On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

Skorohod representation on a given probability space

Let $(\Omega,\mathcal{A},P)Let be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and an arbitrary map, n = 1,2,.... If μ is tight and X _n converges in distribution to μ (in Hoffmann–J?rgensen’s sense), then X∼μ for some S-valued random variable X on . If, in addition, the X _n are measurable and tight, there are S-valued random variables and X, defined on , such that , X∼μ, and a.s. for some subsequence (n _k ). Further, a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X _n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to for some H⊂Ω with P ^*(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken , for some H⊂ (0,1) with outer Lebesgue measure 1, where is the Borel σ-field on (0,1) and m _H the only extension of Lebesgue measure such that m _H (H) = 1. In order to prove the previous results, it is also shown that, if X _n converges in distribution to a separable limit, then X _{n k} converges stably for some subsequence (n _k ).      

Killing spinors on K?hler manifolds

In the paper Kählerian Killing spinors are defined and their basic properties are investigated. Each Kähler manifold that admits a Kählerian Killing spinor is Einstein of odd complex dimension. Kählerian Killing spinors are a special kind of Kählerian twistor spinors. Real Kählerian Killing spinors appear for example, on closed Kähler manifolds with the smallest possible first eigenvalue of the Dirac operator. For the complex projective spaces P ^2l–1 and the complex hyperbolic spaces H ^2l–1 withl>1 the dimension of the space of Kählerian Killing spinors is equal to ( ). It is shown that in complex dimension 3 the complex hyperbolic space H ³ is the only simple connected complete spin Kähler manifold admitting an imaginary Kählerian Killing spinor.     

Normal subgroups of nonstandard symmetric and alternating groups

Let be a nonstandard model of Peano Arithmetic with domain M and let be nonstandard. We study the symmetric and alternating groups S _n and A _n of permutations of the set internal to , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A _n and S _n are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an -valued metric on and (where B _S , B _A are the maximal normal subgroups of S _n and A _n identified earlier) making these groups into topological groups, and by showing that if is -saturated then and are complete with respect to this metric.      

A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇^(α) on a statistical manifold equipped with a pair of conjugate connections and is given as . Here, we develop an expression of curvature R ^(α) for ∇^(α) in relation to those for . Immediately evident from it is that ∇^(α) is equiaffine for any when are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed. The work was conducted when the author was on sabbatical leave as a visiting research scientist at the Mathematical Neuroscience Unit, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan.     

Euler homology

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring of a topological space X. This homology theory Eh _* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh _*, generalizing the equivariant Euler characteristic.     

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces

Let s ^w p be the set of all a ∈ ? such that a ^w (x, D) is Schatten p-operator on L ². Then we prove the following:

• $S(m,g)\subseteq s_p^wLet s ^w p be the set of all a ∈ ℓ such that a ^w (x, D) is Schatten p-operator on L ². Then we prove the following:

  •	iff . Furthermore, when . Consequently, when ;
  •	if , then is symplectically invariantly defined. Moreover, if and is slowly varying (and σ-temperate), then the same is true for G;
  •	a generalization of sharp G?rding's inequality.

  Mathematics Subject Classifications (2000) Primary: 35S05, 47B10, 47L15 Secondary: 32F45, 16W80     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

Extrapolation results for <Emphasis Type="Italic">q</Emphasis> < 1

Given a sublinear operator T such that is bounded, it can be shown that is bounded, with constant C/(1−q), for every 0 < q < 1. In this paper, we study the converse result, not only for sequence spaces, but for general measure spaces proving that, if T : L ^q (μ) → X is bounded, with constant C/(1−q), for every and X is Banach, then T : L log (1/L)(μ) → X is bounded. Moreover, this result is optimal. We also show that things are quite different if the Banach condition on X is dropped. This work has been partially supported by MTM2004-02299 and by 2005SGR00556.     

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., ). We are interested to know under what conditions “typical norm” of the random matrix is of order of 1. An evident necessary condition is , which, essentially, translates to ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is : for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints , where F is quadratic in X = (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices . Research was partly supported by the Binational Science Foundation grant #2002038.     

Evolution of Yamabe constant under Ricci flow

In this note under a crucial technical assumption, we derive a formula for the derivative of Yamabe constant , where g(t) is a solution of Ricci flow on closed manifold. We also give a simple application. Mathematics Subject Classifications (2000): 53C21 and 53C44     

On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary , and let be a covering map, where Y is a closed manifold. Let A _C be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace for N sufficiently large has a complete asymptotic expansion as . In particular, the heat trace has a complete asymptotic expansion as , and the -function has a meromorphic extension to .      

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with ${\nabla=D + S}In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with induced by the one-parameter family of connections given by and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-K?hler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt ^*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt ^*-bundles generalised para-pluriharmonic maps into , respectively, into SO ₀(n,n)/U ^π(C ⁿ ), where U ^π(C ⁿ ) is the para-complex analogue of the unitary group.      

Second-Order Efficient Test for Inhomogeneous Poisson Processes

We observe a realization X ⁽ⁿ⁾ of a Poisson process on the set with intensity function depending on the unknown real parameter . Based on X ⁽ⁿ⁾ we test simple null hypothesis against one sided alternative for given . We improve the level of the well-known locally asymptotically uniformly most powerful (LAUMP) test by using the Edgeworth type expansion for stochastic integral. We show that the improved test is second-order efficient under certain regularity conditions.      

Differential inclusions for differential forms

We study necessary and sufficient conditions for the existence of solutions in of the problem where is a given set. Special attention is given to the case of the curl (i.e. k = 1), particularly in dimension 3. Some applications to the calculus of variations are also stated.     

σk-Scalar curvature and eigenvalues of the Dirac operator

On a four-dimensional closed spin manifold (M ⁴, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ₂-scalar curvature of M ⁴ as follows: Equality implies that (M ⁴, g) is a round sphere and the corresponding eigenspinors are Killing spinors.Dedicated to Professor Wang Guangyin on the occasion of his 80th birthday.     

Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X E and C F, two multifunctions Γ : X → 2 ^X and Ф : X → 2 ^C , and a single-valued map ψ : , find a pair such that , Ф and for all . We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Ф. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213–228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E ^* and ).