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  • 301.
    Duits, Maurice
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Gaussian Free Field in an Interlacing Particle System with Two Jump Rates2013Ingår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, nr 4, s. 600-643Artikel i tidskrift (Refereegranskat)
    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.

  • 302.
    Duits, Maurice
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Painlevé Kernels in Hermitian Matrix Models2014Ingår i: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 39, nr 1, s. 173-196Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4x4 Riemann-Hilbert problem related to the Painlev, II equation. Also an open problem will be presented.

  • 303.
    Duits, Maurice
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Geudens, D.
    A critical phenomenon in the two-matrix model in the quartic/quadratic case2013Ingår i: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 162, nr 8, s. 1383-1462Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a 4×4 Riemann-Hilbert problem related to the Hastings-McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in noncolliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes like a square root at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.

  • 304. Duits, Maurice
    et al.
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Powers of large random unitary matrices and toeplitz determinants2010Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 362, nr 3, s. 1169-1187Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the limiting behavior of Tr U-k(n), where U is an n x n random unitary matrix and k(n) is a natural number that may vary with n in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szego limit theorem for Toeplitz determinants associated to symbols depending on 71 in a particular way. As a consequence of this result, we find that for each fixed m is an element of N, the random variables Tr U-kj(n)/root min(k(j)(n),n), j = 1,..., m, converge to independent standard complex normals.

  • 305.
    Duits, Maurice
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Kuijlaars, A.B.J.
    Mo, M. Y.
    The hermitian two matrix model with an even quartic potential2012Ingår i: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 217, nr 1022, s. 1-118Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the two matrix model with an even quartic potential W(y) = y 4/4+ ay 2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M 1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M 1· Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

  • 306.
    Duse, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On Uniformly Random Discrete Interlacing Systems2015Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis concerns uniformly random discrete interlacing particle sys-tems and their connections to certain random lozenge tiling models. In par-ticular it contains the first derivation of a relatively unknown universal scalinglimit, which we call the Cusp-Airy process, of certain lozenge tiling modelsat a cusp point. In addition it contains a characterization of the geometryof the macroscopic behavior of uniformly random discrete interlaced parti-cle systems that, although not complete, shows many new and interestingfeatures.

  • 307.
    Duse, Erik
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Metcalfe, Anthony
    The Cusp-Airy ProcessManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.

  • 308.
    Duse, Erik
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Metcalfe, Anthony
    Asymptotic geometry of discrete interlaced patterns: Part I2015Ingår i: International Journal of Mathematics, ISSN 0129-167XArtikel i tidskrift (Refereegranskat)
    Abstract [en]

    A discrete Gelfand–Tsetlin pattern is a configuration of particles in ℤ2. The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc., and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand–Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behavior of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic "shape" of such systems. We provide parameterizations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behavior of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the paper [E. Duse and A. Metcalfe, Asymptotic geometry of discrete interlaced patterns: Part II, in preparation], in which we resolve some of the remaining issues. Both of these papers serve as background material for the papers [E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems, in preparation; E. Duse and K. Johansson and A. Metcalfe, Cusp Airy process of discrete interlaced particle systems, in preparation], in which we examine the edge asymptotic behavior.

  • 309.
    Duse, Erik
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Metcalfe, Anthony
    Asymptotic geometry of discrete interlaced patterns: Part IIManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We study the boundary of the liquid region L in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line (x,1), x∈R≡∂H. We assume that the initial particle configuration converges weakly to a limiting density ϕ(x), 0≤ϕ≤1. The liquid region is given by a homeomorphism WL:L→H, the upper half plane, and we consider the extension of W−1L to H¯¯¯. Part of ∂L is given by a curve, the edge E, parametrized by intervals in ∂H, and this corresponds to points where ϕ is identical to 0 or 1. If 0<ϕ<1, the non-trivial support, there are two cases. Either W−1L(w) has the limit (x,1) as w→x non-tangentially and we have a \emph{regular point}, or we have what we call a singular point. In this case W−1L does not extend continuously to H¯¯¯. Singular points give rise to parts of ∂L not given by E and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of ∂L can be very complicated. We expect that on the singular parts of ∂L we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, E and the singular parts of ∂L are shocks of the complex Burgers equation.

