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Optimization in Structure Population Models through the Escalator Boxcar Train
Full text linkThe Escalator Boxcar Train (EBT) is a tool widely used in the study of balance laws motivated by structure population dynamics. This paper proves that the approximate solutions defined through the EBT converge to exact solutions. Moreover, this method is rigorously shown to be effective also in computing optimal controls. As preliminary results, the well posedness of classes of PDEs and of ODEs comprising various biological models is also obtained. A specific application to welfare policies illustrates the whole procedure
An application of Orlicz spaces in partial differential equations
No full textOur purpose is to investigate mathematical properties of some systems of nonlinear partial differential equations where the nonlinear term is monotone and its behaviour - coercivity/growth conditions are given with the help of some general convex function defining Orlicz spaces.
Our first result is the existence of weak solutions to unsteady flows of non-Newtonian incompressible nonhomogeneous (with non-constant density) fluids with nonstandard growth conditions of the stress tensor. We are motivated by the problem of anisotropic behaviour of fluids which are also characterised by rapid shear thickness. Since we are interested in flows with the rheology more general than power-law-type, we describe the growth conditions with the help of an x–dependent convex function and formulate our problem in generalized Orlicz (Musielak-Orlicz) spaces.
As a second result we give a proof of the existence of weak solutions to the problem of the motion of one or several nonhomogenous rigid bodies immersed in a homogenous non-Newtonian fluid. The nonlinear viscous term in the equation is described with the help of a general convex function defining isotropic Orlicz spaces. The main ingredient of the proof is convergence of the nonlinear term achieved with the help of the pressure localisation method.
The third result concerns the existence of weak solutions to the generalized Stokes system with the nonlinear term having growth conditions prescribed by an anisotropic N -function. Our main interest is directed to relaxing the assumptions on the N- function and in particular to capture the shear thinning fluids with rheology close to linear. Additionally, for the purpose of the existence proof, a version of the Sobolev-Korn inequality in Orlicz spaces is proved.
Last but not least, we study also a general class of nonlinear elliptic problems, where the given right-hand side belongs only to the L1 space. Moreover the vector field is monotone with respect to the second variable and satisfies a non-standard growth condition described by an x-dependent convex function that generalizes both Lp(x) and classical Orlicz settings. Using truncation techniques and a generalized Minty method in the functional setting of non reflexive spaces we prove existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established. Sufficient conditions are specified which guarantee that the renormalized solution is already a weak solution to the problem
Fluid model of crystal plasticity - mathematical properties and computer simulations
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Modele Dynamiki Populacyjnej w Przestrzeniach Metrycznych
Full text linkLink archiwalny https://depotuw.ceon.pl/handle/item/38
Modele Dynamiki Populacyjnej w Przestrzeniach Metrycznych
Full text linkThe main goal of this thesis is the analysis of a wide class of structured population models in the space of finite, nonnegative Radon measures equipped with the flat metric. This framework allows a unified approach to a variety of problems providing them with basic well-posedness and stability results. The first result is the existence and uniqueness of measure valued solutions to the one-sex structured population model. A nonlinear semigroup is constructed here by means of the operator splitting algorithm. This technique allows to separate the differential operator from the integral one, which leads to a significant simplification of proofs. Concerning stability, the Lipschitz continuity of solutions with respect to the model coefficients is provided. The next analytical result is the well-posedness of the age-structured two-sex population model. Existence and uniqueness of the measure valued solutions is proved by the regularization technique as well as the stability estimates. A brief discussion on a marriage function, which is the main source of the nonlinearity in this model, is carried out and an example of the marriage function fitting into the considered framework is given. The second part of this thesis is devoted to a development of numerical methods for a particular class of one-sex structured population models. The first method is constructed through the splitting technique and corresponds with a current trend basing on a kinetic approach to the population dynamics problems. Separation of a semigroup induced by the transport operator from a semigroup induced by the nonlocal term allows to keep the solution as a sum of Dirac deltas despite of the regularizing character of the nonlocal boundary condition. As the next step, two alternative methods based on different approximations of the boundary condition are analyzed. These are the Escalator Boxcar Train algorithm and its simplification. Convergence of both methods is proved exploiting the concept of semiflows on metric spaces. Last but not least, the rate of convergence for all schemes mentioned above is provided.Celem naukowym niniejszej rozprawy jest analiza matematyczna dynamiki modeli strukturalnych w przestrzeniach metrycznych. Modele strukturalne opisują ewolucję populacji organizmów, zróżnicowanej ze względu na wybrane cechy. Cechy te zależą od modelowanej populacji, mogą być to, między innymi, wiek lub rozmiar osobnika, dojrzałość pojedynczej komórki, stan jej zróżnicowania lub fenotyp. Przestrzenią metryczną, w której analizujemy równania dynamiki populacyjnej jest przestrzeń skończonych, nieujemnych miar Radona z metryką flat. Nasze wyniki dotyczą między innymi istnienia i jednoznaczności miarowych rozwiązań dla szerokiej klasy modeli ze strukturą. W szczególności, rozpatrujemy modele mające zastosowanie w demografii, biologii i epidemiologii. Otrzymane rezultaty gwarantują także stabilność rozwiązań względem współczynników modelu, co bezpośrednio przekłada się na możliwość tworzenia stabilnych schematów numerycznych. Budowa takich schematów, opartych na metodzie cząstek i algorytmie split-up oraz ich zastosowanie do wyżej wymienionych modeli jest istotnym elementem tejże rozpraw
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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