Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions

We consider a system of the form in Ω with Neumann boundary condition on ∂Ω, where Ω is a smooth bounded domain in and f,g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as ε goes to zero, at a point of the boundary which maximizes the mean curvature of the boundary of Ω.     

A Poincaré-Hopf-type theorem for holomorphic one-forms

We prove that if a holomorphic one-form Ω in a neighborhood of a closed euclidian ball , in the n-dimensional complex affine space, defines a distribution transverse to the boundary sphere , then n is even and Ω admits a sole singularity q∈B²ⁿ. Moreover, this singularity is simple.     

Analytic mappings between noncommutative pencil balls

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems (Helton et al. (2009) [10], de Oliviera et al. (2009) [8]). In the earlier paper (Helton et al. (2009) [9]) we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. In this paper we turn to a more general dimension-free ball BL, called a “pencil ball”, associated with a homogeneous linear pencil

     

Stable manifolds associated to fixed points with linear part equal to identity

We consider maps defined on an open set of having a fixed point whose linear part is the identity. We provide sufficient conditions for the existence of a stable manifold in terms of the nonlinear part of the map.These maps arise naturally in some problems of Celestial Mechanics. We apply the results to prove the existence of parabolic orbits of the spatial elliptic three-body problem.     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

The L resolvents of second-order elliptic operators of divergence form under the Dirichlet condition

Let A be the 2mth-order elliptic operator of divergence form with bounded measurable coefficients defined in a domain Ω of . For 1<p<∞ we regard A as a bounded linear operator from the L^p Sobolev space to H−m,p(Ω). It is known that when , we can construct the resolvent (A−λ)⁻¹ and estimate its operator norm for some λ if the leading coefficients are uniformly continuous. In this paper, we try to extend this result to a general domain. It is successful when m=1 if Ω is the half-space or a domain with C² bounded boundary. For m>1 it is shown that the problem is reduced to the case where Ω is the half-space and A is a homogeneous operator with constant coefficients. We also give a perturbation theorem.     

A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representationr : ∪ Ω → ^ν, where is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space , or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω : → [0, 1]. The value that ω ∈ Ω takes ina ∈ is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a ∨ b) of the join of two orthogonal elementsa, b ∈ is equal tor(a) + r(b). The convex closure of the representation of Σ, the set of atoms of , is equal to the representation of Ω.     

The converse theorem for Minkowski's inequality

Let (Ω, Σ, μ) be a measure space and ?, ψ : (0, ∞) → (0, ∞) some bijective functions. Suppose that the functional _?,ψ defined on class of μ-integrable simple functions χ : Ω → [0, ∞), μ({? : χ(?) > 0} > 0, by the formula

     

Linear sufficiency in a general growth curve model

The notion of linear sufficiency for the whole set of estimable functions in the general Gauss-Markov model is extended to the estimation of any special set of estimable functions in a general growth curve model. Some general results with respect to the concept of linear sufficiency are obtained, from which a necessary and sufficient condition is established for a linear transformation, {F₁,F₂}, of the observation matrix Y to have the property that there exists a linear function of which is the BLUE of the estimable functions .     

Distributions and measures on the boundary of a tree

In this paper, we analyze the space of distributions on the boundary Ω of a tree and its subspace , which was introduced in [Amer. J. Math. 124 (2002) 999-1043] in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if , then μ is a measure which is absolutely continuous with respect to the natural probability measure λ on Ω, but on the other hand there are measures absolutely continuous with respect to λ which are not in . We then give the definition of an absolutely summable distribution and prove that a distribution can be extended to a complex measure on the Borel sets of Ω if and only if it is absolutely summable. This is also equivalent to the condition that the distribution have finite total variation. Finally, we show that for a distribution μ, Ω decomposes into two subspaces. On one of them, a union of intervals A_μ, μ restricted to any finite union of intervals extends to a complex measure and on A_μ we give a version of the Jordan, Hahn, and Lebesgue-Radon-Nikodym decomposition theorems. We also show that there is no interval in the complement of A_μ in which any type of decomposition theorem is possible. All the results in this article can be generalized to results on good (in particular, compact infinite) ultrametric spaces, for example, on the p-adic integers and the p-adic rationals.     

Conditionally positive definite kernels on Euclidean domains

This paper characterizes several classes of conditionally positive definite kernels on a domain Ω of either or . Among the classes is that composed of strictly conditionally positive definite kernels. These kernels are known to be useful in the solution of variational interpolation problems on Ω. Our study covers the case in which Ω is the sphere S^l−1 of or a similar manifold. Among other things, our results imply that the characterization of (strict) conditional positive definiteness on Ω can be obtained from a characterization of (strict) positive definiteness on Ω. The bi-zonal strictly conditionally positive definite kernels on S^l−1, l?3, are described.     

Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and .     

Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient among functions that vanish in a set contained on the boundary ∂Ω of given boundary measure.We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.     

Multifractal spectrum and generic properties of functions monotone in several variables

We study the singularity (multifractal) spectrum of continuous functions monotone in several variables. We find an upper bound valid for all functions of this type, and we prove that this upper bound is reached for generic functions monotone in several variables. Let be the set of points at which f has a pointwise exponent equal to h. For generic monotone functions f:d[0,1]→R, we have that for all h∈[0,1], and in addition, we obtain that the set is empty as soon as h>1. We also investigate the level set structure of such functions.     

An analogue of Hilbert's 10th problem for fields of meromorphic functions over non-Archimedean valued fields

Let K be a complete and algebraically closed valued field of characteristic 0. We prove that the set of rational integers is positive existentially definable in the field of meromorphic functions on K in the language of rings augmented by a constant symbol for the independent variable z and by a symbol for the unary relation “the function x takes the value 0 at 0”. Consequently, we prove that the positive existential theory of in the language is undecidable. In order to obtain these results, we obtain a complete characterization of all analytic projective maps (over K) from an elliptic curve minus a point to , for any elliptic curve defined over the field of constants.     

Uniqueness results for semilinear polyharmonic boundary value problems on conformally contractible domains. I

We study polyharmonic boundary value problems (−Δ)^mu=f(u), , with Dirichlet boundary conditions on bounded and unbounded conformally contractible domains in . Such domains can be contracted to a point (bounded case) or to infinity (unbounded case) by one-parameter groups of conformal maps. The class of star-shaped domain is a subclass. The problem has variational structure. This allows us to derive a sufficient condition for uniqueness by studying the interaction of one-parameter transformation groups with the underlying functional . If the transformation group strictly reduces the values of then uniqueness of the critical point of follows. The proof is inspired by E. Noether's theorem on symmetries and conservation laws. Applications of the uniqueness principle are given in Part II of this paper.     

Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).     

Non-ergodic maps in the tangent family

We consider maps in the tangent family for which the asymptotic values are eventually mapped onto poles. For such functions the Julia set . We prove that for almost all z ∈ J(f) the limit set w(z) is the post-singular set and f is non-ergodic on J(f). We also prove that for such f does not exist a f-invariant measure absolutely continuous with respect to the Lebesgue measure finite on compact subsets of .     

Chambers of arrangements of hyperplanes and Arrow's impossibility theorem

Let A be a nonempty real central arrangement of hyperplanes and Ch be the set of chambers of A. Each hyperplane H defines a half-space H⁺ and the other half-space H⁻. Let B={+,−}. For H∈A, define a map by (if C⊆H⁺) and (if C⊆H⁻). Define . Let Chm=Ch×Ch×?×Ch (m times). Then the maps induce the maps . We will study the admissible maps which are compatible with every . Suppose |A|?3 and m?2. Then we will show that A is indecomposable if and only if every admissible map is a projection to a component. When A is a braid arrangement, which is indecomposable, this result is equivalent to Arrow's impossibility theorem in economics. We also determine the set of admissible maps explicitly for every nonempty real central arrangement.