Geršgorin and Beyond•••

Eigenvalues are useful in various areas of mathematics, such as in testing the critical values of a multi variable function to see if it is a local extrema. One of the more common ways to define eigenvalues is: Definition (1): Given that A is an n by n matrix, λ is an eigenvalue of A if and only if det(A - λIn) = 0. Any nonzero vector in Null(A - λI) is called an eigenvector associated with λ

Shifted linear systems in electromagnetics. Part I: Systems with identical right-hand sides

We consider the solution of multiply shifted linear systems for a single right-hand side. The coefficient matrix is symmetric, complex, and indefinite. The matrix is shifted by different multiples of the identity. Such problems arise in a number of applications, including the electromagnetic simulation in the development of microwave and mm-wave circuits and modules. The properties of microwave circuits can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell's equations with orthogonal grids using the Finite Integration Technique (FIT). Some Krylov subspace methods have been used to solve multiply shifted systems for about the cost of solving just one system. We use the QMR method based on coupled two-term recurrences with polynomial preconditioning

The stability of model ecosystems

Ecologists would like to understand how complexity persists in nature. In this thesis I have taken two fundamentally different routes to study ecosystem stability of model ecosystems: classical community ecology and classical population ecology. In community ecology models, we can study the mathematical mechanisms of stability in general, large model ecosystems. In population ecology models, fewer species are studied but greater detail of species interactions can be incorporated. Within these alternative contexts, this thesis contributes to two consuming issues concerning the stability of ecological systems: the ecosystem stability-complexity debate; and the causes of cyclic population dynamics. One of the major unresolved issues in community ecology is the relationship between ecosystem stability and complexity. In 1958 Charles Elton made the conjecture that the stability of an ecological system was coupled to its complexity and this could be a “wise principle of co-existence between man and nature” with which ecologists could argue the case for the conservation of nature for all species, including man. The earliest and simplest model systems were randomly constructed and exhibited a negative association between stability and complexity. This finding sparked the stability-complexity debate and initiated the search for organising principles that enhanced stability in real ecosystems. One of the universal laws of ecology is that ecosystems contain many rare and few common species. In this thesis, I present analytical arguments and numerical results to show that the stability of an ecosystem can increase with complexity when the abundance distribution is characterized by a skew towards many rare species. This work adds to the growing number of conditions under which the negative stability - complexity relationship can been inverted in theoretical studies. While there is growing evidence that the stability-complexity debate is progressing towards a resolution, community ecology has become increasingly subject to major criticism. A long-standing criticism is the reliance on local stability analysis. There is growing recognition that a global property called permanence is a more satisfactory definition of ecosystem stability because it tests only whether species can coexist. Here I identify and explain a positive correlation between the probability of local stability and permanence, which suggests local stability is a better measure of species coexistence than previously thought. While this offers some relief, remaining issues cause the stability-complexity debate to evade clear resolution and leave community ecology in a poor position to argue for the conservation of natural diversity for the benefit of all species. In classical population ecology, a major unresolved issue is the cause of non-equilibrium population dynamics. In this thesis, I use models to study the drivers of cyclic dynamics in Scottish populations of mountain hares (Lepus timidus), for the first time in this system. Field studies currently favour the hypothesis that parasitism by a nematode Trichostrongylus retortaeformis drives the hare cycles, and theory predicts that the interaction should induce cycling. Initially I used a simple, strategic host-parasite model parameterised using available empirical data to test the superficial concordance between theory and observation. I find that parasitism could not account for hare cycles. This verdict leaves three options: either the parameterisation was inadequate, there were missing important biological details or simply that parasites do not drive host cycles. Regarding the first option, reliable information for some hare-parasite model parameters was lacking. Using a rejection-sampling approach motivated by Bayesian methods, I identify the most likely parameter set to predict observed dynamics. The results imply that the current formulation of the hare-parasite model can only generate realistic dynamics when parasite effects are significantly larger than current empirical estimates, and I conclude it is likely that the model contains an inadequate level of detail. The simple strategic model was mathematically elegant and allowed mathematical concepts to be employed in analysis, but the model was biologically naïve. The second model is the antipode of the first, an individual based model (IBM) steeped in biological reality that can only be studied by simulation. Whilst most highly detailed tactical models are developed as a predictive tool, I instead structurally perturb the IBM to study the ecological processes that may drive population cycles in mountain hares. The model allows delayed responses to life history by linking maternal body size and parasite infection to the future survival and fecundity of offspring. By systematically removing model structure I show that these delayed life history effects are weakly destabilising and allow parameters to lie closer to empirical estimates to generate observed hare population cycles. In a third model I structurally modify the simple strategic host-parasite model to make it spatially explicit by including diffusion of mountain hares and corresponding advection of parasites (transportation with host). From initial simulations I show that the spatially extended host-parasite equations are able to generate periodic travelling waves (PTWs) of hare and parasite abundance. This is a newly documented behaviour in these widely used host-parasite equations. While PTWs are a new potential scenario under which cyclic hare dynamics could be explained, further mathematical development is required to determine whether adding space can generate realistic dynamics with parameters that lie closer to empirical estimates. In the general thesis discussion I deliberate on whether a hare-parasite model has been identified which can be considered the right balance between abstraction and relevant detail for this system