  • 310.
    Ebenfelt, P.
    et al.
    Univ Calif San Diego, Dept Math, La Jolla, CA 92093 USA..
    Khavinson, D.
    Univ S Florida, Dept Math & Stat, Tampa, FL 33620 USA..
    Shapiro, Harold S.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Two-dimensional Shapes and Lemniscates2011Ingår i: COMPLEX ANALYSIS AND DYNAMICAL SYSTEMS IV, PT 1: FUNCTION THEORY AND OPTIMIZATION / [ed] Agranovsky, M BenArtzi, M Galloway, G Karp, L Reich, S Shoikhet, D Weinstein, G Zalcman, L, AMER MATHEMATICAL SOC , 2011, s. 45-+Konferensbidrag (Refereegranskat)
    Abstract [en]

    According to the theorem of A. Kirillov, every smooth, closed Jordan curve in the plane can be represented by its "fingerprint", a diffeomorphism of the unit circle. In this paper we show that the fingerprints of (polynomial) lemniscates of degree n are given by n-th roots of Blaschke products of degree re. The latter are dense in the space of all diffeomorphisms of the unit circle. Moreover, we also prove that every diffeomorphism of the unit circle induced by the n-th root of a Blaschke product with n zeros fingerprints a polynomial lernniscate of the same degree.

  • 311.
    Ebenfelt, Peter
    et al.
    Mathematics, University of California in San Diego.
    Gustafsson, BjörnKTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).Khavinson, DmitryMathematics, University of Arkansas.Putinar, MihaiMathematics, University of California in Santa Barbara.
    Quadrature Domains and Their Applications: The Harold S. Shapiro Anniversary Volume2005Proceedings (redaktörskap) (Refereegranskat)
  • 312.
    Edquist, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Monotonicity formulas and applications in free boundary problems2010Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven.

  • 313.
    Edquist, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lindgren, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Regularity of a parabolic free boundary problem with Hölder continuous coefficientsManuskript (preprint) (Övrigt vetenskapligt)
  • 314.
    Edquist, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Petrosyan, Arshak
    Purdue University.
    A parabolic almost monotonicity formula2008Ingår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 341, nr 2, s. 429-454Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u +/-( x, s) in the strip S-1 = R-n x (-1, 0] satisfying u +/- >= 0, (Delta - partial derivative(S))u +/- >= -1, u(+) center dot u(-) = 0 in S-1. We also establish a localized version of the formula as well as prove one of its variants. At the end of the paper we give an application to a free boundary problem related to the caloric continuation of heat potentials.

  • 315. Eggers, J.
    et al.
    Hoppe, Jens
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Singularity formation for time-like extremal hypersurfaces2009Ingår i: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 680, nr 3, s. 274-278Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We derive self-similar string solutions in a graph representation, near the point of singularity formation, which can be shown to extend to point-like singularities on M(em)branes, as well as to the radially symmetric case. Crown

  • 316.
    Ek, Viktor
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Inre biljarder och tvistavbildningar2012Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Denna rapport diskuterar inre biljarder, som är studiet av rörelsen hos en punktpartikel

    som rör sig inom ett begränsat område och re ekteras mot områdets rand utan att

    tappa i fart. Först introduceras en lämplig parametrisering av rörelsen. Sedan införs

    biljardavbildningen, som beskriver hur man går från en studspunkt till nästa i rörelsen.

    Det visar sig att denna avbildning för vissa former på området tillhör en väl studerad

    typ av avbilding, en så kallad tvistavbildning.

    Dessutom diskuteras vilka periodiska banor som nns när området har olika former

    och egenskaper och hur man kan hitta dessa med hjälp av den så kallade genererande

    funktionen. Vidare bevisas Birkhos sats för periodiska banor, som beskriver existens av

    periodiska banor för konvexa områden med glatt rand och positiv krökning.

