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  • 151.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Graph limits and hereditary properties2016In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 52, p. 321-337Article in journal (Refereed)
    Abstract [en]

    We give a survey of some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.

  • 152.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Maximal clades in random binary search trees2015In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 22, no 1, article id P1.31Article in journal (Refereed)
    Abstract [en]

    We study maximal clades in random phylogenetic trees with the Yule Harding model or, equivalently, in binary search trees. We use probabilistic methods to reprove and extend earlier results on moment asymptotics and asymptotic normality. In particular, we give an explanation of the curious phenomenon observed by Drmota, Fuchs and Lee (2014) that asymptotic normality holds, but one should normalize using half the variance.

  • 153.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    ON DEGENERATE SUMS OF m-DEPENDENT VARIABLES2015In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 52, no 4, p. 1146-1155Article in journal (Refereed)
    Abstract [en]

    It is well known that the central limit theorem holds for partial sums of a stationary sequence (X-i) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(X-i) not equal 0. We show that this happens only in the case when X-i - EXi = Y-i - Y-i for an (m - 1)-dependent stationary sequence (Y-i) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.

  • 154.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On Edge Exchangeable Random Graphs2018In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 173, no 3-4, p. 448-484Article in journal (Refereed)
    Abstract [en]

    We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields convergence to a non-integrable generalized graphon defined on (0, infinity).

  • 155.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On the tails of the limiting Quicksort distribution2015In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 20, p. 1-7, article id UNSP 81Article in journal (Refereed)
    Abstract [en]

    We give asymptotics for the left and right tails of the limiting Quicksort distribution. The results agree with, but are less precise than, earlier non-rigorous results by Knessl and Spankowski.

  • 156.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Patterns in Random Permutations Avoiding the Pattern 1322017In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 26, no 1, p. 24-51Article in journal (Refereed)
    Abstract [en]

    We consider a random permutation drawn from the set of 132-avoiding permutations of length n and show that the number of occurrences of another pattern sigma has a limit distribution, after scaling by n lambda(sigma)/2, where lambda(sigma) is the length of sigma plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

  • 157.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Patterns in random permutations avoiding the pattern 3212019In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, no 2, p. 249-270Article in journal (Refereed)
    Abstract [en]

    We consider a random permutation drawn from the set of 321-avoiding permutations of length n and show that the number of occurrences of another pattern sigma has a limit distribution, after scaling by n(m + l) where m is the length of sigma and l is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.

  • 158.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Renewal theory for asymmetric U-statistics2018In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 23, article id 129Article in journal (Refereed)
    Abstract [en]

    We extend a functional limit theorem for symmetric U-statistics [Miller and Sen, 1972] to asymmetric U-statistics, and use this to show some renewal theory results for asymmetric U-statistics. Some applications are given.

  • 159.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tail bounds for sums of geometric and exponential variables2018In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 135, p. 1-6Article in journal (Refereed)
    Abstract [en]

    We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.

  • 160.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The hiring problem with rank-based strategies2019In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 24, article id 125Article in journal (Refereed)
    Abstract [en]

    The hiring problem is studied for general strategies based only on the relative ranking of the candidates; this includes some well known strategies studied before such as hiring above the median. We give general limit theorems for the number of hired candidates and some other properties, extending previous results. The results exhibit a dichotomy between two classes of rank-based strategies: either the asymptotics of the process are determined by the early events, with a.s. convergence of suitably normalized random variables, or there is a mixing behaviour without long-term memory and with asymptotic normality.

  • 161.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Kaijser, Sten
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Higher moments of Banach space valued random variables2015In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 238, no 1127, p. 1-110Article in journal (Refereed)
    Abstract [en]

    We define the k:th moment of a Banach space valued random variable as the expectation of its k: th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several chapters are devoted to results in special Banach spaces, including Hilbert spaces, C(K) and D[0,1]. The latter space is non-separable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in D[0,1] that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix.

