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  • 1.
    Adamowicz, Tomasz
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Shanmugalingam, Nageswari
    University of Cincinnati, OH USA .
    Prime ends for domains in metric spaces2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 238, p. 459-505Article in journal (Refereed)
    Abstract [en]

    In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

  • 2. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, Pierre
    Double Aztec diamonds and the tacnode process2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 252, p. 518-571Article in journal (Refereed)
    Abstract [en]

    Discrete and continuous non-intersecting random processes have given rise to critical "infinite-dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the tacnode process. It is also-shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily.

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

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

  • 4.
    Alfonseca, M Angeles
    et al.
    N Dakota State University.
    Auscher, Pascal
    University Paris 11.
    Axelsson Rosén, Andreas
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Hofmann, Steve
    University of Missouri.
    Kim, Seick
    Yonsei University.
    Analyticity of layer potentials and L-2 solvability of boundary value problems for divergence form elliptic equations with complex L-infinity coefficients2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 5, p. 4533-4606Article in journal (Refereed)
    Abstract [en]

    We consider divergence form elliptic operators of the form L = -div A (x)del, defined in Rn+1 = {(x, t) is an element of R-n x R}, n andgt;= 2, where the L-infinity coefficient matrix A is (n + 1) x (n + 1), uniformly elliptic, complex and t-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L-2 (R-n) = L-2(partial derivative R-+(n+1)) is stable under complex, L-infinity perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L-2(R-n) whenever A (x) is real and symmetric (and thus, by our stability result, also when A is complex, parallel to A - A(0)parallel to(infinity) is small enough and A(0) is real, symmetric, L-infinity and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L-2 (resp. (L) over dot(1)(2)) data, for small complex perturbations of a real symmetric matrix. Previously, L-2 solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A (j,n+1)= 0 = A(n+1,j), 1 andlt;= j andlt;= n, which corresponds to the Kato square root problem.

  • 5.
    Amini, Nima
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Brändén, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-representable hyperbolic matroids2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 334, p. 417-449Article in journal (Refereed)
    Abstract [en]

    The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The Vamos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the Vamos matroid and the generalized Vamos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre Turan inequality and an inequality due to Jensen. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.

  • 6. Ammann, Bernd
    et al.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Humbert, Emmanuel
    Surgery and harmonic spinors2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 220, no 2, p. 523-539Article in journal (Refereed)
    Abstract [en]

    Let M he a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.

  • 7.
    Andersen, Jorgen Ellegaard
    et al.
    Aarhus Univ, Ctr Quantum Geometry Moduli Spaces QGM, Ny Munkegade 118,Bldg 1530, DK-8000 Aarhus C, Denmark..
    Himpel, Benjamin
    Aarhus Univ, Ctr Quantum Geometry Moduli Spaces QGM, Ny Munkegade 118,Bldg 1530, DK-8000 Aarhus C, Denmark..
    Jørgensen, Søren Fuglede
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Martens, Johan
    Univ Edinburgh, Sch Math, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, Midlothian, Scotland.;Univ Edinburgh, Maxwell Inst Math Sci, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, Midlothian, Scotland..
    McLellan, Brendan
    Harvard Univ, Dept Math, One Oxford St, Cambridge, MA 02138 USA..
    The Witten-Reshetikhin-Turaev invariant for links in finite order mapping tori I2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, p. 131-178Article in journal (Refereed)
    Abstract [en]

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

  • 8.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uraltseva, Nina N.
    Weiss, Georg S.
    Equilibrium points of a singular cooperative system with free boundary2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 280, p. 743-771Article in journal (Refereed)
    Abstract [en]

    In this paper we initiate the study of maps minimising the energy integral(D)(vertical bar del u vertical bar(2) + 2 vertical bar u vertical bar) dx, which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations Delta u = u / vertical bar u vertical bar chi({vertical bar u vertical bar >0}), u = (u(1,) ... ,u(m)) Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

