Dynamic Systems for Biology
Joshua Elkingtonvisibility
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description
4 pages
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1 file
Life is composed of complex networks that manifests itself into the adaptability. Using quasistatic dynamical systems, biology can be generalized into a dynamical system. Quasistatic dynamical systems are processes where the observed system transforms innitesimally due to external inuence to trace a continous path over an innite path. Assuming that biological time states are deterministic and the initial state is random. This assumptions renders the entire network stochastic. In a standard setting with chaos, a biological system could be modeled based on its statistical behavior, which would be determined by a stochastic diusion process. Furthermore, this method could be applied to form networks with unique connection patterns. Modularity of the networks creates a formal semantic that allows dynamics of biology to be predicted.
Related papers
Biocomplexity: adaptive behavior in complex stochastic dynamical systems
Biosystems, 2001
Existing methods of complexity research are capable of describing certain specifics of biosystems over a given narrow range of parameters but often they cannot account for the initial emergence of complex biological systems, their evolution, state changes and sometimes abrupt state transitions. Chaos tools have the potential of reaching to the essential driving mechanisms that organize matter into living substances.
Understanding the Dynamics of Biological Systems
2011
Systems biology aims at integrating processes at various time and spatial scales into a single and coherent formal description to allow analysis and computer simulation. In this context, we focus on rule-based modeling and its integration in the domain-specific language MGS. Through the notions of topological collections and transformations, MGS allows the modeling of biological processes at various levels of description. We validate our approach through the description of various models of a synthetic bacteria designed in the context of the iGEM competition, from a very simple biochemical description of the process to an individual-based model on a Delaunay graph topology. This approach is a first step into providing the requirements for the emerging field of spatial systems biology which integrates spatial properties into systems biology.
Networks of Mobile Elements for Biological Systems
Arxiv preprint q-bio/0509005, 2005
In this paper we present a network model to study the impact of spatial distribution of constituents, coupling between them and diffusive processes in the context of biological situations. The model is in terms of network of mobile elements whose internal dynamics is given by differential equations exhibiting switching and/or oscillatory behaviour. To make the model more consistent with the underlying biological phenomena we incorporate properties like growth and decay into the network. Such a model exhibits a plethora of attributes which are interesting from both the network theory perspective as well as from the point of view of biochemistry and biology. We characterise this network by calculating the usual network measures like network efficiency, entropy growth, vertex degree distribution, geodesic lengths, centrality as well as fractal dimensions and generalised entropy. It is seen that the model can demonstrate the features of both scale free networks as well as small worlds network in different parameter domains. The formation of coherent spatiotemporal patterns is another feature of such networks which makes them appealing for understanding broad qualitative aspects of problems like cell differentiation (e.g. morphogenesis) and synchronization (e.g. quorum sensing mechanisms). One of the key features of any biological system is its response to external attacks. The response of the network to various attack strategies(isolated and multiple) is also studied.
Modelling the dynamics of biosystems
Briefings in Bioinformatics, 2004
The need for a more formal handling of biological information processing with stochastic and mobile process algebras is addressed. Biology can benefit this approach, yielding a better understanding of behavioural properties of cells, and computer science can benefit this approach, obtaining new computational models inspired by nature.
Introduction to Focus Issue: Dynamics in Systems Biology
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2010
The methods of nonlinear systems form an extensive toolbox for the study of biology, and systems biology provides a rich source of motivation for the development of new mathematical techniques and the furthering of understanding of dynamical systems. This Focus Issue collects together a large variety of work which highlights the complementary nature of these two fields, showing what each has to offer the other. While a wide range of subjects is covered, the papers often have common themes such as "rhythms and oscillations," "networks and graph theory," and "switches and decision making." There is a particular emphasis on the links between experimental data and modeling and mathematical analysis.
Structural and dynamical analysis of biological networks
Briefings in functional genomics, 2012
Biological networks are currently being studied with approaches derived from the mathematical and physical sciences. Their structural analysis enables to highlight nodes with special properties that have sometimes been correlated with the biological importance of a gene or a protein. However, biological networks are dynamic both on the evolutionary time-scale, and on the much shorter time-scale of physiological processes. There is therefore no unique network for a given cellular process, but potentially many realizations, each with different properties as a consequence of regulatory mechanisms. Such realizations provide snapshots of a same network in different conditions, enabling the study of condition-dependent structural properties. True dynamical analysis can be obtained through detailed mathematical modeling techniques that are not easily scalable to full network models.
Models Of System Biology
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Emergence of system-level properties in biological networks from cellular automata evolution
2010
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Understanding the dynamics of biological systems: lessons learned from integrative systems biology
2011
Systems biology aims at integrating processes at various time and spatial scales into a single and coherent formal description to allow analysis and computer simulation. In this context, we focus on rule-based modeling and its integration in the domain-specific language MGS. Through the notions of topological collections and transformations, MGS allows the modeling of biological processes at various levels of description. We validate our approach through the description of various models of a synthetic bacteria designed in the context of the iGEM competition, from a very simple biochemical description of the process to an individual-based model on a Delaunay graph topology. This approach is a first step into providing the requirements for the emerging field of spatial systems biology which integrates spatial properties into systems biology.
