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  • 1.
    Adimurthi,
    et al.
    TIFR CAM, PB 6503, Bangalore 560065, Karnataka, India.
    Tintarev, Kyril
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Defect of compactness in spaces of bounded variation2016In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 271, no 1, p. 37-48Article in journal (Refereed)
    Abstract [en]

    Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. Let X be a Banach space continuously imbedded into a Banach space Y, and let D be a group of linear isometric operators on X. A profile decomposition in X, relative to D and Y, for a bounded sequence (x(k))(k is an element of N) subset of X is a sequence (S-k)(k is an element of N), such that (x(k) - S-k)(k is an element of N) is a convergent sequence in Y, and, furthermore, S-k has the particular form S-k = Sigma(n is an element of N)g(k)((n))W((n)) with g(k)((n)) is an element of D and w((n)) is an element of X. This paper extends the profile decomposition proved by Solimini [10] for Sobolev spaces (H) over dot(1,P)(R-N) with 1 < p < N to the non-reflexive case p = 1. Since existence of "concentration profiles" w((n)) relies on weak-star compactness, and the space (H) over dot(1,1) is not a conjugate of a Banach space, we prove a corresponding result for a larger space of functions of bounded variation. The result extends also to spaces of bounded variation on Lie groups.

  • 2. Adimurthi, no first name
    et al.
    Tintarev, Cyril
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On compactness in the Trudinger-Moser inequality2014In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 13, no 2, p. 399-416Article in journal (Refereed)
    Abstract [en]

    We show that the Moser functional J(u) = integral Omega(e(4 pi u2) - 1) dx on the set B = {u is an element of H-0(1)(Omega) : parallel to del u parallel to(2) <= 1}, where Omega subset of R-2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define g(s)w(r) = s(-1/2)w(r(s)) for s > 0. If u(k) -> u in B while lim inf J(u(k)) > J(u), then, with some s(k) -> 0, u(k) = g(sk) [(2 pi)(-1/2) min {1, log1/vertical bar x vertical bar}], up to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.

  • 3.
    Ahlberg, Daniel
    et al.
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden.
    Deijfen, Maria
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Competing first passage percolation on random graphs with finite variance degrees2019In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, no 3, p. 545-559Article in journal (Refereed)
    Abstract [en]

    We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate lambda(1) (lambda(2)) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if lambda(1) = lambda(2), then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V is an element of (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If lambda(1) not equal lambda(2), on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

  • 4.
    Ahlberg, Daniel
    et al.
    Stockholm Univ, Dept Math, Stockholm, Sweden..
    Griffiths, Simon
    Pontificia Univ Catolica Rio de Janeiro, Dept Matemat, Rio de Janeiro, Brazil..
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, Uppsala, Sweden..
    TO FIXATE OR NOT TO FIXATE IN TWO-TYPE ANNIHILATING BRANCHING RANDOM WALKS2021In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 49, no 5, p. 2637-2667Article in journal (Refereed)
    Abstract [en]

    We study a model of competition between two types evolving as branching random walks on Z(d). The two types are represented by red and blue balls, respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of Z(d) contain one ball each which are independently coloured red with probability p and blue otherwise. We address the question of fixation, referring to the sites and eventually settling for a given colour or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for p not equal 1/2 and that every site will change colour infinitely often almost surely for the balanced initial condition p = 1/2.

  • 5.
    Ahlberg, Daniel
    et al.
    Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
    Griffiths, Simon
    PUC Rio, Dept Matemat, BR-22451900 Gavea, RJ, Brazil.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Morris, Robert
    Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
    Competition in growth and urns2019In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 54, no 2, p. 211-227Article in journal (Refereed)
    Abstract [en]

    We study survival among two competing types in two settings: a planar growth model related to two-neighbor bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or infinity, depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls ( but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one-for-one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two-type growth model on Z(2), one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

  • 6.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, SE-75106 Uppsala, Sweden..
    Griffiths, Simon
    Univ Oxford, Dept Stat, Oxford OX1 3TG, England..
    Morris, Robert
    IMPA, Rio De Janeiro, RJ, Brazil..
    Tassion, Vincent
    Univ Geneva, Dept Math, Geneva, Switzerland..
    Quenched Voronoi percolation2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 286, p. 889-911Article in journal (Refereed)
    Abstract [en]

    We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.