  • 9.
    Asekritova, Irina
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kruglyak, Natan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Mastylo, Mieczyslaw
    University of Poznan, Poland.
    Interpolation of Fredholm Operators2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 295, p. 421-496Article in journal (Refereed)
  • 10.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Capogna, Luca
    University of Arkansas.
    Citti, Giovanna
    University of Bologna.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 257, p. 25-65Article in journal (Refereed)
    Abstract [en]

    We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype\begin{equation*}  \partial_tu= -\sum_{i=1}^{m}X_i^\ast ( |\X u|^{p-2} X_i u)\end{equation*}where $p\ge 2$, $ \ \X = (X_1,\ldots, X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $\mu$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $\mu$. Our estimates are derived assuming that (i) the control distance $d$ generated by $\X$ induces the same topology on $\M$; (ii) a doubling condition for the $\mu$-measure of $d-$metric balls and (iii) the validity of a Poincar\'e inequality involving $\X$ and $\mu$. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by H\"ormander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.

  • 11. Averkov, Gennadiy
    et al.
    Krümpelmann, Jan
    Nill, Benjamin
    Stockholm University, Faculty of Science, Department of Mathematics.
    Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 274, p. 118-166Article in journal (Refereed)
    Abstract [en]

    For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.

  • 12. Bao, Zhigang
    et al.
    Erdos, Laszlo
    Schnelli, Kevin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Convergence rate for spectral distribution of addition of random matrices2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 319, p. 251-291Article in journal (Refereed)
  • 13.
    Barmak, Jonathan Ariel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Star clusters in independence complexes of graphs2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 241, p. 33-57Article in journal (Refereed)
    Abstract [en]

    We introduce the notion of star cluster of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from works of Engstrom and Jonsson on the homotopy type of independence complexes of triangle-free graphs and to investigate a large number of examples which appear in the literature. We present an alternative way to study the chromatic and clique numbers of a graph from a homotopical point of view and obtain new results regarding the connectivity of independence complexes.

  • 14. Barnsley, Michael F.
    et al.
    Hutchinson, John E.
    Stenflo, Örjan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    V-variable fractals: Fractals with partial self similarity2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 218, no 6, p. 2051-2088Article in journal (Refereed)
    Abstract [en]

    We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V-variable fractals can also be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems.

  • 15.
    Bauer, Tilman
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Formal plethories2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 254, p. 497-569Article in journal (Refereed)
    Abstract [en]

    Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E* to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where E* is a Prüfer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.

  • 16. Berkesch Zamaere, Christine
    et al.
    Forsgård, Jens
    Stockholm University, Faculty of Science, Department of Mathematics.
    Matusevich, Laura Felicia
    Hypergeometric functions for projective toric curves2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 300, p. 835-867Article in journal (Refereed)
    Abstract [en]

    We produce a decomposition of the parameter space of the A-hypergeometric system associated to a projective monomial curve as a union of an arrangement of lines and its complement, in such a way that the analytic behavior of the solutions of the system is explicitly controlled within each term of the union.

  • 17.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Sharp exponents and a Wiener type condition for boundary regularity of quasiminimizers2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 301, p. 804-819Article in journal (Refereed)
    Abstract [en]

    We obtain a sufficient condition for boundary regularity of quasiminimirers of the p-energy integral in terms of a Wiener type sum. The exponent in the sum is independent of the dimension and is explicitly expressed in terms of p and the quasiminimizing constant. We also show by an example that the exponent is sharp in a certain sense. (C) 2016 Elsevier Inc. All rights reserved.

  • 18.
    Blomgren, Mats
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Chacholski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Farjoun, E. D.
    Segev, Y.
    Idempotent transformations of finite groups2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 233, no 1, p. 56-86Article in journal (Refereed)
    Abstract [en]

    We describe the action of idempotent transformations on finite groups. We show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group. This is done in terms of the first two homology groups. We prove for example that except special linear groups, such an orbit can have at most 7 elements. We also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite.