  • 317.
    Ekedahl, Torsten
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Skjelnes, Roy
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Recovering the good component of the Hilbert scheme2014Ingår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 179, nr 3, s. 805-841Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give an explicit construction, for a flat map X -> S of algebraic spaces, of an ideal in the n'th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemesof the locusof n distinct points.This generalizes Haiman's corresponding result for the affine complex plane.However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory.In the nonflat case we obtain a similar result by replacing the n'th symmetric product by the n'th divided power product.

  • 318.
    Ekholm, Tomas
    et al.
    Lund University.
    Enblom, Andreas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Kovarik, Hynek
    Dipartimento di Matematica, Politecnico di Torino.
    Schrödinger operators on regular metric trees with long range potentials: Weak coupling behavior2010Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 248, nr 4, s. 850-865Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Consider a regular d-dimensional metric tree Γ with root o. Define the Schrödinger operator - Δ - V, where V is a non-negative, symmetric potential, on Γ, with Neumann boundary conditions at o. Provided that V decays like | x |- γ at infinity, where 1 < γ ≤ d ≤ 2, γ ≠ 2, we will determine the weak coupling behavior of the bottom of the spectrum of - Δ - V. In other words, we will describe the asymptotic behavior of inf σ (- Δ - α V) as α → 0 +.

  • 319.
    Ekholm, Tomas
    et al.
    Department of Mathematics, Lund University.
    Frank, Rupert
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lieb-Thirring inequalities on the half-line with critical exponent2008Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, nr 3, s. 739-755Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the operator -d(2)/dr(2) - V in L-2(R+) with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound for any alpha is an element of [0, 1) and gamma >= (1 - alpha)/2. This includes a Lieb-Thirring inequality in the critical endpoint case.

  • 320.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Frank, Rupert
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On Lieb-Thirring inequalities for Schrödinger operators with virtual level2006Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, nr 3, s. 725-740Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the operator H = - Delta- V in L-2(R-d), d >= 3. For the moments of its negative eigenvalues we prove the estimate

    tr H--(gamma) <= C-gamma,C-d integral(Rd) (V(x) - (d-2)(2)/4\x\(2))(gamma+d/2) dx, gamma > 0.

    Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.

  • 321.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Frank, Rupert L.
    Kovarik, Hynek
    Eigenvalue estimates for Schrodinger operators on metric trees2011Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, nr 6, s. 5165-5197Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider Schrodinger operators on radial metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit.

  • 322.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Kovarik, Hynek
    Università degli studi di Brescia.
    Frank, Rupert
    California Institute of Technology.
    Remarks about Hardy inequalities on metric trees: 2008Ingår i: PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, 2008, Vol. 77, s. 369-379Konferensbidrag (Refereegranskat)
    Abstract [en]

    We find sharp conditions on the growth of a rooted regular metric tree Such that the Neumann Laplacian on the tree satisfies a Hardy inequality. In particular, we consider homogeneous metric trees. Moreover, we show that a non-trivial Aharonov-Bohm magnetic field leads to a Hardy inequality on a loop graph.

  • 323.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Kovarik, Hynek
    Laptev, Ari
    Hardy inequalities for p-Laplacians with Robin boundary conditions2015Ingår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 128, s. 365-379Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals ((p-1)/p)(p) whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.

  • 324.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Kovarik, Hynek
    Portmann, Fabian
    Estimates for the lowest eigenvalue of magnetic Laplacians2016Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 439, nr 1, s. 330-346Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded, open, simply connected domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a lower bound for the first magnetic Neumann eigenvalue in the case of constant magnetic field. 

  • 325.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Portmann, Fabian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A magnetic contribution to the Hardy inequality2014Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 55, nr 2, s. 022101-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the quadratic form associated to the kinetic energy operator in the presence of an external magnetic field in d = 3. We show that if the radial component of the magnetic field does not vanish identically, then the classical lower bound given by Hardy is improved by a non-negative potential term depending on properties of the magnetic field.

  • 326.
    Eklund, David
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Algebraic C*-actions and homotopy continuation2008Licentiatavhandling, monografi (Övrigt vetenskapligt)
    Abstract [en]

    Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator.

