Nonnegative sectional curvature hypersurfaces in a real space form

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form M ⁿ⁺¹(c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M ⁿ⁺¹(c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S ^k (a) × S ^n-k (√1 ? a ²), 1 ≤ k ≤ n ? 1, in S ⁿ⁺¹(1) and the Riemannian product H ^k (tanh² r ? 1) × S ^n-k (coth² r ? 1), 1 ≤ k ≤ n ? 1, in H ⁿ⁺¹(?1).     

Latin squares and superqueens

Let L be a Latin square of order n with entries from {0, 1,…, n ? 1}. In addition, L is said to have the (n, k) property if, in each right or left wrap around diagonal, the number of cells with entries smaller than k is exactly k. It is established that a necessary and sufficient condition for the existence of Latin squares having the (n, k) property is that of (2|n ? 2| k) and (3|n ? 3| k). Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.     

Covering tori with squares

We find that an n × n toroidal checkerboard can be covered with $\text{?}\text{n}\text{k}\text{?}\text{n}\text{k}\text{??}$k × k squares, and no fewer. To prove this result we also need to prove that the unit Euclidean torus can be covered with ?α^?1?α^?1?? squares of side α, and no fewer.     

On embedding incomplete symmetric latin squares

Necessary and sufficient conditions are obtained for the extendibility of an r × r symmetric Latin rectangle to an n × n symmetric Latin square. These conditions imply that any incomplete n × n symmetric Latin square can be embedded in a complete symmetric Latin square of order 2n. Also, any incomplete n × n symmetric diagonal Latin square can be embedded in a complete symmetric diagonal Latin square of order 2n + 1.     

Combinatorial constructions for optimal 2-D optical orthogonal codes with AM-OPPTS property

We develop a new one-to-one correspondence between a two-dimensional (m × n, k, ρ) optical orthogonal code (2-D (m × n, k, ρ)-OOC) with AM-OPPTS (at most one-pulse per time slot) property and a certain combinatorial subject, called an n-cyclic holey packing of type m ⁿ . By this link, an upper bound on the size of a 2-D (m × n, k, ρ)-OOC with AM-OPPTS property is derived. Afterwards, we employ combinatorial methods to construct infinitely many 2-D (m × n, k, 1)-OOCs with AM-OPPTS property, whose existence was previously unknown. All these constructions meet the upper bounds with equality and are thus optimal.     

Bounds on the number of autotopisms and subsquares of a Latin square

A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (α, β, γ) such that L is unchanged after the rows are permuted by α, the columns are permuted by β and the symbols are permuted by γ. Let n!(n?1)!R _n be the number of n×n Latin squares. We show that an n×n Latin square has at most n ^{O(log k)} subsquares of order k and admits at most n ^{O(log n)} autotopisms. This enables us to show that {ie11-1} divides R _n for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of R _n , and give a new proof that R _p ≠1 (mod p) for prime p.     

Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types SO(2n)/SO(2k)×SO(2) ^n?k , SO(2n+1)/SO(2k+1)×SO(2) ^n?k , and (S)U(n)/(S)(U(2k)×U(2) ^n?k ) in so(2n), so(2n+1), and (s)u(n), respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit CP ^n?1?u(n).     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

PBIB designs with m associate classes induced by full rank matrices modulo pm

This paper presents series of PBIB designs with m associate classes in which the treatment set is a subset of the Z(p^m)-module of n × 1 vectors over the ring of integers modulo p^m, p any prime. The association scheme of this series of designs is determined by the Fuller canonical form under row equivalence of n × 2 matrices [a,b] for vectors a and b in the treatment set. The blocking procedure utilizes full rank s × n matrices over Z(p^m), 1 ? s ? n ? 2, n ? 3. For m = 2, n = 3, s =1 and for each prime p, each PBIB is regular divisible and yields a finite proper uniform projective Hjelmslev plane with parameters j = p and k = p(p + 1).     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Bounds for ratios of eigenvalues using traces

Let A be an n × n matrix with real eigenvalues λ₁ ? … ? λ_n, and let 1 ? k < l ? n. Bounds involving trA and trA² are introduced for λ_k/λ_l, (λ_k ? λ_l)/(λ_k + λ_l), and {kλ_k + (n ? l + 1)λ_l}²/{kλ²_k + (n ? l + 1)λ²_l}. Also included are conditions for λ_l >; 0 and for λ_k + λ_l > 0.     

On operator norms of submatrices

Both of the following conditions are equivalent to the absoluteness of a norm ν in $\text{C}$ⁿ: (1) for all n×n diagonal matrices D=(d_k), the subordinate operator norm N_ν(D)=max_k|d_k|; (2) for all n×n matrices A, N_ν(A) ?N_ν(|A|). These conditions are modified for partitioned matrices by replacing absolute values with norms of blocks. A generalization of absoluteness is thus obtained.     

Simple product colorings

If A is a set colored with m colors, and B is colored with n colors, the coloring of A × B obtained by coloring (a, b) with the pair (color of a, color of b) will be called an m × n simple product coloring (SPC) of A × B. SPC's of Cartesian products of three or more sets are defined analogously. It is shown that there are 2 × 2, and 2 × 2 × 2 SPC's of $\text{Q}$² and $\text{Q}$³ which forbid the distance one; that there is no 2^k SPC of $\text{Q}$^k forbidding the distance one, for k > 3; and that there is no 2 × 2 SPC of $\text{Q}$ × $\text{Q}$(√15), and thus none of $\text{R}$², forbidding the distance 1.     

A note on computing matrix geometric means

A new definition is introduced for the matrix geometric mean of a set of k positive definite n×n matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n ³ k ²) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando et al. (Linear Algebra Appl 385:305?C334, 2004) have complexity O(n ³ k!2 ^k ). The new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable Riemannian geometry on the set of positive definite matrices. It satisfies most part of the ten properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled.     

On exact recovery of sparse vectors from linear measurements

Let 1 ≤ k ≤ n < N. We say that a vector x ∈ ? ^N is k-sparse if it has at most k nonzero coordinates. Let Φ be an n × N matrix. We consider the problem of recovery of a k-sparse vector x ∈ ? ^N from the vector y = Φx ∈ ? ⁿ . We obtain almost-sharp necessary conditions for k, n, N for this problem to be reduced to that of minimization of the ?₁-norm of vectors z satisfying the condition y = Φz.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Über die Wurzelschranke für das Minimalgewicht von codes

Let d be the minimum distance of an (n, k) code $\text{C}$, invariant under an abelian group acting transitively on the basis of the ambient space over a field F with char F × n. Assume that $\text{C}$ contains the repetition code, that dim($\text{C}$ ∏ $\text{C}$^⊥) = k ? 1 and that the supports of the minimal weight vectors of $\text{C}$ form a 2-design. Then d² ? d + 1 ? n with equality if and only if the design is a projective plane of order d ? 1. The case d² ? d + 1 = n can often be excluded with Hall's multiplier theorem on projective planes, a theorem which follows easily from the tools developed in this paper Moreover, if d² ? d + 1 > n and F = GF(2) then (d ? 1)² ? n. Examples are the generalized quadratic residue codes.     

A Turán-Type Problem on Distances in Graphs

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices pairwise at distance k has at most (n ? k + 1)²/4 pairs of vertices at distance k.     

A theorem on the number of Nash equilibria in a bimatrix game

We show that ify is an odd integer between 1 and 2ⁿ ? 1, there is ann × n bimatrix game with exactlyy Nash equilibria (NE). We conjecture that this 2ⁿ ? 1 is a tight upper bound on the number of NEs in a “nondegenerate”n × n game.