Positive Operator Semigroups

Bátkai, András; Fijavž, Marjeta Kramar; Rhandi, Abdelaziz

doi:10.1007/978-3-319-42813-0

Cited by 89 publications

(63 citation statements)

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“…We begin by recalling some terminology. The following notions are well known and can be found in various sources, including [9,13]. Let X be a real vector lattice with norm · .…”

Section: Preliminariesmentioning

confidence: 99%

Discrete fragmentation systems in weighted $$\pmb {\ell ^{1}}$$ spaces

2020

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We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers), and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted 1 space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted 1 space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.Mathematics Subject Classification: 47D06, 34G10, 80A30, 34D05

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“…We begin by recalling some terminology. The following notions are well known and can be found in various sources, including [9,13]. Let X be a real vector lattice with norm · .…”

Section: Preliminariesmentioning

confidence: 99%

Discrete fragmentation systems in weighted $$\pmb {\ell ^{1}}$$ spaces

2020

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“…The term isolation distance of an eigenvalue λ h of a matrix M means the distance between λ h and the other eigenvalues of M , that is, dist(λ h , σ(M )). It is often convenient to set π 1 = 1 for a PageRank vector π, but Tikhonov's theorem [3] states that norms in a finite-dimensional vector space are equivalent. The r-th derivative of a general function f with a single argument is sometimes denoted as D r f (r ∈ N).…”

Section: Terms and Notationsmentioning

confidence: 99%

The sensitivity of a quantum PageRank

Honda

2020

Japan J. Indust. Appl. Math.

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“…The main result of this paper is best formulated in terms of positive operator semigroups [77]. A positive operator semigroup (P t ) t≥0 is one that preserves the positivity of distributions…”

Section: Acknowledgementsmentioning

confidence: 99%

Effective potential reveals evolutionary trajectories in complex fitness landscapes

Smerlak¹

2019

Preprint

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Growing efforts to measure fitness landscapes in molecular and microbial systems are premised on a tight relationship between landscape topography and evolutionary trajectories. This relationship, however, is far from being straightforward: depending on their mutation rate, Darwinian populations can climb the closest fitness peak (survival of the fittest), settle in lower regions with higher mutational robustness (survival of the flattest), or fail to adapt altogether (error catastrophes). These bifurcations highlight that evolution does not necessarily drive populations "from lower peak to higher peak", as Wright imagined. The problem therefore remains: how exactly does a complex landscape topography constrain evolution, and can we predict where it will go next?Here I introduce a generalization of quasispecies theory which identifies metastable evolutionary states as minima of an effective potential. From this representation I derive a coarse-grained, Markov state model of evolution, which in turn forms a basis for evolutionary predictions across a wide range of mutation rates. Because the effective potential is related to the ground state of a quantum Hamiltonian, my approach could stimulate fruitful interactions between evolutionary dynamics and quantum many-body theory. SIGNIFICANCE STATEMENTThe course of evolution is determined by the relationship between heritable types and their adaptive values, the fitness landscape. Thanks to the explosive development of sequencing technologies, fitness landscapes have now been measured in a diversity of systems from molecules to micro-organisms. How can we turn these data into evolutionary predictions? I show that preferred evolutionary trajectories are revealed when the effects of selection and mutations are blended in a single effective evolutionary force. With this reformulation, the dynamics of selection and mutation becomes Markovian, bringing a wealth of classical visualization and analysis tools to bear on evolutionary dynamics. Among these is a coarse-graining of evolutionary dynamics along its metastable states which greatly reduces the complexity of the prediction problem.

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Positive Operator Semigroups

Cited by 89 publications

References 0 publications

Discrete fragmentation systems in weighted $$\pmb {\ell ^{1}}$$ spaces

Discrete fragmentation systems in weighted $$\pmb {\ell ^{1}}$$ spaces

The sensitivity of a quantum PageRank

Effective potential reveals evolutionary trajectories in complex fitness landscapes

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