Parity-time (PT) symmetry is of great interest. The reciprocal and unidirectional features are intriguing besides the (PT) symmetry phase transition. Recently, the reciprocal transmission, unidirectional reflectionless and invisibility are intensively studied. Here, we show the reciprocal reflection/transmission in (PT)-symmetric system is closely related to the type of (PT) symmetry, that is, ...

Semi-regular locales are extensions of the classical semiregular spaces. We investigate the conditions such that semi-regularization is a functor. We also investigate the conditions such that semi-regularization is a reflection or coreflection.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding infinitesimal objects. MSC 2000: 53C50, 53C35, 53C40

In this paper, we present a mirror reflection invariant descriptor which is inspired from SIFT. While preserving tolerance to scale, rotation and even affine transformation, the proposed descriptor, MIFT, is also invariant to mirror reflection. We analyze the structure of MIFT and show how MIFT outperforms SIFT in the context of mirror reflection while performs as well as SIFT when there is no ...

This paper is to serve as an introduction to the study of symmetric spaces, with the goal of describing Hermitian symmetric spaces of noncompact type. There are three basic types of symmetric space: compact, noncompact, and Euclidean, defined in terms of properties of g, the Lie algebra of its group of isometries. It turns out that a simply connected symmetric space can be described completely ...

3 Convex functions on symmetric spaces 11 3.1 Geometric preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1 Metric spaces with curvature bounds . . . . . . . . . . . . . . 11 3.1.2 Hadamard spaces . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.3 Symmetric spaces of noncompact type . . . . . . . . . . . . . 15 3.1.4 Auxiliary results . . . . . . . . . . . . . . . ....