Convex Chain Research Articles

ON RANDOM CONVEX CHAINS, ORTHOGONAL POLYNOMIALS, PF SEQUENCES AND PROBABILISTIC LIMIT THEOREMS

Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull of $(0,1)$, $(1,0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain $T_n$. A three-term recursion for the probability generating function $G_n$ of the number $f_0(T_n)$ of vertices of $T_n$ is proved. Via the link to orthogonal polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in $(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of probabilistic consequences of this surprising and remarkable fact are discussed in detail.

Impact of Isomeric Dicarboxylate Ligands on the Formation of One-Dimensional Coordination Polymers and Metallocycles: A Novel cis→trans Isomerization.

A series of Co(II), Ni(II) and Cu(II) coordination polymers and dinuclear metallocycles containing 4-aminopyridine (4-ampy) and benzenedicarboxylate ligands, {[M(4-ampy)2(1,4-BDC)]·H2O·CH3CH2OH}n (M = Ni, 1a; Co, 1b, 1,4-H2BDC = benzene-1,4-dicarboxylic acid), {[Ni2(4-ampy)4(1,3-BDC)2]·H2O·CH3CH2OH}n (1,3-H2BDC = benzene-1,3-dicarboxylic acid), 2, [M2(4-ampy)4(1,2-BDC)2] (M = Ni, 3a; Co, 3b, 1,2-H2BDC = benzene-1,2-dicarboxylic acid), [Co(4-ampy)2(1,3-BDC)]n, 4, {[Cu(4-ampy)2(1,4-BDC)] CH3CH2OH}n, 5a, and {[Cu(4-ampy)2(1,4-BDC)]·H2O}n, 5b·H2O, are reported, which were hydrothermally prepared and structurally characterized by using single crystal X-ray diffraction. Complexes 1a and 1b are isomorphous 1D zigzag chains, while 2 displays a concave–convex chain and 3a and 3b are dinuclear metallocycles that differ in the boding modes of the 1,2-BDC2− ligands, forming a 3D and a 2D supramolecular structures with the pcu and sql topologies, respectively. Complex 4 exhibit a 1D helical chain and complexes 5a and 5b·H2O are 1D linear and zigzag chains, in which the Cu2-1,4-BDC2− units adopt the cis and trans configurations, respectively. A novel irreversible structural transformation due to cis→trans isomerization of the Cu2-1,4-BDC2− units was observed in 5b⋅H2O and 5a upon water adsorption of the desolvated product of 5b·H2O.

CC-MUSIC: An Optimization Estimator for Mutual Coupling Correction of L-Shaped Nonuniform Array with Single Snapshot

For the case of the single snapshot, the integrated SNR gain could not be obtained without the multiple snapshots, which degrades the mutual coupling correction performance under the lower SNR case. In this paper, a Convex Chain MUSIC (CC-MUSIC) algorithm is proposed for the mutual coupling correction of the L-shaped nonuniform array with single snapshot. It is an online self-calibration algorithm and does not require the prior knowledge of the correction matrix initialization and the calibration source with the known position. An optimization for the approximation between the no mutual coupling covariance matrix without the interpolated transformation and the covariance matrix with the mutual coupling and the interpolated transformation is derived. A global optimization problem is formed for the mutual coupling correction and the spatial spectrum estimation. Furthermore, the nonconvex optimization problem of this global optimization is transformed as a chain of the convex optimization, which is basically an alternating optimization routine. The simulation results demonstrate the effectiveness of the proposed method, which improve the resolution ability and the estimation accuracy of the multisources with the single snapshot.

