New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

Linear Point Sets and Rédei Type k-blocking Sets in PG(n, q)

In this paper, k-blocking sets in PG(n, q), being of Rédei type, are investigated. A standard method to construct Rédei type k-blocking sets in PG(n, q) is to construct a cone having as base a Rédei type k-blocking set in a subspace of PG(n, q). But also other Rédei type k-blocking sets in PG(n, q), which are not cones, exist. We give in this article a condition on the parameters of a Rédei type k-blocking set of PG(n, q = p ^h ), p a prime power, which guarantees that the Rédei type k-blocking set is a cone. This condition is sharp. We also show that small Rédei type k-blocking sets are linear.     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

Cyclic group actions on odd-dimensional spheres

We show that for any simple (2q−1)-knotk, q>1, and any positive integern, the knot # ₁ ⁿ k is the fixed-point set of aZ _n -action onS ^2q+1. Further, we show that for many values ofn there are examples of (2q−1)-knots,q≥2, which are the fixed-point sets of inequivalentZ _n -actions. This paper was written whilst the first author was in receipt of a Research Grant from the Science Research Council of Great Britain.     

On the Minimum Distances of Non-Binary Cyclic Codes

We deal with the minimum distances of q-ary cyclic codes of length q ^m - 1 generated by products of two distinct minimal polynomials, give a necessary and sufficient condition for the case that the minimum distance is two, show that the minimum distance is at most three if q > 3, and consider also the case q = 3.     

Axisymmetric cellular structures revisited

For an axisymmetric cellular structure of wavenumberq ₀, the amplitude of the first linearly unstable mode is proportional to the Bessel functionJ ₀(q ₀r). If the radius of the structure is much larger thanq ₀ ^–1 , the first mode has a large relative maximum near the center. The nonlinear saturation of this instability gives an amplitude of order ^1/2, being the small growth rate of the linear instability. The saturation amplitude near the center is of order ^1/4. This prediction is confirmed by numerical analysis and numbers are given to make possible a comparison with the Rayleigh-Bénard instability.

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Resumé Pour une structure cellulaire axisymétrique de nombre d'ondeq ₀, l'amplitude du premier mode instable est proportioneile á la fonction de BesselJ ₀(q ₀r). Si le rayon de la structure est très supérieur àq ₀ ^–1 ce premier mode a un grand maximum relatif au centre. La saturation non linéaire de cette instabilité conduit à une amplitude en ^1/2 ( taux de croissance-petit-de l'instabilité linéaire). L'amplitude de saturation au centre est d'ordre ^1/4. Cette prédiction est confirmée par l'analyse numérique, et les nombres nécessaires à la comparaison avec l'instabilité de Rayleigh-Bénard sont donnés.

     

Cyclic codes of length 2 m

In this paper explicit expressions ofm + 1 idempotents in the ring are given. Cyclic codes of length 2 ^m over the finite fieldF _q, of odd characteristic, are defined in terms of their generator polynomials. The exact minimum distance and the dimension of the codes are obtained.     

Cyclic caps in PG(3,q)

This article investigates cyclic completek-caps in PG(3,q). Namely, the different types of completek-capsK in PG(3,q) stabilized by a cyclic projective groupG of orderk, acting regularly on the points ofK, are determined. We show that in PG(3,q),q even, the elliptic quadric is the only cyclic completek-cap. Forq odd, it is shown that besides the elliptic quadric, there also exist cyclick-caps containingk/2 points of two disjoint elliptic quadrics or two disjoint hyperbolic quadrics and that there exist cyclick-caps stabilized by a transitive cyclic groupG fixing precisely one point and one plane of PG(3,q). Concrete examples of such caps, found using AXIOM and CAYLEY, are presented.     

A Characterization of the Complement of a Hyperbolic Quadric in PG (3, q), for q Odd

The incidence structure NQ⁺(3, q) has points the points not on a non-degenerate hyperbolic quadric Q⁺(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q⁺(3, q). It is easy to show that NQ⁺(3, q) is a partial linear space of order (q, q(q−1)/2). If q is odd, then moreover NQ⁺(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q−1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q−1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ⁺(3,q). Let S be a ((q−1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ⁺(3, q).     

The Primitive Idempotents of a Cyclic Group Algebra

Explicit expressions are obtained for the 2n + 1 primitive idempotents in FG, the semisimple group algebra of the cyclic group G of order pⁿ (p an odd prime, n ≥ 1) over the finite field F of prime power order q, when q has order φ(pⁿ)/2 modulo pⁿ.AMS Mathematical Subject Classification (2000): 20C05, 94B05, 12E20, 16S34.     