Artificial Biochemical Networks: Evolving Dynamical Systems to Control Dynamical Systems
IEEE Transactions on Evolutionary Computation, 2014
Biological organisms exist within environments in which complex, non-linear dynamics are ubiquitous. They are coupled to these environments via their own complex, dynamical networks of enzyme-mediated reactions, known as biochemical networks. These networks, in turn, control the growth and behaviour of an organism within its environment. In this paper, we consider computational models whose structure and function are motivated by the organisation of biochemical networks. We refer to these as artificial biochemical networks, and show how they can be evolved to control trajectories within three behaviourally diverse complex dynamical systems: the Lorenz system, Chirikov's standard map, and legged robot locomotion. More generally, we consider the notion of evolving dynamical systems to control dynamical systems, and discuss the advantages and disadvantages of using higher order coupling and configurable dynamical modules (in the form of discrete maps) within artificial biochemical networks. We find both approaches to be advantageous in certain situations, though note that the relative trade-offs between different models of artificial biochemical network strongly depend on the type of dynamical systems being controlled.
❉❨◆❆▼■❈ ❙❨❙❚❊▼❙ ❋❖❘ ❇■❖▲❖●❨
❏❖❙❍❯❆ ❊▲❑■◆●❚❖◆
▲✐❢❡ ✐s ❝♦♠♣♦s❡❞ ♦❢ ❝♦♠♣❧❡① ♥❡t✇♦r❦s t❤❛t ♠❛♥✐❢❡sts ✐ts❡❧❢ ✐♥t♦ t❤❡ ❛❞❛♣t✲
❛❜✐❧✐t②✳ ❯s✐♥❣ q✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠s✱ ❜✐♦❧♦❣② ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ ✐♥t♦ ❛ ❞②♥❛♠✐❝❛❧
s②st❡♠✳ ◗✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠s ❛r❡ ♣r♦❝❡ss❡s ✇❤❡r❡ t❤❡ ♦❜s❡r✈❡❞ s②st❡♠ tr❛♥s✲
❢♦r♠s ✐♥✜♥✐t❡s✐♠❛❧❧② ❞✉❡ t♦ ❡①t❡r♥❛❧ ✐♥✢✉❡♥❝❡ t♦ tr❛❝❡ ❛ ❝♦♥t✐♥♦✉s ♣❛t❤ ♦✈❡r ❛♥ ✐♥✜♥✐t❡
♣❛t❤✳ ❆ss✉♠✐♥❣ t❤❛t ❜✐♦❧♦❣✐❝❛❧ t✐♠❡ st❛t❡s ❛r❡ ❞❡t❡r♠✐♥✐st✐❝ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ✐s
r❛♥❞♦♠✳ ❚❤✐s ❛ss✉♠♣t✐♦♥s r❡♥❞❡rs t❤❡ ❡♥t✐r❡ ♥❡t✇♦r❦ st♦❝❤❛st✐❝✳ ■♥ ❛ st❛♥❞❛r❞ s❡t✲
t✐♥❣ ✇✐t❤ ❝❤❛♦s✱ ❛ ❜✐♦❧♦❣✐❝❛❧ s②st❡♠ ❝♦✉❧❞ ❜❡ ♠♦❞❡❧❡❞ ❜❛s❡❞ ♦♥ ✐ts st❛t✐st✐❝❛❧ ❜❡❤❛✈✐♦r✱
✇❤✐❝❤ ✇♦✉❧❞ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ❛ st♦❝❤❛st✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss✳ ❋✉rt❤❡r♠♦r❡✱ t❤✐s ♠❡t❤♦❞
