A survey of orthogonal arrays of strength two

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Parallel Dixon Matrices by Bracket

It is known that the Dixon matrix can be constructed in parallel either by entry or by diagonal. This paper presents another parallel matrix construction, this time by bracket. The parallel by bracket algorithm is the fastest among the three, but not surprisingly it requires the highest number of processors. The method also shows analytically that the Dixon matrix has a total of m(m+1)²(m+2)n(n+1)²(n+2)/36 brackets but only mn(m+1)(n+1)(mn+2m+2n+1)/6 of them are distinct.     

Weak Covering at Large Cardinals

We show that weakly compact cardinals are the smallest large cardinals k where k⁺ < k⁺ is impossible provided 0^# does not exist. We also show that if k^+Kc < k⁺ for some k being weakly compact (where K^c is the countably complete core model below one strong cardinal), then there is a transitive set M with M ? ZFC + “there is a strong cardinal”.     

On closures verifying that the interior of a closed element is closed

We give significant improvements to the Graham, Knuth & Motzkin result: if R is any binary relation, R ^+c+c+ = R ^+c+c (where P ⁺ denotes the transitive closure of the binary relation P, and P ^c its Boolean complement)     

Positivity and Convexity in Rings of Fractions

Given a commutative ring A equipped with a preordering A⁺ (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A⁺ extends to a preordering R⁺ of R (i.e. with A ∩ R⁺ = A⁺) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R such that A is convex in B with respect to B⁺ : = B ∩ R⁺. Supported by DFG. A short form of this article has been delivered at the conference Carthapos 2006 at Carthago (Tunisia).     

On the Cardinalities of the Row spaces of Non-full Rank Boolean Matrices

n -2 integers 2 ^{n -2}+2 ^{n -3}+2 ^s, where s=0,1,2,..., n-3, in the interval (2 ^{n -2}+2 ^{n -3},2 ^{n -1}] such that these integers are the cardinalities of row spaces R(A) of non-full rank Boolean matrices A of order n. We also show that for each s, where s=0,1,2,..., n-3, there exists A epsilon B _n such that A is non-full rank and the cardinality of R(A) equals 2 ^{n -2}+2 ^{n -3}+2 ^s.     

Congruence-simple subsemirings of ℚ+

Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ⁺. Even the class CongSimp(\mathbb Q⁺)\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+}) of all congruence-simple subsemirings of ℚ⁺ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ⁺. We classify all the maximal elements of CongSimp(\mathbbQ⁺)\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+}) and show that every element of CongSimp(\mathbbQ⁺)\{\mathbbQ⁺}\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\} is contained in at least one of them.     

On a Class of Symmetric Balanced Generalized Weighing Matrices

Let q be a prime power and m a positive integer. A construction method is given to multiply the parametrs of an -circulant BGW(v=1+q+q ₂+·+q ^m , q ^m , q ^m –q ^m–1) over the cyclic group C _n of order n with (q–1)/n being an even integer, by the parameters of a symmetric BGW(1+q ^m+1, q ^m+1, q ^m+1–q ^m ) with zero diagonal over a cyclic group C _vn to generate a symmetric BGW(1+q+·+q ^2m+1,q ^2m+1,q ^2m+1–q ^2m) with zero diagonal, over the cyclic group C _n . Applications include two new infinite classes of strongly regular graphs with parametersSRG(36(1+25+·+25^2m+1),15(25)^2m+1,6(25)^2m+1,6(25)^2m+1), and SRG(36(1+49+·+49^2m+1),21(49)^2m+1,12(49)^2m+1,12(49)^2m+1).     

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

The solvability of the equation n = x ² + y ² + 6pz ² (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm ¹² > p ²¹. As an implication, the solvability of the equation n = x ² + y ² + u ³ + v ³ + z ⁴ + w ¹⁶ + t ^4k+1 for all sufficiently large n is established. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21.     

Lorentzian similarity manifolds

An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝ ^m+2), where G = ℝ ^m+2 ⋊ (O(m+1, 1)×ℝ⁺). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S ^m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.     

Complete and Stable O(p+1)×O(q+1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space ℝ p+q+2

We classify the zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in the euclidean space ℝ ^p+q+2, p,q > 1, analyzing whether they are embedded and stable. The Morse index of the complete hypersurfaces show the existence of embedded, complete and globally stable zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in ℝ ^p+q+2, p+q≥ 7, which are not homeomorphic to ℝ ^p+q+1. Such stable examples provide counter-examples to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature. Mathematics Subject Classifications (2000): 53A10, 53C42,49005.     

Bifurcation in the stable manifold of a chemostat with general polynomial yields

A three‐dimensional chemostat with nth‐ and mth‐order polynomial yields, instead of the particular ones such as A+BS, A+BS², A+BS³, A+BS⁴, A+BS² + CS³, and A+BSⁿ, is proposed. The existence of limit cycles in the two‐dimensional stable manifold, the Hopf bifurcation, and the stability of the periodic solution created by the bifurcation is proved. Copyright © 2009 John Wiley & Sons, Ltd.     

Linear (q+1)-fold Blocking Sets in PG(2, q)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms

We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x ²+5y ². Making use of Ramanujan’s ₁ ψ ₁ summation formula, we establish a new Lambert series identity for $\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}}We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x ²+5y ². Making use of Ramanujan’s ₁ ψ ₁ summation formula, we establish a new Lambert series identity for ?_n,m=-￥^￥q^n2+5m2\sum_{n,m=-\infty }^{\infty}q^{n^{2}+5m^{2}} . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x ²+6y ², 2x ²+3y ², x ²+15y ², 3x ²+5y ², x ²+27y ², x ²+5(y ²+z ²+w ²), 5x ²+y ²+z ²+w ². In the process, we find many new multiplicative eta-quotients and determine their coefficients.     

An Explicit Computation of a Family of Trivializing Étale Covers

We explicitly compute étale covers of the smooth Fermat curves Y _p+1 = Proj k[u, v, w]/(u ^p+1 + v ^p+1 ? w ^p+1) which trivialize the vector bundles Syz(u ², v ², w ²)(3), where k is a field of characteristic p ≥ 3.     

Reducible Families of Curves with Ordinary Multiple Points on Surfaces in IP3 C

For a Coxeter system (G, S) the multi-parametric alternating subalgebra H ⁺(G) of the Hecke algebra and the alternating subgroup ?⁺(G) of the braid group are defined. Two presentations for H ⁺(G) and ?⁺(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.     

Regularity of ultrafilters

If there arek ⁺⁺ eventually functions fromk ⁺ intok or if there arek ⁺⁺ eventually different functions fromk ⁺ then uniform ultrafilters onk ⁺ are (k, k ⁺)-regular. The research of the first author was supported in part by NSF grant. The second author is a Miller’s Fellow at the University of California in Berkeley.     

On the number of lines in 4-dimensional linear spaces

Assuming a weak non-degeneracy condition, we show that a linear spaceL of dimension at least 4 withv=q ⁴+q ³+q ²+q+1 points,q > 1 any positive real number, has at least (q²+1)v lines with equality if and only ifq is a prime power andL = PG(4,q).Dedicated to H. Mäurer on the occasion of his 60th birthday     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).     

Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

We prove the nonexistence of a distance-regular graph with intersection array {74,54,15;1,9,60} and of distance-regular graphs with intersection arrays

{4r³+8r²+6r+1,2r(r+1)(2r+1),2r²+2r+1;1,2r(r+1),(2r+1)(2r²+2r+1)}