Green's theorem and Jordan rectifiable curve

Thesis (M.A.)--Boston UniversityA comprehensive study of proof of Green's theorem is presented. A classical approach to the proof of the theorem has been compared with a more generalized form of the proof. Chapter I is devoted to defining curves and regions. Definitions of a regular curve, a regular region, Jordan curve, and Jordan rectifiable curves are presented [TRUNCATED

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton-Jacobi-Isaacs Equations

In this paper, we consider state and control path-dependent stochastic zero-sum differential games, where the dynamics and the running cost include both state and control paths of the players. Using the notion of nonanticipative strategies, we define lower and upper value functionals, which are functions of the initial state and control paths of the players. We prove that the value functionals satisfy the dynamic programming principle. The associated lower and upper Hamilton-Jacobi-Isaacs (HJI) equations from the dynamic programming principle are state and control path-dependent nonlinear second-order partial differential equations. We apply the functional It\^o calculus to prove that the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact subset of an $\kappa$-H\"older space introduced in \cite{Tang_DCD_2015}. Moreover, we show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. For the state path-dependent case, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations.Comment: 29 page

The Post-Apocalyptic Turn: a Study of Contemporary Apocalyptic and Post-Apocalyptic Narrative

Few periods have witnessed so strong a cultural fixation on apocalyptic calamity as the present. From fictions and comic books to Hollywood films, television shows, and video games, the end of the world is ubiquitous in the form of apocalyptic and post-apocalyptic narratives. Imagining world-changing catastrophes, contemporary apocalyptic and post-apocalyptic narratives force us to face urgent socio-political questions such as danger of globalization, effect of neoliberal capitalist hegemony, ecological disasters, fragility of human civilization, and so on. J. G. Ballard\u27s final fictions, though they do not directly deal with apocalyptic events but evoke apocalyptic mood, portray the bleak landscape of post-political, post-historical, late capitalist society, where extreme boredom generates mindless violence. Unlike Ballard, Margaret Atwood\u27s satirical MaddAddam trilogy not only envisions the real possibility of apocalypse under the current neoliberal tendency but also presents a utopian desire in the form of a religious group that actively resists the hegemony of neoliberalism. James Howard Kunstler\u27s post-apocalyptic fiction focuses on a post-petroleum age, where people lead simple and quotidian lives due to the scarcity of oil. By bringing the sense of scarcity to the fore, Kunstler\u27s novel also formulates one version of realist worldview, in which the scarcity of resources inevitably calls for the strict rule of law. As an ultimate social allegory of anxiety and fear in our times, the global zombie apocalypse envisages the total destruction of civilization, examining the rising necessity of realist attitude that fundamentally negates the traditional belief of progress. Although the scope of contemporary apocalyptic and post-apocalyptic narratives is wide and varied, they share one thing in common: the bold desire to imagine a totally different world by questioning the current order of things