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@ -0,0 +1,54 @@
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PROJECTNAME=$(basename $(wildcard *.tex))
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LIBS=$(notdir $(wildcard libs/*))
|
||||
FIGS=$(notdir $(wildcard figs/*.fig))
|
||||
|
||||
PDFS=$(addsuffix .pdf, $(PROJECTNAME))
|
||||
SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME))
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||||
|
||||
all: $(PROJECTNAME)
|
||||
|
||||
$(PROJECTNAME): $(LIBS) $(FIGS)
|
||||
pdflatex -file-line-error $@.tex
|
||||
pdflatex -file-line-error $@.tex
|
||||
pdflatex -synctex=1 $@.tex
|
||||
|
||||
$(PROJECTNAME).aux: $(LIBS) $(FIGS)
|
||||
pdflatex -file-line-error -draftmode $(PROJECTNAME).tex
|
||||
|
||||
|
||||
$(SYNCTEXS): $(LIBS) $(FIGS)
|
||||
pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@)
|
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|
||||
libs: $(LIBS)
|
||||
|
||||
$(LIBS):
|
||||
ln -fs libs/$@ ./
|
||||
|
||||
figs: $(FIGS)
|
||||
|
||||
$(FIGS):
|
||||
make -C figs/$@
|
||||
for pdf in $$(find figs/$@/ -name '*.pdf'); do ln -fs "$$pdf" ./ ; done
|
||||
|
||||
|
||||
clean-aux: clean-figs-aux
|
||||
rm -f $(addsuffix .aux, $(PROJECTNAME))
|
||||
rm -f $(addsuffix .log, $(PROJECTNAME))
|
||||
rm -f $(addsuffix .out, $(PROJECTNAME))
|
||||
rm -f $(addsuffix .toc, $(PROJECTNAME))
|
||||
|
||||
clean-libs:
|
||||
rm -f $(LIBS)
|
||||
|
||||
clean-figs:
|
||||
$(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; )
|
||||
rm -f $(notdir $(wildcard figs/*.fig/*.pdf))
|
||||
|
||||
clean-figs-aux:
|
||||
$(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; )
|
||||
|
||||
|
||||
clean-tex:
|
||||
rm -f $(PDFS) $(SYNCTEXS)
|
||||
|
||||
clean: clean-aux clean-tex clean-libs clean-figs
|
||||
|
|
@ -0,0 +1,47 @@
|
|||
This directory contains the source files to typeset the article, and generate
|
||||
the figures. This can be accomplished by running
|
||||
make
|
||||
|
||||
A program is bundled with this document to compute solutions to the simple
|
||||
equation numerically. It is located in figs/numerical.figs/simpleq. We will not
|
||||
give a full documentation of the program, which will be provided with a more
|
||||
complete version at some later time. The program is run when running 'make' to
|
||||
compute the data for the figures.
|
||||
|
||||
This document uses a custom class file, located in the 'libs' directory, which
|
||||
defines a number of commands. Most of these are drop-in replacements for those
|
||||
defined in the 'article' class.
|
||||
|
||||
Some extra functionality is provided in custom style files, located in the
|
||||
'libs' directory.
|
||||
|
||||
|
||||
* Dependencies:
|
||||
|
||||
pdflatex
|
||||
TeXlive packages:
|
||||
amsfonts
|
||||
array
|
||||
color
|
||||
dsfont
|
||||
etoolbox
|
||||
graphics
|
||||
hyperref
|
||||
latex
|
||||
marginnote
|
||||
pgf
|
||||
standalone
|
||||
GNU make
|
||||
julia
|
||||
|
||||
* Files:
|
||||
|
||||
Carlen_Jauslin_Lieb_2019.tex:
|
||||
main LaTeX file
|
||||
|
||||
libs:
|
||||
custom LaTeX class file
|
||||
|
||||
figs:
|
||||
source code for the figures
|
||||
|
||||
|
|
@ -0,0 +1,42 @@
|
|||
\bibitem[Bo47]{Bo47}N. Bogolubov - {\it On the theory of superfluidity}, Journal of Physics (USSR), volume\-~11, number\-~1 , pages\-~23-32 (translated from the Russian Izv.Akad.Nauk Ser.Fiz, volume\-~11, pages\-~77-90), 1947.\par\medskip
|
||||
|
||||
\bibitem[CHe]{CHe}E.\-~Carlen, M.\-~Holzmann, I.\-~Jauslin, E.H.\-~Lieb, in preparation.\par\medskip
|
||||
|
||||
\bibitem[Dy57]{Dy57}F.J. Dyson - {\it Ground-State Energy of a Hard-Sphere Gas}, Physical Review, volume\-~106, issue\-~1, pages\-~20-26, 1957,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.106.20}{10.1103/PhysRev.106.20}}.\par\medskip
|
||||
|
||||
\bibitem[FS19]{FS19}S.\-~Fournais, J.P.\-~Solovej - {\it The energy of dilute Bose gases}, 2019,\par\penalty10000
|
||||
arxiv:{\tt\href{http://arxiv.org/abs/1904.06164}{1904.06164}}.\par\medskip
|
||||
|
||||
\bibitem[Ga99]{Ga99}G. Gallavotti - {\it Statistical mechanics, a short treatise}, Springer, 1999.\par\medskip
|
||||
|
||||
\bibitem[LHY57]{LHY57}T.D. Lee, K. Huang, C.N. Yang - {\it Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties}, Physical Review, volume\-~106, issue\-~6, pages\-~1135-1145, 1957,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.106.1135}{10.1103/PhysRev.106.1135}}.\par\medskip
|
||||
|
||||
\bibitem[Le29]{Le29}W. Lenz - {\it Die Wellenfunktion und Geschwindigkeitsverteilung des entarteten Gases}, Zeitschrift f\"ur Physik, volume\-~56, issue\-~11-12, pages\-~778-789, 1929,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1007/BF01340138}{10.1007/BF01340138}}.\par\medskip
|
||||
|
||||
\bibitem[Li63]{Li63}E.H. Lieb - {\it Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas}, Physical Review, volume\-~130, issue\-~6, pages\-~2518-2528, 1963,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.130.2518}{10.1103/PhysRev.130.2518}}.\par\medskip
|
||||
|
||||
\bibitem[LL64]{LL64}E.H. Lieb, W. Liniger - {\it Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model}, Physical Review, volume\-~134, issue\-~2A, pages A312-A315, 1964,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1103/PhysRev.134.A312}{10.1103/PhysRev.134.A312}}.\par\medskip
|
||||
|
||||
\bibitem[LL01]{LL01}E.H. Lieb, M. Loss - {\it Analysis}, Second edition, Graduate studies in mathematics, Americal Mathematical Society, 2001.\par\medskip
|
||||
|
||||
\bibitem[LY98]{LY98}E.H. Lieb, J. Yngvason - {\it Ground State Energy of the Low Density Bose Gas}, Physical Review Letters, volume\-~80, issue\-~12, pages\-~2504-2507, 1998,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1103/PhysRevLett.80.2504}{10.1103/PhysRevLett.80.2504}}, arxiv:{\tt\href{http://arxiv.org/abs/cond-mat/9712138}{cond-mat/9712138}}.\par\medskip
|
||||
|
||||
\bibitem[LY01]{LY01}E.H. Lieb, J. Yngvason - {\it The Ground State Energy of a Dilute Two-Dimensional Bose Gas}, Journal of Statistical Physics, volume\-~103, issue\-~3-4, pages\-~509-526, 2001,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1023/A:1010337215241}{10.1023/A:1010337215241}}, arxiv:{\tt\href{http://arxiv.org/abs/math-ph/0002014}{math-ph/0002014}}.\par\medskip
|
||||
|
||||
\bibitem[PT12]{PT12}E. Pulvirenti, D. Tsagkarogiannis - {\it Cluster Expansion in the Canonical Ensemble}, Communications in Mathematical Physics, volume\-~316, issue\-~2, pages\-~289-306, 2012,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1007/s00220-012-1576-y}{10.1007/s00220-012-1576-y}}, arxiv:{\tt\href{http://arxiv.org/abs/1105.1022}{1105.1022}}.\par\medskip
|
||||
|
||||
\bibitem[RS75]{RS75b}M. Reed, B. Simon - {\it Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness}, second edition, Academic Press, New York, 1975.\par\medskip
|
||||
|
||||
\bibitem[Ru99]{Ru99}D. Ruelle - {\it Statistical mechanics: rigorous results}, Imperial College Press, World Scientific, (first edition: Benjamin, 1969), 1999.\par\medskip
|
||||
|
||||
\bibitem[YY09]{YY09}H. Yau, J. Yin - {\it The Second Order Upper Bound for the Ground Energy of a Bose Gas}, Journal of Statistical Physics, volume\-~136, issue\-~3, pages\-~453-503, 2009,\par\penalty10000
|
||||
doi:{\tt\href{http://dx.doi.org/10.1007/s10955-009-9792-3}{10.1007/s10955-009-9792-3}}, arxiv:{\tt\href{http://arxiv.org/abs/0903.5347}{0903.5347}}.\par\medskip
|
||||
|
||||
|
|
@ -0,0 +1,32 @@
|
|||
PROJECTNAME=bosegas convexity
|
||||
|
||||
DATS=erho.dat ddrhoe.dat
|
||||
PDFS=$(addsuffix .pdf, $(PROJECTNAME))
|
||||
TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
|
||||
|
||||
all: $(PDFS)
|
||||
|
||||
$(PDFS): $(DATS)
|
||||
gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
|
||||
pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
|
||||
|
||||
erho.dat:
|
||||
julia ./simpleq/main.jl -p "tolerance=1e-14;order=100" energy_rho > $@
|
||||
ddrhoe.dat:
|
||||
julia ./simpleq/main.jl -p "tolerance=1e-14;order=100" convexity > $@
|
||||
|
||||
install: $(PDFS)
|
||||
cp $^ $(INSTALLDIR)/
|
||||
|
||||
clean-aux:
|
||||
rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
|
||||
rm -f $(addsuffix .aux, $(PROJECTNAME))
|
||||
rm -f $(addsuffix .log, $(PROJECTNAME))
|
||||
|
||||
clean-dat:
|
||||
rm -f $(DATS)
|
||||
|
||||
clean-tex:
|
||||
rm -f $(PDFS)
|
||||
|
||||
clean: clean-aux clean-tex
|
||||
|
|
@ -0,0 +1,34 @@
|
|||
set ylabel "$\\displaystyle\\frac{e}{4\\pi\\rho}$" norotate offset -1,0
|
||||
set xlabel "$\\rho$"
|
||||
|
||||
set xtics 1e-6, 100, 100
|
||||
set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
|
||||
unset mxtics
|
||||
|
||||
set ytics 0.6, 0.1
|
||||
set mytics 2
|
||||
|
||||
set xrange [0.000001:100]
|
||||
set yrange [0.6:1.05]
|
||||
|
||||
# default output canvas size: 12.5cm x 8.75cm
|
||||
set term lua tikz size 8,6 standalone
|
||||
|
||||
set key off
|
||||
|
||||
|
||||
# set linestyle
|
||||
set style line 1 linetype rgbcolor "#4169E1" linewidth 3
|
||||
set style line 2 linetype rgbcolor "#DC143C" linewidth 3
|
||||
set style line 3 linetype rgbcolor "#32CD32" linewidth 3
|
||||
set style line 4 linetype rgbcolor "#4B0082" linewidth 3
|
||||
set style line 5 linetype rgbcolor "#DAA520" linewidth 3
|
||||
|
||||
set pointsize 1
|
||||
|
||||
set logscale x
|
||||
|
||||
plot "erho.dat" using 1:(0.95*$2/$1/(4*pi)):(1.05*$2/$1/(4*pi)) with filledcurves linetype rgbcolor "#DDDDDD", \
|
||||
"erho.dat" using 1:($2/$1/(4*pi)) with lines linestyle 1, \
|
||||
"holzmann_2019-09-28.dat" using 1:($2/$1/(4*pi)) with points linestyle 2
|
||||
|
||||
|
|
@ -0,0 +1,30 @@
|
|||
set ylabel "$\\frac1{4\\pi}\\partial_\\rho^2(e\\rho)$" offset -4,0 norotate
|
||||
set xlabel "$\\rho$"
|
||||
|
||||
set xtics 1e-6, 100, 100
|
||||
set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
|
||||
unset mxtics
|
||||
set xrange [0.000001:100]
|
||||
|
||||
set ytics 1.2, 0.2
|
||||
set mytics 2
|
||||
set yrange [1.2:2.1]
|
||||
|
||||
# default output canvas size: 12.5cm x 8.75cm
|
||||
set term lua tikz size 8,6 standalone
|
||||
|
||||
# no key
|
||||
set key off
|
||||
|
||||
# set linestyle
|
||||
set style line 1 linetype rgbcolor "#4169E1" linewidth 3
|
||||
set style line 2 linetype rgbcolor "#DC143C" linewidth 3
|
||||
set style line 3 linetype rgbcolor "#000000" linewidth 1
|
||||
set style line 4 linetype rgbcolor "#4B0082" linewidth 3
|
||||
set style line 5 linetype rgbcolor "#DAA520" linewidth 3
|
||||
|
||||
set pointsize 0.6
|
||||
|
||||
set logscale x
|
||||
|
||||
plot "ddrhoe.dat" using 1:($2/(4*pi)) with lines linestyle 1
|
||||
|
|
@ -0,0 +1,9 @@
|
|||
## data from M. Holzmann, 2019-09-22
|
||||
# rho E0 E0+dE0
|
||||
1e-4 8.3500e-4 8.3600e-4
|
||||
1e-3 9.1340e-3 9.1350e-3
|
||||
1e-2 1.0609e-1 1.0610e-1
|
||||
1e-1 1.1930e+0 1.1940e+0
|
||||
1e-0 1.2445e+1 1.2446e+1
|
||||
1e+1 1.2544e+2 1.2545e+2
|
||||
|
||||
|
|
@ -0,0 +1,28 @@
|
|||
# approximate \int_a^b f using Gauss-Legendre quadratures
|
||||
function integrate_legendre(f,a,b,weights)
|
||||
out=0
|
||||
for i in 1:length(weights[1])
|
||||
out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)
|
||||
end
|
||||
return out
|
||||
end
|
||||
# \int f*g where g is sampled at the Legendre nodes
|
||||
function integrate_legendre_sampled(f,g,a,b,weights)
|
||||
out=0
|
||||
for i in 1:length(weights[1])
|
||||
out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)*g[i]
|
||||
end
|
||||
return out
|
||||
end
|
||||
|
||||
|
||||
|
||||
# approximate \int_a^b f/sqrt((b-x)(x-a)) using Gauss-Chebyshev quadratures
|
||||
function integrate_chebyshev(f,a,b,N)
|
||||
out=0
|
||||
for i in 1:N
|
||||
out=out+pi/N*f((b-a)/2*cos((2*i-1)/(2*N)*pi)+(b+a)/2)
|
||||
end
|
||||
return out
|
||||
end
