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  • 1.
    Ameur, Yacin
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hedenmalm, Håkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Makarov, Nikolai
    Berezin Transform in Polynomial Bergman Spaces2010In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 63, no 12, p. 1533-1584Article in journal (Refereed)
    Abstract [en]

    Fix a smooth weight function Q in the plane, subject to a growth condition from below Let K-m,K-n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L-2-norm with respect to the measure e-(mQ) dA Here dA is normalized area measure, and m is a positive real scaling parameter The (polynomial) Berezin measure dB(m,n)(< z0 >) (z) = K-m,K-n(z(0).z(0))(-1) vertical bar K-m,K-n(z.z(0))vertical bar(2)e(-mQ(z)) dA(z) for the point z(0) is a probability measure that defines the (polynomial) Berezin transform B-m,B-n f(z(0)) = integral(C) f dB(m,n)(< z0 >) for continuous f is an element of L-infinity (C). We analyze the semiclassical limit of the Berezin measure (and transform) as m -> +infinity while n = m tau + o(1), where tau is fixed, positive, and real We find that the Berezin measure for z(0) converges weak-star to the unit point mass at the point z(0) provided that Delta Q(z(0)) > 0 and that z(0) is contained in the interior of a compact set f(tau). defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane For points z(0) is an element of C\f(tau), the Berezin measure cannot converge to the point mass at z(0) In the model case Q(z) = vertical bar z vertical bar(2), when f(tau) is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z(0) relative to C\f(tau) Our results have applications to the study of the cigenvalues of random normal matrices The auxiliary results include weighted L-2-estimates for the equation partial derivative u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at infinity.

  • 2. Andersson, John
    et al.
    Lindgren, Erik
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal Regularity for the No-Sign Obstacle Problem2013In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, no 2, p. 245-262Article in journal (Refereed)
    Abstract [en]

    In this paper we prove the optimal C-1,C-1(B-1/2)-regularity for a general obstacle-type problem Delta u = f chi({u not equal 0}) in B-1, under the assumption that f * N is C-1,C-1(B-1), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C-1,C-1-regularity. As a by-product of the C-1,C-1-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x(0), the free boundary is locally a C-1-graph close to x(0) provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades.

  • 3. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Lee-Yang and Polya-Schur Programs. II. Theory of Stable Polynomials and Applications2009In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 62, no 12, p. 1595-1631Article in journal (Refereed)
    Abstract [en]

    In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Polya and Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.

  • 4.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gaussian Free Field in an Interlacing Particle System with Two Jump Rates2013In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, no 4, p. 600-643Article in journal (Refereed)
    Abstract [en]

    We study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two-dimensional lattice. There are two different types of particles, one with a low jump rate and the other with a high jump rate. In the large time limit, the random surface has a deterministic shape. Due to the different jump rates, the limit shape and the domain on which it is defined are not smooth. The main result is that the fluctuations of the random surface are governed by the Gaussian free field.

  • 5.
    Duits, Maurice
    et al.
    Katholieke Univ Leuven, Belgium.
    Kuijlaars, Arno B., J.
    Universality in the two-matrix model: A Riemann-Hilbert Steepest-Descent Analysis2009In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 62, no 8, p. 1076-1153Article in journal (Refereed)
    Abstract [en]

    The eigenvalue statistics of a pair (M(1), M(2)) of n x n Hermitian matrices taken randomly with respect to the measure 1/Z(n) exp (-n Tr(V(M(1)) + W(M(2)) - tau M(1)M(2)))dM(1) dM(2) can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 x 4 matrix-valued Riemann-Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W(y) = y(4)/4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M(1) (when averaged over M(2)) in the global and local regime as n -> infinity in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.

