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In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

CR-lightlike submanifolds of a golden semi-Riemannian manifold are the focus of the research presented in this work, which aims to define and investigate these structures. Under the context of a golden semi-Riemannian manifold, we study the properties of geodesic CR-lightlike submanifolds as well as umbilical CR-lightlike submanifolds. In addition, on a golden semi-Riemannian manifold, we find numerous intriguing findings for entirely geodesic and totally umbilical CR-lightlike subamnifolds. Also, we construct an example of a CR-lightlike submanifold of a golden semi-Riemannian manifold. [ABSTRACT FROM AUTHOR]

In this research, we examine CR-lightlike submanifolds (CRL)subM of a Golden semi-Riemannian manifold (SR)M. We obtain some interesting theorems on CR-lightlike warped product (WP). We also construct an example of CR-lightlike submanifolds. [ABSTRACT FROM AUTHOR]

The main purpose of the current paper is to study certain curvature conditions in Lorentzian para-Kenmotsu n-manifolds (briefly, (LPK)n) admitting ρ-Einstein solitons (ρ-ES). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a bi-slant lightlike submanifold of a golden semi-Riemannian manifold. We provide two examples. We give some characterizations about the geometry of such submanifolds. [ABSTRACT FROM AUTHOR]

In this paper, we define and study CR-submanifolds of a golden Riemannian manifold. We investigate some properties for CR-submanifolds. Moreover, we obtain many interesting results of totally umbilical CR-submanifolds on a golden Riemannian manifold. [ABSTRACT FROM AUTHOR]

Water scarcity is a growing concern around the world. Opportunities to develop new water supplies are declining while new demands for industrial and urban uses and the environment are growing. [...]