Remarks on the <Emphasis Type="Italic">k</Emphasis>-error linear complexity of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic sequences

Recently the first author presented exact formulas for the number of 2 ⁿ -periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2 ⁿ -periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for p ⁿ -periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p ⁿ -periodic sequences over .      

Optimal Decompositions of Translations of L 2-Functions

In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space . Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space , or equivalent the spectral theory of a unitary representation U of the rank-n lattice in . Starting with a non-zero vector , we look for relations among the vectors in the cyclic subspace in generated by ψ. Since these vectors involve infinite “linear combinations,” the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L ²-independence. This refers to infinite linear combinations of integral translates of a fixed function with l ²-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals. Work supported in part by the U.S. National Science Foundation.     

On the discriminant locus of a Lagrangian fibration

Let be an irreducible holomorphic symplectic manifold of dimension 2n fibred over . Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus parametrizing singular fibres. Our main result is a formula for the degree of Δ, leading to bounds on the degree when X is a fourfold.     

A Generalization Of Reifenberg’s Theorem In {\mathbb{R}}^3

In 1960 Reifenberg proved the topological disc property. He showed that a subset of which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-H?lder image of the unit ball in . In this paper we prove that a subset of which is well approximated in the Hausdorff distance sense by one of the three standard area-minimizing cones at each point and at each (small) scale is locally a bi-H?lder deformation of a minimal cone. We also prove an analogous result for more general cones in . Received: July 2006, Revised: August 2007, Accepted: January 2008     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

On Kulikov’s problem

Kulikov has given an étale morphism of degree d > 1 which is surjective modulo codimension two with X simply connected, settling his generalized jacobian problem. His method reduces the problem to finding a hypersurface and a subgroup of index d generated by geometric generators. By contrast we show that if D has simple normal crossings away from a set of codimension three and meets the hyperplane at infinity transversely, then necessarily d = 1. Received: 21 November 2006     

A new family of maximal curves over a finite field

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type and defined over the finite field all have the maximum number of -rational points allowed by the Weil “explicit formulas”, and that these curves are -maximal curves over infinitely many algebraic extensions of . Serre showed that an -rational curve which is -covered by an -maximal curve is also -maximal. This has posed the problem of the existence of -maximal curves other than the Deligne–Lusztig curves and their -subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n ³ with n = p ^r  > 2, p ≥ 2 prime, we give a simple, explicit construction of an -maximal curve that is not -covered by any -maximal Deligne–Lusztig curve. Furthermore, the -automorphism group Aut has size n ³(n ³ + 1)(n ² − 1)(n ² − n + 1). Interestingly, has a very large -automorphism group with respect to its genus . Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni, PRIN 2006–2007.     

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

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.     

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator on locally defined polynomial forms on the unit sphere of ; functions valued in the universal Clifford algebra , here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δ _s g = f. This work was partially supported by NNSF of China (No.10471107) and RFDP of Higher Education (No.20060486001).     

Reconstruction of Function Fields

For k an algebraic closure of the finite field , ℓ prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-ℓ-quotient of its absolute Galois group – in fact already from the second central descending series quotient of . Submitted: July 2004, Revision: October 2005, Final revision: February 2008, Accepted: February 2008     

On the negative cyclic homology of <Emphasis Type="Italic">shc</Emphasis>-algebras

Let be a field of characteristic and S ¹ the unit circle. We prove that the shc-structure on a cochain algebra (A,d _A ) induces an associative product on the negative cyclic homology HC _*⁻ A. When the cochain algebra (A,d _A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in , then HC _*⁻ A is isomorphic as a graded algebra to the S ¹-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S ¹-equivariant cohomology algebras of the free loop space of the complex projective space when n + 1 = 0 [p] and of the even spheres S ²ⁿ when p = 2.      

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

Convexity properties of coverings of 1-convex manifolds

We show the following result: let X be a 1-convex manifold, A its exceptional set, k = dimA and p : Y → X any covering. Then Y can be exhausted by an increasing sequence of smoothly bounded k-convex domains.      

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.