  • 162.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Yuan, Linglong
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Johannes Gutenberg Univ Mainz, Inst Math, Staudingerweg 9, D-55099 Mainz, Germany..
    On a representation theorem for finitely exchangeable random vectors2016In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 442, no 2, p. 703-714Article in journal (Refereed)
    Abstract [en]

    A random vector X = (X1, ... ,X-n) with the X-i taking values in an arbitrary measurable space (S, Sp) is exchangeable if its law is the same as that of (X-sigma(1), ... ,X-sigma(n)) for any permutation a. We give an alternative and shorter proof of the representation result (Jaynes [6] and Kerns and Szekely [9]) stating that the law of X is a mixture of product probability measures with respect to a signed mixing measure. The result is "finitistic" in nature meaning that it is a matter of linear algebra for finite S. The passing from finite S to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof ("canonical mixing measure") by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on S; but for S =, one can choose a mixing measure on R-n.

  • 163.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Linusson, Svante
    Matematiska institutionen, KTH, Stockholm, Sweden.
    Proportionella val inom kommunfullmäktige2019Report (Other academic)
    Abstract [sv]

    Vi diskuterar två olika problem som kan uppstå vid proportionella val i kommunfullmäktige och regionfullmäktige når ett parti försöker en kupp genom att utan samtycke gå i kartell med ett annat parti vid val till nämnd eller styrelse, vilket aktualiserades i åtminstone ett par fall hösten 2018. Det första problemet är vad sådana oönskade valkarteller kan få för effekter, och vilka möjligheter det finns för ett parti att skydda sig från att bli del i en oönskad valkartell. Det andra problemet är att i en sådan valkartell kan ett parti genom att splittra upp sina kandidater strategiskt  på flera olika valsedlar få fler platser i en nämnd är vad som är proportionellt. Detta andra problem bottnar i att lagen om proportionella val stipulerar att Thieles metod skall användas för fördelning inom kartellen. På detta problem finns en enkel matematisk lösning och vi argumenterar för att man skall byta till Phragméns metod som används för motsvarande val till utskott i riksdagen.

  • 164.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Luczak, Malwina
    Queen Mary Univ London, Sch Math Sci, Mile End Rd, London E1 4NS, England..
    Windridge, Peter
    Queen Mary Univ London, Sch Math Sci, Mile End Rd, London E1 4NS, England..
    House, Thomas
    Univ Manchester, Sch Math, Manchester M13 9PL, Lancs, England..
    Near-critical SIR epidemic on a random graph with given degrees2017In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 74, no 4, p. 843-886Article in journal (Refereed)
    Abstract [en]

    Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered epidemic on a random (multi)graph on n vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is , with tending to infinity slowly as the population size, n, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as . As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the 'sparse' regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

  • 165.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Riordan, Oliver
    Univ Oxford, Math Inst, Radcliffe Observ Quarter, Woodstock Rd, Oxford OX2 6GG, England.
    Warnke, Lutz
    Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA;Univ Cambridge Peterhouse, Cambridge CB2 1RD, England.
    Sesqui-type branching processes2018In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 128, no 11, p. 3628-3655Article in journal (Refereed)
    Abstract [en]

    We consider branching processes consisting of particles (individuals) of two types (type L and type S) in which only particles of type L have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.

  • 166.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Sen, Subhabrata
    Harvard Univ, Dept Stat, 1 Oxford St,SC 712, Cambridge, MA 02138 USA.
    Spencer, Joel
    NYU, Courant Inst, Dept Comp Sci, Room 829,251 Mercer St, New York, NY 10012 USA;NYU, Courant Inst, Dept Math, Room 829,251 Mercer St, New York, NY 10012 USA.
    Preferential attachment when stable2019In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 51, no 4, p. 1067-1108Article in journal (Refereed)
    Abstract [en]

    We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the ath power (alpha > 1) of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of 2n - 2 new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower- tail large deviation principle (LDP) for L= Sigma(n)(i=1) (S-i(2)/i(2)), where {S-n : n >= 0} is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