❝♦✉❧❞ ❜❡ ❛♣♣❧✐❡❞ t♦ ❢♦r♠ ♥❡t✇♦r❦s ✇✐t❤ ✉♥✐q✉❡ ❝♦♥♥❡❝t✐♦♥ ♣❛tt❡r♥s✳ ▼♦❞✉❧❛r✐t② ♦❢ t❤❡
♥❡t✇♦r❦s ❝r❡❛t❡s ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝ t❤❛t ❛❧❧♦✇s ❞②♥❛♠✐❝s ♦❢ ❜✐♦❧♦❣② t♦ ❜❡ ♣r❡❞✐❝t❡❞✳
❆❜str❛❝t✳
✶✳
■♥tr♦❞✉❝t✐♦♥
❚❤✐s ♣❛♣❡r st✉❞✐❡s t❤❡ ❞②♥❛♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❜✐♦❧♦❣✐❝❛❧ s②st❡♠s✳ ❚❤❡ ♣❛♣❡r ✉s❡s ♣r♦❜✲
❛❜✐❧✐st✐❝ t❡❝❤♥✐q✉❡s t♦ ♣r♦❜❡ ♥❛t✉r❛❧ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳
✶✳✶✳ ▼♦t✐✈❛t✐♦♥ ❛♥❞ ❛❜str❛❝t s❡t✉♣✳ ❚♦ st✉❞② t❤❡ ❞②♥❛♠✐❝s ♦❢ ❜✐♦❧♦❣②✱ ❛ s②st❡♠ a ✐♥
❝♦♥t❛❝t ✇✐t❤ ❛♥ ❛♠❜✐❡♥t s②st❡♠ E ✇❛s ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ ❥♦✐♥t s②st❡♠ (a, E) ✐s t♦♦ ❝♦♠♣❧❡①
t♦ ❛♥❛❧②③❡ ❞✐r❡❝t❧②✱ ✇❤✐❧❡ a✱ t❤❡ ♦❜s❡r✈❡❞ s✉❜s②st❡♠ ♠♦❞✉❧❡ ❛♠❡♥❛❜❧❡ t♦ ❛♥❛❧②s✐s✳
❚❤❡ ♦❜s❡r✈❡❞ s②st❡♠ a ❛♥❞ t❤❡ ❛♠❜✐❡♥t s②st❡♠ E ✐♥t❡r❛❝t✱ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✲
str❛✐♥ts✿ ✭■✮ t❤❡ ❝❤❛♥❣❡ ♦❢ a ❤❛s ❛ s♠❛❧❧ ❡✛❡❝t ♦♥ E✱ ❛♥❞ ✭■■✮ t❤❡ s②st❡♠s ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞✳
■❢ ❛ str✉❝t✉r❡ r❡♣❡❛ts ✐ts❡❧❢ ❛t ✈❛r✐♦✉s s❝❛❧❡s ♦❢ ❛ ♠♦❞❡❧✱ t❤❡ ♠♦❞❡❧ ✐s ❢♦r♠❛❧✐③❡❞ ✉s✐♥❣
♦♣❡r❛❞s✳ ■❢ t❤❡r❡ ✐s ❛♥ ♦♣❡r❛❞ ♦❢ t❤❡ ♥❡t✇♦r❦✱ t❤❡♥ t❤❡ ♠♦❞❡❧ r❡❝r❡❛t❡s ❛ ♥❡t✇♦r❦ ♦❢
♥❡t✇♦r❦s✳ ❚❤✐s ✜ts ✐♥ ♣❡r❢❡❝t❧② ✇✐t❤ t❤❡ ❢r❛❝t❛❧ ♥❛t✉r❡ ♦❢ ❜✐♦❧♦❣②✳ ❚❤❡r❡ ✐s ❛♥ ♦♣❡r❛❞
♦❢ Y t❤❛t ✐s ❛ ❤✐❡r❛r❝❤✐❝❛❧ ♠♦❞❡❧ ♦❢ Z ✳ ■♥ ♣r❡✈✐♦✉s ✇♦r❦ ❬❄✱ ❄✱ ❄❪✱ ♦♣❡r❛❞s ✇❡r❡ ✉s❡❞ t♦
❞❡s❝r✐❜❡ ❛ ✉♥✐q✉❡ s❡t ♦❢ ♥❡t✇♦r❦s✱ ❝❛❧❧❡❞ ✇✐r✐♥❣ ❞✐❛❣r❛♠s✳ ❚❤✐s ♥❡t✇♦r❦s ✇❡r❡ s❤♦✇♥ t♦
♠♦❞❡❧ ❝♦♠♣❧❡① r❡❧❛t✐♦♥s❤✐♣s ✇✐t❤✐♥ ♥❡t✇♦r❦s✱ ✇❤✐❝❤ ✐s ✉s❡❢✉❧ t♦ ♠♦❞❡❧ ❜✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s✳
❚❤❡s❡ ❞✐❛❣r❛♠s ❛r❡ ✜①❡❞ s❡t ♦❢ ♥♦❞❡s ✇✐t❤ ❞✐r❡❝t❡❞ ❡❞❣❡s ✇❤❡r❡ t❤❡ ♥♦❞❡s ❤❛✈❡ ❞✐✛❡r❡♥t
✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts✳ ▼♦r❡♦✈❡r✱ t❤❡s❡ ♥♦❞❡s ❛r❡ r❡♠♦✈❛❜❧❡ ✐♥t♦ ❛♥❞ ♦✉t ♦❢ ♠❛♥② ❣r❛♣❤s t♦
❡①♣❛♥❞ ❛♥❞ ❝♦♥tr❛❝t ❛♥② ♣❛rt✐❝✉❧❛r ♣♦s✐t✐♦♥✳ ❚❤✐s ✐s s✐♠✐❧❛r t♦ s✐❣♥❛❧❧✐♥❣ ♥❡t✇♦r❦s ✇✐t❤✐♥
❛ ❝❡❧❧ t❤❛t ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥ s✉❜♠♦❞✉❧❛r ❞♦♠❛✐♥s✳ ❚❤❡ st❛t❡s ♦❢ t❤❡ ♥♦❞❡s ❞❡✜♥❡ ❛
❧♦❝❛❧ st❛t❡ ✇❤❡r❡ t❤❡s❡ ❞✐❛❣r❛♠s ♠♦❞❡✲❞❡♣❡♥❞❡♥t ♥❡t✇♦r❦s✳
❚❤❡ ❣❡♥❡r❛❧ ❝♦♥❝❡♣t ✐s t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ a ♦✈❡r ❛ ♣❡r✐♦❞ t✐♠❡ t♦ ♥♦♥✲
tr✐✈✐❛❧❧② ♠♦❞❡❧ ❜✐♦❧♦❣②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛ s✉❜s②st❡♠ a ♠❛② ♥♦t r❡❛❝❤ ❡q✉✐❧✐❜r✐✉♠ ✉♥❞❡r t❤❡