|
||||
|
||||
|
|
@ -0,0 +1,55 @@
|
|||
# \hat u_n
|
||||
function hatun_iteration(e,order,d,v,maxiter)
|
||||
# gauss legendre weights
|
||||
weights=gausslegendre(order)
|
||||
|
||||
# init V and Eta
|
||||
(V,V0,Eta,Eta0)=init_veta(weights,d,v)
|
||||
|
||||
# init u and rho
|
||||
u=Array{Array{Complex{Float64}}}(undef,maxiter+1)
|
||||
u[1]=zeros(Complex{Float64},order)
|
||||
rho=zeros(Complex{Float64},maxiter+1)
|
||||
|
||||
# iterate
|
||||
for n in 1:maxiter
|
||||
u[n+1]=A(e,weights,Eta)\b(u[n],e,rho[n],V)
|
||||
rho[n+1]=rhon(u[n+1],e,weights,V0,Eta0)
|
||||
end
|
||||
|
||||
return (u,rho)
|
||||
end
|
||||
|
||||
# A
|
||||
function A(e,weights,Eta)
|
||||
N=length(weights[1])
|
||||
out=zeros(Complex{Float64},N,N)
|
||||
for i in 1:N
|
||||
k=(1-weights[1][i])/(1+weights[1][i])
|
||||
out[i,i]=k^2+4*e
|
||||
for j in 1:N
|
||||
y=(weights[1][j]+1)/2
|
||||
out[i,j]+=weights[2][j]*(1-y)*Eta[i][j]/(2*(2*pi)^3*y^3)
|
||||
end
|
||||
end
|
||||
return out
|
||||
end
|
||||
|
||||
# b
|
||||
function b(u,e,rho,V)
|
||||
out=zeros(Complex{Float64},length(V))
|
||||
for i in 1:length(V)
|
||||
out[i]=V[i]+2*e*rho*u[i]^2
|
||||
end
|
||||
return out
|
||||
end
|
||||
|
||||
# rho_n
|
||||
function rhon(u,e,weights,V0,Eta0)
|
||||
S=V0
|
||||
for i in 1:length(weights[1])
|
||||
y=(weights[1][i]+1)/2
|
||||
S+=-weights[2][i]*(1-y)*u[i]*Eta0[i]/(2*(2*pi)^3*y^3)
|
||||
end
|
||||
return 2*e/S
|
||||
end
|
||||
|
|
@ -0,0 +1,192 @@
|
|||
using FastGaussQuadrature
|
||||
using Printf
|
||||
using LinearAlgebra
|
||||
include("tools.jl")
|
||||
include("integration.jl")
|
||||
include("simpleq.jl")
|
||||
include("simpleq-newton.jl")
|
||||
include("iteration.jl")
|
||||
|
||||
|
||||
function main()
|
||||
## defaults
|
||||
|
||||
tolerance=1e-14
|
||||
order=100
|
||||
maxiter=21
|
||||
|
||||
rho=1e-6
|
||||
e=1e-4
|
||||
|
||||
d=3
|
||||
v=v_exp3d
|
||||
a0=a0_exp3d
|
||||
|
||||
# plot range when plotting in x
|
||||
xmin=0
|
||||
xmax=100
|
||||
nx=100
|
||||
|
||||
# read cli arguments
|
||||
(params,command)=read_args(ARGS)
|
||||
|
||||
# read params
|
||||
if params!=""
|
||||
for param in split(params,";")
|
||||
terms=split(param,"=")
|
||||
if length(terms)!=2
|
||||
print(stderr,"error: could not read parameter '",param,"'.\n")
|
||||
exit(-1)
|
||||
end
|
||||
lhs=terms[1]
|
||||
rhs=terms[2]
|
||||
if lhs=="rho"
|
||||
rho=parse(Float64,rhs)
|
||||
elseif lhs=="e"
|
||||
e=parse(Float64,rhs)
|
||||
elseif lhs=="tolerance"
|
||||
tolerance=parse(Float64,rhs)
|
||||
elseif lhs=="order"
|
||||
order=parse(Int64,rhs)
|
||||
elseif lhs=="maxiter"
|
||||
maxiter=parse(Int64,rhs)
|
||||
elseif lhs=="xmin"
|
||||
xmin=parse(Float64,rhs)
|
||||
elseif lhs=="xmax"
|
||||
xmax=parse(Float64,rhs)
|
||||
elseif lhs=="nx"
|
||||
nx=parse(Int64,rhs)
|
||||
else
|
||||
print(stderr,"unrecognized parameter '",lhs,"'.\n")
|
||||
exit(-1)
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
## run command
|
||||
# e(rho)
|
||||
if command=="energy_rho"
|
||||
energy_rho(order,a0,d,v,maxiter,tolerance)
|
||||
# d^2(rho*e(rho))
|
||||
elseif command=="convexity"
|
||||
ddrhoe(order,a0,d,v,maxiter,tolerance)
|
||||
# u_n(x)
|
||||
elseif command=="iteration"
|
||||
iteration_ux(order,e,a0,d,v,maxiter)
|
||||
else
|
||||
print(stderr,"unrecognized command '",command,"'.\n")
|
||||
exit(-1)
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
# read cli arguments
|
||||
function read_args(ARGS)
|
||||
# flag
|
||||
flag=""
|
||||
|
||||
# output strings
|
||||
params=""
|
||||
command=""
|
||||
|
||||
# loop over arguments
|
||||
for arg in ARGS
|
||||
# argument that starts with a dash
|
||||
if arg[1]=='-'
|
||||
# go through characters after dash
|
||||
for char in arg[2:length(arg)]
|
||||
|
||||
# set params
|
||||
if char=='p'
|
||||
# raise flag
|
||||
flag="params"
|
||||
else
|
||||
print_usage()
|
||||
exit(-1)
|
||||
end
|
||||
end
|
||||
# arguments that do not start with a dash
|
||||
else
|
||||
if flag=="params"
|
||||
params=arg
|
||||
else
|
||||
command=arg
|
||||
end
|
||||
# reset flag
|
||||
flag=""
|
||||
end
|
||||
end
|
||||
|
||||
if command==""
|
||||
print_usage()
|
||||
exit(-1)
|
||||
end
|
||||
|
||||
return (params,command)
|
||||
end
|
||||
|
||||
# usage
|
||||
function print_usage()
|
||||
print(stderr,"usage: simpleq [-p params] <command>\n")
|
||||
end
|
||||
|
||||
# exponential potential in 3 dimensions
|
||||
function v_exp3d(k)
|
||||
return 8*pi/(1+k^2)^2
|
||||
end
|
||||
a0_exp3d=1.254356410591064819505409291110046864031181245635821944528
|
||||
|
||||
# compute energy as a function of rho
|
||||
function energy_rho(order,a0,d,v,maxiter,tolerance)
|
||||
minlrho=-6
|
||||
maxlrho=2
|
||||
nlrho=100
|
||||
|
||||
for j in 0:nlrho-1
|
||||
rho=10^(minlrho+(maxlrho-minlrho)*j/nlrho)
|
||||
# linear spacing
|
||||
#rho=10.0^minlrho+(10.0^maxlrho-10.0^minlrho)*j/nlrho
|
||||
|
||||
(u,E)=hatu_newton(order,rho,a0,d,v,maxiter,tolerance)
|
||||
|
||||
@printf("% .8e % .8e\n",rho,real(E))
|
||||
|
||||
end
|
||||
end
|
||||
|
||||
# compute \partial^2(e\rho) as a function of rho
|
||||
function ddrhoe(order,a0,d,v,maxiter,tolerance)
|
||||
minlrho=-6
|
||||
maxlrho=2
|
||||
nlrho=100
|
||||
|
||||
for j in 0:nlrho-1
|
||||
rho=10^(minlrho+(maxlrho-minlrho)*j/nlrho)
|
||||
|
||||
# interval
|
||||
drho=rho*1.01
|
||||
|
||||
(u,E)=hatu_newton(order,rho,a0,d,v,maxiter,tolerance)
|
||||
(up,Ep)=hatu_newton(order,rho+drho,a0,d,v,maxiter,tolerance)
|
||||
(um,Em)=hatu_newton(order,rho-drho,a0,d,v,maxiter,tolerance)
|
||||
|
||||
@printf("% .8e % .8e\n",rho,real((rho+drho)*Ep+(rho-drho)*Em-2*rho*E)/drho^2)
|
||||
|
||||
end
|
||||
end
|
||||
|
||||
# compute \int u_n(x) at every step
|
||||
function iteration_ux(order,e,a0,d,v,maxiter)
|
||||
(u,rho)=hatun_iteration(e,order,d,v,maxiter)
|
||||
|
||||
weights=gausslegendre(order)
|
||||
|
||||
intun=0.
