Acta Applicandae Mathematicae

We present the basic notions and results of the geometric theory of second order PDEs in the framework of contact and symplectic manifolds including characteristics, formal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchy problems. Then, we focus our attention to Monge-Ampère equations (MAEs) and discuss a natural class of MAEs arising in Kähler and para-Kähler geometry whose solutions are special Lagrangian submanifolds.

In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. We specifically obtain the Gevrey index of families of invariant curves which is related to the smoothness of area preserving mappings and the exponent of small divisors condition. Moreover, we obtain a Gevrey normal form of area preserving mappings in a neighborhood of the union of the invariant curves.

A new solution to Riordan’s problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated.

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.

Các đường Lagrangian trong \(\mathbb {R}^{4}\) có những mối quan hệ thú vị với biến dạng bậc hai của các đường phẳng dưới nhóm affine đặc biệt và các đường null trong một dạng không gian Lorentz ba chiều. Chúng tôi cung cấp một khung liên hợp phi tuyến tự nhiên cho các đường Lagrangian. Điều này cho phép chúng tôi phân loại các đường Lagrangian có độ cong symplectic không đổi, xây dựng một lớp các torus Lagrangian trong \(\mathbb {R}^{4}\) và xác định các đường geodesic Lagrangian.