✐♥✢✉❡♥❝❡ ♦❢ t❤❡ ❛♠❜✐❡♥t s②st❡♠✱ E✳ ❍♦✇❡✈❡r✱ a ♠❛② ♣❛ss ♥❡❛r ❡q✉✐❧✐❜r✐❛ ❝♦♥t✐♥✉♦✉s❧②✳
❲✐t❤♦✉t ❛♥ ❡q✉✐❧✐❜r✐✉♠ st❛t❡✱ t❤❡ st❛t✐st✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s②st❡♠ r❡♠❛✐♥ ✉♥❞❡✜♥❡❞✳
❚♦ ♠♦❞❡❧ t❤❡ s②st❡♠✱ t❤❡ ♥♦t✐♦♥ ♦❢ ❛ q✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✐s ✉s❡❞✳ ❚❤✐s
✐s ❛ ♠❡t❤♦❞ ❞❡s❝r✐❜✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ s②st❡♠ ✇✐t❤ st♦❝❤❛st✐❝ ❞✐✛✉s✐♦♥ ♣r♦❝❡ss❡s t❤❛t
❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❢♦r ❜✐♦❧♦❣✐❝❛❧ s②st❡♠s✳
✶
✷
❏❖❙❍❯❆ ❊▲❑■◆●❚❖◆
✶✳✷✳ ◗✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳ ❙t❛♥❞❛r❞ ♥♦t❛t✐♦♥ ✐s ✉s❡❞
{u} = u (mod 1) ❛♥❞ ⌊u⌋ = u − {u}
❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❛♥❞ ✐♥t❡❣❡r ❝♦♠♣♦♥❡♥t ♦❢ r❡❛❧ ♥✉♠❜❡r u ≥ 0✱ r❡s♣❡❝t✐✈❡❧②✳
❉❡✜♥✐t✐♦♥ ✶✳✶✳ ▲❡t Y ❜❡ ❛ s❡t ❛♥❞ A ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❡❧❢ r❡❢❡r❡♥❝✐♥❣ ♠❛♣s J : Y → Z
✇✐t❤ ❛ ✉♥✐q✉❡ t♦♣♦❧♦❣②✳ ❲✐t❤ ❛ tr✐❛♥❣❧❡ ❛rr❛②
= {Jn,k ∈ A : 0 ≤ k ≤ n, n ≥ 1}
♦❢ ❡❧❡♠❡♥ts ✇✐t❤ A✳ ■❢ t❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ δ : [0, 1] → M s✉❝❤ t❤❛t
lim Jn,⌊nt⌋ = δt ,
n→∞
t ∈ [0, 1] ,
t❤❡♥ (, δ) ✐s ❛ q✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✭◗❉❙✮ ❬❄❪✳
❉❡✜♥✐t✐♦♥ ✶✳✷✳ Y ✐s t❤❡ st❛t❡ s♣❛❝❡ ❛♥❞ A ✐s t❤❡ s②st❡♠ s♣❛❝❡ ♦❢ t❤❡ q✉❛s✐st❛t✐❝
❞②♥❛♠✐❝❛❧ s②st❡♠✳
❆ q✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✭◗❉❙✮ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❛♥ ♦r❞✐♥❛r② ❞②♥❛♠✐❝❛❧
s②st❡♠✳ ❆ ◗❉❙ ✉s❡s ❛ s✐♥❣♣❧❡ ♠❛♣✿ J : Y → Z ✱ t♦ ❝r❡❛t❡ ❛ ❞✐s❝r❡t❡ s②st❡♠✱ ✇❤✐❝❤ ✐s t❤❡
❞❡❣❡♥❡r❛t❡ ◗❉❙ ✐♥ Jn,k = J ❢♦r ❛❧❧ k ❛♥❞ n✳
❆♥ ❡❧❡♠❡♥t ♦❢ t❤❡ st❛t❡ s♣❛❝❡ Z r❡♣r❡s❡♥ts t❤❡ st❛t❡ ♦❢ t❤❡ ❡♥t✐r❡ s②st❡♠✱ ❛♥❞ t❤❡
tr✐❛♥❣❧❡ ❛rr❛② r❡♣r❡s❡♥ts t❤❡ ❞②♥❛♠✐❝s ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ st❛t❡ y ∈ Y ❛♥❞ ✐♥t❡❣❡rs 1 ≤ k ≤ n✱
❝♦♥❢❡rr✐♥❣ ❛♥ ✐♠❛❣❡ t❤❛t
xn,k = Jn,k ◦ · · · ◦ Jn,1 (x) ∈ X
✐s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠ ❛❢t❡r k st❡♣s ♦♥ nt❤ ❧❡✈❡❧ ♦❢ t❤❡ tr✐❛♥❣❧❡ ❛rr❛② ✳ ❚❤❡ ❝♦♥t✐♥✉♦✉s
t✐♠❡ ♣❛r❛♠❡t❡r t ∈ [0, 1] ✇✐t❤ t❤❡ s❝❛❧✐♥❣ ❢❛❝t♦r k = ⌊nt⌋✱ t❤❛t t❤❡ ❝✉r✈❡ t 7→ Jn,⌊nt⌋
❛♣♣r♦①✐♠❛t❡s δ ✐♥ t❤❡ ♥❡t✇♦r❦ s♣❛❝❡ M✱ ❛♥❞ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❝♦♥✈❡r❣❡s ❛s n → ∞✳
