arXiv research

The paper finds sign-changing solutions for a specific type of elliptic equation.

Study on blow-up behavior of sign-changing solutions for Yamabe equation.

Study finds infinite sign-changing solutions for a specific equation on manifolds.

Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.

Solves Brezis' first open problem on ball solutions.

The paper proves the existence of infinite sign-changing solutions to a Hardy-Sobolev equation on Riemannian manifolds.

New solutions found for Yamabe problem on spheres with foliations.

Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.

Study on compact Kähler surfaces for sign-changing curvatures.

Study optimal partition problem for Q-curvature equations on Einstein manifolds.

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the …

Researchers solve the negative Yamabe case for scalar curvature prescription.

Paper proves no nontrivial solutions to certain elliptic equations on graphs.

We derive a monotonicity formula and classify finite Morse index solutions (positive or sign-changing, radial or not) to the following triharmonic Lane-Emden equation: \begin{equation}\nonumber (-Δ)^3 u=|u|^{p-1}u \hbox{ in } \mathbb{R}^n, \end{equation} where $p$ is below the Joseph-Lundgren exponent. As a byproduct w…

In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our argumen…

Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs

The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preservin…

Study on scalar curvature minimizability loss and saddle point solutions.

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an…

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $-Δu=K(x)e^u, in R^2$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite to…

Paper studies existence of Toda systems with sign-changing functions.

Learning rates in stochastic neural network training are currently determined a priori to training, using expensive manual or automated iterative tuning. This study proposes gradient-only line searches to resolve the learning rate for neural network training algorithms. Stochastic sub-sampling during training decreases…

The lowest eigenvalue of the Schrödinger operator $-Δ+\mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.

Study on optimal partitions and nodal solutions for the Yamabe equation.

The paper proposes a method to test properties of the optimal assortment in multinomial logit models.

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0.$ Let $p_1,\ldots,p_n$ be any set of points at which $f(p_i)=0$ and $D^2f(p_i)$ is non-singular. We prove that for all sufficiently small $λ>…

Extends Toda system existence results to negative functions.

Improved Lasso estimator speeds up variable selection.

This paper focuses on the problem of prescribing mean curvature on the unit ball. Assume that $f$, which is allowed to change sign, satisfies Morse index counting or certain kind of symmetry condition. By using a negative gradient flow method, we then prove that $f$ can be realized as the boundary mean curvature of som…

In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges …

Two spheres found with specific curvature constraints.

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\…

In low dimensions, minimizers for the second conformal eigenvalue do not exist near the round sphere.

Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the …

The study finds multiple conformal metrics with specific curvature properties on compact surfaces.

Study on prescribing curvature on specific manifolds with negative Gauduchon degree.

Optimal kernel improves estimation accuracy in modal statistical methods.

Sharp fractional Sobolev inequalities on closed manifolds identified.

Study on minimizing singular capillary cones with stability and instability results.

The paper derives theorems about curl eigenfields on a 3-sphere using angular momentum theory.

Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a given function $\widetilde s(x)$ as its scalar curvature. In particular, the work…

The study counts critical points of Steklov eigenfunctions on manifolds.

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of itself. The parabolic set itself is detected in this way, and each arc is given …

New method reveals true causal functions in nonlinear time series, not just scores.

Flow approach solves Toda system equations.

This paper explores how rational numbers on the Stern-Brocot diagram map to lines when terms are extended.

The study explores spacetimes with changing spatial curvature, leading to topological transitions.