arXiv research

Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.

There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic co…

Proximal methods avoid local minima in weakly convex problems.

Local minimality proven for stable free-boundary minimal hypersurfaces.

As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy.…

Strict convexity is essential for compact minimal surfaces in curved spaces.

For the minimal graph defined on a convex ring in the space form with nonnegative curvature, we obtain the regularity and the strict convexity about its level sets by the continuity method.

Geometric proof shows regularity of anisotropic minimal surfaces in 2D.

This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.

New boundary condition for Black-Scholes equations in strict local martingale models.

Let (X,L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1,1)-forms in the first Chern class of L, endowed with the Mabuchi metric, is not strictly convex modulo automorphisms. How…

New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.

New method solves subspace optimization problems efficiently.

New method speeds up GAN training by solving saddle point problem.

We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we …

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

Note on minimal maps' uniqueness via singular values.

Study rigidity of minimal disks in specific 3-manifolds.

Improved Frank-Wolfe algorithm for polytopes converges linearly with dimension dependence on optimal face.

New tensor recovery method improves efficiency under strict complementarity.

Paper aims to minimize ruin probability in insurance companies using Sparre Andersen model.

New findings on stable minimal hypersurfaces in curved 4-manifolds.

The paper establishes conditions for strict power concavity in convolutions.

Enumerates knots up to five crossings and describes moves between them.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

It is shown that there are infinitely many compact orientable smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2 chi > 3 |tau|. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems fro…

New insights into Bartnik mass from improvability of dominant energy scalar.

The paper studies strict equivalence in multi-virtual linkoids with new invariants.

We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemi…

New proof shows how to identify DAGs with weakly increasing errors.

Gradient descent-ascent converges to strict local minmax equilibria with a finite timescale separation.

We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm…

Study strict stability of cones with isolated singularities.

We introduce a more restrictive version of the strict $CD(K,\infty)$ -condition, the so-called very strict $CD(K,\infty)$ -condition, and show the existence of optimal maps in very strict $CD(K,\infty)$ -spaces despite the possible lack of uniqueness of optimal plans.

We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Ito process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We al…

Maps Lie 2-groups to Weil algebras, showing cohomology isomorphisms.

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational dire…

Proves integrability of strict Lie 2-algebras using cohomological methods.

Stochastic subgradient descent avoids critical points in definable functions.

Strong geodesic convex function and strong monotone vector field of order $m$ on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order $m$ for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…

We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are i…

We study convergence properties of Stochastic Gradient Descent (SGD) for convex objectives without assumptions on smoothness or strict convexity. We consider the question of establishing that with high probability the objective evaluated at the candidate minimizer returned by SGD is close to the minimal value of the ob…

In an earlier work, we constructed the almost strict Morse $n$-category $\mathcal X$ which extends Cohen $\&$ Jones $\&$ Segal's flow category. In this article, we define two other almost strict $n$-categories $\mathcal V$ and $\mathcal W$ where $\mathcal V$ is based on homomorphisms between real vector spaces and $\ma…

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $ν$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …

Study complex hyperbolic lattices and their relation to strict hyperbolization.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non Kähler) manifolds and in twistor spaces $Z\sp{4n+2}$ over quaternionic Kähler manifolds $Q\sp{4n}$ are minimal. Moreover, we will prove that any Lagrangian submanifold $L$ in a nearly Kähler manifold $M$ splits into a product of two Lagrangian s…

Study compares nodal sets of solutions to the Allen-Cahn equation.