  • 327.
    Eklund, David
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Algebraic C*-actions and inverse kinematicsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    Let X be a smooth quadric of dimension 2m in P2m+1C and let Y,Z X be subvarietiesboth of dimension m which intersect transversely. In this paper we give an algorithm forcomputing the intersection points of Y  Z based on a homotopy method. The homotopyis constructed using a C*-action on X whose fixed points are isolated, which inducesthe so-called Bialynicki-Birula decompositions of X into locally closed invariant subsets.Notably, the homotopy has the optimal number of solution paths. As an applicationwe present a new solution to the inverse kinematics problem of a general six-revoluteserial-link manipulator.

  • 328.
    Eklund, David
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Curves on Heisenberg invariant quartic surfaces in projective 3-spaceManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    This paper is about the family of smooth quartic surfaces X P3 that are invariant under the Heisenberg group H2,2. For a very generic X, we show that the Picard number of X is 16 and determine its Picard group. It turns out that a very generic X contains 320 irreducible conics which generate the Picard group of X.

  • 329.
    Eklund, David
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Topics in computation, numerical methods and algebraic  geometry2010Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers.

    In Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes.

    In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism.

    Paper IV is a study of curves on certain special quartic surfaces in projective 3-space. The surfaces are invariant under the action of a finite group called the level (2,2) Heisenberg group. In the paper, we determine the Picard group of a very general member of this family of quartics. We have found that the general Heisenberg invariant quartic contains 320 smooth conics and we prove that in the very general case, this collection of conics generates the Picard group.

  • 330.
    El Gaidi, Khalid
    et al.
    KTH, Skolan för teknikvetenskaplig kommunikation och lärande (ECE), Lärande.
    Ekholm, Tomas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Contextualizing calculus with everyday examples to enhance conceptual learning2015Ingår i: ASEE Annual Conference and Exposition, Conference Proceedings, 2015, Vol. 122, nr 122nd ASEE Annual Conference and Exposition: Making Value for SocietyKonferensbidrag (Refereegranskat)
    Abstract [en]

    Many engineering students in Sweden have difficulty passing the initial mathematics courses. Teachers complain that students are ill-prepared in pre-calculus and even the best students can only solve standard problems using standard procedures. The aim of teaching in mathematics at university is to develop deep understanding of the subject and to produce conceptually trained students who can then creatively solve unforeseen problems. But how should we educate such students’ We hypothesize that the root of the problem lies in decontextualized abstract teaching. The approach adopted in this study is to introduce contextualized teaching of mathematics through concrete examples and to focus students’ attention on the role of definitions in mathematics in order to scaffold their development of conceptual understanding. The general idea is to start from the most concrete, everyday examples and work towards more abstract mathematics. By everyday examples, we mean those that can instantly be understood by the students with reference solely to their life experience. Each new concept in the course is introduced verbally, numerically, graphically, and algebraically when applicable. Everyday examples are introduced in the verbal phase of the sequence. Application examples are also used to boost the students’ motivation. Quantitative data were gathered from both a diagnostic test and the exam, and qualitative data come from a student questionnaire. The exam results show that the students in the intervention group succeeded better than the control groups. They also became more motivated and had a better grasp of abstract thinking in mathematics.

  • 331. El-Zanati, S.
    et al.
    Heden, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Seelinger, G.
    Sissokho, P.
    Spence, L.
    Vanden Eynden, C.
    Partitions of the 8-Dimensional Vector Space Over GF(2)2010Ingår i: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 18, nr 6, s. 462-474Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let V=V(n,q) denote the vector space of dimension n over GF(q). A set of subspaces of V is called a partition of V if every nonzero vector in V is contained in exactly one subspace of V. Given a. partition P of V with exactly a(i) subspaces of dimension i for 1 <= i <= n, we have Sigma(n)(i=1) a(i)(q(i)-1) = q(n)-1, and we call the n-tuple (a(n), a(n-1), ..., a(1)) the type of P. In this article we identify all 8-tuples (a(8), a(7), ..., a(2), 0) that are the types of partitions of V(8,2).

  • 332. Emamizadeh, Behrouz
    et al.
    Prajapat, Jyotshana V.
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A Two Phase Free Boundary Problem Related to Quadrature Domains2011Ingår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 34, nr 2, s. 119-138Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution to -Delta u=(mu(+)-lambda(+))(chi{u>0}) - (mu(-) - lambda(-))(chi{u<0}) in IRN. where lambda(+/-) >0 are given constants and mu(+/-), are non-negative bounded Radon measures, with compact support. Our primary concern is to discuss existence and geometric properties of solutions.

