On the well-definedness of the order of an ordinary differential equation

It is proved that if the differential equations y ^{( n )}=f(x, y, y′, …, y ^{( n ?1 )}) and y ^{( m )}=g(x, y, y′, …, y ^{( n ?1 )}) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems.     

Ganea comonads

We construct for all topological space X and all n∈N a natural section e ⁿ _X :G ⁿ X→G ⁿ G ⁿ X of the Ganea projection :G ⁿ G ⁿ X→G ⁿ X and show that the triple (G ⁿ ,g ⁿ ,e ⁿ ) is a comonad on Top ^*. Received: 6 March 2000     

The pinched Veronese is Koszul

In this paper we prove that the coordinate ring of the pinched Veronese, k[X ³,X ² Y,XY ²,Y ³,X ² Z,Y ² Z,XZ ²,YZ ²,Z ³], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.     

Cylindricity of tangent bundles of strongly parabolic metrics and strongly parabolic surfaces

It is proved that if the intrinsic zero-index of the Sasaki metric of a tangent bundleTM ⁿ isk, thenk is even andM ⁿ is the metric product of a Riemannian manifoldM ^n–k/2 by a Euclidean spaceE ^k/2, whileTM ⁿ is the metric product ofTM ^n–k/2 byE ^k . An expression is obtained for the second fundamental forms of the imbeddingTF ^l TM ⁿ in terms of the second fundamental forms of the imbeddingF ^l M ⁿ and the curvature tensor ofM ⁿ . It is proved thatTF ^l is totally geodesic inTM ⁿ if and only ifF ^l is totally geodesic inM ⁿ .Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 12–32.     

Stability of semilinear ill-posed problems with a prescribed energy bound

In the present paper we discuss the stability of semilinear problems of the form A^αu + G^α(u) = ? under assumption of an a priori bound for an energy functional E^α(u) ? E, where α is a parameter in a metric space M. Following [11] the problem A^αu + G^α(u) = ?, E^α(u) ? E is called stable in a Hilbert space H at a point α ? M if for any ??H, E, ? > 0 there exists δ > 0 such that for any functions u^α1, u^α2 satisfying A^αju^αj + G^αj(u^αj) = ?^αj, E^αj(u^αj) ? E, j = 1,2 we have ‖u^α1 ? u^α2_H ? ? provided ρ_M(α_j, α) ? δ, ‖?^αj ? ?‖_H ? δ, j = 1,2. In the present paper we obtain stability conditions for the problem A^αu + G^α(u) = ?, E^α(u) ? E.     

关于环Fpm+uFpm+u2Fpm上循环码的注记

本文研究了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码分类.通过建立环F_p^m+uF_p^m+u²F_p^m到环F_p^m+uF_p^m的同态,给出了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码的新分类方法.应用这种方法,得到了环F_p^m+uF_p^m+u²F_p^m长度为p^s的循环码的码词数.     

Dense Subsets of H1/2(S2, S1)

We prove that the maps from S ² intoS ¹ having a finite number of isolated singularities ofdegree ±1 are dense for the strong topology inH ^1/2(S ², S ¹). We also prove that smooth maps are densein H ^1/2(S ², S ¹)for the sequentially weak topology andthat this is no more the case in H ^s (S ², S ¹) for s> 1/2.     

The space Ω m p (R d ) and some properties

Let m be a v-moderate function defined on R ^d and let g ∈ L ²(R ^d ). In this work, we defineΩ _m ^p (R ^d ) to be the vector space of f ∈ L _m ² (R ^d ) such that the Gabor transform V ^f belongs to L ^p (R ^2d ), where 1 ≤ p < ∞. We equip it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of Ω _m ^p (R ^d ). We also discuss inclusion properties and obtain the dual space of Ω _m ^p (R ^d ). At the end of this work, we study multipliers from L _w ¹ (R ^d ) into Ω _w ^p (R ^d ) and from Ω _w ^p (R ^d ) into L _w−1 ^∞ (R ^d ), where w is the Beurling weight function. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 139–145, January, 2006.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

NONLINEAR HOMOTOPY AND PRE-POINCARÉ LEMMAS

Abstract

Most homotopies considered in the literature are linear homotopies of the form h ⁱ (λ) = λx ⁱ + (1—λ)y ⁱ , 0 ≤ λ ≤ 1. Although these prove to be adequate in most instances, they lack direct geometric significance because {h ⁱ (λ) | 0 ≤ λ ≤ 1} are not orbits of a vector field. On the other hand, the nonlinear homotopy g ⁱ (s) = e ^s x ⁱ + (1—e ^s )y ⁱ ,—∞ ≤ s ≤ 0, are orbits of a vector field (i.e., dg ⁱ /ds = g ⁱ —y ⁱ , g ⁱ (0) = x ⁱ ), and thus have direct geometric significance. This suggests that useful results can be obtained by replacing linear homotopy by transport along flows of smooth vector fields. The purpose of this paper is to elaborate on this simple idea. We define prehomotopy operators induced by vector fields on a manifold. These allow us to obtain finite transport relations and pre-Poincaré lemmas that generalize the classical results. They are shown to reproduce the classical results as asymptotic limits and to obtain representations of all solutions of complete systems of exterior differential equations on a star shaped region of a manifold.     

