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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.
    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.

  • 4.
    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.

  • 5.
    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.

  • 6.
    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.

  • 7.
    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.

  • 8.
    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.

  • 9.
    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
  • 10.
    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
  • 11.
    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.

  • 12.
    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
  • 13.
    Auscher, Pascal
    et al.
    Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.
    Egert, Moritz
    Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.
    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 coefficients2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Article 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.

  • 14. 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 form2016In: 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.

  • 15.
    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.

  • 16.
    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.

  • 17.
    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. 

  • 18.
    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.

  • 19.
    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.

  • 20.
    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.

  • 21.
    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.

  • 22.
    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.

  • 23.
    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.

  • 24.
    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.

  • 25.
    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.

  • 26.
    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.

  • 27.
    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.

  • 28.
    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.

  • 29.
    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.

  • 30.
    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
  • 31.
    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.

  • 32.
    Berg, Jens
    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.
    A unified deep artificial neural network approach to partial differential equations in complex geometries2018In: Neurocomputing, ISSN 0925-2312, E-ISSN 1872-8286, Vol. 317, p. 28-41Article in journal (Refereed)
    Abstract [en]

    In this paper, we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

  • 33.
    Berg, Jens
    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.
    Data-driven discovery of PDEs in complex datasets2018In: Article in journal (Refereed)
    Abstract [en]

    Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution

  • 34.
    Berg, Jens
    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 network augmented inverse problems for PDEs2017In: Article in journal (Refereed)
    Abstract [en]

    In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a parametric container for the coefficient to be estimated from noisy data. Neural networks are global, smooth function approximators and as such they do not require regularization of the error functional to recover smooth solutions and coefficients. We give detailed examples using the Poisson equation in 1, 2, and 3 space dimensions and show that the neural network augmentation is robust with respect to noisy data, mesh, and geometry.

  • 35.
    Bergström, Jonas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pricing American Options using Lévy Processes and Monte Carlo Simulations2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 36.
    Bergström, Jonas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 37. Betancor, Jorge J.
    et al.
    Castro, Alejandro J.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Farina, Juan C.
    Rodriguez-Mesa, L.
    Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions2015In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 431, no 1, p. 440-470Article in journal (Refereed)
    Abstract [en]

    We consider the Weinstein type equation L(lambda)u = 0 on (0, infinity) X (0, infinity), where L-lambda= delta(2)(t) + delta-lambda(lambda-1)/x(2), In this paper we characterize the solutions of L(lambda)u = = 0 on (0, infinity) x (0, infinity) representable by Bessel-Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.

  • 38.
    Betancor, Jorge J.
    et al.
    Univ La Laguna, Dept Anal Matemat, Campus Anchieta, San Cristobal la Laguna 38271, Sta Cruz De Ten, Spain..
    Castro, Alejandro J.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Farina, Juan C.
    Univ La Laguna, Dept Anal Matemat, Campus Anchieta, San Cristobal la Laguna 38271, Sta Cruz De Ten, Spain..
    Rodriguez-Mesa, Lourdes
    Univ La Laguna, Dept Anal Matemat, Campus Anchieta, San Cristobal la Laguna 38271, Sta Cruz De Ten, Spain..
    UMD Banach spaces and square functions associated with heat semigroups for Schrödinger, Hermite and Laguerre operators2016In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, no 4, p. 410-435Article in journal (Refereed)
    Abstract [en]

    In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schrodinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L-p-boundedness properties for the square functions to our Banach valued setting by using gamma-radonifying operators. We also prove that these L-p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.

  • 39.
    Betancor, Jorge J.
    et al.
    Univ La Laguna, Dept Anal Matemat, Campus Anchieta, San Cristobal la Laguna 38271, Sta Cruz De Ten, Spain..
    Castro, Alejandro J.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Rodriguez-Mesa, L.
    Univ La Laguna, Dept Anal Matemat, Campus Anchieta, San Cristobal la Laguna 38271, Sta Cruz De Ten, Spain..
    Square Functions And Spectral Multipliers For Bessel Operators In Umd Spaces2016In: Banach Journal of Mathematical Analysis, ISSN 1735-8787, Vol. 10, no 2, p. 338-384Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner Lebesgue space L-p((0, infinity), B), where B is a UMD Banach space. As special cases, we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator Delta(lambda) = -x(-lambda) d/dxx(2 lambda)d/dxx(-lambda), lambda > 0. We characterize the UMD property for a Banach space I; by using L-p((0, infinity), B)-boundedness properties of g-functions defined by Bessel Poisson semigroups. As a by-product, we prove that the fact that the imaginary power Delta(iw)(lambda), w is an element of R \ {0}, of the Bessel operator Delta(lambda) is bounded in L-p((0, infinity), In), 1 < p < infinity, characterizes the UMD property for the Banach space B. As applications of our results for square functions, we establish the boundedness in L-p((0, infinity), B) of spectral multipliers m(Delta(lambda)) of Bessel operators defined by functions m which are holomorphic in sectors Sigma(v).

  • 40. Betancor, Jorge J.
    et al.
    Castro, Alejandro J.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Rodriguez-Mesa, Lourdes
    UMD-Valued Square Functions Associated with Bessel Operators in Hardy and BMO Spaces2015In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 81, no 3, p. 319-374Article in journal (Refereed)
    Abstract [en]

    We consider Banach valued Hardy and BMO spaces in the Bessel setting. Square functions associated with Poisson semigroups for Bessel operators are defined by using fractional derivatives. If is a UMD Banach space we obtain for -valued Hardy and BMO spaces equivalent norms involving gamma-radonifying operators and square functions. We also establish characterizations of UMD Banach spaces by means of Hardy and BMO-boundedness properties of g-functions associated to Bessel-Poisson semigroup.

  • 41.
    Björklund, Johan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Falgas-Ravry, Victor
    Holmgren, Cecilia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On percolation in one-dimensional stable Poisson graphs2015In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 20Article in journal (Refereed)
    Abstract [en]

    Equip each point x of a homogeneous Poisson point process P on R with D-x edge stubs, where the D-x are i.i.d. positive integer-valued random variables with distribution given by mu. Following the stable multi-matching scheme introduced by Deijfen, Haggstrom and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph G on the vertex set P. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Haggstrom and Holroyd [1] on percolation (the existence of an infinite connected component) in G. We prove that percolation may occur a.s. even if mu has support over odd integers. Furthermore, we show that for any epsilon > 0, there exists a distribution mu such that mu ({1}) > 1 - epsilon, but percolation still occurs a.s..

  • 42.
    Bliatsios, George
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Financial Modeling Under Incomplete Information2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 43.
    Bollobas, Bela
    et al.
    Dept Pure Math & Math Stat, Wilberforce Rd, Cambridge CB3 0WB, England.;Univ Memphis, Dept Math Sci, Memphis, TN 38152 USA.;London Inst Math Sci, 35a South St, London W1K 2XF, England..
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Scott, Alex
    Univ Oxford, Radcliffe Observ Quarter, Math Inst, Andrew Wiles Bldg,Woodstock Rd, Oxford OX2 6GG, England..
    Packing Random Graphs and Hypergraphs2017In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 51, no 1, p. 3-13Article in journal (Refereed)
    Abstract [en]

    We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have different densities. In the graph case, we prove a stronger result, on packing a random graph with a fixed graph.

  • 44. Bollobas, Bela
    et al.
    Smith, Paul
    Uzzell, Andrew
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Monotone Cellular Automata in a Random Environment2015In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 24, no 4, p. 687-722Article in journal (Refereed)
    Abstract [en]

    In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in Z(d) with random initial configurations. Formally, we are given a set U = {X-1,...,X-m} of finite subsets of Z(d) \ {0}, and an initial set A(0) subset of Z(d) of 'infected' sites, which we take to be random according to the product measure with density p. At time t is an element of N, the set of infected sites A(t) is the union of A(t-1) and the set of all x is an element of Z(d) such that x + X is an element of A(t-1) for some X is an element of U. Our model may alternatively be thought of as bootstrap percolation on Z(d) with arbitrary update rules, and for this reason we call it U-bootstrap percolation. In two dimensions, we give a classification of U-bootstrap percolation models into three classes -supercritical, critical and subcritical - and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on (Z/nZ)(2) is (log n)(-Theta(1)) for all models in the critical class, and that it is n(-Theta(1)) for all models in the supercritical class. The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on Z(d).

  • 45.
    Braojos Peláes, Marta Rita
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Optimal exercise of an American Option under drift uncertainty2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 46.
    Brightwell, Graham
    et al.
    London Sch Econ, Dept Math, London WC2A 2AE, England..
    Janson, Svante
    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, London, England..
    The Greedy Independent Set in a Random Graph with Given Degrees2017In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 51, no 4, p. 565-586Article in journal (Refereed)
    Abstract [en]

    We analyse the size of an independent set in a random graph on n vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as n (the jamming constant), expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges in probability to the jamming constant as n. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model.

  • 47.
    Broman, Erik I.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Stochastic Ordering of Infinite Geometric Galton-Watson Trees2016In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 29, no 3, p. 1069-1082Article in journal (Refereed)
    Abstract [en]

    We consider Galton-Watson trees with Geom(p) offspring distribution. We let T-infinity (p) denote such a tree conditioned on being infinite. We prove that for any 1/2 <= p(1) <= p2 <= 1, there exists a coupling between T-infinity (p(1)) and T-infinity (p(2)) such that P(T-infinity(p(1)) subset of T-infinity(p(2))) = 1.

  • 48.
    Broman, Erik I.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tykesson, Johan
    Chalmers, Dept Math, Gothenburg, Sweden.;Gothenburg Univ, S-41124 Gothenburg, Sweden..
    Connectedness of Poisson cylinders in Euclidean space2016In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 52, no 1, p. 102-126Article in journal (Refereed)
    Abstract [en]

    We consider the Poisson cylinder model in R-d, d >= 3. We show that given any two cylinders c(1) and c(2) in the process, there is a sequence of at most d - 2 other cylinders creating a connection between c(1) and c(2). In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in (Probab. Theory Related Fields 154 (2012) 165-191). We also show that there are cylinders in the process that are not connected by a sequence of at most d - 3 other cylinders. Thus, the diameter of the cluster of cylinders equals d - 2.

  • 49.
    Broman, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tykesson, Johan
    Poisson cylinders in hyperbolic space2015In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, p. 1-25Article in journal (Refereed)
    Abstract [en]

    We consider the Poisson cylinder model in d-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.

  • 50.
    Cai, Xing Shi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Non-fringe subtrees in conditioned Galton-Watson trees2018In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 25, no 3, article id P3.40Article in journal (Refereed)
    Abstract [en]

    We study S(T-n), the number of subtrees in a conditioned Galton-Watson tree of size n. With two very different methods, we show that log(S(T-n)) has a Central Limit Law and that the moments of S(T-n) are of exponential scale.

123456 1 - 50 of 258
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