Representations of algebraic groups of type <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in characteristic 2 with small weight multiplicities

Lower estimates for the mazimal weight multiplicities in irreducible representations of algebraic groups of type D _n in characteristic 2 are found. lf n ≥ 8, then either such a multiplicity is at least n—4— [n]₄, where [n]₄ is the residue of n modulo 4, or all of its weight multiplicities arc equal to 1. For groups of types B _n and D _n in odd characteristic and of type C _n , in characteristics greater than 7, similar results were obtained earlier. Bibliography: 11 titles.     

Closure for the property of having a hamiltonian prism

We prove that a graph G of order n has a hamiltonian prism if and only if the graph Cl_4n/3–4/3(G) has a hamiltonian prism where Cl_4n/3–4/3(G) is the graph obtained from G by sequential adding edges between non‐adjacent vertices whose degree sum is at least 4n/3–4/3. We show that this cannot be improved to less than 4n/3–5. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 209–220, 2007     

Vertex Partitions of K4,4-Minor Free Graphs

 We prove that a 4-connected K _4,4-minor free graph on n vertices has at most 4n−8 edges and we use this result to show that every K _4,4-minor free graph has vertex-arboricity at most 4. This improves the case (n,m)=(7,3) of the following conjecture of Woodall: the vertex set of a graph without a K _n -minor and without a -minor can be partitioned in n−m+1 subgraphs without a K _m -minor and without a -minor. Received: January 7, 1998 Final version received: May 17, 1999     

On weak and Moments Convergence of Randomly Indexed Sums

Let {X_n, n ? 1) be a sequence of independent random variables such that EX_n = a_n, E(X_n ? a_n)² = σ, n ? 1. Let {N_n, n ? 1} be a sequence of positive integer-valued random variables. Let us put In this paper we present necessary and sufficient conditions for weak and moments convergence of the sequence {(S-L_n)/s_n, n ? 1}, as n → ∞. Hermite polinomial type limit theorems are also considered. The obtained results extend the main theorem of M. Finkelstein and H. G. Tucker (1989).     

Exact Solution of the Six‐Vertex Model with Domain Wall Boundary Conditions: Antiferroelectric Phase

We obtain the large‐n asymptotics of the partition function Z_n of the six‐vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights a = sinh(γ ? t), b = sinh(γ + t), c = sinh(2γ), |t| < γ. We prove the conjecture of Zinn‐Justin, that as n → ∞, Z_n = C?₄(nω)F [1 + O(n^?1)], where ω and F are given by explicit expressions in γ and t, and ?₄(z) is the Jacobi theta function. The proof is based on the Riemann‐Hilbert approach to the large‐n asymptotic expansion of the underlying discrete orthogonal polynomials and on the Deift‐Zhou nonlinear steepest‐descent method. © 2009 Wiley Periodicals, Inc.     

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C₀(X) into C₀(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ_n δ ?h_n for a (possibly finite) sequence {x_n }_n of distinct points in X and a norm null sequence {h_n }_n of mutually disjoint functions in C₀(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Overlarge sets of disjoint Steiner quadruple systems

In this article, we construct overlarge sets of disjoint S(3, 4, 3ⁿ − 1) and overlarge sets of disjoint S(3, 4, 3ⁿ + 1) for all n ≥ 2. Up to now, the only known infinite sequence of overlarge sets of disjoint S(3, 4, v) were the overlarge sets of disjoint S(3, 4, 2ⁿ) obtained from the oval conics of desarguesian projective planes of order 2ⁿ. © 1999 John Wiley & Sons, Inc. J Combin Design 7: 311–315, 1999     

Almost Resolvable Maximum Packings of Complete Graphs with 4-Cycles

If the complete graph K _n has vertex set X, a maximum packing of K _n with 4-cycles, (X, C, L), is an edge-disjoint decomposition of K _n into a collection C of 4-cycles so that the unused edges (the set L) is as small a set as possible. Maximum packings of K _n with 4-cycles were shown to exist by Sch?nheim and Bialostocki (Can. Math. Bull. 18:703–708, 1975). An almost parallel class of a maximum packing (X, C, L) of K _n with 4-cycles is a largest possible collection of vertex disjoint 4-cycles (so with ?/4?{\lfloor/4\rfloor} 4-cycles in it). In this paper, for all orders n, except 9, which does not exist, and possibly 23, 41 and 57, we exhibit a maximum packing of K _n with 4-cycles so that the 4-cycles in the packing are resolvable into almost parallel classes, with any remaining 4-cycles being vertex disjoint. [Note: The three missing orders have now been found, and appear in Billington et al. (to appear).]     

