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 its 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 is revealed after you answer it, whether you were right or wrong. Tap an unlocked card to review it.
Study the graph and tap Yes or No. Hint opens a unit-disc control: drag its center and adjust the radius to see a lifted circle and its center on the surface. No mean value or mean-height marker is shown during the question.
Ordinary mode asks whether u is subharmonic. In the −log mode, the graph shows a positive u, but your answer concerns v = −log(u). The graph still shows u after you answer.
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!
The answer colors the graph according to the complex Hessian of the function being tested (u or v). Red is negative, white is zero and blue is positive. Stronger colors indicate larger absolute values within that question.
For non-smooth functions we color a numerical circle-mean difference J(a,r) = average over the circle − value at its center. Its color depends on the chosen radius: r(a) = min(0.08, 0.45(1−|a|)). It is NOT the complex Hessian, and the coloring is only a numerical illustration, not a proof. Near the boundary the radius becomes smaller.
All answers are based on mathematical reasoning, not numerical guesses. Advanced graphs such as Green functions or lacunary series are drawn using finite approximations.
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The vertical scale is adjusted for visibility. Rotate to inspect the graph.
Keyboard: S = Yes / 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 your saved progress. You can revisit them in the gallery.