DSpace Community:
http://hdl.handle.net/2381/445
2017-10-20T08:49:41ZTowards characterisation of chaotic attractors in terms of embedded coherent structures
http://hdl.handle.net/2381/40289
Title: Towards characterisation of chaotic attractors in terms of embedded coherent structures
Authors: Crane, Daniel Lewis
Abstract: The central theme of this thesis is the development of general methods for the modelling of the dynamics on chaotic attractors by a coarse-grained representation constructed through the use of embedded periodic orbits & other coherent structures. Our aim is to develop tools for constructing two types of reduced representations of chaotic attractors: Markov-type models, and symbolic dynamics. For Markov models, we present construction of a minimal cover of chaotic attractors of maps and high-dimensional flows by embedded coherent structures such as periodic orbits from which a Markov chain of the dynamics can be constructed. For the symbolic dynamics, we investigate the utility of unstable periodic orbits for the construction of an approximate generating partition of a chaotic attractor.
In the first section of Part 1 we present an original method by which chaotic attractors of discrete-time dynamical systems can be covered using a small set of unstable periodic orbits (UPOs) following an iterative selection algorithm that only chooses those UPOs that provide additional covering of the attractor to be included into the cover. We then show how this representation can be used to represent trajectories in the system as a series of transition between cover elements, using which as a basis for the construction of a Markov chain representation of the dynamics. In the second section we extend this method to continuous-time dynamical systems, introducing methods by which covers of high-dimensional attractors can be constructed in low dimensional projections with as little information loss as possible, and also giving an example of how group symmetries of the system can be dealt with.
In Part 2 we change our focus to the construction of symbolic dynamics of discrete-time systems, presenting an extension to an existing method for the computational construction of approximate generating partitions that increases the applicability of the method to a wider range of systems, and also significantly improving the results for more complex maps.2017-08-30T14:44:44ZComputational diagnosis and risk evaluation for canine lymphoma.
http://hdl.handle.net/2381/40283
Title: Computational diagnosis and risk evaluation for canine lymphoma.
Authors: Mirkes, E. M.; Alexandrakis, I.; Slater, K.; Tuli, R.; Gorban, A. N.
Abstract: The canine lymphoma blood test detects the levels of two biomarkers, the acute phase proteins (C-Reactive Protein and Haptoglobin). This test can be used for diagnostics, for screening, and for remission monitoring as well. We analyze clinical data, test various machine learning methods and select the best approach to these oblems. Three families of methods, decision trees, kNN (including advanced and adaptive kNN) and probability density evaluation with radial basis functions, are used for classification and risk estimation. Several pre-processing approaches were implemented and compared. The best of them are used to create the diagnostic system. For the differential diagnosis the best solution gives the sensitivity and specificity of 83.5% and 77%, respectively (using three input features, CRP, Haptoglobin and standard clinical symptom). For the screening task, the decision tree method provides the best result, with sensitivity and specificity of 81.4% and >99%, respectively (using the same input features). If the clinical symptoms (Lymphadenopathy) are considered as unknown then a decision tree with CRP and Hapt only provides sensitivity 69% and specificity 83.5%. The lymphoma risk evaluation problem is formulated and solved. The best models are selected as the system for computational lymphoma diagnosis and evaluation of the risk of lymphoma as well. These methods are implemented into a special web-accessed software and are applied to the problem of monitoring dogs with lymphoma after treatment. It detects recurrence of lymphoma up to two months prior to the appearance of clinical signs. The risk map visualization provides a friendly tool for exploratory data analysis.2017-08-29T13:10:13ZWhat could have tipped the EU referendum result in favour of Remain
http://hdl.handle.net/2381/40225
Title: What could have tipped the EU referendum result in favour of Remain
Authors: Zhang, Aihua
Abstract: [First paragraphs] Much of the analysis about why the UK voted to leave the European Union in June 2016 has been done by looking at individual factors in isolation, or using opinion poll data from both before and after the vote.
In a new research paper, I applied two statistical analyses to the actual referendum voting data obtained from the Electoral Commission and the UK’s latest census data. I found that while voters’ level of higher education was the most important factor, the gender of voters and the turnout level also had parts to play in the victory for the Leave campaign.
Description: This article was written for The Conversation by invitation2017-08-23T09:08:40ZThe Lusternik-Schnirelmann Category for a Differentiable Stack
http://hdl.handle.net/2381/40216
Title: The Lusternik-Schnirelmann Category for a Differentiable Stack
Authors: Alsulami, Samirah; Colman, Hellen; Neumann, Frank
Abstract: We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids. This extends the notion of Lusternik-Schnirelmann category for smooth manifolds and orbifolds.
Description: The file associated with this record is under embargo while permission to archive is sought from the publisher. The full text may be available through the publisher links provided above.2017-08-22T12:17:53ZLong and short range multi-locus QTL interactions in a complex trait of yeast
http://hdl.handle.net/2381/40203
Title: Long and short range multi-locus QTL interactions in a complex trait of yeast
Authors: Mirkes, Evgeny M.; Walsh, Thomas; Louis, Edward J.; Gorban, Alexander N.
