On a characterization of Clarke's tangent cone

For a subsetM of a normed vector spaceX, it is shown that, in the characterization given by Elster and Thierfelder of Clarke's tangent cone toM at a pointx ⁰X, there is a requirement which becomes superfluous whenx ⁰ belongs to the closure ofM.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

On certain classes of curve singulariries with reduced tangent cone

We study a class of rational curves with an ordinary singular point, which was introduced in [GO]. We find some conditions under which the tangent cone is reduced and we show that the tangent cone is not always reduced. We construct another class of rational curves with an ordinary singular point satisfying the condition required in [GO] and whose tangent cone is always reduced.     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

关于切锥与切导数的等价命题

给出集值映射导数的若干等价命题,这有助于加深对集值映射的导数概念的理解,并给使用带来方便.     

On theA-integral representation of the Hilbert transform and conjugate function

Рассматривается воп рос о представлении о ператора Гильберта и сопряжен ной функцииA-интегралом. Доказывается следую щая Теорема. Если ? - такая неотрицательная фун кция на [0, ∞), что х^?1?(х) монотонно не убывает на (0, ∞) и для н екоторого Н> 0 \(\mathop \smallint \limits_H^\infty \varphi ^{ - 1} (x)dx< \infty\) , а определенная на R функ ция fε?∩?(?), то почти всюду оператор Гильберта $$\tilde f(x) = - \frac{1}{\pi }(A)\mathop \smallint \limits_0^\infty \frac{{f(x + t) - f(x - t)}}{t}dt$$ . Из данной теоремы сле дует, что для функций и з ?^p, 1<р<#x221E;, оператор Гильберта и сопряженная функция представляютсяA-инте гралом. Что для функций из ?¹ п одобное утверждение неверно, показывает следующа я теорема. Теорема.Существует т акая суммируемая на R ф ункция f≧0, что почти всюду $$\mathop {\lim sup}\limits_{n \to \infty } \mathop \smallint \limits_0^\infty \left[ {\frac{{f(x + t) - f(x - t)}}{t}} \right]_n dt = \infty$$ .     

Large deviation and the tangent cone at infinity of a crystal lattice

We discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as we shall observe, has remarkable combinatorial features, and shows up also in the Gromov-Hausdorff limit of a crystal lattice.To the memory of our late friend Robert Brooks     

On the depth of the storage access function

We consider the synthesis problem for the storage access function, which often appears as a component of integrated circuits and is also used in theoretical investigations. The precise value of the depth of the storage access function depending on n address variables, where 1 ⩽ n ⩽ 5 or n ⩾ 20, is established over the standard basis under the assumption that the conjunction and disjunction gates have the depth 1 and that the negation gates have the depth 0. For the rest of the values of n the upper and the lower bounds for the specified depth are proven.     

Algebraic properties of the tangent cone to a quasimetric space with dilations

Doklady Mathematics -     

On local cohomology and Hilbert function of powers of ideals

Let I be a homogeneous ideal in a polynomial ring R=K[x₁,,x_m], where K is an infinite field. In this article we investigate the asymptotic behaviour of the a_i-invariants and the regularity index of the ring R/Iⁿ when n»0.Both authors were supported by the Academy of Finland, Project 48556. The first named author was also supported in part by the National Basic Research Program (Vietnam)Mathematics Subject Classification (2000):13D40, 13D45     

Generalized tangent cone and an optimization problem in a normed space

Most abstract multiplier rules in the literature are based on the tangential approximation at a point to some set in a Banach space. The present paper is concerned with the study of a generalized tangent cone, which is a tangential approximation to that set at a common point of two sets. The new notion of tangent cone generalizes previous concepts of tangent cones. This generalized tangent cone is used to characterize the optimality conditions for a simultaneous maximization and minimization problem. The paper is of theoretical character; practical applications are not found so far.     

On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces

Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of P ⁿ with one point as support and with high regularity index.     

A characterization of Hyers-Lang differentiability in terms of a tangent cone

We introduce the notion of Hyers-Lang tangent cone to a subset of a topological vector space, and give a characterization, for the case of locally convex topological vector spaces, of Hyers-Lang differentiability in terms of the Hyers-Lang tangent cone to the graph.     

On the method of tangent hyperbolas

In this study we provide new convergence theorems for the method of tangent hyperbolas under standard Newton kantorovich-type assumptions. We also show how to improve on the results obtained by Safiev, Yamamoto and Kanno.     

On the geometry of tangent bundles

The main aim of this survey paper is to write a detailed and unified presentation of some of the best known results on the geometry of tangent bundles of Riemannian manifolds.     

Separating sets, metric tangent cone and applications for complex algebraic germs

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.     

On the curature of the tangent bundle

Summary The tangent bundle is regarded as an almost product manifold. Connections adapted to this structure are introduced. Use is made of the theory of submersions, Entrata in Redazione il 23 agosto 1968.