HOLOMORPHONAUTICA · MINI GAME
Subharmonic?
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A graph of an upper semicontinuous function u on the unit disc Δ := {z ∈ ℂ | |z| < 1} is shown. Is it subharmonic? Tap Hint to explore circles on the graph...!?
解答後も観察を続けられます。予想と数学的な理由を見比べてみてください。Keep exploring after answering and compare your prediction with the mathematics.
難易度は、グラフを見て判定する難しさの目安です。数式や理論の高度さとは異なります。Levels indicate the difficulty of judging the graph, rather than the sophistication of its formula or theory.
60 seconds; wrong answers cost 5 seconds. Answers advance automatically, with no explanations. No repeats until the pool is exhausted.
A function becomes visible after you answer it, correctly or not. Tap a revealed card to revisit the graph and its explanation.
Study the graph and choose Yes or No. Hint opens a unit disc: drag the center and adjust the radius to see the lifted circle and its center on the surface. Neither mean values nor mean-height markers are displayed. You can continue exploring after answering, alongside an optional orange counterexample circle.
Ordinary mode asks if u is subharmonic. In −log mode, you see the graph of a positive function u, but decide if v = −log(u) is subharmonic. The displayed surface is always the graph of u.
Why −log? For a positive continuous function u, consider the region “below” its graph as a domain in ℂ²: each height u(z) is the radius of a disc in the w-plane.
This Hartogs domain is pseudoconvex if and only if −log(u) is subharmonic. In other words, this mode lets you explore pseudoconvexity through the graph!
After answering, the graph is colored by the complex Hessian of the TESTED function (u or v). Red means negative, white zero, and blue positive. Color intensity uses a nonlinear contrast scale within each question, so relatively small nonzero values remain easier to see. It is not a linear scale.
For non-smooth and Green-type functions, coloring is based on a numerical circle-mean difference J(a,r) = circle average − center value. We use r(a) = min(0.08, 0.45(1−|a|)). The radius decreases near the boundary to keep circles inside Δ. Here w denotes the function being tested (u or −log u). This is NOT the complex Hessian, and the colors are only numerical evidence, not a proof. Numerically spurious negative values are suppressed for mathematically certified subharmonic problems. Nonlinear contrast makes small signs more noticeable.
Correct answers come from mathematical arguments, not the numerical colors. Advanced graphs such as Green functions and lacunary series are displayed using finite approximations.
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The vertical scale is adjusted for visibility. Rotate to inspect the graph.
Keyboard: S = Yes (subharmonic) / N = No
The surface is only slightly transparent while circles are visible. Rotate to inspect.
Orange: counterexample circle · Yellow: its center
The orange circle violates the mean-value inequality for −log(u); both circles are drawn on the graph of u.
Tap or drag the disc to move the center a.
The circle stays inside Δ. Rotate the 3D graph freely.
Only submitted answers count toward saved progress. Revisit answered questions in the gallery.