|
||||
for n in 2:maxiter+1
|
||||
# compute \hat u_n(0)=1/(2*rho_n)+rho_{n-1}/2*\hat u_{n-1}(0)^2
|
||||
intun=real(1/(2*rho[n])+rho[n-1]/2*intun^2)
|
||||
@printf("%2d % .15e\n",n-1,abs(intun-1/rho[maxiter+1]))
|
||||
end
|
||||
end
|
||||
|
||||
main()
|
||||
|
|
@ -0,0 +1,95 @@
|
|||
# \hat u(k) computed using Newton algorithm
|
||||
function hatu_newton(order,rho,a0,d,v,maxiter,tolerance)
|
||||
|
||||
# compute gaussian quadrature weights
|
||||
weights=gausslegendre(order)
|
||||
|
||||
# initialize V and Eta
|
||||
(V,V0,Eta,Eta0)=init_veta(weights,d,v)
|
||||
|
||||
# initialize u, V and Eta
|
||||
u=zeros(Complex{Float64},order)
|
||||
for j in 1:order
|
||||
# transformed k
|
||||
k=(1-weights[1][j])/(1+weights[1][j])
|
||||
if d==2
|
||||
u[j]=2*pi*a0*rho/k
|
||||
elseif d==3
|
||||
u[j]=4*pi*a0*rho/k^2
|
||||
end
|
||||
end
|
||||
|
||||
# iterate
|
||||
for i in 1:maxiter-1
|
||||
new=u-inv(dXi(u,V,V0,Eta,Eta0,weights,rho,d))*Xi(u,V,V0,Eta,Eta0,weights,rho,d)
|
||||
|
||||
if(norm(new-u)/norm(u)<tolerance)
|
||||
u=new
|
||||
break
|
||||
end
|
||||
u=new
|
||||
end
|
||||
|
||||
return (u,en(u,V0,Eta0,rho,weights,d)*rho/2)
|
||||
end
|
||||
|
||||
# Xi(u)=0 is equivalent to the linear equation
|
||||
function Xi(u,V,V0,Eta,Eta0,weights,rho,d)
|
||||
order=length(weights[1])
|
||||
|
||||
# init
|
||||
out=zeros(Complex{Float64},order)
|
||||
|
||||
# compute E before running the loop
|
||||
E=en(u,V0,Eta0,rho,weights,d)
|
||||
|
||||
for i in 1:order
|
||||
# k_i
|
||||
k=(1-weights[1][i])/(1+weights[1][i])
|
||||
# S_i
|
||||
S=V[i]-1/(rho*(2*pi)^d)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights)
|
||||
# X_i
|
||||
X=k^2/(2*E*rho)
|
||||
|
||||
# U_i
|
||||
out[i]=u[i]-S/(2*E*(X+1))*Phi(S/(E*(X+1)^2))
|
||||
end
|
||||
|
||||
return out
|
||||
end
|
||||
|
||||
# derivative of Xi (for Newton)
|
||||
function dXi(u,V,V0,Eta,Eta0,weights,rho,d)
|
||||
order=length(weights[1])
|
||||
|
||||
# init
|
||||
out=zeros(Complex{Float64},order,order)
|
||||
|
||||
# compute E before the loop
|
||||
E=en(u,V0,Eta0,rho,weights,d)
|
||||
|
||||
for i in 1:order
|
||||
# k_i
|
||||
k=(1-weights[1][i])/(1+weights[1][i])
|
||||
# S_i
|
||||
S=V[i]-1/(rho*(2*pi)^d)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights)
|
||||
# X_i
|
||||
X=k^2/(2*E*rho)
|
||||
|
||||
for j in 1:order
|
||||
y=(weights[1][j]+1)/2
|
||||
dS=-1/rho*(1-y)*Eta[i][j]/(2*(2*pi)^d*y^3)*weights[2][j]
|
||||
dE=-1/rho*(1-y)*Eta0[j]/(2*(2*pi)^d*y^3)*weights[2][j]
|
||||
dX=-k^2/(2*E^2*rho)*dE
|
||||
out[i,j]=(i==j ? 1 : 0)-(dS-S*dE/E-S*dX/(X+1))/(2*E*(X+1))*Phi(S/(E*(X+1)^2))-S/(2*E^2*(X+1)^3)*(dS-S*dE/E-2*S*dX/(X+1))*dPhi(S/(E*(X+1)^2))
|
||||
end
|
||||
end
|
||||
|
||||
return out
|
||||
end
|
||||
|
||||
# energy
|
||||
function en(u,V0,Eta0,rho,weights,d)
|
||||
return V0-1/(rho*(2*pi)^d)*integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*u,0,1,weights)
|
||||
end
|
||||
|
||||
|
|
@ -0,0 +1,40 @@
|
|||
# \eta
|
||||
function eta(x,t,weights,d,v)
|
||||
if d==2
|
||||
return integrate_chebyshev(y->4*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y))/sqrt(((x+t)*y+abs(x-t)*(2-y))*((x+t)*(1+y)+abs(x-t)*(1-y))),0,1,length(weights))
|
||||
elseif d==3
|
||||
return (x>t ? 2*t/x : 2)* integrate_legendre(y->2*pi*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y)),0,1,weights)
|
||||
end
|
||||
end
|
||||
|
||||
# initialize V and Eta
|
||||
function init_veta(weights,d,v)
|
||||
order=length(weights[1])
|
||||
V=Array{Complex{Float64}}(undef,order)
|
||||
Eta=Array{Array{Complex{Float64}}}(undef,order)
|
||||
Eta0=Array{Complex{Float64}}(undef,order)
|
||||
V0=v(0)
|
||||
for i in 1:order
|
||||
k=(1-weights[1][i])/(1+weights[1][i])
|
||||
V[i]=v(k)
|
||||
Eta[i]=Array{Complex{Float64}}(undef,order)
|
||||
for j in 1:order
|
||||
y=(weights[1][j]+1)/2
|
||||
Eta[i][j]=eta(k,(1-y)/y,weights,d,v)
|
||||
end
|
||||
y=(weights[1][i]+1)/2
|
||||
Eta0[i]=eta(0,(1-y)/y,weights,d,v)
|
||||
end
|
||||
return(V,V0,Eta,Eta0)
|
||||
end
|
||||
|
||||
# inverse Fourier transform
|
||||
function u_x(x,u,weights,d)
|
||||
order=length(weights[1])
|
||||
if d==2
|
||||
out=integrate_legendre_sampled(y->(1-y)/y^3*besselj(0,x*(1-y)/y)/(2*pi),u,0,1,weights)
|
||||
elseif d==3
|
||||
out=integrate_legendre_sampled(y->(1-y)/y^3*sin(x*(1-y)/y)/x/(2*pi^2),u,0,1,weights)
|
||||
end
|
||||
return out
|
||||
end
|
||||
|
|
@ -0,0 +1,20 @@
|
|||
# \Phi(x)=2*(1-sqrt(1-x))/x
|
||||
function Phi(x)
|
||||
if abs(x)>1e-5
|
||||
return 2*(1-sqrt(1-x))/x
|
||||
else
|
||||
return 1+x/4+x^2/8+5*x^3/64+7*x^4/128+21*x^5/512
|
||||
end
|
||||
end
|
||||
# \partial\Phi
|
||||
function dPhi(x)
|
||||
#if abs(x-1)<1e-5
|
||||
# @printf(stderr,"warning: dPhi is singular at 1, and evaluating it at (% .8e+i% .8e)\n",real(x),imag(x))
|
||||
#end
|
||||
if abs(x)>1e-5
|
||||
return 1/(sqrt(1-x)*x)-2*(1-sqrt(1-x))/x^2
|
||||
else
|
||||
return 1/4+x/4+15*x^2/64+7*x^3/32+105*x^4/512+99*x^5/512
|
||||
end
|
||||
end
|
||||
|
||||
|
|
@ -0,0 +1,673 @@
|
|||
%%
|
||||
%% Ian's class file
|
||||
%%
|
||||
|
||||
%% TeX format
|
||||
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
|
||||
|
||||
%% class name
|
||||
\ProvidesClass{ian}[2017/09/29]
|
||||
|
||||
%% boolean to signal that this class is being used
|
||||
\newif\ifianclass
|
||||
|
||||
%% options
|
||||
% no section numbering in equations
|
||||
\DeclareOption{section_in_eq}{\sectionsineqtrue}
|
||||
\DeclareOption{section_in_fig}{\sectionsinfigtrue}
|
||||
\DeclareOption{section_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
|
||||
\DeclareOption{section_in_all}{\sectionsineqtrue\sectionsinfigtrue\PassOptionsToPackage{section_in_theo}{iantheo}}
|
||||
\DeclareOption{subsection_in_eq}{\subsectionsineqtrue}
|
||||
\DeclareOption{subsection_in_fig}{\subsectionsinfigtrue}
|
||||
\DeclareOption{subsection_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
|
||||
\DeclareOption{subsection_in_all}{\subsectionsineqtrue\subsectionsinfigtrue\PassOptionsToPackage{subsection_in_theo}{iantheo}}
|
||||
\DeclareOption{no_section_in_eq}{\sectionsineqfalse}
|
||||
\DeclareOption{no_section_in_fig}{\sectionsinfigfalse}
|
||||
\DeclareOption{no_section_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
|
||||
\DeclareOption{no_section_in_all}{\sectionsineqfalse\sectionsinfigfalse\PassOptionsToPackage{no_section_in_theo}{iantheo}}
|
||||
\DeclareOption{no_subsection_in_eq}{\subsectionsineqfalse}
|
||||
\DeclareOption{no_subsection_in_fig}{\subsectionsinfigfalse}
|
||||
\DeclareOption{no_subsection_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
|
||||
