We deal with a stationary problem of a reaction–diffusion system with a conservation law under the Neu-mann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coef-ficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling with κthe asymptotic profile exhibits a parabola in the nonvanishing region. We also prove the existence of an unstable monotone solution when the mass is small.

Let 1 p+. We show that the positive part of the closed unit ball of a non-commutative L p-space, as a metric space, is a complete Jordan-invariant for the underlying von Neumann algebra.

We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with A_k = A^{2^k} not explicitly computed but in the recursive form A_k = A^{2^k}, and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension <i>n</i>, the resulting algorithms are of an efficient <i>O</i>(<i>n</i>) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.

We study BPS states of 5d N=1 SU(2) Yang-Mills theory on S^1×R^4. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface F_0. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d N=2 Seiberg-Witten theory, recovering spectral networks in the limit.

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks$N$_{$u,v$}in the disk that correspond to the choice of a pair ($u, v$) of Coxeter elements in the symmetric group$S$_{$n$}and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for$N$_{$u,v$}represent elements of the Coxeter double Bruhat cell$G$^{$u,v$}⊂GL_{$n$}. The Cartan subgroup$H$acts on$G$^{$u,v$}by conjugation. The standard Poisson structure on the space of weights of$N$_{$u,v$}induces a Poisson structure on$G$^{$u,v$}, and hence on the quotient$G$^{$u,v$}/$H$, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for$N$_{$u,v$}. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows.