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  • 151.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Attractors in the quasi-periodically perturbed quadratic family2012Ingår i: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, nr 5, s. 1537-1545Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a geometric description of an attractor arising in quasi-periodically perturbed maps T x [0, 1] (sic) (theta, x) bar right arrow (theta + omega, c(theta)x(1 - x)) is an element of T x [0, 1] for certain choices of smooth c : T -> [1.5, 4] and Diophantine omega. The existence of the 'strange' attractor was established in Bjerklov 2009 Commun. Math. Phys. 286 137.

  • 152.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On some generalizations of skew-shifts on T-22019Ingår i: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 39, s. 19-61Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we investigate maps of the two-torus T-2 of the form T (x, y) = (x + omega, g(x) + f (y)) for Diophantine omega is an element of T and for a class of maps f, g : T -> T, where each g is strictly monotone and of degree 2 and each f is an orientation-preserving circle homeomorphism. For our class of f and g, we show that T is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two T-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in T-2. Only a low-regularity assumption (Lipschitz) is needed on the maps f and g, and the results are robust with respect to Lipschitz-small perturbations of f and g.

  • 153.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrodinger equations2005Ingår i: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 25, s. 1015-1045Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the discrete quasi-periodic Schrodinger equation -(u(n+1) + u(n-1)) + lambda V(theta + n omega)u(n) = Eu-n with a non-constant C-1 potential function V : T -> R. We prove that for sufficiently large k there is a set Omega subset of T of frequencies omega, whose measure tends to 1 as lambda -> infinity, with the following property. For each w e Q there is a 'large' (in measure) set of energies E, all lying in the spectrum of the associated Schrodinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrodinger cocycle is minimal but not ergodic.

  • 154.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Positive Lyapunov exponents for continuous quasiperiodic Schrodinger equations2006Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 47, nr 2Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove that the continuous one-dimensional Schrodinger equation with an analytic quasi-periodic potential has positive Lyapunov exponents in the bottom of the spectrum for large couplings.

  • 155.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Quasi-periodic kicking of circle diffeomorphisms having unique fixed points2019Ingår i: Moscow Mathematical Journal, ISSN 1609-3321, E-ISSN 1609-4514, Vol. 19, nr 2, s. 189-216Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We investigate the dynamics of certain homeomorphisms F: T-2 -> T-2 of the form F(x, y) = (x + omega , h(x)+ f (y)), where omega is an element of R\Q, f: T -> T is a circle diffeomorphism with a unique (and thus neutral) fixed point and h: T -> T is a function which is zero outside a small interval. We show that such a map can display a non-uniformly hyperbolic behavior: (small) negative fibred Lyapunov exponents for a.e. (x, y) and an attracting non-continuous invariant graph. We apply this result to (projective) SL(2, R)-cocycles G: (x, u) bar right arrow (x + omega, A(x)u) with A(x) = R phi(x)B, where R-theta is a rotation matrix and B is a parabolic matrix, to get exam ples of non-uniformly hyperbolic cocycles (homotopic to the identity) with perturbatively small Lyapunov exponents.

  • 156.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Quasi-periodic perturbation of unimodal maps exhibiting an attracting 3-cycle2012Ingår i: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, nr 3, s. 683-741Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study a class of smooth maps Phi : T x [0, 1]. T x [0, 1] of the form theta bar right arrow theta + omega x bar right arrow c(theta)h(x) where h : [0, 1] --> [0, 1] is a unimodal map exhibiting an attracting periodic point of prime period 3, and omega is irrational (T = R/Z). We show that the following phenomenon can occur for certain h and c : T --> R: There exists a single measurable function psi : T --> [0, 1] whose graph attracts (exponentially fast) a.e. (theta, x) is an element of T x [0, 1] under forward iterations of the map Phi. Moreover, the graph of psi is dense in a cylinder M subset of T x [0, 1]. Furthermore, for every integer n >= 1 there exists n distinct repelling continuous curves Gamma(k) : (theta, phi(k)(theta))(theta is an element of T), all lying in M, such that Phi(Gamma(k)) = Gamma(k+1) (k < n) and Phi(Gamma(n)) = Gamma(1). We give concrete examples where both c(theta) and h(x) are real-analytic, but in the analysis we only need that they are C-1. In our setting the function c(theta) will be very close to 1 for all theta outside a tiny interval; on the interval c(theta) > 1 makes a small bump. Thus we cause the perturbation of h by rare quasi-periodic kicking.