\DeclareOption{no_subsection_in_all}{\subsectionsineqfalse\subsectionsinfigfalse\PassOptionsToPackage{no_subsection_in_theo}{iantheo}}
|
||||
% reset point
|
||||
\DeclareOption{point_reset_at_section}{\PassOptionsToPackage{reset_at_section}{point}}
|
||||
\DeclareOption{point_no_reset_at_section}{\PassOptionsToPackage{no_reset_at_section}{point}}
|
||||
\DeclareOption{point_reset_at_theo}{\PassOptionsToPackage{reset_at_theo}{point}}
|
||||
\DeclareOption{point_no_reset_at_theo}{\PassOptionsToPackage{no_reset_at_theo}{point}}
|
||||
|
||||
\def\ian@defaultoptions{
|
||||
\ExecuteOptions{section_in_all, no_subsection_in_all}
|
||||
\ProcessOptions
|
||||
|
||||
%% required packages
|
||||
\RequirePackage{iantheo}
|
||||
\RequirePackage{point}
|
||||
\RequirePackage{color}
|
||||
\RequirePackage{marginnote}
|
||||
\RequirePackage{amssymb}
|
||||
\PassOptionsToPackage{hidelinks}{hyperref}
|
||||
\RequirePackage{hyperref}
|
||||
}
|
||||
|
||||
%% paper dimensions
|
||||
\setlength\paperheight{297mm}
|
||||
\setlength\paperwidth{210mm}
|
||||
|
||||
%% fonts
|
||||
\input{size11.clo}
|
||||
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
|
||||
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
|
||||
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
|
||||
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
|
||||
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
|
||||
|
||||
%% text dimensions
|
||||
\hoffset=-50pt
|
||||
\voffset=-72pt
|
||||
\textwidth=460pt
|
||||
\textheight=704pt
|
||||
|
||||
|
||||
%% remove default indentation
|
||||
\parindent=0pt
|
||||
%% indent command
|
||||
\def\indent{\hskip20pt}
|
||||
|
||||
%% something is wrong with \thepage, redefine it
|
||||
\gdef\thepage{\the\c@page}
|
||||
|
||||
%% array lines (to use the array environment)
|
||||
\setlength\arraycolsep{5\p@}
|
||||
\setlength\arrayrulewidth{.4\p@}
|
||||
|
||||
|
||||
%% correct vertical alignment at the end of a document
|
||||
\AtEndDocument{
|
||||
\vfill
|
||||
\eject
|
||||
}
|
||||
|
||||
|
||||
%% hyperlinks
|
||||
% hyperlinkcounter
|
||||
\newcounter{lncount}
|
||||
% hyperref anchor
|
||||
\def\hrefanchor{%
|
||||
\stepcounter{lncount}%
|
||||
\hypertarget{ln.\thelncount}{}%
|
||||
}
|
||||
|
||||
%% define a command and write it to aux file
|
||||
\def\outdef#1#2{%
|
||||
% define command%
|
||||
\expandafter\xdef\csname #1\endcsname{#2}%
|
||||
% hyperlink number%
|
||||
\expandafter\xdef\csname #1@hl\endcsname{\thelncount}%
|
||||
% write command to aux%
|
||||
\immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1\endcsname{\csname #1\endcsname}}%
|
||||
\immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1@hl\endcsname{\thelncount}}%
|
||||
}
|
||||
|
||||
%% can call commands even when they are not defined
|
||||
\def\safe#1{%
|
||||
\ifdefined#1%
|
||||
#1%
|
||||
\else%
|
||||
{\color{red}\bf?}%
|
||||
\fi%
|
||||
}
|
||||
|
||||
%% define a label for the latest tag
|
||||
%% label defines a command containing the string stored in \tag
|
||||
\def\deflabel{
|
||||
\def\label##1{\expandafter\outdef{label@##1}{\safe\tag}}
|
||||
|
||||
\def\ref##1{%
|
||||
% check whether the label is defined (hyperlink runs into errors if this check is omitted)
|
||||
\ifcsname label@##1@hl\endcsname%
|
||||
\hyperlink{ln.\csname label@##1@hl\endcsname}{{\color{blue}\safe\csname label@##1\endcsname}}%
|
||||
\else%
|
||||
\ifcsname label@##1\endcsname%
|
||||
{\color{blue}\csname ##1\endcsname}%
|
||||
\else%
|
||||
{\bf ??}%
|
||||
\fi%
|
||||
\fi%
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
%% make a custom link at any given location in the document
|
||||
\def\makelink#1#2{%
|
||||
\hrefanchor%
|
||||
\outdef{label@#1}{#2}%
|
||||
}
|
||||
|
||||
|
||||
%% section command
|
||||
% counter
|
||||
\newcounter{sectioncount}
|
||||
% space before section
|
||||
\newlength\secskip
|
||||
\setlength\secskip{40pt}
|
||||
% a prefix to put before the section number, e.g. A for appendices
|
||||
\def\sectionprefix{}
|
||||
% define some lengths
|
||||
\newlength\secnumwidth
|
||||
\newlength\sectitlewidth
|
||||
\def\section#1{
|
||||
% reset counters
|
||||
\stepcounter{sectioncount}
|
||||
\setcounter{subsectioncount}{0}
|
||||
\ifsectionsineq
|
||||
\setcounter{seqcount}0
|
||||
\fi
|
||||
\ifsectionsinfig
|
||||
\setcounter{figcount}0
|
||||
\fi
|
||||
|
||||
% space before section (if not first)
|
||||
\ifnum\thesectioncount>1
|
||||
\vskip\secskip
|
||||
\penalty-1000
|
||||
\fi
|
||||
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\sectionprefix\thesectioncount}
|
||||
|
||||
% get widths
|
||||
\def\@secnum{{\bf\Large\sectionprefix\thesectioncount.\hskip10pt}}
|
||||
\settowidth\secnumwidth{\@secnum}
|
||||
\setlength\sectitlewidth\textwidth
|
||||
\addtolength\sectitlewidth{-\secnumwidth}
|
||||
|
||||
% print name
|
||||
\parbox{\textwidth}{
|
||||
\@secnum
|
||||
\parbox[t]{\sectitlewidth}{\Large\bf #1}}
|
||||
|
||||
% write to table of contents
|
||||
\iftoc
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@sec.\thesectioncount\endcsname{\thelncount}}
|
||||
\write\tocoutput{\noexpand\tocsection{#1}{\thepage}}
|
||||
\fi
|
||||
|
||||
%space
|
||||
\par\penalty10000
|
||||
\bigskip\penalty10000
|
||||
}
|
||||
|
||||
%% subsection
|
||||
% counter
|
||||
\newcounter{subsectioncount}
|
||||
% space before subsection
|
||||
\newlength\subsecskip
|
||||
\setlength\subsecskip{30pt}
|
||||
\def\subsection#1{
|
||||
% counters
|
||||
\stepcounter{subsectioncount}
|
||||
\setcounter{subsubsectioncount}{0}
|
||||
\ifsubsectionsineq
|
||||
\setcounter{seqcount}0
|
||||
\fi
|
||||
\ifsubsectionsinfig
|
||||
\setcounter{figcount}0
|
||||
\fi
|
||||
|
||||
% space before subsection (if not first)
|
||||
\ifnum\thesubsectioncount>1
|
||||
\vskip\subsecskip
|
||||
\penalty-500
|
||||
\fi
|
||||
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount}
|
||||
|
||||
% get widths
|
||||
\def\@secnum{{\bf\large\hskip.5cm\sectionprefix\thesectioncount.\thesubsectioncount.\hskip5pt}}
|
||||
\settowidth\secnumwidth{\@secnum}
|
||||
\setlength\sectitlewidth\textwidth
|
||||
\addtolength\sectitlewidth{-\secnumwidth}
|
||||
% print name
|
||||
\parbox{\textwidth}{
|
||||
\@secnum
|
||||
\parbox[t]{\sectitlewidth}{\large\bf #1}}
|
||||
|
||||
% write to table of contents
|
||||
\iftoc
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsec.\thesectioncount.\thesubsectioncount\endcsname{\thelncount}}
|
||||
\write\tocoutput{\noexpand\tocsubsection{#1}{\thepage}}
|
||||
\fi
|
||||
|
||||
% space
|
||||
\par\penalty10000
|
||||
\medskip\penalty10000
|
||||
}
|
||||
|
||||
%% subsubsection
|
||||
% counter
|
||||
\newcounter{subsubsectioncount}