❚❤❡ ♥❡t✇♦r❦ s②st❡♠ Jn,n ≈ δ1 ✐s ❣❡♥❡r❛❧❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ s②st❡♠ Jn,1 ≈ δ0 ✳
❚❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ Jn,⌊nt⌋ t♦ δt ✇✐❧❧ ❞❡t❡r♠✐♥❡ t❤❛t ❛ttr✐❜✉t❡s ♦❢ t❤❡ s②st❡♠✳
❉❡t❡r♠✐♥✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ t❤❡ ❝✉r✈❡ δ ❛♥❞ t❤❡ tr✐❛♥❣❧❡ ❛rr❛② ✳
❆ s②st❡♠ ❝❛♥♥♦t ❜❡ st✉❞✐❡❞ ❞✐r❡❝t❧② ❢♦r ❧❡tt✐♥❣ ✜rst n → ∞ ❞✉❡ t♦ t❤❡ ❡♥❞❧❡ss ✐t❡r❛t✐♥❣
t❤❡ ♠❛♣ δ0 ✳ ❙♦ t❤❡ st❛t✐st✐❝❛❧ ❛ttr✐❜✉t❡s ♦❢ t❤❡ s②st❡♠ ❛t ❧❡✈❡❧ n ❜❡❤❛✈❡ ✉♥✐q✉❡❧② ✐♥ t❤❡
❧✐♠✐t✳ ❚t♦ ✐♥✐t✐❛t❡ ❛ st✉❞② ♦❢ ❜✐♦❧♦❣✐❝❛❧ s②st❡♠s ✉s✐♥❣ ◗❉❙s t♦ ❞❡✈❡❧♦♣ ❛ ❝♦♠♣r❡❤❡♥s✐✈❡
❜✐♦❧♦❣✐❝❛❧ t❤❡♦r②✳
❚❤❡ ❝✉r✈❡ δ ♠♦❞❡❧s t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛♥② ♦❜s❡r✈❡❞ s②st❡♠ a✱ ❛s t❤❡ ❛♠❜✐❡♥t s②st❡♠ E
❢♦r❝❡s a t♦ ❝❤❛♥❣❡✳ ❚❤❡ ❝✉r✈❡ ❥✉♠♣s ✐♥ δ ❡♥❛❜❧❡ ❛❜r✉♣t ❛♥❞ ❝❤❛♦t✐❝ ❡✈❡♥ts✳
■♥ t❤❡ ♠♦❞❡❧✱ q✉❛s✐st❛t✐❝ r❡♣r❡s❡♥ts ❛♥ ✐❞❡❛❧✐③❡❞ ♣r♦❝❡ss ✐♥ ✇❤✐❝❤ t❤❡ ♦❜s❡r✈❡❞ s②st❡♠
tr❛♥s❢♦r♠s ✐♥✜♥✐t❡s✐♠❛❧❧② s❧♦✇❧② ❞✉❡ t♦ ❛♥ ♦✉ts✐❞❡ ✐♥✢✉❡♥❝❡✳ ❚❤✐s t②♣❡ ♦❢ s②st❡♠ ✐s ✐♥
t❤❡r♠♦❞②♥❛♠✐❝ ❡q✉✐❧✐❜r✐✉♠ ❛t ❛♥② ❣✐✈❡♥ t✐♠❡✳ ❨❡t t❤❡ s②st❡♠ ♠❛♣s ❛ ♣❛t❤ ❝♦♥t✐♥✉♦✉s❧②
♦✈❡r ♠❛♥② ❡q✉✐❧✐❜r✐❛ ♦✈❡r ❛♥ ✐♥✜♥✐t❡❧② ❧♦♥❣ t✐♠❡✳
❆ ❝❧❛ss ♦❢ ◗❉❙✿
❊①❛♠♣❧❡ ✶✳✸ ✭▼❛♣s✮✳ ▲❡t (Uu )∞
u=0 ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s✱ ✇❤❡r❡ d(Uu−1 , Uu ) < ε
❤♦❧❞s ✉♥✐❢♦r♠❧② ❢♦r s♦♠❡ s♠❛❧❧ ε > 0✳ ❙♦ d ✐s ❛ ❞✐st❛♥❝❡ ♦♥ t❤❡ ♠❛♣ U ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s✳
❚❤❡♥ WUu ,Uu−1 ✐s t❤❡ ❞②♥❛♠✐❝s ❜❡t✇❡❡♥ t❤❡ (u − 1)t❤ ❛♥❞ ut❤✱ ✇❤❡r❡ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❤❛s
❝❤❛♥❣❡❞ ❜② ❛ ❞✐st❛♥❝❡ < ε✱ ❢r♦♠ Uu−1 t♦ Uu ✱ ❛♥❞ t❤❡ ❝♦♠♣♦s✐t✐♦♥s WUu ,Uu−1 ◦· · ·◦WU1 ,U0 ❛r❡
t❤❡ ❞②♥❛♠✐❝s ❢♦r s❝❛tt❡r❡rs ✇✐t❤ ❛ s♣❡❡❞ ♦❢ < ε✳ ❚❤✐s r❡♣r❡s❡♥ts ❛ q✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧
s②st❡♠ ✐♥ t❤❡ ❧✐♠✐t ε → 0 ♦❢ ✐♥✜♥✐t❡s✐♠❛❧❧② s❧♦✇❧② ♠♦✈✐♥❣ s❝❛tt❡r❡rs✳ ❈♦♥s✐❞❡r ❛ tr✐❛♥❣❧❡
❛rr❛② ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s✿ {Un,u ∈ U : 0 ≤ k ≤ n, n ≥ 1} ✇✐t❤ d(Un,u−1 , Un,u ) < εn ✱ ✇❤❡r❡
❉❨◆❆▼■❈ ❙❨❙❚❊▼❙ ❋❖❘ ❇■❖▲❖●❨
✸
limn→∞ εn = 0✳ ❚❤❡r❡ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ δ U : [0, 1] → U ✇❤❡r❡ limn→∞ Un,⌊nt⌋ = δtU ✳
❲✐t❤ Jn,u = WUn,u ,Un,u−1 ❢♦r 1 ≤ k ≤ n ❛♥❞ n ≥ 1 r❡s✉❧ts ✐♥ ❛ q✉❛s✐❞②♥❛♠✐❝ s②st❡♠✳ ❬❄❪✳
●❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥✿
ϕr,s
n = Jn,⌊r+s⌋ ◦ · · · ◦ Jn,⌊r⌋+1 ,
0≤r ≤r+s≤n .
❈♦♥✈❡r❣❡♥❝❡ ✐♥ ❉❡✜♥✐t✐♦♥ ❄❄ ❜❡❝♦♠❡s
ϕnt−1,1
= Jn,⌊nt⌋ → δt ,
n
❬❄❪✳
❋♦r ♠❛♥② ♥❡t✇♦r❦s✱ t❤❡ ❝♦♥♥❡❝t✐♦♥ ♣❛tt❡r♥ ✈❛r✐❡s ✐♥ t✐♠❡✳ ❚❤✐s ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤❛♥❣✲
✐♥❣ st❛t❡ ♦❢ t❤❡ ♠♦❞✉❧❡s ♦❢ t❤❡ ♥❡t✇♦r❦✳ ❚❤❡ ♠❡t❤♦❞ ❞❡✈❡❧♦♣s ❛ ❛♥❛❧②s✐s ❢♦r ❜✐♦❧♦❣✐❝❛❧
♥❡t✇♦r❦s ❞❡♣❡♥❞❡♥t ♦♥ ❛♥② ♠♦❞❡✳ ❊❛❝❤ ♠♦❞✉❧❡ ✇✐t❤✐♥ t❤❡ ♥❡t✇♦r❦ r❡♣r❡s❡♥ts ❛ ❞②♥❛♠✲
✐❝❛❧ s②st❡♠✳ ❚❤❡ ❢♦r♠❛♥❧ s❡♠❛♥t✐❝s ♣r❡s❡♥t❡❞ s❤♦✇s t❤❛t ❡❛❝❤ ♠♦❞✉❧❡ ❝❛♥ ❜❡ ✉s❡❞ ❛s
❝❛t❡❣♦r② ❢r❛♠❡✇♦r❦ ❢♦r ❜✐♦❧♦❣② ❬❄❪✳
✯❆❝❦♥♦✇❧❡❞❣♠❡♥ts
❚❤❛♥❦s ❣♦ t♦ ▲✉❝✐♥❞❛✳
❘❡❢❡r❡♥❝❡s
❬✶❪ ❆❧❜❡rt✱ ❘✳❀ ❏❡♦♥❣✱ ❍✳❀ ❇❛r❛❜❛s✐ ❆✳ ▲✳ ✭✷✵✵✵✮ ✏❊rr♦r ❛♥❞ ❛tt❛❝❦ t♦❧❡r❛♥❝❡ ♦❢ ❝♦♠♣❧❡① ♥❡t✇♦r❦s✑✳
◆❛t✉r❡ ✹✵✻✱ ♣♣✳ ✸✼✽✲✽✷✳
❬✷❪ ❆✇♦❞❡②✱ ❙✳ ✭✷✵✶✵✮ ❈❛t❡❣♦r② t❤❡♦r②✳ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ❖①❢♦r❞ ▲♦❣✐❝ ●✉✐❞❡s✱ ✺✷✳ ❖①❢♦r❞ ❯♥✐✈❡rs✐t②
Pr❡ss✱ ❖①❢♦r❞✳
❬✸❪ ❇❛❡③✱ ❏✳❈✳❀ ❉♦❧❛♥✱ ❏✳ ✭✶✾✾✽✮ ✏❍✐❣❤❡r✲❉✐♠❡♥s✐♦♥❛❧ ❆❧❣❡❜r❛ ■■■✿ ♥✲❈❛t❡❣♦r✐❡s ❛♥❞ t❤❡ ❆❧❣❡❜r❛ ♦❢
❖♣❡t♦♣❡s✧✳ ❆❞✈✳ ▼❛t❤✳ ✶✸✺✱ ♥♦✳ ✷✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴q✲❛❧❣✴✾✼✵✷✵✶✹✳
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴
❬✹❪ ❇❛❡③✱ ❏✳❈✳❀ ❊r❜❡❧❡✱ ❏✳ ✭✷✵✶✹✮ ✏❈❛t❡❣♦r✐❡s ✐♥ ❈♦♥tr♦❧✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
✶✹✵✺✳✻✽✽✶✳
❬✺❪ ❇❛❡③✱ ❏✳❈✳❀ ❋♦♥❣✱ ❇✳ ✭✷✵✶✹✮ ✏◗✉❛♥t✉♠ ❚❡❝❤♥✐q✉❡s ❢♦r ❙t✉❞②✐♥❣ ❊q✉✐❧✐❜r✐✉♠ ✐♥ ❘❡❛❝t✐♦♥ ◆❡t✲
✇♦r❦s✧✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣❧❡① ◆❡t✇♦r❦s✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✸✵✻✳✸✹✺✶✳
❬✻❪ ❇♦✉r❜❛❦✐✱ ◆✳ ✭✶✾✼✷✮ ✏❚❤é♦r✐❡ ❞❡s t♦♣♦s ❡t ❝♦❤♦♠♦❧♦❣✐❡ ét❛❧❡ ❞❡s s❝❤é♠❛s✳ ❚♦♠❡ ✶✿ ❚❤é♦r✐❡ ❞❡s
t♦♣♦s✱✧ ❙é♠✐♥❛✐r❡ ❞❡ ●é♦♠étr✐❡ ❆❧❣é❜r✐q✉❡ ❞✉ ❇♦✐s✲▼❛r✐❡ ✶✾✻✸Ð✶✾✻✹ ✭❙●❆ ✹✮✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥
▼❛t❤❡♠❛t✐❝s ✷✻✾✱ ❙♣r✐♥❣❡r✳
❬✼❪ ❇r♦②✱ ▼✳ ✭✶✾✾✼✮ ✏❈♦♠♣♦s✐t✐♦♥❛❧ r❡✜♥❡♠❡♥t ♦❢ ✐♥t❡r❛❝t✐✈❡ s②st❡♠s✧✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❈▼✳