  • 167.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Shcherbakov, Vadim
    Royal Holloway Univ London, Egham, Surrey, England.
    Volkov, Stanislav
    Lund Univ, Lund, Sweden.
    Long Term Behaviour of a Reversible System of Interacting Random Walks2019In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 175, no 1, p. 71-96Article in journal (Refereed)
    Abstract [en]

    This paper studies the long-term behaviour of a system of interacting random walks labelled by vertices of a finite graph. We show that the system undergoes phase transitions, with different behaviour in various regions, depending on model parameters and properties of the underlying graph. We provide the complete classification of the long-term behaviour of the corresponding continuous time Markov chain, identifying whether it is null recurrent, positive recurrent, or transient. The proofs are partially based on the reversibility of the model, which allows us to use the method of electric networks. We also provide some alternative proofs (based on the Lyapunov function method and the renewal theory), which are of interest in their own right, since they do not require reversibility and can be applied to more general situations.

  • 168.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Sos, Vera T.
    More on quasi-random graphs, subgraph counts and graph limits2015In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 46, p. 134-160Article in journal (Refereed)
    Abstract [en]

    We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases. The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.

  • 169.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Stefánsson, Sigurdur Örn
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Scaling limits of random planar maps with a unique large face2015In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 43, no 3, p. 1045-1081Article in journal (Refereed)
    Abstract [en]

    We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n(-1/2) is described by a Brownian excursion. The planar maps, with the graph metric resealed by n(-1/2), are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

  • 170.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Uzzell, Andrew J.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    On String Graph Limits and the Structure of a Typical String Graph2017In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 84, no 4, p. 386-407Article in journal (Refereed)
    Abstract [en]

    We study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.

  • 171.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Viola, Alfredo
    Univ Republica, Montevideo, Uruguay..
    A Unified Approach to Linear Probing Hashing with Buckets2016In: Algorithmica, ISSN 0178-4617, E-ISSN 1432-0541, Vol. 75, no 4, p. 724-781Article in journal (Refereed)
    Abstract [en]

    We give a unified analysis of linear probing hashing with a general bucket size. We use both a combinatorial approach, giving exact formulas for generating functions, and a probabilistic approach, giving simple derivations of asymptotic results. Both approaches complement nicely, and give a good insight in the relation between linear probing and random walks. A key methodological contribution, at the core of Analytic Combinatorics, is the use of the symbolic method (based on q-calculus) to directly derive the generating functions to analyze.

  • 172.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Warnke, Lutz
    Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA.;Univ Cambridge Peterhouse, Cambridge CB2 1RD, England..
    On the critical probability in percolation2018In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 23, article id 1Article in journal (Refereed)
    Abstract [en]

    For percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erdos-Renyi random graph G(n,p), and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window p = 1/n + Theta(n(-4/3)), and (ii) the inverse of its maximum value coincides with the Theta(n(-4/3))-width of the critical window. We also prove that the maximizer is not located at p = 1/n or p = 1/(n - 1), refuting a speculation of Peres.

  • 173.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Warnke, Lutz
    Univ Cambridge, Dept Pure Math & Math Stat, Cambridge CB3 0WB, England..
    The lower tail: Poisson approximation revisited2016In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 48, no 2, p. 219-246Article in journal (Refereed)
    Abstract [en]

    The well-known Janson's inequality gives Poisson-like upper bounds for the lower tail probability P(X(1-epsilon)EX) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case epsilon=1) of Janson, uczak and Ruciski.

  • 174.
    Janson, Svante
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A piecewise contractive dynamical system and Phragmèn's election method2019In: Bulletin de la Société Mathématique de France, ISSN 0037-9484, E-ISSN 2102-622X, Vol. 147, no 3, p. 395-441Article in journal (Refereed)
    Abstract [en]

    We prove some basic results for a dynamical system given by a piece-wise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases, and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter is shown to have Hausdorff-dimension one. We also study the invariant sets and the limit sets; these are sometimes different and there are several cases to consider. In addition, we prove the existence of a unique invariant measure. We apply some of our results for the dynamical system, involving a study of rational and irrational rotation numbers, to a combinatorial problem involving an election method suggested by Phragmen, and we show that the proportion of elected seats for each party converges to a limit, which is a rational number except for a very small exceptional set of parameters.

