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In-phase synchronization is a stable state of identical Kuramoto oscillators coupled on a network with identical positive connections, regardless of network topology. However, this fact does not mean that the networks always synchronize in-phase because other attractors besides the stable state may exist. The critical connectivity μ c is defined as the network connectivity above which only the in-phase state is stable for all the networks. In other words, below μ c , one can find at least one network that has a stable state besides the in-phase sync. The best known evaluation of the value so far is 0.6828 ... ≤ μ c ≤ 0.7889. In this paper, focusing on the twisted states of the circulant networks, we provide a method to systematically analyze the linear stability of all possible twisted states on all possible circulant networks. This method using integer programming enables us to find the densest circulant network having a stable twisted state besides the in-phase sync, which breaks a record of the lower bound of the μ c from 0.6828 ... to 0.6838 .... We confirm the validity of the theory by numerical simulations of the networks not converging to the in-phase state. [ABSTRACT FROM AUTHOR]

A family of non-Hermitian real and tridiagonal-matrix candidates H (N) (λ) = H 0 (N) + λ W (N) (λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum of H 0 (N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be P T -symmetric]. Despite that, the "physical" parametric domain D [ N ] is (constructively) shown to exist, guaranteeing that in its interior, the spectrum remains real and non-degenerate, rendering the quantum evolution unitary. Among the non-Hermitian degeneracies occurring at the boundary ∂ D [ N ] of the domain of stability, our main attention is paid to their extreme version corresponding to Kato's exceptional point of order N (EPN). The localization of the EPNs and, in their vicinity, of the quantum-phase-transition boundaries ∂ D [ N ] is found feasible, at the not too large N, using computer-assisted symbolic manipulations, including, in particular, the Gröbner-basis elimination and the high-precision arithmetics. [ABSTRACT FROM AUTHOR]

In the present work, we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation (− Δ) p s u + a | u | p − 2 u + λ (ln | ⋅ | * | u | p ) | u | p − 2 u = f (u) i n R N , where N = sp, s ∈ (0, 1), p > 2, a > 0, λ > 0, and f : R → R is a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Moreover, when f has subcritical growth, we prove the existence of infinitely many solutions via genus theory. [ABSTRACT FROM AUTHOR]