A Relative Filtration Index and Fibers of Normal Primes in Extensions of Finite Type

For an extension R A of commutative Noetherian rings the behavior of the associated morphism of topological spaces Spec A Spec R is often measured by its behavior on each of its fibers. Specifically, one studies the 'splitting' (or 'branching') and the 'ramification' that occurs in each fiber. In the classical constructions of faithfully flat analytic extensions (e.g., completion or Henselization) of excellent local rings the splitting and ramification properties are fairly well understood; see EGA IV [6, 18.10], Nagata [13, Sect. 37] or Raynaud [15, Ch. IX]. The strongest results are usually achieved for fibers over a 'normal point' of Spec R, that is, over p Spec R such that R/p is a normal domain [e.g., the property of a normal prime p in a local ring to be 'unibranched', i.e., the Henselization of R/p is a (normal) domain].     

Analytic deviation of ideals and intersection theory of analytic spaces

In this note a stratification of analytic subspaces by analytic deviation of ideals is introduced and applied to define embedded components of an intersection (not necessarily proper) of complex analytic subspaces. Using the method of compact semi-analytic Stein neighbourhoods, a pointwise defined intersection multiplicity is proved to be constant along a dense Zariski open subset of such embedded components. For complex subspaces of a projective space one obtains precisely the distinguished varieties of intersection in the sense of Fulton and their intersection numbers.     

Analytic density in Lie groups

A subgroupH of an analytic groupG is said to beanalytically dense if the only analytic subgroup ofG containingH isG itself. The main purpose of this paper is to give sufficient conditions onG (analogous to those of [8], [9], and [7] in the case of Zariski density) which guarantee the analytic density of cofinite volume subgroupsH. First we consider the case of arbitrary cofinite volume subgroups (Theorem 5 and its corollaries). Then we specialize to lattices, and prove the following result (Theorem 8):Let G be an analytic group whose radical is simply connected and whose Levi factor has no compact part and a finite center. Then any lattice in G is analytically dense. In proving this use is made of a result of Montgomery which also implies that for any simply connected solvable group, cocompactness of a closed subgroup implies analytic density. In the case of a solvable group with real roots this means analytic density and cocompactness are equivalent and thus completes a circle of ideas raised in Saito [13]. In Corollary 9 we deal with a related local condition. Finally in Theorem 10 and its corollaries we apply these considerations to prove a homomorphism extension theorem and an isomorphism theorem for 1-dimensional cohomology.     

The presence of symplectic strata improves the Gevrey regularity for sums of squares

We consider a class of operators of the type sum of squares of real analytic vector fields satisfying the Hörmander bracket condition. The Poisson-Treves stratification is associated to the vector fields. We show that if the deepest stratum in the stratification, i.e., the stratum associated to the longest commutators, is symplectic, then the Gevrey regularity of the solution is better than the minimal Gevrey regularity given by the Derridj-Zuily theorem.     

On the variety of rational curves inP n

Summary In this note one defines a stratification of the variety of rational curves inP ⁿ of given degree in terms of the decomposition of the normal bundle (see [E-vdV], [G-S] for the casen=3). The strata are showed to be irreducible and their dimension is computed.     

On the singularities of weakly normal varieties

Among the weakly normal varieties (in the sense of Andreotti and Bombieri, [1]) are of particular interest those varieties such that the normalization morphism is unramified outside a subvariety of codimension not less than 2. We describe the singularities of these varieties (called here WN1) by means of analytic equations, tangent cones, analytic branches and we show that any irreducible projective variety is birationally equivalent to a WN1 hypersur face and that a Gorenstein variety is weakly normal if and only if it is WN1.This research was done when the authors were members of G.N.S.A.G.A. of the C.N.R.     

Deformations and contractions of algebraic structures

We describe the basic notions of versal deformation theory of algebraic structures and compare it with the analytic theory. As a special case, we consider the notion of versal deformation used by Arnold. With the help of versal deformation we get a stratification of the moduli space into projective orbifolds. We compare this with Arnold’s stratification in the case of similarity of matrices. The other notion we discuss is the opposite notion of contraction.     