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  • 7.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.;Uppsala Univ, Dept Math, SE-75106 Uppsala, Sweden..
    Steif, Jeffrey E.
    Univ Gothenburg, Chalmers Univ Technol, Math Sci, SE-41296 Gothenburg, Sweden..
    Pete, Gabor
    Hungarian Acad Sci, Renyi Inst, 13-15 Realtanoda U, H-1053 Budapest, Hungary.;Budapest Univ Technol & Econ, Inst Math, 1 Egry Jozsef U, H-1111 Budapest, Hungary..
    Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?2017In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 53, no 4, p. 2135-2161Article in journal (Refereed)
    Abstract [en]

    Consider a monotone Boolean function f : {0, 1}(n) -> {0, 1} and the canonical monotone coupling {eta(p) : p is an element of [0, 1]} of an element in {0, 1}(n) chosen according to product measure with intensity p is an element of [0, 1]. The random point p is an element of [0, 1] where f (eta(p)) flips from 0 to 1 is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large n, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majority and percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.

  • 8.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Matematica Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
    Tassion, Vincent
    Univ Geneva, 2-4 Rue Lievre, CH-1211 Geneva, Switzerland.
    Teixeira, Augusto
    Inst Matematica Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
    Sharpness of the phase transition for continuum percolation in R22018In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 172, no 1-2, p. 525-581Article in journal (Refereed)
    Abstract [en]

    We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter.c. Moreover, For. <.c, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution. At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius. For. >.c, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast. The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

  • 9.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Nacl Matemat Pura & Aplicada, Rio De Janeiro, RJ, Brazil;Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden;Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
    Tykesson, Johan
    Chalmers Univ Technol, Dept Math, SE-41296 Gothenburg, Sweden;Univ Gothenburg, Gothenburg, Sweden.
    Gilbert´s disc model with geostatical marking2018In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 50, no 4, p. 1075-1094Article in journal (Refereed)
    Abstract [en]

    We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in R-2 with radii determined by an underlying stationary and ergodic random field phi: R-2 -> [0, infinity), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of R(2 )does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

  • 10.
    Alam, Mahbub
    et al.
    Tata Inst Fundamental Res, Sch Math, Mumbai 400005, Maharashtra, India.
    Ghosh, Anish
    Tata Inst Fundamental Res, Sch Math, Mumbai 400005, Maharashtra, India.
    Yu, Shucheng
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Quantitative Diophantine approximation with congruence conditions2021In: Journal de Théorie des Nombres de Bordeaux, ISSN 1246-7405, E-ISSN 2118-8572, Vol. 33, no 1, p. 261-271Article in journal (Refereed)
    Abstract [en]

    In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.

  • 11.
    Albert, Michael
    et al.
    Univ Otago, Dept Comp Sci, Dunedin, New Zealand.
    Holmgren, Cecilia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Johansson, Tony
    Stockholm Univ, Dept Math, Stockholm, Sweden.
    Skerman, Fiona
    Masaryk Univ, Fac Informat, Brno, Czech Republic.
    Embedding Small Digraphs and Permutations in Binary Trees and Split Trees2020In: Algorithmica, ISSN 0178-4617, E-ISSN 1432-0541, Vol. 82, no 3, p. 589-615Article in journal (Refereed)
    Abstract [en]

    We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335-364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409-432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

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  • 12.
    Alm, Sven Erick
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Deijfen, Maria
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden..
    First Passage Percolation on \(\mathbb {Z}^2\): A Simulation Study2015In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 161, no 3, p. 657-678Article in journal (Refereed)
    Abstract [en]

    First passage percolation on is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges. In this paper, the speed of the growth and the shape of the infected set is studied by aid of large-scale computer simulations, with focus on continuous passage time distributions. It is found that the most important quantity for determining the value of the time constant, which indicates the inverse asymptotic speed of the growth, is , where are i.i.d. passage time variables. The relation is linear for a large class of passage time distributions. Furthermore, the directional time constants are seen to be increasing when moving from the axis towards the diagonal, so that the limiting shape is contained in a circle with radius defined by the speed along the axes. The shape comes closer to the circle for distributions with larger variability.

