Study on volume growth of metric balls on unbounded hypersurfaces in complex space.
problem Understanding volume growth rates of metric balls on unbounded model hypersurfaces.
method Analyzing the Carnot-Carathéodory metric on unbounded model hypersurfaces in C 2 \mathbb{C}^2 C 2 . result Metric balls of radius δ ≫ 1 δ\gg 1 δ ≫ 1 have volume on the order of δ 3 δ^3 δ 3 or δ 4 δ^4 δ 4 depending on the hypersurface structure. The paper proves a Willmore-type inequality for unbounded convex sets.
problem Proving a Willmore-type inequality for unbounded convex sets.
method Analytical proof involving hypersurfaces, contact angle conditions, and asymptotic volume ratio.
result The Willmore-type inequality holds for unbounded closed convex sets with certain conditions.
Study shows non-compactness of minimal hypersurfaces on certain manifolds.
problem Analyzing non-compactness of minimal hypersurfaces on specific manifolds.
method Using min-max method and properties of bumpy metrics with positive Ricci curvature.
result Found that the space of minimal hypersurfaces is non-compact under certain conditions.
Researchers solved a Calabi-Yau conjecture for specific 3D hypersurfaces in 4D space.
problem Whether complete minimal hypersurfaces must be unbounded or proper.
method Analyzing bounded second fundamental form and finite second Betti number.
result Resolved the Calabi-Yau conjecture for specific 3D hypersurfaces in 4D space.
We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisation…
We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of R n \R^n R n which are homeomorphic to R n − 1 \R^{n-1} R n − 1 . In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonic…
New findings on bendable hypersurfaces in higher dimensions.
problem Characterizing infinitesimal bendable complete hypersurfaces.
method Analyzing non-flat infinitesimal bendable hypersurfaces in Euclidean spaces.
result Infinitesimal bendable complete hypersurfaces are only along ruled strips.
We characterize embedded $\C^1$ hypersurfaces of R n \R^n R n as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m < 3 / 2 m<3/2 m < 3/2 . It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform r…
Maximal hypersurfaces in spacetimes with translational symmetry are characterized and their properties studied.
problem Characterizing maximal hypersurfaces in spacetimes with translational symmetry.
method Analyzing quotient spacetimes and using properties of maximal hypersurfaces.
result Complete noncompact maximal hypersurfaces in quotient spacetimes are either cylinders or conformal to the Euclidean plane.
The study finds large area minimal surfaces in certain manifolds.
problem Existence of minimal surfaces with large area in manifolds.
method Analyzes bumpy closed Riemannian n-manifolds for n between 3 and 7.
result Proves the existence of a sequence of minimal surfaces with unbounded area.
Static potentials on flat manifolds imply zero boundary values or non-compact area minimizers.
problem Characterizing static potentials on asymptotically flat manifolds.
method Analyzing the properties of static potentials and area minimizing hypersurfaces.
result Asymptotically flat manifolds with static potentials either have zero boundary values or contain non-compact area minimizers.
The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.
problem Properness of complete minimal hypersurfaces in higher dimensions.
method Chord-arc estimates and gluing techniques.
result Construction of a complete, improperly embedded minimal hypersurface in R n + 1 \mathbb{R}^{n+1} R n + 1 for every n ≥ 3 n\ge 3 n ≥ 3 . New mathematical surfaces without boundaries found.
problem Existence of nonlocal free boundary minimal surfaces.
method Fractional perimeter critical points with invariant boundary.
result Existence of nonlocal free boundary minimal surfaces without boundaries.
Smooth convergence to an enveloping cylinder proved for mean curvature flow of complete graphical hypersurfaces.
problem Proving smooth convergence of mean curvature flow to an enveloping cylinder.
method Analyzing mean curvature flow of complete graphical hypersurfaces over domains Ω t Ω_{t} Ω t , proving convergence under certain circumstances. result Smooth convergence of M t − h e n + 1 M_{t}-h\,e_{n+1} M t − h e n + 1 to the enveloping cylinder under specific conditions. The paper proves the existence of certain minimal surfaces in specific manifolds.
problem Existence of minimal surfaces with specific properties in Riemannian manifolds.
method Proof of existence using Morse index, bumpy metrics, and cyclic coverings.
result Connected, immersed Morse index one, closed minimal hypersurfaces with unbounded volumes.
