Δ-tame quasi-hereditary algebras

Let (K, M, H) be an upper triangular biomodule problem. Brüstle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K, M, H) is quasi-hereditary, and there is an equivalence between the category of Δ-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of Δ-tame representation type, then the category F(Δ) has the homogeneous property, i.e. almost all modules in F(Δ) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M, H) is an upper triangular bipartite bimodule problem, then A is of Δ-tame representation type if and only if F(Δ) is homogeneous.     

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

Images of contractive projections on operator algebras

It is shown that if P is a weak^∗-continuous contractive projection on a JBW^∗-triple M, then P(M) is of type I or semifinite, respectively, if M is of the corresponding type. We also show that P(M) has no infinite spin part if M is a type I von Neumann algebra.     

Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.     

Schatten class membership of Hankel operators on the unit sphere

Let H²(S) be the Hardy space on the unit sphere S in Cn, n?2. Consider the Hankel operator Hf=(1−P)Mf|H²(S), where the symbol function f is allowed to be arbitrary in L²(S,dσ). We show that for p>2n, Hf is in the Schatten class Cp if and only if f−Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if” part. We also show that the membership Hf∈C2n implies f−Pf=0, i.e., Hf=0.     

A finite volume method for solving Navier-Stokes problems

We develop a finite volume method for solving the Navier-Stokes equations on a triangular mesh. We prove that the unique solution of the finite volume method converges to the true solution with optimal order for velocity and for pressure in discrete H¹ norm and L² norm respectively.     

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.     

A note on factorizations of singular M-matrices

Supposing that M is a singular M-matrix, we show that there exists a permutation matrix P such that PMP^T = LU, where L is a lower triangular M-matrix and U is an upper triangular singular M-matrix. An example is given to illustrate that the above result is the best possible one.     

Periods of principal homogeneous spaces of algebraic tori

A generic torsor of an algebraic torus S over a field F is the generic fiber of a S-torsor P → T, where P is a quasi-trivial torus containing S as a subgroup and T = P/S. The period of a generic S-torsor over a field extension K/F, i.e., the order of the class of the torsor in the group H ^p1(K, S) does not depend on the choice of a generic torsor. In the paper we compute the period of a generic torsor of S in terms of the character lattice of the torus S.     

Simultaneous iteration for the matrix eigenvalue problem

Let B be a given positive definite Hermitian matrix, and assume the matrix P satisfies the “normality” condition PB^?1P^HB=B^?1P^HBP, where P^H denotes the Hermitian of P. In this paper, we develop an accelerated version of simultaneous iteration for partial solution of the eigenproblem Px=λx. Convergence together with sharp error bounds is obtained. The results are then applied to the solution of the symmetric eigenproblem Ax=λBx, where the algorithms are shown to be improvements over existing techniques.     

On interpolation of interpolating sequences

Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces H^P(σ) and the H^P(σ) interpolating sequences S in the p-spectrum M_p of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence S ⊂ M_p is H^P(σ) interpolating, then it is H^s(σ) interpolating for s < p. In the special case of the unit ball B of ?ⁿ this answers a natural question asked in [8].     

LIE SUPER-BIALGEBRA STRUCTURES ON GENERALIZED SUPER-VIRASORO ALGEBRAS

In this article,Lie super-bialgebra structures on generalized super-Virasoro algebras L are considered.It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if H1(L,LL)=0.     

Martin Boundary of a Fine Domain and a Fatou-Naïm-Doob Theorem for Finely Superharmonic Functions

Let Γ be a graph with the doubling property for the volume of balls and P a reversible random walk on Γ. We introduce H¹ Hardy spaces of functions and 1-forms adapted to P and prove various characterizations of these spaces. We also characterize the dual space of H¹ as a BMO-type space adapted to P. As an application, we establish H¹ and H¹- L¹ boundedness of the Riesz transform.     

Estimates in the Hardy-Sobolev space of the annulus and stability result

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space H ^k,∞; k ∈ ?* of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L.Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S.Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces H ^k,∞, C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.     

Equivariant gerbes over compact simple Lie groups

Using groupoid S¹-central extensions, we present, for a compact simple Lie group G, an infinite dimensional model of S¹-gerbe over the differential stack G/G whose Dixmier–Douady class corresponds to the canonical generator of the equivariant cohomology H_G³(G). To cite this article: K. Behrend et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Factorization of symmetric singular M-matrices

Suppose A is a symmetric, singular M-matrix. A sufficient condition for A to have a triangular, singular M-matrix factorization is given, and it is shown that PAP^T always has such a factorization for a particular permutation matrix P.     

Spectre et homologie des variétés hyperboliques complexes de congruenceSpectrum and homology of congruence complex hyperbolic manifolds

In the first part of this Note, we show that the first non-zero eigenvalue of the Laplace operator on 1-forms of a standard congruence arithmetic complex hyperbolic n-manifold is always $\text{?}\text{10n?11}\text{25}$. The following parts of this Note concern homological applications of this result. We prove, in particular, that if Sh⁰H?Sh⁰G are two Shimura varieties of type U(n?1,1) and U(n,1), the natural map H_2n?3(Sh⁰H)→H_2n?3(Sh⁰G) is injective, first step of a “Lefschetz theorem” for Shimura varieties. To cite this article: N. Bergeron, L. Clozel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 995–998.     

Groups generated by analytic Schrödinger operators

If the potential in a two-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has an analytic continuation H(φ) which is not normal. In case the potential is local and belongs to suitable $\text{L}$^p-spaces, there is a bounded operator P(0, φ) projecting onto the continuous subspace of H(φ). This paper shows that P(0, φ) H(φ) e^?2iφ generates a strongly differentiable group. It is proved that P(0, φ) H(φ) is spectral, and details of the spectral projection operators are presented. The reasoning is based on the Paley-Wiener theorem for functions in a strip. It applies to larger systems provided the resolvent of the multiparticle operator H(φ) satisfies certain regularity conditions that come from the theory of smooth operators. There are no smallness conditions on the potential.     

Extrapolation cascadic multigrid method on piecewise uniform grid

The triangular linear fnite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed.Global superconvergence in discrete H1-norm and global extrapolation in discrete L2-norm are proved.Based on these global estimates the conjugate gradient method(CG)is efective,which is applied to extrapolation cascadic multigrid method(EXCMG).The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.