  • 157.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    SNA's in the Quasi-Periodic Quadratic Family2009Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, nr 1, s. 137-161Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We rigorously show that there can exist Strange Nonchaotic Attractors (SNA) in the quasi-periodically forced quadratic ( or logistic) map (theta, x) -> (theta + omega, c(theta)x(1 - x)) for certain choices of c : T bar right arrow [3/2, 4] and Diophantine omega.

  • 158.
    Bjerklöv, Kristian
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles2015Ingår i: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 16, nr 4, s. 961-1031Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let f : T -> R be a Morse function of class C-2 with exactly two critical points, let omega is an element of T be Diopharitine, and let lambda > 0 be sufficiently large (depending on f and omega). For any value of the parameter E is an element of R, we make a careful analysis of the dynamics of the skew-product map Phi(E)(theta, r) = (theta + omega, lambda f(theta) - E - 1/r), acting on the "torus" T x (R) over cap. Here, (R) over cap denotes the projective space R boolean OR {infinity}. The map Phi(E) is intimately related to the quasi-periodic Schrodinger cocycle (omega, A(E)) : T x R-2 -> T x R-2, (theta, x) -> (theta + omega, A(E)(theta) . x), where A(E) : T -> SL(2, R) is given by A(E)(theta) = ((0)(-1) 1(lambda f(theta) - E)), E is an element of R. More precisely, (omega, A(E)) naturally acts on the space T x (R) over cap, and Phi(E) is the map thus obtained. The cocycle (omega, A(E)) arises when investigating the eigenvalue equation H(theta)u = Eu, where H-theta is the quasi-periodic Schrodinger operator (H(theta)u)(n) = -(u(n+1) + u(n-1)) + lambda f (theta + (n - 1)omega)u(n), (1) The (maximal) Lyapunov exponent of the Schrodinger cocycle (omega, A(E)) is greater than or similar to log lambda, uniformly in E is an element of R. This implies that the map PE has exactly two ergodic probability measures for all E is an element of R; (2) If E is on the edge of an open gap in the spectrum sigma(H), then there exist a phase 0 is an element of T and a vector u is an element of l(2)(Z), exponentially decaying at +/-infinity, such that H(theta)u = Eu;acting on the space l(2) (Z). It is well known that the spectrum of H-theta, sigma(H), is independent of the phase theta is an element of T. Under our assumptions on f, omega and lambda, Sinai (in J Stat Phys 46(5-6):861-909, 1987) has shown that sigma(H) is a Cantor set, and the operator H-theta has a pure-point spectrum, with exponentially decaying eig,enfunctions, for a.e. theta is an element of T The analysis of Phi(E) allows us to derive three main results: (3) The map Phi(E) is minimal iff E E is an element of sigma(H)\ {edges of open gaps}. In particular, Phi(E) is minimal for all E is an element of R for which the fibered rotation number alpha(E) associated with (omega, A(E)) is irrational with respect to omega.

  • 159. Bjerklöv, Kristian
    et al.
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Universal asymptotics in hyperbolicity breakdown2008Ingår i: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 21, nr 3, s. 557-586Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study a scenario for the disappearance of hyperbolicity of invariant tori in a class of quasi-periodic systems. In this scenario, the system loses hyperbolicity because two invariant directions come close to each other, losing their regularity. In a recent paper, based on numerical results, Haro and de la Llave (2006 Chaos 16 013120) discovered a quantitative universality in this scenario, namely, that the minimal angle between the two invariant directions has a power law dependence on the parameters and the exponents of the power law are universal. We present an analytic proof of this result.

  • 160.
    Björnberg, Jakob Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Critical Value of the Quantum Ising Model on Star-Like Graphs2009Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 135, nr 3, s. 571-583Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a rigorous determination of the critical value of the ground-state quantum Ising model in a transverse field, on a class of planar graphs which we call star-like. These include the junction of several copies of a"currency sign at a single point. Our approach is to use the graphical, or fk-, representation of the model, and the probabilistic and geometric tools associated with it.

  • 161.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A cell complex in number theory2011Ingår i: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 46, nr 1-4, s. 71-85Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let Delta(n) be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system. In this paper we study the asymptotic behavior of the individual Betti numbers beta(k)(Delta(n)) and of their sum. We show that Delta(n) has the homotopy type of a wedge of spheres, and that as n -> infinity S beta(k)(Delta(n)) = 2n/pi(2) + O(n(theta)), for all theta > 17/54, Furthermore, for fixed k, beta k(Delta(n)) similar to n/2logn (log log n)(k)/k!. As a number-theoretic byproduct we obtain inequalities partial derivative(k)(sigma(odd)(k+1)(n)) infinity S beta k((Delta) over tilde (n)) = n/3 + O(n(theta)), for all theta > 22/27. 

  • 162.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A comparison theorem for f-vectors of simplicial polytopes2007Ingår i: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 3, nr 1, s. 347-356Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let f(i)(P) denote the number of i-dimensional faces of a convex polytope P. Furthermore, let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Our main result is that for every simplicial d-polytope P, if f(r) (S (n(1), d)) <= f(r) (P) <= f(r) (C (n(2), d)) for some integers n(1), n(2) and r, then f(s) (S (n(1), d)) <= f(s) (P) <= f(s) (C (n(2), d)) for all s such that r < s. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain comparison theorem for f-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.

  • 163.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A {$q$}-analogue of the {FKG} inequality and some applications2011Ingår i: Combinatorica, ISSN 0209-9683, E-ISSN 1439-6912, Vol. 31, nr 2, s. 151-164Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let L be a finite distributive lattice and mu: L -> R(+) a log-supermodular function k: L -> R(+) let [GRAPHICS] We prove for any pair g, h: L -> R(+) of monotonely increasing functions, that E mu(g; q) . E mu(h; q) << E mu(1; q) . E mu(gh; q), where "<<" denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to q=1. The polynomial FKG inequality has applications to f-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of Schubert varieties, and to correlation-type inequalities for a class of power series weighted by Young tableaux. This class contains series involving Plancherel measure for the symmetric groups and its poissonization.

  • 164.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    NOTE: RANDOM-TO-FRONT SHUFFLES ON TREES2009Ingår i: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 14, s. 36-41Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.

  • 165.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries2008Ingår i: Building bridges: The Lovász Festschrift / [ed] Grötschel and G. O. H. Katona, Berlin: Springer Berlin/Heidelberg, 2008, s. 165-203Kapitel i bok, del av antologi (Refereegranskat)
    Abstract [en]

    The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves.We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markow chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula.Background material needed for understanding the walks is reviewed in rather great detail.

  • 166.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Ekedahl, Torsten
    On the shape of Bruhat intervals2009Ingår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, nr 2, s. 799-817Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic subgroup W-J. For w is an element of W-J, let f(i)(w,J) denote the number of elements of length i below w in Bruhat order on W-J (with notation simplified to f(i)(w) in the case when W-J = W). We show that 0 <= i < j <= l(w)-i implies f(i)(w,J) <= f(j)(w,J). Also, the case of equalities f(i)(w) = f(l(w)-i)(w) for i = 1,..., k is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial P-e,P-w (q). We show that if W is finite then the number sequence f(0)(w), f(1)(w),... f(l(w))(w) cannot grow too rapidly. Further, in the finite case, for any given k >= 1 and any w is an element of W of sufficiently great length (with respect to k), we show f(l(w)-k)(w) >= f(l(w)-k+1)(w) >= ... >= f(l(w))(w). The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if (X) over bar (w) is a Schubert variety of dimension d = l(w), and lambda = c(1) (L) is an element of H-2 ((X) over bar (w)) is the restriction to (X) over bar (w) of the Chem class of an ample line bundle, then (lambda(k)) . : Hd-k((X) over bar (w)) -> Hd+k((X) over bar (w)) is injective for all k >= 0.

  • 167.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Farley, J. D.
    Chain polynomials of distributive lattices are 75% unimodal2005Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 12, nr 1Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    It is shown that the numbers c(i) of chains of length i in the proper part L\{0, 1} of a distributive lattice L of length l + 2 satisfy the inequalities c(0) <...< c([l/2]) and c([3l.4]) >... > c(l). This proves 75% of the inequalities implied by the Neggers unimodality conjecture.

  • 168.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Goodarzi, A.
    On codimension one embedding of simplicial complexes2017Ingår i: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer International Publishing , 2017, s. 207-219Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    We study d-dimensional simplicial complexes that are PL embeddable in Rd+1. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.

  • 169.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Matoušek, J.
    Ziegler, G. M.
    Using brouwer’s fixed point theorem2017Ingår i: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer International Publishing , 2017, s. 221-271Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

    Brouwer’s fixed point theorem from 1911 is a basic result in topology- with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain their applications to the fascinating (and still not fully solved) evasiveness problem. © Springer International Publishing AG 2017.

  • 170.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Paffenholz, A.
    Sjöstrand, J.
    Ziegler, G. M.
    Bier spheres and posets2005Ingår i: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 34, nr 1, s. 71-86Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n -2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that cut across an ideal. Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular generalized Bier spheres. In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields many shellable spheres, most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.

  • 171.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Peeva, I.
    Sidman, J.
    Subspace arrangements defined by products of linear forms2005Ingår i: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 71, s. 273-288Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in P-2 and lines in P-3 are also considered.

  • 172.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sagan, Bruce E.
    Rationality of the Mobius function of a composition poset2006Ingår i: Theoretical Computer Science, ISSN 0304-3975, E-ISSN 1879-2294, Vol. 359, nr 3-Jan, s. 282-298Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the zeta and Mobius functions of a partial order on integer compositions first studied by Bergeron, Bousquet-Melou, and Dulucq. The Mobius function of this poset was determined by Sagan and Vatter. We prove rationality of various formal power series in noncommuting variables whose coefficients are evaluations of the zeta function, zeta, and the Mobius function, mu. The proofs are either directly from the definitions or by constructing finite-state automata. We also obtain explicit expressions for generating functions obtained by specializing the variables to commutative ones. We reprove Sagan and Vatter's formula for it using this machinery. These results are closely related to those of Bjorner and Reutenauer about subword order, and we discuss a common generalization.

  • 173.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Stanley, Richard P.
    Massachusetts Institute of Technology.
    A combinatorial miscellany2010Bok (Refereegranskat)
  • 174.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Tancer, Martin
    Note: Combinatorial Alexander Duality-A Short and Elementary Proof2009Ingår i: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 42, nr 4, s. 586-593Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X* = {sigma subset of V vertical bar V \ sigma is not an element of X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (vertical bar V vertical bar - i - 3) th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.

  • 175.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Vorwerk, Kathrin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Connectivity of chamber graphs of buildings and related complexes2010Ingår i: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 31, nr 8, s. 2149-2160Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let Delta be a thick and locally finite building with the property that no edge of the associated Coxerer diagram has label "infinity". The chamber graph G(Delta), whose edges are the pairs of adjacent chambers in Delta is known to be q-regular for a certain number q = q(Delta). Our main result is that G(Delta) is q-connected in the sense of graph theory. In the language of building theory this means that every pair of chambers of Delta is connected by q pairwise disjoint galleries. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.

  • 176.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Vorwerk, Kathrin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On the connectivity of manifold graphs2015Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, nr 10, s. 4123-4132Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.

  • 177.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wachs, M. L.
    Welker, V.
    Poset fiber theorems2005Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 357, nr 5, s. 1877-1899Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Suppose that f : P --> Q is a poset map whose fibers f(-1)(Qless than or equal to(q)) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some applications are discussed.

  • 178.
    Björner, Anders
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Wachs, Michelle
    Welker, Volkmar
    ON SEQUENTIALLY COHEN-MACAULAY COMPLEXES AND POSETS2009Ingår i: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 169, nr 1, s. 295-316Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra.

  • 179.
    Björner, Anders.
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Welker, V.
    Segre and Rees products of posets, with ring-theoretic applications2005Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 198, nr 3-Jan, s. 43-55Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce (weighted) Segre and Rees products for posets and show that these constructions preserve the Cohen-Macaulay property over a field k and homotopically. As an application we show that the weighted Segre product of two affine semigroup rings that are Koszul is again Koszul. This result generalizes previous results by Crona on weighted Segre products of polynomial rings. We also give a new proof of the fact that the Rees ring of a Koszul affine semigroup ring is again Koszul. The paper ends with a list of some open problems in the area.

  • 180.
    Blomgren, Martin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Fibrations and Idempotent Functors2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis consists of two articles. Both articles concern homotopical algebra. In Paper I we study functors indexed by a small category into a model category whose value at each morphism is a weak equivalence. We show that the category of such functors can be understood as a certain mapping space. Specializing to topological spaces, this result is used to reprove a classical theorem that classifies fibrations with a fixed base and homotopy fiber. In Paper II we study augmented idempotent functors, i.e., co-localizations, operating on the category of groups. We relate these functors to cellular coverings of groups and show that a number of properties, such as finiteness, nilpotency etc., are preserved by such functors. Furthermore, we classify the values that such functors can take upon finite simple groups and give an explicit construction of such values.

  • 181.
    Blomgren, Martin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Chachólski, Wojciech
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    On the classification of fibrations2015Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, nr 1, s. 519-557Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.

  • 182.
    Blomgren, Martin
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Chachólski, Wojciech
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Farjoun, Emmanuel D.
    Hebrew University of Jerusalem.
    Segev, Yoav
    Ben-Gurion University.
    Idempotent deformations of finite groupsManuskript (preprint) (Övrigt vetenskapligt)
  • 183.
    Blomgren, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Chacholski, Wojciech
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Farjoun, E. D.
    Segev, Y.
    Idempotent transformations of finite groups2013Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 233, nr 1, s. 56-86Artikel i tidskrift (Refereegranskat)
    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.

  • 184.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Carlini, Enrico
    Geramita, Anthony V.
    Monomials as sums of powers: The real binary case2011Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 139, nr 9, s. 3039-3043Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We generalize an example, due to Sylvester, and prove that any monomial of degree d in R[x(0), x(1)], which is not a power of a variable, cannot be written as a linear combination of fewer than d powers of linear forms.

  • 185.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Fløystad, Gunnar
    The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two2011Ingår i: Combinatorial Aspects Of Commutative Algebra And Algebraic Geometry: The Abel Symposium 2009 / [ed] Floystad, G; Johnsen, T; Knutsen, AL, Springer Berlin/Heidelberg, 2011, Vol. 6, s. 1-16Konferensbidrag (Refereegranskat)
    Abstract [en]

    We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences, p and q, of these total degrees are relatively prime, the extremal rays are parametrized by order ideals in N-2 contained in the region px + qy < (p - 1)(q - 1). We also consider some examples concerning artinian modules of codimension three.

  • 186.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Fröberg, R.
    Lundqvist, S.
    Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections2018Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 495, s. 1-14Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Given an ideal I=(f1,…,fr) in C[x1,…,xn] generated by forms of degree d, and an integer k&gt;1, how large can the ideal Ik be, i.e., how small can the Hilbert function of C[x1,…,xn]/Ik be? If r≤n the smallest Hilbert function is achieved by any complete intersection, but for r&gt;n, the question is in general very hard to answer. We study the problem for r=n+1, where the result is known for k=1. We also study a closely related problem, the Weak Lefschetz property, for S/Ik, where I is the ideal generated by the d'th powers of the variables.

  • 187.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Geramita, Anthony
    Notes on Diagonal Coinvariants of the Dihedral Group2010Ingår i: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 53, nr 4, s. 602-613Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The bigraded Hilbert function and the minimal free resolutions for the diagonal coinvariants of the dihedral groups are exhibited, as well as for all their bigraded invariant Gorenstein quotients.

  • 188.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Iarrobino, Anthony
    Reducible family of height three level algebras2009Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 321, nr 1, s. 86-104Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let R = k[x(1),.....X-r] be the polynomial ring in r variables over an infinite field k, and let M be the maximal ideal of R. Here a level algebra will be a graded Artinian quotient A of R having socle Soc(A) = 0 : M in a single degree j. The Hilbert function H(A) = (h(0), h(1)..... h(j)) gives the dimension h(i) = dim(k) A(i) of each degree-i graded piece of A for 0 <= i <= j. The embedding dimension of A is h(1), and the type of A is dim(k) Soc(A), here h(j). The family LevAlg(H) of level algebra quotients of R having Hilbert function H forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions H (1, 3, 4, 4) and H-2 = (1. 3. 6, 8, 9. 3) the family LevAlg(H) has several irreducible components (Theorems 2.3(A), 2.4). We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points (Theorem 2.3(B)). We show that the second example is the first in an infinite sequence of examples of type three Hilbert functions H(c) in which also the number of components gets arbitrarily large (Theorem 2.10). The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors (unpublished) and also by J.O. Kleppe.

  • 189.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Migliore, J.
    Miró-Roig, R. M.
    Nagel, U.
    Zanello, F.
    On the Weak Lefschetz Property for artinian Gorenstein algebras of codimension three2014Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 403, s. 48-68Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1, 3, 6, 6, 3, 1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from P2 to P3, and Hesse configurations in P2.

  • 190.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Migliore, Juan C.
    Univ Notre Dame, Notre Dame, IN, USA.
    Miro-Roig, Rosa M.
    Univ Barcelona, Barcelona, Spain.
    Nagel, Uwe
    Univ Kentucky, Lexington, KY USA.
    Zanello, Fabrizio
    Michigan Technol Univ, Houghton, MI USA ; MIT, Cambridge, MA 02139 USA .
    On the Shape of a Pure O-sequence2012Ingår i: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 218, nr 1024, s. 1-+Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A monomial order ideal is a finite collection X of (monic) monomials such that, whenever M is an element of X and N divides M, then N is an element of X. Hence X is a poset, where the partial order is given by divisibility. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, (h) under bar = (h(0) = 1, h(1), ..., h(e)), counting the monomials of X in each degree. Equivalently, pure O-sequences can be characterized as the f-vectors of pure multicomplexes, or, in the language of commutative algebra, as the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of the early works of Stanley's in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their f-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, which makes an extensive use of both algebraic and combinatorial techniques, in particular includes: (i) A characterization of the first half of a pure O-sequence, which yields the exact converse to a g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, a natural bijection between integer partitions and type 1 pure O-sequences, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) A study of the Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) full characterization; (v) A pithy connection of the ICP with Stanley's conjecture on the h-vectors of matroid complexes; (vi) A more specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 over a field of characteristic zero. As an immediate corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over an arbitrary field). (vii) An analysis, from a commutative algebra viewpoint, of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras. (viii) Some observations about pure f-vectors, an important special case of pure O-sequences.

  • 191.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Söderberg, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case2012Ingår i: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 6, nr 3, s. 437-454Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We use results of Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination of Betti diagrams of modules with a pure resolution. This implies the multiplicity conjecture of Herzog, Huneke, and Srinivasan for modules that are not necessarily Cohen-Macaulay and also implies a generalized version of these inequalities. We also give a combinatorial proof of the convexity of the simplicial fan spanned by pure diagrams.

  • 192.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Zanello, Fabrizio
    Level algebras with bad properties2007Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 135, nr 9, s. 2713-2722Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property ( WLP). In fact, for e >> 0, we will construct a codimension three, type two h- vector of socle degree e such that all the level algebras with that h-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each e >> 0. 2). There exist reduced level sets of points in P-3 of type two whose artinian reductions all fail to have theWLP. Indeed, the examples constructed here have the same h- vectors we mentioned in 1). 3). For any integer r >= 3, there exist non- unimodal monomial artinian level algebras of codimension r. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the abovementioned preprint, Theorem 4.3) that, for any r >= 3, there exist reduced level sets of points in P-r whose artinian reductions are non- unimodal.

  • 193.
    Boij, Mats
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Zanello, Fabrizio
    Some algebraic consequences of Green's hyperplane restriction theorems2010Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 214, nr 7, s. 1263-1270Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We discuss Green's paper [11] from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine a new infinite class of symmetric h-vectors that cannot be Gorenstein h-vectors, which was left open in the recent work [19]. This includes the smallest example, previously unknown, h = (1, 10, 9, 10, 1). As M. Green's results depend heavily on the characteristic of the base field, so will ours. The Appendix contains a new argument, kindly provided to us by M. Green, for Theorems 3 and 4 of [11], since we had found a gap in the original proof of those results during the preparation of this manuscript.

  • 194. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products2008Ingår i: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 143, nr 2, s. 205-223Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For (n x n)-matrices A and B, define eta(A, B) = Sigma(S)det(A[S])det(B[S']), where the summation is over all subsets of {1,..., n}, S' is the complement of S', and A [S] is the principal submatrix of A with rows and columns indexed by S. We prove that if A >= 0 and B is Hermitian, then (1) the polynomial eta(zA, -B) has all real roots; (2) the latter polynomial has as many positive, negative, and zero roots (counting multiplicities) as suggested by the inertia of B if A > 0; and (3) for 1 <= i <= n, the roots of eta(zA[{1}'], -B[{i}']) interlace those of eta(zA, -B). Assertions (1) - (3) solve three important conjectures proposed by C. R. Johnson in the mid-1980s in [20, pp. 169, 170], [21]. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process, we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials, and as an application, we derive similar properties for symmetrized Fischer products of positive-definite matrices. We also obtain Laguerre-type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices which considerably generalize a certain subset of the Hadamard., Fischer, and Koteljanskii inequalities for principal minors of positive-definite matrices. Finally, we propose Lax-type problems for real stable polynomials and mixed determinants.

  • 195. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lee-Yang Problems and the Geometry of Multivariate Polynomials2008Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 86, nr 1, s. 53-61Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Polya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory.

  • 196. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Polya-Schur master theorems for circular domains and their boundaries2009Ingår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, nr 1, s. 465-492Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Omega subset of C for arbitrary closed circular domains Omega (i.e., images of the closed unit disk under a Mobius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for Omega = R our results settle open questions that go back to Laguerre and Polya-Schur.

  • 197. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The Lee-Yang and Polya-Schur programs. I. Linear operators preserving stability2009Ingår i: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 177, nr 3, s. 541-569Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Polya-Schur on univariate polynomials with such properties.

  • 198. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The Lee-Yang and Polya-Schur Programs. II. Theory of Stable Polynomials and Applications2009Ingår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 62, nr 12, s. 1595-1631Artikel i tidskrift (Refereegranskat)
    Abstract [en]

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

  • 199. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Liggett, Thomas M.
    NEGATIVE DEPENDENCE AND THE GEOMETRY OF POLYNOMIALS2009Ingår i: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 22, nr 2, s. 521-567Artikel i tidskrift (Refereegranskat)
  • 200. Borichev, Alexander
    et al.
    Hedenmalm, Håkan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Weighted integrability of polyharmonic functions2014Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 264, s. 464-505Artikel i tidskrift (Refereegranskat)
    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.

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