  • 6.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Liu, J.-G.
    Numerical methods for oscillatory solutions to hyperbolic problems1993In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 46, no 10, p. 1327-1361Article in journal (Refereed)
    Abstract [en]

    Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths

  • 7.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Majda, A
    Radiation boundary conditions for acoustic and elastic wave calculations1979In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 32, no 3, p. 314-358Article in journal (Refereed)
    Abstract [en]

    A technique for developing radiating boundary conditions for artificial computational boundaries is described and applied to a class of problems typical in exploration seismology involving acoustic and elastic wave equations. First, one considers a constant coefficient scalar wave equation where the artificial boundary is one edge of a rectangular domain. By using continued fraction expansions, a systematic sequence of stable highly absorbing boundary conditions with successively better absorbing properties as the order of the boundary conditions increases is obtained. There follows a systematic derivation of a hierarchy of local radiating boundary conditions for the elastic wave equation. A theoretical procedure to guarantee stability at corners of the rectangular domain is worked out. A technique for fitting the discrete radiating boundary conditions directly to the difference scheme itself is proposed.

  • 8.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Weinan, E
    Large time behavior and homogenization of solutions of two-dimensional conservation laws1993In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 46, no 1, p. 1-26Article in journal (Refereed)
    Abstract [en]

    We study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data. Under a very weak nonlinearity condition, we prove that the solutions converge to constants as time goes to infinity. Even in one space dimension our results improve the earlier ones since we only require the fluxes to be nonlinear in a neighborhood of the mean value of the initial data. We then use these results to study the homogenization problem for scalar conservation laws with oscillatory initial data.

  • 9.
    Engquist, Björn
    et al.
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Ying, Lexing
    Department of Mathematics, The University of Texas at Austin, Austin, USA.
    Sweeping preconditioner for the helmholtz equation: hierarchical matrix representation2011In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 64, no 5, p. 697-735Article in journal (Refereed)
    Abstract [en]

    The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable-coefficient Helmholtz equation including very-high-frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three-dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.

  • 10. Johnson, Claes
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Adaptive finite element methods for conservation laws based on a posteriori error estimates1995In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 48, no 3, p. 199-234Article in journal (Refereed)
    Abstract [en]

    We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way.

  • 11. Kaloshin, Vadim
    et al.
    Levi, Mark
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Arnol ' d Diffusion in a Pendulum Lattice2014In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 67, no 5, p. 748-775Article in journal (Refereed)
    Abstract [en]

    The main model studied in this paper is a lattice of pendula with a nearest-neighbor coupling. If the coupling is weak, then the system is near-integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way.

  • 12. Lee, K. A.
    et al.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations2001In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 54, no 1, p. 43-56Article in journal (Refereed)
    Abstract [en]

    Our objective in this paper is to analyze the regularity of the boundary of a set Omega2 that admits a solution u to the following overdetermined problem: F(D(2)u) = chi (Omega) in B, u = \ delu \ = 0 in B \ Omega. Here F is a convex, uniformly elliptic operator of homogeneity one, B is the unit ball in R-n, and the equation is satisfied in the viscosity sense. We also consider the case of concave F in R-2.

  • 13.
    Ringström, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in T-3-Gowdy2006In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 59, no 7, p. 977-1041Article in journal (Refereed)
    Abstract [en]

    This is the first of two papers that together prove strong cosmic censorship in T-3-Gowdy space-times. In the end, we prove that there is a set of initial data, open with respect to the C-2 X C-1 topology and dense with respect to the C-infinity topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracted with itself, blows up in the incomplete direction. In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions. In this paper, we shall, however, focus on the concept of asymptotic velocity. Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint. The target of the wave map is the hyperbolic plane. There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy. We define the asymptotic velocity v(infinity) to be the nonnegative square root of the limit of the kinetic energy density. The asymptotic velocity has some very important properties. In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v(infinity). It also has properties such that if 0 < v(infinity)(theta(0)) < 1, then v(infinity) is smooth in a neighborhood of theta(0). Furthermore, if v(infinity)(theta(0)) > 1 and v(infinity) is continuous at theta(0), then v(infinity) is smooth in a neighborhood of theta(0). Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to the C-2 X C-1 topology on initial data.