Abstract: We analyse interactions of Quantitative Trait Loci (QTL) in heat selected yeast by comparing them to an unselected pool of random individuals. Here we re-examine data on individual F12 progeny selected for heat tolerance, which have been genotyped at 25 locations identified by sequencing a selected pool [Parts, L., Cubillos, F. A., Warringer, J., Jain, K., Salinas, F., Bumpstead, S. J., Molin, M., Zia, A., Simpson, J. T., Quail, M. A., Moses, A., Louis, E. J., Durbin, R., and Liti, G. (2011). Genome research, 21(7), 1131-1138]. 960 individuals were genotyped at these locations and multi-locus genotype frequencies were compared to 172 sequenced individuals from the original unselected pool (a control group). Various non-random associations were found across the genome, both within chromosomes and between chromosomes. Some of the non-random associations are likely due to retention of linkage disequilibrium in the F12 population, however many, including the inter-chromosomal interactions, must be due to genetic interactions in heat tolerance. One region of particular interest involves 3 linked loci on chromosome IV where the central variant responsible for heat tolerance is antagonistic, coming from the heat sensitive parent and the flanking ones are from the more heat tolerant parent. The 3-locus haplotypes in the selected individuals represent a highly biased sample of the population haplotypes with rare double recombinants in high frequency. These were missed in the original analysis and would never be seen without the multigenerational approach. We show that a statistical analysis of entropy and information gain in genotypes of a selected population can reveal further interactions than previously seen. Importantly this must be done in comparison to the unselected population's genotypes to account for inherent biases in the original population.2017-08-21T10:03:01ZStochastic Separation Theorems
http://hdl.handle.net/2381/40202
Title: Stochastic Separation Theorems
Authors: Gorban, A. N.; Tyukin, I. Y.
Abstract: The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear discriminants. Surprisingly, separation of a new image from a very large set of known images is almost always possible even in moderately high dimensions by linear functionals, and coefficients of these functionals can be found explicitly. Based on fundamental properties of measure concentration, we show that for $M<a\exp(b{n})$ random $M$-element sets in $\mathbb{R}^n$ are linearly separable with probability $p$, $p>1-\vartheta$, where $1>\vartheta>0$ is a given small constant. Exact values of $a,b>0$ depend on the probability distribution that determines how the random $M$-element sets are drawn, and on the constant $\vartheta$. These {\em stochastic separation theorems} provide a new instrument for the development, analysis, and assessment of machine learning methods and algorithms in high dimension. Theoretical statements are illustrated with numerical examples.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2017-08-21T09:52:59ZMaximal zero product subrings and inner ideals of simple rings
http://hdl.handle.net/2381/40198
Title: Maximal zero product subrings and inner ideals of simple rings
Authors: Baranov, Alexander; Fernández López, Antonio
Abstract: Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S^2=0. We classify all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.
Description: MSC classes: 16D30, 17B602017-08-18T09:54:27ZA posteriori error estimates for leap-frog and cosine methods for second order evolution problems
http://hdl.handle.net/2381/40197
Title: A posteriori error estimates for leap-frog and cosine methods for second order evolution problems
Authors: Georgoulis, Emmanuil H.; Lakkis, Omar; Makridakis, Charalambos G.; Virtanen, Juha M.
Abstract: We consider second order explicit and implicit two-step time-discrete schemes for wave-type equations. We derive optimal order a posteriori estimates controlling the time discretization error. Our analysis has been motivated by the need to provide a posteriori estimates for the popular leap-frog method (also known as Verlet's method in the molecular dynamics literature); it is extended, however, to general cosine-type second order methods. The estimators are based on a novel reconstruction of the time-dependent component of the approximation. Numerical experiments confirm similarity of the convergence rates of the proposed estimators and the theoretical convergence rate of the true error.
Description: AMS Subject Headings 35L05, 37M05, 37M15, 65M60, 65N502017-08-17T10:27:51ZModelling Share Prices as a Random Walk on a Markov Chain
http://hdl.handle.net/2381/40129
Title: Modelling Share Prices as a Random Walk on a Markov Chain
Authors: Samci Karadeniz, Rukiye
Abstract: In the financial area, a simple but also realistic means of modelling real data is very important. Several approaches are considered to model and analyse the data presented herein. We start by considering a random walk on an additive functional of a discrete time Markov chain perturbed by Gaussian noise as a model for the data as working with a continuous time model is more convenient for option prices. Therefore, we consider the renowned (and open) embedding problem for Markov chains: not every discrete time Markov chain has an underlying continuous time Markov chain. One of the main goals of this research is to analyse whether the discrete time model permits extension or embedding to the continuous time model. In addition, the volatility of share price data is estimated and analysed by the same procedure as for share price processes. This part of the research is an extensive case study on the embedding problem for financial data and its volatility.
Another approach to modelling share price data is to consider a random walk on the lamplighter group. Specifically, we model data as a Markov chain with a hidden random walk on the lamplighter group Z3 and on the tensor product of groups Z2 ⊗ Z2. The lamplighter group has a specific structure where the hidden information is actually explicit. We assume that the positions of the lamplighters are known, but we do not know the status of the lamps. This is referred to as a hidden random walk on the lamplighter group. A biased random walk is constructed to fit the data. Monte Carlo simulations are used to find the best fit for smallest trace norm difference of the transition matrices for the tensor product of the original transition matrices from the (appropriately split) data.
In addition, splitting data is a key method for both our first and second models. The tensor product structure comes from the split of the data. This requires us to deal with the missing data. We apply a variety of statistical techniques such as Expectation- Maximization Algorithm and Machine Learning Algorithm (C4.5).
In this work we also analyse the quantum data and compute option prices for the binomial model via quantum data.2017-08-02T14:10:44ZAlgebras and varieties
http://hdl.handle.net/2381/40125
Title: Algebras and varieties
Authors: Green, Edward L.; Hille, Lutz; Schroll, Sibylle
Abstract: In this paper we introduce new affine algebraic varieties whose points correspond to quotients of paths algebras. We show that the algebras within a variety share many important homological properties. The case of finite dimensional algebras as well as that of graded algebras arise as classes of subvarieties of the varieties we define.
Description: 20 pages2017-08-02T13:25:47ZOn extensions for gentle algebras
http://hdl.handle.net/2381/40124
Title: On extensions for gentle algebras
Authors: Canakci, Ilke; Pauksztello, David; Schroll, Sibylle
Abstract: We develop an algorithmic method for determining the cohomology of homotopy string and band complexes in the derived category of a gentle algebra. We then use this to give a complete description of a basis of the extensions between string and quasi-simple band modules in the module category of a gentle algebra.
