400 755928ABAcBAzAAbBA+ 446400 755928IBAyAAMBAJBAbBAGBA+ 496800 755928HBAzAA+ 518400 755928JBAKBAbBA)paps_exec +(+ 36000 743904IBAqBAJBAqBAGBAbBAvAA)paps_exec +paps_eop +showpage +%%Trailer +%%Pages: 3 +%%EOF diff --git a/hw/12/bookprobs_othoulrich_hw12.pdf b/hw/12/bookprobs_othoulrich_hw12.pdf new file mode 100644 index 0000000..2c62453 Binary files /dev/null and b/hw/12/bookprobs_othoulrich_hw12.pdf differ diff --git a/hw/12/othoulrich_hw12.pdf b/hw/12/othoulrich_hw12.pdf new file mode 100644 index 0000000..17c0193 Binary files /dev/null and b/hw/12/othoulrich_hw12.pdf differ diff --git a/hw/hw14.pdf b/hw/hw14.pdf new file mode 100644 index 0000000..bb47b77 Binary files /dev/null and b/hw/hw14.pdf differ diff --git a/presentation/.ipynb_checkpoints/Entanglement-checkpoint.ipynb b/presentation/.ipynb_checkpoints/Entanglement-checkpoint.ipynb new file mode 100644 index 0000000..c192fea --- /dev/null +++ b/presentation/.ipynb_checkpoints/Entanglement-checkpoint.ipynb @@ -0,0 +1,349 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "# EPR Paradox Example\n", + "\n", + "# A neutral pi meson (pi0) decays into an electron/positron (e-/e+) \n", + "# pair.\n", + "\n", + "# pi0 --> e+ + e- (electron-positron pair)\n", + "# pi0 has angular momentum l = s = 0\n", + "\n", + "# Align electron and positron detectors in opposite directions.\n", + "\n", + "# | pi0 |\n", + "# |e- <------ ------> e+|\n", + "\n", + "# Where hbar = 1, the measurement of the spin in some direction is \n", + "# +/-1 with spin state [1 0] (up) or [0 1] (down).\n", + "\n", + "# The Pauli exclusion principle with conserved angular momentum 0\n", + "# dictates this system must be in the singlet state \n", + "# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)].\n", + "\n", + "# In this state, if the positron is measured to have spin [1 0], the \n", + "# electron must have spin [0 1], or vice versa. There is an equal \n", + "# probability to find either state during the first measurement.\n", + "\n", + "# This view is consistent with the realist view. The realist view could \n", + "# hold that the electron and position had those angular momenta \n", + "# from creation.\n", + "\n", + "# EPR assumes influences cannot propagate faster than the speed of \n", + "# light. \"Wave function collapse\" is apparently instantaneous, however.\n", + "\n", + "\n", + "import numpy as np\n", + "import matplotlib\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.patches as mpatches\n", + "%matplotlib inline " + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# If information about the measurement of the wave function propagated\n", + "# at a finite speed, the particles could conceivably be measured such\n", + "# that both are equally likely to hold either spin up or spin down.\n", + "\n", + "# What would happen if the measurements were uncorrelated?\n", + "\n", + "plt.hist(np.random.randint(0,2,1000),bins=4)\n", + "plt.hist(np.random.randint(3,5,1000),bins=4)\n", + "elepatch = mpatches.Patch(color='blue', label='e-')\n", + "pospatch = mpatches.Patch(color='orange', label='e+')\n", + "#plt.legend(handles=[elepatch,pospatch])\n", + "plt.text(0.5,565,\"e-\",size=20)\n", + "plt.text(3.5,565,\"e+\",size=20)\n", + "\n", + "plt.suptitle(\"Uncorrelated Spins\",fontsize=20)\n", + "plt.ylim([400,600])\n", + "plt.xlim([-1,5])\n", + "plt.xticks([0.125,0.85,3.125,3.85],[\"down\",\"up\",\"down\",\"up\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": "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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# How many violations of angular momentum would be measured?\n", + "\n", + "# From running several simulations, it's evident a violation in the \n", + "# conservation of angular momentum would be measured half of the time.\n", + "# We can conclude that the information that the entangled particles are in\n", + "# orthogonal spin states is instantaneously agreed once a measurement\n", + "# is made.\n", + "\n", + "violations = 0\n", + "\n", + "for trial in range(0,1000):\n", + " elespin = np.random.randint(0,2)\n", + " posspin = np.random.randint(0,2)\n", + " if elespin == 0:\n", + " elespin = np.matrix('0 1')\n", + " else:\n", + " elespin = np.matrix('1 0')\n", + " if posspin == 0:\n", + " posspin = np.matrix('0 1')\n", + " else:\n", + " posspin = np.matrix('1 0')\n", + " \n", + " elespin.transpose()\n", + " posspin.transpose()\n", + " chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1')\n", + " \n", + " if chi_squared == 0:\n", + " violations = violations + 1\n", + "\n", + "zeroes = np.zeros(violations,dtype=int)\n", + "ones = np.full((1000-violations),1,dtype=int)\n", + "result = np.concatenate((ones,zeroes))\n", + "\n", + "plt.ylim([400,600])\n", + "plt.xlim([-1,2])\n", + "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.suptitle(\"Conservation Violations\",fontsize=20)\n", + "plt.hist([result],bins=4)\n", + "plt.figure()" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Bell's Experiment took this a step further, to rule out locality \n", + "# completely. Establish the detectors to \"float\" such that they \n", + "# measure the components of the spins of the electron and positron \n", + "# along a unit vector a or b, with angles phi_a and phi_b, \n", + "# respectively. Compute a product P of the spins in units of hbar/2. \n", + "# This will give +/-1. \n", + "\n", + "# / pi0 \\\n", + "# /e- <------ ------> e+\\\n", + "\n", + "# QM predicts P(a,b) = -a dot b, the expectation value of the product \n", + "# of the spins.\n", + "\n", + "# In 1964, Bell derived the Bell inequality for a local hidden variable\n", + "# theory: abs(P(a,b) - P(a,c)) <= 1 + P(b,c)\n", + "\n", + "# For any local hidden variable theory, the Bell inequality must hold. \n", + "# It introduces a third unit vector c, which is any other unit vector \n", + "# than a or b.\n", + "\n", + "# Does the quantum mechanical prediction violate the Bell inequality?\n", + "# Testing several detector configurations in a plane, systematically \n", + "# from 0 to pi, we determine whether the QM prediction is consistent \n", + "# with a local hidden variable theory.