❬✽❪ ❋✐❡❧❞✱ ▼✳ ✭✷✵✶✺✮ ✏❆s②♥❝❤r♦♥♦✉s ♥❡t✇♦r❦s ❛♥❞ ❡✈❡♥t ❞r✐✈❡♥ ❞②♥❛♠✐❝s✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
✴✴✇✇✇❢✳✐♠♣❡r✐❛❧✳❛❝✳✉❦✴⑦♠❥❢✐❡❧❞✶✴♣❛♣❡rs✴■♠♣◆♦t❡s✳♣❞❢✳
❬✾❪ ❍❛r❞②✱ ▲✳ ✭✷✵✶✸✮ ✏❖♥ t❤❡ t❤❡♦r② ♦❢ ❝♦♠♣♦s✐t✐♦♥ ✐♥ ♣❤②s✐❝s✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
♦r❣✴❛❜s✴✶✸✵✸✳✶✺✸✼✳
❤tt♣✿
❤tt♣✿✴✴❛r①✐✈✳
❬✶✵❪ ■s❜❡❧❧✱ ❏✳ ✭✶✾✻✾✮ ✏❖♥ ❝♦❤❡r❡♥t ❛❧❣❡❜r❛s ❛♥❞ str✐❝t ❛❧❣❡❜r❛s✧✳ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶✸
❬✶✶❪ ❏♦②❛❧✱ ❆✳❀ ❙tr❡❡t✱ ❘✳❀ ❱❡r✐t②✱ ❉✳ ✭✶✾✾✻✮ ✏❚r❛❝❡❞ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r✐❡s✧✳ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❝❡❡❞✐♥❣s
♦❢ t❤❡ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s♦♣❤✐❝❛❧ ❙♦❝✐❡t② ✶✶✾✱ ♥♦✳ ✸✱ ♣♣✳ ✹✹✼✕✹✻✽✳
❬✶✷❪ ❑✐✈❡❧❛✱ ▼✳❀ ❆r❡♥❛s ❆✳❀ ❇❛rt❤❡❧❡♠②✱ ▼✳❀ ●❧❡❡s♦♥✱ ❏✳ P✳❀ ▼♦r❡♥♦✱ ❨❛♠✐r❀ P♦rt❡r✱ ▼✳ ❆✳ ✭✷✵✶✹✮
✏▼✉❧t✐❧❛②❡r ♥❡t✇♦r❦s✑✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣❧❡① ◆❡t✇♦r❦s ✷✱ ♣♣✳ ✷✵✸✕✷✼✶✳
❬✶✸❪ ▲❡✐♥st❡r✱ ❚✳ ✭✷✵✵✹✮ ❍✐❣❤❡r ♦♣❡r❛❞s✱ ❤✐❣❤❡r ❝❛t❡❣♦r✐❡s✳ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ▲❡❝t✉r❡ ◆♦t❡
❙❡r✐❡s✱ ✷✾✽✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳
❬✶✹❪ ▼❛❝ ▲❛♥❡✱ ❙✳ ✭✶✾✾✽✮ ❈❛t❡❣♦r✐❡s ❢♦r t❤❡ ✇♦r❦✐♥❣ ♠❛t❤❡♠❛t✐❝✐❛♥✳ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✳ ●r❛❞✉❛t❡ ❚❡①ts
✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✺✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳
❬✶✺❪ ◆❡✇♠❛♥✱ ▼✳ ✭✷✵✵✸✮ ✏❚❤❡ str✉❝t✉r❡ ❛♥❞ ❢✉♥❝t✐♦♥ ♦❢ ❝♦♠♣❧❡① ♥❡t✇♦r❦s✳✑ ❙■❆▼ r❡✈✐❡✇ ✹✺✳✷✱
♣♣✳ ✶✻✼✕✷✺✻✳
❬✶✻❪ ❘❡✐s✱ ❙✳ ❉✳ ❙✳❀ ❍✉✱ ❨❛♥q✐♥❣❀ ❇❛❜✐♥♦✱ ❆♥❞r❡s❀ ❆♥❞r❛❞❡✱ ❏♦s❡ ❙✳ ❏r✳❀ ❈❛♥❛❧s✱ ❙✳❀ ❙✐❣♠❛♥✱ ▼✳❀ ▼❛❦s❡✱
❍✳ ❆✳ ✭✷✵✶✹✮ ✏❆✈♦✐❞✐♥❣ ❝❛t❛str♦♣❤✐❝ ❢❛✐❧✉r❡ ✐♥ ❝♦rr❡❧❛t❡❞ ♥❡t✇♦r❦s ♦❢ ♥❡t✇♦r❦s✑✱ ◆❛t✉r❡ P❤②s✐❝s
✶✵✱ ♣♣✳ ✼✻✷✲✼✻✼✳
❬✶✼❪ ❘✉♣❡❧✱ ❉✳❀ ❙♣✐✈❛❦✱ ❉✳■✳ ✭✷✵✶✸✮ ✏❚❤❡ ♦♣❡r❛❞ ♦❢ t❡♠♣♦r❛❧ ✇✐r✐♥❣ ❞✐❛❣r❛♠s✿ ❢♦r♠❛❧✐③✐♥❣ ❛ ❣r❛♣❤✐❝❛❧