Description: 32 pages, comments welcome2017-08-02T13:21:14ZOn the representation dimension of monomial and self-injective special multiserial algebras
http://hdl.handle.net/2381/40091
Title: On the representation dimension of monomial and self-injective special multiserial algebras
Authors: Schroll, Sibylle
Abstract: For a monomial special multiserial algebra, which in general is of wild representation type, we construct a radical embedding into an algebra of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less than or equal to three.
Description: 5 pages, this version corrects a mistake in the previous version2017-07-27T16:26:36ZAlmost gentle algebras and their trivial extensions
http://hdl.handle.net/2381/40090
Title: Almost gentle algebras and their trivial extensions
Authors: Green, Edward L.; Schroll, Sibylle
Abstract: In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that admissible cuts of Brauer configuration algebras give rise to gentle algebras and as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one, is the trivial extension of an almost gentle algebra.
Description: 2010 Mathematics Subject Classification. 16G20,2017-07-27T16:20:58ZBrauer configuration algebras: A generalization of Brauer graph algebras
http://hdl.handle.net/2381/40089
Title: Brauer configuration algebras: A generalization of Brauer graph algebras
Authors: Green, Edward L.; Schroll, Sibylle
Abstract: In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying and analysing structural properties of Brauer configurations and Brauer configuration algebras, we show that a Brauer configuration algebra is multiserial; that is, its Jacobson radical is a sum of uniserial modules whose pairwise intersection is either zero or a simple module. The paper ends with a detailed study of the relationship between radical cubed zero Brauer configuration algebras, symmetric matrices with non-negative integer entries, finite graphs and associated symmetric radical cubed zero algebras.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.; MSC
16G20; 16D502017-07-27T15:42:55ZSpecial multiserial algebras, Brauer configuration algebras and more : a survey
http://hdl.handle.net/2381/40088
Title: Special multiserial algebras, Brauer configuration algebras and more : a survey
Authors: Green, Edward L.; Schroll, Sibylle
Abstract: We survey results on multiserial algebras, special multiserial algebras and Brauer configuration algebras. A structural property of modules over a special multiserial algebra is presented. Almost gentle algebras are introduced and we describe some results related to this class of algebras. We also report on the structure of radical cubed zero symmetric algebras.
Description: 2010 Mathematics Subject Classification. 16S37, 14M99,16W602017-07-27T15:03:27ZBrauer graph algebras
http://hdl.handle.net/2381/40087
Title: Brauer graph algebras
Authors: Schroll, Sibylle
Abstract: These lecture notes on Brauer graph algebras are the result of a series of four lectures given at the CIMPA research school in Mar del Plata, Argentina, in March 2016. After motivating the study of Brauer graph algebras by relating them to special biserial algebras, the definition of Brauer graph algebras is given in great detail with many examples to illustrate the concepts. This is followed by a short section on the interpretation of Brauer graphs as decorated ribbon graphs. A section on gentle algebras and their graphs, trivial extensions of gentle algebras, admissible cuts of Brauer graph algebras and a first connection of Brauer graph algebras with Jacobian algebras associated to triangulations of marked oriented surfaces follows. The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras with derived equivalences of frozen Jacobian algebras is the topic of the next section. In the last section, after defining Green's walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.
Description: 55 pages, many figures and examples throughout, comments and suggestions welcome, minor corrections2017-07-27T14:38:44ZAdaptivity and blow-up detection for nonlinear evolution problems
http://hdl.handle.net/2381/40080
Title: Adaptivity and blow-up detection for nonlinear evolution problems
Authors: Cangiani, Andrea; Georgoulis, Emmanuil H.; Kyza, Irene; Metcalfe, Stephen
Abstract: This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posteriori error bounds are derived in the $L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time implicit-explicit interior penalty discontinuous Galerkin in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time-stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.2017-07-13T13:08:01ZVirtual Element Methods
http://hdl.handle.net/2381/39955
Title: Virtual Element Methods
Authors: Sutton, Oliver James
Abstract: In this thesis we study the Virtual Element Method, a recent generalisation of the standard conforming Finite Element Method offering high order approximation spaces on computational meshes consisting of general polygonal or polyhedral elements. Our particular focus is on developing the tools required to use the method as the foundation of mesh adaptive algorithms which are able to take advantage of the flexibility offered by such general meshes.
We design virtual element discretisations of certain classes of linear second order elliptic and parabolic partial differential equations, and present a detailed exposition of their implementation aspects. An outcome of this is a concise and usable 50-line MATLAB implementation of a virtual element method for solving a model elliptic problem on general polygonal meshes, the code for which is included as an appendix. Optimal order convergence rates in the H1 and L2 norms are proven for the discretisation of elliptic problems. Alongside these, we derive fully computable residual-type a posteriori estimates of the error measured in the H1 and L2 norms for the methods we develop for elliptic problems, and in the L2(0; T;H1) and L∞(0; T;L2) norms for parabolic problems. In deriving the L∞(0; T;L2) error estimate, we introduce a new technique (which translates naturally back into the setting of conventional finite element methods) to produce estimates with effectivities which become constant for long time simulations. Mesh adaptive algorithms, designed around these methods and computable error estimates, are proposed and numerically assessed in a variety of challenging stationary and time-dependent scenarios.