  • 19.
    Borcea, Julius
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Boegvad, Rikard
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shapiro, Boris
    Stockholm University, Faculty of Science, Department of Mathematics.
    On rational approximation of algebraic functions2006In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 204, no 2, p. 448-480Article in journal (Refereed)
    Abstract [en]

    We construct a new scheme of approximation of any multivalued algebraic function f (z) by a sequence {r(n)(z)}(n is an element of N) of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Pade approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Pade Conjecture and Nuttall's Conjecture for the sequence {r(n)(z)}(n is an element of N) in the complement CP1\D-f, where D-f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r(n)(z)}(n is an element of N). As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis

  • 20. Borichev, Alexander
    et al.
    Hedenmalm, Håkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weighted integrability of polyharmonic functions2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 264, p. 464-505Article in journal (Refereed)
    Abstract [en]

    To address the uniqueness issues associated with the Dirichlet problem for the N-harmonic equation on the unit disk D in the plane, we investigate the L-P integrability of N-harmonic functions with respect to the standard weights (1 vertical bar z vertical bar(2))(alpha). The question at hand is the following. If u solves Delta(N)u = 0 in D, where Delta stands for the Laplacian, and integral(D)vertical bar u(Z)vertical bar(p)(1 - vertical bar z vertical bar(2))(alpha)dA(z) < +infinity, must then u(z) 0? Here, N is a positive integer, alpha is real, and 0 < p < +infinity; dA is the usual area element. The answer will, generally speaking, depend on the triple (N, p, alpha). The most interesting case is 0 < p < 1. For a given N, we find an explicit critical curve p bar right arrow beta(N, p) - a piecewise affine function - such that for alpha > beta(N, p) there exist nontrivial functions u with Delta Nu = 0 of the given integrability, while for alpha <= beta(N, p), only u(z) 0 is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in PHN, alpha p (D) when this space is nontrivial. We find a new structural decomposition of the polyharmonic functions - the cellular decomposition - which decomposes the polyharmonic weighted LP space in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the (p, alpha) plane into cells. The above uniqueness for the Dirichlet problem may be considered for any elliptic operator of order 2N. However, the above-mentioned critical integrability curve will depend rather strongly on the given elliptic operator, even in the constant coefficient case, for N > 1.

  • 21. Brasco, L.
    et al.
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 304, p. 300-354Article in journal (Refereed)
    Abstract [en]

    We prove that for p≥2, solutions of equations modeled by the fractional p-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in Wloc 1,p and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1. © 2016 Elsevier Inc.

  • 22. Brasco, Lorenzo
    et al.
    Lindgren, Erik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Schikorra, Armin
    Higher Holder regularity for the fractional p-Laplacian in the superquadratic case2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 338, p. 782-846Article in journal (Refereed)
    Abstract [en]

    We prove higher Holder regularity for solutions of equations involving the fractional p-Laplacian of order s, when p >= 2 and 0 < s < 1. In particular, we provide an explicit Holder exponent for solutions of the non-homogeneous equation with data in L-q and q > N/(sp), which is almost sharp whenever sp <= (p 1) + N/q. The result is new already for the homogeneous equation.

  • 23. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 265, p. 441-484Article in journal (Refereed)
    Abstract [en]

    We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers under fairly weak conditions on the underlying measure mu. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.

  • 24. Bränden, Petter
    Polynomials with the half-plane property and matroid theory2007In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 216, no 1, p. 302-320Article in journal (Refereed)
    Abstract [en]

    A polynomial f is said to have the half-plane property if there is an open half-plane H subset of C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory. (1) We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. (2) We prove that a multivariate multi-affine polynomial f is an element of R[z(1),..., z(n)] has the half-plane property (with respect to the upper half-plane) if and only if partial derivative f/partial derivative(zi)(x)center dot partial derivative f/partial derivative(zj)(x)-partial derivative(2)f/partial derivative(zi)partial derivative(zj)(x)center dot f(x)>= 0 for all x is an element of R-n and 1 <= i, j <= n. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. (3) We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.

  • 25.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    Obstructions to determinantal representability2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 2, p. 1202-1212Article in journal (Refereed)
    Abstract [en]

    There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.

  • 26.
    Brändén, Petter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Obstructions to determinantal representability2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 2, p. 1202-1212Article in journal (Refereed)
    Abstract [en]

    There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.