  • 175. Johansson, Anders
    et al.
    Öberg, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pollicott, Mark
    Ergodic Theory of Kusuoka Measures2017In: Journal of Fractal Geometry, ISSN 2308-1309, Vol. 4, no 2, p. 185-214Article in journal (Refereed)
    Abstract [en]

    In the analysis on self-similar fractal sets, the Kusuoka measure plays an important role (cf. \cite{kusuoka2}, \cite{kajino}, \cite{str3}). Here we investigate the Kusuoka measure from an ergodic theoretic viewpoint, seen as an invariant measure on a symbolic space. Our investigation shows that the Kusuoka measure generalizes Bernoulli measures and their properties to higher dimensions of an underlying finite dimensional vector space. Our main result is that the transfer operator on functions has a spectral gap when restricted to a certain Banach space that contains the H\"older continuous functions, as well as the highly discontinuous $g$-function associated to the Kusuoka measure. As a consequence, we obtain exponential decay of correlations. In addition, we provide some explicit rates of convergence for a family of generalized Sierpi\'nski gaskets.

  • 176.
    Järvenpää, Esa
    et al.
    Univ Oulu, Dept Math Sci, POB 3000, Oulu 90014, Finland..
    Järvenpää, Maarit
    Univ Oulu, Dept Math Sci, POB 3000, Oulu 90014, Finland..
    Li, Bing
    Univ Oulu, Dept Math Sci, POB 3000, Oulu 90014, Finland.;South China Univ Technol, Dept Math, Guangzhou 510641, Guangdong, Peoples R China..
    Stenflo, Örjan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Random affine code tree fractals and Falconer-Sloan condition2016In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 36, no 5, p. 1516-1533Article in journal (Refereed)
    Abstract [en]

    We calculate the almost sure dimension for a general class of random affine code tree fractals in R-d. The result is based on a probabilistic version of the Falconer-Sloan condition C(s) introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413-427]. We verify that, in general, systems having a small number of maps do not satisfy condition C(s). However, there exists a natural number n such that for typical systems the family of all iterates up to level n satisfies condition C(s).

  • 177.
    Kaj, Ingemar
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Predicting pathogenicity behavior in Escherichia coli population through a state dependent model and TRS profiling2018In: PloS Computational Biology, ISSN 1553-734X, E-ISSN 1553-7358, Vol. 14, no 1, article id e1005931Article in journal (Refereed)
    Abstract [en]

    The Binary State Speciation and Extinction (BiSSE) model is a branching process based model that allows the diversification rates to be controlled by a binary trait. We develop a general approach, based on the BiSSE model, for predicting pathogenicity in bacterial populations from microsatellites profiling data. A comprehensive approach for predicting pathogenicity in E. coli populations is proposed using the state-dependent branching process model combined with microsatellites TRS-PCR profiling. Additionally, we have evaluated the possibility of using the BiSSE model for estimating parameters from genetic data. We analyzed a real dataset (from 251 E. coli strains) and confirmed previous biological observations demonstrating a prevalence of some virulence traits in specific bacterial sub-groups. The method may be used to predict pathogenicity of other bacterial taxa.

  • 178.
    Kaj, Ingemar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Konane, Victorien
    Uppsala University, Disciplinary Domain of Science and Technology, För teknisk-naturvetenskapliga fakulteten gemensamma enheter, International Science Programme (ISP). Univ Ouagadougou, Dept Math, Ouagadougou, Burkina Faso.
    Modeling battery cells under discharge using kinetic and stochastic battery models2016In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 40, no 17-18, p. 7901-7915Article in journal (Refereed)
    Abstract [en]

