Unified Characterizations of Minuscule Kac–Moody Representations Built from Colored Posets

R.M. Green described structural properties that "doubly infinite" colored posets should possess so that they can be used to construct representations of most affine Kac–Moody algebras. These representations are analogs of the minuscule representations of the semisimple Lie algebras, and his posets ("full heaps") are analogs of the finite minuscule posets. Here only simply laced Kac–Moody algebras are considered. Working with their derived subalgebras, we provide a converse to Green's theorem. Smaller collections of colored structural properties are also shown to be necessary and sufficient for such poset-built representations to be produced for smaller subalgebras, especially the "Borel derived" subalgebra. These developments lead to the formulation of unified definitions of finite and infinite colored minuscule and $d$-complete posets. This paper launches a program that seeks to extend the notion of "minuscule representation" to Kac–Moody algebras, and to classify such representations.

Realization of Memory Effect on Hysteresis Lobe Area of the TiO2 Based HP Memristor

In this paper, the memory effect on the hysteresis lobe area of TiO2 based memristive model is studied. The Green’s theorem is used to derive the novel general formula for the area of hysteresis lobe of HP memristor model. Further the memory and boundary values are derived mathematically, where the memory needs to be stable. It is analyzed that in the initial state this nanoscale non-volatile memristor retains its memory. The relation of memory with the lobe area and memristance are also established respectively. The analytical results mentioned above are demonstrated by numerical simulations and graphical representations.

Green's theorem in seismic imaging across the scales

The earthquake seismology and seismic exploration communities have developed a variety of seismic imaging methods for passive- and active-source data. Despite the seemingly different approaches and underlying principles, many of those methods are based in some way or another on Green's theorem. The aim of this paper is to discuss a variety of imaging methods in a systematic way, using a specific form of Green's theorem (the homogeneous Green's function representation) as a common starting point. The imaging methods we cover are time-reversal acoustics, seismic interferometry, back propagation, source–receiver redatuming and imaging by double focusing. We review classical approaches and discuss recent developments that fully account for multiple scattering, using the Marchenko method. We briefly indicate new applications for monitoring and forecasting of responses to induced seismic sources, which are discussed in detail in a companion paper.

Green's theorem in seismic imaging across the scales

The solid earth and exploration communities independently developed a variety of seismic imaging methods for passive- and active-source data. Despite the seemingly different approaches and underlying principles, many of those methods are based in some way or another on Green's theorem. The aim of this paper is to discuss a variety of imaging methods in a systematic way, using a specific form of Green's theorem (the homogeneous Green's function representation) as the common starting point. The imaging methods we cover are time-reversal acoustics, seismic interferometry, back propagation, source-receiver redatuming and imaging by double focusing. We review classical approaches and discuss recent developments that fully account for multiple scattering, using the Marchenko method. We briefly indicate new applications for monitoring and forecasting of responses to induced seismic sources, which are discussed in detail in a companion paper.