  • 27. Chhita, Sunil
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Domino statistics of the two-periodic Aztec diamond2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 294, p. 37-149Article in journal (Refereed)
    Abstract [en]

    Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary.

  • 28.
    Chhita, Sunil
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Young, Benjamin
    Coupling functions for domino tilings of Aztec diamonds2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 259, p. 173-251Article in journal (Refereed)
    Abstract [en]

    The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, that we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q(vol) weighting which corresponds to multiplying the product of tile weights by q if we add a 'box' to the height function and the two-periodic weighting which exhibits a flat region with defects in the center.

  • 29.
    Cianchi, Andrea
    et al.
    University of Florence, Italy.
    Mazya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. University of Liverpool, England.
    Sobolev inequalities in arbitrary domains2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 293, p. 644-696Article in journal (Refereed)
    Abstract [en]

    A theory of Sobolev inequalities in arbitrary open sets in R-n is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities exhibit the same critical exponents as in the classical framework. Moreover, they involve constants independent of the geometry of the domain, and hence yield genuinely new results even in the case when just smooth domains are considered. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. (C) 2016 Elsevier Inc. All rights reserved.

  • 30.
    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, p. 1511-1585Article 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).

  • 31.
    Coulembier, Kevin
    et al.
    Univ Ghent, Dept Math Anal, B-9000 Ghent, Belgium..
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Some homological properties of category O. III2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 283, p. 204-231Article in journal (Refereed)
    Abstract [en]

    We prove that thick category O associated to a semi-simple complex finite dimensional Lie algebra is extension full in the category of all modules. We also prove the weak Alexandru conjecture both for regular blocks of thick category O and the associated categories of Harish-Chandra bimodules, but disprove it for singular blocks.

  • 32. Dickenstein, Alicia
    et al.
    Di Rocco, Sandra
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Piene, Ragni
    Classifying smooth lattice polytopes via toric fibrations2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 222, no 1, p. 240-254Article in journal (Refereed)
    Abstract [en]

    We show that any smooth Q-normal lattice polytope P of dimension it and degree d is a strict Cayley polytope if n >= 2d + 1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.

  • 33. Dress, A.
    et al.
    Huber, K. T.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Moulton, Vincent
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    An explicit computation of the injective hull of certain finite metric spaces in terms of their associated Buneman complex2002In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 168, no 1, p. 1-28Article in journal (Refereed)
  • 34. Eckhardt, Caleb
    et al.
    Raum, Sven
    Stockholm University, Faculty of Science, Department of Mathematics.
    C*-superrigidity of 2-step nilpotent groups2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 338, p. 175-195Article in journal (Refereed)
    Abstract [en]

    We show that torsion-free finitely generated nilpotent groups are characterised by their group C*-algebras and we additionally recover their nilpotency class as well as the subquotients of the upper central series. We then use a C*-bundle decomposition and apply K-theoretic methods based on noncommutative tori to prove that every torsion-free finitely generated 2-step nilpotent group can be recovered from its group C*-algebra.

  • 35.
    Ekholm, Tomas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Frank, Rupert L.
    Kovarik, Hynek
    Eigenvalue estimates for Schrodinger operators on metric trees2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 6, p. 5165-5197Article in journal (Refereed)
    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.

  • 36.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Schweigert, Christoph
    Hamburg University.
    Consistent systems of correlators in non-semisimple conformal field theory2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 307, p. 598-639Article in journal (Refereed)
    Abstract [en]

    Based on the modular functor associated with a -- not necessarily semisimple -- finite non-degenerate ribbon category D, we present a definition of a consistent system of bulk field correlators for a conformal field theory which comprises invariance under mapping class group actions and compatibility with the sewing of surfaces. We show that when restricting to surfaces of genus zero such systems are in bijection with commutative symmetric Frobenius algebras in D, while for surfaces of any genus they are in bijection with modular Frobenius algebras in D. This provides additional insight into structures familiar from rational conformal field theories and extends them to rigid logarithmic conformal field theories.