\n", + "\n", + "violations = 0\n", + "trials = 0\n", + "for step_a in range(0,6):\n", + " for step_b in range (0,6):\n", + " phi_a = step_a/6*np.pi\n", + " phi_b = step_b/6*np.pi\n", + " phi_c = phi_a - 0.5*phi_b\n", + " P_ab = -1 * np.cos(phi_a - phi_b)\n", + " P_ac = -1 * np.cos(phi_a - phi_c)\n", + " P_bc = -1 * np.cos(phi_b - phi_c)\n", + " \n", + " bell_lhs = np.abs(P_ab - P_ac)\n", + " bell_rhs = 1 + P_bc\n", + " \n", + " if bell_lhs > bell_rhs:\n", + " violations = violations + 1\n", + " \n", + " trials = trials + 1\n", + " \n", + "zeroes = np.zeros(violations,dtype=int)\n", + "ones = np.full((trials-violations),1,dtype=int)\n", + "result = np.concatenate((ones,zeroes))\n", + "\n", + "plt.ylim([trials/2-10,trials/2+10])\n", + "plt.xlim([-1,2])\n", + "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n", + "plt.hist([result],bins=4)\n", + "plt.figure()" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# It is seen that the QM prediction disagrees with a local hidden \n", + "# variable theory in a significant number of configurations.\n", + "\n", + "# On average, for random orientations between 0 and pi, how often? \n", + "# After running several trials, it appears to be about half of the, \n", + "# which is what one would expect from the quantum mechanical\n", + "# prediction.\n", + "\n", + "violations = 0\n", + "trials = 0\n", + "for rand_a in range(0,10):\n", + " for rand_b in range (0,10):\n", + " phi_a = np.random.rand(1)*np.pi\n", + " phi_b = np.random.rand(1)*np.pi\n", + " phi_c = phi_a - 0.5*phi_b\n", + " P_ab = -1 * np.cos(phi_a - phi_b)\n", + " P_ac = -1 * np.cos(phi_a - phi_c)\n", + " P_bc = -1 * np.cos(phi_b - phi_c)\n", + " \n", + " bell_lhs = np.abs(P_ab - P_ac)\n", + " bell_rhs = 1 + P_bc\n", + " \n", + " if bell_lhs > bell_rhs:\n", + " violations = violations + 1\n", + " \n", + " trials = trials + 1\n", + "\n", + "zeroes = np.zeros(violations,dtype=int)\n", + "ones = np.full((trials-violations),1,dtype=int)\n", + "result = np.concatenate((ones,zeroes))\n", + "\n", + "plt.ylim([trials/2-10,trials/2+10])\n", + "plt.xlim([-1,2])\n", + "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n", + "plt.hist([result],bins=4)\n", + "plt.figure()" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "# The prediction (Bell's inequality) made by assuming a local hidden\n", + "# variable is violated by a significant number of the possibile \n", + "# bborientations. This simulation cannot determine which theory is\n", + "# correct, but the QM prediction has been confirmed through experiment.\n", + "# No hidden local variable holds actionable information about the \n", + "# state. Entangled states retain their entanglement in a non-local \n", + "# nature." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.4.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/presentation/Document.docx b/presentation/Document.docx new file mode 100644 index 0000000..3333244 Binary files /dev/null and b/presentation/Document.docx differ diff --git a/presentation/Entanglement.ipynb b/presentation/Entanglement.ipynb new file mode 100644 index 0000000..966b384 --- /dev/null +++ b/presentation/Entanglement.ipynb @@ -0,0 +1,387 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "# EPR Paradox Example\n", + "\n", + "# A neutral pi meson (pi0) decays into an electron/positron (e-/e+) \n", + "# pair.\n", + "\n", + "# pi0 --> e+ + e- (electron-positron pair)\n", + "# pi0 has angular momentum l = s = 0\n", + "\n", + "# Align electron and positron detectors in opposite directions.\n", + "\n", + "# | pi0 |\n", + "# |e- <------ ------> e+|\n", + "\n", + "# Where hbar = 1, the measurement of the spin in some direction is \n", + "# +/-1 with spin state [1 0] (up) or [0 1] (down).\n", + "\n", + "# The Pauli exclusion principle with conserved angular momentum 0\n", + "# dictates this system must be in the singlet state \n", + "# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)].\n", + "\n", + "# In this state, if the positron is measured to have spin [1 0], the \n", + "# electron must have spin [0 1], or vice versa. There is an equal \n", + "# probability to find either state during the first measurement.\n", + "\n", + "# This view is consistent with the realist view. The realist view could \n", + "# hold that the electron and position had those angular momenta \n", + "# from creation.\n", + "\n", + "# EPR assumes influences cannot propagate faster than the speed of \n", + "# light. \"Wave function collapse\" is apparently instantaneous, however.\n", + "\n", + "\n", + "import numpy as np\n", + "import matplotlib\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.patches as mpatches\n", + "%matplotlib inline " + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# If information about the measurement of the wave function propagated\n", + "# at a finite speed, the particles could conceivably be measured such\n", + "# that both are equally likely to hold either spin up or spin down.\n", + "\n", + "# What would happen if the measurements were uncorrelated?\n", + "\n", + "plt.hist(np.random.randint(0,2,1000),bins=4)\n", + "plt.hist(np.random.randint(3,5,1000),bins=4)\n", + "elepatch = mpatches.Patch(color='blue', label='e-')\n", + "pospatch = mpatches.Patch(color='orange', label='e+')\n", + "#plt.legend(handles=[elepatch,pospatch])\n", + "plt.text(0.5,565,\"e-\",size=20)\n", + "plt.text(3.5,565,\"e+\",size=20)\n", + "\n", + "plt.suptitle(\"Uncorrelated Spins\",fontsize=20)\n", + "plt.ylim([400,600])\n", + "plt.xlim([-1,5])\n", + "plt.xticks([0.125,0.85,3.125,3.85],[\"down\",\"up\",\"down\",\"up\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": "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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# How many violations of angular momentum would be measured?\n", + "\n", + "# From running several simulations, it's evident a violation in the \n", + "# conservation of angular momentum would be measured half of the time.\n", + "# We can conclude that the information that the entangled particles are in\n", + "# orthogonal spin states is instantaneously agreed once a measurement\n", + "# is made.