Mixed norm estimate for Radon transform on weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We will discuss about the mapping property of Radon transform on L ^p spaces with power weight. It will be shown that the Pitt’s inequality together with the weighted version of Hardy-Littlewood-Sobolev lemma imply weighted inequality for the Radon transform.     

Product of functions in BMO and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> in non-homogeneous spaces

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular BMO and the atomic block H ¹ in the sense of distribution, and show that this product may be split into two parts, one in L ¹ and the other in some Hardy-Orlicz space.     

Singular hyperbolicity for transitive attractors with singular points of 3-dimensional <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-flows

In the context of C^r-flows on 3-manifolds (r ≥ 1), the notion of singular hyperbolicity, inspired on the Lorenz Attractor, is the right generalization of hyperbolicity (in the sense of Smale) for C¹-robustly transitive sets with singularities. We estabish conditions (on the associated linear Poincaré flow and on the nature of the singular set) under which a transitive attractor with singularities of a C²-flow on a 3-manifold is singular hyperbolic.     

The negative gradient flow for the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-integral of Ricci curvature

 In this paper, the negative gradient flow for the L ²-integral of Ricci curvature on a 3-manifold is considered. It is not known whether the solution to this fourth order geometric evolution equation exists, and whether it will develop singularities in finite time. Based on the trick of De Turck and the idea of Hamilton on the flow of Ricci curvature, the local existence on any compact Riemannian manifold is obtained. In addition, the conditions for the occurences of singularities in finite time during the evolution and the asymptotic behavior of the flow on a 3-manifold are discussed. Received: 22 April 2002 / Revised version: 5 January 2003 Published online: 24 April 2003 This work is partially supported by the Foundations of NNSF of China, the Foundation for University Key Teacher by MEC and Shanghai Priority Academic Discipline.     

On a control problem for the wave equation in R<Superscript>3</Superscript>

Solutions of the wave equation (waves) initiated by infinitely distant sources (controls) are considered, and the L₂-completeness of reachable sets consisting of such waves is stidued. This problem is a natural analog of the control problem for a bounded domain where the completeness (local approximate controllability) in subdomains filled with waves generated by boundary controls occurs. It is shown that, in contrast to the latter case, the reachable sets formed by waves incoming from infinity are not complete in filled subdomains and describe the associated defect. Next, extending the class of controls to a set of special polynomials, the completeness is gained. A transform defined by jumps that arise in projecting functions to reachable sets is introduced. Its relevance to the Radon transform is clarified. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 19–37.     

Singularities of the Radon Transform of a Class of Piecewise Smooth Functions in R2

The Radon transform is the mathematical foundation of Computerized Tomography[1](CT).Its important applications includes medical CT,noninvasive test and etc.If one is specially interested in the places at which the image function changed largely such as the interfaces of two different tissues,tissue and ill tissue and the interfaces of two difierent matters,and want to reconstruct the outlines of the interfaces,one should reconstruct the singularities of the image function.The exact inversion of the Radon transform is valid only for smooth function[2].The singularity places of the reconstructed function should be studied specially.The research includes the propagation and inversion of singularity of the Radon transform.If one use convolutionbackprojection method to reconstruct the image function,the reconstructed function become blurring at the singularity places of the original function.M.Jiang and etc[3]developed a blind deconvolution method deblurring reconstructed image.By[4]and following research,we see that one can use a neighborhood data of the singularities of the Radon transform to inverse the singularity of the Radon transform,and therefore the reconstruction is available for some incomplete data reconstructions.     

A singular value decomposition for the radon transform in n-dimensional euclidean space

We construct the singular value decomposition of the Radon transform when the Radon transform is restricted to functions which are either square integrable on the unit disc in IR ⁿ with respect to one of the weights (1-r ²)^n/2-λ: or square integrable on IR ⁿ with respect to exp(r ²). An application to calculating mollifiers for approximate inversion of the sampled Radon transform is discussed.     

Atomic hardy-type spaces between <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> on metric spaces with non-doubling measures

Let ( Y,d,dl )\left( {\mathcal{Y},d,d\lambda } \right) be (ℝ ⁿ , |·|, μ), where |·| is the Euclidean distance, μ is a nonnegative Radon measure on ℝ ⁿ satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ ⁿ , |·|, d _λ ), or the space (S, d, ρ), where S ≡ ℝ ⁿ ⋉ ℝ⁺ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces { X_s ( Y ) }_{0 < s \leqslant ￥}\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty } and the BMO-type spaces { BMO( Y, s ) }_{0 < s \leqslant ￥}\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }. Let H ¹ ( Y )\left( \mathcal{Y} \right) be the known atomic Hardy space and L ₀¹ ( Y )\left( \mathcal{Y} \right) the subspace of f ∈ L ¹ ( Y )\left( \mathcal{Y} \right) with integral 0. The authors prove that the dual space of X _s ( Y )\left( \mathcal{Y} \right) is BMO( Y,s )BMO\left( {\mathcal{Y},s} \right) when s ∈ (0,∞), X _s ( Y )\left( \mathcal{Y} \right) = H ¹ ( Y )\left( \mathcal{Y} \right) when s ∈ (0, 1], and X _∞ ( Y )\left( \mathcal{Y} \right) = L ₀¹ ( Y )\left( \mathcal{Y} \right) (or L ¹ ( Y )\left( \mathcal{Y} \right)). As applications, the authors show that if T is a linear operator bounded from H ¹ ( Y )\left( \mathcal{Y} \right) to L ¹ ( Y )\left( \mathcal{Y} \right) and from L ¹ ( Y )\left( \mathcal{Y} \right) to L ^1,∞ ( Y )\left( \mathcal{Y} \right), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from X _r ( Y )\left( \mathcal{Y} \right) to the Lorentz space L ^1,s ( Y )\left( \mathcal{Y} \right), which applies to the Calderón-Zygmund operator on (ℝ ⁿ , |·|, μ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ ⁿ , |·|, d _γ ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.     