  • 14. Ringström, Hans
    On a wave map equation arising in general relativity2004In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 57, no 5, p. 657-703Article in journal (Refereed)
    Abstract [en]

    We consider a class of space-times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t --> infinity. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t(-1/2) as t --> infinity. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half-plane (after applying an isometry of hyperbolic space if necessary): (1) The solution converges to a point. (2) The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). (3) The solution goes to infinity along a curve y = const. (4) The solution oscillates around a circle inside the upper half-plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space-times. For instance, one obtains the leading-order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness.

  • 15.
    Shahgholian, Henrik
    KTH, Superseded Departments, Mathematics.
    C-1,C-1 regularity in semilinear elliptic problems2003In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 56, no 2, p. 278-281Article in journal (Refereed)
    Abstract [en]

    In this paper we give an astonishingly simple proof of C-1,C-1 regularity in elliptic theory. Our technique yields both new simple proofs of old results as well as new optimal results. The setting we'll consider is the following. Let it be a solution to Deltau = f (x, u) in B, where B is the unit ball in R-n, f (x, t) is a bounded Lipschitz function in x, and f(t)' is bounded from below. Then we prove that u is an element of C-1,C-1 (B-1/2). Our method is a simple corollary to a recent monotonicity argument due to Caffarelli, Jerison, and Kenig.

  • 16.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    On the stability of nite element methods for shock waves1992In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, p. 923-946Article in journal (Refereed)
  • 17.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    On the stability of finite element methods for shock waves1992In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 45, no 8, p. 923-946Article in journal (Refereed)
    Abstract [en]

    this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form (1:1) u t + f(u) x = u xx ; x 2 R I ; t ? 0 ; u 2 R I u(\Delta; 0) = u 0 (\Delta) : We treat systems which are strictly hyperbolic. Such systems possess a smooth travelling wave solution, which is called a viscous p-shock wave solution, u(x; t) = OE(x \Gamma oet) x!\Sigma1 OE(x) = u \Sigma ; provided that the shock strength ffl j ju + \Gamma u \Gamma j is small [19], the constant states u \Sigma and the wave speed oe are related by the Rankine-Hugoniot condition (1:3a) f(u \Gamma ) \Gamma f(u+ ) = oe(u \Gamma \Gamma u+ )

  • 18.
    Szepessy, Anders
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zouraris, G. E.
    Adaptive weak approximation of stochastic differential equations2001In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 54, no 10, p. 1169-1214Article in journal (Refereed)
    Abstract [en]

    Adaptive time-stepping methods based on the Monte Carlo Euler method for weak approximation of Ito stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading-order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps.

  • 19. Tornberg, Anna-Karin
    et al.
    Engquist, Björn
    The segment projection method for interface tracking2003In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 56, no 1, p. 47-79Article in journal (Refereed)
    Abstract [en]

    There has recently been important progress in the development of front tracking and level set methods for the numerical simulation of moving interfaces. The segment projection method is a new technique for computational geometry. It can be seen as a compromise between front tracking and level set methods. It is based on the regular mathematical representation of a manifold as an atlas of charts. Each chart or segment is evolved independently by a partial differential equation that is discretized on an Euldrian grid. The connectivity of the segments is handled by an appropriate data structure and by numerical interpolation. The method is presented and its properties are analyzed. Applications to multiphase flow, epitaxial growth, and high-frequency wave propagation are given.

  • 20. Weinan, E
    et al.
    Engquist, Björn
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Blowup of solutions of the unsteady Prandtl's equation1997In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 50, p. 1287-1293Article in journal (Refereed)
    Abstract [en]

    We prove that for certain class of compactly supported C˜ initial data, smooth solutions of the unsteady Prandtl's equation blow up in nite time

1 - 20 of 20
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