Partitioning Sets of Triples into Small Planes

We study partitions of the set of all ₃ ^v triples chosen from a v-set intopairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2,2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions)or copies of some planes of each type (mixed partitions).We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in severalcases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We constructsuch partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, andan affine partition for v = 18. Using these as starter partitions, we prove that Fano partitionsexist for v = 7 ⁿ + 1, 13 ⁿ + 1,27 ⁿ + 1, and affine partitions for v = 8 ⁿ + 1,9 ⁿ + 1, 17 ⁿ + 1. In particular, both Fano and affine partitionsexist for v = 3⁶ⁿ + 1. Using properties of 3-wise balanced designs, weextend these results to show that affine partitions also exist for v = 3²ⁿ .Similarly, mixed partitions are shown to exist for v = 8 ⁿ ,9 ⁿ , 11 ⁿ + 1.     

A bijection between certain non-crossing partitions and sequences

We present a bijection between non-crossing partitions of the set [2n+1] into n+1 blocks such that no block contains two consecutive integers, and the set of sequences such that 1?s_i?i, and if s_i=j, then s_i-r?j-r for 1?r?j-1.     

Enumeration of M-partitions

An M-partition of a positive integer m is a partition of m with as few parts as possible such that every positive integer less than m can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahon's perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers m, the number of M-partitions of m is equal to the number of binary partitions of 2ⁿ⁺¹-1-m, where . In this note we extend O’Shea's result to cover all numbers m.     

Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian

Let G be a simple algebraic group defined over C and T be a maximal torus of G. For a dominant coweight λ of G, the T-fixed point subscheme of the Schubert variety in the affine Grassmannian GrG is a finite scheme. We prove that for all such λ if G is of type A or D and for many of them if G is of type E, there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to λ and the ring of functions (twisted by certain line bundle on GrG) of . We use this fact to give a geometrical proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of A,D,E type and lattice vertex algebras.     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

R-analogue of the Burnside ring of profinite groups and free Lie algebras

Let R be a finite-dimensional torsion-free special λ-ring. In this paper we generalize the results in Dress and Siebeneicher (Adv. in Math. 70 (1988) 89; 78 (1989) 1) by constructing R-analogue of the Burnside ring of profinite groups . In particular, we remark that the (Grothendieck) Lie-module denominator identity of free Lie algebras in Oh (Necklace rings and logarithmic functions, preprint, KIAS, 2003) is closely related to the canonical isomorphism between and Grothendieck's ring of formal power series with coefficients in R and constant term 1.     

Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say (p₁,p₂,…,p_k are distinct odd primes and a_i>0 for each i?1), there are t distinct reflexible regular embeddings of the complete bipartite graph K_n,n up to isomorphism, where t=1 if a=0, t=2^k if a=1, t=2^k+1 if a=2, and t=3·2^k+1 if a?3. And, there are s distinct self-Petrie dual regular embeddings of K_n,n up to isomorphism, where s=1 if a=0, s=2^k if a=1, s=2^k+1 if a=2, and s=2^k+2 if a?3.     

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph Γ with diameter D?4, valency k?3, intersection numbers b_i,c_i, distance matrices A_i, and eigenvalues θ₀>θ₁>?>θ_D. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat_X(C) generated by , where A=A₁ and denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever for 0?i?D. By the endpoint of W we mean . Assume W is thin with endpoint 2. Observe is a one-dimensional eigenspace for ; let η denote the corresponding eigenvalue. It is known where , and d=⌊D/2⌋. To describe the structure of W we distinguish four cases: (i) ; (ii) D is odd and ; (iii) D is even and ; (iv) . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D-1-e where e=1 in case (iii) and e=0 in case (iv). Let v denote a nonzero vector in . We show W has a basis , where E_i denotes the primitive idempotent of A associated with θ_i and where the set S is {1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.     

Groupoid normalizers of tensor products

We consider an inclusion B⊆M of finite von Neumann algebras satisfying B^′∩M⊆B. A partial isometry v∈M is called a groupoid normalizer if vBv^∗,v^∗Bv⊆B. Given two such inclusions Bi⊆Mi, i=1,2, we find approximations to the groupoid normalizers of in , from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis , i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v∈M satisfying vBv^∗⊆B and v^∗v,vv^∗∈B.     

A cyclic group action on resolutions of quadruple systems

We prove that the necessary conditions for a quadruple system with a cyclic resolution to exist are sufficient for all possible v and all .     

A Borsuk-Ulam type theorem for sphere bundles over stunted projective spaces

A CW complex B is described as I-trivial if there does not exist a Z₂-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. For n>1, we determine all positive integers m for which the stunted projective space is I-trivial, where F=R,C or H.     

Clique partitions of complements of forests and bounded degree graphs

In this paper, we prove that for any forest F⊂K_n, the edges of E(K_n)?E(F) can be partitioned into O(nlogn) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in K_n.In the second part of the paper, we show that for n sufficiently large and any ε∈(0,1], if a graph G has maximum degree O(n^1-ε), then the edges of E(K_n)?E(G) can be partitioned into cliques provided there exist certain Steiner systems. Furthermore, we show that there are such graphs G for which Ω(ε²n^2-2ε) cliques are required in every clique partition of E(K_n)?E(G).     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).