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"content": "Investigate the existence and regularity of a minimal surface spanning a prescribed closed curve in a Riemannian manifold whose sectional curvature varies between positive and negative values, and determine conditions under which the surface is uniquely determined by its boundary data?",
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"content": "Consider a high‑dimensional elliptic PDE with random coefficients modeled by a Karhunen–Loève expansion; construct a sparse polynomial chaos surrogate and quantify the error in the Sobolev norm as a function of truncation level?",
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"content": "<think>\nThis is a comp... | {
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"content": "Consider a sequence of finite, regular graphs whose adjacency matrices converge in spectral distribution to a limiting measure; formulate a condition on the growth of girth ensuring the limiting spectral measure has a nontrivial absolutely continuous component?",
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"c... | {
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"content": "Given a compact Riemannian manifold with boundary, formulate the inverse boundary value problem of determining the metric from the Dirichlet-to-Neumann map for the Laplace–Beltrami operator, and discuss uniqueness under curvature constraints?",
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"content": "<think>\n... | {
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"content": "For a multi-class queueing network with state-dependent arrival rates and service disciplines governed by a stochastic differential game, derive the Hamilton‑Jacobi‑Bellman equations that characterize the optimal control policy minimizing a long‑run average cost subject to stability constraints?",... | {
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"content": "Consider a self‑similar fractal generated by an iterated function system of four affine maps with contraction ratios r1=0.4, r2=0.3, r3=0.2, r4=0.1. Derive the exact Hausdorff dimension and discuss its dependence on the open set condition?",
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"content": "<think>\nThi... | {
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[
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"content": "Given a compact Riemannian manifold with boundary, formulate an inverse problem to recover the metric from the Dirichlet-to-Neumann map for the Laplace–Beltrami operator, and discuss uniqueness conditions involving curvature bounds and boundary convexity?",
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"content... | {
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"content": "Consider a self-similar fractal generated by iterating a non-linear conformal map on the complex plane with a scaling factor that is an algebraic irrational number; compute its Hausdorff dimension and describe the spectrum of its Laplacian operator?",
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"content": "<t... | {
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"content": "In a stochastic differential game with continuous‑time controls, where the state dynamics follow a controlled Itô diffusion on a compact manifold and the payoff is a discounted integral of a non‑convex cost, establish the existence of a value and characterize optimal feedback strategies via a visc... | {
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"content": "In a high-dimensional linear data assimilation system, how does the choice of observation operator covariance scaling with dimension affect the stability of the Kalman filter error covariance under random Gaussian observation noise?",
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"content": "<think>\nThe user i... | {
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