We further propose a virtual element discretisation and computable coarsening/refinement indicator for a system of semilinear parabolic partial differential equations which we apply to a Lotka-Volterra type model of interacting species. These components form the basis of an adaptive method which we use to reveal a variety of new pattern-forming mechanisms in the cyclic competition model.2017-06-26T13:44:08ZDynamics of a Two Subpopulations System Including Immigration
http://hdl.handle.net/2381/39942
Title: Dynamics of a Two Subpopulations System Including Immigration
Authors: Zincenko, A.; Petrovskii, Sergei
Abstract: The phenomenon of replacement migration into declining population prompts development of multicomponent models in population dynamics. We propose a simple model of population including resident and migrant components with migration flow as an external input. The main assumption is that offspring that are born to migrants will have the same vital rates as the resident population. The proposed model is based on partial differential equation to take into account the age structure of the population. The formulae for exact solutions are derived. We focus on the case when native population declines in the absence of migration. Assuming a sufficiently large constant growth rate of migration we obtain asymptotic solutions as t → â. Using the asymptotic solutions, we have calculated "critical" value of growth rate of migrant inflow that is the value that provides, as time tends to infinity, the equal number of residents and migrants in the population. We provide numerical illustrations using demographic data for Germany in 2010.2017-06-22T14:22:55ZWhich Random Walk is Faster? Methods to Compare Different Step Length Distributions in Individual Animal Movement
http://hdl.handle.net/2381/39940
Title: Which Random Walk is Faster? Methods to Compare Different Step Length Distributions in Individual Animal Movement
Authors: Choules, J. D.; Petrovskii, Sergei
Abstract: Good understanding of individual animal movement is needed in the context of epidemiology in order to predict the rate of spread of infectious diseases. It is also required for problems arising in nature conservation, biological invasion, pest monitoring, etc. A question that often appears in the centre of the movement studies is which movement pattern is 'faster' or more efficient. For instance, it is widely believed that the pattern quantified by a power law distribution of movement steps is faster than the Brownian motion. Here we show that the answer to this question may be not so straightforward and depends on the way how different step length distributions are compared.2017-06-22T14:04:28ZSpectral Sequences for Hochschild cohomology and graded centers of derived categories
http://hdl.handle.net/2381/39908
Title: Spectral Sequences for Hochschild cohomology and graded centers of derived categories
Authors: Neumann, Frank; Szymik, Markus
Abstract: The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general. To illustrate this, we discuss modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-06-14T15:08:22ZAdaptive radial basis function interpolation using an error indicator
http://hdl.handle.net/2381/39810
Title: Adaptive radial basis function interpolation using an error indicator
Authors: Zhang, Qi; Zhao, Yangzhang; Levesley, Jeremy
Abstract: In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.2017-05-17T15:34:19ZA Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation
http://hdl.handle.net/2381/39785
Title: A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation
Authors: Banjai, Lehel; Georgoulis, Emmanuil H.; Lijoka, Oluwaseun
Abstract: A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339-363; Comput. Methods Appl. Mech. Engrg., 84 (1990), pp. 327-348]. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behavior of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions d = 1 and d = 2. Numerical experiments highlight the effectivness of the Trefftz method in problems with energy at high frequencies.2017-05-15T13:54:03ZOn the non-parallel instability of the rotating-sphere boundary layer
http://hdl.handle.net/2381/39778
Title: On the non-parallel instability of the rotating-sphere boundary layer
Authors: Segalini, Antonio; Garrett, Stephen J.
Abstract: We present a new solution for the steady boundary-layer flow over the rotating sphere that also accounts for the eruption of the boundary layer at the equator and other higher-order viscous effects. Non-parallel corrections to the local type I and type II convective instability modes of this flow are also computed as a function of spin rate. Our instability results are associated with the previously observed spiral vortices and remarkable agreement between our predictions of the number of vortices and experimental observations is found. Vortices travelling at 70 %–80 % of the local surface speed are found to be the most amplified for sufficient spin rates, also consistent with prior experimental observations.2017-05-15T10:38:27ZApplications of Quaternionic Holomorphic Geometry to minimal surfaces
http://hdl.handle.net/2381/39775
Title: Applications of Quaternionic Holomorphic Geometry to minimal surfaces
Authors: Leschke, K.; Moriya, K.
Abstract: In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface.2017-05-12T15:29:05ZInteraction of human migration and wealth distribution
http://hdl.handle.net/2381/39760
Title: Interaction of human migration and wealth distribution
Authors: Volpert, V.; Petrovskii, Sergei; Zincenko, A.
Abstract: Dynamics of human populations depends on various economical and social factors. Their migration is partially determined by the economical conditions and it can also influence these conditions. This work is devoted to the analysis of the interaction of human migration and wealth distribution. The model consists of a system of equations for the population density and for the wealth distribution with conventional diffusion terms and with cross diffusion terms describing human migration determined by the wealth gradient and wealth flux determined by human migration. Wealth production and consumption depend on the population density while the natality and mortality rates depend on the level of wealth. In the absence of cross diffusion terms, dynamics of solutions is described by travelling wave solutions of the corresponding reaction-diffusion systems of equations. We show persistence of such solutions for sufficiently small cross diffusion coefficients. This result is based on the perturbation methods and on the spectral properties of the linearized operators.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above. Following the embargo period the above license will apply.2017-05-10T15:05:00ZFortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations
http://hdl.handle.net/2381/39759
Title: Fortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations
Authors: Potts, Jonathan R.; Petrovskii, Sergei V.
