Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.     

Remarks on Normal Generation of Special Line Bundles on Algebraic Curves

Let L be a very ample line bundle with h ¹(L) ≥2 on a curve of genus g. We prove that L is normally generated if deg(L) ≥2g ? 1 ? 4h ¹(L) for large enough genus g.     

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L ¹-norm and with order 1/2 w.r.t. the L ²-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.     

<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">h</Emphasis>, 1, 1)-labeling of outerplanar graphs

An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, . . . , λ} to the nodes of the graph such that adjacent nodes are assigned integers of at least distance h ≥ 1 apart and all nodes of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λ _{h, 1, 1} and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n ³), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h, 1, 1)-labeling for outerplanar graphs that is within additive constants of the optimum values. This research is partially supported by the European Research Project Algorithmic Principles for Building Efficient Overlay Computers (AEOLUS) and was done during the visit of Richard B. Tan at the Department of Computer Science, University of Rome “Sapienza”, supported by a visiting fellowship from the University of Rome “Sapienza”.     

On polarized 3-folds (X, L) withg(L)=q(X)+1 andh 0(L)≥4

Let (X, L) be a polarized 3-fold over the complex number field. In [Fk3], we proved thatg(L)≥q(X) ifh ⁰(L)≥2 and moreover we classified (X, L) withh ⁰(L)≥3 andg(L)=q(X), whereg(L) is the sectional genus of (X, L) andq(X)=dimH ¹(O _X ) the irregularity ofX. In this paper we will classify polarized 3-folds (X, L) withh ⁰(L)≥4 andg(L)=q(X)+1 by the method of [Fk3].     

Best one-sided L 1-approximation by harmonic functions

Let , where B is the open unit ball in (), and let denote the collection of functions h in which are harmonic on B and satisfy on . A function h ^* in is called a best harmonic one-sided L ¹-approximant to f if for all h in . This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L ¹-approximation without a one-sidedness condition. Received: 25 September 1997     

Hessian-free metric-based mesh adaptation via geometry of interpolation error

The article presents analysis of a new methodology for generating meshes minimizing L ^p -norms of the interpolation error or its gradient, p > 0. The key element of the methodology is the construction of a metric from node-based and edge-based values of a given function. For a mesh with N _h triangles, we demonstrate numerically that L ^∞-norm of the interpolation error is proportional to N _h ⁻¹ and L ^∞-norm of the gradient of the interpolation error is proportional to N _h ^−1/2. The methodology can be applied to adaptive solution of PDEs provided that edge-based a posteriori error estimates are available.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> error estimate of the finite volume element methods on quadrilateral meshes

In this paper the optimal L ² error estimates of the finite volume element methods (FVEM) for Poisson equation are discussed on quadrilateral meshes. The trial function space is taken as isoparametric bilinear finite element space on quadrilateral partition, and the test function space is defined as piecewise constant space on dual partition. Under the assumption that all elements on quadrilateral meshes are O(h ²) quasi-parallel quadrilateral elements, we prove convergence rate to be O(h ²) in L ² norm.     

A remark on Weierstrass points and linear series on multiple covering curves

Here we study multiple coverings of rational and irational curves. We give a theorem about the non-gap sequence on m-gonal curves. We then study general irrational covering f ： X→ C, and say when h^0（X, f^＊（L）） = h^0（C,L） for L line bundle on C.     

A finite element method for surface PDEs: matrix properties

We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces. The main idea of this method is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a problem in an outer domain that contains the surface, for example, two-phase flow problems. It has been proved that the method has optimal order of convergence both in the H ¹ and in the L ²-norm. In this paper, we address linear algebra aspects of this new finite element method. In particular the conditioning of the mass and stiffness matrix is investigated. For the two-dimensional case we present an analysis which proves that the (effective) spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h ⁻³| ln h| and h ⁻²| ln h|, respectively, where h is the mesh size of the outer triangulation.     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001     

L 2 estimates for the finite element method for the Dirichlet problem with singular data

Summary In this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous Dirichlet condition on the boundary. In these problems the data are Borel measures. Using a quasiuniform mesh of finite elements and polynomials of degree 1, we prove that in two dimensions the convergence is of orderh inL ² and in three dimensions of orderh ^1/2.     

