BOCK optimizes Bayesian Optimization by transforming the search space to reduce boundary evaluations.
problem Bayesian Optimization struggles with boundary issues, wasting evaluations near the search space boundary.
method BOCK uses a cylindrical transformation to redirect Gaussian Process efforts away from the boundary and towards the center of the search space.
result BOCK achieves better accuracy and efficiency, scaling to high-dimensional problems and optimizing neural network layers and hyperparameters.
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
Transforming cylindrical packings into bicontinuous surfaces.
problem Understanding the early development of bicontinuous structures in plant plastids.
method Geometric modeling and computational simulations of cylinder packings.
result Specific cylinder packings with cubic symmetry transform into TPMS.
We establish the basics of the analysis of operators on coverings of manifolds with cylindrical ends with a group of deck transformations Γ. We prove the Γ-analogue of the Atiyah-Patodi-Singer formula for Dirac operators on such coverings.
In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class [g0] of the Riemannian product S1×Sn−1 : a circle of length T crossed with the (n-1)-dimension…
We develop a new method to solve complex physics equations more accurately and efficiently.
problem Challenges in solving functional differential equations due to high computational costs and inaccurate approximations.
method Combining physics-informed neural networks (PINNs) with cylindrical approximation to handle functional derivatives.
result Our method achieves typical L1 relative error orders of PINNs of ∼10−3 on two FDEs. Study cylindrical symmetric Finsler metrics with vanishing Douglas curvature.
problem Understanding Finsler metrics with specific curvature properties.
method Analyzing cylindrically symmetric Finsler metrics and solving differential equations.
result Differential equations for cylindrically symmetric Finsler metrics with vanishing Douglas curvature.
In this study, we consider the notion of similar ruled surface for timelike and spacelike ruled surfaces in Minkowski 3-space. We obtain some properties of these special surfaces in E_1^3 and we show that developable ruled surfaces in E_1^3 form a family of similar ruled surfaces if and only if the striction curves of …
Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindri…
Study gluing and deformations of special Lagrangian submanifolds.
problem Deforming special Lagrangian submanifolds in Calabi-Yau manifolds.
method Proves well-defined gluing map and local diffeomorphism for deformations.
result Gluing map defines local diffeomorphism between deformations.
Study resolves flow through cylindrical singularities, proving nonfattening.
problem Analyzing free boundary flow through cylindrical singularities.
method Foundational results for free boundary Brakke flows and classification of ancient flows.
result Proves all cylindrical singularities have a mean-convex neighborhood, leading to well-posed flow.
Study cylindrical symmetric Finsler metrics that are projectively flat.
problem Characterize Finsler metrics that are projectively flat.
method Solve the system of differential equations for cylindrical symmetric Finsler metrics.
result Provide a family of solutions for the projectively flat Finsler metrics.
This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
In an earlier paper, we proved that, under certain hypotheses, the moduli space of an asymptotically cylindrical special Lagrangian submanifold with fixed boundary of an asymptotically cylindrical Calabi-Yau 3-fold is a smooth manifold. Here we prove the analogous result for an asymptotically cylindrical special Lagran…
Constructs geodesics near intersection points of Lagrangian submanifolds.
problem Geodesics of positive Lagrangian submanifolds near intersection points.
method Cylindrical transform and C1,1 regularity proof. result First examples of C1,1 geodesics in arbitrary dimensions. Shows uniqueness of cylindrical blowups in mean curvature flow.
problem Uniqueness of cylindrical blowups in mean curvature flow in higher codimension.
method Developed new methods to prove uniqueness of cylindrical blowups.
result Implication of regularity of the singular set for the system.
Finite number of eigenvalues found in cylindrical surface.
problem Finding eigenvalues in manifolds with cylindrical ends.
method Constructing a surface with a cylindrical end attached to a hyperbolic torus.
result First example of a finite and nonzero number of embedded eigenvalues in a cylindrical manifold.
Study proves existence and uniqueness of Yang-Mills flow on cylindrical 4-manifolds.
problem Existence and uniqueness of Yang-Mills flow on specific 4-manifolds.
method Established through mathematical proofs and hypotheses on underlying data.
result Long-time existence and infinite-time convergence of the Yang-Mills flow.
Study on parabolic points and cylindrical surfaces in Euclidean 3-space.
problem Characterizing parabolic points and their geometric properties.
method Introducing contact cylindrical surfaces and analyzing their properties.
result Characterization of A-singularity through projections. We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstr…
Cylindrical contact homology studied for a specific class of 3-manifolds.
problem Calculating the cylindrical contact homology for a specific class of 3-manifolds.
method Using cylindrical contact homology, a relatively simple version of symplectic field theory.
result Complete description of cylindrical contact homology for a specific class of 3-manifolds.
Proves uniqueness of cylindrical tangent flows in mean curvature flow.
problem Proving uniqueness of cylindrical singularity models in mean curvature flow.
method Inspired by Székelyhidi's approach, uses a different method to prove uniqueness.
result Proves uniqueness of cylindrical tangent flows.
