PHANTOM MAPS AND SPACES OF THE SAME n-TYPE FOR ALL n

1.IntroductionLetXbeaconnectedCWcomplexandX.denoteitsn-skeleton.Amapf:X-YiscalledaphantommapifitsrestrictiontoeachskeletonXuisnullhomotopic.ItiseasytoknowthateveryphantommapfromafiniteCWcomplexortoaspacewithonlyfinitenootrivialhomotopygroupsisnecessarilytrivialuptohomotopy.HenceessentialphantommapcanoccuronlywhenthedomainXisaninfinitedimensionalspaceorthetargetisaspacewithinfinitenontrivialhomotopygroups.Suchmapsappeartobenullhomotopicfromanumberofdifferentpoilltsofview;e.g.,theyinducethe…     

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

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{\tf S}$^{\bold 1}${\td X}{\tf S}$^{\bold 1}$-INDEX THEORY ON PRODUCT SPACE

A new index is constructed by use of the canonical representation of $S^1\times S^1$ 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.     

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 $n\leq x$.     

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 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.     

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.     

Injective Strong Commutativity Preservers on {\mathcal{T}_{\infty}(F)}

We describe strong commutativity preservers on the algebra of infinite upper triangular matrices over a field F such that char\({(F) \neq 2}\). We show that every such map is some kind of a sum of a few sorts of maps. We also discuss the form of the maps that preserve commutativity in both directions.     

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.     

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.     

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.     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

Localization,completion and detecting equivariant maps on skeletons

In the category of equivariant spaces with base point, we prove the injectivity of the induced map between homotopy sets under some conditions. We study some relations between the localization and the completion. By using these results, we characterize continuous maps which are homotopic on skeletons, and obtain a generalization of the theory of phantom maps.     

Asymptotic solution for EI Ni$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{n} $$o-southern oscillation of nonlinear model

A class of nonlinear coupled system for EI Nino-Southern Oscillation （ENSO） model is considered. Using the asymptotic theory and method of variational iteration, the asymptotic expansion of the solution for ENSO models is obtained.     

Lognormal {\star} -scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.     

The {\bar{\partial}} -Neumann problem on product domains in {\mathbb{C}^{n}}

Let ${\Omega=\Omega_{1}\times\cdots\times\Omega_{n}\subset\mathbb{C}^{n}}$ , where ${\Omega_{j}\subset\mathbb{C}}$ is a bounded domain with smooth boundary. We study the solution operator to the ${\overline\partial}$ -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the ${\overline\partial}$ -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the ${\overline\partial}$ -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.     

$${\cal K}$$-Convexity1 in $$\Re^{n}$$

We generalize the concept of K-convexity to an n-dimensional Euclidean space. The resulting concept of -convexity is useful in addressing production and inventory problems where there are individual product setup costs and/or joint setup costs. We derive some basic properties of -convex functions. We conclude the paper with some suggestions for future research. Support from Columbia University and University of Texas at Dallas is gratefully acknowledged. Helpful comments from Qi Feng are appreciated.     

Compactified Picard stacks over $${\overline{\mathcal M}_g}$$

We study algebraic (Artin) stacks over [`(M)]<sub>g</sub>{\overline{\mathcal M}_{g}} giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get Deligne–Mumford stacks strongly representable over[`(M)]_g{\overline{\mathcal M}_{g}} .     

Extremal effective divisors on \overline{\mathcal M}_{1,n}

For every \(n\ge 3\) , we exhibit infinitely many extremal effective divisors on \(\overline{\mathcal M}_{1,n}\) , the Deligne-Mumford moduli space parameterizing stable genus one curves with \(n\) ordered marked points.