CLIFFORD MARTINGALES Φ-EQUIVALENCE BETWEEN S（f） AND f^＊

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GLOBAL TOPOLOGICAL PROPERTIES OF HOMOGENEOUS VECTOR FIELDS IN {\tf R}$^{^{\hbox{\normal 3}}}$

In this paper, the authors prove that the flows of homogeneous vector field $Q(x)$ at infinity are topologically equivalent to the flows of the tangent vector field $Q_T(u)$ $(u\in S^2)$ on the sphere $S^2$, and show the theorems for the global topological classification of $Q(x)$. They derive the necessary and sufficient conditions for the global asymptotic stability and the boundedness of vector field $Q(x),$ and obtain the criterion for the global topological equivalence of two homogeneous vector fields.     

PHANTOM MAPS AND SPACES OF THE SAME {\tf n}-TYPE FOR ALL {\tf n}

After reviewing the theory of phantom maps and SNT, the author gives several general results which relate the theory of phantom maps and SNT and which extend that of Harper and Roitberg.     

DARBOUX EQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITH NONNEGATIVE CURVATURE IN {\tf R}$^{^{\hbox{\td 3}}}$

The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate on the boundary is contained. As some applications the smooth isometric embeddings of positively and nonnegatively curved disks into $R^3$ are constructed.     

DENSENESS OF RADIAL-BASIS FUNCTIONS IN L^2（R^n） AND ITS APPLICATIONS IN NEURAL NETWORKS

The authors discuss problems of approximation to functions in L2(Rn) and operators fromL2(Rn1) to L2(Rn2) by Radial-Basis Functions. The results obtained solve the problem ofcapability of RBF neural networks, a basic problem in neural networks.     

THE FUNCTION {\tf d}$_{{\boldkey k}}$({\tf n}) AT CONSECUTIVE INTEGERS

Let $d_{k}(n)$ denote the $k$-fold iterated divisor function $(k\geq 2)$. It is proved that for sufficiently large $x$, $d_{k}(n)=d_{k}(n+1)$ holds for $\gg x(\log\log x)^{-3}$ integers $n\leq x$.     

Signature and Spectral Flow of J-Unitary {\mathbb{S}^1} -Fredholm Operators

Bijective operators conserving the indefinite scalar product on a Krein space ${(\mathcal{K}, J)}$ are called J-unitary. Such an operator T is defined to be ${\mathbb{S}^1}$ -Fredholm if T?z 1 is Fredholm for all z on the unit circle ${\mathbb{S}^1}$ , and essentially ${\mathbb{S}^1}$ -gapped if there is only discrete spectrum on ${\mathbb{S}^1}$ . For paths in the ${\mathbb{S}^1}$ -Fredholm operators an intersection index similar to the Conley–Zehnder index is introduced. The strict subclass of essentially ${\mathbb{S}^1}$ -gapped operators has a countable number of components which can be distinguished by a homotopy invariant given by the signature of J restricted to the eigenspace of all eigenvalues on ${\mathbb{S}^1}$ . These concepts are illustrated by several examples.     

Rigidity of certain polyhedra in $ {\bold R}^3 $

We extend the Cauchy theorem stating rigidity of convex polyhedra in . We do not require that the polyhedron be convex nor embedded, only that the realization of the polyhedron in be linear and isometric on each face. We also extend the topology of the surfaces to include the projective plane in addition to the sphere. Our approach is to choose a convenient normal to each face in such a way that as we go around the star of a vertex the chosen normals are the vertices of a convex polygon on the unit sphere. When we can make such a choice at each vertex we obtain rigidity. For example, we can prove that the heptahedron is rigid. Received: March 3, 1999; revised: December 7, 1999.     

THE EXISTENCE OF {\tf J}-HOLOMORPHIC CURVES AND APPLICATIONS TO THE WEINSTEIN CONJECTURE

The author first proves the existences of $J-$holomorphic curves in the symplectizations of Legendre fibrations and then as an application confirms the Weinstein conjectures on contact manifolds of Legendre fibrations. As a corollary, a new proof on the theorem due to Hofer, Viterbo, Gluck, Ziller, Weinstein and Ljusternik-Fet Theorem is provided, which is quite different from their original proofs.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

Strict {\Pi^1_1}-reflection in bounded arithmetic

We prove two conservation results involving a generalization of the principle of strict P¹₁{\Pi^1_1}-reflection, in the context of bounded arithmetic. In this context a separation between the concepts of bounded set and binary sequence seems to emerge as fundamental.     

Period functions for $ {\mathcal{C}^0} $- and $ {\mathcal{C}^1} $-flows

Let F:M ×\mathbbR ? M {\mathbf{F}}:M \times \mathbb{R} \to M be a continuous flow on a manifold M, let V ⊂ M be an open subset, and let x:V ? \mathbbR \xi :V \to \mathbb{R} be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is topologically conjugate to some C¹ {\mathcal{C}^1} -flow. It is shown in this paper that, in this case, the period functions of F satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.     

The spin $^{\bf c}$ Dirac operator on high tensor powers of a line bundle

We study the asymptotic of the spectrum of the Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10]. Received: 6 June 2001; in final form: 18 September 2001 / Published online: 28 February 2002     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

Isometric Embeddings Between the Near Polygons {\mathbb{H}_n} and {\mathbb{G}_n}

Let ${n \in \mathbb{N}\backslash \{0, 1, 2\}}$ . We prove that there exists up to equivalence one and up to isomorphism (n+1)(2n+1) isometric embeddings of the near 2n-gon ${\mathbb{H}_n}$ into the near 2n-gon ${\mathbb{G}_n}$ .     

Torified varieties and their geometries over {\mathbb{F}_1}

This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over ${\mathbb Z}$ that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes?CConsani and an object in the sense of Soulé and show that both are varieties over ${\mathbb{F}_1}$ in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over ${\mathbb{F}_1}$ in the literature so far. Furthermore, we compare Connes?CConsani??s geometry, Soulé??s geometry and Deitmar??s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over ${\mathbb{F}_1}$ in the given categories.     

S^1*S^1-INDEX THEORY ON PRODUCT SPACE

A new index is constructed by use of the canonical representation of S1 x S1 group over a product space. This index satisfies the general properties of the usual index but does not satifsy the dimension property. As an application, two abstract critical point theorems are given.     

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.     

The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.