Two-dimensional quotients of C<Superscript>n</Superscript> are isomorphic to C<Superscript>2</Superscript>/Γ

We will prove that any two-dimensional quotient of an affine space modulo a reductive algebraic group is isomorphic to a quotient of C² modulo a finite group. The proof uses some new results due to Koras and Russell on contractible surfaces with at most quotient singularities and also several results about reductive group actions on affine varieties.     

Equivariant Classification of <Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-symplectic Surfaces

Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of b^m-Poisson structures which can be also visualized using differential forms with singularities as b^m-symplectic structures. In this paper we extend the classification scheme in [24] for b^m-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct b^m-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.     

The geometric complexity of special Lagrangian <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript>-cones

We prove a number of results on the geometric complexity of special Lagrangian (SLG) T²-cones in ³. Every SLG T²-cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T²-cone gives a lower bound for its geometric complexity, i.e. the area, the stability index and the Legendrian index of any SLG T²-cone are all bounded below by explicit linearly growing functions of the spectral curve genus. We prove that the cone on the Clifford torus (which has spectral curve genus zero) in S⁵ is the unique SLG T²-cone with the smallest possible Legendrian index and hence that it is the unique stable SLG T²-cone. This leads to a classification of all rigid index 1 SLG cone types in dimension three. For cones with spectral curve genus two we give refined lower bounds for the area, the Legendrian index and the stability index. One consequence of these bounds is that there exist S¹-invariant SLG torus cones of arbitrarily large area, Legendrian and stability indices. We explain some consequences of our results for the programme (due to Joyce) to understand the most common three-dimensional isolated singularities of generic families of SLG submanifolds in almost Calabi-Yau manifolds. Mathematics Subject Classification (1991) 53C38, 53C43     

Exemples de Lattès et domaines faiblement sphériques de ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 We describe the attracting basins of the origin in ℂ ^k+1 for the polynomial lifts of Lattès examples. We show that the boundary of these bounded pseudoconvex domains is a quotient of a compact spherical hypersurface, and we describe the singularities that appear. These domains are surprising, because they are very close to the ball, and admit non injective proper holomorphic self-maps. We also explicit some Lattès examples in dimension 2. Received: 17 October 2002 / Revised version: 7 February 2003 Published online: 19 May 2003 Mathematics Subject Classification (2000): 14K25, 32S25, 32T99, 32H50     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Integrability,Supports of Fourier Transforms and Uniqueness for Convolution Equations

It is proved that a nonzero function is not in L^p(Rⁿ) with p \le 2n/d if its Fourier transform is supported by a d-dimensional submanifold. It is shown that the assertion fails for p > 2n/d and d \ge n/2. The result is applied for obtaining uniqueness theorems for convolution equations in L^p-spaces.     

Traces of functions with a dominating mixed derivative in ℝ<Superscript>3</Superscript>

We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in ℝ³, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to p = 2. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.     

On the extension of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> holomorphic functions VII: Hypersurfaces with isolated singularities

An extension theorem for holomorphic functions with L ² growth condition is strengthened for the case of the extension from hypersurfaces with isolated singularities.     

Some results concerning cryptographically significant mappings over GF(2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper we investigate the existence of permutation polynomials of the form F(x) = x ^d  + L(x) over GF(2 ⁿ ), L being a linear polynomial. The results we derive have a certain impact on the long-term open problem on the nonexistence of APN permutations over GF(2 ⁿ ), when n is even. It is shown that certain choices of exponent d cannot yield APN permutations for even n. When n is odd, an infinite class of APN permutations may be derived from Gold mapping x ³ in a recursive manner, that is starting with a specific APN permutation on GF(2 ^k ), k odd, APN permutations are derived over GF(2 ^k+2i ) for any i ≥ 1. But it is demonstrated that these classes of functions are simply affine permutations of the inverse coset of the Gold mapping x ³. This essentially excludes the possibility of deriving new EA-inequivalent classes of APN functions by applying the method of Berveglieri et al. (approach proposed at Asiacrypt 2004, see [3]) to arbitrary APN functions.     

Some results on 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2 ^n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 ^n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2 ^n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4 ^m 2 ⁿ designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4 ^m 2 ⁿ designs under the clear effects criterion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10571093, 10671099 and 10771123), the Research Foundation for Doctor Programme (Grant No. 20050055038) and the Natural Science Foundation of Shandong Province of China (Grant No. Q2007A05). Zhang’s research was also supported by the Visiting Scholar Program at Chern Institute of Mathematics.     

State spaces of <Emphasis Type="Italic">JB</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-triples<Superscript>*</Superscript>

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^*-triple, and up to complete isometry, of one-sided ideals in C^*-algebras.Mathematics Subject Classification (2000):17C65, 46L07Both authors are supported by NSF grant DMS-0101153     

Interpolation Scheme for Planar Cubic <Emphasis Type="Italic">G</Emphasis><Superscript>2</Superscript> Spline Curves

In this paper a method for interpolating planar data points by cubic G ² splines is presented. A spline is composed of polynomial segments that interpolate two data points, tangent directions and curvatures at these points. Necessary and sufficient, purely geometric conditions for the existence of such a polynomial interpolant are derived. The obtained results are extended to the case when the derivative directions and curvatures are not prescribed as data, but are obtained by some local approximation or implied by shape requirements. As a result, the G ² spline is constructed entirely locally.     

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.