\n", + "\n", + "violations = 0\n", + "\n", + "for trial in range(0,1000):\n", + " elespin = np.random.randint(0,2)\n", + " posspin = np.random.randint(0,2)\n", + " if elespin == 0:\n", + " elespin = np.matrix('0 1')\n", + " else:\n", + " elespin = np.matrix('1 0')\n", + " if posspin == 0:\n", + " posspin = np.matrix('0 1')\n", + " else:\n", + " posspin = np.matrix('1 0')\n", + " \n", + " elespin.transpose()\n", + " posspin.transpose()\n", + " chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1')\n", + " \n", + " if chi_squared == 0:\n", + " violations = violations + 1\n", + "\n", + "zeroes = np.zeros(violations,dtype=int)\n", + "ones = np.full((1000-violations),1,dtype=int)\n", + "result = np.concatenate((ones,zeroes))\n", + "\n", + "plt.ylim([400,600])\n", + "plt.xlim([-1,2])\n", + "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.suptitle(\"Conservation Violations\",fontsize=20)\n", + "plt.hist([result],bins=4)\n", + "plt.figure()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": "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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# Bell's Experiment took this a step further, to rule out locality \n", + "# completely. Establish the detectors to \"float\" such that they \n", + "# measure the components of the spins of the electron and positron \n", + "# along a unit vector a or b, with angles phi_a and phi_b, \n", + "# respectively. Compute a product P of the spins in units of hbar/2. \n", + "# This will give +/-1. \n", + "\n", + "# / pi0 \\\n", + "# /e- <------ ------> e+\\\n", + "\n", + "# QM predicts P(a,b) = -a dot b, the expectation value of the product \n", + "# of the spins.\n", + "\n", + "# In 1964, Bell derived the Bell inequality for a local hidden variable\n", + "# theory: abs(P(a,b) - P(a,c)) <= 1 + P(b,c)\n", + "\n", + "# For any local hidden variable theory, the Bell inequality must hold. \n", + "# It introduces a third unit vector c, which is any other unit vector \n", + "# than a or b.\n", + "\n", + "# Does the quantum mechanical prediction violate the Bell inequality?\n", + "# Testing several detector configurations in a plane, systematically \n", + "# from 0 to pi, we determine whether the QM prediction is consistent \n", + "# with a local hidden variable theory.\n", + "\n", + "violations = 0\n", + "trials = 0\n", + "for step_a in range(0,6):\n", + " for step_b in range (0,6):\n", + " phi_a = step_a/6*np.pi\n", + " phi_b = step_b/6*np.pi\n", + " phi_c = phi_a - 0.5*phi_b\n", + " P_ab = -1 * np.cos(phi_a - phi_b)\n", + " P_ac = -1 * np.cos(phi_a - phi_c)\n", + " P_bc = -1 * np.cos(phi_b - phi_c)\n", + " \n", + " bell_lhs = np.abs(P_ab - P_ac)\n", + " bell_rhs = 1 + P_bc\n", + " \n", + " if bell_lhs > bell_rhs:\n", + " violations = violations + 1\n", + " \n", + " trials = trials + 1\n", + " \n", + "zeroes = np.zeros(violations,dtype=int)\n", + "ones = np.full((trials-violations),1,dtype=int)\n", + "result = np.concatenate((ones,zeroes))\n", + "\n", + "plt.ylim([trials/2-10,trials/2+10])\n", + "plt.xlim([-1,2])\n", + "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n", + "plt.hist([result],bins=4)\n", + "plt.figure()" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": "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\n", + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# It is seen that the QM prediction disagrees with a local hidden \n", + "# variable theory in a significant number of configurations.\n", + "\n", + "# On average, for random orientations between 0 and pi, how often? \n", + "# After running several trials, it appears to be about half of the, \n", + "# which is what one would expect from the quantum mechanical\n", + "# prediction.\n", + "\n", + "violations = 0\n", + "trials = 0\n", + "for rand_a in range(0,10):\n", + " for rand_b in range (0,10):\n", + " phi_a = np.random.rand(1)*np.pi\n", + " phi_b = np.random.rand(1)*np.pi\n", + " phi_c = phi_a - 0.5*phi_b\n", + " P_ab = -1 * np.cos(phi_a - phi_b)\n", + " P_ac = -1 * np.cos(phi_a - phi_c)\n", + " P_bc = -1 * np.cos(phi_b - phi_c)\n", + " \n", + " bell_lhs = np.abs(P_ab - P_ac)\n", + " bell_rhs = 1 + P_bc\n", + " \n", + " if bell_lhs > bell_rhs:\n", + " violations = violations + 1\n", + " \n", + " trials = trials + 1\n", + "\n", + "zeroes = np.zeros(violations,dtype=int)\n", + "ones = np.full((trials-violations),1,dtype=int)\n", + "result = np.concatenate((ones,zeroes))\n", + "\n", + "plt.ylim([trials/2-10,trials/2+10])\n", + "plt.xlim([-1,2])\n", + "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n", + "plt.tick_params(axis='both',labelsize=15)\n", + "plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n", + "plt.hist([result],bins=4)\n", + "plt.figure()" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "# The prediction (Bell's inequality) made by assuming a local hidden\n", + "# variable is violated by a significant number of the possibile \n", + "# bborientations. This simulation cannot determine which theory is\n", + "# correct, but the QM prediction has been confirmed through experiment.\n", + "# No hidden local variable holds actionable information about the \n", + "# state. Entangled states retain their entanglement in a non-local \n", + "# nature." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.4.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/presentation/entanglement.cpp b/presentation/entanglement.cpp new file mode 100644 index 0000000..dc9e933 --- /dev/null +++ b/presentation/entanglement.cpp @@ -0,0 +1,43 @@ +#include + +struct spin {bool up; bool down;} +struct entangled_spin {bool up_1; bool up_2; bool down_1; bool down_2; } + +struct state { double prob_up; double prob_down; } +struct entangled_state {} + + + + +spin measure (particle measured) { + +} + + +int main(int argc, char const *argv[]) +{ + + std::vector v; + spin electron_spin; + spin positron_spin; + + + \\ uncorrelated measurements + for (int trial=1; trial<=1000; trial++) { + particle positron; + particle electron; + measure_spin(positron) + } + + \\ entangled measurements + for (int trial=1; trial<=1000; trial++) { + entangled_pair pair = new entangled_pair( + particle position, + particle electron); + measure_spin(positron) + } + + return 0; + + +//============================================================================= \ No newline at end of file diff --git a/presentation/notebook.tex b/presentation/notebook.tex new file mode 100644 index 0000000..7a25aa7 --- /dev/null +++ b/presentation/notebook.tex @@ -0,0 +1,564 @@ + +% Default to the notebook output style + + + + +% Inherit from the specified cell style. + + + + + +\documentclass[11pt]{article} + + + + \usepackage[T1]{fontenc} + % Nicer default font (+ math font) than Computer Modern for most use cases + \usepackage{mathpazo} + + % Basic figure setup, for now with no caption control since it's done + % automatically by Pandoc (which extracts ![](path) syntax from Markdown). + \usepackage{graphicx} + % We will generate all images so they have a width \maxwidth. This means + % that they will get their normal width if they fit onto the page, but + % are scaled down if they would overflow