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Dissertations / Theses on the topic 'Random walks'

Author: Grafiati

Published: 4 June 2021

Last updated: 26 July 2025

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1

Bowditch, Adam. "Biased randomly trapped random walks and applications to random walks on Galton-Watson trees." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/97359/.

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In this thesis we study biased randomly trapped random walks. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive. This application was initially considered model in its own right. We prove conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. We also study the regime in which the walk is sub-ballistic; in this case we prove convergence to a stable subordinator. Furthermore, we study the

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2

Forghani, Behrang. "Transformed Random Walks." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32538.

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We consider transformations of a given random walk on a countable group determined by Markov stopping times. We prove that these transformations preserve the Poisson boundary. Moreover, under some mild conditions, the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory.

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3

Gnacik, Michal. "Quantum random walks." Thesis, Lancaster University, 2014. http://eprints.lancs.ac.uk/69946/.

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In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree represent

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4

Phetpradap, Parkpoom. "Intersections of random walks." Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.548100.

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We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we

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5

Windisch, David. "Random walks, disconnection and random interlacements /." [S.l.] : [s.n.], 2009. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=18343.

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6

Oosthuizen, Joubert. "Random walks on graphs." Thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/86244.

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Thesis (MSc)--Stellenbosch University, 2014.<br>ENGLISH ABSTRACT: We study random walks on nite graphs. The reader is introduced to general Markov chains before we move on more specifically to random walks on graphs. A random walk on a graph is just a Markov chain that is time-reversible. The main parameters we study are the hitting time, commute time and cover time. We nd novel formulas for the cover time of the subdivided star graph and broom graph before looking at the trees with extremal cover times. Lastly we look at a connection between random walks on graphs and electrical netw

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7

Montgomery, Aaron. "Topics in Random Walks." Thesis, University of Oregon, 2013. http://hdl.handle.net/1794/13335.

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We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem.

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8

Buckley, Stephen Philip. "Problems in random walks in random environments." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:06a12be2-b831-4c2a-87b1-f0abccfb9b8b.

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Recent years have seen progress in the analysis of the heat kernel for certain reversible random walks in random environments. In particular the work of Barlow(2004) showed that the heat kernel for the random walk on the infinite component of supercritical bond percolation behaves in a Gaussian fashion. This heat kernel control was then used to prove a quenched functional central limit theorem. Following this work several examples have been analysed with anomalous heat kernel behaviour and, in some cases, anomalous scaling limits. We begin by generalizing the first result - looking for suffici

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9

Ngoc, Anh Do Hoang. "Anomalous diffusion and random walks on random fractals." Doctoral thesis, Universitätsbibliothek Chemnitz, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201000205.

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The purpose of this research is to investigate properties of diffusion processes in porous media. Porous media are modelled by random Sierpinski carpets, each carpet is constructed by mixing two different generators with the same linear size. Diffusion on porous media is studied by performing random walks on random Sierpinski carpets and is characterized by the random walk dimension $d_w$. In the first part of this work we study $d_w$ as a function of the ratio of constituents in a mixture. The simulation results show that the resulting $d_w$ can be the same as, higher or lower than $d_w$ of c

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10

Gabucci, Ilenia. "Random walks classici e quantistici." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/17759/.

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Questa tesi si propone di fornire uno studio sul random walk. Partendo da un approfondimento sulla teoria della probabilita` alla base di tale processo ed in particolare le distribuzioni Binomiale e Gaussiana, si `e potuto studiare il caso del random walk classico, sia nel caso del reticolo monodimensionale che per reticoli a piu` dimensioni. Sempre nell’ambito classico si sono analizzati anche i processi stocastici dipendenti dal tempo, detti Processi di Markov, e il moto browniano, per cui si sono ricavate le equazioni del moto della particella browniana, ovvero le equazioni di Langevin. Si

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11

Codling, Edward Alexander. "Biased random walks in biology." Thesis, University of Leeds, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275673.

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12

Janse, van Rensburg Esaias Johannes. "Field theory and random walks." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.328723.

