Reverse bubbling of currents and harmonic heat flows with prescribed singular set

We consider the harmonic heat flow for maps u between B³, the unit ball of , and its boundary S². For a class of possibly singular initial and boundary data we show the existence of a weak solution whose singular set is a fixed discrete set on the vertical axis arbitrarily prescribed. Other examples including isolated singularities moving along a prescribed trajectory are given.Received: 23 October 2002, Accepted: 12 March 2003, Published online: 4 September 2003     

On positive scalar curvature on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

We consider the problem of prescribing scalar curvature on the standard 2-sphere S ². It is proved that any positive smooth function on S ² is the scalar curvature of some pointwise conformal metric, if an associated map has non-zero degree. As a result we improve some previous important results and give some completely new ones.Received: 21 January 2003, Accepted: 12 March 2003, Published online: 6 June 2003Mathematics Subject Classification (2000): 35J60, 58G03Min Ji: Supported in part by the NSF grant 19725102 and the 973 grant G1999075107 of China     

A lower bound on the number of Semi‐Boolean quadruple systems

A Steiner quadruple system of order 2ⁿ is Semi‐Boolean (SBQS(2ⁿ) in short) if all its derived triple systems are isomorphic to the point‐line design associated with the projective geometry PG(n?1, 2). We prove by means of explicit constructions that for any n, up to isomorphism, there exist at least 2^{? 3(n?4)/2?} regular and resolvable SBQS(2ⁿ). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 229–239, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10050     

2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

The number of k‐SAT functions

We study the number SAT(k; n) of Boolean functions of n variables that can be expressed by a k‐SAT formula. Equivalently, we study the number of subsets of the n‐cube 2ⁿ that can be represented as the union of (n ? k)‐subcubes. In The number of 2‐SAT functions (Isr J Math, 133 (2003), 45–60) the authors and Imre Leader studied SAT(k; n) for k ≤ n/2, with emphasis on the case k = 2. Here, we prove bounds on SAT(k; n) for k ≥ n/2; we see a variety of different types of behavior. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 227–247, 2003     

A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like

Some more 5‐GDDs and optimal (v), 5, 1)‐packings

In this paper, we continue to investigate the existence of 5‐GDDs with types g⁵ m¹ and g^u. As an application, the known results on the existence of optimal packings with block size 5 are also improved. © 2003 Wiley Periodicals, Inc.     

Designs arising from Buekenhout‐Metz unitals

An affine 2–(q³,q², q + 1) design is constructed from a Buekenhout‐Metz unital of the affine plane AG(2,q²), with q > 2. It is also shown that such a design is isomorphic to the point‐plane design of the affine space AG(3,q). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 79–88, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10010     

Classes of Maxwell Spaces Admitting Translations

A Maxwell space is a pair (M ⁴, F), where M ⁴ is a four-dimensional Minkowski space or a domain in it and F is a closed exterior differential 2-form on M ⁴. We describe classes of Maxwell spaces admitting translation groups of dimension 1–4.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Two Conditions for Infinite Groups to Satisfy Certain Laws

Let G be an infinite group and m {2^k | k N^*}. In this paper, we prove that G satisfies the law [x_m, y_m] = 1 if and only if in any two infinite subsets X and Y of G, there exist a X and b Y such that [a^m,b^m] = 1. We also prove that G satisfies the law (x₁^mx₂^m x_n^m)² = 1 if and only if in any n infinite subsets X₁, X₂,..., X_n, there exist a_i X_i (i = 1,..., n) such that (a₁^ma₂^m a_n^m)² = 1.2000 Mathematics Subject Classification: 20F99     

Structure of Degenerate Block Algebras

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map : A A F, we define a Lie algebra = (A, ) over F with basis {e_x | x A/{0}} and Lie product [e_x,e_y] = (x,y)e_x+y. We show that is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der of is a complete Lie algebra. We describe the double extension D( , T) of by T, where T is spanned by the locally finite derivations of , and determine the second cohomology group H²(D( , T),F) using anti-derivations related to the form on D( , T). Finally, we compute the second Leibniz cohomology groups HL²( , F) and HL²(D( , T), F).2000 Mathematics Subject Classification: 17B05, 17B30This work was supported by the NNSF of China (19971044), the Doctoral Programme Foundation of Institution of Higher Education (97005511), and the Foundation of Jiangsu Educational Committee.     

风险函数与非负分布尾部的刻划

Some equivalent conditions on the classes of lighted-tailed and heavily heavy-tailed and lightly heavy-tailed d. f. s are introduced. The limit behavior of and are discussed. Some properties of the subclass D K ^c and subclass D K ^c are obtained. Supported by the National Natural Science Foundation of China (10271087).     

Lp error estimates and superconvergence for covolume or finite volume element methods

We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the L^p norm, 2 ≤ p ≤ ∞, are derived. We also show second‐order convergence in the L^p norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the “supercloseness” results in Chou and Li [Math Comp 69(229) (2000), 103–120] to the L^p based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463–486, 2003     

Smoothed analysis of termination of linear programming algorithms

 We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar's condition number by Dunagan, Spielman and Teng (http://arxiv.org/abs/cs.DS/0302011) we show that the smoothed complexity of interior-point algorithms for linear programming is O(m ³ log(m/Σ)). In contrast, the best known bound on the worst-case complexity of linear programming is O(m ³ L), where L could be as large as m. We include an introduction to smoothed analysis and a tutorial on proof techniques that have been useful in smoothed analyses. Received: December 10, 2002 / Accepted: April 28, 2003 Published online: June 5, 2003 Key words. smoothed analysis – linear programming – interior-point algorithms – condition numbers Mathematics Subject Classification (1991): 90C05, 90C51, 68Q25     

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.     

Spectra of Compact Composition Operators Over Bounded Symmetric Domains

Let H²(D) denote the Hardy space of a bounded symmetric domain in its standard Harish-Chandra realization, and let be the weighted Bergman space with and where is a critical value depending on D. Suppose that is holomorphic. We show that if the composition operator defined by is compact (or, more generally, power-compact) on H²(D) or then has a unique fixed point z₀ in D. We then prove that the spectrum of as an operator on these function spaces is precisely the set consisting of 0, 1, and all possible products of eigenvalues of These results extend previous work by Caughran/Schwartz and MacCluer. As a corollary, we now have that MacCluers previous spectrum results on the unit ball B_n extend to H^p(n) (not only for p = 2 but for all p > 1) and (for p 1), where ⁿ is the polydisk in      

<Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology

We define K-homology groups K _* () for small C ^* -categories in terms of Hilbert modules over the C ^* -category . We also define a functor A ^f from the category of small C ^* -categories into the category of C ^* -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C ^* -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C ^* -algebras A. It then follows that the composition A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.     

3‐Designs of PSL(2, 2n) With block sizes 4 and 5

We determine the distribution of 3‐designs among the orbits of 4‐ and 5‐element subsets under the action of PSL(2,2ⁿ) on the projective line. Thus we give complete information on all Kramer–Mesner matrices for the action of PSL(2,2ⁿ) on 3‐sets versus 4‐ and 5‐sets. As a consequence, all 3‐designs with block sizes 4 and 5 and automorphism group PSL(2,2ⁿ) can immediately be obtained. © 2003 Wiley Periodicals, Inc.