arXiv research

New formula simplifies interior polynomial calculation.

The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a par…

The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a famous link invariant with many known properties. For example, the HOMFLY polynom…

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polyno…

IPMs struggle with hyperbolic spaces due to polynomially growing barrier parameters.

We present new rectification theorems of degenerate quasi-conformal structures that give a meaning to quotients of Riemann surfaces with empty interior "fundamental domains". These techniques are used to define the unique renormalization of polynomials with Cantor set Julia sets.

Solving polynomial equations finds circle packings on surfaces.

Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying $\Box_g ψ- μψ=0$ on the interior of Anti-de Sitter (AdS) black holes. We prescribe initial data on a spacelike hypersurface of a Reissner--Nordström--AdS black hole and impose…

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

New geometric object for polynomials simplifies complex data.

Paper derives estimates for Hessian equations under concavity assumptions.

Estimates for special Lagrangian curvature equations in critical and convex cases.

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $σ_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate…

In 1985, Barnsley and Harrington defined a ``Mandelbrot Set'' $\mathcal{M}$ for pairs of similarities --- this is the set of complex numbers $z$ with $0<|z|<1$ for which the limit set of the semigroup generated by the similarities $x \mapsto zx$ and $x \mapsto z(x-1)+1$ is connected. Equivalently, $\mathcal{M}$ is the …

Develops fractional de Rham theory for Maxwell equations.

Study interior estimates for solutions of Poisson equation on Riemann surfaces.

This paper compiles formulas involving differential operators and interior products.

Proves interior singular set dimension for area-minimizing currents in smooth submanifolds.

This paper reverses a construction by merging boundary critical points into an interior one.

The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.

Interior $C^{2}$ estimates for sum Hessian quotient equations on Riemannian manifolds

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.

We prove a priori interior C2 estimate for σ_2 = f in R3, which generalizes Warren-Yuan's result.

We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.

We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.

We use elementary methods to construct a minimal lamination of the interior of a positive cone in R3.

In this paper we prove the interior regularity for the solution to the Abreu equation in any dimension assuming the existence of the $C^0$ estimate.

We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.

In this paper, we consider the Dirichlet problem of a complex Monge-Ampère equation on a ball in $\mathbb C^n$. With $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) data, we prove an interior $\mathcal C^{1,α}$ (resp. $\mathcal C^{0,α}$) estimate for the solution. These estimates are generalized versions of the Bedford-T…

The study provides interior estimates for $Q_k$-flows and translators in $\mathbb{R}^{n+1}$.

We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.

In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.

New examples show flat singular sets can be arbitrarily complex.

The study confirms conjectures about normals to convex polytopes in 3D space.

The notions of the interior and truncated connections of a nonholonomic manifold are introduced. A class of extended truncated connections is distinguished. For the case of a contact space with a Finsler metric, it is shown that there exists a unique extended truncated connection that satisfies additional properties. T…

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

The paper examines metrics on foliated manifolds that have special geometric properties.

Develops a first-order interior-point method for solving constrained variational inequalities.

Study shows instability of naked singularities in scalar field models.

Sharp estimate for flow in any dimension.

In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of…

Let $(M, g)$ be an $n$-dimensional complete Riemannian manifold with $Ric(M)\geq-(n-1)Q$, where $Q\geq0$ is a constant. We obtain an interior gradient bound for minimal graphs in $M\times R$ under some technical assumptions. For details, see Theorem 2.

In this paper, the notion of an almost contact Kählerian structure is introduced. The interior geometry of almost contact Kählerian spaces is investigated. On the zero-curvature distribution of an almost contact metric structure, as on the total space of a vector bundle, an almost contact Kählerian structure is obtaine…

In this paper, we study reinforcement learning (RL) algorithms to solve real-world decision problems with the objective of maximizing the long-term reward as well as satisfying cumulative constraints. We propose a novel first-order policy optimization method, Interior-point Policy Optimization (IPO), which augments the…

The paper connects two skein algebras and characterizes their representations.

In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…

New method solves optimization problems with stochastic objectives and constraints.