Coherent State Transforms and Abelian Varieties

We extend the coherent state transform (CST) of Hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group K=U(1)^g. We show that this transform, applied to appropriate distributions on K, gives all classical theta functions, and that, by defining on this space of theta functions an inner product related to the K-averaged heat kernel, the unitarity of the CST transform is still preserved.     

A variant of the Johnson-Lindenstrauss lemma for circulant matrices

We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in Hinrichs and Vybíral (in press) [7]. We reduce the bound on k from k=Ω(ε−2log³n) proven there to k=Ω(ε−2log²n). Our technique differs essentially from the one used in Hinrichs and Vybíral (in press) [7]. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

Active redundancy allocation for a k-out-of-n:F system of dependent components

We characterize active redundancy through compensator transform and use the reverse rule of order 2 (RR₂) property between compensator processes to investigate the problem of where to allocate a spare in a k-out-of-n:F system of dependent components through active redundancy.     

Highly connected monochromatic subgraphs

We conjecture that for n>4(k-1) every 2-coloring of the edges of the complete graph K_n contains a k-connected monochromatic subgraph with at least n-2(k-1) vertices. This conjecture, if true, is best possible. Here we prove it for k=2, and show how to reduce it to the case n<7k-6. We prove the following result as well: for n>16k every 2-colored K_n contains a k-connected monochromatic subgraph with at least n-12k vertices.     

On the uniform edge-partition of a tree

We study the problem of uniformly partitioning the edge set of a tree with n edges into k connected components, where k?n. The objective is to minimize the ratio of the maximum to the minimum number of edges of the subgraphs in the partition. We show that, for any tree and k?4, there exists a k-split with ratio at most two. For general k, we propose a simple algorithm that finds a k-split with ratio at most three in O(nlogk) time. Experimental results on random trees are also shown.     

On the chromatic number of certain highly symmetric graphs

For 1 ⩽ k ⩽ n, let V(n, k) denote the set of n(n − 1) … (n − k + 1) sequences of k distinct elements from {1, …, n}. Let us define the graph Γ(n, k) on the vertex set V(n, k) by joining two sequences if they differ at exactly one place. We investigate the chromatic number and another related parameter of these graphs. We give a sharp answer for some infinite families, using the theory of sharply transitive permutation groups. The problems discussed are related to a question of Henkin, Monk and Tarski in mathematical logic.     

A note on k-plane integral transforms

Let Π be a k-dimensional subspace of Rⁿ, n ? 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π^⊥. The k-plane transform of a measurable function ? in the direction Π at the point x″ is defined by $\text{L?(Π, x″) = ∝Π?(x′, x″) dx′}$. In this article certain a priori inequalities are established which show in particular that if $\text{? ? L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{)}$, $\text{1 ? p}\text{\$}\text{?}\text{n}\text{k}$, then ? is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces L^p(Rⁿ), p ? 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rⁿ are also obtained.     

Cauchy transforms of self-similar measures: the Laurent coefficients

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in . Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {a_n}_n=0^∞ of F. We give asymptotic formulas for {a_n}_n=0^∞ and for F^(k)(z) near the support of μ, hence the precise growth rates on |a_n| and |F^(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series.     

Homology representations arising from the half cube

We construct a CW decomposition Cn of the n-dimensional half cube in a manner compatible with its structure as a polytope. For each 3?k?n, the complex Cn has a subcomplex Cn,k, which coincides with the clique complex of the half cube graph if k=4. The homology of Cn,k is concentrated in degree k−1 and furthermore, the (k−1)st Betti number of Cn,k is equal to the (k−2)nd Betti number of the complement of the k-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type Dn act naturally on the complexes Cn,k, and thus on the associated homology groups.     

Note on the size of binary Armstrong codes

We show for binary Armstrong codes Arm(2, k, n) that asymptotically n/k ≤ 1.224, while such a code is shown to exist whenever n/k ≤ 1.12. We also construct an Arm(2, n ? 2, n) and Arm(2, n ? 3, n) for all admissible n.     

Further gossip problems

n people have distinct bits of information. They can communicate via k-party conference calls. How many such calls are needed to inform everyone of everyone else's information? Let f(n,k) be this minimum number. Then we give a simple proof that f(n,k)= [(n?k)(k?1)]+[nk] for 1?n?k², f(n,k)=2[(n?k)(k?1)] for n>k².In the 2-party case we consider the case in which certain of the calls may permit information flow in only one direction. We show that any 2n-4 call scheme that conveys everone's information to all must contain a 4-cycle, each of whose calls is “two way”, along with some other results. The method follows that of Bumby who first proved the 4-cycle conjecture.     