the margins. + \makeatletter + \def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth + \else\Gin@nat@width\fi} + \makeatother + \let\Oldincludegraphics\includegraphics + % Set max figure width to be 80% of text width, for now hardcoded. + \renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}} + % Ensure that by default, figures have no caption (until we provide a + % proper Figure object with a Caption API and a way to capture that + % in the conversion process - todo). + \usepackage{caption} + \DeclareCaptionLabelFormat{nolabel}{} + \captionsetup{labelformat=nolabel} + + \usepackage{adjustbox} % Used to constrain images to a maximum size + \usepackage{xcolor} % Allow colors to be defined + \usepackage{enumerate} % Needed for markdown enumerations to work + \usepackage{geometry} % Used to adjust the document margins + \usepackage{amsmath} % Equations + \usepackage{amssymb} % Equations + \usepackage{textcomp} % defines textquotesingle + % Hack from http://tex.stackexchange.com/a/47451/13684: + \AtBeginDocument{% + \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code + } + \usepackage{upquote} % Upright quotes for verbatim code + \usepackage{eurosym} % defines \euro + \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support + \usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document + \usepackage{fancyvrb} % verbatim replacement that allows latex + \usepackage{grffile} % extends the file name processing of package graphics + % to support a larger range + % The hyperref package gives us a pdf with properly built + % internal navigation ('pdf bookmarks' for the table of contents, + % internal cross-reference links, web links for URLs, etc.) + \usepackage{hyperref} + \usepackage{longtable} % longtable support required by pandoc >1.10 + \usepackage{booktabs} % table support for pandoc > 1.12.2 + \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment) + \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout) + % normalem makes italics be italics, not underlines + + + + + % Colors for the hyperref package + \definecolor{urlcolor}{rgb}{0,.145,.698} + \definecolor{linkcolor}{rgb}{.71,0.21,0.01} + \definecolor{citecolor}{rgb}{.12,.54,.11} + + % ANSI colors + \definecolor{ansi-black}{HTML}{3E424D} + \definecolor{ansi-black-intense}{HTML}{282C36} + \definecolor{ansi-red}{HTML}{E75C58} + \definecolor{ansi-red-intense}{HTML}{B22B31} + \definecolor{ansi-green}{HTML}{00A250} + \definecolor{ansi-green-intense}{HTML}{007427} + \definecolor{ansi-yellow}{HTML}{DDB62B} + \definecolor{ansi-yellow-intense}{HTML}{B27D12} + \definecolor{ansi-blue}{HTML}{208FFB} + \definecolor{ansi-blue-intense}{HTML}{0065CA} + \definecolor{ansi-magenta}{HTML}{D160C4} + \definecolor{ansi-magenta-intense}{HTML}{A03196} + \definecolor{ansi-cyan}{HTML}{60C6C8} + \definecolor{ansi-cyan-intense}{HTML}{258F8F} + \definecolor{ansi-white}{HTML}{C5C1B4} + \definecolor{ansi-white-intense}{HTML}{A1A6B2} + + % commands and environments needed by pandoc snippets + % extracted from the output of `pandoc -s` + \providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} + \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}} + % Add ',fontsize=\small' for more characters per line + \newenvironment{Shaded}{}{} + \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}} + \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}} + \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}} + \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}} + \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}} + \newcommand{\RegionMarkerTok}[1]{{#1}} + \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}} + \newcommand{\NormalTok}[1]{{#1}} + + % Additional commands for more recent versions of Pandoc + \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}} + \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}} + \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}} + \newcommand{\ImportTok}[1]{{#1}} + \newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}} + \newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}} + \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}} + \newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}} + \newcommand{\BuiltInTok}[1]{{#1}} + \newcommand{\ExtensionTok}[1]{{#1}} + \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}} + \newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}} + \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}} + + + % Define a nice break command that doesn't care if a line doesn't already + % exist. + \def\br{\hspace*{\fill} \\* } + % Math Jax compatability definitions + \def\gt{>} + \def\lt{<} + % Document parameters + \title{Entanglement} + + + + + % Pygments definitions + +\makeatletter +\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax% + \let\PY@ul=\relax \let\PY@tc=\relax% + \let\PY@bc=\relax \let\PY@ff=\relax} +\def\PY@tok#1{\csname PY@tok@#1\endcsname} +\def\PY@toks#1+{\ifx\relax#1\empty\else% + \PY@tok{#1}\expandafter\PY@toks\fi} +\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{% + \PY@it{\PY@bf{\PY@ff{#1}}}}}}} +\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}} + +\expandafter\def\csname PY@tok@ss\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}} +\expandafter\def\csname PY@tok@vm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@gd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}} +\expandafter\def\csname PY@tok@nv\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf} +\expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}} +\expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}} +\expandafter\def\csname PY@tok@gi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.63,0.00}{##1}}} +\expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@nl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.63,0.00}{##1}}} +\expandafter\def\csname PY@tok@m\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@na\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.49,0.56,0.16}{##1}}} +\expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@kc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@nd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@nf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@kr\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@mb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@cpf\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@no\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@kn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@nb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@ne\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.82,0.25,0.23}{##1}}} +\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}} +\expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}} +\expandafter\def\csname