Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems

In this paper, we consider the validity of the strong maximum principle for weakly coupled, degenerate and cooperative elliptic systems in a bounded domain. In particular, we are interested in the viscosity solutions of elliptic systems with fully nonlinear degenerated principal symbol. Applying the method of viscosity solutions, introduced by Crandall, Ishii and Lions in 1992, we prove the validity of strong interior and boundary maximum principle for semi-continuous viscosity sub- and super-solutions of such nonlinear systems. For the first time in the literature, the strong maximum principle is considered for viscosity solutions to nonlinear elliptic systems. As a consequence of the strong interior maximum principle, we derive comparison principle for viscosity sub- and super-solutions in case when on of them is a classical one. The main novelty of this work is the reduction of the smoothness of the solution. In the literature the strong maximum principle is proved for classical C2 or generalized C1 solutions, while we prove it for semi-continuous ones.

Inverse Problem for the Yang–Mills Equations

We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang–Mills equations in Minkowski space $${\mathbb {R}}^{1+3}$$ R 1 + 3 . Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. The principal symbol analysis of the interaction is based on a delicate calculation that involves the structure of the Lie algebra under consideration and the final result holds for any compact Lie group.

Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

On the Origins of the Cartouche and Encircling Symbolism in Old Kingdom Pyramids

On the Origins of the Cartouche and Encircling Symbolism in Old Kingdom Pyramids is a treatise on the subject of encircling symbolism in pharaonic monumental tomb architecture. The study focuses on the Early Dynastic Period and the Old Kingdom of ancient Egypt; from the first dynasty through the sixth. During that time, encircling symbolism was developed most significantly and became most influential. The cartouche also became the principal symbol of the pharaoh for the first time. This work demonstrates how the development of the cartouche was closely related to the monumental encircling symbolism incorporated into the architectural designs of the Old Kingdom pyramids. By employing a new architectural style, the pyramid, and a new iconographic symbol, the cartouche, the pharaoh sought to elevate his status above that of the members of his powerful court. These iconic new emblems emphasized and protected the pharaoh in life, and were retained in the afterlife. By studying the available evidence, the new and meaningful link between the two artistic media; iconographic and architectural, is catalogued, understood, and traced out through time.

Inverse scattering for a random potential

In this paper, we consider an inverse problem for the [Formula: see text]-dimensional random Schrödinger equation [Formula: see text]. We study the scattering of plane waves in the presence of a potential [Formula: see text] which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential [Formula: see text], we uniquely determine the principal symbol of the covariance operator of [Formula: see text]. Especially, for [Formula: see text] this result is obtained for the full nonlinear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance.

King Philip II of Spain as a symbol of ‘Tyranny’ in Spinoza’s Political Writings

The highly abstract style of Spinoza’s philosophy has encouraged some interpretations of him as a thinker with little immediate connection with the whirl of social and cultural affairs around him. This article shows that all three major Western revolts - those of the Netherlands, Portugal and Aragon - against Philip II (his principal symbol and embodiment of tyranny, arbitrary and illicit governance, intolerance and repression of basic liberties) became in some sense internationally entwined and were intensely present in his life, which helps to understand that Spinoza was indeed a revolutionary.