Šurov komplement i teorija H-matrica

This thesis studies subclasses of the class of H-matrices and their applications, with emphasis on the investigation of the Schur complement properties. The contributions of the thesis are new nonsingularity results, bounds for the maximum norm of the inverse matrix, closure properties of some matrix classes under taking Schur complements, as well as results on localization and separation of the eigenvalues of the Schur complement based on the entries of the original matrix.Докторска дисертација изучава поткласе класе Х-матрица и њихове примене, првенствено у истраживању својстава Шуровог комплемента. Оригиналан допринос тезе представљају нови услови за регуларност матрица, оцене максимум норме инверзне матрице, резултати о затворености појединих класа матрица на Шуров комплемент, као и резултати о локализацији и сепарацији карактеристичних корена Шуровог комплемента на основу елемената полазне матрице.Doktorska disertacija izučava potklase klase H-matrica i njihove primene, prvenstveno u istraživanju svojstava Šurovog komplementa. Originalan doprinos teze predstavljaju novi uslovi za regularnost matrica, ocene maksimum norme inverzne matrice, rezultati o zatvorenosti pojedinih klasa matrica na Šurov komplement, kao i rezultati o lokalizaciji i separaciji karakterističnih korena Šurovog komplementa na osnovu elemenata polazne matrice

Algoritmi za računanje optimalnih lokalizacija Geršgorinovog tipa

There are numerous ways to localize eigenvalues. One of the best known results is that the spectrum of a given matrix ACn,n is a subset of a union of discs centered at diagonal elements whose radii equal to the sum of the absolute values of the off-diagonal elements of a corresponding row in the matrix. This result (Geršgorin's theorem, 1931) is one of the most important and elegant ways of eigenvalues localization ([63]). Among all Geršgorintype sets, the minimal Geršgorin set gives the sharpest and the most precise localization of the spectrum ([39]). In this thesis, new algorithms for computing an efficient and accurate approximation of the minimal Geršgorin set are presented.Постоје бројни начини за локализацију карактеристичних корена. Један од најчувенијих резултата је да се спектар дате матрице АCn,n налази у скупу који представља унију кругова са центрима у дијагоналним елементима матрице и полупречницима који су једнаки суми модула вандијагоналних елемената одговарајуће врсте у матрици. Овај резултат (Гершгоринова теорема, 1931.), сматра се једним од најзначајнијих и најелегантнијих начина за локализацију карактеристичних корена ([61]). Међу свим локализацијама Гершгориновог типа, минимални Гершгоринов скуп даје најпрецизнију локализацију спектра ([39]). У овој дисертацији, приказани су нови алгоритми за одређивање тачне и поуздане апроксимације минималног Гершгориновог скупа.Postoje brojni načini za lokalizaciju karakterističnih korena. Jedan od najčuvenijih rezultata je da se spektar date matrice ACn,n nalazi u skupu koji predstavlja uniju krugova sa centrima u dijagonalnim elementima matrice i poluprečnicima koji su jednaki sumi modula vandijagonalnih elemenata odgovarajuće vrste u matrici. Ovaj rezultat (Geršgorinova teorema, 1931.), smatra se jednim od najznačajnijih i najelegantnijih načina za lokalizaciju karakterističnih korena ([61]). Među svim lokalizacijama Geršgorinovog tipa, minimalni Geršgorinov skup daje najprecizniju lokalizaciju spektra ([39]). U ovoj disertaciji, prikazani su novi algoritmi za određivanje tačne i pouzdane aproksimacije minimalnog Geršgorinovog skupa

The cost of segregation in (social) networks

This paper investigates the welfare effect of income redistribution in the private provision of public goods on networks. We first show that the welfare effect of income redistribution is determined by Bonacich centrality. Then we develop an index based on the network structure of interactions, which, roughly speaking, measures the welfare effect of income redistribution confined to a component of contributors. The proposed index vanishes when applied to large components of contributors that display special segregated interactions, which suggests an “asymptotic neutrality” of redistributive policies. JEL classification C72; D31; H4

Scaling Approaches to Quantum Many-Body Problems

In the present thesis, we will focus on a less studied aspect of Thomas-Fermi theory: the highly non-trivial scaling relations following from it. The main objective of this thesis is to introduce this scaling approach, not as a method to solve the many-body problem, but as an efficient way of organizing the information contained in its solution in order to extract yet more – sometimes non-trivial – information. To this goal we apply the scaling approach to a wide range of system, from nanostructures (quantum dots) to atoms and atomic ions.Our main findings can be summarized as follows: (i) the obtainment of scaling relations for the correlation energy of quantum dots and atomic ions, respectively. This allows us to extend our scaling approach to complex quantities that are beyond mean-field methods; (ii) the obtainment of scaling relations for the chemical potentials and addition energies of two-dimensional quantum dots, which allows us to compare our results to experimental data; and (iii) the obtainment of scaling relations for the ground-state energy, chemical potentials, and addition energies of three-dimensional quantum dots, which allows us to explore the dimensionality effects on the scaling relations.In all cases, we not only showed the functional form of the scaling relations, but we also provided explicit analytical expressions for the scaled quantities. Such expressions are not simple by-products of the approach, but approximations that can be used for estimating relevant quantities with practically no computational cost. Furthermore, the obtained scaling relation may serve as a starting point for the improvement of more elaborated theories, for example, in the optimization of density functionals within density functional theory.The above results are reported in four publications which constitute the basis of the thesis