Showing papers in "Arkiv för Matematik in 2010"
TL;DR: In this paper, the authors consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set.
Abstract: We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.
120 citations
TL;DR: In this article, the transition probability of the Markov chain (G(i,1),...,G(m,n))i is given by a determinant of a special form.
Abstract: We show that the transition probability of the Markov chain (G(i,1),...,G(i,n))i≥1, where the G(i,j)’s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for G(m,n). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.
38 citations
TL;DR: In this article, the authors provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite distance, and weighted transfinite radius for sets in ℂN.
Abstract: We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂN. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann.337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂN. Our results extend to certain weights and measures defined on cones in ℝN.
33 citations
TL;DR: In this article, a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety was given, which generalizes a result by A. Tsikh for the case of a reduced complete intersection.
Abstract: We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A. Tsikh for the case of a reduced complete intersection.
29 citations
TL;DR: In this article, the tropical lines contained in smooth tropical quadric surfaces were studied and two one-dimensional families of tropical lines were found, like in classical algebraic geometry for smooth tropical surfaces of any degree with infinitely many tropical lines.
Abstract: We study the tropical lines contained in smooth tropical surfaces in ℝ3 On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines
25 citations
TL;DR: In this paper, the finiteness properties of the kernel and the cokernel of the natural map f were studied in a Noetherian ring, a an ideal of R, M an R-module and n a non-negative integer.
Abstract: Let R be a Noetherian ring, a an ideal of R, M an R-module and n a non-negative integer. In this paper we first study the finiteness properties of the kernel and the cokernel of the natural map f: ...
24 citations
TL;DR: In this paper, the authors construct a 2-dimensional complex manifold X which is the increasing union of proper subdomains that are biholomorphic to ℂ2, but X is not Stein.
Abstract: We construct a 2-dimensional complex manifold X which is the increasing union of proper subdomains that are biholomorphic to ℂ2, but X is not Stein.
20 citations
TL;DR: In this article, the authors prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems for two-dimensional spectral analysis.
Abstract: We prove that spherical spectral analysis and synthesis hold in Damek–Ricci spaces and derive two-radius theorems.
20 citations
TL;DR: In this paper, the same authors derived the same estimate for the harmonic map heat flow from a compact Kahler manifold N with nonnegative holomorphic bisectional curvature, using an inequality due to Sobolev, and used it to establish the uniformity properties of the harmonic heat flow.
Abstract: A Łojasiewicz-type estimate is a powerful tool in studying the rigidity properties of the harmonic map heat flow. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$
to
$\mathbb{S}^{2}$
(J. Differential Geom. 45 (1997), 593–610). In this note, using an inequality due to Sobolev, we will derive the same estimate for maps from $\mathbb{S}^{2}$
to a compact Kahler manifold N with nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$
to N, which generalizes Topping’s result.
17 citations
TL;DR: In this paper, Rong et al. showed the existence of quasi-parabolic manifolds for transformations of Cn under some extra assumptions, and showed that these manifolds can be used to obtain quasi-Parabolic curves.
Abstract: In [Rong, F., Quasi-parabolic analytic transformations of Cn, J. Math. Anal. Appl.343 (2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of Cn. Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.
14 citations
TL;DR: In this paper, the authors introduced the arithmetic-geometric scaling of continued fraction digits with the denominators of the corresponding approximants, and determined its multifractal spectrum by means of a number-theoretical free-energy function and showed that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform.
Abstract: To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number-theoretical free-energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free-energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued-fraction digits exceeding a given number which tends to infinity.
TL;DR: The T-equivariant Schubert calculus on G(k,n), T being an n-dimensional torus, through derivations on the exterior algebra of a free A-module of rank n is described in this paper.
Abstract: We describe T-equivariant Schubert calculus on G(k,n), T being an n-dimensional torus, through derivations on the exterior algebra of a free A-module of rank n, where A is the T-equivariant cohomology of a point. In particular, T-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Quart. 2 (2006), 699–717).
TL;DR: In this paper, a moment problem is presented for a class of signed measures which are termed pseudo-positive, and a characterization of determinacy in the class of equivalent pseudo positive representation measures is discussed.
Abstract: A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure. The second main result is a characterization of determinacy in the class of equivalent pseudo-positive representation measures. Finally the corresponding truncated moment problem is discussed.
TL;DR: In this article, it was shown that an L1(ℝd)-function f belongs to the Hardy space associated with L if sup t>0|Ktf| belongs to L 1 (for some p>d/2).
