In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms and . These operators arise when one studies the interior problem of tomography. The diagonalization of has been previously obtained, but only asymptotically when . We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

First we introduce the two tau-functions which appeared either as the -function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large -matrix integrals to (higher) order free probability. Within this context of dynamically evolving contours, we determine a class of driving functions for controlled Loewner-Kufarev equations, which enables us to give a continuity estimate for the solution of such an equation when embedded into the Segal-Wilson Grassmannian.

In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity : from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of . It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity . An important implication of our results is the following important observation. Note that one can regard the value of as the natural 'size' (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of . Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number . We show even more, that for the values there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing . Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

We take a closer look at the Riemann-Hilbert problem associated to one-gap solutions of the Korteweg-de Vries equation. To gain more insight, we reformulate it as a scalar Riemann-Hilbert problem on the torus. This enables us to derive deductively the model vector-valued and singular matrix-valued solutions in terms of Jacobi theta functions. We compare our results with those obtained in recent literature.

For a unitary operator on a separable complex Hilbert space , we describe the set of all conjugations (antilinear, isometric, and involutive maps) on for which . As this set might be empty, we also show that if and only if is unitarily equivalent to .

If is a unitary operator on a separable complex Hilbert space , an application of the spectral theorem says there is a conjugation on (an antilinear, involutive, isometry on ) for which In this paper, we fix a unitary operator and describe of the conjugations which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for if and only if it is invariant for any conjugation for which .

We show that the analytic content is neither subadditive nor semiadditive. To be precise, for compact sets in the complex plane, is the -uniform distance from the complex conjugation to the algebra of all rational functions with poles outside . Thus, given any integer , it is proven that each compactum can be decomposed as the union of two new compact sets and with for . Moreover, we also show that no compactum with positive analytic content can be decomposed as the countable union of compact sets of zero analytic content.