  • 333.
    Engardt, Max
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Structure Learning of Bayesian Networks with Bounded Treewidth Using Mixed Integer Linear Programming2014Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [en]

    When given a Bayesian network, a common use of it is calculating conditional probabilities. This is known as inference. In order to be able to infer effectively, the structure of the Bayesian network is required to have low treewidth. Therefore, the problem of learning the structure of Bayesian networks with bounded treewidth is studied in this thesis. This is solved by reducing the problem to a mixed integer linear problem using several formulation for the structure of the Bayesian network as well as for bounding the treewidth of the structure. Solving the problem in this way gives an algorithm known as an anytime algorithm which can be aborted during the run and return a solution as well as an upper bound for the value of the best possible solution. Tests show that several of these formulations are of practical use as implementations of them prove solutions to be optimal or nearly optimal for several data sets.

  • 334.
    Engström, Alexander
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Discrete Morse Functions from Fourier Transforms2009Ingår i: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 18, nr 1, s. 45-53Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A discrete Morse function on a simplicial complex describes how to construct a homotopy-equivalent CW-complex with possibly fewer cells. We associate a Boolean function with a given simplicial complex and construct a discrete Morse function using its Fourier transform. Methods from theoretical computer science by O'Donnell, Saks, Schramm, and Servedio, together with experimental data on complexes from Hachimori's library and on chessboard complexes, provide some evidence that the constructed discrete Morse functions are efficient.

  • 335.
    Engström, Alexander
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Norén, Patrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Polytopes from subgraph statistics2011Artikel i tidskrift (Övrigt vetenskapligt)
  • 336.
    Engström, Alexander
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Norén, Patrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Tverbergs´s theorem and graph coloring2011Artikel i tidskrift (Övrigt vetenskapligt)
  • 337.
    Engström, Stefan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Random acyclicorientations of graphs2013Självständigt arbete på avancerad nivå (magisterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
  • 338. Ericsson, S.
    et al.
    Grip, N.
    Johansson, E.
    Persson, L. E.
    Sjoberg, R.
    Strömberg, Jan-Olov
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Towards automatic detection of local bearing defects in rotating machines2005Ingår i: Mechanical systems and signal processing, ISSN 0888-3270, E-ISSN 1096-1216, Vol. 19, nr 3, s. 509-535Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we derive and compare several different vibration analysis techniques for automatic detection of local defects in bearings. Based on a signal model and a discussion on to what extent a good bearing monitoring method should trust it, we present several analysis tools for bearing condition monitoring and conclude that wavelets are especially well suited for this task. Then we describe a large-scale evaluation of several different automatic bearing monitoring methods using 103 laboratory and industrial environment test signals for which the true condition of the bearing is known from visual inspection. We describe the four best performing methods in detail (two wavelet-based, and two based on envelope and periodisation techniques). In our basic implementation, without using historical data or adapting the methods to (roughly) known machine or signal parameters, the four best methods had 9-13% error rate and are all good candidates for further fine-tuning and optimisation. Especially for the wavelet-based methods, there are several potentially performance improving additions, which we finally summarise into a guiding list of suggestion.

  • 339. Eriksen, Niklas
    et al.
    Hultman, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Expected reflection distance in G(r, 1, n) after a fixed number of reflections2005Ingår i: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 9, nr 1, s. 21-33Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Extending to r > 1 a formula of the authors, we compute the expected reflection distance of a product of t random reflections in the complex reflection group G (r, 1, n). The result relies on an explicit decomposition of the reflection distance function into irreducible G (r, 1, n) characters and on the eigenvalues of certain adjacency matrices.

  • 340. Eriksen, Niklas
    et al.
    Sjöstrand, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Equidistributed Statistics on Matchings and Permutations2014Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 21, nr 4, artikel-id P4.43Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization. As a special case we obtain proofs of two conjectures of Claesson and Linusson. Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.