Abstract: Animal movement is a key mechanism for shaping population dynamics. The effect of interactions between competing animals on a population's survival has been studied for many decades. However, interactions also affect an animal's subsequent movement decisions. Despite this, the indirect effect of these decisions on animal survival is much less well-understood. Here, we incorporate movement responses to foreign animals into a model of two competing populations, where inter-specific competition is greater than intra-specific competition. When movement is diffusive, the travelling wave moves from the stronger population to the weaker. However, by incorporating behaviourally induced directed movement towards the stronger population, the weaker one can slow the travelling wave down, even reversing its direction. Hence movement responses can switch the predictions of traditional mechanistic models. Furthermore, when environmental heterogeneity is combined with aggressive movement strategies, it is possible for spatially segregated co-existence to emerge. In this situation, the spatial patterns of the competing populations have the unusual feature that they are slightly out-of-phase with the environmental patterns. Finally, incorporating dynamic movement responses can also enable stable co-existence in a homogeneous environment, giving a new mechanism for spatially segregated co-existence.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above. Following the embargo period the above license applies.2017-05-10T14:56:57ZBabuška-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshes
http://hdl.handle.net/2381/39708
Title: Babuška-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshes
Authors: Georgoulis, Emmanuil; Makridakis, Charalambos; Pryer, Tristan
Abstract: We prove the inf-sup stability of the interior penalty class of discontinuous Galerkin schemes in unbalanced mesh-dependent norms, under a mesh condition allowing for a general class of meshes, which includes many examples of geometrically graded element neighbourhoods. The inf-sup condition results in the stability of the interior penalty Ritz projection in $L^2$ as well as, for the first time, quasi-best approximations in the $L^2$-norm which in turn imply a priori error estimates that do not depend on the global maximum meshsize in that norm. Some numerical experiments are also given.2017-04-27T14:22:23ZSmall Cocycles, Fine Torus Fibrations, and a Z^2 Subshift with Neither
http://hdl.handle.net/2381/39673
Title: Small Cocycles, Fine Torus Fibrations, and a Z^2 Subshift with Neither
Authors: Clark, Alex; Sadun, Lorenzo
Abstract: Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free ZdZd actions on Cantor sets admit “small cocycles.” These represent classes in H1H1 that are mapped to small vectors in RdRd by the Ruelle–Sullivan (RS) map. We show that there exist Z2Z2 actions where no such small cocycles exist, and where the image of H1H1 under RS is Z2Z2 . Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of “virtual eigenvalues,” i.e., elements of RdRd that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles.2017-04-19T15:45:19ZCalculating Exceedance Probabilities Using a Distributionally Robust Method
http://hdl.handle.net/2381/39669
Title: Calculating Exceedance Probabilities Using a Distributionally Robust Method
Authors: Faridafshin, Farzad; Grechuk, Bogdan; Naess, Arvid
Abstract: Calculation of exceedance probabilities or the inverse problem of finding the level corresponding to a given exceedance probability occurs in many practical applications. For instance, it is often of interest in offshore engineering to evaluate the wind, wave, current, and sea ice properties with annual exceedance probabilities of, e.g., 10−1, 10−2, and 10−3, or so-called 10-year, 100-year, and 1000-year values. A methodology is provided in this article to calculate a tight upper bound of the exceedance probability, given any probability distribution from a wide range of commonly used distributions. The approach is based on a generalization of the Chebyshev inequality for the class of distributions with a logarithmically concave cumulative distribution function, and has the potential to relieve the often-debated exercise of determining an appropriate probability distribution function based on limited data, particularly in terms of tail behavior. Two numerical examples are provided for illustration.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-04-19T14:49:39ZRegimes of electrostatic collapse of a highly charged polyelectrolyte in a poor solvent
http://hdl.handle.net/2381/39651
Title: Regimes of electrostatic collapse of a highly charged polyelectrolyte in a poor solvent
Authors: Tom, Anvy Moly; Vemparala, Satyavani; Rajesh, R.; Brilliantov, Nikolai V.
Abstract: We perform extensive molecular dynamics simulations of a highly charged, collapsed, flexible polyelectrolyte chain in a poor solvent for the case when the electrostatic interactions, characterized by the reduced Bjerrum length ℲB, are strong. We find the existence of several sub-regimes in the dependence of the gyration radius of the chain Rg on ℲB characterized by Rg ∼ Ⅎ−γB. In contrast to a good solvent, the exponent γ for a poor solvent crucially depends on the size and valency of the counterions. To explain the different sub-regimes, we generalize the existing counterion fluctuation theory by including a more complete account of all possible volume interactions in the free energy of the polyelectrolyte chain. We also show that the presence of condensed counterions modifies the effective attraction among the chain monomers and modulates the sign of the second virial coefficient under poor solvent conditions.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-04-11T11:12:38ZBeyond Navier–Stokes equations: capillarity of ideal gas
http://hdl.handle.net/2381/39650
Title: Beyond Navier–Stokes equations: capillarity of ideal gas
Authors: Gorban, Alexander N.; Karlin, I. V.
Abstract: The system of Navier–Stokes–Fourier equations is one of the most celebrated systems of equations in modern science. It describes dynamics of fluids in the limit when gradients of density, velocity and temperature are sufficiently small, and loses its applicability when the flux becomes so non-equilibrium that the changes of velocity, density or temperature on the length compatible with the mean free path are non-negligible. The question is: how to model such fluxes? This problem is still open. (Despite the fact that the first ‘final equations of motion’ modified for analysis of thermal creep in rarefied gas were proposed by Maxwell in 1879.) There are, at least, three possible answers: (i) use molecular dynamics with individual particles, (ii) use kinetic equations, like Boltzmann’s equation or (iii) find a new system of equations for description of fluid dynamics with better accounting of non-equilibrium effects. These three approaches work at different scales. We explore the third possibility using the recent findings of capillarity of internal layers in ideal gases and of saturation effect in dissipation (there is a limiting attenuation rate for very short waves in ideal gas and it cannot increase infinitely). One candidate equation is discussed in more detail, the Korteweg system proposed in 1901. The main ideas and approaches are illustrated by a kinetic system for which the problem of reduction of kinetics to fluid dynamics is analytically solvable.
Description: The file associated with this record is embargoed until 12 months after the date of publication. The final published version may be available through the links above.2017-04-11T11:06:09ZModern Mathematical Methods for Actuarial Sciences
http://hdl.handle.net/2381/39613
Title: Modern Mathematical Methods for Actuarial Sciences
Authors: Kaya, Ahmet
Abstract: In the ruin theory, premium income and outgoing claims play an important role. We introduce several ruin type mathematical models and apply various mathematical methods to find optimal premium price for the insurance companies. Quantum theory is one of the significant novel approaches to compute the finite time non-ruin probability. More exactly, we apply the discrete space Quantum mechanics formalism (see main thesis for formalism) and continuous space Quantum mechanics formalism (see main thesis for formalism) with the appropriately chosen Hamiltonians.
Several particular examples are treated via the traditional basis and quantum mechanics formalism with the different eigenvector basis. The numerical results are also obtained using the path calculation method and compared with the stochastic modeling results.
