"""CC0. Reproduce the exact sphere geometry and labeled Fourier model samples.""" from pathlib import Path from tempfile import TemporaryDirectory import json, os ROOT = Path(__file__).resolve().parent with TemporaryDirectory(prefix="an02-own252-mpl-") as temporary: os.environ["MPLCONFIGDIR"] = temporary import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt from matplotlib.patches import Circle import numpy as np import mpmath as mp mp.mp.dps = 70 plt.rcParams.update({ "font.family": "DejaVu Sans", "font.size": 11, "svg.hashsalt": "radial-boundary-252", "axes.spines.top": False, "axes.spines.right": False }) out = ROOT/"figures" out.mkdir(exist_ok=True) def save(fig, name): fig.savefig(out/(name+".png"), dpi=160, bbox_inches="tight", pad_inches=.15, metadata={"Software": "Open Mathematics Courses"}) fig.savefig(out/(name+".svg"), bbox_inches="tight", pad_inches=.15, metadata={"Date": None, "Creator": "Open Mathematics Courses"}) plt.close(fig) fig, axes = plt.subplots(1, 2, figsize=(13.8, 5.4), layout="constrained") center = np.array([1., -.5]) theta = np.array([.6, .8]) radius = 1.5 point = center+radius*theta for ax, sign, title in zip(axes, [1, -1], ["Kernel: singular circle a + (3/2) S¹", "Compact parametrix: reflected circle −a + (3/2) S¹"]): c = sign*center p = sign*point direction = sign*theta ax.add_patch(Circle(c, radius, fill=False, lw=2.2, edgecolor="#205a98")) ax.plot([c[0], p[0]], [c[1], p[1]], "--", color="#79848a", lw=1.5) ax.scatter([c[0]], [c[1]], color="#79848a", s=35) ax.scatter([p[0]], [p[1]], color="#b44c2e", s=45, zorder=5) ax.annotate("", xy=p+.55*direction, xytext=p, arrowprops={"arrowstyle": "->", "color": "#b44c2e", "lw": 1.7}) ax.annotate("a = (1, −1/2)" if sign==1 else "−a = (−1, 1/2)", xy=c, xytext=(c[0], c[1]-.5*sign), ha="center", fontsize=10) ax.annotate("p = (19/10, 7/10)" if sign==1 else "−p = (−19/10, −7/10)", xy=p, xytext=(1.3*sign, 1.65*sign), ha="center", fontsize=10) ax.text(*(c+.45*radius*direction+np.array([-.25,.1])), "radius 3/2", fontsize=10, rotation=53 if sign==1 else 53) ax.text(*(p+.82*direction+sign*np.array([.02,.12])), "θ" if sign==1 else "−θ", color="#b44c2e", fontsize=12) ax.axhline(0, lw=.65, color="#a1a7ad") ax.axvline(0, lw=.65, color="#a1a7ad") ax.set(xlim=(-3.2,3.2), ylim=(-2.7,2.7), xlabel="x₁", ylabel="x₂", title=title) ax.set_aspect("equal") ax.grid(alpha=.15) save(fig, "translated-sphere-and-reflected-parametrix") def g2(r): return 1j*mp.pi**2*mp.hankel2(0, r) C2 = 1j*mp.pi*mp.sqrt(2*mp.pi)*mp.exp(1j*mp.pi/4) real_samples = [] for r_float in np.geomspace(1., 64., 65): r = mp.mpf(str(r_float)) W2 = mp.exp(1j*r)*mp.sqrt(r)*g2(r)/C2 scaled = 8*r*abs(W2-1) assert scaled <= 1+mp.mpf("1e-50") real_samples.append(dict(r=str(r), dimension2_scaled_error=str(scaled), dimension5_scaled_error="1", dimension3_relative_error="0")) profile_samples = [] for R in [16, 256, 65536]: for tau_float in np.linspace(-2, 2, 81): tau = mp.mpf(str(tau_float)) r = R+1j*tau*mp.log(R) value = mp.log(abs(g2(r)))/mp.log(R) profile_samples.append(dict(R=R, tau=str(tau), model_log_profile=str(value), proved_limit=str(tau-mp.mpf(".5")))) fig, axes = plt.subplots(1, 2, figsize=(13.8, 5.3), layout="constrained") xs = [float(x["r"]) for x in real_samples] ys = [float(x["dimension2_scaled_error"]) for x in real_samples] axes[0].plot(xs, ys, color="#205a98", lw=2, label="n = 2: sampled 8r |W₂ − 1|") axes[0].axhline(1, color="#9d632c", ls="--", lw=1.6, label="proved bound 1; n = 5: r |W₅ − 1| = 1") axes[0].set(xscale="log", xlabel="Positive real frequency r", ylabel="Scaled relative error", ylim=(.69,1.035), title="Gaussian bound and exact odd-dimensional error") axes[0].text(1.25,.72,"n = 3: W₃ − 1 = 0 exactly",fontsize=10) axes[0].legend(loc="lower right", fontsize=9) axes[0].grid(alpha=.18) for R, color in zip([16, 256, 65536], ["#a16836","#547b46","#205a98"]): items = [x for x in profile_samples if x["R"]==R] axes[1].plot([float(x["tau"]) for x in items], [float(x["model_log_profile"]) for x in items], color=color, label=f"global model samples, R = {R}") taus = np.linspace(-2,2,81) axes[1].plot(taus,taus-.5,"--",color="#46354e",lw=1.7, label="proved compact limit: τ − 1/2") axes[1].set(xlabel="τ, with r = R + iτ log R", ylabel="log |G₂(r)| / log R", title="Direction θ = e₁: logarithmic profiles") axes[1].legend(loc="upper left", fontsize=9) axes[1].grid(alpha=.18) save(fig,"radial-phase-errors-and-logarithmic-profiles") geometry = dict( schema="AN02-radial-geometry252/v1", exact_sphere_geometry=dict( dimension=2, center=["1","-1/2"], radius="3/2", unit_direction=["3/5","4/5"], singular_point=["19/10","7/10"], reflected_center=["-1","1/2"], reflected_point=["-19/10","-7/10"], singular_sets_are_circles_not_filled_disks=True, higher_dimension_picture_is_section=True, proof="Proposition 10.1; exact reflection in equation (10.2)"), relative_error_definition="W_n=exp(ir)*r^((n-1)/2)*G_n/C_n", proved_bounds=dict(dimension2="8r*abs(W_2-1)<=1 for real r>0", dimension3="W_3-1=0", dimension5="W_5-1=-i/r"), real_samples=real_samples, profile_samples=profile_samples, profiles_are_global_model_samples_not_cutoff_transform_samples=True, compact_limit_proof="Theorem 8.1 and Proposition 9.1", numerical_precision_decimal_digits=70 ) (out/"geometry252.json").write_text(json.dumps(geometry,indent=2)+"\n",encoding="utf-8") print(json.dumps(dict(figures=2,real_samples=len(real_samples), profile_samples=len(profile_samples), owned_MPL_temporary_directory_removed_on_exit=True)))