  • 13.
    Andersen, Jorgen Ellegaard
    et al.
    Aarhus Univ, Ctr Quantum Geometry Moduli Spaces QGM, Ny Munkegade 118,Bldg 1530, DK-8000 Aarhus C, Denmark..
    Himpel, Benjamin
    Aarhus Univ, Ctr Quantum Geometry Moduli Spaces QGM, Ny Munkegade 118,Bldg 1530, DK-8000 Aarhus C, Denmark..
    Jørgensen, Søren Fuglede
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Martens, Johan
    Univ Edinburgh, Sch Math, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, Midlothian, Scotland.;Univ Edinburgh, Maxwell Inst Math Sci, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, Midlothian, Scotland..
    McLellan, Brendan
    Harvard Univ, Dept Math, One Oxford St, Cambridge, MA 02138 USA..
    The Witten-Reshetikhin-Turaev invariant for links in finite order mapping tori I2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, p. 131-178Article in journal (Refereed)
    Abstract [en]

    We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automor-phisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno. (C) 2016 Elsevier Inc. All rights reserved.

  • 14.
    Andersen, Jørgen Ellegaard
    et al.
    Centre for Quantum Geometry of Moduli Spaces, Aarhus University.
    Jørgensen, Søren Fuglede
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On the Witten–Reshetikhin–Turaev invariants of torus bundles2015In: Journal of knot theory and its ramifications, ISSN 0218-2165, Vol. 24, no 11, article id 1550055Article in journal (Refereed)
    Abstract [en]

    By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys. 147 (1992) 563-604], we compute the quantum SU(N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining - in combination with Jeffrey's results - a proof of the asymptotic expansion conjecture for all torus bundles.

  • 15.
    Andersson, Rasmus
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Cavalieris indivisibler2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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  • 16.
    Andriantiana, Eric O. D.
    et al.
    Rhodes Univ, Dept Math Pure & Appl, POB 94, ZA-6140 Grahamstown, South Africa..
    Misanantenaina, Valisoa Razanajatovo
    Stellenbosch Univ, Dept Math Sci, Private Bag X1, ZA-7602 Matieland, South Africa..
    Wagner, Stephan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Stellenbosch Univ, Dept Math Sci, Private Bag X1, ZA-7602 Matieland, South Africa.
    Extremal trees with fixed degree sequence2021In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 28, no 1, article id P1.1Article in journal (Refereed)
    Abstract [en]

    The greedy tree G(D) and the M-tree M(D) are known to be extremal among trees with degree sequence D with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which G(D) or M(D) is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence D to the set of trees whose degree sequence is majorised by a given sequence D, which also has a number of applications.

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  • 17.
    Andrén, Dag
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Om oändliga tal2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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  • 18.
    Anema, Jason A.
    et al.
    Univ Illinois, Dept Math, Urbana, IL 61801 USA..
    Tsougkas, Konstantinos
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Counting spanning trees on fractal graphs and their asymptotic complexity2016In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 49, no 35, article id 355101Article in journal (Refereed)
    Abstract [en]

    Using the method of spectral decimation and a modified version of Kirchhoff's matrix-tree theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in theorem 3.4. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski gasket, a non-post critically finite analog of the Sierpinski gasket, the Diamond fractal, and the hexagasket. For each example, the asymptotic complexity constant is found.