Analytic stratifications and the cut locus of a class of distance functions

We study the analytic singularities of viscosity solutions of equations of eikonal type and obtain that the analytic singular support of these functions has an analytic stratification. The singular support can be identified with the cut locus of the distance to the boundary of an open set, when the interior is equipped with a degenerate Riemannian metric. We apply the result to elliptic equations as well as to model operators of Grušin type.     

A maximal inequality for stochastic convolution integrals on hilbert spaces and space-time regularity of linear stochastic partial differential equations

We consider optimal control problems for one-dimensional diffusion processes [ILM0001] where the control processes υ_t are increasing, positive, and adapted. Several types of expected cost structures associated with each policy υ(.) are adopted, e.g. discounted cost, long term average cost and time average cost. Our work is related to [2,6,12,14,16 and 21], where diffusions are allowed to evolve in the whole space, and to [13] and [20], where diffusions evolve only in bounded regions. We shall present some analytic results about value functions, mainly their characterizations, by simple dynamic programming arguments. Several simple examples are explicitly solved to illustrate the singular behaviour of our problems.     

Ordinary differential operator and some of its applications to certain meromorphically p-valent functions

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out.     

KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω ₀ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω ₀ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω ₀. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.     

On some applications of the Briot-Bouquet differential subordination

Recently Srivastava et al. [J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003) 7-18; J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1-13; Y.C. Kim, H.M. Srivastava, Fractional integral and other linear operators associated with the Gaussian hypergeometric function, Complex Var. Theory Appl. 34 (1997) 293-312] introduced and studied a class of analytic functions associated with the generalized hypergeometric function. In the present paper, by using the Briot-Bouquet differential subordination, new results in this class are obtained.     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

A separation theorem for the hyperext

Summary We introduce, on the hyperext spaces associated to complexes of sheaves with coherent cohomology on an analytic space, a natural topology that extends the natural topology defined by Verdier on the hypercohomology (cf. [13]). For a holomorphically convex space we prove that this topology is separated, which generalizes a result of Ramis [10].

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Riassunto Si introduce, sugli spazi iperext associati ai complessi di fasci a coomologia coerente su uno spazio analitico, una topologia naturale che estende la topologia naturale definita da Verdier sull'ipercoomologia (cf. [13]). Per uno spazio olomorficamente convesso si dimostra che questa topologia è separata e generalizza un risultato di Ramis [10].

     

On Flatness Properties of Torsion Free Right Rees Factor Acts

In this paper, we investigate the monoids over which all torsion free right Rees factor acts satisfy some properties that follow from projectivity(such as (weak) flatness, strong flatness, condition (P), etc.). These results answer the questions in [1].     

Weierstrass formula and zero-finding methods

Summary. Classical Weierstrass' formula [29] has been often the subject of investigation of many authors. In this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions. We are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros. Conditions for the safe convergence of Weierstrass' method, depending only on initial approximations, are given. In particular, we study polynomials with interval coefficients. Using an interval version of Weierstrass' method enclosures in the form of disks for the complex-valued set containing all zeros of a polynomial with varying coefficients are obtained. We also present Weierstrass-like algorithm for approximating, simultaneously, all zeros of a class of analytic functions in a given closed region. To demonstrate the proposed algorithms, three numerical examples are included. Received September 13, 1993     

FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand [10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch [1]), improving the constantC _0,∞ of the results of Hargé-Lebeau [13] and Sjöstrand-Zworski [13] valid in C^∞ The works [3], [13] and [10] feature an adapted complex scaling for convex obstacles, but in [10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.     

The local flow-box theorem for discretizations the analytic case

The discretization counterpart of the C^w local flow-box theorem, a C^w normal form result for one-step discretizations of ordinary differential equations in the vicinity of nonequilibria is presented. The very same problem in the less smoother function class c^k, k∞ has been investigated in [lo]. The remaining analytic case requires completely different techniques. The proof is based on the parametrized version of a Nash-Moser type implicit function theorem by Belitskii and Tkachenko [5,6]. Connections to results on structural stability under discrctization and backward error analysis are also investigated.