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ch5.mp
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% CH5.mp
% MetaPost input file with chapter five pictures.
%
% HISTORY
% 2001-Apr-15 Jim Hefferon jim@joshua.smcvt.edu Written
verbatimtex
%&latex
\documentclass{book}
\usepackage{bookjh}
\usepackage{linalgjh}
\begin{document}
etex
input jh
input arrow
defaultscale := 9pt/fontsize defaultfont;
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.04in; v:=u; w:=v;
%--------------------------------
% section 5.3
% Map from a space to itself, iterated.
%
beginfig(1); % map from a space to itself
%numeric u; %scaling factor
%numeric v; %vertical scaling factor
%numeric w; %horizontal scaling factor
numeric a, b; a=.8in; b=1.1a; % a is width of set, b is hgt
set_pic(a,b);
z1=(.4a,.9b);
drawpoint(z1); label.lft(btex $\vec{v}$ etex,z1);
z2=(.5a,.55b);
drawpoint(z2); label.lft(btex $t(\vec{v}\,)$ etex,z2);
z3=(.6a,.15b);
drawpoint(z3); label.lft(btex $t^2(\vec{v}\,)$ etex,z3);
%mapsto(z1,.5[z1,z2]+(.1a,0),z2);
%mapsto(z2,.5[z2,z3]+(.1a,0),z3);
draw_action_arrow(z1..(.5[z1,z2]+(.1a,0))..z2);
draw_action_arrow(z2..(.5[z2,z3]+(.1a,0))..z3);
endfig;
%--------------------------------
% section 5.3
% Graph of rank and nullity, on iteration of a transformation.
%
beginfig(2);
%numeric u; %scaling factor
%numeric v; %vertical scaling factor
%numeric w; %horizontal scaling factor
numeric a, b; % a is width, b is height
a=2.5in; b=1.25in;
draw begingraph(a,b);
setrange(origin,whatever,whatever);
gdraw "ranknullity.d" plot drawn_point; % btex$\circ$etex;
% glabel.lft(btex \begin{tabular}{r}
% nullity of $t^j$ (above) \\
% and \\
% rank of $t^j$ (below)
% \end{tabular} etex,OUT);
glabel.bot(btex \small Power $j$ of the transformation etex,OUT);
otick.bot(btex \small $0$etex,1);
otick.bot(btex \small $1$etex,2);
otick.bot(btex \small $2$etex,3);
otick.bot(btex \small $j$etex,5);
otick.bot(btex{\ }etex,8);
grid.bot(btex$n$etex,9) withcolor .85white;
otick.bot(btex{\ }etex,10);
grid.lft(btex \small $n$etex,8) withcolor .85white;
otick.lft(btex{\ }etex,7);
otick.lft(btex{\ }etex,6);
otick.lft(btex$\small \text{rank}(t^j)$etex,3);
otick.lft(btex{\ }etex,1);
otick.lft(btex{\ }etex,1);
otick.lft(btex{\ }etex,1);
frame.llft;
endgraph;
endfig;
%--------------------------------
% section 5.3
% Graph of rank and nullity, on iteration of a transformation.
