Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for closed locally c…
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13 results for “tripods”
Counting tripods on a flat torus using lattice point counting.
problem Counting finite BPS webs in flat torus geometry.
method Lattice point counting techniques in .
result Asymptotic counting result for tripods on the torus.
We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an -dimensional plane at …
Tripod spiders' energy control analyzed for Hooke and Coulomb potentials.
problem Control of tripod spiders' energy configurations.
method Morse theory for Hooke potential, stationary charges for Coulomb energy.
result For positive charges in a regular triangle, the domain of robust control is non-void.
Reduces conjecture for Artin groups to simpler cases.
problem Proving for Artin groups with specific spherical parabolics.
method Reduces to simpler cases, uses injective metric spaces, combinatorial convexity, and Bestvina-type inequalities.
result Deduces conjecture for specific Artin groups.
Distribution-free binary classification: prediction sets, confidence intervals and calibrationstat.ML
Study three types of uncertainty quantification for binary classification without distributional assumptions.
problem Uncertainty quantification for binary classification in a distribution-free setting.
method Established theorems connecting calibration, confidence intervals, and prediction sets for score-based classifiers.
result Distribution-free calibration is only possible using scoring functions that partition feature space into countably many sets.
Study polynomial cubic differentials on Riemann surfaces using spectral networks.
problem Characterize polynomial cubic differentials with saddle connections or critical tripods.
method Introduced spectral core, refined classical core concept, and applied Gaiotto-Moore-Neitzke's algorithm.
result Completely characterized polynomial cubic differentials up to degree 3, including wall-and-chamber structure.
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose into cells of prescribed (positive) Gaussian measure when , is to use a "simplicial cluster", obtained from the Voronoi cells of equidistant points. Moreover, we prove that…
We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice and are consistent on the multidimensional lattice . Our list includes the most prominent representatives of this class, the discrete KP equation and its S…
Let be a countable group that splits as a free product of groups of the form , where is a finitely generated free group. We identify the closure of the outer space for the axes topology with the space of projective minimal, \emph{very small} …
Study of automorphisms and splittings of special groups, showing infinite groups under certain conditions.
problem Understanding the structure and automorphisms of special groups .
method Constructing and analyzing non-small, stable -actions on -trees.
result Conditions for the existence of infinite-order automorphisms and splittings.
New framework for analyzing line fields on surfaces, proving stability under specific conditions.
problem Understanding structural stability and generic transitions of line fields on surfaces.
method Developed a new topological framework and introduced representations of complete invariants for line fields and their transitions.
result Line fields with 1-prong and 3-prong singularities are generic under an incompressibility condition.
Study compares Cox model and RSF for predicting patient survival, finding RSF superior in certain scenarios.
problem Comparing predictive accuracy of Cox proportional hazards model and Random Survival Forest for patient-specific survival probabilities.
method Conducted a comprehensive comparison study using simulation scenarios and real-world datasets.
result RSF outperforms Cox model in nonproportional hazards settings and with treatment-covariate interactions.