$Assumptions={n>0,d>1,m>0,h>0,Σ>0,t>0,β>0,γ>0,ν>0,ℳ>0,H>0,𝒮>0,α>0,δ>0,X>0};

f=Σ (n^((d-1)/d))-h m n;

sol=Solve[D[f,n]==0,n]

ff=f/.sol[[1]]//FullSimplify

ff=H m(((H m))^(-d))Σ(((Σ))^(d-1))

ff=ff/.{m->(t^β)ℳ,Σ->(t^μ)𝒮}/.μ->{-ν+γ+2β}//FullSimplify

FullSimplify[ff//.{ν->(2-α)/d,γ->2-α-2β,δ->(2-α)/β-1,H->X (t^(β δ))}]

Plot[Exp[-1/x],{x,0,1},PlotStyle->{Thick,Black},Frame->True,LabelStyle->{FontFamily->"CMS Concrete",Black}]

Plot[ Exp[-1/x],{x,0,1},PlotStyle->{Thick,Black},Frame->True,LabelStyle->{FontFamily->"CMS Concrete",Black}]

sol=Solve[D[t (x^2)+u (x^4)-h x,x]==0,x];

Plot[Evaluate[Flatten[{Re@#}&@x/.sol/.{u->1,t->Cos[θ],h->Sin[θ]}]],{θ,0,2π},Axes->False]

Plot[Evaluate[Flatten[{Arg@#}&@t (x^2)+u (x^4)-h x/.sol/.{u->1,t->Cos[θ],h->Sin[θ]}]],{θ,0,2π},Axes->False]

Plot[Evaluate[Flatten[{#}&@D[t (x^2)+u (x^4)-h x,{x,2}]/.sol/.{u->1,t->Cos[θ],h->Sin[θ]}]],{θ,0,2π}]

p1=Plot[Piecewise[{{0,t<0},{I,t>0}}],{t,-2,2},Frame->True,FrameLabel->{t, H},LabelStyle->{Black,FontFamily->"CMU Concrete"},PlotStyle->{Black,Thickness[0.03]},AspectRatio->1,ImageSize->398.3386 .4,Epilog->{red,PointSize[0.07],Point[{0,0}]}]

Export["~/doc/research/sethna/pres/aps_mm_2017/figs/fig3.pdf",p1]

p2=Plot[{Piecewise[{{((Abs[t])^(1/8)),t<0},{I,t>0}}],Piecewise[{{-((Abs[t])^(1/8)),t<0},{I,t>0}}]},{t,-2,2},Frame->True,FrameLabel->{t, M},LabelStyle->{Black,FontFamily->"CMU Concrete"},PlotStyle->{{Black,Thickness[0.03]},{Black,Thickness[0.03]}},AspectRatio->1,ImageSize->398.3386 .4,Epilog->{red,PointSize[0.07],Point[{0,0}]},PlotPoints->60000,Filling->Axis,FillingStyle->grey]

Export["~/doc/research/sethna/pres/aps_mm_2017/figs/fig4.pdf",p2]

p3=Plot[Piecewise[{{0,t<0},{I,t>0}}],{t,-2,2},Frame->True,FrameLabel->{t, H},LabelStyle->{Black,FontFamily->"CMU Concrete"},PlotStyle->{Black,Thickness[0.03]},AspectRatio->1,ImageSize->398.3386 .4,Epilog->{{red,PointSize[0.07],Point[{0,0}]},{dgrey,PointSize[0.07],Point[{-1,0.5}]}}]

Export["~/doc/research/sethna/pres/aps_mm_2017/figs/fig11.pdf",p3]

Solve[D[-1 (M^2)+(M^4)+.2M,M]==0,M]

-1 (M^2)+(M^4)+.2M/.%

p4=Plot[-1 (M^2)+(M^4)-.5M,{M,-1.3,1.3},Frame->True,FrameLabel->{M, F},LabelStyle->{Black,FontFamily->"CMU Concrete"},PlotStyle->{Black,Thick},AspectRatio->.36/.6,ImageSize->398.3386 .6,Epilog->{{dgrey,PointSize[0.04],Point[{0.809017,-0.630636}]}}]

Export["~/doc/research/sethna/pres/aps_mm_2017/figs/fig12.pdf",p4]

p5=Plot[Piecewise[{{0,t<0},{I,t>0}}],{t,-2,2},Frame->True,FrameLabel->{t, H},LabelStyle->{Black,FontFamily->"CMU Concrete"},PlotStyle->{Black,Thickness[0.03]},AspectRatio->1,ImageSize->398.3386 .4,Epilog->{{red,PointSize[0.07],Point[{0,0}]},{dgrey,PointSize[0.07],Point[{-1,0.5}]},{Dashed,dgrey,Thickness[0.02],Arrowheads[.1],Arrow[{{-1,0.5},{-1,-0.2}}]}}]

Export["~/doc/research/sethna/pres/aps_mm_2017/figs/fig13.pdf",p5]

p6=Plot[-1 (M^2)+(M^4)+.2M,{M,-1.3,1.3},Frame->True,FrameLabel->{M, F},LabelStyle->{Black,FontFamily->"CMU Concrete"},PlotStyle->{Black,Thick},AspectRatio->.36/.6,ImageSize->398.3386 .6,Epilog->{{dgrey,PointSize[0.04],Point[{0.650487994155197`,-0.11399411614674623`}]}}]

Export["~/doc/research/sethna/pres/aps_mm_2017/figs/fig14.pdf",p6]

398 .4

red=RGBColor[204/255,0,0];

grey=RGBColor[243/255,243/255,243/255];

dgrey=RGBColor[217/255,217/255,217/255];