Exact envelope-soliton solutions of a two-dimensional nonlinear wave equation

ExactN-envelope-soliton solutions are obtained, by extending Hirota's procedure, for the twodimensional nonlinear wave Eqn. (1) withq>0, which describes the evolution of the envelope of a train of surface gravity waves on deep water. They are shown to propagate in directions making an angle greater than tan^–1/2 with the propagation direction of the underlying carrier waves. We also point out and discuss the limitations of Hirota's procedure for generating solitonsolutions to problems of more than one spatial dimensions. Envelope-soliton solutions to Eqn. (1) withq<0 are also discussed.

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Sommaire On généralise la méthode de Hirota pour obtenir des solutions exactes àN-solitons et on l'applique à l'équation des ondes nonlinéaires à deux dimensions (1) avecq>0, qui décrit l'évolution de l'enveloppe d'un train d'ondes de gravitation dans un fluide de grande profondeur. Ces solutions se propagent en directions formant un angle plus grand que tan^–1/2 avec la direction de propagation des ondes porteuses fondamentales. On montre aussi que la méthode de Hirota n'est pas capable de produire, en deux dimensions, des solutions exactes aussi générales que dans le cas d'une seule dimension. Enfin on étudie les solutions de l'équation (1) avecq<0.

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Research supported by the Natural Sciences and Engineering Research Council of Canada. The author thanks Dr. G. Tenti for valuable discussions.     

A characterisation of spreads ovally-derived from Desarguesian spreads

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding ovally-derived planes are generalised André planes, of order 2 ^N , although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq ² can be ovally-derived from a Desarguesian spread if and only ifq is a square.     

A footnote to mean square value of DirichletL-series

Following appropriate use of approximate functional equation for Hurwitz Zeta function, we obtain upper bounds for } Here fors = σ + it, L(s,x) denotes DirichletL-series for character x(modq). In particular, we obtain S(1/2 +it) ≪q logqt + t^5/8 q^−1/8, which is an improvement in the range q |t| < q^11/7, on hitherto best known result. This incidentally gives S(1/2+ it)≪ q log3q for |t|q^9/5.     

Generalized Bernstein Polynomials

We introduce polynomials B ⁿ _i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B ⁿ _i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Cyclic Difference Packing and Covering Arrays

Let n and k be positive integers. Let C_q be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (a_ij) with entries a_ij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from C_q such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of C_q occurs in the difference list at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right.     

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

Global flow of the anisotropic Kepler problem on negative energy levels whenμ>9/8

In a previous study of the orbits of the anisotropic Kepler problem for>9/8 it had been shown that there are orbits which visit both theq ₂>0 and theq ₂<0 half-plane infinite times and that, denoting byc _n the number of times an orbit crosses theq ₂ axis before the half-plane where it is to be found changes forn-th time, every sequence of positive integersc _n is realized by at least one orbit. In this paper it is shown that these crossings do not occur in arbitrary regions of theq ₂ axis and the spatial order to which they obey is found: both half-axes,q ₂>0 andq ₂<0, may be divided into a sequence of contiguous segmentsp _n such that, for every family ofc _k succesive crossings, thei-th crossing occurs atp _i, fori going from 1 toc _k.

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Resumé Ce travail compléte la caractérization des orbites du probléme anisotropique de Kepler pour>9/8 et prend comme point de départ l'existence déjá connue d'orbites qui visitent les deux semiplansq ₂>0 etq ₂<0 une infinité de fois et qui réalisent toute successionc _n de nombres entiers positifs, désignant parc _n le nombre de fois que l'orbite coupe l'axe desq ₂ avant de changer de semiplan pour lan-éme fois. Nous démontrons que ces intersections de l'axeq ₂ sont ordonnées de la façon suivante: on peut définir une division de chaque semiaxeq ₂>0 etq ₂<0 en une succession de ségments contigusp _n telle que, pour toute famille dec _k croisements succéssifs et pouri entre 1 etc _k, lei-éme croisement coupe le ségment p_i.

     

New relationships between steenrod operations for certain spaces

In this paper, we show that under a certain technical condition, if a space has no 2-torsion, then eitherS q ² n x≠0 or there exists somey withS q ² n y=S q ² n ⁺¹ x, if for somen≥3S q ² n ⁺¹ x≠0. The proof uses relations between Steenrod operations and operations in connective realK-Theory. This research was partially supported by NSERC.