    In this paper we review several approaches to mathematical modeling of simple batterycells and develop these ideas further with emphasis on charge recovery and the responsebehavior of batteries to given external load. We focus on models which use few param-eters and basic battery data, rather than detailed reaction and material characteristicsof a specific battery cell chemistry, starting with the coupled ODE linear dynamics ofthe kinetic battery model. We show that a related system of PDE with Robin typeboundary conditions arises in the limiting regime of a spatial kinetic battery model,and provide a new probabilistic representation of the solution in terms of Brownianmotion with drift reflected at the boundaries on both sides of a finite interval. Tocompare linear and nonlinear dynamics in kinetic and stochastic battery models westudy Markov chains with states representing available and remaining capacities of thebattery. A natural scaling limit leads to a class of nonlinear ODE, which can be solvedexplicitly and compared with the capacities obtained for the linear models. To indicatethe potential use of the modeling we discuss briefly comparison of discharge profilesand effects on battery performance.

  • 179.
    Kaj, Ingemar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Mugal, Carina F.
    Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Ecology and Genetics, Evolutionary Biology.
    The non-equilibrium allele frequency spectrum in a Poisson random field framework2016In: Theoretical Population Biology, ISSN 0040-5809, E-ISSN 1096-0325, Vol. 111, p. 51-64Article in journal (Refereed)
    Abstract [en]

    In population genetic studies, the allele frequency spectrum (AFS) efficiently summarizes genome-wide polymorphism data and shapes a variety of allele frequency-based summary statistics. While existing theory typically features equilibrium conditions, emerging methodology requires an analytical understanding of the build-up of the allele frequencies over time. In this work, we use the framework of Poisson random fields to derive new representations of the non-equilibrium AFS for the case of a Wright-Fisher population model with selection. In our approach, the AFS is a scaling-limit of the expectation of a Poisson stochastic integral and the representation of the non-equilibrium AFS arises in terms of a fixation time probability distribution. The known duality between the Wright-Fisher diffusion process and a birth and death process generalizing Kingman's coalescent yields an additional representation. The results carry over to the setting of a random sample drawn from the population and provide the non-equilibrium behavior of sample statistics. Our findings are consistent with and extend a previous approach where the non-equilibrium AFS solves a partial differential forward equation with a non-traditional boundary condition. Moreover, we provide a bridge to previous coalescent-based work, and hence tie several frameworks together. Since frequency-based summary statistics are widely used in population genetics, for example, to identify candidate loci of adaptive evolution, to infer the demographic history of a population, or to improve our understanding of the underlying mechanics of speciation events, the presented results are potentially useful for a broad range of topics.

  • 180.
    Karakhanyan, Aram L.
    et al.
    Univ Edinburgh, Sch Math, Edinburgh, Midlothian, Scotland..
    Strömqvist, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization2016In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 6Article in journal (Refereed)
    Abstract [en]

    We consider the intersection of a convex surface Gamma with a periodic perforation of R-d, which looks like a sieve, given by T epsilon = boolean OR(d)(k is an element of Z) {epsilon k + a epsilon T} where T is a given compact set and a epsilon << epsilon is the size of the perforation in the epsilon-cell (0, epsilon)(d) subset of R-d. When epsilon tends to zero we establish uniform estimates for p- capacity, 1 < p < d, of the set Gamma n T-epsilon. Additionally, we prove that the intersections Gamma boolean AND {epsilon k + a(epsilon)T}(k) are uniformly distributed over Gamma and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of Gamma and the perforations, in a given bounded domain, is homogenizable when p < 1+ d/4. This result is new even for the classical Laplace operator.

  • 181.
    Karlsson, Anders
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Friedli, Fabien
    University of Geneva, Mathematics department.
    Spectral zeta functions of graphs and the Riemann zeta function in the critical strip2016In: Tohoku mathematical journal, ISSN 0040-8735Article in journal (Refereed)
    Abstract [en]

    We initiate the study of spectral zeta functions ζX for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function ζ(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of ζ(s). We relate ζZ to Euler's beta integral and show how to complete it giving the functional equation ξZ(1−s)=ξZ(s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of ζZd(s) to the whole plane and identify the poles. From our aymptotics several known special values of ζ(s) are derived as well as its non-vanishing on the line Re(s)=1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F1 via an Euler-type integral formula due to Picard.

