The paper proves unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.
problem Proving unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.
method Explicit local calculations combined with covering arguments.
result Proves unboundedness above and below of the Donaldson-Hitchin functionals on G2 and tG2 forms.
Short note shows unbounded dimensions in Fano K-moduli spaces.
problem Unboundedness of Fano K-moduli space dimensions.
method Analyzes Fano varieties and their K-moduli spaces.
result Dimension of K-moduli space can be unbounded.
The paper proves unboundedness of a functional on G2 forms and describes manifold limits.
problem Proving unboundedness of a functional on G2 forms and describing manifold limits.
method Scaling arguments, geometric estimates, collapsing theorem for orbifolds.
result Explicit descriptions of large volume limits of two G2 manifolds.
Study spectral invariants over integers, discovering unboundedness and field-dependence.
problem Dependence of spectral invariants on integer coefficients.
method Floer homology theory, focusing on Hamiltonian Floer homology.
result Spectral norm is unbounded over integers for complex projective spaces.
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 Rn+1 for every n≥3. Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.
problem Validation of Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.
method Introduced infinite dimensional Hilbertian H-type groups with weak, graded, left invariant Riemannian metrics and proved the vanishing of geodesic distance and local unboundedness of sectional curvature.
result Validation of Michor-Mumford conjecture linking geodesic distance vanishing to local unboundedness of sectional curvature.
This article finds a structure of singular sets on compact Kahler surfaces, which Taubes introduced in the studies of the asymptotic analysis of solutions to the Kapustin-Witten equations and the Vafa-Witten ones originally on smooth four-manifolds. These equations can be seen as real four-dimensional analogues of the …
Paper proves unbounded stabilization heights of fiber surfaces using Hopf invariant.
problem Unbounded stabilization heights of fiber surfaces.
method Alternative proof using Hopf invariant.
result Proves unbounded stabilization heights of fiber surfaces.
In this paper, we study Euler classes in groups of homeomorphisms of Seifert fibered 3-manifolds. We show that, in contrast to the familiar Euler class for Homeo0(S1)δ, these Euler classes for Homeo0(M3)δ are unbounded classes. In fact, we give examples of flat topological M bundles over a g…
New distances defined on Legendrian spaces without positive loops.
problem Defining distances on Legendrian spaces without positive loops.
method Constructing unbounded invariant distances on Legendrian isotopy classes.
result Invariant distances on Legendrian isotopy classes are discrete.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character analysis.
result Low-regularity spacetimes with unbounded curvature.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character maximizers.
result Low-regularity inextendibility linked to unbounded curvature.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
In this note we partially answer a question posed by Colbois, Dryden, and El Soufi. Consider the space of constant-volume Riemannian metrics on a connected manifold M which are invariant under the action of a discrete Lie group G. We show that the first eigenvalue of the Laplacian is not bounded above on this space, pr…
This paper extends quasimorphism results to nonorientable surfaces.
problem Understanding quasimorphisms on nonorientable surface diffeomorphism groups.
method Constructing infinitely many quasimorphisms on the identity component of nonorientable surface diffeomorphism groups.
result The space of nontrivial quasimorphisms on the identity component of the diffeomorphism group of a closed nonorientable surface is infinite-dimensional.
Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half-space. We establish e…
The paper improves convergence for linear systems using entropic mirror descent with Polyak stepsizes.
problem Convergence analysis for linear systems with unbounded domain.
method Entropic mirror descent with Polyak stepsizes, sublinear and linear convergence results.
result Generalized convergence result for arbitrary convex functions.
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Beth…
The paper studies quasimorphisms and distortion in homeomorphism groups of manifolds.
problem Identifying quasimorphisms that extend continuously to homeomorphism groups.
method Analyzing Gambaudo-Ghys and Polterovich quasimorphisms, extending them continuously, and presenting applications.
result Conditions for undistorted homeomorphisms and unbounded bi-invariant metrics.
The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curv…
Operator-theoretic analysis of nonlinear dynamical systems has attracted much attention in a variety of engineering and scientific fields, endowed with practical estimation methods using data such as dynamic mode decomposition. In this paper, we address a lifted representation of nonlinear dynamical systems with random…
Paper tackles unbounded density ratio estimation for covariate shift adaptation.
problem Understudied challenge in statistical learning: unbounded density ratios.
method Three-step estimation method: relative density ratio, truncation, and transformation.
result Established rigorous convergence guarantees for density ratio and regression estimators.
The support vector machine (SVM) is one of the most successful learning methods for solving classification problems. Despite its popularity, SVM has a serious drawback, that is sensitivity to outliers in training samples. The penalty on misclassification is defined by a convex loss called the hinge loss, and the unboun…
Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.
problem Geodesics on SL(n) with Hilbert-Schmidt metric.
method Analysis of geodesics, use of Virial-identity-based criterion, study of explicit families of solutions, classification of geodesics.
result Complex dynamics in higher dimensions, existence of bounded geodesic motions in even dimensions, instability of swirling and shear flows in even dimensions.
Study improves robust nonparametric regression in heavy-tailed noise.
problem Robust nonparametric regression with heavy-tailed noise and unbounded functions.
method Huber regression in reproducing kernel Hilbert spaces (RKHS), probabilistic effective hypothesis space, new comparison theorems.
result Explicit finite-sample error bounds and convergence rates for Huber regression in RKHS under heavy-tailed noise.
The paper provides stability guarantees for non-parametric maximum likelihood estimation using statistical mechanics.
problem Non-parametric maximum likelihood estimation and Gaussian mixture models.
method Statistical mechanics analysis to establish stability guarantees for NPMLE.
result High probability upper bounds on the Kullback-Leibler divergence between NPMLE estimators and the true density.
Novel oracle-type inequality for logistic loss in DNNs achieves sharp convergence rates.
problem Generalization analysis for binary classification with DNNs and logistic loss.
method Established an oracle-type inequality to handle the boundedness of the target function.
result Optimal convergence rates for fully connected ReLU DNN classifiers trained with logistic loss.