Generalized hexagons and Singer geometries

In this paper, we consider a set of lines of with the properties that (1) every plane contains 0, 1 or q + 1 elements of , (2) every solid contains no more than q ² + q + 1 and no less than q + 1 elements of , and (3) every point of is on q + 1 members of , and we show that, whenever (4) q ≠ 2 (respectively, q = 2) and the lines of through some point are contained in a solid (respectively, a plane), then is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in , with q even. We present examples of such sets not satisfying (4) based on a Singer cycle in , for all q.      

Homogeneous factorisations of Johnson graphs

For a graph Γ, subgroups , and an edge partition of Γ, the pair is a (G, M)-homogeneous factorisation if M is vertex-transitive on Γ and fixes setwise each part of , while G permutes the parts of transitively. A classification is given of all homogeneous factorisations of finite Johnson graphs. There are three infinite families and nine sporadic examples. This paper forms part of an ARC Discovery grant of the last two authors. The second author holds an Australian Research Council Australian Research Fellowship.     

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .      

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

<Emphasis Type="Italic">h</Emphasis>-Principles for curves and knots of constant curvature

We prove that curves of constant curvature satisfy, in the sense of Gromov, the relative -dense h-principle in the space of immersed curves in Euclidean space R ^{n ≥ 3}. In particular, in the isotopy class of any given knot f there exists a knot f͂ of constant curvature which is -close to f. More importantly, we show that if f is , then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull. The author’s research was supported in part by NSF CAREER award DMS-0332333.     

A Note on the Green Invariants in Finite Group Modular Representative Theory

As in Finite Group Modular Representation Theory, let be a commutative complete noetherian ring with an algebraically closed residue field k. Let G be a finite group and let N be a normal subgroup of G. First suppose that V is an indecomposable -module, so that Inf ^G _G/N (V) is an indecomposable G-module. We relate the Green invariants of V as an -module to those of Inf ^G _G/N (V) as an G-module. Secondly, let V and W be indecomposable G-modules. Assume that W is an endo-permutation lattice and that is also an indecomposable G-module. We relate the Green invariants of the G-modules V and . (This situation arises under important Morita equivalences.) Received: December 11, 2006. Revised: August 22, 2007.     

Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check (LDPC) codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4,q) to find information about the LDPC codes defined by Q (4,q), and . For , and , we are able to describe small weight codewords geometrically. For , q odd, and for , we improve the best known lower bounds on the minimum distance, again only using geometric arguments. Similar results are also presented for the LDPC codes LU(3,q) given in [Kim, (2004) IEEE Trans. Inform. Theory, Vol. 50: 2378–2388]     

The Fine Intersection Problem for Steiner Triple Systems

The intersection of two Steiner triple systems and is the set . The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection satisfies and . We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v ³), where previous results only imply that |I(v)| = Ω(v ²). Received: January 23, 2006. Final Version received: September 2, 2006     

Elliptic near-MDS codes over $${\mathbb{F}}_5$$

Let Γ₆ be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding

Embedding FD(ω) into {\mathcal{P}_s} densely

Let be the lattice of degrees of non-empty subsets of 2 ^ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in . Cenzer and Hinman proved that is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any , we can lattice embed FD(ω) into strictly between and . We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made. Thanks to my adviser Peter Cholak for his guidance in my research. I also wish to thank the anonymous referee for helpful comments and suggestions. My research was partially supported by NSF grants DMS-0245167 and RTG-0353748 and a Schmitt Fellowship at the University of Notre Dame.     

Maximal partial packings of $${Z_2^n}$$ with perfect codes

A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of , n = 2 ^m −1, satisfies . It is also proved that for any n equal to 2 ^m −1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2 ^m −1, . All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

Scales for co-compact embeddings of virtually free groups

Let Γ be a group which is virtually free of rank at least 2 and let be the family of totally disconnected, locally compact groups containing Γ as a co-compact lattice. We prove that the values of the scale function with respect to groups in evaluated on the subset Γ have only finitely many prime divisors. This can be thought of as a uniform property of the family .      

Quasiperiodic Group Factorizations

If is any ring or semi-ring (e.g., ) and G is a finite abelian group, two elements a, b of the group (semi-)ring are said to form a factorization of G if ab = rΣ _g∈G g for some . A factorization is called quasiperiodic if there is some element g ∈ G of order m > 1 such that either a or b – say b – can be written as a sum b ₀ + ... + b _m−1 of m elements of such that ab _h = g ^h ab ₀ for h = 0, ... , m − 1. Hajós [5] conjectured that all factorizations are quasiperiodic when and r = 1 but Sands [15] found a counterexample for the group . Here we show however that all factorizations of abelian groups are quasiperiodic when and that all factorizations of cyclic groups or of groups of the type are quasiperiodic when . We also give some new examples of non-quasiperiodic factorizations with for the smaller groups and . Received: May 12, 2006. Revised: October 3, 2007.     

Normal generation and Clifford index on algebraic curves

For a smooth curve C it is known that a very ample line bundle on C is normally generated if Cliff() < Cliff(C) and there exist extremal line bundles (:non-normally generated very ample line bundle with Cliff() = Cliff(C)) with . However it has been unknown whether there exists an extremal line bundle with . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle with . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle with . More generally, if C has a line bundle computing the Clifford index c of C with , then C has such an extremal line bundle . For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch Promotion Fund)(KRF-2005-070-C00005).     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

Uniform central limit theorems for kernel density estimators

Let be the classical kernel density estimator based on a kernel K and n independent random vectors X _i each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h _n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.      

The nonexistence of near-extremal formally self-dual codes

A code is called formally self-dual if and have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over , and . These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang’s systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8|n was dealt with by Han and Lee.      

Γ-limits and relaxations for rate-independent evolutionary problems

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals and the dissipation distance . For sequences and we address the question under which conditions the limits q _∞ of solutions satisfy a suitable limit problem with limit functionals and , which are the corresponding Γ-limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q _∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model. Research partially supported by LC06052 (MŠMT), MSM21620839 (MŠMT), A1077402 (GAČR), by the Deutsche Forschungsgemeinschaft under MATHEON C18 and under SFB404 C7, by the European Union under HPRN-CT-2002-00284 Smart Systems, and by the Alexander von Humboldt-Stiftung. Both, TR and US gratefully acknowledge the kind hospitality of the WIAS, where this research was initiated.