| 1. |
- Wang, Siyang, et al.
(författare)
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CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS
- 2018
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 87:314, s. 2737-2763
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Tidskriftsartikel (refereegranskat)abstract
- When using a finite difference method to solve an initial-boundary-value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.
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| 2. |
- Foda, O., et al.
(författare)
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Overlaps of partial Neel states and Bethe states
- 2016
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Ingår i: Journal of Statistical Mechanics. - : IOP Publishing. - 1742-5468 .- 1742-5468.
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Tidskriftsartikel (refereegranskat)abstract
- Partial Neel states are generalizations of the ordinary Neel (classical anti-ferromagnet) state that can have arbitrary integer spin. We study overlaps of these states with Bethe states. We first identify this overlap with a partial version of reflecting-boundary domain-wall partition function, and then derive various determinant representations for off-shell and on-shell Bethe states.
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| 3. |
- Liu, Y., et al.
(författare)
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Side Reactions Do Not Completely Disrupt Linear Self-Replicating Chemical Reaction Systems
- 2020
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Ingår i: Artificial Life. - : MIT Press - Journals. - 1064-5462 .- 1530-9185. ; 26:3, s. 327-337
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Tidskriftsartikel (refereegranskat)abstract
- A crucial question within the fields of origins of life and metabolic networks is whether or not a self-replicating chemical reaction system is able to persist in the presence of side reactions. Due to the strong nonlinear effects involved in such systems, they are often difficult to study analytically. There are however certain conditions that allow for a wide range of these reaction systems to be well described by a set of linear ordinary differential equations. In this article, we elucidate these conditions and present a method to construct and solve such equations. For those linear self-replicating systems, we quantitatively find that the growth rate of the system is simply proportional to the sum of all the rate constants of the reactions that constitute the system (but is nontrivially determined by the relative values). We also give quantitative descriptions of how strongly side reactions need to be coupled with the system in order to completely disrupt the system.
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| 4. |
- Gyllingberg, Linnéa, et al.
(författare)
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Finding analytical approximations for discrete, stochastic, individual-based models of ecology
- 2023
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Ingår i: Mathematical Biosciences. - : Elsevier. - 0025-5564 .- 1879-3134. ; 365
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Tidskriftsartikel (refereegranskat)abstract
- Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.
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| 5. |
- Gyllingberg, Linnéa
(författare)
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Mathematical models of biological interactions
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Mathematical models are used to describe and analyse different types of biological interactions. From self-propelled particle models capturing the collective motion of fish schools to models in mathematical neuroscience describing the interactions between neurons to individual-based models of ecological interactions. A question that arises for all such models is how we scale from one level to another. How do we scale from fish interactions to the movement of the school of fish? How do we scale from neuronal interactions to the functioning of the brain? How do we scale from animal competition to population dynamics? It is approaches to this question that we study in this thesis for two different systems. In paper I, we study a class of spatially explicit individual-based models with contest competition. Based on measures of the spatial statistics, we develop two new approximate descriptions of the spatial population dynamics. The first is based on local interactions of the individuals and approximates the individual-based model well for small dispersal distances. The second approximates the long-range interactions of the individual-based model. Both approximations incorporate the demographic stochasticity from the individual-based model and show that dispersal stabilizes the population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the classical mean field approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Taken together, our results deepen the understanding of spatial population dynamics and introduces new approximate analytical descriptions.In paper II, we propose a model of social burst and glide motion in pairs of fish by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. Our model, in which visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, captures a rich set of swimming dynamics found in many species of fish. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and pendulum-like tit-for-tat turn taking. Unlike self-propelled particle models, which assume that fish move at a constant speed, the model produces realistic motion of individual fish. Moreover, unlike previous studies where a random component is used for leadership switching to occur, we show that leadership switching, both periodic and chaotic, can be the result of from a deterministic interaction. We give several empirically testable predictions on how fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.
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| 6. |
- Curic, Vladimir, et al.
(författare)
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Salience adaptive structuring elements
- 2012
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Ingår i: IEEE Journal on Selected Topics in Signal Processing. - 1932-4553 .- 1941-0484. ; 6:7, s. 809-819
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Tidskriftsartikel (refereegranskat)abstract
- Spatially adaptive structuring elements adjust their shape to the local structures in the image, and are often defined by a ball in a geodesic distance or gray-weighted distance metric space. This paper introduces salience adaptive structuring elements as spatially variant structuring elements that modify not only their shape, but also their size according to the salience of the edges in the image. Morphological operators with salience adaptive structuring elements shift edges with high salience to a less extent than those with low salience. Salience adaptive structuring elements are less flexible than morphological amoebas and their shape is less affected by noise in the image. Consequently, morphological operators using salience adaptive structuring elements have better properties.
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| 7. |
- Curic, Vladimir, et al.
(författare)
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Salience-Based Parabolic Structuring Functions
- 2013
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Ingår i: Mathematical Morphology and Its Applications to Signal and Image Processing. - Berlin, Heidelberg : Springer Berlin/Heidelberg. - 9783642382932 ; 7883, s. 183-194
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Konferensbidrag (refereegranskat)abstract
- It has been shown that the use of the salience map based on the salience distance transform can be useful for the construction of spatially adaptive structuring elements. In this paper, we propose salience-based parabolic structuring functions that are defined for a fixed, predefined spatial support, and have low computational complexity. In addition, we discuss how to properly define adjunct morphological operators using the new spatially adaptive structuring functions. It is also possible to obtain flat adaptive structuring elements by thresholding the salience-based parabolic structuring functions.
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| 8. |
- Luengo, Cris, et al.
(författare)
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Adaptive structuring elements based on salience information
- 2012
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Ingår i: Lecture Notes in Computer Science. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 0302-9743 .- 1611-3349. ; 7594, s. 321-328, s. 321-328
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Konferensbidrag (refereegranskat)abstract
- Adaptive structuring elements modify their shape and size according to the image content and may outperform fixed structuring elements. Without any restrictions, they suffer from a high computational complexity, which is often higher than linear with respect to the number of pixels in the image. This paper introduces adaptive structuring elements that have predefined shape, but where the size is adjusted to the local image structures. The size of adaptive structuring elements is determined by the salience map that corresponds to the salience of the edges in the image, which can be computed in linear time. We illustrate the difference between the new adaptive structuring elements and morphological amoebas. As an example of its usefulness, we show how the new adaptive morphological operations can isolate the text in historical documents.
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| 9. |
- Andersson, Ola, et al.
(författare)
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Robustness to strategic uncertainty
- 2014
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Ingår i: Games and Economic Behavior. - : Elsevier BV. - 0899-8256 .- 1090-2473. ; 85:1, s. 272-288
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.
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| 10. |
- Andersson, Ola, et al.
(författare)
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Robustness to strategic uncertainty in the Nash demand game
- 2018
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Ingår i: Mathematical Social Sciences. - : Elsevier. - 0165-4896 .- 1879-3118. ; 91, s. 1-5
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Tidskriftsartikel (refereegranskat)abstract
- This paper studies the role of strategic uncertainty in the Nash demand game. A player's uncertainty about another player's strategy is modeled as an atomless probability distribution over that player's strategy set. A strategy profile is robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others (Andersson et al., 2014). In the context of the Nash demand game, we show that robustness to symmetric (asymmetric) strategic uncertainty singles out the (generalized) Nash bargaining solution. The least uncertain party obtains the bigger share. (C) 2017 Elsevier B.V. All rights reserved.
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