We study Ornstein–Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The associated semigroups are symmetric, analytic and positivity preserving; the Dirichlet semigroup is sub-Markovian, while the Neumann semigroup is Markovian and admits the Gaussian measure as its unique invariant measure up to scalar multiples. We prove compactness of the resolvent and derive linear eigenvalue asymptotics. For regular rooted trees, we adapt the Naimark–Solomyak decomposition to the Gaussian weighted setting, reducing the operators to one-dimensional half-line problems and obtaining refined spectral localisation and lower bounds.

This paper investigates the well-posedness and positivity of solutions to a class of delayed transport equations on a network. The material flow is delayed at the vertices and along the edges. The problem is reformulated as an abstract boundary delay equation, and well-posedness is proved by using the Staffans-Weiss theory. We also establish spectral theory for the associated delay operators and provide conditions for the positivity of the semigroup.

We consider transport processes on metric graphs with time-dependent velocities and show that, under continuity assumption of the velocity coefficients, the corresponding non-autonomous abstract Cauchy problem is well-posed by means of evolution families and evolution semigroups.

For given square matrices A and B we denote by Y = AB -BA and by Z = AY -YA. It is well known that if A and Y commute, i.e., if Z = 0, then Y is a nilpotent matrix. In this note we show that the same is true if YZ = ZY . We also generalize this result by using commutators of higher order.

In this note we study the generation of C0-semigroups by first order differential operators on Lp(R+,Cℓ)×Lp([0,1],Cm) with general boundary conditions. In many cases we are able to characterize the generation property in terms of the invertibility of a matrix associated to the boundary conditions. The abstract results are used to study well-posedness of transport equations on non-compact metric graphs.

A Correction to this paper has been published: 10.1007/s00028-021-00715-0

We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order reduction. We study forward and backward well-posedness; furthermore, we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first-order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied.

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $ {\mathrm{L}}^{\infty} $-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.

We prove a Hille–Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on ℝ+. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded, relatively uniformly continuous, positive semigroup.

We present the operator semigroups approach to the first- and second-order dynamical systems taking place on metric graphs. We briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, we show two applications to biological models. This article is part of the theme issue ‘Semigroup applications everywhere’.

We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order reduction. We study forward and backward well-posedness; furthermore, we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first-order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied.

This study present an efficient graph-theoretical method for automatic design of district metered areas (DMAs) in water distribution networks (WDNs). The proposed method consists of two main parts, WDN partitioning and DMA connection, and is tested on a real-life WDN for which three spectral partitioning methods, multiple weight cases, and two clustering criteria are compared. The efficiency of the proposed DMA connection algorithm with respect to the traditional combinatorial approach is shown for different numbers of established DMAs. The final solution is selected according to a multicriteria evaluation model, which was developed in order to reduce the subjective influence in the selection process and considers hydraulic, cost, and topological criteria. The results show that all three tested spectral partitioning methods, i.e., the ratio cut, normalized cut, and newly proposed generalized normalized cut, are suitable for WDN partitioning and that the quality of the obtained solutions can be further improved by considering appropriate topological and cost-based WDN information in the partitioning process.

We prove well-posedness for very general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various different conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on ${\mathrm{L}}^p({\mathbb{R}_+},{\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$ for $1\le p<+\infty$.

We present an efficient graph-theoretical method for partitioning water distribution networks (WDNs) into district metered areas (DMAs). The proposed algorithm consists of two main parts, namely WDN partitioning and DMA connection, and is tested on a real-life WDN, for which different weight cases are compared. The efficiency of the proposed DMA connection algorithm, in regard to the traditional combinatorics approach, is shown for various numbers of established DMAs. The final solution is selected according to the multi-criteria evaluation model, which was developed in order to reduce the subjective influence in the selection process and considers hydraulic, cost, and topological criteria. The results show that the newly proposed spectral partitioning method, namely generalized normalized cut, is appropriate for WDN partitioning and that we can further improve the quality of the obtained solutions by considering appropriate topological and cost-based WDN information in the partitioning process. * Stik / Correspondence: daniel.kozelj@fgg.uni-lj.si © Zevnik J. et al.; Vsebina tega članka se sme uporabljati v skladu s pogoji licence Creative Commons Priznanje avtorstva – Nekomercialno – Deljenje pod enakimi pogoji 4.0. © Zevnik J. et al.; This is an open-access article distributed under the terms of the Creative Commons Attribution – Non Commercial – ShareAlike 4.0 Licence. https://doi.org/10.15292/acta.hydro.2018.03 Zevnik J. et al.: Učinkovita particija vodovodnih omrežij na merilna območja z uporabo teorije grafov – Efficient partitioning of water distribution networks using a graph-theoretical approach Acta hydrotechnica 31/54 (2018), 35‒50, Ljubljana 36

For two unbounded linear operators A and B on a Banach space X it is not always evident how to define in a reasonable way their sum A+B.

In this chapter we give an application of positive semigroup theory to linear transport equations. This is a wonderful piece of mathematics modeling neutron transport in a reactor which uses much of the theory we developed in this text.

Many applications of positive semigroups occur in mathematical biology or chemistry. In the finite-dimensional part of our text we have already discussed a very simple discrete-time population model, called the Leslie model (see Section 6.3). In this chapter we present a time-continuous age-structured population model with spatial diffusion. We present a rather advanced model in order to show the reader some generalizations and applications.

We characterize the space of all exactly reachable states of an abstract boundary control system using a semigroup approach. Moreover, we study the case when the controls of the system are constrained to be positive. The abstract results are then applied to study flows in networks with static as well as dynamic boundary conditions.

We have seen that, in general, the growth bound ω 0(T) of a C 0-semigroup (T(t)) t≥0 and the spectral bound s(A) of its generator A do not coincide, even if positivity is assumed. It turns out that in Hilbert spaces a deeper analysis is possible using the boundedness of the resolvent. This has the consequence that for a positive semigroup T(t)) t≥0 on a Hilbert space the equality s(A) = ω 0(T) holds. This is the most important result of Section 15.2.