arXiv research

New cylindrical solutions found for Grushin-type problem.

The paper proves rigidity results for Serrin-type problems in manifolds.

Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.

Proves product metrics are Yamabe metrics under small flat torus conditions.

Proves rotational symmetry for Serrin-type problems in doubly connected domains.

Study solves inverse problems for real principal type operators using unique data sets and ray transforms.

In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restric…

We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equatio…

Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization o…

In this paper we present a correlation inequality with respect to Cauchy type measures. To prove our inequality, we transport the problem onto the Riemannian sphere then state and solve some special cases for a spherical correlation problem. This method, as we shall explain, opens up a new class of interesting problems…

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

Study finds solutions for complex problems on non-compact manifolds.

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

New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.

Solves a specific Dirichlet problem on Hermitian manifolds.

Sharp bounds found for Steklov-type eigenvalues on surfaces.

Study solves inverse problems for equations with fractional nonlinearities.

Paper solves curvature equations in Minkowski space for non-convex domains.

We explain how to relate the problem of finding a mirror manifold for a Calabi-Yau manifold to the problem of characterizing the rational homotopy types of closed Kähler manifolds.

Sigmoid-type networks avoid vanishing gradients with regularization and rescaling.

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was…

Study of Dirac equation with non-local nonlinearity on spheres.

Paper classifies critical points in half-space with new distance function.

In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property …

A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem

This paper addresses the so-called conformal capacities in $\mathbb R^n$, $n\ge 3$, through comparing three existing definitions (due to Betsakos, Colesanti-Cuoghi, Anderson-Vamananmurthy-Fuglede respectively) and studying their associated iso-capacitary inequalities with connection to half-diameter, mean-width, mean-c…

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations whi…

Finite type invariants separate PL links in 3D space.

We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In the different cases, the lower bounds obtained for the number of solutions are related to the winding number of a curve in the plane, the homology of a space in $\R^3$, the knot type of a …

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-ty…

Study on homological Dehn functions of groups of type $FP_2$.

Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.

Study on bandit problem with fixed number of arm types, achieving optimal regret bounds.

New algorithm for countable bandits with optimal regret.

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on t…

The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $Ω$ is 4, then the equivalence under diffeomorphisms of $Ω$ is reduced to the equivalence of a symplectic linear frame functorially attached to $Ω$. As the equi…

Study Neumann problem for special Lagrangian type equations.

Study new Willmore-type variational problem for foliated hypersurfaces.

New non-quadratic hypersurfaces found for higher dimensions.

The paper solves Serrin-type problems on Riemannian manifolds using new inequalities and identities.

We consider the problem of a firm seeking to use personalized pricing to sell an exogenously given stock of a product over a finite selling horizon to different consumer types. We assume that the type of an arriving consumer can be observed but the demand function associated with each type is initially unknown. The fir…

In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and of fundamentally important. In this paper, we prove that every $δ$(3)-ideal nul…

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.

For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.

Study on scheduling jobs with unknown types, achieving sublinear excess cost.

Study rigidity of geodesic balls on manifolds with boundary.