Abstract: Let L=−Δ+V be a Schrodinger operator on ℝd, d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to Lp(ℝd), for some p>d/2. Let Kt be the semigroup generated by −L. We say that an L1(ℝd)-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup t>0|Ktf| belongs to L1(ℝd). We prove that \(f\in H^{1}_{L}\) if and only if Rjf∈L1(ℝd) for j=1,…,d, where Rj=(∂/∂xj)L−1/2 are the Riesz transforms associated with L.
TL;DR: In this paper, it was shown that multiplicative bijections are implemented by diffeomorphisms and they are automatically algebra isomorphisms, which confirms a conjecture by Mrcun and Semrl.
Abstract: We study bijections between algebras of smooth functions preserving certain parts of its structure. In particular, we show that multiplicative bijections are implemented by diffeomorphisms and they are automatically algebra isomorphisms. This confirms a conjecture by Mrcun and Semrl.
TL;DR: In this paper, the authors define linear forms on C-n and O-X of the coordinate ring of a linear form defined on C n and X = C n\boolean OR(m)(i=1) V(alpha(i)), where V(α(i)) is an element of C n : alpha(i)(p)=0}.
Abstract: Let alpha(1), alpha(2),..., alpha(m) be linear forms defined on C-n and X = C-n\boolean OR(m)(i=1) V(alpha(i)), where V(alpha(i))={p is an element of C-n : alpha(i)(p)=0}. The coordinate ring O-X o ...
TL;DR: In this article, the main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions, where φα(B)=VolD(B∩V)diam(B)α-D, where VolD denotes the D-dimensional volume.
Abstract: The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions \(\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)\), where \(\mathcal{B}_E\) is the family of all balls centred on E and α is a real parameter. With mild assumptions on φα, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ℝD, these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ∂V. We stress that these dimensions depend on the choice of φα. Two determining functions are considered, φα(B)=VolD(B∩V)diam(B)α-D and φα(B)=VolD(B∩V)α/D, where VolD denotes the D-dimensional volume.
TL;DR: In this paper, the equivalence classes of matrix subspaces and Grassmannians are discussed. But the problem of characterizing the inverses and related nonlinear matrix geometries is not addressed.
Abstract: Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.
TL;DR: For every two-dimensional random walk on the square lattice Z2 having zero mean and finite variance, this paper obtained fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site (s, 0) when it is started at a far from both (0, s) and the origin.
Abstract: For every two-dimensional random walk on the square lattice Z2 having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site (s,0), when it is started at a site far from both (0,s) and the origin.
TL;DR: In this article, a sufficient condition on a Levy measure μ which ensures that the generator L of the corresponding pure jump Levy process is (locally) hypoelliptic was given.
Abstract: We give a sufficient condition on a Levy measure μ which ensures that the generator L of the corresponding pure jump Levy process is (locally) hypoelliptic, i.e., \(\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu\) for all admissible u. In particular, we assume that \(\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})\) . We also show that this condition is necessary provided that \(\mathop {\mathrm {supp}}\mu\) is compact.
TL;DR: In this article, it was shown that if M is an invariant subspace for T such that dim M ⊖ TM<∞, then every maximal invariants subspace of M is of codimension 1 in M.
Abstract: In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and \(1-TT^{*}\in\mathcal{S}_{p}\) for some p≥1. It is shown that if M is an invariant subspace for T such that dim M ⊖ TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim M ⊖ zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.
TL;DR: In this article, Moskowitz et al. established general finiteness results concerning lattices Γ in connected Lie groups G which possess certain "density" properties, such as bounded displacement and bounded cocycles, and investigated analogous questions for the automorphism group Aut(G) proving that StabAut(G)(Γ) is discrete.
Abstract: In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups G which possess certain “density” properties (see Moskowitz, M., On the density theorems of Borel and Furstenberg, Ark. Mat. 16 (1978), 11–27, and Moskowitz, M., Some results on automorphisms of bounded displacement and bounded cocycles, Monatsh. Math.
85 (1978), 323–336). For such groups we show that Γ always has finite index in its normalizer N
G
(Γ). We then investigate analogous questions for the automorphism group Aut(G) proving, under appropriate conditions, that StabAut(G)(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$
, of Aut(G) has finite index in StabAut(G)(Γ). We test the limits of our results with various examples and counterexamples.
TL;DR: In this paper, the question of completeness of compact Hausdorff spaces was discussed and a kernel function on X×X, enjoying additional properties, naturally defines a semi-inner product structure on certain subspaces of all finite signed Borel measures.
Abstract: Let X be a compact Hausdorff space. A kernel function on X×X, enjoying additional properties, naturally defines a semi-inner product structure on certain subspaces of all finite signed Borel measures on X. This paper discusses the question of completeness of such spaces.