Three novel Cd(II) coordination polymers: Synthesis, crystal structures, fluorescence and thermal properties

Three novel Cd(II) coordination polymers, {[Cd(C 4BIm)(N 3)(OAc)]·C 2H 5OH} n ( 1), [Cd(C 4BIm)(H 2O) 3(SO 4)] n ( 2) (C 4BIm = 1,4-bis(benzimidazolyl)butane) and [Cd(bbbm) 1.5(NO 3) 2] n ( 3) (bbbm = 1,1′-(1,4-butanediy)bis-1H-benzimidazole) have been prepared and characterized spectroscopically and crystallographically. In polymer 1, C 4BIm–Cd chains and N 3–Cd chains criss-cross to a layer structure. Polymer 2 possesses a concave–convex chain structure. Polymer 3 exhibits a two-dimensional (2-D) rhomboid grid network, the dimensions of the grid is 25.807 × 13.771 Å, and the diagonal-to-diagonal distances are 28.608 × 21.145 Å. The fluorescence properties and the thermal stabilities of the three polymers were investigated.

Longest convex chains

AbstractAssume Xn is a random sample of n uniform, independent points from a triangle T. The longest convex chain, Y, of Xn is defined naturally (see the next paragraph). The length |Y| of Y is a random variable, denoted by Ln. In this article, we determine the order of magnitude of the expectation of Ln. We show further that Ln is highly concentrated around its mean, and that the longest convex chains have a limit shape. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

The Exact Distribution of the Number of Vertices of a Random Convex Chain

Assume that n points P1,…,Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1,…,Pn, and (1, 0). The vertices of the convex hull form a convex chain. Let be the probability that the convex chain consists – apart from the points (0, 1) and (1, 0) – of exactly k of the points P1,…,Pn. Bárány, Rote, Steiger, and Zhang [3] proved that . The values of are determined for k = 1,…,n − 1, and thus the distribution of the number of vertices of a random convex chain is obtained. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability.

Multipolytopes and convex chains

For a simple complete multipolytope \(\mathcal{P}\) in ℝn, Hattori and Masuda defined a locally constant function \(DH_\mathcal{P} \) on ℝn minus the union of hyperplanes associated with \(\mathcal{P}\), which agrees with the density function of an equivariant complex line bundle over a Duistermaat-Heckman measure when \(\mathcal{P}\) arises from a moment map of a torus manifold. We improve the definition of \(DH_\mathcal{P} \) and construct a convex chain \(\overline {DH_\mathcal{P} } \) on ℝn. The well-definiteness of this convex chain is equivalent to the semicompleteness of the multipolytope \(\mathcal{P}\). Generalizations of the Pukhlikov-Khovanskii formula and an Ehrhart polynomial for a simple lattice multipolytope are given as corollaries. The constructed correspondence ⨑ub;simple semicomplete multipolytopes⫂ub; →; ⨑ub;convex chains⫂ub; is surjective but not injective. We will study its “kernel.”

On the development of the intersection of a plane with a polytope

Define a “slice” curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex “openings” of a planar convex chain.

Metrizability, monotone normality, and other strong properties in trees

Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFC-independent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.

Some chain visibility problems in a simple polygon

In this paper, the notions of convex chain visibility and reflex chain visibility of a simple polygonP are introduced, and some optimal algorithms concerned with convex- and reflex-chain visibility problems are described. For a convex-chain visibility problem, two linear-time algorithms are exhibited for determining whether or notP is visible from a given convex chain; one is the turn-checking approach and the other is the decomposition approach based on checking edge visibilities. We also present a linear-time algorithm for finding, if any, all maximal convex chains of a given polygonP from whichP is visible, where a maximal convex chain is a convex chain which does not properly include any other convex chains. It can be made by showing that there can be at most four visible maximal convex chains inP with an empty kernel. By similar arguments, we show that the same problems for reflex chain visibility can be easily solved in linear time.

Convex polyhedral chains: a representation for geometric data

A representation scheme for general polyhedra in arbitrary dimensions is presented. Each polyhedron is represented as a convex chain, i.e. an algebraic sum of convex polyhedra (cells). Each cell in turn is represented as the intersection of halfspaces and encoded in a vector. The notion of vertices is abandoned completely. It is shown how this approach allows the decomposition of set operations (such as intersection) on polyhedra into two independent steps. The first step consists of a collection of vector operations; the second step is a garbage collection where vectors that represent empty cells are eliminated. No special treatment of singular intersection cases is needed. This approach to set operations is significantly different from algorithms that have been proposed previously.