|
||||
% space before subsubsection
|
||||
\newlength\subsubsecskip
|
||||
\setlength\subsubsecskip{20pt}
|
||||
\def\subsubsection#1{
|
||||
% counters
|
||||
\stepcounter{subsubsectioncount}
|
||||
|
||||
% space before subsubsection (if not first)
|
||||
\ifnum\thesubsubsectioncount>1
|
||||
\vskip\subsubsecskip
|
||||
\penalty-500
|
||||
\fi
|
||||
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount}
|
||||
|
||||
% get widths
|
||||
\def\@secnum{{\bf\hskip1.cm\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount.\hskip5pt}}
|
||||
\settowidth\secnumwidth{\@secnum}
|
||||
\setlength\sectitlewidth\textwidth
|
||||
\addtolength\sectitlewidth{-\secnumwidth}
|
||||
% print name
|
||||
\parbox{\textwidth}{
|
||||
\@secnum
|
||||
\parbox[t]{\sectitlewidth}{\large\bf #1}}
|
||||
|
||||
% write to table of contents
|
||||
\iftoc
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsubsec.\thesectioncount.\thesubsectioncount.\thesubsubsectioncount\endcsname{\thelncount}}
|
||||
\write\tocoutput{\noexpand\tocsubsubsection{#1}{\thepage}}
|
||||
\fi
|
||||
|
||||
% space
|
||||
\par\penalty10000
|
||||
\medskip\penalty10000
|
||||
}
|
||||
|
||||
%% itemize
|
||||
\newlength\itemizeskip
|
||||
% left margin for items
|
||||
\setlength\itemizeskip{20pt}
|
||||
\newlength\itemizeseparator
|
||||
% space between the item symbol and the text
|
||||
\setlength\itemizeseparator{5pt}
|
||||
% penalty preceding an itemize
|
||||
\newcount\itemizepenalty
|
||||
\itemizepenalty=0
|
||||
% counter counting the itemize level
|
||||
\newcounter{itemizecount}
|
||||
|
||||
% item symbol
|
||||
\def\itemizept#1{
|
||||
\ifnum#1=1
|
||||
\textbullet
|
||||
\else
|
||||
$\scriptstyle\blacktriangleright$
|
||||
\fi
|
||||
}
|
||||
|
||||
|
||||
\newlength\current@itemizeskip
|
||||
\setlength\current@itemizeskip{0pt}
|
||||
\def\itemize{%
|
||||
\par\expandafter\penalty\the\itemizepenalty\medskip\expandafter\penalty\the\itemizepenalty%
|
||||
\addtocounter{itemizecount}{1}%
|
||||
\addtolength\current@itemizeskip{\itemizeskip}%
|
||||
\leftskip\current@itemizeskip%
|
||||
}
|
||||
\def\enditemize{%
|
||||
\addtocounter{itemizecount}{-1}%
|
||||
\addtolength\current@itemizeskip{-\itemizeskip}%
|
||||
\par\expandafter\penalty\the\itemizepenalty\leftskip\current@itemizeskip%
|
||||
\medskip\expandafter\penalty\the\itemizepenalty%
|
||||
}
|
||||
|
||||
% item, with optional argument to specify the item point
|
||||
% @itemarg is set to true when there is an optional argument
|
||||
\newif\if@itemarg
|
||||
\def\item{%
|
||||
% check whether there is an optional argument (if there is none, add on empty '[]')
|
||||
\@ifnextchar [{\@itemargtrue\@itemx}{\@itemargfalse\@itemx[]}%
|
||||
}
|
||||
\newlength\itempt@total
|
||||
\def\@itemx[#1]{
|
||||
\if@itemarg
|
||||
\settowidth\itempt@total{#1}
|
||||
\else
|
||||
\settowidth\itempt@total{\itemizept\theitemizecount}
|
||||
\fi
|
||||
\addtolength\itempt@total{\itemizeseparator}
|
||||
\par
|
||||
\medskip
|
||||
\if@itemarg
|
||||
\hskip-\itempt@total#1\hskip\itemizeseparator
|
||||
\else
|
||||
\hskip-\itempt@total\itemizept\theitemizecount\hskip\itemizeseparator
|
||||
\fi
|
||||
}
|
||||
|
||||
%% prevent page breaks after itemize
|
||||
\newcount\previtemizepenalty
|
||||
\def\nopagebreakafteritemize{
|
||||
\previtemizepenalty=\itemizepenalty
|
||||
\itemizepenalty=10000
|
||||
}
|
||||
%% back to previous value
|
||||
\def\restorepagebreakafteritemize{
|
||||
\itemizepenalty=\previtemizepenalty
|
||||
}
|
||||
|
||||
%% enumerate
|
||||
\newcounter{enumerate@count}
|
||||
\def\enumerate{
|
||||
\setcounter{enumerate@count}0
|
||||
\let\olditem\item
|
||||
\let\olditemizept\itemizept
|
||||
\def\item{
|
||||
% counter
|
||||
\stepcounter{enumerate@count}
|
||||
% set header
|
||||
\def\itemizept{\theenumerate@count.}
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\theenumerate@count}
|
||||
\olditem
|
||||
}
|
||||
\itemize
|
||||
}
|
||||
\def\endenumerate{
|
||||
\enditemize
|
||||
\let\item\olditem
|
||||
\let\itemizept\olditemizept
|
||||
}
|
||||
|
||||
|
||||
%% equation numbering
|
||||
% counter
|
||||
\newcounter{seqcount}
|
||||
% booleans (write section or subsection in equation number)
|
||||
\newif\ifsectionsineq
|
||||
\newif\ifsubsectionsineq
|
||||
\def\seqcount{
|
||||
\stepcounter{seqcount}
|
||||
% the output
|
||||
\edef\seqformat{\theseqcount}
|
||||
% add subsection number
|
||||
\ifsubsectionsineq
|
||||
\let\tmp\seqformat
|
||||
\edef\seqformat{\thesubsectioncount.\tmp}
|
||||
\fi
|
||||
% add section number
|
||||
\ifsectionsineq
|
||||
\let\tmp\seqformat
|
||||
\edef\seqformat{\sectionprefix\thesectioncount.\tmp}
|
||||
\fi
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\seqformat}
|
||||
% write number
|
||||
\marginnote{\hfill(\seqformat)}
|
||||
}
|
||||
%% equation environment compatibility
|
||||
\def\equation{\hrefanchor$$\seqcount}
|
||||
\def\endequation{$$\@ignoretrue}
|
||||
|
||||
|
||||
%% figures
|
||||
% counter
|
||||
\newcounter{figcount}
|
||||
% booleans (write section or subsection in equation number)
|
||||
\newif\ifsectionsinfig
|
||||
\newif\ifsubsectionsinfig
|
||||
% width of figures
|
||||
\newlength\figwidth
|
||||
\setlength\figwidth\textwidth
|
||||
\addtolength\figwidth{-2.5cm}
|
||||
% caption
|
||||
\def\defcaption{
|
||||
\long\def\caption##1{
|
||||
\stepcounter{figcount}
|
||||
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
|
||||
% the number of the figure
|
||||
\edef\figformat{\thefigcount}
|
||||
% add subsection number
|
||||
\ifsubsectionsinfig
|
||||
\let\tmp\figformat
|
||||
\edef\figformat{\thesubsectioncount.\tmp}
|
||||
\fi
|
||||
% add section number
|
||||
\ifsectionsinfig
|
||||
\let\tmp\figformat
|
||||
\edef\figformat{\sectionprefix\thesectioncount.\tmp}
|
||||
\fi
|
||||
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\figformat}
|
||||
|
||||
% write
|
||||
\hfil fig \figformat: \parbox[t]{\figwidth}{\leavevmode\small##1}
|
||||
|
||||
% space
|
||||
\par\bigskip
|
||||
}
|
||||
}
|
||||
%% short caption: centered
|
||||
\def\captionshort#1{
|
||||
\stepcounter{figcount}
|
||||
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
|
||||
% the number of the figure
|
||||
\edef\figformat{\thefigcount}
|
||||
% add section number
|
||||
\ifsectionsinfig
|
||||
\let\tmp\figformat
|
||||
\edef\figformat{\sectionprefix\thesectioncount.\tmp}
|
||||
\fi
|
||||
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\figformat}
|
||||
|
||||
% write
|
||||
\hfil fig \figformat: {\small#1}
|
||||
|
||||
%space
|
||||
\par\bigskip
|
||||
}
|
||||
|
||||
%% environment
|
||||
\def\figure{
|
||||