In addition, we also construct various models with interest rate. For these models, optimal premium prices are stochastically calculated for independent and dependent claims with different dependence levels by using the Frank copula method.2017-04-03T08:09:18ZAdaptive large-scale mantle convection simulations
http://hdl.handle.net/2381/39571
Title: Adaptive large-scale mantle convection simulations
Authors: Cox, Samuel Peter
Abstract: The long-term motion of the Earth's mantle is of considerable interest to geologists and geodynamists in explaining the evolution of the planet and its internal and surface history. The inaccessible nature of the mantle necessitates the use of computer simulations to further our understanding of the processes underlying the motion of tectonic plates.
Numerical methods employed to solve the equations describing this motion lead to linear systems of a size which stretch the current capabilities of supercomputers to their limits. Progress towards the satisfactory simulation of this process is dependent upon the use of new mathematical and computational ideas in order to bring the largest problems within the reach of current computer architectures.
In this thesis we present an implementation of the discontinuous Galerkin method, coupled to a more traditional finite element method, for the simulation of this system. We also present an a posteriori error estimate for the convection-diffusion equation without reaction, using an exponential fitting technique and artificial reaction to relax the restrictions upon the derivative of the convection field that are usually imposed within the existing literature. This error bound is used as the basis of an h-adaptive mesh refinement strategy. We present an implementation of the calculation of this bound alongside the simulation and the indicator, in a parallelised C++ code, suitable for use in a distributed computing setting.
Finally, we present an implementation of the discontinuous Galerkin method into the community code ASPECT, along with an adaptivity indicator based upon the proven a posteriori error bound. We furnish both implementations with numerical examples to explore the applicability of these methods to a number of circumstances, with the aim of reducing the computational cost of large mantle convection simulations.2017-03-27T10:20:14ZA dissipative force between colliding viscoelastic bodies: Rigorous approach
http://hdl.handle.net/2381/39544
Title: A dissipative force between colliding viscoelastic bodies: Rigorous approach
Authors: Brilliantov, Nikolay V.; Pimenova, Anastasiya V.; Goldobin, Denis S.
Abstract: A collision of viscoelastic bodies is analysed within a mathematically rigorous approach. We develop a perturbation scheme to solve continuum mechanics equation, which deals simultaneously with strain and strain rate in the bulk of the bodies' material. We derive dissipative force that acts between particles and express it in terms of particles' deformation, deformation rate and material parameters. It differs noticeably from the currently used dissipative force, found within the quasi-static approximation and does not suffer from inconsistencies of this approximation. The proposed approach may be used for other continuum mechanics problems where the bulk dissipation is addressed.2017-03-22T13:48:38ZA Generalist Predator Regulating Spread of a Wildlife Disease: Exploring Two Infection Transmission Scenarios
http://hdl.handle.net/2381/39543
Title: A Generalist Predator Regulating Spread of a Wildlife Disease: Exploring Two Infection Transmission Scenarios
Authors: Sen, M.; Banerjee, M.; Morozov, A.
Abstract: Ecoepidemiology is a well-developed branch of theoretical ecology, which explores interplay between the trophic interactions and the disease spread. In most ecoepidemiological models, however, the authors assume the predator to be a specialist, which consumes only a single prey species. In few existing papers, in which the predator was suggested to be a generalist, the alternative food supply was always considered to be constant. This is obviously a simplification of reality, since predators can often choose between a number of different prey. Consumption of these alternative prey can dramatically change their densities and strongly influence the model predictions. In this paper, we try to bridge the gap and explore a generic eco-epidemiological system with a generalist predator, where the densities of all prey are dynamical variables. The model consists of two prey species, one of which is subject to an infectious disease, and a predator, which consumes both prey species. We investigate two main scenarios of infection transmission mode: (i) the disease transmission rate is predator independent and (ii) the transmission rate is a function of predator density. For both scenarios we fulfil an extensive bifurcation analysis. We show that including a second dynamical prey in the system can drastically change the dynamics of the single prey case. In particular, the presence of a second prey impedes disease spread by decreasing the basic reproduction number and can result in a substantial drop of the disease prevalence. We demonstrate that with efficient consumption of the second prey species by the predator, the predator-dependent disease transmission can not destabilize interactions, as in the case with a specialist predator. Interestingly, even if the population of the second prey eventually vanishes and only one prey species finally remains, the system with two prey species may exhibit different properties to those of the single prey system.2017-03-22T13:39:47ZModelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue
http://hdl.handle.net/2381/39542
Title: Modelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue
Authors: Morozov, A.2017-03-22T10:03:33ZNoise-Produced Patterns in Images Constructed from Magnetic Flux Leakage Data
http://hdl.handle.net/2381/39541
Title: Noise-Produced Patterns in Images Constructed from Magnetic Flux Leakage Data
Authors: Goldobin, D. S.; Pimenova, A. V.; Levesley, J.; Elkington, P.; Bacciarelli, M.
Abstract: Magnetic flux leakage measurements help identify the position, size and shape of corrosion-related defects in steel casings used to protect boreholes drilled into oil and gas reservoirs. Images constructed from magnetic flux leakage data contain patterns related to noise inherent in the method. We investigate the patterns and their scaling properties for the case of delta-correlated input noise, and consider the implications for the method’s ability to resolve defects. The analytical evaluation of the noise-produced patterns is made possible by model reduction facilitated by large-scale approximation. With appropriate modification, the approach can be employed to analyze noise-produced patterns in other situations where the data of interest are not measured directly, but are related to the measured data by a complex linear transform involving integrations with respect to spatial coordinates.
Description: Mathematics Subject Classification: 78A30 / 78M34 / 60G602017-03-22T09:58:52ZMagnetic Flux Leakage Method: Large-Scale Approximation
http://hdl.handle.net/2381/39540
Title: Magnetic Flux Leakage Method: Large-Scale Approximation
Authors: Pimenova, A. V.; Goldobin, D. S.; Levesley, J.; Ivantsov, A. O.; Elkington, P.; Bacciarelli, M.