Books Versus triangles

A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ?n₂/4?; + 1 edges. Rademacher proved that G contains at least ?n/2?; triangles, and several authors proved that G contains a book of size at least n/6. We prove the following “linear combination” of these two results. Suppose that and the maximum size of a book in G is less than αn/2. Then G contains at least triangles as n→?. This is asymptotically sharp. On the other hand, for every , there exists β>0 such that G contains at least βn₃ triangles. It remains an open problem to determine the largest possible β in terms of α. Our short proof uses the triangle removal lemma, although there is another approach which avoids this. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Maximal Rectilinear Crossing of Cycles

We address the following question: In drawing a cycle on n vertices (or a graph all of whose degrees are 2) in the plane with straight line arcs, how many crossings can there be? A complete answer is given; namely, if n is odd, the number of crossings can be anything up to n(n?3)/2 except n(n?3)/2?1. For n even, the number of crossings in one cycle can be any integer up to n(n?4)/2+1; general bivalent even graphs can achieve any integer up to n(2n?7)/4 as the number of crossings.     

Complete Riemannian manifolds with pointwise pinched curvature

An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold M ⁿ . We prove that if n≥4 and the curvature operator of M ⁿ is pointwise pinched, or if n=3 and the Ricci curvature of M ³ is pointwise pinched, then M ⁿ is compact. Oblatum 4-II-1999 & 10-XI-1999?Published online: 21 February 2000     

Local Solutions of Semilinear Wave Equations in Hs+1

The Cauchy problem for the wave equation with power type non-linearity ±∣u∣u and data in H^s+1(ℝⁿ)×H^s(ℝⁿ) is considered, where 0<s<(n/2)−1 and n≥3. Under the growth restriction σ_*⩽4/(n−2−2s) in many cases the existence of a local solution with u(t)∈H^s+1(ℝⁿ) is shown which is unique in a closely related class.     

A Note on the Multiplicative Semigroups Z n Whose Bi-ideals Are Quasi-ideals

We show that for any positive integer n, the multiplicative semigroup Z_n has the property that the set of bi-ideals and the set of quasi-ideals coincide if and only if either n = 4 or n is square-free.AMS Subject Classification (1991): 20M10     

Congruences for generalized Frobenius partitions with 4 colors

We present some congruences involving the functions c?₄(n) and which denote, respectively, the number of generalized Frobenius partitions of n with 4 colors and 4-order generalized Frobenius partitions of n with 4 colors.     

On non-Kählerianity of nonuniform lattices¶in SO(n, 1)(n≥ 4)

In this note, we will show that no nonuniform lattice of SO(n, 1)(n≥ 4) is the fundamental group of a quasi-compact K?hler manifold Received: 18 July 2000 / Revised version: 4 September 2000     

Optimal holey packings OHP4(2, 4, n ,3)'s

An optimal holey packing OHP_d(2, k, n, g) is equivalent to a maximal (g + 1)‐ary (n, k, d) constant weight code. In this paper, we provide some recursive constructions for OHP_d(2, k, n, g)'s and use them to investigate the existence of an OHP₄(2, 4, n, 3) for n ≡ 2, 3 (mod 4). Combining this with Wu's result ( 18 ), we prove that the necessary condition for the existence of an OHP₄(2, 4, n, 3), namely, n ≥ 5 is also sufficient, except for n ∈ {6, 7} and except possibly for n = 26. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 111–123, 2006     

The threshold of effective damping for semilinear wave equations

In this paper, we obtain the global existence of small data solutions to the Cauchy problem in space dimension n ≥ 1, for p > 1 + 2 ∕ n, where μ is sufficiently large. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. In particular, for μ ≥ 2 + n, the damping term is effective with respect to the L¹ ? L² low‐frequency estimates for the solution and its energy. In this case, we may prove global existence in any space dimension n ≥ 3, by assuming smallness of the initial data in some weighted energy space. In space dimension n = 1,2, we only assume smallness of the data in some Sobolev spaces, and we introduce an approach based on fractional Sobolev embedding to improve the threshold for global existence to μ ≥ 5 ∕ 3 in space dimension n = 1 and to μ ≥ 3 in space dimension n = 2. Copyright © 2014 John Wiley & Sons, Ltd.     

m‐ary Search trees when m ≥ 27: A strong asymptotics for the space requirements

It is known that the joint distribution of the number of nodes of each type of an m‐ary search tree is asymptotically multivariate normal when m ≤ 26. When m ≥ 27, we show the following strong asymptotics of the random vector X_n = ^t(X, … , X), where X denotes the number of nodes containing i ? 1 keys after having introduced n ? 1 keys in the tree: There exist (nonrandom) vectors X, C, and S and random variables ρ and φ such that (X_n ? nX)/n ? ρ(C cos(τ₂log n + φ) + S sin(τ₂log n + φ)) →_n→∞ 0 almost surely and in L²; σ₂ and τ₂ denote the real and imaginary parts of one of the eigenvalues of the transition matrix, having the second greatest real part. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

The determining number of a Cartesian product

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G=G□?□G is the prime factor decomposition of a connected graph then Det(G)=max{Det(G)}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q_n)=?log₂n?+1 which matches the lower bound, and that Det(K)=?log₃(2n+1)?+1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(Hⁿ)=Θ(logn). © 2009 Wiley Periodicals, Inc. J Graph Theory