Regular polygons in taxicab geometry

A polygon of n sides will be called regular in taxicab geometry if it has n equal angles and n sides of equal taxicab length. This paper will show that there are no regular taxicab triangles and no regular taxicab pentagons. The sets of taxicab rectangles and taxicab squares will be shown to be the same, respectively, as the sets of Euclidean rectangles and Euclidean squares. A method of construction for a regular taxicab 2n-gon for any n will be demonstrated.     

On the calculation of extended max and min operations between convex fuzzy sets of the real line

In this paper we will identify the sets of so-called sub- and pseudo-highest intersection points of convex fuzzy sets of the real line and will explore their properties. Based on the properties of these sets, an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions, not necessarily continuous, is proposed.     

Covering and packing in {\mathbb Z^n} and {\mathbb R^n}. (II)

In the present part (II) we will deal with the group \mathbb G = \mathbb Zⁿ{\mathbb G = \mathbb Z^n} , and we will study the effect of linear transformations on minimal covering and maximal packing densities of finite sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} . As a consequence, we will be able to show that the set of all densities for sets A{\mathcal A} of given cardinality is closed, and to characterize four-element sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} which are “tiles”. The present work will be largely independent of the first part (I) presented in [4].     

Linear Elliptic Boundary Value Problems with Non‐Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non‐smooth data L^∞‐coefficients, sets with Lipschitz boundary, regular sets, non‐homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev‐Campanato spaces of functions and functionals, respectively.     

On Pseudocomplemented and Stone Ordered Sets

The aim of this paper is to characterize both the pseudocomplemented and Stone ordered sets in a manner similar to that used previously for Boolean and distributive ordered sets. The sublattice G(A) of the Dedekind–Mac Neille completion DM(A) of an ordered set A generated by A is said to be the characteristic lattice of A. We will show that there are distributive pseudocomplemented ordered sets whose characteristic lattices are not pseudocomplemented. We can define a stronger notion of pseudocomplementedness by demanding that both A and G(A) be pseudocomplemented. It turns out that the two concepts are the same for finite and Stone ordered sets.     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

Piercing Numbers for Balanced and Unbalanced Families

Given a finite family F\mathcal{F} of convex sets in ℝ ^d , we say that F\mathcal{F} has the (p,q) _r property if for any p convex sets in F\mathcal{F} there are at least r q-tuples that have nonempty intersection. The piercing number of F\mathcal{F} is the minimum number of points we need to intersect all the sets in F\mathcal{F}. In this paper we will find some bounds for the piercing number of families of convex sets with (p,q) _r properties.     

Fast generalized cross validation using Krylov subspace methods

The task of fitting smoothing spline surfaces to meteorological data such as temperature or rainfall observations is computationally intensive. The generalized cross validation (GCV) smoothing algorithm, if implemented using direct matrix techniques, is O(n ³) computationally, and memory requirements are O(n ²). Thus, for data sets larger than a few hundred observations, the algorithm is prohibitively slow. The core of the algorithm consists of solving series of shifted linear systems, and iterative techniques have been used to lower the computational complexity and facilitate implementation on a variety of supercomputer architectures. For large data sets though, the execution time is still quite high. In this paper we describe a Lanczos based approach that avoids explicitly solving the linear systems and dramatically reduces the amount of time required to fit surfaces to sets of data.      

A planar minimax alogrithm for analysis of coordinate measurements

We characterize the solution of the following problem and describe an algorithm for numerically solving it. Two sets ofN points in the plane, labeled 1,...,N, are given: a fixed set of nominal points and a set of measured points. We wish to transform the messured points as a whole, by translation and rotation, so that the maximal distance between corresponding points in the two sets is minimized. This algorthm provides an accept-reject criterion that may be used together with a coordinate measuring machine to determine if two mating parts will fit, or if a part is sufficiently close to its ideal measurements. A weighted version, suitable for point-dependent tolerances, is also discussed, as is optimal joint scalling of the data.     

Constructions of Spherical 3-Designs

 Spherical t-designs are Chebyshev-type averaging sets on the d-sphere which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size of such designs, in particular, that the number of points in a 3-design on S ^d must be at least . In this paper we give explicit constructions for spherical 3-designs on S ^d consisting of n points for d=1 and ; d=2 and ; d=3 and ; d=4 and ; and odd or even. We also provide some evidence that 3-designs of other sizes do not exist. We will introduce and apply a concept from additive number theory generalizing the classical Sidon-sequences. Namely, we study sets of integers S for which the congruence mod n, where and , only holds in the trivial cases. We call such sets Sidon-type sets of strength t, and denote their maximum cardinality by s(n, t). We find a lower bound for s(n, 3), and show how Sidon-type sets of strength 3 can be used to construct spherical 3-designs. We also conjecture that our lower bound gives the true value of s(n, 3) (this has been verified for n≤125). Received: June 19, 1996     

Categorical data clustering with automatic selection of cluster number

In this paper, we investigate the problem of determining the number of clusters in the k-modes based categorical data clustering process. We propose a new categorical data clustering algorithm with automatic selection of k. The new algorithm extends the k-modes clustering algorithm by introducing a penalty term to the objective function to make more clusters compete for objects. In the new objective function, we employ a regularization parameter to control the number of clusters in a clustering process. Instead of finding k directly, we choose a suitable value of regularization parameter such that the corresponding clustering result is the most stable one among all the generated clustering results. Experimental results on synthetic data sets and the real data sets are used to demonstrate the effectiveness of the proposed algorithm.     

