Rigidity of Schubert classes in orthogonal Grassmannians

A Schubert class σ in the cohomology of a homogeneous variety X is called rigid if the only projective subvarieties of X representing σ are Schubert varieties. A Schubert class σ is called multi rigid if the only projective subvarieties representing positive integral multiples of σ are unions of Schubert varieties. In this paper, we discuss the rigidity and multi rigidity of Schubert classes in orthogonal Grassmannians. For a large set of non-rigid classes, we provide explicit deformations of Schubert varieties using combinatorially defined varieties called restriction varieties. We characterize rigid and multi rigid Schubert classes of Grassmannian and quadric type. We also characterize all the rigid classes in OG(2, n) if n > 8.     

Resolvable Coverings of 2-Paths by Cycles

 Let K _n be the complete graph on n vertices. A C(n,k,λ) design is a multiset of k-cycles in K _n in which each 2-path (path of length 2) of K _n occurs exactly λ times. A C(lk,k,1) design is resolvable if its k-cycles can be partitioned into classes so that every vertex appears exactly once in each class. A C(n,n,1) design gives a solution of Dudeney's round table problem. It is known that there exists a C(n,n,1) design when n is even and there exists a C(n,n,2) design when n is odd. In general the problem of constructing a C(n,n,1) design is still open when n is odd. Necessary and sufficient conditions for the existence of C(n,k,λ) designs and resolvable C(lk,k,1) designs are known when k=3,4. In this paper, we construct a resolvable C(n,k,1) design when n=p ^e +1 ( p is a prime number and e≥1) and k is any divisor of n with k≠1,2. Received: October, 2001 Final version received: September 4, 2002 RID="*" ID="*" This research was supported in part by Grant-in-Aid for Scientific Research (C) Japan     

Gray code enumeration of families of integer partitions

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes P_δ(n, k) and D(n, k) of partitions of n into parts of size at most k in which, for P_δ(n, k), the parts are congruent to one modulo δ and, for D(n, k), the parts are distinct. It is shown that the elements of these classes can be listed so that the only change between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n, k), δ = 1.     

The structure of digraphs associated with the congruence <Emphasis Type="Italic">x</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> ≡ <Emphasis Type="Italic">y</Emphasis> (mod <Emphasis Type="Italic">n</Emphasis>)

We assign to each pair of positive integers n and k ⩾ 2 a digraph G(n, k) whose set of vertices is H = {0, 1, ..., n − 1} and for which there is a directed edge from a ∈ H to b ∈ H if a ^k ≡ b (mod n). We investigate the structure of G(n, k). In particular, upper bounds are given for the longest cycle in G(n, k). We find subdigraphs of G(n, k), called fundamental constituents of G(n, k), for which all trees attached to cycle vertices are isomorphic.     

Asymptotic results on infinite tandem queueing networks

We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k. We establish a duality property by which w(n, k) and the “idle times”y(n, k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider two versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0,k), k∈ℕ} and {t(n, 0), n∈ℕ}, are given. In the half-plane version, the sequence {t(n, 0), n∈ℕ} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergodic results for both versions of the model. For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n∈ℕ} converges weakly with k to a limiting distribution, which is invariant for the queueing operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1/n∑ _k=1 ⁿ w(0, k) converges almost surely and in L ₁ to a finite constant. This constitutes a first step in trying to prove that {t(n,k), n∈ℤ} converges weakly with k to an invariant limiting distribution. Received: 23 March 1999 / Revised version: 5 January 2000 / Published online: 12 October 2000     

Digraphs that have at most one walk of a given length with the same endpoints

Let Θ(n,k) be the set of digraphs of order n that have at most one walk of length k with the same endpoints. Let θ(n,k) be the maximum number of arcs of a digraph in Θ(n,k). We prove that if n≥5 and k≥n−1 then θ(n,k)=n(n−1)/2 and this maximum number is attained at D if and only if D is a transitive tournament. θ(n,n−2) and θ(n,n−3) are also determined.     

The Cohomology Ring of Weight Varieties and Polygon Spaces

We use a theorem of S. Tolman and J. Weitsman (The cohomology rings of Abelian symplectic quotients, math. DG/9807173) to find explicit formulæ for the rational cohomology rings of the symplectic reduction of flag varieties in ⁿ, or generic coadjoint orbits of SU(n), by (maximal) torus actions. We also calculate the cohomology ring of the moduli space of n points in P^k, which is isomorphic to the Grassmannian of k planes in ⁿ, by realizing it as a degenerate coadjoint orbit.     

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X ₁₁, . . . , X _nn , T) be the characteristic polynomial of the generic matrix (X _ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.     

On free semilattice-ordered semigroups satisfying x n =x

Let B(k,0,n) denote the group with k generators which is free in the group variety defined by the identity x ⁿ =1. Let B _slo (k,1,n) denote the semilattice-ordered semigroup with k generators which is free in the semilattice-ordered semigroup variety defined by the identity x ⁿ =x. We prove a generalization of the Green-Rees theorem: B _slo (k,1,n) is finite for all k≥1 if and only if B(k,0,n−1) is finite for all k≥1. We find a formula for card(B _slo (1,1,n)). We construct B _slo (k,1,n) for some concrete values of k and n.     

On the Range of Possible Integrities of Graphs G(n, k)

We discuss the range of values for the integrity of a graphs G(n, k) where G(n, k) denotes a simple graph with n vertices and k edges. Let I _max(n, k) and I _min(n, k) be the maximal and minimal value for the integrity of all possible G(n, k) graphs and let the difference be D(n, k) = I _max(n, k) − I _min(n, k). In this paper we give some exact values and several lower bounds of D(n, k) for various values of n and k. For some special values of n and for s < n ^1/4 we construct examples of graphs G _n  = G _n (n, n + s) with a maximal integrity of I(G _n ) = I(C _n ) + s where C _n is the cycle with n vertices. We show that for k = n ²/6 the value of D(n, n ²/6) is at least \frac?6-13n{\frac{\sqrt{6}-1}{3}n} for large n.     

On the size of graphs labeled with a condition at distance two

A labeling of graph G with a condition at distance two is an integer labeling of V(G) such that adjacent vertices have labels that differ by at least two, and vertices distance two apart have labels that differ by at least one. The lambda-number of G, λ(G), is the minimum span over all labelings of G with a condition at distance two. Let G(n, k) denote the set of all graphs with order n and lambda-number k. In this paper, we examine the sizes of graphs in G(n, k). We modify Chvàtal's result on non-hamiltonian graphs to obtain a formula for the minimum size of a graph in G(n, k), and we use an algorithmic approach to obtain a formula for the maximum size. Finally, we show that for any integer j between the maximum and minimum sizes there exists a graph with size j in G(n, k). © 1996 John Wiley & Sons, Inc.     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

Ideal einstein,conformally flat and semi-symmetric immersions

Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR ^m (ε) of constant sectional curvature ε satisfies a basic inequality δ(n ₁,…,n _k )≤c(n ₁,…,n _k )H ²+b(n ₁,…,n _k )ε, whereH is the mean curvature of the immersion, andc(n ₁,…,n _k ) andb(n ₁,…,n _k ) are constants depending only onn ₁,…,n _k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n ₁,…,n _k ). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n ₁+…+n _k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n ₁+…+n _k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms. The author was supported by the NSFC and RFDP.     

Rang maximal pourT_{P^n }

Letk be an algebraically closed field,P ⁿ the n-dimensional projective space overk andT _P ⁿ the tangent vector bundle ofP ⁿ . In this paper I prove the following result: for every integerl, for every non-negative integers, ifZ _s is the union ofs points in sufficiently general position inP ⁿ , then the restriction mapH ⁰(P ⁿ ,T _P ⁿ (l)) →H ⁰(Z _s,T _P ⁿ (l)_{|z s} ) has maximal rank. This result implies that the last non-trivial term of the minimal free resolution of the homogeneous ideal ofZ _s is the conjectured one by the Minimal Resolution Conjecture of Anna Lorenzini (cf. [Lo]).     

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k ₁, ..., k _r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.     

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.     

Orders on Multisets and Discrete Cones

We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees, words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei, Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.      

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

Cocycle Twisting of E(n)-Module Algebras and Applications to the Brauer Group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group BQ(k,E(n)) of E(n) that are isomorphic. For any triangular structure R on E(n) we prove that the subgroup BM(k,E(n),R) of BQ(k,E(n)) arising from R is isomorphic to a direct product of BW(k), the Brauer-Wall group of the ground field k, and Sym_n(k), the group of n × n symmetric matrices under addition. For a general quasi-triangular structure R on E(n) we construct a split short exact sequence having BM(k,E(n),R⁾ as a middle term and as kernel a central extension of the group of symmetric matrices of order r < n (r depending on R). We finally describe how the image of the Hopf automorphism group inside BQ(k,E(n)) acts on Sym_n (k).     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.