Abstract: We consider the application of the magnetic flux leakage (MFL) method to the detection of defects in ferromagnetic (steel) tubulars. The problem setup corresponds to the cases where the distance from the casing and the point where the magnetic field is measured is small compared to the curvature radius of the undamaged casing and the scale of inhomogeneity of the magnetic field in the defect-free case. Mathematically this corresponds to the planar ferromagnetic layer in a uniform magnetic field oriented along this layer. Defects in the layer surface result in a strong deformation of the magnetic field, which provides opportunities for the reconstruction of the surface profile from measurements of the magnetic field. We deal with large-scale defects whose depth is small compared to their longitudinal sizes—these being typical of corrosive damage. Within the framework of large-scale approximation, analytical relations between the casing thickness profile and the measured magnetic field can be derived.
Description: Mathematics Subject Classification: 78A30 / 78M34 / 78A552017-03-22T09:55:42ZThree Waves of Chemical Dynamics
http://hdl.handle.net/2381/39539
Title: Three Waves of Chemical Dynamics
Authors: Gorban, A. N.; Yablonsky, G. S.
Abstract: Three epochs in development of chemical dynamics are presented. We try to understand the modern research programs in the light of classical works.2017-03-22T09:46:50ZGeneration of mechanical force by grafted polyelectrolytes in an electric field
http://hdl.handle.net/2381/39530
Title: Generation of mechanical force by grafted polyelectrolytes in an electric field
Authors: Brilliantov, N. V.; Budkov, Yu. A.; Seidel, C.
Abstract: We study theoretically and by means of molecular dynamics (MD) simulations the generation of mechanical force by grafted polyelectrolytes in an external electric field, which favors its adsorption on the grafting plane. The force arises in deformable bodies linked to the free end of the chain. Varying the field, one controls the length of the nonadsorbed part of the chain and hence the deformation of the target body, i.e., the arising force too. We consider target bodies with a linear force-deformation relation and with a Hertzian one. While the first relation models a coiled Gaussian chain, the second one describes the force response of a squeezed colloidal particle. The theoretical dependences of generated force and compression of the target body on an applied field agree very well with the results of MD simulations. The analyzed phenomenon may play an important role in future nanomachinery, e.g., it may be used to design nanovices to fix nanosized objects.2017-03-21T10:25:12ZForward-Invariant Peeling in Chemical Dynamics: a Simple Case Study
http://hdl.handle.net/2381/39528
Title: Forward-Invariant Peeling in Chemical Dynamics: a Simple Case Study
Authors: Gorban, A. N.
Abstract: Forward-invariant peeling aims to produce forward-invariant subset from a given set in phase space. The structure of chemical kinetic equations allows us to describe the general operations of the forward-invariant peeling. For example, we study a simple reaction network with three components A1,A2,A3 and reactions A1 → A2 → A3 → A1, 2A1 ⇌ 3A2 (without any stoichiometric conservation law). We assume that kinetics obey the classical mass action law and reaction rate constants are positive intervals 0 <ki min ≤ ki ≤ ki max< ∞. Kinetics of this system is described by a system of differential inclusions. We produce forward-invariant sets for these kinetic inclusions from the sets { c | ci ≥ 0, ∑ ci ≥ ε } by the forward-invariant peeling (for sufficiently small ε> 0). In particular, this construction proves persistence of this kinetic system (a positive solution cannot approach the origin even asymptotically, as t → ∞).
Description: Mathematics Subject Classification: 37C10, 34D20, 93D052017-03-21T10:12:37ZGeneralized Mass Action Law and Thermodynamics of Nonlinear Markov Processes
http://hdl.handle.net/2381/39527
Title: Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes
Authors: Gorban, A. N.; Kolokoltsov, V. N.
Abstract: The nonlinear Markov processes are measure-valued dynamical systems which preserve positivity. They can be represented as the law of large numbers limits of general Markov models of interacting particles. In physics, the kinetic equations allow Lyapunov functionals (entropy, free energy, etc.). This may be considered as a sort of inheritance of the Lyapunov functionals from the microscopic master equations. We study nonlinear Markov processes that inherit thermodynamic properties from the microscopic linear Markov processes. We develop the thermodynamics of nonlinear Markov processes and analyze the asymptotic assumption, which are sufficient for this inheritance.
Description: Mathematics Subject Classification: 80A30 / 60J25 / 60J60 / 60J75 / 82B402017-03-21T09:59:51ZThe Impact of Fragmented Habitat's Size and Shape on Populations with Allee Effect
http://hdl.handle.net/2381/39524
Title: The Impact of Fragmented Habitat's Size and Shape on Populations with Allee Effect
Authors: Alharbi, W. G.; Petrovskii, S. V.
Abstract: This study aims to explore the ways in which population dynamics are affected by the shape and size of fragmented habitats. Habitat fragmentation has become a key concern in ecology over the past 20 years as it is thought to increase the threat of extinction for a number of plant and animal species; particularly those close to the fragment edge. In this study, we consider this issue using mathematical modelling and computer simulations in several domains of various shape and with different strength of the Allee effect. A two-dimensional reaction-diffusion equation (taking the Allee effect into account) is used as a model. Extensive simulations are performed in order to determine how the boundaries impact the population persistence. Our results indicate the following: (i) for domains of simple shape (e.g. rectangle), the effect of the critical patch size (amplified by the Allee effect) is similar to what is observed in 1D space, in particular, the likelihood of population survival is determined by the interplay between the domain size and thee strength of the Allee effect; (ii) in domains of complicated shape, for the population to survive, the domain area needs to be larger than the area of the corresponding rectangle. Hence, it can be concluded that domain size and shape both have crucial effect on population survival.