\par
|
||||
\vfil\penalty100\vfilneg
|
||||
\bigskip
|
||||
}
|
||||
\def\endfigure{
|
||||
\par
|
||||
\bigskip
|
||||
}
|
||||
|
||||
|
||||
%% start appendices
|
||||
\def\appendix{
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
% counter
|
||||
\setcounter{sectioncount}0
|
||||
|
||||
% prefix
|
||||
\def\sectionprefix{A}
|
||||
|
||||
% write
|
||||
{\bf \LARGE Appendices}\par\penalty10000\bigskip\penalty10000
|
||||
|
||||
% add a mention in the table of contents
|
||||
\iftoc
|
||||
\immediate\write\tocoutput{\noexpand\tocappendices}\penalty10000
|
||||
\fi
|
||||
|
||||
%% uncomment for new page for each appendix
|
||||
%\def\seqskip{\vfill\pagebreak}
|
||||
}
|
||||
|
||||
|
||||
%% bibliography
|
||||
% size of header
|
||||
\newlength\bibheader
|
||||
\def\thebibliography#1{
|
||||
\hrefanchor
|
||||
|
||||
% add a mention in the table of contents
|
||||
\iftoc
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@references\endcsname{\thelncount}}
|
||||
\write\tocoutput{\noexpand\tocreferences{\thepage}}\penalty10000
|
||||
\fi
|
||||
|
||||
% write
|
||||
{\bf \LARGE References}\par\penalty10000\bigskip\penalty10000
|
||||
% width of header
|
||||
\settowidth\bibheader{[#1]}
|
||||
\leftskip\bibheader
|
||||
}
|
||||
% end environment
|
||||
\def\endthebibliography{
|
||||
\par\leftskip0pt
|
||||
}
|
||||
|
||||
%% bibitem command
|
||||
\def\bibitem[#1]#2{%
|
||||
\hrefanchor%
|
||||
\outdef{label@cite#2}{#1}%
|
||||
\hskip-\bibheader%
|
||||
\makebox[\bibheader]{\cite{#2}\hfill}%
|
||||
}
|
||||
|
||||
%% cite command (adapted from latex.ltx)
|
||||
% @tempswa is set to true when there is an optional argument
|
||||
\newif\@tempswa
|
||||
\def\cite{%
|
||||
% check whether there is an optional argument (if there is none, add on empty '[]')
|
||||
\@ifnextchar [{\@tempswatrue\@citex}{\@tempswafalse\@citex[]}%
|
||||
}
|
||||
% command with optional argument
|
||||
\def\@citex[#1]#2{\leavevmode%
|
||||
% initialize loop
|
||||
\let\@citea\@empty%
|
||||
% format
|
||||
\@cite{%
|
||||
% loop over ',' separated list
|
||||
\@for\@citeb:=#2\do{%
|
||||
% text to add at each iteration of the loop (separator between citations)
|
||||
\@citea\def\@citea{,\ }%
|
||||
% add entry to citelist
|
||||
\@writecitation{\@citeb}%
|
||||
\ref{cite\@citeb}%
|
||||
}%
|
||||
}%
|
||||
% add optional argument text (as an argument to '\@cite')
|
||||
{#1}%
|
||||
}
|
||||
\def\@cite#1#2{%
|
||||
[#1\if@tempswa , #2\fi]%
|
||||
}
|
||||
%% add entry to citelist after checking it has not already been added
|
||||
\def\@writecitation#1{%
|
||||
\ifcsname if#1cited\endcsname%
|
||||
\else%
|
||||
\expandafter\newif\csname if#1cited\endcsname%
|
||||
\immediate\write\@auxout{\string\citation{#1}}%
|
||||
\fi%
|
||||
}
|
||||
|
||||
%% table of contents
|
||||
% boolean
|
||||
\newif\iftoc
|
||||
\def\tableofcontents{
|
||||
{\bf \large Table of contents:}\par\penalty10000\bigskip\penalty10000
|
||||
|
||||
% copy content from file
|
||||
\IfFileExists{\jobname.toc}{\input{\jobname.toc}}{{\tt error: table of contents missing}}
|
||||
|
||||
% open new toc
|
||||
\newwrite\tocoutput
|
||||
\immediate\openout\tocoutput=\jobname.toc
|
||||
|
||||
\toctrue
|
||||
}
|
||||
%% close file
|
||||
\AtEndDocument{
|
||||
% close toc
|
||||
\iftoc
|
||||
\immediate\closeout\tocoutput
|
||||
\fi
|
||||
}
|
||||
|
||||
|
||||
%% fill line with dots
|
||||
\def\leaderfill{\leaders\hbox to 1em {\hss. \hss}\hfill}
|
||||
|
||||
%% same as sectionprefix
|
||||
\def\tocsectionprefix{}
|
||||
|
||||
%% toc formats
|
||||
\newcounter{tocsectioncount}
|
||||
\def\tocsection #1#2{
|
||||
\stepcounter{tocsectioncount}
|
||||
\setcounter{tocsubsectioncount}{0}
|
||||
\setcounter{tocsubsubsectioncount}{0}
|
||||
% write
|
||||
\smallskip\hyperlink{ln.\csname toc@sec.\thetocsectioncount\endcsname}{{\bf \tocsectionprefix\thetocsectioncount}.\hskip5pt {\color{blue}#1}\leaderfill#2}\par
|
||||
}
|
||||
\newcounter{tocsubsectioncount}
|
||||
\def\tocsubsection #1#2{
|
||||
\stepcounter{tocsubsectioncount}
|
||||
\setcounter{tocsubsubsectioncount}{0}
|
||||
% write
|
||||
{\hskip10pt\hyperlink{ln.\csname toc@subsec.\thetocsectioncount.\thetocsubsectioncount\endcsname}{{\bf \thetocsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par
|
||||
}
|
||||
\newcounter{tocsubsubsectioncount}
|
||||
\def\tocsubsubsection #1#2{
|
||||
\stepcounter{tocsubsubsectioncount}
|
||||
% write
|
||||
{\hskip20pt\hyperlink{ln.\csname toc@subsubsec.\thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount\endcsname}{{\bf \thetocsubsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par
|
||||
}
|
||||
\def\tocappendices{
|
||||
\medskip
|
||||
\setcounter{tocsectioncount}0
|
||||
{\bf Appendices}\par
|
||||
\smallskip
|
||||
\def\tocsectionprefix{A}
|
||||
}
|
||||
\def\tocreferences#1{
|
||||
\medskip
|
||||
{\hyperlink{ln.\csname toc@references\endcsname}{{\color{blue}\bf References}\leaderfill#1}}\par
|
||||
\smallskip
|
||||
}
|
||||
|
||||
|
||||
%% definitions that must be loaded at begin document
|
||||
\let\ian@olddocument\document
|
||||
\def\document{
|
||||
\ian@olddocument
|
||||
|
||||
\deflabel
|
||||
\defcaption
|
||||
}
|
||||
|
||||
%% end
|
||||
\ian@defaultoptions
|
||||
\endinput
|
||||
|
|
@ -0,0 +1,162 @@
|
|||
%%
|
||||
%% iantheorem package:
|
||||
%% Ian's customized theorem command
|
||||
%%
|
||||
|
||||
%% boolean to signal that this package was loaded
|
||||
\newif\ifiantheo
|
||||
|
||||
%% TeX format
|
||||
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
|
||||
|
||||
%% package name
|
||||
\ProvidesPackage{iantheo}[2016/11/10]
|
||||
|
||||
%% options
|
||||
\newif\ifsectionintheo
|
||||
\DeclareOption{section_in_theo}{\sectionintheotrue}
|
||||
\DeclareOption{no_section_in_theo}{\sectionintheofalse}
|
||||
\newif\ifsubsectionintheo
|
||||
\DeclareOption{subsection_in_theo}{\subsectionintheotrue}
|
||||
\DeclareOption{no_subsection_in_theo}{\subsectionintheofalse}
|
||||
|
||||
\def\iantheo@defaultoptions{
|
||||
\ExecuteOptions{section_in_theo, no_subsection_in_theo}
|
||||
\ProcessOptions
|
||||
|
||||
%%% reset at every new section
|
||||
\ifsectionintheo
|
||||
\let\iantheo@oldsection\section
|
||||
\gdef\section{\setcounter{theocount}{0}\iantheo@oldsection}
|
||||
\fi
|
||||
|
||||
%% reset at every new subsection
|
||||
\ifsubsectionintheo
|
||||
\let\iantheo@oldsubsection\subsection
|
||||
\gdef\subsection{\setcounter{theocount}{0}\iantheo@oldsubsection}
|
||||
\fi
|
||||
}
|
||||
|
||||
|
||||
%% delimiters
|
||||
\def\delimtitle#1{
|
||||
\par%
|
||||
\leavevmode%
|
||||
\raise.3em\hbox to\hsize{%
|
||||