  • 341.
    Eriksson, Henrik
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    Eriksson, Kimmo
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wästlund, Johan
    Dense packing of patterns in a permutation2007Ingår i: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 11, nr 3-4, s. 459-470Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the length L-k of the shortest permutation containing all patterns of length k. We establish the bounds e(-2)k(2) < L-k <= (2/3 + o(1))k(2). We also prove that as k there are permutations of length (1/4+o(1))k(2) containing almost all patterns of length k.

  • 342. Eriksson, Kimmo
    et al.
    Sjöstrand, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On two theorems of Quinzii and rent controlled housing allocation in Sweden2007Ingår i: International Journal of Game Theory, ISSN 0020-7276, E-ISSN 1432-1270, Vol. 3, nr 9, s. 515-526Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Swedish rent control system creates a white market for swapping rental contracts and a black market for selling rental contracts. Empirical data suggests that in this black-and-white market some people act according to utility functions that are both discontinuous and locally decreasing in money. We discuss Quinzii's theorem for the nonemptiness of the core of generalized house-swapping games, and show how it can be extended to cover the Swedish game. In a second part, we show how this theorem of Quinzii and her second theorem on nonemptiness of the core in two-sided models are both special cases of a more general theorem.

  • 343.
    Ernvall-Hytönen, Anna Maria
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Matomäki, K.
    Haukkanen, P.
    Merikoski, J. K.
    Formulas for the number of gridlines2011Ingår i: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 164, nr 2, s. 157-170Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let l(n) be the number of lines through at least two points of an n × n rectangular grid. We prove recursive and asymptotic formulas for it using respectively combinatorial and number theoretic methods. We also study the ratio l(n)/l(n - 1). All this originates from Mustonen's experimental results.

  • 344.
    Ernvall-Hytönen, Anne-Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On certain exponential sums related to GL(3) cusp forms2010Ingår i: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 348, nr 1-2, s. 5-8Artikel i tidskrift (Refereegranskat)
  • 345. Exner, Pavel
    et al.
    Frank, Rupert L.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Absolute continuity of the spectrum for periodically modulated leaky wires in R-32007Ingår i: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 8, nr 2, s. 241-263Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.

  • 346.
    Faber, Carel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A remark on a conjecture of Hain and Looijenga2011Ingår i: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 61, nr 7, s. 2745-2750Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the natural generalization of a conjecture of Hain and Looijenga to the case of pointed curves holds for all g and n if and only if the tautological rings of the moduli spaces of curves with rational tails and of stable curves are Gorenstein.

  • 347.
    Faber, Carel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Pandharipande, R.
    Relative maps and tautological classes2005Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 7, nr 1, s. 13-49Artikel i tidskrift (Refereegranskat)
  • 348.
    Faber, Carel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    van der Geer, GerardKdV-Institute, University of Amsterdam.Looijenga, EduardDepartment of Mathematics, Utrecht University.
    Classification of algebraic varieties2011Proceedings (redaktörskap) (Refereegranskat)
  • 349.
    Fahlgren, Anton
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Combining Acoustic Echo Cancellation and Voice Activity Detection in Social Robotics2019Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Vi ger en teoretisk introduktion till grundläggande koncept inom kontinuerlig och diskret signalhantering som Fourier-transformen, linjära tidsinvarianta system och spektralanalys av slumpsignaler. En andra del behandla teori och tillämpningar för ekokansellering och röstdetektion i så kallad social robotik. Existerande metoder presenteras tillsammans med nya specialiserade metoder och samtliga utvärderas

  • 350. Fainsilber, L.
    et al.
    Kurlberg, Pär
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wennberg, B.
    Lattice points on circles and discrete velocity models for the Boltzmann equation2006Ingår i: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 37, nr 6, s. 1903-1922Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. However, by showing that lattice points on most circles are equidistributed we find that the collision operator can indeed be approximated as a sum over lattice points in the two-dimensional case. The proof uses a weak form of the Halberstam-Richert inequality for multiplicative functions (a proof is given in the paper), and estimates for the angular distribution of Gaussian primes. For higher dimensions, this result has already been obtained by Palczewski, Schneider, and Bobylev [SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883].

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