  • 19.
    Angel, Omer
    et al.
    Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada.
    van Der Hofstad, Remco
    Eindhoven Univ Technol, Dept Math & Comp Sci, POB 513, NL-5600 MB Eindhoven, Netherlands.
    Holmgren, Cecilia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Limit laws for self-loops and multiple edges in the configuration model2019In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 55, no 3, p. 1509-1530Article in journal (Refereed)
    Abstract [en]

    We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a uniform random graph with prescribed degrees. Simplicity corresponds to the absence of self-loops and multiple edges. We show that the number of self-loops and multiple edges converges in distribution to two independent Poisson random variables when the second moment of the empirical degree distribution converges. We also provide estimations on the total variation distance between the numbers of self-loops and multiple edges and their limits, as well as between the sum of these values and the Poisson random variable to which this sum converges to. This revisits previous works of Bollobas, of Janson, of Wormald and others. The error estimates also imply sharp asymptotics for the number of simple graphs with prescribed degrees. The error estimates follow from an application of the Stein-Chen method for Poisson convergence, which is a novel method for this problem. The asymptotic independence of self-loops and multiple edges follows from a Poisson version of the Cramer-Wold device using thinning, which is of independent interest. When the degree distribution has infinite second moment, our general results break down. We can, however, prove a central limit theorem for the number of self-loops, and for the multiple edges between vertices of degrees much smaller than the square root of the size of the graph. Our results and proofs easily extend to directed and bipartite configuration models.

  • 20.
    Aptekarev, Alexander, I
    et al.
    Russian Acad Sci, Keldysh Inst Appl Math, Miusskaya Pl 4, Moscow 125047, Russia..
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a Nevai class2020In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 255, article id 105409Article in journal (Refereed)
    Abstract [en]

    A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equations become ordinary. For Angelesco systems, the result is illustrated numerically. 

  • 21.
    Archibald, Margaret
    et al.
    Univ Witwatersrand, Johannesburg, South Africa..
    Blecher, Aubrey
    Univ Witwatersrand, Johannesburg, South Africa..
    Brennan, Charlotte
    Univ Witwatersrand, Johannesburg, South Africa..
    Knopfmacher, Arnold
    Univ Witwatersrand, Johannesburg, South Africa..
    Wagner, Stephan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Stellenbosch Univ, Stellenbosch, South Africa..
    Ward, Mark Daniel
    Purdue Univ, Dept Stat, W Lafayette, IN 47907 USA..
    The number of distinct adjacent pairs in geometrically distributed words2021In: Discrete Mathematics & Theoretical Computer Science, ISSN 1462-7264, E-ISSN 1365-8050, Vol. 22, no 4, article id 10Article in journal (Refereed)
    Abstract [en]

    A sequence of geometric random variables of length n is a sequence of n independent and identically distributed geometric random variables (Gamma(1), Gamma(2), ..., Gamma(n)) where P (Gamma(j) = i) = pq(i-1) for 1 <= j <= n with p + q = 1. We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length n. We also obtain the asymptotics for the expected number as n -> infinity.

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  • 22.
    Ashraf, Pouya
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pathological functions and the Baire category theorem2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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  • 23.
    Ataei, Alireza
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Egert, Moritz
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The Kato square root problem for weighted parabolic operators2023In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206XArticle in journal (Refereed)
    Abstract [en]

    We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on the nowadays classical reduction to a quadratic estimate and a Carleson-type inequality. The precise organization of the estimates is different from earlier works. In particular, we succeed in separating space and time variables almost completely despite the non-autonomous character of the operator. Hence, we can allow for degenerate ellipticity dictated by a spatial A2-weight, which has not been treated before in this context.

  • 24.
    Ataei, Alireza
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A note on fundamental solutions and Gaussian bounds for degenerate parabolic equations with time-dependent coefficients2023In: Article in journal (Refereed)
    Abstract [en]

    We consider second order degenerate parabolic equations with real, measurable, and time-dependent coefficients. We allow for degenerate ellipticity dictated by a spatial A_2-weight. We prove the existence of a fundamental solution and derive Gaussian bounds. Our construction is based on the original work of Kato \cite{Kato}.

  • 25.
    Ataei, Alireza
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The Kato square root problem for parabolic operators with an anti-symmetric part in BMO2022In: Article in journal (Refereed)
    Abstract [en]

    We solve the Kato square root problem for parabolic operators whose coefficients can be written as the sum of a complex part, which is elliptic, and a real anti-symmetric part which is in BMO. In particular, we allow for unbounded coefficients.

  • 26.
    Auscher, Pascal
    et al.
    Univ Paris Saclay, CNRS, Lab Math Orsay, F-91405 Orsay, France..
    Egert, Moritz
    Univ Paris Saclay, CNRS, Lab Math Orsay, F-91405 Orsay, France..
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients2020In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 22, no 9, p. 2943-3058Article in journal (Refereed)
    Abstract [en]

    We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.

  • 27. Auscher, Pascal
    et al.
    Egert, Moritz
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The Dirichlet problem for second order parabolic operators in divergence form2018In: Journal de l'École polytechnique — MathématiquesArticle in journal (Refereed)
    Abstract [en]

    We study parabolic operators $\cH = \partial_t-\div_{\lambda,x} A(x,t)\nabla_{\lambda,x}$ in the parabolic upper half space $\mathbb R^{n+2}_+=\{(\lambda,x,t):\ \lambda>0\}$. We assume that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. We prove that the associated parabolic measure is  absolutely continuous with respect to the surface measure on $\mathbb R^{n+1}$  in the sense defined by $A_\infty(\mathrm{d} x\d t)$. Our argument also gives a simplified proof of the corresponding result for elliptic measure.

  • 28.
    Avelin, Benny
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundary behavior of solutions to the parabolic p-Laplace equation II2020In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 279, no 1, article id 108515Article in journal (Refereed)
    Abstract [en]

    This paper is the second installment in a series of papers concerning the boundary behavior of solutions to the p-parabolic equations. In this paper we are interested in the short time behavior of the solutions, which is in contrast with much of the literature, where all results require a waiting time. We prove a dichotomy about the decay-rate of non-negative solutions vanishing on the lateral boundary in a cylindrical C-1,C-1 domain. Furthermore we connect this dichotomy to the support of the boundary type Riesz measure related to the p-parabolic equation in NTA-domains, which has consequences for the continuation of solutions.

  • 29.
    Avelin, Benny
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Jyvaskyla, Dept Math & Stat, POB 35, Jyvaskyla 40014, Finland.;Aalto Univ, Inst Math, POB 11100, Aalto 00076, Finland..
    On time dependent domains for the degenerate p-parabolic equation: Carleson estimate and Holder continuity2016In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 364, no 1-2, p. 667-686Article in journal (Refereed)
    Abstract [en]

    In this paper we propose a definition of "parabolic NTA" for solutions to the degenerate p-parabolic equation. Given this definition we prove the Carleson estimate, originally proved for this equation in Avelin et al. (J Eur Math Soc, 2015) for cylindrical domains. Moreover we study a non-optimal, stronger "outer corkscrew" condition, such that we obtain Holder continuity up to the boundary, for non-negative solutions vanishing on a part of the boundary.

  • 30.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Gianazza, Ugo
    Dipartimento di Matematica "F. Casorati", Università di Pavia.
    Salsa, Sandro
    Dipartimento di Matematica "F. Brioschi", Politecnico di Milano.
    Boundary Estimates for Certain Degenerate and Singular Parabolic Equations2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 2, p. 381-424Article in journal (Refereed)
    Abstract [en]

    We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

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  • 31.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Hed, Lisa
    Persson, Håkan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed)
    Abstract [en]

    We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 184 1987) beyond Lipschitz regularity. 

  • 32.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Jyvaskyla, Dept Math & Stat, Jyvaskyla 40014, Finland.
    Hed, Lisa
    Umeå University.
    Persson, Håkan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Approximation of plurisubharmonic functions2016In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed)
    Abstract [en]

    We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

  • 33. Avelin, Benny
    et al.
    Hou, Mingyi
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions2023In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077Article in journal (Refereed)
    Abstract [en]

    In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain (0,T)×D×Rd, where D is either Td or Rd. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{Alb+21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.

