Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

In this paper, we establish lower bounds for n-term approximations in the metric of L ²⁽I ² ) of characteristic functions of plane convex subsets of the square I ² with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .     

Some greedyt-intersecting families of finite sequences

Letn, s ₁,s ₂, ... ands _n be positive integers. Assume is an integer for eachi}. For , , and , denotes _p (a)={j|1jn,a _j p}, , and . is called anI _t ^p -intersecting family if, for any a,b ,a _i b _i =min(a _i ,b _i )p for at leastt i's. is called a greedyI _t ^P -intersecting family if is anI _t ^p -intersecting family andW _p (A)W _p (B+A ^c ) for anyAS _p ( ) and any with |B|=t–1.In this paper, we obtain a sharp upper bound of | | for greedyI _t ^p -intersecting families in for the case 2ps _i (1in) ands ₁>s ₂>...>s _n .This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China.     

q-Pseudoconvexity and Regularity at the Boundary for Solutions of the \bar \partial -problem

For a domain of we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the -complex for forms with -coefficients in degree . All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the L ²-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-L ² and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.     

Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions f with positive exponent of smoothness s > 0 given on a domain . The norms on these spaces are defined via integral norms of the difference of the function f of order m > s treated as a function of the point of the domain and of the difference increment. For an arbitrary domain , we characterize these spaces in terms of the local approximations of the function by polynomials of degree m – 1.     

L 2 harmonic forms and stability of hypersurfaces with constant mean curvature

We prove that a complete noncompact oriented strongly stable hypersurfaceM ⁿ with cmc (constant mean curvature)H in a complete oriented manifoldN ⁿ⁺¹ with bi-Ricci curvature, satisfying alongM, admits no nontrivialL ² harmonic 1-forms. This implies ifM ⁿ (2n4) is a complete noncompact strongly stable hypersurface in hyperbolic spaceH ⁿ⁺¹(–1) with cmc , there exist no nontrivialL ² harmonic 1-forms onM. We also classify complete oriented strongly stable surfaces with cmcH in a complete oriented manifoldN ³ with scalar curvature satisfying .     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

Spectral Properties of Abelian C*-Algebras

Let be an Abelian unital C ^*-algebra and let denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of to be unitarily equivalent to a representation in which the elements of act multiplicatively, by their Gelfand transforms, on a space L ²( ,), where is a positive measure on the Baire sets of . We also compare these conditions with the multiplicity-free property of a representation.     

A priori estimates of strong solutions of semilinear parabolic equations

We study an initial boundary value problem for the semilinear parabolic equation

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Limit theorems for compact two-point homogeneous spaces of large dimensions

Let be the field , , or of real dimension . For each dimensiond2, we study isotropic random walks(Y ₁)₁0 on the projective space with natural metricD where the random walk starts at some with jumps at each step of a size depending ond. Then the random variablesX ₁ ^d :=cosD(Y ₁ ^d ,x ₀ ^d ) form a Markov chain on [–1, 1] whose transition probabilities are related to Jacobi convolutions on [–1, 1]. We prove that, ford, the random variables (vd/2)(X _l(d) ^d +1) tend in distribution to a noncentral ²-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for as well as thed-spheresS ^d ford.     

An Algorithm for l∞ Regression with Quadratic Complexity

Using a few very basic observations, we proposed recently a direct and finite algorithm for the computation of the l regression line on a discrete set under the assumption that In this paper, we extend the algorithm to the case with at least one, possibly multiple y-values for each distinct x_i. Our algorithm finds all the regression lines in O(n ²) operations in the worst-case scenario and improves the existing best-known computational complexity result for this problem. Numerical results on random problems are included.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

Representable K-Theory for σ-C*-algebras

We define a version of K-theory on the category of -C ^*-algebras (countable inverse limits of C ^*-algebras). Our theory is homotopy invariant, has long exact sequences and a Milnor sequence, and satisfies Bott periodicity. On C ^*-algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X. (We do not claim that our theory is in general a representable functor.) We also define an equivariant version, and discuss several related groups.Partially supported by a National Science Foundation Postdoctoral Fellowship.     

Codes with the Same Weight Distributions as the Goethals Codes and the Delsarte-Goethals Codes

The Goethals code is a binary nonlinear code of length 2^m+1 which has codewords and minimum Hamming distance 8 for any odd . Recently, Hammons et. al. showed that codes with the same weight distribution can be obtained via the Gray map from a linear code over Z ₄of length 2^m and Lee distance 8. The Gray map of the dual of the corresponding Z ₄ code is a Delsarte-Goethals code. We construct codes over Z ₄ such that their Gray maps lead to codes with the same weight distribution as the Goethals codes and the Delsarte-Goethals codes.