λ-fold indecomposable large sets of Steiner triple systems

A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).     

Embedding of a pseudo-residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let v, t and λ₁, 0 ? i ? t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ₀, λ₁,…, λ_t; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λ_t blocks, 0 ? i ? t, and (3) for every block A in $\text{U}$, |A| ?K. It is well-known that if K consists of a singleton k, then λ₀,…, λ_{t ? 1} can be determined from υ, t, k and λ_t. Hence, we shall denote a (λ₀,…, λ_t; {k}, υ)t-design by S_λ(t, k, υ), where λ = λ_t. A Möbius plane M is an S₁(3, q + 1, q² + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ₀ = q³ + q ? 1, λ₁ = q² + q, λ₂ = q + 1, λ₃ = 1, K = {q + 1, q, q ? 1}, and υ = q² ? 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition.     

The spectrum for overlarge sets of hybrid triple systems

A hybrid triple system of order v and index λ,denoted by HTS(v,λ),is a pair(X,B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X,such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v,λ),denoted by OLHTS(v,λ),is a collection {(Y \{y},Ai)}i,such that Y is a(v+1)-set,each(Y \{y},Ai) is an HTS(v,λ) and all Ais form a partition of all cyclic triples and transitive triples on Y.In this paper,we shall discuss the existence problem of OLHTS(v,λ) and give the following conclusion: there exists an OLHTS(v,λ) if and only if λ=1,2,4,v ≡ 0,1(mod 3) and v≥4.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Some infinite families of large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection {(X, B_i) : 1 ≤ i ≤ v ? 2}, where every (X,B_i) is a KTS(v) and all B_i form a partition of all triples on X. Many researchers have studied the existence of LKTS(v) for a long time. In [13], the author introduced a concept—large set of generalized Kirkman systems (LGKS), which plays an important role in the discussion of LKTS. In this article, we give a new construction for LGKS and obtain some new results of LKTS, that is, there exists an LKTS(6u + 3) for u = qⁿ, where n ≥ 1, q ≡ 7 (mod 12) and q is a prime power. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 202–212, 2008     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x'= A(t)x + f(t, x, λ), t≠τi and x(τ+i) = Bix(τi) + gi(x(τi), λ), i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x'= A(t)x, t≠τi and x(τ+i) = Bix(τi), i ∈ N admits a nonuniform (μ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.     

Characterization of unigraphic and unidigraphic integer-pair sequences

Given a graph (digraph) G with edge (arc) set E(G) = {(u₁}, υ₁), (u₂, υ₂),?,(u_q, υ_q, where q = |E(G)|, we can associate with it an integer-pair sequence S_G = ((a₁, b₁), (a₂, b₂),?, (a_q, b_q)) where a_i, b_i are the degrees (indegrees) of u_i, υ_i respectively. An integer- pair sequence S is said to be graphic (digraphic) if there exists a graph (digraph) G such that _SG = S. In this paper we characterize unigraphic and unidigraphic integer-pair sequences.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

Caps in PG(n,q),q even,n⩾3

Letm′₂(3,q) be the largest value ofk(k<q ²+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′₂(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4.     

Incidence graphs and subdesigns of semisymmetric designs

A semisymmetric design is a connected incidence structure satisfying; two points (blocks) are on 0 or λ blocks (points). Every block (point) is incident with k points (blocks). Properties of the incidence graph of these structures are investigated, leading to bounds on its diameter (d?k if λ = 2, d?[2k/(λ + 1)]+ 1 if λ > 2), and the number of points of these structures (υ?2^k-1 if λ = 2, υ?k2^{[2k/(λ + 1)]} if λ > 2). Bounds are also found for semisymmetric designs containing a subdesign. We give characterizations of semisymmetric designs with λ = 2 (semibiplanes) which contain a subdesign and achieve the bounds. This leads to a construction for a semibiplane with parameters υ = 2^r-1 (q²?1), k = q+q₁+?+q_r, where q_r is aprime power, q_i = q²_i+1 and q=q²₁.     

Strip Bundles and the Highest Weight Crystals Over Quantum Infinite Rank Affine Algebras

We give new realizations of the highest weight crystals B(λ) over the quantum infinite rank affine algebras U _q (A _∞), U _q (B _∞), U _q (C _∞), and U _q (D _∞) using strip bundles, which are related with Nakajima monomials.     

On the space Ext for the group SL(2, q)

We consider the space Ext ^r (A,B) = Ext _KG ^r (A, B), where G = SL(2, q), q = p ⁿ , K is an algebraically closed field of characteristic p, A and B are irreducible KG-modules, and r ? 1. Carlson [6] described a basis of Ext ^r (A, B) in arithmetical terms. However, there are certain difficulties concerning the dimension of such a space. In the present article, we find the dimension of Ext ^r (A, B) for r = 1, 2 (in the above-mentioned article, Carlson presents the corresponding assertions without proofs; moreover, there are errors in their formulations). As a corollary, we find the dimension of the space H ^r (G, A), where A is an irreducible KG-module. This result can be used in studying nonsplit extensions of L ₂(q).     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

Stochastic gradient algorithm with random truncations

Let f : R^d × R^d → R be a Borel-measurable function which satisfies ∫_R^d′|f(θ, x) < ∞, ∨θ ϵ R^d, where q₀(·) is a probability measure on (R^d′, B^d′). The problem of minimization of the function f₀(θ) = ∫_R^d′(θ, x)q₀(d), θ ϵ R^d, is considered for the case when the probability measure q₀(·) is unknown, but a realization of a non-stationary random process {X_n}_n⩾1 whose single probability measures in a certain sense tend to q₀(·), is available. The random process {X_n}_n⩾1 is defined on a common probability space, R^′-valued, correlated and satisfies certain uniform mix conditions. The function f(·, ·) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of f₀(·), and its almost sure convergence is proved.     

Hall parameters of complete and complete bipartite graphs

Given a graph G, for each υ ∈V(G) let L(υ) be a list assignment to G. The well‐known choice number c(G) is the least integer j such that if |L(υ)| ≥j for all υ ∈V(G), then G has a proper vertex colouring ? with ?(υ) ∈ L (υ) (?υ ∈V(G)). The Hall number h(G) is like the choice number, except that an extra non‐triviality condition, called Hall's condition, has to be satisfied by the list assignment. The edge‐analogue of the Hall number is called the Hall index, h′(G), and the total analogue is called the total Hall number, h″(G), of G. If the stock of colours from which L(υ) is selected is restricted to a set of size k, then the analogous numbers are called k‐restricted, or restricted, Hall parameters, and are denoted by h_k(G), h′_k(G) and h″_k(G). Our main object in this article is to determine, or closely bound, h′(K), h″(K_n), h′(K_m,n) and h′_k(K_m,n). We also answer some hitherto unresolved questions about Hall parameters. We show in particular that there are examples of graphs G with h′(G)?h′(G ? e)>1. We show that there are examples of graphs G and induced subgraphs H with h_k(G)<h_k(H) [this phenomenon cannot occur with unrestricted Hall numbers]. We also give an example of a graph G and an integer k such that h_k(G)<χ(G)<h(G). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 208–237, 2002     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.