Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.
problem Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
method Riemannian geometry of Bergman metrics and smoothness of families of isometries.
result Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
The study shows how strictly convex domains in Euclidean spaces are rigid.
problem Understanding the rigidity of strictly convex domains in Euclidean spaces.
method Proved a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
result Smooth strictly convex domains in Euclidean spaces are rigid.
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi di…
Method approximates Lipschitz domains with smoother shapes.
problem Approximating bounded Lipschitz domains.
method Sequence of smooth, bounded domains with weak curvatures.
result Uniform isocapacitary estimates for approximating sets.
Estimates time-varying network connections using multi-stage smoothing.
problem Estimating edge probabilities of time-varying networks.
method Multi-stage smoothing: temporal local smoothing followed by node-domain smoothing.
result Captures both smooth temporal evolution and structural patterns in connectivity.
Diffusion models adapt to data geometry through log-domain smoothing.
problem Understanding why diffusion models generalize well across diverse domains.
method Investigating the role of score matching and log-domain smoothing in diffusion models.
result Log-domain smoothing adapts the diffusion model to the data manifold.
New approach to solving minimal surface system Dirichlet problem on smooth domains.
problem Solving Dirichlet problem for minimal surface system on smooth domains.
method Using mean curvature flow (MCF) with boundary conditions and non-negative Ricci curvature assumption.
result Existence of long-time mean curvature flow and existence result for exterior Dirichlet problem.
Boundary of fiber convex domains is a cohomological sphere.
problem Understanding the boundary properties of fiber convex domains.
method Analyzing smooth fiber convex domains with smooth boundaries.
result The boundary is a cohomological sphere.
Corners can be identified by a drum's sound spectrum.
problem Determining the presence of corners in a drum's shape from its sound.
method Proving spectral invariance of corners in domains with Lipschitz, piecewise smooth boundaries.
result Corners are uniquely determined by a drum's spectrum among domains with fixed genus.
Paper solves a complex equation for smooth domains.
problem Investigates a specific type of parabolic equation in complex domains.
method Uses J-functional to prove solution convergence.
result Proves the convergence of solutions to the equation.
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…
We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry …
The paper improves generalization bounds for domain adaptation.
problem Improving generalization bounds for domain adaptation under practical conditions.
method Derives generalization bounds for domain adaptation based on finitely many moments and smoothness conditions.
result Obtains generalization bounds for domain adaptation.
The study proves Gromov hyperbolicity for certain complex domains.
problem Characterizing Gromov hyperbolicity for complex domains.
method Analyzing domains in C2 with finite d'Angelo type and using automorphisms. result Domains in C2 with finite d'Angelo type are Gromov hyperbolic. New constraints rule out some optimal domains for helicity maximisation.
problem Finding a smooth domain of fixed volume that maximizes helicity.
method Established additional geometric constraints on optimal domains.
result Ruled out the optimality of a broad class of solid tori.
A simple proof shows standard billiard for certain convex domains.
problem Characterizing billiards in convex domains that are both projective and Minkowski.
method Direct simple proof in C1-smoothness, semi-local and local versions proved. result Standard Euclidean billiard in an appropriate structure.
Smoothly bounded domains have special functions that are plurisubharmonic.
problem Finding smooth functions that are plurisubharmonic on bounded domains.
method Proving existence of smooth defining functions that are p-plurisubharmonic. result Smooth domains with smooth p-convex boundaries admit smooth defining functions that are p-plurisubharmonic. We construct open domains in Euclidean 3-space which do not admit complete properly immersed minimal surfaces with an annular end. These domains can not be smooth by a recent result of Martin and Morales
Paper improves privacy and utility of SGD with bounded domain and smooth losses.
problem Lack of tight privacy bounds and practical assumptions in DPSGD.
method Rigorous privacy characterization for DPSGD with general L-smooth and non-convex loss functions, tracking privacy loss over iterations.
result Privacy loss converges without convexity assumption for bounded domain, improving utility.
Recent unsupervised approaches to domain adaptation primarily focus on minimizing the gap between the source and the target domains through refining the feature generator, in order to learn a better alignment between the two domains. This minimization can be achieved via a domain classifier to detect target-domain feat…
Proves a generalized Minkowski inequality for starshaped domains.
problem Proving a generalized Minkowski inequality for smooth, (k−1)-convex starshaped domains. method Solvability of the degenerate k-Hessian equation on the exterior domain Rn∖Ω. result Generalized Minkowski inequality holds for smooth, (k−1)-convex, starshaped domains. Paper improves neural network robustness certification with tighter radii estimates.
problem Certifying neural networks' robustness against adversarial attacks.
method Advanced algorithms for discrete and continuous domains, optimizing sample size, standard deviation, and temperature.
result Significant improvement in certified test-set accuracy with tighter certified radii bounds.
Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for …
Convex domains have a unique boundary property related to normal vectors.
problem Characterizing convex domains using boundary properties and inequalities.
method Proving an inequality involving boundary normal vectors and distances.
result A constant cn exists such that the inequality holds for convex domains, with equality if and only if the domain is convex. We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth …
For a bounded domain Ω⊂Rm,m≥2, of class C0, the properties are studied of fields of `good directions', that is the directions with respect to which ∂Ω can be locally represented as the graph of a continuous function. For any such domain there is a canonical smooth field of good direct…
For a given bounded domain Ω⊂Rn with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as t→0+. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…
Local vanishing theorems for complex spaces with smooth boundaries.
problem Vanishing of cohomology groups for complex spaces with smooth boundaries.
method Local vanishing theorem for Dolbeault cohomology groups.
result Vanishing of L2 and L2,loc Dolbeault cohomology groups for q>0. By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…
The paper classifies fiber structures of discontinuity domains for Anosov representations.
problem Understanding the topology of discontinuity domains for Anosov representations.
method Explicitly working out a smooth version of Fintushel's classification theorem for S1-actions on 4-manifolds. result The action on the fiber is equivalent to a circle action on a Hirzebruch surface.
Criterion for lifting smooth contact maps between Carnot groups to central extensions.
problem Existence of smooth contact map lifts between Carnot groups and their central extensions.
method Criterion using pullbacks and Lie algebra cohomology classes.
result Necessary and sufficient conditions for lifting are formulated.
Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.
problem Invertibility of layer potentials for generalized Stokes operators on smooth domains.
method Developed algebra toolkit to handle layer operators' limit and jump relations; proved Fredholm property and invertibility.
result Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.
It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in Rn, continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the s…
Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.
problem Characterizing real analytic functions on closed subanalytic domains.
method Analyzing functions defined on closed uniformly polynomially cuspidal sets in Rn using composites with polynomial curves. result Conditions for a function to be real analytic are effectively related to the regularity of the boundary of the domain.
Study shows Bergman metric is non-Einstein for certain domains.
problem Characterizing the Bergman metric of specific domains.
method Analyzing pseudoconvex domains with strongly pseudoconvex polyhedral boundaries.
result Bergman metric is not Einstein for the studied domains.
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on s…
Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani.
Let p:X→Y be a surjective holomorphic mapping between Kähler manifolds. Let D be a bounded smooth domain in X such that every generic fiber Dy:=D∩p−1(y) for y∈Y is a strongly pseudoconvex domain in Xy:=p−1(y), which admits the complete Kähler-Einstein metric. This family of Kähler-…
Paper shows regularization improves robustness in domain generalization.
problem Improving robustness in domain generalization.
method Derives novel theoretical analysis to control representation smoothness and proposes a regularization method.
result Regularization improves robustness in domain generalization.
In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.
This paper describes a method to construct standard 4-balls from homotopy 4-balls in C2.
problem The problem is whether every homotopy 4-ball in S4 is standard. method The approach is to use Stein surfaces and pseudoconvex domains to construct a diffeomorphic domain that is the union of three pseudoconvex domains, ensuring it is a standard 4-ball.
result The construction method ensures that the domain is a standard 4-ball, providing a compelling reimbedding construction for homotopy 4-balls in C2. SmoothFool efficiently computes smooth adversarial perturbations for deep networks.
problem Vulnerability of deep neural networks to adversarial attacks with specific statistical properties.
method SmoothFool: a general and computationally efficient framework for computing smooth adversarial perturbations.
result Smoothness significantly enhances robustness against adversarial attacks and improves transferability.
Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.
We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with …
Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.
problem Characterizing domains with specific metric properties.
method Analyzing Carathéodory and Bergman metrics on strictly pseudoconvex domains.
result Domains with specific metric properties are biholomorphically equivalent to balls.
Study smooth mappings between manifolds and their properties.
problem Understanding mappings between manifolds with various dimensions and boundaries.
method Analyzes mappings Cα in terms of iterated directional derivatives and smooth structures. result Establishes a canonical smooth manifold structure for mappings under certain conditions.
New BdryMatérn GP model for reliable boundary integration on irregular domains.
problem Incorporating boundary information in Gaussian process models for complex phenomena.
method Proposes a novel BdryMatérn GP framework with a new covariance kernel derived via path integral and stochastic PDE.
result Sample paths from the BdryMatérn GP satisfy desired boundaries with smoothness control on derivatives.
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with c…