\lower0.3em\hbox{\vrule height0.3em}%
|
||||
\hrulefill%
|
||||
\ \lower.3em\hbox{#1}\ %
|
||||
\hrulefill%
|
||||
\lower0.3em\hbox{\vrule height0.3em}%
|
||||
}%
|
||||
\par\penalty10000%
|
||||
}
|
||||
|
||||
%% callable by ref
|
||||
\def\delimtitleref#1{
|
||||
\par%
|
||||
%
|
||||
\ifdefined\ianclass%
|
||||
% hyperref anchor%
|
||||
\hrefanchor%
|
||||
% define tag (for \label)%
|
||||
\xdef\tag{#1}%
|
||||
\fi%
|
||||
%
|
||||
\leavevmode%
|
||||
\raise.3em\hbox to\hsize{%
|
||||
\lower0.3em\hbox{\vrule height0.3em}%
|
||||
\hrulefill%
|
||||
\ \lower.3em\hbox{\bf #1}\ %
|
||||
\hrulefill%
|
||||
\lower0.3em\hbox{\vrule height0.3em}%
|
||||
}%
|
||||
\par\penalty10000%
|
||||
}
|
||||
|
||||
%% no title
|
||||
\def\delim{
|
||||
\par%
|
||||
\leavevmode\raise.3em\hbox to\hsize{%
|
||||
\lower0.3em\hbox{\vrule height0.3em}%
|
||||
\hrulefill%
|
||||
\lower0.3em\hbox{\vrule height0.3em}%
|
||||
}%
|
||||
\par\penalty10000%
|
||||
}
|
||||
|
||||
%% end delim
|
||||
\def\enddelim{
|
||||
\par\penalty10000%
|
||||
\leavevmode%
|
||||
\raise.3em\hbox to\hsize{%
|
||||
\vrule height0.3em\hrulefill\vrule height0.3em%
|
||||
}%
|
||||
\par%
|
||||
}
|
||||
|
||||
|
||||
%% theorem
|
||||
% counter
|
||||
\newcounter{theocount}
|
||||
% booleans (write section or subsection in equation number)
|
||||
\def\theo#1{
|
||||
\stepcounter{theocount}
|
||||
\ifdefined\ianclass
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
\fi
|
||||
% the number
|
||||
\def\formattheo{\thetheocount}
|
||||
% add subsection number
|
||||
\ifsubsectionintheo
|
||||
\let\tmp\formattheo
|
||||
\edef\formattheo{\thesubsectioncount.\tmp}
|
||||
\fi
|
||||
% add section number
|
||||
\ifsectionintheo
|
||||
\let\tmp\formattheo
|
||||
\edef\formattheo{\sectionprefix\thesectioncount.\tmp}
|
||||
\fi
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\formattheo}
|
||||
% write
|
||||
\delimtitle{\bf #1 \formattheo}
|
||||
}
|
||||
\let\endtheo\enddelim
|
||||
%% theorem headers with name
|
||||
\def\theoname#1#2{
|
||||
\theo{#1}\hfil({\it #2})\par\penalty10000\medskip%
|
||||
}
|
||||
|
||||
|
||||
%% qed symbol
|
||||
\def\qedsymbol{$\square$}
|
||||
\def\qed{\penalty10000\hfill\penalty10000\qedsymbol}
|
||||
|
||||
|
||||
%% compatibility with article class
|
||||
\ifdefined\ianclasstrue
|
||||
\relax
|
||||
\else
|
||||
\def\thesectioncount{\thesection}
|
||||
\def\thesubsectioncount{\thesubsection}
|
||||
\def\sectionprefix{}
|
||||
\fi
|
||||
|
||||
|
||||
%% prevent page breaks after displayed equations
|
||||
\newcount\prevpostdisplaypenalty
|
||||
\def\nopagebreakaftereq{
|
||||
\prevpostdisplaypenalty=\postdisplaypenalty
|
||||
\postdisplaypenalty=10000
|
||||
}
|
||||
%% back to previous value
|
||||
\def\restorepagebreakaftereq{
|
||||
\postdisplaypenalty=\prevpostdisplaypenalty
|
||||
}
|
||||
|
||||
|
||||
%% end
|
||||
\iantheo@defaultoptions
|
||||
\endinput
|
||||
|
|
@ -0,0 +1,19 @@
|
|||
%%
|
||||
%% largearray package:
|
||||
%% Array spanning the entire line
|
||||
%%
|
||||
|
||||
%% TeX format
|
||||
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
|
||||
|
||||
%% package name
|
||||
\ProvidesPackage{largearray}[2016/11/10]
|
||||
|
||||
\RequirePackage{array}
|
||||
|
||||
%% array spanning the entire line
|
||||
\newlength\largearray@width
|
||||
\setlength\largearray@width\textwidth
|
||||
\addtolength\largearray@width{-10pt}
|
||||
\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\largearray@width}}\\[-.5cm]}
|
||||
\def\endlargearray{\end{array}}
|
||||
|
|
@ -0,0 +1,107 @@
|
|||
%%
|
||||
%% Points package:
|
||||
%% \point commands
|
||||
%%
|
||||
|
||||
%% TeX format
|
||||
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
|
||||
|
||||
%% package name
|
||||
\ProvidesPackage{point}[2017/06/13]
|
||||
|
||||
%% options
|
||||
\newif\ifresetatsection
|
||||
\DeclareOption{reset_at_section}{\resetatsectiontrue}
|
||||
\DeclareOption{no_reset_at_section}{\resetatsectionfalse}
|
||||
\newif\ifresetatsubsection
|
||||
\DeclareOption{reset_at_subsection}{\resetatsubsectiontrue}
|
||||
\DeclareOption{no_reset_at_subsection}{\resetatsubsectionfalse}
|
||||
\newif\ifresetattheo
|
||||
\DeclareOption{reset_at_theo}{\resetattheotrue}
|
||||
\DeclareOption{no_reset_at_theo}{\resetattheofalse}
|
||||
|
||||
\def\point@defaultoptions{
|
||||
\ExecuteOptions{reset_at_section, reset_at_subsection, no_reset_at_theo}
|
||||
\ProcessOptions
|
||||
|
||||
%% reset at every new section
|
||||
\ifresetatsection
|
||||
\let\point@oldsection\section
|
||||
\gdef\section{\resetpointcounter\point@oldsection}
|
||||
\fi
|
||||
%% reset at every new subsection
|
||||
\ifresetatsubsection
|
||||
\let\point@oldsubsection\subsection
|
||||
\gdef\subsection{\resetpointcounter\point@oldsubsection}
|
||||
\fi
|
||||
|
||||
%% reset at every new theorem
|
||||
\ifresetattheo
|
||||
\ifdefined\iantheotrue
|
||||
\let\point@oldtheo\theo
|
||||
\gdef\theo{\resetpointcounter\point@oldtheo}
|
||||
\fi
|
||||
\fi
|
||||
}
|
||||
|
||||
|
||||
%% point
|
||||
% counter
|
||||
\newcounter{pointcount}
|
||||
\def\point{
|
||||
\stepcounter{pointcount}
|
||||
\setcounter{subpointcount}{0}
|
||||
% hyperref anchor (only if the class is 'ian')
|
||||
\ifdefined\ifianclass
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\arabic{pointcount}}
|
||||
\fi
|
||||
% header
|
||||
\indent{\bf \arabic{pointcount}\ - }
|
||||
}
|
||||
|
||||
%% subpoint
|
||||
% counter
|
||||
\newcounter{subpointcount}
|
||||
\def\subpoint{
|
||||
\stepcounter{subpointcount}
|
||||
\setcounter{subsubpointcount}0
|
||||
% hyperref anchor (only if the class is 'ian')
|
||||
\ifdefined\ifianclass
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\arabic{pointcount}-\arabic{subpointcount}}
|
||||
\fi
|
||||
% header
|
||||
\indent\hskip.5cm{\bf \arabic{pointcount}-\arabic{subpointcount}\ - }
|
||||
}
|
||||
|
||||
%% subsubpoint
|
||||
% counter
|
||||
\newcounter{subsubpointcount}
|
||||
\def\subsubpoint{
|
||||
\stepcounter{subsubpointcount}
|
||||
% hyperref anchor (only if the class is 'ian')
|
||||
\ifdefined\ifianclass
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\arabic{pointcount}-\arabic{subpointcount}-\arabic{subsubpointcount}}
|
||||
\fi
|
||||
\indent\hskip1cm{\bf
|
||||
\arabic{pointcount}-\arabic{subpointcount}-\arabic{subsubpointcount}\ - }
|
||||
}
|
||||
|
||||
|
||||
%% reset point counters
|
||||
\def\resetpointcounter{
|
||||
\setcounter{pointcount}{0}
|
||||
\setcounter{subpointcount}{0}
|
||||
\setcounter{subsubpointcount}{0}
|
||||
}
|
||||
|
||||
|
||||
|
||||
%% end
|
||||
\point@defaultoptions
|
||||
\endinput
|
||||
Loading…
Reference in New Issue