%
beginfig(3); %graph of rank and nullity on iteration
%numeric u; %scaling factor
%numeric v; %vertical scaling factor
%numeric w; %horizontal scaling factor
numeric a, b; % a is width, b is height
a=2.5in; b=1.25in;
pickup pensquare scaled line_width_light;
draw (0,b)--(0,0)--(a,0);
label.lft(btex \begin{tabular}{r}
nullity of $t^j$ (above) \\
and \\
rank of $t^j$ (below)
\end{tabular} etex,.5[(0,b),(0,0)]);
label.bot(btex Power $j$ of the transformation etex,.5[(0,0),(a,0)]);
endfig;
beginfig(4) % equiv relation; row-equiv mats from ch1.27, further split
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=1in; v:=u; w:=v; % was: 1in
path p[]; partition; % dotlabels.ulft(5,6,7,8,9,10,11,12,13);
label(btex {\small \ldots} etex,z13);
x14=.8[x0,x9]; y14=.7[y10,y9]; drawpoint(z14);
label.rt(btex {\small $S$} etex,z14);
x15=.85[x0,x9]; y15=.3[y10,y9]; drawpoint(z15);
label.rt(btex {\small $T$} etex,z15);
pickup pencircle scaled line_width_light;
% split part in center
z20=.5[z6,z8]; x21=.8[x20,x14]; y21=y20;
z22=(z20--(3[z20,z21])) intersectionpoint p3;
p20 = z20--z22;
draw p20 dashed evenly scaled .5;
z23 = point .6 of p1;
z24 = whatever[z20,z22]; x24 = x23;
p21 = z23--z24;
draw p21 dashed evenly scaled .5;
% split part in upper left
z30 = point .25 of p1;
z31 = whatever[z0,z3]; y31 = y30;
p30 = z30--z31;
draw p30 dashed evenly scaled .5;
% split part in upper right
z40 = point .55 of p4;
z41 = point .85 of p4;
z42 = whatever[z2,z3]; x42 = x40;
p40 = z40--z42;
z43 = whatever[z2,z3]; x43 = x41;
p41 = z41--z43;
draw p40 dashed evenly scaled .5;
draw p41 dashed evenly scaled .5;
endfig;
beginfig(5) % equiv relation; finer partition needs more reps
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=1in; v:=u; w:=v; % was: 1in
path p[]; partition; % dotlabels.ulft(5,6,7,8,9,10,11,12,13);
label(btex {\small \ldots} etex,z13);
x14=.8[x0,x9]; y14=.7[y10,y9]; %drawpoint(z14);
label.rt(btex {\small $\star$} etex,z14);
x15=.75[x0,x9]; y15=.35[y10,y9]; %drawpoint(z15);
label.rt(btex {\small $\star$} etex,z15);
pickup pencircle scaled line_width_light;
% split part in center
z20=.5[z6,z8]; x21=.8[x20,x14]; y21=y20;
z22=(z20--(3[z20,z21])) intersectionpoint p3;
p20 = z20--z22;
draw p20 dashed evenly scaled .5;
z23 = point .6 of p1;
z24 = whatever[z20,z22]; x24 = x23;
p21 = z23--z24;
draw p21 dashed evenly scaled .5;
z25=.55[z20,z24]; z26=.4[z24,z23];
label(btex {\small $\star$} etex,(x25,y26));
% split part in upper left
z30 = point .25 of p1;
z31 = whatever[z0,z3]; y31 = y30;
p30 = z30--z31;
draw p30 dashed evenly scaled .5;
z32=.35[z31,z30]; z33=.5[z6,z31];
label(btex {\small $\star$} etex,(x32,y33));
z34=.55[z31,z30]; z35=.5[z3,z31];
label(btex {\small $\star$} etex,(x34,y35));
% split part in upper right
z40 = point .55 of p4;
z41 = point .85 of p4;
z42 = whatever[z2,z3]; x42 = x40;
p40 = z40--z42;
z43 = whatever[z2,z3]; x43 = x41;
p41 = z41--z43;
draw p40 dashed evenly scaled .5;
draw p41 dashed evenly scaled .5;
z44=.5[z5,z43]; z45=.45[z41,z43];
label(btex {\small $\star$} etex,(x44,y45));
z46=.5[z43,z42]; z47=.55[z41,z43];
label(btex {\small $\star$} etex,(x46,y47));
z48=.45[z42,z11]; z49=.65[z40,z42];
label(btex {\small $\star$} etex,(x48,y49));
% then lower left
z50=.4[z0,z7]; z51=.4[z0,z8];
label(btex {\small $\star$} etex,(x50,y51));
%dotlabels.top(20,21,22,23,24,30,31,40,41,42,43);
endfig;
% ==================== Linear Recurrences ===============
beginfig(6) % Tower of Hanoi (three disks at start)
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v; %
save disk_length, disk_width, disk_hgt;
numeric disk_length, disk_width, disk_hgt;
disk_length = 8u; disk_width = .4disk_length; disk_hgt = .2u;
save needle_shift; pair needle_shift; needle_shift = (1.2disk_length,0v);
save needle_length, needle_width, needle_hgt;
numeric needle_length, needle_width, needle_hgt; % how thick is body of bee?
needle_length = .4u; needle_width = .4needle_length; needle_hgt = 4u;
% left needle
drawdisk(disk_length,disk_width,disk_hgt);
drawdisk(.9disk_length,.9disk_width,disk_hgt) shifted (0,1.5disk_hgt);
drawdisk(.8disk_length,.8disk_width,disk_hgt) shifted (0,3disk_hgt);
drawdisk(needle_length,needle_width,needle_hgt-3disk_hgt)
shifted (0,needle_hgt);
% middle needle
drawdisk(needle_length,needle_width,needle_hgt)
shifted (0,needle_hgt) shifted needle_shift;
% right needle
drawdisk(needle_length,needle_width,needle_hgt)
shifted (0,needle_hgt) shifted 2needle_shift;
endfig;
beginfig(7) % Tower of Hanoi (three disks at start)
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v; %
save disk_length, disk_width, disk_hgt;
numeric disk_length, disk_width, disk_hgt;
disk_length = 8u; disk_width = .4disk_length; disk_hgt = .2u;
save needle_shift; pair needle_shift; needle_shift = (1.2disk_length,0v);
save needle_length, needle_width, needle_hgt;
numeric needle_length, needle_width, needle_hgt; % how thick is body of bee?
needle_length = .4u; needle_width = .4needle_length; needle_hgt = 4u;
% left needle
drawdisk(disk_length,disk_width,disk_hgt);
drawdisk(needle_length,needle_width,needle_hgt-disk_hgt)
shifted (0,needle_hgt);
% middle needle
drawdisk(.9disk_length,.9disk_width,disk_hgt)
shifted needle_shift;
drawdisk(.8disk_length,.8disk_width,disk_hgt)
shifted (0,1.5disk_hgt) shifted needle_shift;
drawdisk(needle_length,needle_width,needle_hgt-2disk_hgt)
shifted (0,needle_hgt) shifted needle_shift;
% right needle
drawdisk(needle_length,needle_width,needle_hgt)
shifted (0,needle_hgt) shifted 2needle_shift;
endfig;
beginfig(8) % graph of rank falling and nullity rising
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v; %
% axes
z0=(0w,0v);
z1=(+.5w,0v); z2=(21.5w,0v); % x axis
z3=(0w,+.5v); z4=(0w,9.5v); % y axis
pickup pensquare scaled line_width_light;
draw z1--z2;
draw z3--z4;
% label.lrt(btex \small power etex,z2);
% label.ulft(btex \small rank etex,z4);
% point of interest
z10=(10w,3.75v);
% ticks
pickup updown_tick;
for i = 1 upto 3:
drawdot (i*w,0v);
endfor
label.bot(btex \small $0$ etex,(1w,0v));
label.bot(btex \small $1$ etex,(2w,0v));
label.bot(btex \small $2$ etex,(3w,0v));
drawdot (x10,0v);
label.bot(btex \small $j$ etex,(10w,0v));
for i = 17 upto 21:
drawdot (i*w,0v);
endfor
% label.bot(btex \small $k$ etex,(18w,0v));
label.bot(btex \small $n$ etex,(20w,0v));
% label.bot(btex \small $k+1$ etex,(14w,0v));
pickup sidetoside_tick;
for i = 9 upto 9:
drawdot (0w,i*v);
endfor
label.lft(btex \small $n$ etex,(0w,9v));
% drawdot (0w,y10);
for i = 1 upto 1:
drawdot (0w,i*v);
endfor
label.lft(btex \small $0$ etex,(0w,1v));
% The points
numeric whisker_width; whisker_width=.4v;
pickup pencircle scaled line_width_light;
drawpoint(z10);
pickup pensquare scaled line_width_light;
draw (x10,y10+0.5v)--(x10,y4-0.5v) withcolor shading_color; % vert line above dot
draw (x10-.5whisker_width,y4-0.5v)--(x10+.5whisker_width,y4-0.5v) withcolor shading_color; % whisker at top
draw (x10-.5whisker_width,y10+0.5v)--(x10+.5whisker_width,y10+0.5v) withcolor shading_color; % whisker at bot
label.rt(btex \small $\nullity (t^j)$ etex,(x10,.5[y10,y4]));
draw (x10,y10-0.5v)--(x10,1v) withcolor shading_color; % vert line below dot
draw (x10-.5whisker_width,y10-0.5v)--(x10+.5whisker_width,y10-0.5v) withcolor shading_color; % whisker at top
draw (x10-.5whisker_width,1v)--(x10+.5whisker_width,1v) withcolor shading_color; % whisker at bot
label.lft(btex \small $\rank (t^j)$ etex,(x10,.55[y10,1v]));
pickup pencircle scaled line_width_light;
drawpoint((1w,9v));
drawpoint((2w,7v));
drawpoint((3w,6v));
drawpoint((17w,2.2v));
drawpoint((18w,2v));
drawpoint((19w,2v));
drawpoint((20w,2v));
% drawpoint((21w,2v));
label(btex \small $\ldots$ etex,(21w,2v));
% Generalized spaces
z20=(20w,2v); % were to anchor the line for generalized spaces
pickup pensquare scaled line_width_light;
draw (x20,y20+0.5v)--(x20,y4-0.5v) withcolor shading_color; % vert line above dot
draw (x20-.5whisker_width,y4-0.5v)--(x20+.5whisker_width,y4-0.5v) withcolor shading_color; % whisker at top
draw (x20-.5whisker_width,y20+0.5v)--(x20+.5whisker_width,y20+0.5v) withcolor shading_color; % whisker at bot
label.rt(btex \small $\dim(\gennullspace{t})$ etex,(x20,.5[y20,y4]));
draw (x20,y20-0.5v)--(x20,1v) withcolor medgray; % vert line below dot
draw (x20-.5whisker_width,y20-0.5v)--(x20+.5whisker_width,y20-0.5v) withcolor shading_color; % whisker at top
draw (x20-.5whisker_width,1v)--(x20+.5whisker_width,1v) withcolor shading_color; % whisker at bot
label.rt(btex \small $\dim(\genrangespace{t})$ etex,(x20,.8[y20,1v]));
endfig;
%--------------------------------
% Topic: Page Rank
% Graph illustrating page links
%
beginfig(9); % m
%numeric u; %scaling factor
%numeric v; %vertical scaling factor
%numeric w; %horizontal scaling factor
numeric circlescale; circlescale=19pt;
path node; node=fullcircle scaled circlescale;
path n[]; % node paths
pickup pencircle scaled line_width_light;
z1=(0w,6v);
n1=node shifted z1;
draw n1; label(btex \small $p_1$ etex,z1);
z2=(9w,y1);
n2=node shifted z2;
draw n2; label(btex \small $p_2$ etex,z2);
z3=(x2,0v);
n3=node shifted z3;
draw n3; label(btex \small $p_3$ etex,z3);
z4=(x1,y3);
n4=node shifted z4;
draw n4; label(btex \small $p_4$ etex,z4);
path p[], q[];
pair times[]; % intersection times
% arrow from p1 to p2
p12=z1--z2;
times121=p12 intersectiontimes n1;
times122=p12 intersectiontimes n2;
drawarrow subpath(xpart(times121)+.05,xpart(times122)-.05) of p12;
% arrow from p2 to p3
p23=z2--z3;
times232=p23 intersectiontimes n2;
times233=p23 intersectiontimes n3;
drawdblarrow subpath(xpart(times232)+.05,xpart(times233)-.05) of p23;
% arrow from p3 to p1
p31=z3--z1;
times313=p31 intersectiontimes n3;
times311=p31 intersectiontimes n1;
drawarrow subpath(xpart(times313)+.05,xpart(times311)-.05) of p31;
% arrow from p3 to p4
p34=z3--z4;
times343=p34 intersectiontimes n3;
times344=p34 intersectiontimes n4;
drawarrow subpath(xpart(times343)+.05,xpart(times344)-.05) of p34;
endfig;
% Exercise from KURT BRYAN AND TANYA LEISE
%
beginfig(10); % m
%numeric u; %scaling factor
%numeric v; %vertical scaling factor
%numeric w; %horizontal scaling factor
numeric circlescale; circlescale=19pt;
path node; node=fullcircle scaled circlescale;
path n[]; % node paths
pickup pencircle scaled line_width_light;
z1=(0w,6v);
n1=node shifted z1;
draw n1; label(btex \small $p_1$ etex,z1);
z2=(9w,y1);
n2=node shifted z2;
draw n2; label(btex \small $p_2$ etex,z2);
z3=(x2,0v);
n3=node shifted z3;
draw n3; label(btex \small $p_3$ etex,z3);
z4=(x1,y3);
n4=node shifted z4;
draw n4; label(btex \small $p_4$ etex,z4);
path p[], q[];
pair times[]; % intersection times
% arrow from p1 to p2
p12=z1--z2;
times121=p12 intersectiontimes n1;
times122=p12 intersectiontimes n2;
drawarrow subpath(xpart(times121)+.05,xpart(times122)-.05) of p12;
% arrow from p1 to p3
p13=z1--z3;
times131=p13 intersectiontimes n1;
times133=p13 intersectiontimes n3;
drawdblarrow subpath(xpart(times131)+.05,xpart(times133)-.05) of p13;
% arrow from p1 to p4
p14=z1--z4;
times141=p14 intersectiontimes n1;
times144=p14 intersectiontimes n4;
drawdblarrow subpath(xpart(times141)+.05,xpart(times144)-.05) of p14;
% arrow from p2 to p3
p23=z2--z3;
times232=p23 intersectiontimes n2;
times233=p23 intersectiontimes n3;
drawarrow subpath(xpart(times232)+.05,xpart(times233)-.05) of p23;
% arrow from p2 to p4
p24=z2--z4;
times242=p24 intersectiontimes n2;
times244=p24 intersectiontimes n4;
drawarrow subpath(xpart(times242)+.05,xpart(times244)-.05) of p24;
% arrow from p4 to p5
p43=z4--z3;
times433=p43 intersectiontimes n4;
times434=p43 intersectiontimes n3;
drawarrow subpath(xpart(times433)+.05,xpart(times434)-.05) of p43;
endfig;
% Search: every site points in a circle
%
beginfig(11); %
%numeric u; %scaling factor
%numeric v; %vertical scaling factor
%numeric w; %horizontal scaling factor
numeric circlescale; circlescale=19pt;
path node; node=fullcircle scaled circlescale;
path n[]; % node paths
pickup pencircle scaled line_width_light;
z1=(0w,6v);
n1=node shifted z1;
draw n1; label(btex \small $p_1$ etex,z1);
z2=(9w,y1);
n2=node shifted z2;
draw n2; label(btex \small $p_2$ etex,z2);
z3=(x2,0v);
n3=node shifted z3;
draw n3; label(btex \small $p_3$ etex,z3);
z4=(x1,y3);
n4=node shifted z4;
draw n4; label(btex \small $p_4$ etex,z4);
path p[], q[];
pair times[]; % intersection times
% arrow from p1 to p2
p12=z1--z2;
times121=p12 intersectiontimes n1;
times122=p12 intersectiontimes n2;
drawarrow subpath(xpart(times121)+.05,xpart(times122)-.05) of p12;
% arrow from p2 to p3
p23=z2--z3;
times232=p23 intersectiontimes n2;
times233=p23 intersectiontimes n3;
drawarrow subpath(xpart(times232)+.05,xpart(times233)-.05) of p23;
% arrow from p3 to p4
p34=z3--z4;
times343=p34 intersectiontimes n3;
times344=p34 intersectiontimes n4;
drawarrow subpath(xpart(times343)+.05,xpart(times344)-.05) of p34;
% arrow from p4 to p1
p41=z4--z1;
times414=p41 intersectiontimes n4;
times411=p41 intersectiontimes n1;
drawarrow subpath(xpart(times414)+.05,xpart(times411)-.05) of p41;
endfig;
end