PY@tok@sx\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@s2\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@dl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@kt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}} +\expandafter\def\csname PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@gt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}} +\expandafter\def\csname PY@tok@bp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@vc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@vg\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@gh\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@kp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@nc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@err\endcsname{\def\PY@bc##1{\setlength{\fboxsep}{0pt}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}} +\expandafter\def\csname PY@tok@nn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@nt\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@ch\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@sh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@fm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@ow\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\expandafter\def\csname PY@tok@s\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit} +\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@kd\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}} +\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@sa\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} + +\def\PYZbs{\char`\\} +\def\PYZus{\char`\_} +\def\PYZob{\char`\{} +\def\PYZcb{\char`\}} +\def\PYZca{\char`\^} +\def\PYZam{\char`\&} +\def\PYZlt{\char`\<} +\def\PYZgt{\char`\>} +\def\PYZsh{\char`\#} +\def\PYZpc{\char`\%} +\def\PYZdl{\char`\$} +\def\PYZhy{\char`\-} +\def\PYZsq{\char`\'} +\def\PYZdq{\char`\"} +\def\PYZti{\char`\~} +% for compatibility with earlier versions +\def\PYZat{@} +\def\PYZlb{[} +\def\PYZrb{]} +\makeatother + + + % Exact colors from NB + \definecolor{incolor}{rgb}{0.0, 0.0, 0.5} + \definecolor{outcolor}{rgb}{0.545, 0.0, 0.0} + + + + + % Prevent overflowing lines due to hard-to-break entities + \sloppy + % Setup hyperref package + \hypersetup{ + breaklinks=true, % so long urls are correctly broken across lines + colorlinks=true, + urlcolor=urlcolor, + linkcolor=linkcolor, + citecolor=citecolor, + } + % Slightly bigger margins than the latex defaults + + \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} + + + + \begin{document} + + + \maketitle + + + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}1}]:} \PY{c+c1}{\PYZsh{} EPR Paradox Example} + + \PY{c+c1}{\PYZsh{} A neutral pi meson (pi0) decays into an electron/positron (e\PYZhy{}/e+) } + \PY{c+c1}{\PYZsh{} pair.} + + \PY{c+c1}{\PYZsh{} pi0 \PYZhy{}\PYZhy{}\PYZgt{} e+ + e\PYZhy{} (electron\PYZhy{}positron pair)} + \PY{c+c1}{\PYZsh{} pi0 has angular momentum l = s = 0} + + \PY{c+c1}{\PYZsh{} Align electron and positron detectors in opposite directions.} + + \PY{c+c1}{\PYZsh{} | pi0 |} + \PY{c+c1}{\PYZsh{} |e\PYZhy{} \PYZlt{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{} \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZgt{} e+|} + + \PY{c+c1}{\PYZsh{} Where hbar = 1, the measurement of the spin in some direction is } + \PY{c+c1}{\PYZsh{} +/\PYZhy{}1 with spin state [1 0] (up) or [0 1] (down).} + + \PY{c+c1}{\PYZsh{} The Pauli exclusion principle with conserved angular momentum 0} + \PY{c+c1}{\PYZsh{} dictates this system must be in the singlet state } + \PY{c+c1}{\PYZsh{} chi = [1/sqrt(2) (|up+\PYZgt{}|down\PYZhy{}\PYZgt{} \PYZhy{} |down+\PYZgt{}|up\PYZhy{}\PYZgt{})].} + + \PY{c+c1}{\PYZsh{} In this state, if the positron is measured to have spin [1 0], the } + \PY{c+c1}{\PYZsh{} electron must have spin [0 1], or vice versa. There is an equal } + \PY{c+c1}{\PYZsh{} probability to find either state during the first measurement.} + + \PY{c+c1}{\PYZsh{} This view is consistent with the realist view. The realist view could } + \PY{c+c1}{\PYZsh{} hold that the electron and position had those angular momenta } + \PY{c+c1}{\PYZsh{} from creation.} + + \PY{c+c1}{\PYZsh{} EPR assumes influences cannot propagate faster than the speed of } + \PY{c+c1}{\PYZsh{} light. \PYZdq{}Wave function collapse\PYZdq{} is apparently instantaneous, however.} + + + \PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np} + \PY{k+kn}{import} \PY{n+nn}{matplotlib} + \PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt} + \PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{patches} \PY{k}{as} \PY{n+nn}{mpatches} + \PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline +\end{Verbatim} + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}2}]:} \PY{c+c1}{\PYZsh{} If information about the measurement of the wave function propagated} + \PY{c+c1}{\PYZsh{} at a finite speed, the particles could conceivably be measured such} + \PY{c+c1}{\PYZsh{} that both are equally likely to hold either spin up or spin down.} + + \PY{c+c1}{\PYZsh{} What would happen if the measurements were uncorrelated?} + + \PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)} + \PY{n}{elepatch} \PY{o}{=} \PY{n}{mpatches}\PY{o}{.}\PY{n}{Patch}\PY{p}{(}\PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{blue}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{e\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + \PY{n}{pospatch} \PY{o}{=} \PY{n}{mpatches}\PY{o}{.}\PY{n}{Patch}\PY{p}{(}\PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{orange}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{e+}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + \PY{c+c1}{\PYZsh{}plt.legend(handles=[elepatch,pospatch])} + \PY{n}{plt}\PY{o}{.}\PY{n}{text}\PY{p}{(}\PY{l+m+mf}{0.5}\PY{p}{,}\PY{l+m+mi}{565}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{e\PYZhy{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{size}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{text}\PY{p}{(}\PY{l+m+mf}{3.5}\PY{p}{,}\PY{l+m+mi}{565}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{e+}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{size}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)} + + \PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Uncorrelated Spins}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{600}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{,}\PY{l+m+mf}{3.125}\PY{p}{,}\PY{l+m+mf}{3.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{down}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{up}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{down}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{up}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)} +\end{Verbatim} + + + \begin{center} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_1_0.png} + \end{center} + { \hspace*{\fill} \\} + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}3}]:} \PY{c+c1}{\PYZsh{} How many violations of angular momentum would be measured?} + + \PY{c+c1}{\PYZsh{} From running several simulations, it\PYZsq{}s evident a violation in the } + \PY{c+c1}{\PYZsh{} conservation of angular momentum would be measured half of the time.} + \PY{c+c1}{\PYZsh{} We can conclude that the information that the entangled particles are in} + \PY{c+c1}{\PYZsh{} orthogonal spin states is instantaneously agreed once a measurement} + \PY{c+c1}{\PYZsh{} is made.} + + \PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0} + + \PY{k}{for} \PY{n}{trial} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{:} + \PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} + \PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} + \PY{k}{if} \PY{n}{elespin} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} + \PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + \PY{k}{else}\PY{p}{:} + \PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + \PY{k}{if} \PY{n}{posspin} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} + \PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + \PY{k}{else}\PY{p}{:} + \PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + + \PY{n}{elespin}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)} + \PY{n}{posspin}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)} + \PY{n}{chi\PYZus{}squared} \PY{o}{=} \PY{n}{elespin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1; 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{o}{*}\PY{n}{posspin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0; 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{posspin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1; 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{o}{*}\PY{n}{elespin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0; 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} + + \PY{k}{if} \PY{n}{chi\PYZus{}squared} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} + \PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1} + + \PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)} + \PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{l+m+mi}{1000}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)} + \PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)} + + \PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{600}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Conservation Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)} +\end{Verbatim} + + +\begin{Verbatim}[commandchars=\\\{\}] +{\color{outcolor}Out[{\color{outcolor}3}]:} +\end{Verbatim} + + \begin{center} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_2_1.png} + \end{center} + { \hspace*{\fill} \\} + + + \begin{verbatim} + + \end{verbatim} + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}4}]:} \PY{c+c1}{\PYZsh{} Bell\PYZsq{}s Experiment took this a step further, to rule out locality } + \PY{c+c1}{\PYZsh{} completely. Establish the detectors to \PYZdq{}float\PYZdq{} such that they } + \PY{c+c1}{\PYZsh{} measure the components of the spins of the electron and positron } + \PY{c+c1}{\PYZsh{} along a unit vector a or b, with angles phi\PYZus{}a and phi\PYZus{}b, } + \PY{c+c1}{\PYZsh{} respectively. Compute a product P of the spins in units of hbar/2. } + \PY{c+c1}{\PYZsh{} This will give +/\PYZhy{}1. } + + \PY{c+c1}{\PYZsh{} / pi0 \PYZbs{}} + \PY{c+c1}{\PYZsh{} /e\PYZhy{} \PYZlt{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{} \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZgt{} e+\PYZbs{}} + + \PY{c+c1}{\PYZsh{} QM predicts P(a,b) = \PYZhy{}a dot b, the expectation value of the product } + \PY{c+c1}{\PYZsh{} of the spins.} + + \PY{c+c1}{\PYZsh{} In 1964, Bell derived the Bell inequality for a local hidden variable} + \PY{c+c1}{\PYZsh{} theory: abs(P(a,b) \PYZhy{} P(a,c)) \PYZlt{}= 1 + P(b,c)} + + \PY{c+c1}{\PYZsh{} For any local hidden variable theory, the Bell inequality must hold. } + \PY{c+c1}{\PYZsh{} It introduces a third unit vector c, which is any other unit vector } + \PY{c+c1}{\PYZsh{} than a or b.} + + \PY{c+c1}{\PYZsh{} Does the quantum mechanical prediction violate the Bell inequality?} + \PY{c+c1}{\PYZsh{} Testing several detector configurations in a plane, systematically } + \PY{c+c1}{\PYZsh{} from 0 to pi, we determine whether the QM prediction is consistent } + \PY{c+c1}{\PYZsh{} with a local hidden variable theory.} + + \PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0} + \PY{n}{trials} \PY{o}{=} \PY{l+m+mi}{0} + \PY{k}{for} \PY{n}{step\PYZus{}a} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{:} + \PY{k}{for} \PY{n}{step\PYZus{}b} \PY{o+ow}{in} \PY{n+nb}{range} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{:} + \PY{n}{phi\PYZus{}a} \PY{o}{=} \PY{n}{step\PYZus{}a}\PY{o}{/}\PY{l+m+mi}{6}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi} + \PY{n}{phi\PYZus{}b} \PY{o}{=} \PY{n}{step\PYZus{}b}\PY{o}{/}\PY{l+m+mi}{6}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi} + \PY{n}{phi\PYZus{}c} \PY{o}{=} \PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{phi\PYZus{}b} + \PY{n}{P\PYZus{}ab} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}b}\PY{p}{)} + \PY{n}{P\PYZus{}ac} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)} + \PY{n}{P\PYZus{}bc} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}b} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)} + + \PY{n}{bell\PYZus{}lhs} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{P\PYZus{}ab} \PY{o}{\PYZhy{}} \PY{n}{P\PYZus{}ac}\PY{p}{)} + \PY{n}{bell\PYZus{}rhs} \PY{o}{=} \PY{l+m+mi}{1} \PY{o}{+} \PY{n}{P\PYZus{}bc} + + \PY{k}{if} \PY{n}{bell\PYZus{}lhs} \PY{o}{\PYZgt{}} \PY{n}{bell\PYZus{}rhs}\PY{p}{:} + \PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1} + + \PY{n}{trials} \PY{o}{=} \PY{n}{trials} \PY{o}{+} \PY{l+m+mi}{1} + + \PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)} + \PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{n}{trials}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)} + \PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)} + + \PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{p}{,}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{10}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Hidden