16 research outputs found
Bounds for the expected value of one-step processes
Mean-field models are often used to approximate Markov processes with large
state-spaces. One-step processes, also known as birth-death processes, are an
important class of such processes and are processes with state space
and where each transition is of size one. We derive explicit
bounds on the expected value of such a process, bracketing it between the
mean-field model and another simple ODE. Our bounds require that the Markov
transition rates are density dependent polynomials that satisfy a sign
condition. We illustrate the tightness of our bounds on the SIS epidemic
process and the voter model.Comment: 14 pages, 4 figures, revise
Dynamic control of modern, network-based epidemic models
In this paper we make the first steps to bridge the gap between classic control theory and modern, network-based epidemic models. In particular, we apply nonlinear model predictive control (NMPC) to a pairwise ODE model which we use to model a susceptible-infectious-susceptible (SIS) epidemic on nontrivial contact structures. While classic control of epidemics concentrates on aspects such as vaccination, quarantine, and fast diagnosis, our novel setup allows us to deliver control by altering the contact network within the population. Moreover, the ideal outcome of control is to eradicate the disease while keeping the network well connected. The paper gives a thorough and detailed numerical investigation of the impact and interaction of system and control parameters on the controllability of the system. For a certain combination of parameters, we used our method to identify the critical control bounds above which the system is controllable. We foresee that our approach can be extended to even more realistic or simulation-based models with the aim of applying these to real-world situations
Leibniz seminorms in probability spaces
In this paper we study the (strong) Leibniz property of centered moments of bounded random variables. We shall answer a question raised by M. Rieffel on the non-commutative standard deviation
Parciális differenciálegyenletek
A jegyzet betekintést kíván nyújtani a másodrendű lineáris
parciális differenciálegyenletek elméletébe. Az első részben röviden
összefoglaljuk a későbbi fejezetek megértéséhez szükséges előismereteket. A
második részben fizikai példákat mutatunk parciális differenciálegyenletek
előfordulására, majd részletesen tanulmányozzuk a hővezetési és a Laplace-egyenletet
klasszikus elméletét. Ezt követően a disztribúcióelmélettel foglalkozunk,
és alkalmazzuk Cauchy-feladatok megoldására. Az utolsó részben
bevezetjük a Szoboljev-féle függvénytereket és értelmezzük elliptikus, illetve
időfüggő feladatok gyenge megoldásainak fogalmát. Minden fejezet végén önálló
gondolkodásra kitűzött feladatok találhatók, amelyek egy részéhez megoldást
is adunk a jegyzet végén
Completely positive mappings and mean matrices
Some functions f: R + → R + induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define a linear mapping (J f D)−1: Mn → Mn on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of (J f D)−1 in the case of a few concrete functions f. This problem has been motivated by applications in quantum information