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13

Dou, Carl C. Z. (Carl Changzhu). "Studies of random walks on groups and random graphs." Thesis, Massachusetts Institute of Technology, 1992. http://hdl.handle.net/1721.1/13243.

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14

Bertacchi, D., and Andreas Cap@esi ac at. "Random Walks on Diestel--Leader Graphs." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1004.ps.

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15

Dykiel, Patrik. "Asymptotic properties of coalescing random walks." Thesis, Uppsala University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121369.

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16

He, Mu. "The Torsion Angle of Random Walks." TopSCHOLAR®, 2013. http://digitalcommons.wku.edu/theses/1242.

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In this thesis, we study the expected mean of the torsion angle of an n-stepequilateral random walk in 3D. We consider the random walk is generated within a confining sphere or without a confining sphere: given three consecutive vectors →e1 , →e2 , and →e3 of the random walk then the vectors →e1 and →e2 define a plane and the vectors →e2 and →e3 define a second plane. The angle between the two planes is called the torsion angle of the three vectors. Algorithms are described to generate random walks which are used in a particular space (both without and with confinement). The torsion angle is e

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17

Deligiannidis, Georgios. "Some results associated with random walks." Thesis, University of Nottingham, 2010. http://eprints.nottingham.ac.uk/13104/.

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In this thesis we treat three problems from the theory and applications of random walks. The first question we tackle is from the theory of the optimal stopping of random walks. We solve the infinite-horizon optimal stopping problem for a class of reward functions admitting a representation introduced in Boyarchenko and Levendorskii [1], and obtain closed expressions for the expected reward and optimal stopping time. Our methodology is a generalization of an early paper by Darling et al. [2] and is based on probabilistic techniques: in particular a path decomposition related to the Wiener-Hopf

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18

Xu, Chang. "Convex hulls of planar random walks." Thesis, University of Strathclyde, 2017. http://digitool.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=28164.

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For the perimeter length Ln and the area An of the convex hull of the first n steps of a planar random walk, this thesis study n â mean and variance asymptotics and establish distributional limits. The results apply to random walks both with drift (the mean of random walk increments) and with no drift under mild moments assumptions on the increments. Assuming increments of the random walk have finite second moment and non zero mean, Snyder and Steele showed that nâ1Ln converges almost surely to a deterministic limit, and proved an upper bound on the variance Var[Ln] = O(n).We show that nâ

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19

Lapeyre, Gerald John. "Random walks on a fluctuating lattice." Diss., The University of Arizona, 2001. http://hdl.handle.net/10150/298790.

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In recent years, studies of diffusion in random media have been extended to include the effects of media in which the defects fluctuate randomly in time. Typically, the diffusive motion of particles in a static medium persists when the medium is allowed to fluctuate, with the diffusivity (diffusion constant) D depending on the character of the fluctuations. In the present work, we study random walks on lattices in which the bonds connecting vertices open and close randomly in time, and the walker is not allowed to cross a closed bond. Variations of the model studied here have been used to mod

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20

Jiang, Jianping. "Random Walks and Their Scaling Limits." Diss., The University of Arizona, 2015. http://hdl.handle.net/10150/556605.

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This dissertation deals with two-dimensional random walks and their conformally invariant scaling limits. More precisely, we study two kinds of random walks: sum of independently identically distributed random variables (with Brownian motion as the scaling limit) and self-interacting random walks (with Schramm-Loewner Evolution processes as their scaling limits). We organize our main results in three parts. In the first part, we study two types of probability measures on Brownian paths: fixed time ensemble and fixed endpoints ensemble. We prove a relationship between those two ensembles. The r

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21

Tokushige, Yuki. "Random Walks on random trees and hyperbolic groups: trace processes on boundaries at infinity and the speed of biased random walks." Kyoto University, 2019. http://hdl.handle.net/2433/242580.

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22

Carigi, Giulia. "On the recurrence of random walks in Lévy random environments." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10088/.

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This thesis investigates one-dimensional random walks in random environment whose transition probabilities might have an infinite variance. The ergodicity of the dynamical system ''from the point of view of the particle'' is proved under the assumptions of transitivity and existence of an absolutely continuous steady state on the space of the environments. We show that, if the average of the local drift over the environments is summable and null, then the RWRE is recurrent. We provide an example satisfying all the hypotheses.

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23

Zimmermann, Jochen [Verfasser], and Andreas [Akademischer Betreuer] Buchleitner. "Random walks with nonlinear interactions on heterogeneous networks = Random Walk mit nichtlinearen Wechselwirkungen auf heterogenen Netzwerken." Freiburg : Universität, 2015. http://d-nb.info/1123482381/34.

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24

fr, kaimanov@univ-rennes1. "Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1085.ps.

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25

Sylvester, John A. "Random walks, effective resistance and neighbourhood statistics in binomial random graphs." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/106467/.

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The binomial random graph model G(n; p), along with its near-twin sibling G(n; m), were the starting point for the entire study of random graphs and even probabilistic combinatorics as a whole. The key properties of these models are woven into the fabric of the field and their behaviour serves as a benchmark to compare any other model of random structure. In this thesis we contribute to the already rich literature on G(n; p) in a number of directions. Firstly, vertex to vertex hitting times of random walks in G(n; p) are considered via their interpretation as potential differences in an electr

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26

Bui, Thi Thuy. "Limit theorems for branching random walks and products of random matrices." Thesis, Lorient, 2020. https://tel.archives-ouvertes.fr/tel-03261556.

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L’objectif du sujet de ma thèse est d’établir des théorèmes limites pour des marches aléatoires avec branchement gouvernées par des produits de matrices aléatoires, en profitant des progrès récents sur les produits de matrices aléatoires et en y établissant de nouveaux résultats adaptés au besoin. La première partie concerne le modèle classique d'une marche aléatoire avec branchement sur la droite réelle. Nous établissons une borne Berry-Esseen et une asymptotique précise de déviation modérée de type Cramér pour la mesure de comptage qui compte le nombre de particules de n-ième génération situ

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27

Maddalena, Daniela. "Stationary states in random walks on networks." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10170/.

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In this thesis we dealt with the problem of describing a transportation network in which the objects in movement were subject to both finite transportation capacity and finite accomodation capacity. The movements across such a system are realistically of a simultaneous nature which poses some challenges when formulating a mathematical description. We tried to derive such a general modellization from one posed on a simplified problem based on asyncronicity in particle transitions. We did so considering one-step processes based on the assumption that the system could be describable through discr

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28

Schubert, Sven. "Random walks in complex systems anomalous relaxation /." Doctoral thesis, [S.l. : s.n.], 1999. http://deposit.ddb.de/cgi-bin/dokserv?idn=956664911.

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29

Bertacchi, Daniela, Fabio Zucca, and Andreas Cap@esi ac at. "Classification on the Average of Random Walks." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1026.ps.

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30

at, Klaus Schmidt@univie ac. "Growth and Recurrence of Stationary Random Walks." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1071.ps.

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31

Wallén, Daniel. "Cover times of random walks on graphs." Thesis, Uppsala University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-125278.

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32

Schulzky, Christian Berthold. "Anomalous Diffusion and Random Walks on Fractals." Doctoral thesis, Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000705.

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In dieser Arbeit werden verschieden Ansätze diskutiert, die zum Verständnis und zur Beschreibung anomalen Diffusionsverhaltens beitragen, wobei insbesondere zwei unterschiedliche Aspekte hervorgehoben werden. Zum einen wird das Entropieproduktions-Paradoxon beschrieben, welches bei der Analyse der Entropieproduktion bei der anomalen Diffusion, beschrieben durch fraktionale Diffusionsgleichungen auftritt. Andererseits wird ein detaillierter Vergleich zwischen Lösungen verallgemeinerter Diffusionsgleichungen mit numerischen Daten präsentiert, die durch Iteration der Mastergleichung auf verschied

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33

Sbihi, Amine M. (Amine Mohammed). "Covering times for random walks on graphs." Thesis, McGill University, 1990. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74538.

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This thesis is a contribution to the covering times problems for random walks on graphs. By considering uniform random walks on finite connected graphs, the covering time is defined as the time (number of steps) taken by the random walk to visit every vertex. The motivating problem of this thesis is to find bounds for the expected covering times. We provide explicit bounds that are uniformly valid over all starting points and over large classes of graphs. In some cases the asymptotic distribution of the suitably normalized covering time is given as well.

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34

Folz, Matthew Bryan. "Adapted metrics and random walks on graphs." Thesis, University of British Columbia, 2013. http://hdl.handle.net/2429/44947.

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This thesis discusses various aspects of continuous-time simple random walks on measure weighted graphs, with a focus on behaviors related to large-scale geometric properties of the underlying graph. In contrast to previous work in this area, the majority of the results presented here are applicable to random walks with unbounded generators. A recurring theme in this research is the use of novel distance functions for graphs known as adapted metrics, which are demonstrated to be a powerful tool for studying random walks on graphs. Chapter 2 provides an overview of the relevant probabilist

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35

Böhm, Walter, and Kurt Hornik. "On Two-Periodic Random Walks with Boundaries." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2008. http://epub.wu.ac.at/936/1/document.pdf.

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Two-periodic random walks are models for the one-dimensional motion of particles in which the jump probabilities depend on the parity of the currently occupied state. Such processes have interesting applications, for instance in chemical physics where they arise as embedded random walk of a special queueing problem. In this paper we discuss in some detail first passage time problems of two-periodic walks, the distribution of their maximum and the transition functions when the motion of the particle is restricted by one or two absorbing boundaries. As particular applications we show how our res

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36

Volkov, Oleksii. "Random Walks on Products of Hyperbolic Groups." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/41955.

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The subject area of this thesis is the theory of random walks on groups. First, we study random walks on products of hyperbolic groups and show that the Poisson boundary can be identified with an appropriate geometric boundary (the skeleton). Second, we show that in the particular case of free and free-product factors, the Hausdorff dimension of the conditional measures on product fibers of the Poisson boundary is related to the asymptotic entropy and the rate of escape of the corresponding conditional random walks via a generalized entropy-dimension formula.

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37

Schwalbe, Lehtihet André, and Bulancea Oscar Lindvall. "Quantum Random Walks with Perturbing Potential Barriers." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210864.

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With a recent interest in quantum computers, the properties of quantum mechanicalcounterparts to classical algorithms have been studied in the hope of providing efficientalgorithms for quantum computers. Because of the success of classical random walks inproviding good algorithms on classical computers, attention has been turned to quantumrandom walks, since they may similarly be used to construct efficient probabilisticalgorithms on quantum computers. In this thesis we examine properties of the quantumwalk on the line, in particular the standard deviation and the shape of the probabilitydistr

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38

Huang, Tsongjy. "Random walks on randomly partitioned lattices with applications toward protein fluctuations." Diss., The University of Arizona, 1995. http://hdl.handle.net/10150/187406.

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Random walks on state space partitions provide an abstract generic picture for the description of macroscopic fluctuations in complex systems like proteins. We first determine the average residence probability and the average distribution of residence times in a particular macroscopic state for the ensemble of random partitions of a one-dimensional state space. We then extend our study to the Bethe lattice and also the 2-, 3- and higher dimensional lattices. Our treatment involves both extensive analytical and numerical analyses. Finally, we compare the solution of our model on the Bethe latti

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39

Nakashima, Makoto. "Almost sure central limit theorem for branching random walks in random environment." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157736.

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40

Carlsund, Anna. "Cover Times, Sign-dependent Random Walks, and Maxima." Doctoral thesis, KTH, Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3624.

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41

Devore, Lucas Clay. "Random Walks with Elastic and Reflective Lower Boundaries." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/134.

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42

Alharbi, Manal. "Random Walks on Free Products of Cyclic Groups." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/37494.

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In this thesis, we investigate examples of random walks on free products of cyclic groups. Free products are groups that contain words constructed by concatenation with possible simplifications[20]. Mairesse in [17] proved that the harmonic measure on the boundary of these random walks has a Markovian Multiplicative structure (this is a class of Markov measures which requires fewer parameters than the usual Markov measures for its description ), and also showed how in the case of the harmonic measure these parameters can be found from Traffic Equations. Then Mairesse and Math ́eus in [20] cont

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43

Jones, Owen Dafydd. "Random walks on pre-fractals and branching processes." Thesis, University of Cambridge, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.388440.