The Theory of Constructive Signal Analysis

Let h(x) = e^?αxk(x), where and λ0=0. The closure theorem, V_h = L¹(?), is proved for various α and k (V_h is the L¹-closed variety generated by h). The Tauberian condition, |?| > 0, is not used, since generally this condition is difficult to compute directly. The functions h arise naturally in time series and analytic number theory. The technique of proof is constructive and depends on the semigroup {γ_j} generated by {λ_j}. The semigroup theory which consolidates and completes the results herein will be developed separately as “A closure problem for signals in semigroup invariant systems.”     

The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves

LetX be an integral projective curve andL ∃ Pic^a(X),M ∃ Pic^b (X) with h¹(X, L)= h¹(X, M) = 0 andL, M general. Here we study the rank of the multiplication map μ _L,M :H ⁰(X,L)⊗H ⁰(X,M)→H ⁰(X,L⊗M). We also study the same problem whenL andM are rank 1 torsion free sheaves onX. Most of our results are forX with only nodes as singularities.     

Error bounds for multistep methods revisited

For linear multistep methods with constant stepsize we consider error bounds in terms of weightedL ₂-norms ofh ^px^(p) rather than ofh ^px^(p+1). The bounds apply to stiff systemsx'=Ax+f(t,x) where the spectrum ofA lies in a sector andf is of moderate size.     

Estimates for the Second Order Derivatives of Eigenvectors in Thin Anisotropic Plates with Variable Thickness

For the second order derivatives of eigenvectors in a thin anisotropic heterogeneous plate Ω_h, we derive estimates of their weighted L₂-norms with majorants whose dependence on the plate thickness h and on the eigenvalue number is expressed explicitly. These estimates maintain the asymptotic sharpness throughout the entire spectrum, whereas inside its low-frequency band the majorants remain bounded as h → +0. The latter is a rather unexpected fact, because for the first eigenfunction u¹ of a similar boundary-value problem for a scalar second order differential operator with variable coefficients, the norm ‖∇ _x ² u⁰; L₂(Ω_h)‖ is of order h⁻¹ and grows as h tends to zero. Bibliography: 35 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 161–180.     

Linear series onk-gonal curves

Here we prove the following result. Theorem 1.1.Let X be an integral projective curve of arithmetic genus g and k≧ ≧4 an integer. Assume the existence of L ∈ Pic^k (X) with h ⁰ (X, L)=2 and L spanned. Fix a rank 1 torsion free sheaf M on X with h ⁰(X,M)=r+1≧2, h¹ (X, M)≧2 and M spanned by its global sections. Set d≔deg(M) and s≔max {n≧0:h ⁰ (X, M ⊗(L^*)^⊗n)>0}. Then one of the following cases occur:

Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations

In this article, we investigate interior penalty discontinuous Galerkin (IPDG) methods for solving a class of two‐dimensional nonlinear parabolic equations. For semi‐discrete IPDG schemes on a quasi‐uniform family of meshes, we obtain a priori bounds on solutions measured in the L² norm and in the broken Sobolev norm. The fully discrete IPDG schemes considered are based on the approximation by forward Euler difference in time and broken Sobolev space. Under a restriction related to the mesh size and time step, an hp ‐version of an a priori l^∞(L²) and l²(H¹) error estimate is derived and numerical experiments are presented.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 288–311, 2012     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.     

Comparison of Spaces of Hardy Type for the Ornstein–Uhlenbeck Operator

Denote by γ the Gauss measure on ℝ ⁿ and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} to L ¹γ. This result is in sharp contrast both with the fact that (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is bounded from H ¹γ to L ¹γ, where H ¹γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007), and with the fact that in the Euclidean case (rI-D)^iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space \mathfrakh¹\mathbbRⁿ{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to L ¹ℝ ⁿ . We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T{\mathcal{T}}, bounded on L ² μ, and with a kernel satisfying certain analytic assumptions, is bounded from H ¹ μ to L ¹ μ if and only if it is bounded from \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} to L ¹ μ. Here H ¹ μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009). The case of translation invariant operators on homogeneous trees is also considered.