Remarks on Propagation of Singularities in Thermoelasticity

The propagation of high order weak singularities for the system of homogeneous thermoelasticity in one space variable is studied by using paralinearization and a new decoupling technique introduced by the author (Microlocal analysis in nonlinear thermoelasticity, to appear). For the linear system, one shows that the nonsmooth initial data for the parabolic part lead to singularities in the hyperbolic part of solutions, even when the initial data for that part are identically zero. Both the Cauchy problem and the problem inside of a domain for the semilinear system are considered. It is shown that the propagation of high order singularities is essentially dominated by the hyperbolic operator in the system of thermoelasticity.     

解析函数在本性奇点的一些研究

用正规族理论,研究解析函数在本性奇点附近性质,得到解析函数及其导数在孤立本性奇点附近的一些结果.     

On the asymptotic behaviour of the discrete spectrum in buckling problems for thin plates

We consider the buckling problem for a family of thin plates with thickness parameter ε. This involves finding the least positive multiple λ of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non‐compact operator. We show that under certain assumptions on the load, we have λ = ??(ε²). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non‐compact operator. We provide numerical computations illustrating some of our theoretical results. Copyright © 2005 John Wiley & Sons, Ltd.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693     

Analytic Wave Front Set for Solutions to Schrödinger Equations II – Long Range Perturbations

This paper is a continuation of [9 Martinez , A. , Nakamura , S. , Sordoni , V. ( 2009 ). Analytic wave front set for solutions to Schrödinger equations . Adv. Math. 222 : 1277 – 1307 .[Crossref], [Web of Science ®] , [Google Scholar]], where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of [9 Martinez , A. , Nakamura , S. , Sordoni , V. ( 2009 ). Analytic wave front set for solutions to Schrödinger equations . Adv. Math. 222 : 1277 – 1307 .[Crossref], [Web of Science ®] , [Google Scholar]] to long-range perturbations (in particular, we can allow potentials growing like ?x?^2?? at infinity). More precisely, we construct a modified quantum free evolution G ₀(?s, hD _z ) acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution e ^?itH u ₀ of the Schrödinger equation, in terms of the semiclassical exponential decay of G ₀(?th ^?1, hD _z )T u ₀, where T stands for the Bargmann-transform. The result is valid for t < 0 near the forward non trapping points, and for t > 0 near the backward non trapping points. It is an extension of [12 Nakamura , S. ( 2009 ). Semiclassical singularities propagation properties for the Schrödinger equations . J. Math. Soc. Japan 61 : 177 – 211 . [Google Scholar]] to the analytic framework.     

Simple Cohen–Macaulay Codimension 2 Singularities

In this article, we provide a complete list of simple isolated Cohen–Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.     

算子的Kato本质谱与Browder本质谱的连通分支

利用算子谱的精密结构的分析方法,给出算子S和T拟相似时Kato本质谱σK(S)的一个连通分支与σK(T)相交的充分条件和充要条件,同时证明了Browder本质谱σB(S)的每一个连通分支与Kato本质谱σK(T)的某些子集的交集是非空的.     

Resolution of singularities of real-analytic vector fields in dimension three

Let χ be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of χ can be made elementary by a finite number of blowing-ups in the ambient space. This work has been partially supported by the CNPq/Brasil Grant 205904/2003-5 and Fapesp Grant 02/03769-9.     

A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : W_r W, obtained as a composition of monoidal transformations,so that if X_r W_r denotes the strict transform of X W then: (1) the morphism : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

解析Toeplitz代数的本质换位及其相关问题

在本文中,我们决定出多复变Ｈａｒｄｙ空间Ｈ２上解析Ｔｏｅｐｌｉｔｚ代数的本质换位．即一个算子与所有解析Ｔｏｅｐｌｉｔｚ算子本质可换,当且仅当它是符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的紧扰动．由此,符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（Ａｃ）在Ｃａｌｋｉｎ代数中的像是极大可换闭代数,这导致了Ｌ．Ｃｏｂｕｒｎ正合列的极大扩充．从这个事实,证明了符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的本质谱是连通的,这大大改进了Ｃ－Ｓ最近的工作．从本文的主要定理,证明了Ｔｏｅｐｌｉｔｚ代数Ｆ（Ｌ∞）的本质换位和本质中心是由符号属于ＱＣ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（ＱＣ）,这些结果又导致了对代数Ｆ（Ｈ∞）＋Ｋ自同构群的确定．     

A Characterization of Quasi-Homogeneous Purely Elliptic Complete Intersection Singularities

It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. In this paper we prove that this characterization extends to isolated purely elliptic complete intersection singularities, with bounds on neither the embedding codimension nor the dimension of the singularity.     

算子的左本质谱与Browder本质谱

设算子S和T拟相似,应用算子谱的精密结构的分析,证明了Browder本质谱σB(S)的连通分支与σB(T)的某些子集的相交关系以及左本质谱σle(S)的连通分支与本质谱σe(T)的某些子集的相交关系,给出左本质谱σle(S)的连通分支与σle(T)相交的充分条件和充要条件.     

Perturbation of semi-Browder operators and stability of Browder's essential defect and approximate point spectrum

In the present paper we investigate the stability of closed densely defined semi-Browder operators under operator perturbations that belong to a perturbation class related to compact operators. Furthermore, we apply the obtained results to give a characterization and to study the stability of Browder's essential approximate point spectrum and Browder's essential defect spectrum.     

一类无界算子矩阵的本质谱

本文研究了次对角占优的无界算子矩阵M=(ABCD)的左本质谱和本质谱.利用分析方法和分块算子的性质,得到了整个算子矩阵的本质谱(左本质谱)与其内部元素的本质谱(左本质谱)之间的关系.     

一类具指数系数的对称微分算子谱的离散性

研究了具指数函数系数的2n阶实系数微分算式生成的对称微分算子,利用算子的直和分解法及不等式估计得到此类微分算子谱是离散的充分条件.     

Analytic wave front set for solutions to Schrödinger equations

This paper is a continuation of [A. Martinez, S. Nakamura, V. Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. LIX (2006) 1330–1351], where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian H₀ on . In this paper, we consider short-range type perturbations H of the Laplacian on , and we characterize the analytic wave front set of the solution to the Schrödinger equation: e^−itHf, in terms of that of the free solution: e^−itH₀f, for t<0 in the forward non-trapping region. The same result holds for t>0 in the backward non-trapping region. This result is an analytic analogue of results by Hassel and Wunsch [A. Hassel, J. Wunsch, The Schrödinger propagator for scattering metrics, Ann. of Math. 162 (2005) 487–523] and Nakamura [S. Nakamura, Wave front set for solutions to Schrödinger equations, J. Funct. Anal. 256 (2009) 1299–1309].