Locality Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)} +\end{Verbatim} + + +\begin{Verbatim}[commandchars=\\\{\}] +{\color{outcolor}Out[{\color{outcolor}4}]:} +\end{Verbatim} + + \begin{center} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_3_1.png} + \end{center} + { \hspace*{\fill} \\} + + + \begin{verbatim} + + \end{verbatim} + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}5}]:} \PY{c+c1}{\PYZsh{} It is seen that the QM prediction disagrees with a local hidden } + \PY{c+c1}{\PYZsh{} variable theory in a significant number of configurations.} + + \PY{c+c1}{\PYZsh{} On average, for random orientations between 0 and pi, how often? } + \PY{c+c1}{\PYZsh{} After running several trials, it appears to be about half of the, } + \PY{c+c1}{\PYZsh{} which is what one would expect from the quantum mechanical} + \PY{c+c1}{\PYZsh{} prediction.} + + \PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0} + \PY{n}{trials} \PY{o}{=} \PY{l+m+mi}{0} + \PY{k}{for} \PY{n}{rand\PYZus{}a} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)}\PY{p}{:} + \PY{k}{for} \PY{n}{rand\PYZus{}b} \PY{o+ow}{in} \PY{n+nb}{range} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)}\PY{p}{:} + \PY{n}{phi\PYZus{}a} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{rand}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi} + \PY{n}{phi\PYZus{}b} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{rand}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi} + \PY{n}{phi\PYZus{}c} \PY{o}{=} \PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{phi\PYZus{}b} + \PY{n}{P\PYZus{}ab} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}b}\PY{p}{)} + \PY{n}{P\PYZus{}ac} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)} + \PY{n}{P\PYZus{}bc} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}b} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)} + + \PY{n}{bell\PYZus{}lhs} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{P\PYZus{}ab} \PY{o}{\PYZhy{}} \PY{n}{P\PYZus{}ac}\PY{p}{)} + \PY{n}{bell\PYZus{}rhs} \PY{o}{=} \PY{l+m+mi}{1} \PY{o}{+} \PY{n}{P\PYZus{}bc} + + \PY{k}{if} \PY{n}{bell\PYZus{}lhs} \PY{o}{\PYZgt{}} \PY{n}{bell\PYZus{}rhs}\PY{p}{:} + \PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1} + + \PY{n}{trials} \PY{o}{=} \PY{n}{trials} \PY{o}{+} \PY{l+m+mi}{1} + + \PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)} + \PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{n}{trials}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)} + \PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)} + + \PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{p}{,}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{10}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Hidden Locality Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)} + \PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)} +\end{Verbatim} + + +\begin{Verbatim}[commandchars=\\\{\}] +{\color{outcolor}Out[{\color{outcolor}5}]:} +\end{Verbatim} + + \begin{center} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_4_1.png} + \end{center} + { \hspace*{\fill} \\} + + + \begin{verbatim} + + \end{verbatim} + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}6}]:} \PY{c+c1}{\PYZsh{} The prediction (Bell\PYZsq{}s inequality) made by assuming a local hidden} + \PY{c+c1}{\PYZsh{} variable is violated by a significant number of the possibile } + \PY{c+c1}{\PYZsh{} bborientations. This simulation cannot determine which theory is} + \PY{c+c1}{\PYZsh{} correct, but the QM prediction has been confirmed through experiment.} + \PY{c+c1}{\PYZsh{} No hidden local variable holds actionable information about the } + \PY{c+c1}{\PYZsh{} state. Entangled states retain their entanglement in a non\PYZhy{}local } + \PY{c+c1}{\PYZsh{} nature.} +\end{Verbatim} + + + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}7}]:} \PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)} +\end{Verbatim} + + + + % Add a bibliography block to the postdoc + + + + \end{document} diff --git a/presentation/output_1_0.png b/presentation/output_1_0.png new file mode 100644 index 0000000..5966448 Binary files /dev/null and b/presentation/output_1_0.png differ diff --git a/presentation/output_2_1.png b/presentation/output_2_1.png new file mode 100644 index 0000000..01005d4 Binary files /dev/null and b/presentation/output_2_1.png differ diff --git a/presentation/output_3_1.png b/presentation/output_3_1.png new file mode 100644 index 0000000..263ff63 Binary files /dev/null and b/presentation/output_3_1.png differ diff --git a/presentation/output_4_1.png b/presentation/output_4_1.png new file mode 100644 index 0000000..e18b0c1 Binary files /dev/null and b/presentation/output_4_1.png differ diff --git a/presentation/sim_notes.motes b/presentation/sim_notes.motes new file mode 100644 index 0000000..be9cf79 --- /dev/null +++ b/presentation/sim_notes.motes @@ -0,0 +1,70 @@ +Three viewpoints: realist, orthodox, agnostic + + + +// |phi_1> |phi_2> + + + +For parallel detectors: + +P(a,b) = -1 + + +arbitrary orientation: + +P(a,b) = -a⋅b + +Our understanding of entanglement is consistent with the idea that modern local variables + + + +A(a,λ) = ±1. +B(b,λ) = ±1. + +If detectors are aligned: + +A(a,λ) = -B(b,λ). + +Average of product of measurements + +P(a,b) = ∫ ρ(λ) A(a,λ) B(b,λ) dλ + + but since A(a,λ) = -B(b,λ), + + P(a,b) = - ∫ ρ(λ) A(a,λ) A(b,λ) dλ + +c is any other unit vector... + +P(a,b) - P(a,c) = - ∫ ρ(λ) [ A(a,λ) A(b,λ) - A(a,λ) A(c,λ) ] dλ + + = - ∫ ρ(λ) [ 1 - A(a,λ) A(c,λ) ] A(a,λ) A(b,λ) dλ + +Because A(a,λ) = ±1 and B(b,λ) = ±1, + + -1 ≤ [A(a,λ) A(b,λ)] ≤ +1. + +ρ(λ) [1 - A(b,λ) A(c,λ)] ≥ 0, so + + │P(a,b) - P(a,c)│ ≤ ∫ ρ(λ) [1 - A(B,λ) A(c,λ)] dλ. + + + + +│P(a,b) - P(a,c)│ ≤ 1 + P(b,c) + + + +simulation: + + +pion decays, leaving two particles + +each particle has a spin state + + + + + + + diff --git a/presentation/simulation.py b/presentation/simulation.py new file mode 100644 index 0000000..934b7b8 --- /dev/null +++ b/presentation/simulation.py @@ -0,0 +1,207 @@ +# EPR Paradox Example + +# A pion (pi0) decays into an electron/positron (e-/e+) pair. + +# pi0 --> e+ + e- (electron-positron pair) +# pi0 has angular momentum l = 0 + +# Align electron and positron detectors in opposite directions. + +# | pi0 | +# |e- <------ ------> e+| + +# Where hbar = 1, the measurement of the spin in some direction is either +# [1 0] or [0 1]. + +# Pauli exclusion principle with conserved angular momentum l=0 says this +# system must be in the singlet state +# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)]. + +# In this state, if the positron is measured to have spin up, the electron +# must have spin down, or vice versa. There is an equal probability to +# measure either spin by the first measurement. + +# This view is consistent with the realist view. The realist view could hold +# that the electron and position had those angular momenta from creation. + +# EPR assumes influences cannot propagate faster than the speed of light. +# "Wave function collapse" is instantaneous. + + +import numpy as np +import matplotlib +import matplotlib.pyplot as plt +import matplotlib.patches as mpatches + + + +# If information about the measurement of the wave function propagated +# at a finite speed, the particles could conceivably be measured such +# that both are equally likely to hold either spin up or spin down. + +# What would happen if the measurements were uncorrelated? + +plt.ylim([400,600]) +plt.xlim([-1,5]) +plt.xticks([0.125,0.85,3.125,3.85],["down","up","down","up"]) +plt.tick_params(axis='both',labelsize=15) +plt.hist(np.random.randint(0,2,1000),bins=4) +plt.hist(np.random.randint(3,5,1000),bins=4) +elepatch = mpatches.Patch(color='blue', label='e-') +pospatch = mpatches.Patch(color='orange', label='e+') +plt.legend(handles=[elepatch,pospatch]) +plt.figure() + + + +# On average, how many violations of angular momentum would be measured? + +# Clearly, a violation in angular momentum would be measured half of the time. +# We can conclude that the information that the entangled particles are in +# the opposite spin states of eachother is instantaneously known once +# a measurement is made. + +violations = 0 + +for trial in range(0,1000): + elespin = np.random.randint(0,2) + posspin = np.random.randint(0,2) + if elespin == 0: + elespin = np.matrix('0 1') + else: + elespin = np.matrix('1 0') + if posspin == 0: + posspin = np.matrix('0 1') + else: + posspin = np.matrix('1 0') + + elespin.transpose() + posspin.transpose() + chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1') + + if chi_squared == 0: + violations = violations + 1 + +zeroes = np.zeros(violations,dtype=int) +ones = np.full((1000-violations),1,dtype=int) +result = np.concatenate((ones,zeroes)) + +plt.ylim([400,600]) +plt.xlim([-1,2]) +plt.xticks([0.125,0.85],["violation","adherence"]) +plt.tick_params(axis='both',labelsize=15) +plt.suptitle("Conservation Violations",fontsize=20) +plt.hist([result],bins=4) +plt.figure() + + + + + +# Bell's Experiment took this a step further, to rule out locality completely. +# Establish the detectors to "float" such that they measure the components of +# the spins of the electron and positron along a unit vector a or b, with +# angles phi_a and phi_b, respectively. Compute a product P of the spins in +# units of hbar/2. This will give +/-1. + +# / pi0 \ +# /e- <------ ------> e+\ + +# QM predicts P(a,b) = -a dot b, the expectation value of the product of +# the spins. + +# In 1964, Bell derived the Bell inequality for a local hidden variable +# theory: abs(P(a,b) - P(a,c)) <= 1 + P(b,c) + +# For any local hidden variable theory, the Bell inequality must hold. It +# introduces a third unit vector c, which is any other unit vector than +# a or b. + +# Does the quantum mechanical prediction violate the Bell inequality? Testing +# several detector configurations in a plane, systematically from 0 to pi, +# we determine whether the QM prediction is consistent with a local hidden +# variable theory. + +violations = 0 +trials = 0 +for step_a in range(0,6): + for step_b in range (0,6): + phi_a = step_a/6*np.pi + phi_b = step_b/6*np.pi + phi_c = phi_a - 0.5*phi_b + P_ab = -1 * np.cos(phi_a - phi_b) + P_ac = -1 * np.cos(phi_a - phi_c) + P_bc = -1 * np.cos(phi_b - phi_c) + + bell_lhs = np.abs(P_ab - P_ac) + bell_rhs = 1 + P_bc + + if bell_lhs > bell_rhs: + violations = violations + 1 + + trials = trials + 1 + +zeroes = np.zeros(violations,dtype=int) +ones = np.full((trials-violations),1,dtype=int) +result = np.concatenate((ones,zeroes)) + +plt.ylim([trials/2-10,trials/2+10]) +plt.xlim([-1,2]) +plt.xticks([0.125,0.85],["violation","adherence"]) +plt.tick_params(axis='both',labelsize=15) +plt.suptitle("Hidden Locality Violations",fontsize=20) +plt.hist([result],bins=4) +plt.figure() + + + + +# It is seen that the QM prediction disagrees with a local hidden variable +# theory in a significant number of configurations. + +# On average, for random orientations between 0 and pi, how often? After running several trials, +# it appears to be about half of the time. + +violations = 0 +trials = 0 +for rand_a in range(0,10): + for rand_b in range (0,10): + phi_a = np.random.rand(1)*np.pi + phi_b = np.random.rand(1)*np.pi + phi_c = phi_a - 0.5*phi_b + P_ab = -1 * np.cos(phi_a - phi_b) + P_ac = -1 * np.cos(phi_a - phi_c) + P_bc = -1 * np.cos(phi_b - phi_c) + + bell_lhs = np.abs(P_ab - P_ac) + bell_rhs = 1 + P_bc + + if bell_lhs > bell_rhs: + violations = violations + 1 + + trials = trials + 1 + +zeroes = np.zeros(violations,dtype=int) +ones = np.full((trials-violations),1,dtype=int) +result = np.concatenate((ones,zeroes)) + +plt.ylim([trials/2-10,trials/2+10]) +plt.xlim([-1,2]) +plt.xticks([0.125,0.85],["violation","adherence"]) +plt.tick_params(axis='both',labelsize=15) +plt.suptitle("Hidden Locality violations",fontsize=20) +plt.hist([result],bins=4) +plt.figure() + + + +# The prediction (Bell's inequality) made by assuming a local hidden +# variable is violated by a significant number of the possibile orientations. +# This simulation cannot determine which theory is correct, but the QM +# prediction has been confirmed through experiment. No hidden local variable +# holds actionable information about the state. Entangled states retain +# their entanglement in a non-local nature. + + + + diff --git a/recursion.pdf b/recursion.pdf new file mode 100644 index 0000000..8f1404d Binary files /dev/null and b/recursion.pdf differ