Symmetry

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, the existence and uniqueness are proven by application of the Leray–Schauder nonlinear alternative and Banach’s contraction principle, respectively. In addition, we discuss the Ulam–Hyers stability and generalized Ulam–Hyers stability of the results, and illustrative examples are also presented to demonstrate their correctness and effectiveness.

This paper extends previous work on echo chambers modeled by an Ising-like system at zero temperature (Spontaneous Symmetry Breaking, Group Decision Making and Beyond 1: Echo Chambers and random Polarization, Symmetry 2024, 16(12), 1566). There, polarization emerged as a spontaneous symmetry-breaking process with a randomly selected direction. Here using a mean-field analysis and Monte Carlo simulations I show that this mechanism is highly vulnerable to minimal distortions. An external symmetry-breaking field, even very small, suffices to impose a global direction and suppress opposite domains, producing distorted full polarization. In contrast, a handful of quenched local fields with zero average do not erase polarization but reorganize it into opposing domains. Remarkably, as few as two opposed fields, if placed at tipping sites, can redirect the entire system. These fragile sites, indistinguishable from others, act as hidden tipping points that amplify microscopic biases into macroscopic outcomes. The difference in local field proportions is found to be instrumental to guaranteeing a winning majority. The results highlight how minimal, strategically placed interventions can override autonomous self-organization. The results could, if applicable to social media platforms, question their presumed democratic nature of consensus.

It has been shown that there is a planar graph without 3-cycles which is not -colorable for any given ${\Delta }_1,{\Delta }_2$. This inspires many research to obtain sufficient conditions for planar graphs without 4-cycles and other cycles to be -colorable. For example, planar graphs without k-cycles and 4-cycles are -colorable for each and $({\Delta }_1,{\Delta }_2)=\{(2,6),(3,4)\}$. In this work, we study the values of ${\Delta }_1$ and ${\Delta }_2$ that make a planar graph without only 4-cycles -colorable. We begin with ${\Delta }_1=6$ and ${\Delta }_2=6$. Planar graphs without 4-cycles are -colorable. Two colors in a -coloring, where ${\Delta }_1={\Delta }_2$ are switchable, thus reflect the symmetry of the resulting coloring. Furthermore, some proof techniques are using the symmetry of these two colors.