We revisit several classical matrix inequalities related to the celebrated Thompson triangle inequality. We then present the second author's version of Thompson's inequality and related results. Finally, we survey a number of recent and elegant matrix triangle inequalities that may be viewed as cousins of Thompson's inequality. Taken together, these results illustrate the central role played by various notions of modulus in the noncommutative setting. Their interplay opens new perspectives in matrix analysis and operator theory.
This is a streamlined introduction to the global (infrared) completion of Maxwell-type higher gauge fields (as in the higher gauge sectors of higher-dimensional supergravity and its brane probes) by electromagnetic flux quantization in differential nonabelian cohomology, using cohesive homotopy theory. Applications include D/NS-brane charge in (unstable) K-theory, M-brane charge in unstable Cohomotopy, and geometric engineering of topological quantum order on probe M5-branes.
This paper investigates the [Formula: see text]-boundedness of wave operators [Formula: see text] associated with discrete fourth-order Schrödinger operators [Formula: see text] on the lattice [Formula: see text], where [Formula: see text] and [Formula: see text] is a real-valued potential on [Formula: see text]. Under suitable decay assumptions on [Formula: see text] (depending on the types of zero resonance of [Formula: see text]), we show that the wave operators [Formula: see text] are bounded on [Formula: see text] for all [Formula: see text]: [Formula: see text] In particular, if both thresholds [Formula: see text] and [Formula: see text] are regular points of [Formula: see text], we prove that [Formula: see text] are neither bounded on the endpoint spaces [Formula: see text] nor on [Formula: see text]. We remark that the proof of these bounds relies fundamentally on the asymptotic expansions of the resolvent of [Formula: see text] near the two spectral thresholds, and on the theory of discrete singular integrals on the lattice. As an application, we derive the following [Formula: see text] decay estimates for solutions to the discrete beam equation with a parameter [Formula: see text] on the lattice [Formula: see text]: [Formula: see text] where [Formula: see text], [Formula: see text] is the conjugate exponent of [Formula: see text] and [Formula: see text] denotes the spectral projection onto the absolutely continuous spectral subspace of [Formula: see text].
We review the basic features of a logarithmic conformal field theory that arise in the context of the scaling limit of lattice models. The theory of interest is the symplectic fermions, whose central charge is [Formula: see text]. We provide an explicit construction of its space of fields as a logarithmic Fock space, and discuss its logarithmic structure as a representation of the Virasoro algebra. The construction of the correlation functions (in simply-connected domains of the plane) is presented following the ideas of the bootstrap approach. The text aims to be accessible to readers with little or no expertise in conformal field theory.
In the space & Ropf;(3), for Robin parameter L > 0, it is shown that there is a family of Schr & ouml;dinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the Gamma-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.
