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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, 37-48 p.Article 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, 399-416 p.Article 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, 889-911 p.Article 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.
    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, 657-678 p.Article 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.

  • 5.
    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, 131-178 p.Article 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.

  • 6.
    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, 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.

  • 7.
    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
  • 8.
    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, 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.

  • 9.
    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
  • 10.
    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.
    Boundary value problems for parabolic systems via a first order approach2016Article 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.

  • 11. 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 form2016Article 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.

  • 12.
    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, 667-686 p.Article 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.

  • 13.
    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, 381-424 p.Article 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.

  • 14.
    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, 531-545 p.Article 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. 

  • 15.
    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, 23-28 p.Article 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.

  • 16.
    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, 3176-3215 p.Article 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.

  • 17.
    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 equation2015Article 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 H\"older 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.

  • 18.
    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, 573-594 p.Article 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.

  • 19.
    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, 3475-3486 p.Article 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.

  • 20.
    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, 827-848 p.Article 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.

  • 21.
    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 domains2015In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Article in journal (Refereed)
    Abstract [en]

    We show that if $\om \subset \ree$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\om$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $\om$ 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.

  • 22.
    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, 463-473 p.Article 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.

  • 23.
    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, 803-846 p.Article 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.

  • 24.
    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, 206-222 p.Article 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.

  • 25.
    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, 1395-1412 p.Article 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.

  • 26.
    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 StochasticOperators Acting on L12016In: Bulletin of the Malaysian Mathematical Sciences SocietyArticle 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.

  • 27.
    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
  • 28.
    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, 271-319 p.Article 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.

  • 29.
    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
  • 30.
    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
  • 31. 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, 440-470 p.Article 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.

  • 32.
    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, 410-435 p.Article 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.

  • 33.
    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, 338-384 p.Article 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).

  • 34. 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, 319-374 p.Article 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.

  • 35.
    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..

  • 36.
    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
  • 37.
    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, 3-13 p.Article 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.

  • 38. 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, 687-722 p.Article 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).

  • 39.
    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
  • 40.
    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, 1069-1082 p.Article 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.

  • 41.
    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, 102-126 p.Article 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.

  • 42.
    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, Vol. 20, 1-25 p.Article 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.

  • 43.
    Casanova, Adrian Gonzalez
    et al.
    Tech Univ Berlin, Inst Math, Str 17 Juni 136, D-10623 Berlin, Germany..
    Kurt, Noemi
    Tech Univ Berlin, Inst Math, Str 17 Juni 136, D-10623 Berlin, Germany..
    Wakolbinger, Anton
    Goethe Univ Frankfurt, Math Inst, Box 111932, D-60054 Frankfurt, Germany..
    Yuan, Linglong
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    An individual-based model for the Lenski experiment, and the deceleration of the relative fitness2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 8, 2211-2252 p.Article in journal (Refereed)
    Abstract [en]

    The Lenski experiment investigates the long-term evolution of bacterial populations. In this paper we present an individual-based probabilistic model that captures essential features of the experimental design, and whose mechanism does not include epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part. We prove that under some assumptions excluding clonal interference, the rescaled relative fitness process converges in the large population limit to a power law function, similar to the one obtained by Wiser et al. (2013), there attributed to effects of clonal interference and epistasis.

  • 44.
    Cassar, Peter
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Forecasting of a Loan Book Using Monte Carlo Methods2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 45.
    Castro, Alejandro J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Hytonen, Tuomas P.
    Univ Helsinki, Gustaf Hallstromin Katu 2b, FIN-00014 Helsinki, Finland..
    Bounds for partial derivatives: necessity of UMD and sharp constants2016In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 282, no 3-4, 635-650 p.Article in journal (Refereed)
    Abstract [en]

    We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of bounds between different partial derivatives of . In particular, we show that the estimate characterizes the UMD property, and the best constant K is equal to one half of the UMD constant. This precise value of K seems to be new even for scalar-valued functions.

  • 46.
    Castro, Alejandro J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Nowak, Adam
    Polish Acad Sci, Inst Math, Sniadeckich 8, PL-00656 Warsaw, Poland..
    Szarek, Tomasz Z.
    Polish Acad Sci, Inst Math, Sniadeckich 8, PL-00656 Warsaw, Poland..
    Riesz-Jacobi Transforms as Principal Value Integrals2016In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 22, no 3, 493-541 p.Article in journal (Refereed)
    Abstract [en]

    We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz-Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz-Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.

  • 47.
    Castro, Alejandro, J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Rodríguez-López, Salvador
    Staubach, Wolfgang
    L2 Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with time-independent Hölder-continuous coefficientsIn: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed)
    Abstract [en]

    We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with time-independent H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in $\mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a systematic transference scheme which yields suitable parabolic Rellich-type estimates.

  • 48.
    Castro, Alejandro
    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.
    Sande, Olow
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients2016In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 5, 124Article in journal (Refereed)
    Abstract [en]

    We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is an $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates (Theorem \ref{th0}), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two  crucial estimates in the case of real, symmetric operators (Theorem \ref{th2}). Our results are crucial when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.

  • 49.
    Colesanti, Andrea
    et al.
    U.Dini, Firenze, Italy.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Salani, Paolo
    U.Dini, Firenze, Italy.
    Xiao, Jie
    Memorial University of Newfoundland, St. John's, Canada.
    Yang, Deane
    Polytechnic Institute of New York University, New York, USA.
    Zhang, Gaoyong
    Polytechnic Institute of New York University, New York, USA.
    The Hadamard variational formula and the Minkowski problem for p-Capacity2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 285, 1511-1585 p.Article in journal (Refereed)
    Abstract [en]

    A Hadamard variational formula for p-capacity of convex bodies in R-n is established when 1 < p < n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge Ampere type equation. zkiniqueness for the Minkowski problem for p-capacity is established when 1 < p < n and existence and regularity when 1 < p < 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).

  • 50.
    Coletti, Cristian F.
    et al.
    Univ Fed ABC, Sao Paulo, Brazil..
    Miranda, Daniel
    Univ Fed ABC, Sao Paulo, Brazil..
    Mussini, Filipe
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Invariance Under Quasi-isometries of Subcritical and Supercritical Behavior in the Boolean Model of Percolation2016In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 162, no 3, 685-700 p.Article in journal (Refereed)
    Abstract [en]

    In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. More precisely, we prove that if a metric space M is mm-quasi-isometric to another metric space N and the Poisson Boolean model in M exhibits any of the following: (a) a subcritical phase; (b) a supercritical phase; or (c) a phase transition, then respectively so does the Poisson Boolean model of percolation in N. Then we use these results in order to understand the phase transition phenomenon in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.

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