  • 37.
    Fuchs, Jürgen
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Schweigert, Christoph
    Hamburg University.
    Stigner, Carl
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Higher genus mapping class group invariants from factorizable Hopf algebras2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 250, p. 285-319Article in journal (Refereed)
    Abstract [en]

    Lyubashenko's construction associates representations of mapping class groups Map_{g,n} of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Map_{g,n} for all values of g and n. More generally, we obtain such invariants for any pair (H,omega), where omega is a ribbon automorphism of H. Our results are motivated by the quest to understand correlation functions of bulk fields in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple, so-called logarithmic conformal field theories.

  • 38.
    Goldring, Wushi
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Koskivirta, Jean-Stefan
    Quasi-constant characters: Motivation, classification and applications2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 339, p. 336-366Article in journal (Refereed)
    Abstract [en]

    In [131, initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of minuscule character which we termed quasi-constant. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if mu is a quasi-constant cocharacter of an F-p-group G, then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack G-Zip(mu), without any restrictions on p, even if the pair (G, mu) is not of Hodge type and even if mu is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc. cit.

  • 39.
    Graham, C. R.
    et al.
    University of Washington Seattle.
    Juhl, Andreas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Holographic formula for Q-curvature2007In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 216, no 2, p. 841-853Article in journal (Refereed)
  • 40. Granville, Andrew
    et al.
    Kurlberg, Pär
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Poisson statistics via the Chinese Remainder Theorem2008In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 218, no 6, p. 2013-2042Article in journal (Refereed)
    Abstract [en]

    We consider the distribution of spacings between consecutive elements in subsets of Z/qZ, where q is highly composite and the subsets are defined via the Chinese Remainder Theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q has Poisson spacings. We also study the spacings of subsets of Z/q(1)q(2)Z that are created via the Chinese Remainder Theorem from subsets of Z/q(1)Z and Z/q(2)Z (for q(1), q(2) coprime), and give criteria for when the spacings modulo q(1)q(2) are Poisson. Moreover, we also give some examples when the spacings modulo q(1)q(2) are not Poisson, even though the spacings modulo q(1) and modulo q(2) are both Poisson.

  • 41.
    Gustafsson, Björn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Putinar, Mihai
    Saff, Edward B.
    Stylianopoulos, Nikos
    Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction2009In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 222, no 4, p. 1405-1460Article in journal (Refereed)
    Abstract [en]

    Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains ( called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago vertical bar z(m)-1 vertical bar < r(m), 0 < r < 1, which consists of m islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szego orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.

  • 42. Hall, J.
    et al.
    Rydh, David
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Hilbert stack2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 253, p. 194-233Article in journal (Refereed)
    Abstract [en]

    Let π : X → S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack-the Hilbert stack-parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.

  • 43.
    Hedenmalm, Håkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Saint Petersburg State University,Russian Federation.
    Bloch functions and asymptotic tail variance2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 313, p. 947-990Article in journal (Refereed)
    Abstract [en]

    Let P denote the Bergman projection on the unit disk D, P mu(z) := integral(D) mu(w)/(1-z (w) over bar)(2) dA(w), z is an element of D, where dA is normalized area measure. We prove that if vertical bar mu(z)vertical bar <= 1 on D, then the integral I(mu()a,r) := integral(2 pi)(0) exp {a r(4)vertical bar P mu(re(i theta))vertical bar(2)/log 1/1-r(2)}d theta/2 pi, 0 < r < 1, has the bound I-mu(a,r) <= C(a) := 10(1-a)(-3/2) for 0 < a < 1, irrespective of the choice of the function mu. Moreover, for a > 1, no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection P-mu (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0 < a < pi(2)/64 = 0.1542.... We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k-quasiconformal extension, for 0 < k < 1. The estimate reads, for t is an element of C and 0 < k < 1, B(k,t) <= {1/4 k(2)vertical bar t vertical bar(2)(1 + 7k)(2), for vertical bar t vertical bar <= 2/k(1 + 7k)(2), k vertical bar t vertical bar - 1/(1 + 7k)(2), for vertical bar t vertical bar >= 2/k(1 + 7k)(2), which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with vertical bar t vertical bar <= 2/k, we should have B(k, t) = 1/4k(2)t(2).

