Mathematica

Plot Tricks

Unified style function

(*for many lines, e.g. stochastic trajectories*)
Styles = {
   PlotStyle -> {Thickness[0.005], Opacity[0.4]},
   LabelStyle -> 
    Directive[FontFamily -> "Helvetica", Black, FontSize -> 15],
   FrameStyle -> Directive[Black, FontSize -> 15, Thickness[.002]]
   };

Evaluate@Styles

Styles = {PlotTheme -> {"Scientific", "DashedLines"},
   PlotTheme -> {"Scientific", "DashedLines"},
   LabelStyle -> 
    Directive[FontFamily -> "Helvetica", Black, FontSize -> 15], 
   FrameStyle -> Directive[Black, FontSize -> 15, Thickness[.002]]};

Theme with linestyle

PlotTheme -> {"Scientific", "DashedLines"}

Vertical lines in a plot

Epilog -> { Line[ {{x0,y0}, {x1,y1}} ] }

ListPlot: dots -> line

Joined -> True

Legend size

LabelStyle -> 
 Directive[FontFamily -> "Helvetica", Black, FontSize -> 15]

Colorbar Legend

PlotLegends -> BarLegend[Automatic, LegendLabel -> "z"]

Frame ticks and labels size

FrameStyle -> Directive[Black, FontSize -> 15, Thick]

Unify and resize fig

Rasterize[,RasterSize -> 15]

Latex input

<< MaTeX`
MaTeX["\sin{x}"]

Multiple figures in a single plot

Column@{
Row@{
}
}

Plot only on a certain region

Method 1: ImplicitRegion

{x0, y0} \[Element] ImplicitRegion[.7 <= x0^2 + y0^2 <= 1, {x0, y0}]

Method 2: RegionFunction

RegionFunction -> 
 Function[{x, y}, .64 < x^2 + y^2 < 1]

Stream and contour plot

With[
  {reactions = { , },
  Show[
   StreamPlot[
    Evaluate[
     {
      reactions[[1]], reactions[[2]]
      }
     ],
    {r, 0, 1}, {s, 0, 1}
    ],
   ContourPlot[
    Evaluate[
     {
      0 == reactions[[1]],
      0 == reactions[[2]]
      }
     ],
    {r, 0, 1}, {s, 0, 1}
    ]
   ]
  ],

Animation

plots = Table[
   Plot[
    ], {t, 0, 30, .2}];
Export["DecayingWave.gif",plots,"AnimationRepetitions"-> \
\[Infinity]]
ListAnimate[plots]

### 3D plot without light
```Mathematica
PlotStyle -> Directive[Black, Glow[RGBColor["#1f78b4"]], Opacity[.6]]
```

### 3D plot without clipping
```Mathematica
ClippingStyle -> None
```

Functional programming

Basic use of function

f[x_,y_] := 

Map and apply

f[x_,y_] :=
f@@{x,y}

f[x_] :=
f/@x

For loop

For[With{},..,..]

Math and Models

Uniform unicycle Laplacian operator

cycle[n_] := 
 Table[If[i == j, -kf - kb, 
   If[i == Mod[(j + 1), n] || i == j + 1, kb, 
    If[Mod[i + 1, n] == j || i + 1 == j, kf, 0]]], {i, n}, {j, n}]

Basin of attraction

tmax = 10;
tol = 0.01;
(*Solution to ODE that maps t to {x[t],y[t]}*)
sol[x0_?NumericQ, y0_?NumericQ, Req_, \[Eta]_] := 
  First@NDSolve[
     Evaluate[{x[0] == x0, y[0] == y0, 
       x'[t] == \[Eta] x[t]^2 ((1 - y[t] - x[t]) - Req x[t]) - x[t], 
       y'[t] == \[Eta] y[t]^2 ((1 - y[t] - x[t]) - Req y[t]) - 
         y[t]}], {x, y}, {t, 0, tmax}] /. 
   HoldPattern[{x -> xi_, y -> yi_}] :> Function[{t}, {xi[t], yi[t]}];
BasinAssign := Function[{f}, Norm[f[tmax]]];
Manipulate[
 Show[DensityPlot[
   BasinAssign[sol[x0, y0, 0.5, 10^log\[Eta]]], {x0, 0, 1}, {y0, 0, 
    1}, PlotPoints -> 20, PlotLegends -> Automatic, 
   PlotRange -> Full], 
  StreamPlot[{\[Eta] l^2 ((1 - d - l) - 0.5 l) - 
      l, \[Eta] d^2 ((1 - d - l) - 0.5 d) - d} /. \[Eta] -> 10^
     log\[Eta], {l, 0, 1}, {d, 0, 1}, FrameLabel -> {"[B]", "[C]"}], 
  ContourPlot[
   Evaluate[{0 == \[Eta] l^2 ((1 - d - l) - 0.5 l) - l, 
      0 == \[Eta] d^2 ((1 - d - l) - 0.5 d) - d} /. \[Eta] -> 10^
      log\[Eta]], {l, 0, 1}, {d, 0, 1}, PlotPoints -> 100, 
   MaxRecursion -> 2]], {{log\[Eta], 1}, .7, 2, 
  Appearance ->  "Labeled"}, {{Req, 0.5}, 0, 1, 
  Appearance ->  "Labeled"}]

General tricks

Rule and RuleDelayed

The “rule” replacement replace the variables after evaluated, however, the “ruledelayed” replace the variables and then the evaluation starts.

In[3]:= {x, x, x} /. x :> RandomReal[]
{x, x, x} /. x -> RandomReal[]

Out[3]= {0.424537, 0.500212, 0.799267}

Out[4]= {0.541989, 0.541989, 0.541989}

Joint string and variables

dTlist = Table[StringForm["\[CapitalDelta]T=``", i ], {i, 1, 3}];