In this paper, we study asymptotic expansion at infinity and symmetry of zero mean curvature equations gradient graph in dimension 2, which include the Monge--Amp\`ere equation, inverse harmonic Hessian equation special Lagrangian equation. This refines research behavior, gives precise gap between exterior minimal entire case, extends classification results equations.

We present an algorithm for minimizing a sum of functions that combines the computational efficiency of stochastic gradient descent (SGD) with the second order curvature information leveraged by quasi-Newton methods. We unify these disparate approaches by maintaining an independent Hessian approximation for each contributing function in the sum. We maintain computational tractability and limit ...

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand p...

In this paper‎, ‎the‎ ‎authors prove that a strictly Kähler-Berwald manifold with‎ ‎nonzero constant holomorphic sectional curvature must be a‎ Kähler manifold‎. 

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq using only terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing analogue Kulkarni-Nomizu product, we provide simple formula such terms. also give an application Bochner technique to closed $4$-manifolds with indefinite intersection form $\sec>0$ or $\sec\geq0$, obtaini...

Optimization techniques used in Machine Learning play an important role in the training of the Neural Network in regression and classification tasks. Predominantly, first order optimization methods such as Gradient Descent have been used in the training of Neural Networks, since second order methods, such as Newton’s method, are computationally infeasible. However, second order methods show muc...

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of ...