  • 44. Heluani, Reimundo
    et al.
    Zabzine, Maxim
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy.
    Generalized Calabi-Yau manifolds and the chiral de Rham complex2010In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 223, no 5, p. 1815-1844Article in journal (Refereed)
    Abstract [en]

    We show that the chiral de Rham complex of a generalized Calabi-Yau manifold carries N = 2 supersymmetry. We discuss the corresponding topological twist for this N = 2 algebra. We interpret this as an algeboid version of the super-Sugawara or Kac-Todorov construction.

  • 45. Hepworth, R.
    et al.
    Lahtinen, Anssi
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On string topology of classifying spaces2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 281, p. 394-507Article in journal (Refereed)
    Abstract [en]

    Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.

  • 46.
    Hultman, Axel
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Criteria for rational smoothness of some symmetric orbit closures2012In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 229, no 1, p. 183-200Article in journal (Refereed)
    Abstract [en]

    Let G be a connected reductive linear algebraic group over C with an involution theta. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety G/B are indexed by the twisted identities t = {theta(omega(-1))omega | omega is an element of W} in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph" whose vertices form a subset of t. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on t is rank symmetric. less thanbrgreater than less thanbrgreater thanIn the special case K = Sp(2n) (C), G = SL(2n) (C), we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one", needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.

  • 47. Hultman, Axel
    Fixed points of involutive automorphisms of the Bruhat order2005In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 195, no 1, p. 283-296Article in journal (Refereed)
    Abstract [en]

    Applying a classical theorem of Smith, we show that the poset property of being Gorenstein* over Z(2) is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right.

  • 48.
    Jaye, Benjamin J
    et al.
    Kent State University, OH 44240 USA .
    Maz´ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Verbitsky, Igor E
    University of Missouri, MO 65211 USA .
    Quasilinear elliptic equations and weighted Sobolev-Poincare inequalities with distributional weights2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 232, no 1, p. 513-542Article in journal (Refereed)
    Abstract [en]

    We introduce a class of weak solutions to the quasilinear equation -Delta(p)u = sigma vertical bar u vertical bar(P-2)u in an open set Omega subset of R-n with p andgt; 1, where Delta(p)u = del. (vertical bar del u vertical bar(p-2)del u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients sigma which satisfy the inequality less thanbrgreater than less thanbrgreater than-Lambda integral(Omega) vertical bar del h vertical bar(p)dx andlt;= andlt;vertical bar h vertical bar(p), sigma andgt; andlt;= lambda integral(Omega) vertical bar del h vertical bar(p)dx, less thanbrgreater than less thanbrgreater thanfor all h is an element of C-0(infinity)(Omega). Here 0 andlt; Lambda andlt; +infinity-, and less thanbrgreater than less thanbrgreater than0 andlt; lambda andlt; (p - 1)(2-p) if p andgt;= 2, or 0 andlt; lambda andlt; 1 if 0 andlt; p andlt; 2. less thanbrgreater than less thanbrgreater thanAs we shall demonstrate, these conditions on lambda are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on sigma. less thanbrgreater than less thanbrgreater thanWe also study weak solutions of the closely related equation -Delta p nu = (p - 1)vertical bar del nu vertical bar(p) +sigma, under the same conditions on . Our results for this latter equation will allow us to characterize the class of sigma satisfying the above inequality for positive lambda and Lambda. thereby extending earlier results on the form boundedness problem for the Schrodinger operator to p not equal 2.

  • 49.
    Jochemko, Katharina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sanyal, Raman
    Combinatorial mixed valuations2017In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 319, p. 630-652Article in journal (Refereed)
  • 50.
    Juhl, Andreas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Holographic formula for Q-curvature. II2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 4, p. 3409-3425Article in journal (Refereed)
    Abstract [en]

    We extend the holographic formula for the critical Q-curvature in Graham and Juhl (2007) [9] to all Q-curvatures. Moreover, we confirm a conjecture of Juhl (2009) [11].

12 1 - 50 of 74
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