On blocking sets of quadrics

We determine the three smallest blocking sets with respect to lines of the quadric Q(2n, q) withn 3 and the two smallest blocking sets with respect to lines of the quadric Q⁺(2n+1,q) withn 2. These results will be used in a forthcoming paper for determining the smallest blocking sets with respect to higher dimensional subspaces in the quadrics Q(2n, q) and Q⁺(2n+ 1, q).     

On the existence of efficient points

In this note we will study the existence ofefficient points of a set X with respect to a coneC by introducing the so-called quasi-C-bounded sets; relationships among C-compactness, C-boundedness and quasi-C-boundedness are studied in a systematic way, and new existence theorems are obtained which generalize some known results and which allow us to find existence conditions for vector optimization problem.     

A Study of Downward Sets with p-Functions

In this paper, we study downward sets and increasing functions in a topological vector space and their similarities to the convex sets and convex functions. It will be shown that a very special increasing function, namely, the p-function, can give a geometric interpretation for separating downward sets from outside points. Also, this function can be used to approximate topical functions in the framework of abstract convexity.     

Rigid extensions of algebraic frames

An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each ${h \in H}An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each h ? H{h \in H} there exists a g ? G{g \in G} such that h ^⊥⊥ = g ^⊥⊥. In this paper, we will define rigid extensions and some other generalizations in the context of algebraic frames satisfying the FIP. One of the main results is a characterization of rigid extensions using d-elements of the frame. We also show that a rigid extension between two algebraic frames satisfying the FIP will induce a homeomorphism between their corresponding minimal prime spaces with respect to both the hull-kernel topology and the inverse topology. Moreover, basic open sets map to basic open sets.     

On concrete characterization of universal hypergraphic automata

In this paper, we consider structured automata without output signals whose state sets are endowed with an algebraic structure of hypergraphs. The main result of the paper is a theorem where we obtain necessary and sufficient conditions for the possibility of defining on the state set of some automaton A a structure of a hypergraph H such that the automaton A will be the universal hypergraphic automaton.     

On Genericity and Ershov's Hierarchy

It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeffer [9] and Jockusch and Posner [6] by looking at genericity notions within the Δ₂ sets. Different equivalent characterisations of 1‐genericity suggest different ways in which the definition might be generalised. There are two natural ways of casting the notion of 1‐genericity: in terms of sets of strings and in terms of density functions, as we will see here. While these definitions coincide at the first level of the difference hierarchy, they turn out to differ at other levels. Furthermore, these differences remain when the remainder of the Δ⁰₂ sets are considered. While the string characterization of 1‐genericity collapses at the second level of the difference hierarchy to 2‐genericity, the density function definition gives a very interesting hierarchy at level w and above. Both of these results point towards the deep similarities exhibited by the n‐c.e. degrees for n ≥ 2.     

A Structure Preserving Generalization of Independence,Domination and Irredundance

Abstract

The now famous inequality chain ir≤γ≤i≤β ≤ Γ ≤ IR, where ir and IR denote the lower and upper irredundance numbers of a graph, γ and Γ the lower and upper domination numbers, i the independent domination number and β the independence number of a graph, may be seen as the culmination of a process by which we start with independence (a hereditary property of vertex sets); we characterize maximal independence by domination (a superhereditary property of vertex sets), and then characterize minimal domination by irredundance (again a hereditary property). In this paper we generalize independent, dominating and irredundant sets of a graph G to what we will call s-dominating, s-independent and s-irredundant functions (for s a positive integer), which are functions of the type f : V (G) → N, in such a way that the maximal 1-independent, the minimal 1- dominating and the maximal 1-irredundant functions are the characteristic functions of the maximal independent, the minimal dominating and the maximal irredundant sets of G respectively. In addition, we would want to preserve those properties of and relationships between independence, domination and irredundance needed to extend the inequality chain ir≤γ≤i≤β ≤ Γ ≤ IR to one for s-dominating, s-independent and s-irredundant functions by a process similar to that described above.     

S-convex,monotone, and positive interpolation with rational bicubic splines of C2-continuity

In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC ² which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure.     

ζ μ -sets in generalized topological spaces

We define ζ _μ -sets, (ζ,μ)-closed sets and generalized ζ _μ -sets in a generalized topological space and investigate properties of several low separation axioms of generalized topologies constructed by the families of these sets. Characterizations of some properties of (ζ,μ)-R ₀ and (ζ,μ)-R ₁ generalized topological spaces will be given.