  • 34.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland.
    Julin, Vesa
    Univ Jyvaskyla, Dept Math & Stat, POB 35, Jyvaskyla 40014, Finland..
    A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term2017In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 8, p. 3176-3215Article in journal (Refereed)
    Abstract [en]

    This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in [26], to prove a generalized Carleson estimate. We also prove boundary Holder continuity and a boundary Harnack type inequality.

  • 35.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Kuusi, Tuomo
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundary behavior of solutions to the parabolic p-Laplace equation2019In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 12, no 1, p. 1-42Article in journal (Refereed)
    Abstract [en]

    We establish boundary estimates for non-negative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

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  • 36.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Lukkari, Teemu
    Aalto Univ, Finland.
    A comparison principle for the porous medium equation and its consequences2017In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 2, p. 573-594Article in journal (Refereed)
    Abstract [en]

    We prove a comparison principle for the porous medium equation in more general open sets in Rn+1 than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

  • 37.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Lukkari, Teemu
    Lower semicontinuity of weak supersolutions to the porous medium equation2015In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 8, p. 3475-3486Article in journal (Refereed)
    Abstract [en]

    Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that non-negative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.

  • 38.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Neural ODEs as the deep limit of ResNets with constant weights2021In: Analysis and Applications, ISSN 0219-5305, E-ISSN 1793-6861 , Vol. 19, no 03, p. 397-437Article in journal (Refereed)
    Abstract [en]

    In this paper, we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker-Planck equations.

  • 39.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..
    Saari, Olli
    Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..
    Characterizations of interior polar sets for the degenerate p-parabolic equation2017In: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 17, no 2, p. 827-848Article in journal (Refereed)
    Abstract [en]

    This paper deals with different characterizations of sets of nonlinear parabolic capacity zero, with respect to the parabolic p-Laplace equation. Specifically we prove that certain interior polar sets can be characterized by sets of zero nonlinear parabolic capacity. Furthermore we prove that zero capacity sets are removable for bounded supersolutions and that sets of zero capacity have a relation to a certain parabolic Hausdorff measure.

  • 40.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Viitasaari, L.
    Aalto Univ Sch Business, Dept Informat & Serv Management, Helsinki, Finland..
    ON EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS2021In: THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS, ISSN 0094-9000, Vol. 104, p. 49-60Article in journal (Refereed)
    Abstract [en]

    In this article we consider existence and uniqueness of the solutions to a large class of stochastic partial differential equations of the form partial derivative(t)u = L(x)u+ b(t, u)+ s(t, u) (over dot(W)), driven by a Gaussian noise. W, white in time, and with spatial correlations given by a generic covariance gamma. We provide natural conditions under which classical Picard iteration procedure provides a unique solution. We illustrate the applicability of our general result by providing several interesting particular choices for the operator L-x under which our existence and uniqueness results hold. In particular, we show that Dalang condition given in [5] is sufficient in the case of many parabolic and hypoelliptic operators L-x.

  • 41.
    Azzam, Jonas
    et al.
    University of Washington, Seattle, USA.
    Hofmann, Steve
    University of Missouri, Columbia, USA.
    Martell, Jose Maria
    Instituto de Ciencias Matematicas, Madrid, Spain.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Toro, Tatiana
    University of Washington, Seattle, USA.
    A new characterization of chord-arc domains2017In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 4, p. 967-981Article in journal (Refereed)
    Abstract [en]

    We show that if Ω⊂Rn+1, n≥1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems.

  • 42.
    Backlund, Ulf
    et al.
    Danderyds Gymnasium, Danderyd, Sweden..
    Carlsson, Linus
    Malardalen Univ, Acad Culture & Commun, Vasteras, Sweden..
    Fallström, Anders
    Umea Univ, Dept Math & Math Stat, Umea, Sweden..
    Persson, Håkan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Semi-Bloch Functions in Several Complex Variables2016In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 26, no 1, p. 463-473Article in journal (Refereed)
    Abstract [en]

    Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.