  • 182.
    Kart, Özlem
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A Historical Survey of the Development of Classical Probability Theory2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 183.
    Killip, Rowan
    et al.
    Univ Calif Los Angeles, Dept Math, Box 951555, Los Angeles, CA 90095 USA..
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Matrix Models and Eigenvalue Statistics for Truncations of Classical Ensembles of Random Unitary Matrices2017In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 349, no 3, p. 991-1027Article in journal (Refereed)
    Abstract [en]

    We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of , not merely .

  • 184.
    Klimek, Maciej
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Personal reflections on Jozef Siciak's mathematical journey2019In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 123, no 1, p. 1-7Article in journal (Refereed)
    Abstract [en]

    The recent passing of Professor Jozef Siciak inevitably brings about reflections on his legacy, not just in terms of mathematical results he had obtained or inspired, but also in terms of shaping the way mathematics is being developed internally and as a part of science in general. Based on several decades of close personal contacts, the author attempts to outline Professor Siciak's views concerning these matters.

  • 185.
    Klimek, Maciej
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Kosek, Marta
    Jagiellonian Univ, Fac Math & Comp Sci, Inst Math, Krakow, Poland.
    On the metric space of pluriregular sets2018In: Dolomites Research Notes on Approximation, ISSN 2035-6803, Vol. 11, no Special issue, p. 51-61Article in journal (Refereed)
    Abstract [en]

    The metric space of pluriregular sets was introduced over two decades ago but to this day most of its topological properties remain a mystery. The purpose of this short survey is to present the current state of knowledge concerning this space.

  • 186.
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity2013In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 439, no 9, p. 2651-2658Article in journal (Refereed)
    Abstract [en]

    We prove the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.

  • 187.
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A review of Burke's theorem for Brownian motion2016In: Queueing systems, ISSN 0257-0130, E-ISSN 1572-9443, Vol. 83, no 1-2, p. 1-12Article, review/survey (Refereed)
    Abstract [en]

    Burke's theorem is a well-known fundamental result in queueing theory, stating that a stationary M/M/1 queue has a departure process that is identical in law to the arrival process and, moreover, for each time t, the following three random objects are independent: the queue length at time t, the arrival process after t and the departure process before t. Burke's theorem also holds for a stationary Brownian queue. In particular, it implies that a certain "complicated" functional derived from two independent Brownian motions is also a Brownian motion. The aim of this overview paper is to present an independent complete explanation of this phenomenon.

  • 188.
    Konstantopoulos, Takis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Gabrysch, Katja
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Convergence to the Tracy-Widom distribution for longest paths in a directed random graph2013In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 10, no 2, p. 711-730Article in journal (Refereed)
    Abstract [en]

    We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i(1), i(2)) to (j(1), j(2)), whenever i(1) <= j(1), i(2) <= j(2), with probability p, independently for each such pair of vertices. Let L-n,L-m denote the maximum length of all paths contained in an n x m rectangle. We show that there is a positive exponent a, such that, if m/n(a) -> 1, as n -> infinity, then a properly centered/rescaled version of L-n,L-m converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.

  • 189.
    Konstantopoulos, Takis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Liu, Zhenxia
    Linkoping Univ, Dept Math, SE-58183 Linkoping, Sweden..
    Yang, Xiangfeng
    Linkoping Univ, Dept Math, SE-58183 Linkoping, Sweden..
    Laplace Transform Asymptotics and Large Deviation Principles for Longest Success Runs in Bernoulli Trials2016In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, no 3, p. 747-764Article in journal (Refereed)
    Abstract [en]

    The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log(1/p) n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

  • 190.
    Konstantopoulos, Takis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Trinajstić, Katja
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Convergence to the Tracy-Widom distribution for longest paths in a directed random graph2013In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 10, no 2, p. 711-730Article in journal (Refereed)
    Abstract [en]

    We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i1,i2) to (j1,j2), whenever i1 ≤ j1, i2 ≤ j2, with probability p, independently for each such pair of vertices. Let Ln,m denote the maximum length of all paths contained in an n×m rectangle. We show that there is a positive exponent a, such that, if m/na→1, as n→∞, then a properly centered/rescaled version of Ln,m converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.