On the number of cycles possible in digraphs with large girth

We consider directed graphs which have no short cycles. In particular, if n is the number of vertices in a graph which has no cycles of length less than n ? k, for some constant k < ?n, then we show that the graph has no more than 3^k cycles. In addition, we show that for k ≤ ½n, there are graphs with exactly 3^k cycles. We thus are able to show that it is possible to bound the number of cycles possible in a graph which has no cycles of length less than f(n) by a polynomial in n if and only if f(n) ≥ n ? rlog(n) for some r.     

On the topology of two partition posets with forbidden block sizes

We study two subposets of the partition lattice obtained by restricting block sizes. The first consists of set partitions of {1,…,n} with block size at most k, for k≤n−2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2k+2≥n and n=3k+2. For 2k+2>n, the posets in fact have the same S_n−1-homotopy type as the order complex of Π_n−1, and the S_n-homology representation is the “tree representation” of Robinson and Whitehouse. We present similar results for the subposet of Π_n in which a unique block size k≥3 is forbidden. For 2k≥n, the order complex has the homotopy type of a wedge of (n−4)-spheres. The homology representation of S_n can be simply described in terms of the Whitehouse lifting of the homology representation of Π_n−1.     

On the Lusternik-Schnirelmann category of Stiefel manifolds

We determine the Lusternik-Schnirelmann category of real Stiefel manifolds Vn,k and quaternionic Stiefel manifolds Xn,k for n?2k which is equal to the cup-length of the mod 2 cohomology of Vn,k and the integer cohomology of Xn,k, respectively.     

Monomial patterns in the sequence Akb

We consider the pattern of zero and nonzero elements in the sequence A^kb, where A is an n × n nonnegative matrix and b is an n × 1 nonnegative column vector. We establish a tight bound of k < n for the first occurrence of a given monomial pattern, and we give a graph theoretic characterization of triples (A, b, i) such that there exists a k, k ⩾ n, for which A^kb is an i-monomial. The appearance of monomial patterns with a single nonzero entry is linked to controllability of discrete n-dimensional linear dynamic systems with positivity constraints on the state and control.     

Decomposition of the complete directed graph into k-circuits

We study the decomposition of $\text{K}{}_{\mathrm{n}}{}^1$ (the complete directed graph with n vertices) into arc-disjoint elementary k-circuits, primarily for the case k even. We solve the problem for many values of (n, k) and in particular for all n in the cases k = 4, 6, 8, and 16.     

Binomial transforms and integer partitions into parts of <Emphasis Type="Italic">k</Emphasis> different magnitudes

A relationship between the general linear group of degree n over a finite field and the integer partitions of n into parts of k different magnitudes was investigated recently by the author. In this paper, we use a variation of the classical binomial transform to derive a new connection between partitions into parts of k different magnitudes and another finite classical group, namely the symplectic group Sp. New identities involving the number of partitions of n into parts of k different magnitudes are introduced in this context.     

Matrices with maximum kth local exponent in the class of doubly symmetric primitive matrices

Let A be a primitive matrix of order n, and let k be an integer with 1?k?n. The kth local exponent of A, is the smallest power of A for which there are k rows with no zero entry. We have recently obtained the maximum value for the kth local exponent of doubly symmetric primitive matrices of order n with 1?k?n. In this paper, we use the graph theoretical method to give a complete characterization of those doubly symmetric primitive matrices whose kth local exponent actually attain the maximum value.     

The level polynomials of the free distributive lattices

We show that there exist a set of polynomials {L_k?k = 0, 1?} such that L_k(n) is the number of elements of rank k in the free distributive lattice on n generators. L₀(n) = L₁(n) = 1 for all n and the degree of L_k is k?1 for k?1. We show that the coefficients of the L_k can be calculated using another family of polynomials, P_j. We show how to calculate L_k for k = 1,…,16 and P_j for j = 0,…,10. These calculations are enough to determine the number of elements of each rank in the free distributive lattice on 5 generators a result first obtained by Church [2]. We also calculate the asymptotic behavior of the L_k's and P_j's.