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Jones, Elinor Mair. "Large deviations of random walks and levy processes." Thesis, University of Manchester, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.491853.

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45

Katzenbeisser, Walter, and Wolfgang Panny. "The Maximal Height of Simple Random Walks Revisited." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1998. http://epub.wu.ac.at/1126/1/document.pdf.

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In a recent paper Katzenbeisser and Panny (1996) derived distributional results for a number of so called simple random walk statistics defined on a simple random walk in the sense of Cox and Miller (1968) starting at zero and leading to state 1 after n steps, where 1 is arbitrary, but fix. In the present paper the random walk statistics Dn = the one-sided maximum deviation and Qn = the number of times where the maximum is achieved, are considered and distributional results are presented, when it is irrespective, where the random walk terminates after n steps. Thus, the results can be seen as

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46

Slutsky, Michael. "Protein-DNA interaction, random walks and polymer statistics." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/32295.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.<br>Includes bibliographical references (p. 112-124).<br>In Part I of the thesis, a general physical framework describing the kinetics of protein- DNA interaction is developed. Recognition and binding of specific sites on DNA by proteins is central for many cellular functions such as transcription, replication, and recombination. In the process of recognition, a protein rapidly searches for its specific site on a long DNA molecule and then strongly binds this site. Earlier studies have suggested that rapid search in

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47

Abdullah, Mohammed. "The cover time of random walks on graph." Thesis, King's College London (University of London), 2012. https://kclpure.kcl.ac.uk/portal/en/theses/the-cover-time-of-random-walks-on-graph(c23c303f-a6a2-4489-a059-4ade7c118106).html.

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A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex. Central to this thesis is the cover time of the walk, that is, the expectation of the number of steps required to visit every vertex, maximised over all starting vertices. In our rst contribution, we establish a relation between the cover times of a pair of graphs, and the cover time of their Cartesian product. This extends previous work on special cases of the C

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48

Jamshidpey, Arash. "Population Dynamics in Random Environment, Random Walks on Symmetric Group, and Phylogeny Reconstruction." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/34623.

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This thesis concerns applications of some probabilistic tools to phylogeny reconstruction and population genetics. Modelling the evolution of species by continuous-time random walks on the signed permutation groups, we study the asymptotic medians of a set of random permutations sampled from simple random walks at time 0.25cn, for c> 0. Running k independent random walks all starting at identity, we prove that the medians approximate the ancestor (identity permutation) up to time 0.25n, while there exists a constant c>1 after which the medians loose credibility as an estimator. We study the me

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49

Izsak, Alexander. "The second eigenvalue and random walks in random regular graphs with increasing girth." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/12649.

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The goal of this thesis is to upper bound the expected value of the second largest eigenvalue in magnitude of random regular graphs with a given minimum girth. Having a small upper bound implies such random graphs are likely to be expanders and thus have several combinatorial properties useful in various fields of computer science. The best possible upper bound asymptotically on the second eigenvalue has already been proven for random regular graphs without conditions on the girth. Finding this upper bound though required long and complicated analysis due to tangles, which are certain small su

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50

Ahn, Sung Won. "Oscillation of quenched slowdown asymptotics of random walks in random environment in Z." Thesis, Purdue University, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10170588.

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<p> We consider a one dimensional random walk in a random environment (RWRE) with a positive speed lim<i><sub>n</sub></i><sub>→∞</sub> (<i>X<sub>n</sub>/</i>) = &upsi;<sub>α</sub> > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities <i>P</i><sub> ω</sub>(<i>X<sub>n</sub></i> < <i>xn</i>) with <i> x</i>∈ (0,&upsi;<sub>α</sub>) decay approximately like exp{-<i> n</i><sup>1-1/</sup><i><sup>s</sup></i>} for a deterministic <i> s</i> > 1. More precisely, they showed that <i>

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