We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reve…
Study compares nodal sets of solutions to the Allen-Cahn equation.
problem Comparing nodal sets of solutions to the Allen-Cahn equation with conical asymptotics.
method Maximum principle for linearized operator on unbounded domains.
result Positive phase uniquely determines the solution and enforces global ordering.
Two new algorithms improve performance in adversarial bandits with unbounded losses.
problem Adversarial Multi-Armed Bandits with unbounded losses.
method Developed UMAB-NN and UMAB-G for non-negative and general unbounded losses respectively.
result UMAB-NN achieves the first adaptive and scale-free regret bound for non-negative unbounded losses.
Constructs unbounded Kasparov product for sphere embeddings into Euclidean space.
problem Embedding spheres into Euclidean space and their associated Kasparov cycles.
method Constructs unbounded Kasparov cycles, equips with connections, computes unbounded Kasparov product with Dirac operator, identifies index cycles.
result Spectral triple for algebra C ( S n ) C(\mathbb S^n) C ( S n ) differs from round sphere Dirac operator by index cycle. Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
Study utility maximization in financial markets with bounded and unbounded payoffs.
problem Utility maximization in financial markets with constraints and unbounded payoffs.
method Combines quadratic backward stochastic differential equations and convex duality.
result Established utility indifference valuation, regime switching, and consumption-investment problems in unbounded markets.
The study finds equivalent properties for CD inequalities with unbounded Laplacians.
problem Implying gradient estimates for laplace operator on graphs with unbounded Laplacians.
method Investigates equivalent properties of CD(K,∞) and CD(K,n) inequalities with unbounded Laplacians.
result Concludes equivalent properties of CD(K,∞) and CD(K,n) inequalities.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
Let W n \mathcal{W}^{n} W n be the class of C ∞ C^{\infty } C ∞ complete simply connected n − n- n − dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let % W\in \mathcal{W}^{n} and let A A A be a subset of W W W . This article aims at characterization and bu…
Paper derives Li-Yau inequality for unbounded Laplacian on graphs.
problem Deriving Li-Yau inequality for unbounded Laplacian on graphs.
method Assumption of curvature-dimension inequality C D E ′ ( n , K ) CDE'(n,K) C D E ′ ( n , K ) and derivation of Li-Yau inequality. result First results on Li-Yau inequality for unbounded Laplacian on graphs.
Golden L surface has unbounded bunching of saddle connections
problem Unbounded bunching of saddle connections on translation surfaces
method Translation surface with golden ratio
result Every positive integer K has a ball containing at least K saddle connection periods
New manifolds show Riesz transform unbounded for p > 2.
problem Understanding Riesz transform behavior on manifolds.
method Constructing Riemannian manifolds with specific properties.
result Riesz transform unbounded on L p ( M ) L^p(M) L p ( M ) for all p > 2 p > 2 p > 2 . The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.
problem Characterizing quadrics among affine hyperspheres based on section centroid collinearity.
method Extending Meyer and Reisner's theorem to unbounded convex sets and identifying additional assumptions.
result Ellipsoids, paraboloids, and one sheet of a two-sheeted hyperboloid are the only quadrics satisfying the centroid collinearity condition.
New inequalities for unbounded functions improve denoising score matching.
problem Statistical error bounds for denoising score matching with unbounded objective functions.
method Derive new concentration inequalities using McDiarmid's inequality and Rademacher complexity bounds.
result Improved statistical error bounds for denoising score matching.
It is shown that the compactly supported identity component of the diffeomorphism group of the 2-dimensional punctured torus T p 2 \mathbb T^2_p T p 2 is an unbounded group. It follows that the fragmentation norm of T p 2 \mathbb T^2_p T p 2 is unbounded.
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.
problem Understanding ancient solutions on graphs with unbounded Laplacians.
method Generalizing Colding and Minicozzi's theorem and Hua's result to graphs with unbounded Laplacians.
result The dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth.