  • 191.
    Konstantopoulos, Takis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Yuan, Linglong
    Xian Jiaotong Liverpool Univ, Dept Math Sci, 111 Renai Rd, Suzhou 215123, Peoples R China..
    A PROBABILISTIC INTERPRETATION OF THE GAUSSIAN BINOMIAL COEFFICIENTS2017In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 54, no 4, p. 1295-1298Article in journal (Refereed)
    Abstract [en]

    We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.

  • 192.
    Konstantopoulos, Takis
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Yuan, Linglong
    Xian Jiaotong Liverpool Univ, Dept Math Sci, Suzhou, Peoples R China.
    Zazanis, Michael A.
    Athens Univ Econ & Business, Dept Stat, Athens, Greece.
    A fully stochastic approach to limit theorems for iterates of Bernstein operators2018In: Expositiones mathematicae, ISSN 0723-0869, E-ISSN 1878-0792, Vol. 36, no 2, p. 143-165Article in journal (Refereed)
    Abstract [en]

    This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator B-n taking a continuous function f is an element of C[0, 1] to a degree-n polynomial when the number of iterations k tends to infinity and n is kept fixed or when n tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of B-n a number of times k = k(n) to a polynomial f when k(n)/n tends to a constant.

  • 193.
    Kovachev, Yavor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Calibration of stochastic volatility models2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 194.
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Finite range perturbations of finite gap Jacobi and CMV operators2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 301, p. 204-226Article in journal (Refereed)
    Abstract [en]

    Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators solves an open problem of Simon [28, D.2.7]. We also solve the inverse resonance problem: it is shown that an operator is completely determined by the set of its eigen-values and resonances, and we provide necessary and sufficient conditions on their configuration for such an operator to exist.

  • 195.
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Rank One Non-Hermitian Perturbations of Hermitian β-Ensembles of Random Matrices2017In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 168, no 1, p. 92-108Article in journal (Refereed)
    Abstract [en]

    We provide a tridiagonal matrix model and compute the joint eigenvalue density of a rank one non-Hermitian perturbation of a random matrix from the Gaussian or Laguerre beta-ensemble.

  • 196.
    Krouthén, Johannes
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Extreme joint dependencies with copulasA new approach for the structure of C-Vines2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 197.
    Kungsman, Jimmy
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Melgaard, Michael
    Complex absorbing potential method for Dirac operators: clusters of resoances2014In: Journal of operator theory, ISSN 0379-4024, E-ISSN 1841-7744, Vol. 71, no 1, p. 259-283Article in journal (Refereed)
    Abstract [en]

    For both nonrelativistic and relativistic Hamiltonians, the complex absorbing potential (CAP) method has been applied extensively to cal- culate resonances in physics and chemistry. We study clusters of resonances for the perturbed Dirac operator near the real axis and, in the semiclassical limit, we establish the CAP method rigorously by showing that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa.

  • 198.
    Kungsman, Jimmy
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Melgaard, Michael
    Existence of Dirac resonances in the semi-classical limit2014In: Dynamics of Partial Differential Equations, ISSN 1548-159X, E-ISSN 2163-7873, Vol. 11, no 4, p. 381-395Article in journal (Refereed)
    Abstract [en]

    We study the existence of quantum resonances of the three-dimensional semiclassical Dirac operator perturbed by smooth, bounded and real-valued scalar potentials V decaying like < x >(-delta) at infinity for some delta > 0. By studying analytic singularities of a certain distribution related to V and by combining two trace formulas, we prove that the perturbed Dirac operators possess resonances near sup V + 1 and inf V - 1. We also provide a lower bound for the number of resonances near these points expressed in terms of the semiclassical parameter.

  • 199.
    Landstedt, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Parametric Geometry of Numbers and Exponents of Diophantine Approximation2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 200.
    Landström, Julia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The infinity-Laplacian and its properties2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
1234567 151 - 200 of 303
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