Description: Mathematics Subject Classification: 92D40 / 35B36 / 35Q92 / 37N252017-03-21T09:34:56ZModelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue
http://hdl.handle.net/2381/39523
Title: Modelling in Ecology, Epidemiology and Ecoepidemiology: Introduction to the Special Issue
Authors: Morozov, A.; Petrovskii, S.2017-03-20T17:05:07ZFast Sampling of Evolving Systems with Periodic Trajectories
http://hdl.handle.net/2381/39522
Title: Fast Sampling of Evolving Systems with Periodic Trajectories
Authors: Tyukin, I. Yu.; Gorban, A. N.; Tyukina, T. A.; Al-Ameri, J. M.; Korablev, Yu. A.
Abstract: We propose a novel method for fast and scalable evaluation of periodic solutions of systems of ordinary differential equations for a given set of parameter values and initial conditions. The equations governing the system dynamics are supposed to be of a special class, albeit admitting nonlinear parametrization and nonlinearities. The method enables to represent a given periodic solution as sums of computable integrals and functions that are explicitly dependent on parameters of interest and initial conditions. This allows invoking parallel computational streams in order to increase speed of calculations. Performance and practical implications of the method are illustrated with examples including classical predator-prey system and models of neuronal cells.
Description: Mathematics Subject Classification: 93B30 / 34A05 / 92B99 / 93B152017-03-20T16:55:54ZTri-trophic Plankton Models Revised: Importance of Space, Food Web Structure and Functional Response Parametrisation
http://hdl.handle.net/2381/39519
Title: Tri-trophic Plankton Models Revised: Importance of Space, Food Web Structure and Functional Response Parametrisation
Authors: Egilmez, H. I.; Morozov, A. Yu.
Abstract: Revealing mechanisms of efficient top-down control in eutrophic ecosystems remains a long term challenge in theoretical ecology. In this paper, we revisit the role of environmental heterogeneity, food web structure and shape of the predator functional response in persistence and stabilization of a planktonic system with high nutrient supply. We consider a 1D vertically resolved tri-trophic planktonic food web composed of a primary producer, an intermediate predator and a highly mobile top predator, such as a system of phytoplankton, microzooplankton and copepods. We explore the realistic scenario in which the top predator is omnivorous, i.e. when copepods can feed both on phytoplankton and microzooplankton. We show that the interplay between heterogeneity of the environment due to for instance, a light gradient in the water column, and trophic interaction between species can result in an efficient top-down control which would otherwise be impossible in the corresponding well-mixed system. We also find that allowing the top predator to feed on the primary producer may dramatically impede the coexistence of the three trophic levels, with only two levels generally surviving. The coexistence of all three trophic levels within a wide range of parameters becomes possible only when the top predator exhibits active food source switching behaviour. We also show the phenomenon of bistability in the considered tri-trophic food web: a small initial amount of the top predator should lead to its extinction whereas introduction of a supercritical initial amount will eventually result in establishment of the population. The demonstrated mechanism of top-down control seems to be rather generic and might be a good candidate to explain stability in some other non-planktonic tri-trophic ecosystems.
Description: Mathematics Subject Classification: 47A75 / 45K05 / 92D402017-03-20T16:19:32ZPreface. Bifurcations and Pattern Formation in Biological Applications
http://hdl.handle.net/2381/39517
Title: Preface. Bifurcations and Pattern Formation in Biological Applications
Authors: Morozov, A.; Ptashnyk, M.; Volpert, V.
Abstract: In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena.2017-03-20T16:05:40ZA random walk description of individual animal movement accounting for periods of rest
http://hdl.handle.net/2381/39516
Title: A random walk description of individual animal movement accounting for periods of rest
Authors: Tilles, Paulo F. C.; Petrovskii, Sergei V.; Natti, Paulo L.
Abstract: Animals do not move all the time but alternate the period of actual movement (foraging) with periods of rest (e.g. eating or sleeping). Although the existence of rest times is widely acknowledged in the literature and has even become a focus of increased attention recently, the theoretical approaches to describe animal movement by calculating the dispersal kernel and/or the mean squared displacement (MSD) rarely take rests into account. In this study, we aim to bridge this gap. We consider a composite stochastic process where the periods of active dispersal or ‘bouts’ (described by a certain baseline probability density function (pdf) of animal dispersal) alternate with periods of immobility. For this process, we derive a general equation that determines the pdf of this composite movement. The equation is analysed in detail in two special but important cases such as the standard Brownian motion described by a Gaussian kernel and the Levy flight described by a Cauchy distribution. For the Brownian motion, we show that in the large-time asymptotics the effect of rests results in a rescaling of the diffusion coefficient. The movement occurs as a subdiffusive transition between the two diffusive asymptotics. Interestingly, the Levy flight case shows similar properties, which indicates a certain universality of our findings.2017-03-20T14:54:02ZCatching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization.
http://hdl.handle.net/2381/39515
Title: Catching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization.
Authors: Petrovskaya, Natalia; Petrovskii, Sergei
Abstract: Synchronization of population dynamics in different habitats is a frequently observed phenomenon. A common mathematical tool to reveal synchronization is the (cross)correlation coefficient between time courses of values of the population size of a given species where the population size is evaluated from spatial sampling data. The corresponding sampling net or grid is often coarse, i.e. it does not resolve all details of the spatial configuration, and the evaluation error-i.e. the difference between the true value of the population size and its estimated value-can be considerable. We show that this estimation error can make the value of the correlation coefficient very inaccurate or even irrelevant. We consider several population models to show that the value of the correlation coefficient calculated on a coarse sampling grid rarely exceeds 0.5, even if the true value is close to 1, so that the synchronization is effectively lost. We also observe 'ghost synchronization' when the correlation coefficient calculated on a coarse sampling grid is close to 1 but in reality the dynamics are not correlated. Finally, we suggest a simple test to check the sampling grid coarseness and hence to distinguish between the true and artifactual values of the correlation coefficient.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2017-03-20T14:47:07Z