Study on surfaces with unbounded mean curvature and singularities.
problem Understanding surfaces with unbounded mean curvature and singularities.
method Analyzing surfaces given by Kenmotsu-type formula.
result Characterization of singularities in surfaces with unbounded mean curvature.
The study finds conditions for certain graphs to be slices.
problem Conditions for entire constant mean curvature Killing graphs to be slices.
method Analyzing conditions for entire unbounded constant mean curvature Killing graphs.
result Conditions for entire constant mean curvature Killing graphs to be slices.
Study on minimizing perimeter in unbounded convex bodies without boundary regularity.
problem Minimizing perimeter under volume constraint in unbounded convex bodies.
method Introducing uniform geometry, asymptotic cylinders, approximation, and approximation argument.
result Existence of isoperimetric regions in generalized sense and strict concavity of isoperimetric profile.
Paper tackles online control of linear systems with unbounded noise.
problem Online control of linear systems under unbounded noise with unknown convex cost functions.
method Developed an algorithm achieving i l d e O ( T ) ilde{O}(\sqrt{T}) i l d e O ( T ) high-probability regret under unbounded noise, and established O ( m p o l y ( log T ) ) O({
m poly} (\log T)) O ( m p o l y ( log T )) regret bound for strongly convex costs and sub-Gaussian noise. result Achieved i l d e O ( T ) ilde{O}(\sqrt{T}) i l d e O ( T ) high-probability regret under unbounded noise, and O ( m p o l y ( log T ) ) O({
m poly} (\log T)) O ( m p o l y ( log T )) regret bound for specific noise and cost conditions. Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
Study solves wealth maximization problem with unbounded mean and volatility.
problem Maximizing terminal wealth with unbounded mean and volatility under Knightian uncertainty.
method Solves utility maximization problem explicitly with Ornstein-Uhlenbeck and GARCH(1) processes.
result First work on unbounded mean and volatility with Knightian uncertainty and nondominated priors.
We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.
New approach finds solutions to games with unbounded controls.
problem Existence of equilibrium in mean-field games with unbounded controls.
method Weak formulation and new existence/stability results for quadratic-growth generalized McKean-Vlasov BSDEs.
result Existence of equilibrium result for non-Markovian mean-field games with unbounded control space.
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
New aggregation strategy handles unbounded losses with regret bounds.
problem Online optimization with unbounded loss functions.
method Follow The Regularized Leader (FTRL) with φ-divergence.
result Worst regret bound for unbounded losses with alternative divergences.
Gradient estimates for unbounded graph Laplacians under Bakry-Emery curvature.
problem Gradient estimates for unbounded graph Laplacians.
method Proving gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians with ellipticity assumption.
result Gradient estimates and applications to completeness and finiteness of stochastically complete graphs.
Improved sampling from Gaussian distributions with privacy constraints.
problem Sampling from unbounded Gaussian distributions with differential privacy.
method First $\widetilde{\mathcal{O}}\left(d
ight)$ -sample algorithm for unbounded Gaussians under $\left(\varepsilon, δ
ight)$ -differential privacy.
result A quadratic improvement over previous results, settling an open question.
New PAC-Bayes bounds for unbounded loss functions.
problem Generalization bounds for learning problems with unbounded loss functions.
method Introducing HYPE, a new notion for loss range, and deriving a novel PAC-Bayesian generalization bound.
result PAC-Bayes framework extended to unbounded loss functions.
Geometric model of unbounded sl3 laminations with tropical coordinates.
problem Modeling unbounded laminations in cluster varieties.
method Introducing tropical cluster coordinates and geometric gluing procedures.
result Established a geometric gluing procedure for unbounded sl3 laminations.
The paper extends geometric flow results to Riemannian manifolds of higher dimensions.
problem Analyzing high order geometric flows of submanifolds in Riemannian manifolds.
method Extending Mantegazza's results to Riemannian manifolds with higher dimensions and sectional curvature constraints.
result If certain conditions are met, the flow of submanifolds in a Riemannian manifold of higher dimensions will be unbounded in time.
Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
problem Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
method Assumption of conformally equivalent initial metric to a complete background metric with bounded scalar curvature and positive Yamabe invariant.
result Global existence of Yamabe flow without requiring initial curvature bounds.