  • 43.
    Bandyopadhyay, Antar
    et al.
    Indian Stat Inst, Delhi, India.;Indian Stat Inst, Kolkata, India.;Indian Stat Inst, Theoret Stat & Math Unit, Delhi Ctr, 7 SJS Sansanwal Marg, New Delhi 110016, India..
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Thacker, Debleena
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Strong convergence of infinite color balanced urns under uniform ergodicity2020In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 57, no 3, p. 853-865, article id PII S0021900220000376Article in journal (Refereed)
    Abstract [en]

    We consider the generalization of the Polya urn scheme with possibly infinitely many colors, as introduced in [37], [4], [5], and [6]. For countably many colors, we prove almost sure convergence of the urn configuration under theuniform ergodicityassumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with abranching Markov chainon a weightedrandom recursive treeas described in [6], [31], and [26]. Using this coupling we estimate the covariance between any two selected colors. In particular, we re-prove the limit theorem for the classical urn models with finitely many colors.

  • 44.
    Baroni, Paolo
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Riesz potential estimates for a general class of quasilinear equations2015In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 53, no 3-4, p. 803-846Article in journal (Refereed)
    Abstract [en]

    We consider solutions to nonlinear elliptic equations with measure data and general growth and ellipticity conditions of degenerate type, as considered in Lieberman (Commun Partial Differ Equ 16:311-361, 1991); we prove pointwise gradient bounds for solutions in terms of linear Riesz potentials. As a direct consequence, we get optimal conditions for the continuity of the gradient.

  • 45.
    Baroni, Paolo
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Colombo, Maria
    Mingione, Giuseppe
    Harnack inequalities for double phase functionals2015In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 121, p. 206-222Article in journal (Refereed)
    Abstract [en]

    We prove a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions. They are characterised by the fact that their energy density switches between two types of different degenerate phases.

  • 46.
    Bartoszek, Krzysztof
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Bartoszek, Wojciech
    Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland..
    A Noether theorem for stochastic operators on Schatten classes2017In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, no 2, p. 1395-1412Article in journal (Refereed)
    Abstract [en]

    We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

  • 47.
    Bartoszek, Krzysztof
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Linkoping Univ, Dept Comp & Informat Sci, S-58183 Linkoping, Sweden.
    Domsta, Joachim
    State Univ Appl Sci Elblag, Krzysztof Brzeski Inst Appl Informat, Ul Wojska Polskiego 1, PL-82300 Elblag, Poland.
    Pulka, Malgorzata
    Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland.
    Weak Stability of Centred Quadratic Stochastic Operators2019In: BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, ISSN 0126-6705, Vol. 42, no 4, p. 1813-1830Article in journal (Refereed)
    Abstract [en]

    We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring's trait is equal to an additively perturbed arithmetic mean of the parents' traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.

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  • 48.
    Bartoszek, Krzysztof
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pułka, Małgorzata
    Gdansk University of Technology.
    Prevalence Problem in the Set of Quadratic Stochastic Operators Acting on L12018In: Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, Vol. 41, no 1, p. 159-173Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to the study of the problem of prevalence in the classof quadratic stochastic operators acting on the L1 space for the uniform topology.We obtain that the set of norm quasi-mixing quadratic stochastic operators is a denseand open set in the topology induced by a very natural metric. This shows the typicallong-term behaviour of iterates of quadratic stochastic operators.

  • 49.
    Belyaeva, Elena
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On a new logistic regression model for bankruptcy prediction in the IT branch2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
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  • 50.
    Benes, Christian
    et al.
    CUNY Brooklyn Coll, Brooklyn, NY 11210 USA..
    Lawler, Gregory F.
    Univ Chicago, Chicago, IL 60637 USA..
    Viklund, Fredrik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. KTH Royal Inst Technol, Stockholm, Sweden.
    Scaling limit of the loop-erased random walk Green's function2016In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 166, no 1-2, p. 271-319Article in journal (Refereed)
    Abstract [en]

    We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a "spinor" observable for random walk started from one of the vertices of the marked edge.

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