❧❛♥❣✉❛❣❡ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ ♣r♦❝❡ss❡s✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✸✵✼✳✻✽✾✹
❬✶✽❪ ❙✐♠♦♥✱ ❍✳❆✳ ✭✶✾✼✸✮ ✏❚❤❡ ♦r❣❛♥✐③❛t✐♦♥ ♦❢ ❝♦♠♣❧❡① s②st❡♠s✧ ✐♥ ❍✐❡r❛r❝❤② t❤❡♦r②✿ t❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢
❝♦♠♣❧❡① s②st❡♠s✱ ❍✳❍✳ P❛tt❡❡ ✭❡❞✳✮✳
✹
❏❖❙❍❯❆ ❊▲❑■◆●❚❖◆
❬✶✾❪ ❙❝♦tt✱ ❉✳ ✭✶✾✼✶✮ ✏❚❤❡ ❧❛tt✐❝❡ ♦❢ ✢♦✇ ❞✐❛❣r❛♠s✧✳ ❙②♠♣♦s✐✉♠ ♦♥ s❡♠❛♥t✐❝s ♦❢ ❛❧❣♦r✐t❤♠✐❝ ❧❛♥❣✉❛❣❡s✳
❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✳
❬✷✵❪ ❙❤✉❧♠❛♥✱ ▼✳ ✭✷✵✵✽✮ ✏❙❡t t❤❡♦r② ❢♦r ❝❛t❡❣♦r② t❤❡♦r②✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
✵✽✶✵✳✶✷✼✾✳
❬✷✶❪ ❙♣✐✈❛❦✱ ❉✳■✳ ✭✷✵✶✸✮ ✏❚❤❡ ♦♣❡r❛❞ ♦❢ ✇✐r✐♥❣ ❞✐❛❣r❛♠s✿
❞❛t❛❜❛s❡s✱ r❡❝✉rs✐♦♥✱ ❛♥❞ ♣❧✉❣✲❛♥❞✲♣❧❛② ❝✐r❝✉✐ts✧✳
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴
❢♦r♠❛❧✐③✐♥❣ ❛ ❣r❛♣❤✐❝❛❧ ❧❛♥❣✉❛❣❡ ❢♦r
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✸✵✺✳✵✷✾✼
❬✷✷❪ ❙♣✐✈❛❦✱ ❉✳■✳ ✭✷✵✶✹✮ ❈❛t❡❣♦r② ❚❤❡♦r② ❢♦r t❤❡ ❙❝✐❡♥❝❡s✳ ▼■❚ Pr❡ss✳
❬✷✸❪ ❙♣✐✈❛❦✱ ❉✳■✳❀ ❙❝❤✉❧t③✱ P✳❀ ❘✉♣❡❧✱ ❉✳ ✭✷✵✶✺✮ ✏❙tr✐♥❣ ❞✐❛❣r❛♠s ❢♦r tr❛❝❡❞ ❝❛t❡❣♦r✐❡s ❛r❡ ♦r✐❡♥t❡❞
✶✲❝♦❜♦r❞✐s♠s✧✳ ■♥ ♣r❡♣❛r❛t✐♦♥✳
❬✷✹❪ ❱❛❣♥❡r✱ ❉✳❀ ❙♣✐✈❛❦✱ ❉✳■✳❀ ▲❡r♠❛♥✱ ❊✳ ✭✷✵✶✹✮ ✏❆❧❣❡❜r❛s ♦❢ ❖♣❡♥ ❉②♥❛♠✐❝ ❙②st❡♠s ♦♥ t❤❡ ❖♣❡r❛❞
♦❢ ❲✐r✐♥❣ ❉✐❛❣r❛♠s✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✹✵✽✳✶✺✾✽✳
❬✷✺❪ ❙♣✐✈❛❦✱ ❉✳■✳❀ ❚❛♥ ❏✳❩✳ ✭✷✵✶✺✮ ✏◆❡st✐♥❣ ♦❢ ❞②♥❛♠✐❝ s②st❡♠s ❛♥❞ ♠♦❞❡✲❞❡♣❡♥❞❡♥t ♥❡t✇♦r❦s✧✳ ❆✈❛✐❧✲
❛❜❧❡ ♦♥❧✐♥❡✿
❤tt♣✿✴✴❛r①✐✈✳♦r❣✴❛❜s✴✶✺✵✷✳✵✼✸✽✵✳
❬✷✻❪ ❉♦❜❜s✱ ◆✳❀ ❙t❡♥❧✉♥❞✱ ▼✳ ✭✷✵✶✺✮ ✏◗✉❛s✐st❛t✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠s✧✳ ❆✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡✿
❛r①✐✈✳♦r❣✴❛❜s✴✶✺✵✹✳✵✶✾✷✻✳
❤tt♣✿✴✴
✭❏♦s❤✉❛ ❊❧❦✐♥❣t♦♥✮ ❉❡♣❛rt♠❡♥t ♦❢ ▼♦❧❡❝✉❧❛r ❛♥❞ ❈❡❧❧ ❇✐♦❧♦❣②✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧✐❢♦r✲
♥✐❛✱ ❇❡r❦❡❧❡②✳ ▼❛❣✐❝ ▲❛❜
❊✲♠❛✐❧ ❛❞❞r❡ss ✿
❯❈❇✿ ❡❧❦✐♥❣t♦♥❅❜❡r❦❡❧❡②✳❡❞✉ ▼▲✿ ❥①❡❧❦✐♥❣t♦♥❅❣♠❛✐❧✳❝♦♠
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