Aggregate and fractal tessellations

Consider a sequence of stationary tessellations {Θ ⁿ }, n=0,1,…, of ℝ ^d consisting of cells {C ⁿ (x _i ⁿ )}with the nuclei {x _i ⁿ }. An aggregate cell of level one, C ₀ ¹(x _i ⁰), is the result of merging the cells of Θ¹ whose nuclei lie in C ⁰(x _i ⁰). An aggregate tessellation Θ₀ ⁿ consists of the aggregate cells of level n, C ₀ ⁿ (x _i ⁰), defined recursively by merging those cells of Θ ⁿ whose nuclei lie in C ^{n −1}(x _i ⁰). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as n→∞ and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {Θ ⁿ }. Received: 3 June 1999 / Revised version: 22 November 2000 / Published online: 24 July 2001     

Compensated Compactness, Paracommutators, and Hardy Spaces

LetB₁: ⁿ× ^N₁→ ^m₁,B₂: ⁿ× ^N₂→ ^m₂andQ: ^m₂→ ^m₁be bilinear forms which are related as follows: ifμandνsatisfyB₁(ξ, μ)=0 andB₂(ξ, ν)=0 for someξ≠0, thenμ^τQν=0. Supposep⁻¹+q⁻¹=1. Coifman, Lions, Meyer and Semmes proved that, ifuL^p( ⁿ) andvL^q( ⁿ), and the first order systemsB₁(D, u)=0,B₂(D, v)=0 hold, thenu^τQvbelongs to the Hardy spaceH¹( ⁿ), provided that both (i)p=q=2, and (ii) the ranks of the linear mapsB_j(ξ, ·) : ^N_j→ ^m₁are constant. We apply the theory of paracommutators to show that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition whenp=q=2 involves the use of a deep result of Lojasiewicz from singularity theory.     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·)

We study the Riesz potentials I_αf on the generalized Lebesgue spaces L^p(·)(?^d), where 0 < α < d and I_αf(x) ? ∫equation/tex2gif-inf-3.gif |f(y)| |x – y|^{α – d} dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that I_α : L^p(·)(?^d) → L^p?(·)(?^d), where . If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to on ?^d such that there exists a bounded linear extension operator ? : W^1,p(·)(Ω) ? (?^d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings W^k,p(·)(?^d) ?L^p*(·)(R^d) with and W^1,p(·)(Ω) ? L^p*(·)(Ω) for k = 1. We show compactness of the embeddings W^1,p(·)(Ω) ? L^q(·)(Ω), whenever q(x) ≤ p*(x) – ε for some ε > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

On the integral of mean curvature of a flattened convex body in space forms

Under the assumptions that E _λ ⁿ is an n-dimensional, simply connected Riemannian manifold of constant sectional curvature λ and L _λ ^r is an r-dimensional, totally geodesic submanifold of E _λ ⁿ , the paper investigates the q-th integral of the mean curvature M _q ⁿ of a convex body K ^r in E _λ ⁿ and gives the expression of M _q ⁿ in the terms of M _p ^r , where M _p ^r is the p-th integral of the mean curvature of K ^r > in L _λ ^r . A result of L. A. Santaló [2] holds in particular.     

On the Cardinalities of the Row spaces of Non-full Rank Boolean Matrices

n -2 integers 2 ^{n -2}+2 ^{n -3}+2 ^s, where s=0,1,2,..., n-3, in the interval (2 ^{n -2}+2 ^{n -3},2 ^{n -1}] such that these integers are the cardinalities of row spaces R(A) of non-full rank Boolean matrices A of order n. We also show that for each s, where s=0,1,2,..., n-3, there exists A epsilon B _n such that A is non-full rank and the cardinality of R(A) equals 2 ^{n -2}+2 ^{n -3}+2 ^s.     

Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L...     

Regularity properties of isometric immersions

We show that an isometric immersion y from a two-dimensional domain S with C^1,α boundary to ℝ³ which belongs to the critical Sobolev space W^2,2 is C¹ up to the boundary. More generally C¹ regularity up to the boundary holds for all scalar functions V ∈ W^2,2(S) which satisfy det ∇²V=0. If S has only Lipschitz boundary we show such V can be approximated in W^2,2 by functions V_k ∈ W^1,∞∩W^2,2 with det ∇²V_k=0.     

Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces.