On the X-ray number of almost smooth convex bodies and of convex bodies of constant width

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6

Rigid ball-polyhedra in Euclidean 3-space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex-edge-face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.Comment: 11 pages, 2 figure

Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.Comment: 17 page

Two generalizations of Kohonen clustering

The relationship between the sequential hard c-means (SHCM), learning vector quantization (LVQ), and fuzzy c-means (FCM) clustering algorithms is discussed. LVQ and SHCM suffer from several major problems. For example, they depend heavily on initialization. If the initial values of the cluster centers are outside the convex hull of the input data, such algorithms, even if they terminate, may not produce meaningful results in terms of prototypes for cluster representation. This is due in part to the fact that they update only the winning prototype for every input vector. The impact and interaction of these two families with Kohonen's self-organizing feature mapping (SOFM), which is not a clustering method, but which often leads ideas to clustering algorithms is discussed. Then two generalizations of LVQ that are explicitly designed as clustering algorithms are presented; these algorithms are referred to as generalized LVQ = GLVQ; and fuzzy LVQ = FLVQ. Learning rules are derived to optimize an objective function whose goal is to produce 'good clusters'. GLVQ/FLVQ (may) update every node in the clustering net for each input vector. Neither GLVQ nor FLVQ depends upon a choice for the update neighborhood or learning rate distribution - these are taken care of automatically. Segmentation of a gray tone image is used as a typical application of these algorithms to illustrate the performance of GLVQ/FLVQ

Contact numbers for congruent sphere packings in Euclidean 3-space

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.Comment: 18 page

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(\alpha^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(\omega)$ we discovered for any $k$-stress $\omega$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(\alpha^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr

Parasitoids (Hymenoptera) of leaf-spinning moths (Lepidoptera) feeding on Vaccinium uliginosum L. along an ecological gradient in central European peat bogs

Parasitoids of leaf-spinning Lepidoptera associated with two isolated central European peat bogs were investigated. Five families of parasitoid Hymenoptera (Braconidae, Ichneumonidae, Eulophidae, Pteromalidae and Encyrtidae) were recorded. Three categories were recognised: (1) primary parasitoids, (2) facultative hyperparasitoids and (3) obligatory hyperparasitoids. Ten species of Braconidae, five species and seven marked morphospecies among Ichneumonidae, and three species of Chalcidoidea were identified. Despite of some niche-specific (but less host-specific) parasitoids, all these hymenopterans are likely to be generalists and none of them were confirmed to be habitat and/or host specialists. Unlike their eurytopic (opportunistic tyrphoneutral) parasitoids, the Lepidoptera hosts associated with peat bogs are partially highly stenotopic (tyrphobionts and tyrphophiles). The occurrence of parasitoids compared to their potential hosts was structured along an ecological (mesoclimatic) gradient, so most parasitoids were recorded from margins while stenotopic (narrow habitat adaptation) moths were mostly distributed near the centre of the bog habitat

An augmented Lagrangian decomposition method for quasi-separable problems in MDO

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem, and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to KKT points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems. I

FSL-BM: Fuzzy Supervised Learning with Binary Meta-Feature for Classification

This paper introduces a novel real-time Fuzzy Supervised Learning with Binary Meta-Feature (FSL-BM) for big data classification task. The study of real-time algorithms addresses several major concerns, which are namely: accuracy, memory consumption, and ability to stretch assumptions and time complexity. Attaining a fast computational model providing fuzzy logic and supervised learning is one of the main challenges in the machine learning. In this research paper, we present FSL-BM algorithm as an efficient solution of supervised learning with fuzzy logic processing using binary meta-feature representation using Hamming Distance and Hash function to relax assumptions. While many studies focused on reducing time complexity and increasing accuracy during the last decade, the novel contribution of this proposed solution comes through integration of Hamming Distance, Hash function, binary meta-features, binary classification to provide real time supervised method. Hash Tables (HT) component gives a fast access to existing indices; and therefore, the generation of new indices in a constant time complexity, which supersedes existing fuzzy supervised algorithms with better or comparable results. To summarize, the main contribution of this technique for real-time Fuzzy Supervised Learning is to represent hypothesis through binary input as meta-feature space and creating the Fuzzy Supervised Hash table to train and validate model.Comment: FICC201