Academic literature on the topic 'Strong comparison principle'

AbstractIn the following, we show the strong comparison principle for the fractional p-Laplacian, i.e. we analyze\quad\left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p}v+q(x)\lvert v\rvert% ^{p-2}v&\displaystyle\geq 0&&\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle(-\Delta)^{s}_{p}w+q(x)\lvert w\rvert^{p-2}w&\displaystyle\leq 0&% &\displaystyle\phantom{}\text{in ${D}$},\\ \displaystyle v&\displaystyle\geq w&&\displaystyle\phantom{}\text{in ${\mathbb% {R}^{N}}$},\end{aligned}\right.where {s\in(0,1)}, {p>1}, {D\subset\mathbb{R}^{N}} is an open set, and {q\in L^{\infty}(\mathbb{R}^{N})} is a nonnegative function. Under suitable conditions on s, p and some regularity assumptions on v, w, we show that either {v\equiv w} in {\mathbb{R}^{N}} or {v>w} in D. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.

Due to the emergence of technological solutions that allow creating systems based on strong artificial intelligence (AI), and the high probability of developing super-strong AI in the near future, special презумпцию невиновности человека attention has recently been paid to the legal problems of regulating the interaction of humans and society with new intelligent agents. The purpose of the research is to analyze modern legal principles of interaction between humans and artificial intelligence, neural networks and intelligent robots, which are the necessary basis for forming modern legislation in this area in Russia and foreign countries. Methods: empirical methods of comparison, description, interpretation; theoretical methods of formal and dialectical logic are used. Special scientific methods are used: comparative-legal, legal-dogmatic. Results: the study allows us to determine a number of relevant legal principles of human interaction with AI as well as with robots and other objects of robotics, based on a comparative legal analysis of the current legislation. They are the principle of humanism; the principle of justice; the prohibition of discrimination when using AI, robots and objects of robotics; the presumption of human innocence; the principle of human innocence; the principle of respect for human dignity; the principle of confidentiality; the principle of disclosure of information on the development, production and use of robots and artificial intelligence; the principle of autonomy of will when using systems equipped with AI; the principle of the presumption of consent; the principle of informed consent to the use (impact) of systems equipped with AI.