Extends ASD connection existence to 4-manifolds with cylindrical ends.
problem Existence of ASD connections on 4-manifolds with specific ends.
method Extends gluing theorems to cylindrical ends, using mASD connections.
result Establishes existence of ASD connections on 4-manifolds with cylindrical ends.
Researchers glue and deform Calabi-Yau 3-folds, proving local diffeomorphisms.
problem Deforming and gluing asymptotically cylindrical Calabi-Yau 3-folds.
method Developed new proofs for gluing and deformation, extending results to the asymptotically cylindrical case.
result Proved the gluing map is a local diffeomorphism, connecting moduli spaces of Calabi-Yau structures.
We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any …
Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G_2 structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically…
New special Lagrangian submanifolds with cylindrical tangent cones are constructed.
problem Constructing special Lagrangian submanifolds with specific geometric properties.
method Constructing examples in a neighborhood of the origin with an isolated singularity and cylindrical tangent cone.
result Existence of special Lagrangian submanifolds with cylindrical tangent cones, including examples with transverse planes.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
Develops a new method for financial term structure modeling.
problem Analyzing financial term structures with discontinuities.
method Cylindrical stochastic integration approach.
result Establishes a Heath-Jarrow-Morton framework.
Cylindrical contact homology computed for links of simple singularities.
problem Computing cylindrical contact homology for links of simple singularities.
method Perturbing degenerate contact form on S3/G with an invariant Morse function, achieving nondegeneracy up to an action threshold. Recovers cylindrical contact homology via direct limit of action filtered homology groups. result Ranks of cylindrical contact homology groups are given in terms of ∣extConj(G)∣, demonstrating a form of the McKay correspondence. Only cylindrical surfaces in Euclidean space have constant Gaussian curvature.
problem Surfaces with constant Gaussian curvature in Euclidean space.
method Analyzing surfaces constructed by the sum of two space curves.
result Only cylindrical surfaces have constant Gaussian curvature.
Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.
problem Constructing Calderon projector in a specific geometric setting.
method Using resolvent estimates and b-calculus framework.
result Extend a result on adiabatic limits of Cauchy data spaces.
Resolves conjecture on cylindrical mean curvature flows in all dimensions.
problem Mean Convex Neighborhood Conjecture for cylindrical singularities.
method Complete classification of ancient, asymptotically cylindrical flows; refined asymptotic analysis; leading mode condition; induction over thresholds.
result Establishes mean-convex neighborhood for cylindrical singularities; provides local models and canonical families.
In this paper we prove that any asymptotically cylindrical gradient shrinking Ricci soliton is isometric to a cylinder.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
We construct solutions of the vacuum vector constraint equations on manifolds with cylindrical ends.
Researchers found cylindrical steady gradient solitons in 3D.
problem Finding steady gradient solitons in 3D with specific symmetries.
method Constructed a two-parameter family of solitons with SO(2)imesR symmetry. result Found a family of solitons with asymptotic power-law or exponential decay.
Estimates for hypersurfaces in CROSSes show cylindrical profiles near singularities.
problem Analyzing the mean curvature flow of hypersurfaces in complex or quaternionic projective spaces.
method Proving apriori estimates on principal curvatures under a pinching assumption.
result The asymptotic profile near a singularity is either strictly convex or cylindrical.
We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of…
The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
problem Understanding the topology and symmetry of cylindrical handlebody-knots of genus two.
method Analysis of Thurston's hyperbolization theorem and investigation of unknotting annuli.
result The symmetry group is trivial if the unknotting annulus is unique and of type 2. Study shows generic surfaces avoid complex flow patterns.
problem Understanding flow patterns of surfaces in 3D space.
method Analyzes mean curvature flow of closed surfaces in R3. result Non-cylindrical self-shrinkers cannot arise generically.
Study proves uniqueness and rigidity of cylindrical self-shrinkers using Łojasiewicz inequalities.
problem Uniqueness and rigidity of cylindrical self-shrinkers in mean curvature flow.
method Direct perturbative analysis of the shrinker mean curvature and Łojasiewicz inequalities.
result Uniqueness and rigidity of cylindrical self-shrinkers, including round cylinders and cylinders over Abresch-Langer curves.
Study on cylindrical handlebody-knots with symmetry and rigidity properties.
problem Characterizing symmetry groups of cylindrical handlebody-knots of genus two.
method Classification of essential annuli and analysis of symmetry groups based on Koda-Ozawa theorem.
result Most exteriors of genus two cylindrical handlebody-knots contain no essential disks or tori, and type 3-3 annuli are often unique up to isotopy. The paper analyzes high codimension mean curvature flow and proves cylindrical estimates near singularities.
problem Analyzing singularity formation in high codimension mean curvature flow.
method Directly proving a pointwise gradient estimate and showing cylindrical behavior near singularities.
result Near singularities, the surface is quantitatively cylindrical.