Usual Distance Functions Research Articles

Mattila–Sjölin Type Functions: A Finite Field Model

Let \(\phi (x, y)\colon \mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a function. We say ϕ is a Mattila–Sjölin type function of index γ if γ is the smallest number satisfying the property that for any compact set \(E\subset \mathbb {R}^{d}\), ϕ(E,E) has a non-empty interior whenever \(\dim _{H}(E)>\gamma \). The usual distance function, ϕ(x,y) = |x − y|, is conjectured to be a Mattila–Sjölin type function of index \(\frac {d}{2}\). In the setting of finite fields \(\mathbb {F}_{q}\), this definition is equivalent to the statement that \(\phi (E, E)=\mathbb {F}_{q}\) whenever |E|≫ qγ. The main purpose of this paper is to prove the existence of such functions with index \(\frac {d}{2}\) in the vector space \(\mathbb {F}_{q}^{d}\).

Contraction Mapping Theorem in Partial Fuzzy Metric Spaces

ABSTRACT Contraction mapping theorem and its application is often encountered different fields of pure and applied mathematics. In this work, three aims are achieved, which are first of introducing and presenting the basic and fundamental necessary concepts for partial fuzzy metric spaces. The second aim is to state and prove two fuzzy contraction mapping theorems in fuzzy partial metric spaces based on the contractive mapping theorem by using the suggested metric function that is defined over fuzzy points rather than fuzzy sets. Also, we present in this paper, as the third aim, the relationship between the usual distance function defined in fuzzy metric spaces and the distance function defined over partial metric space, which are defined using fuzzy points.

On Perfect Clustering of High Dimension, Low Sample Size Data.

Popular clustering algorithms based on usual distance functions (e.g., the Euclidean distance) often suffer in high dimension, low sample size (HDLSS) situations, where concentration of pairwise distances and violation of neighborhood structure have adverse effects on their performance. In this article, we use a new data-driven dissimilarity measure, called MADD, which takes care of these problems. MADD uses the distance concentration phenomenon to its advantage, and as a result, clustering algorithms based on MADD usually perform well for high dimensional data. We establish it using theoretical as well as numerical studies. We also address the problem of estimating the number of clusters. This is a challenging problem in cluster analysis, and several algorithms are available for it. We show that many of these existing algorithms have superior performance in high dimensions when they are constructed using MADD. We also construct a new estimator based on a penalized version of the Dunn index and prove its consistency in the HDLSS asymptotic regime. Several simulated and real data sets are analyzed to demonstrate the usefulness of MADD for cluster analysis of high dimensional data.

High dimensional nearest neighbor classification based on mean absolute differences of inter-point distances

Traditional nearest neighbor classifiers based on usual distance functions (e.g., Euclidean distance) often suffer in high dimension low sample size (HDLSS) situations, where phenomena like concentration of pairwise distances, violation of cluster assumptions and presence of hubs often have adverse effects on their performance. In order to cope with such problems, instead of usual distance functions, in this article we use a dissimilarity measure based on average of absolute differences between inter-point distances. Our proposed nearest neighbor classifier uses concentration of pairwise distances to its advantage, and it usually yield better performance in high dimension when such concentration occurs. Under appropriate regularity conditions, we proved the optimality of the misclassification probability of the proposed classifier in HDLSS asymptotic regime, where the training sample size remains fixed, and the dimension grows to infinity. Usefulness of the proposed method has also been demonstrated using several simulated and benchmark data sets.

Multiplicative distance: a method to alleviate distance instability for high-dimensional data

Recently, it has been shown that under a broad set of conditions, the commonly used distance functions will become unstable in high-dimensional data space; i.e., the distance to the farthest data point approaches the distance to the nearest data point of a given query point with increasing dimensionality. It has been shown that if dimensions are independently distributed, and normalized to have zero mean and unit variance, instability happens. In this paper, it is shown that the normalization condition is not necessary, but all appropriate moments must be finite. Furthermore, a new distance function, namely multiplicative distance, is introduced. It is theoretically proved that this function is stable for data with independent dimensions (with identical or nonidentical distribution). In contrast to usual distance functions which are based on the summation of distances over all dimensions (distance components), the multiplicative distance is based on the multiplication of distance components. Experimental results show the stability of the multiplicative distance for data with independent and correlated dimensions in the high-dimensional space and the superiority of the multiplicative distance over the norm distances for the high-dimensional data.

Edge-disjoint spanners in tori

A spanning subgraph S = ( V , E ′ ) of a connected graph G = ( V , E ) is an ( x + c ) -spanner if for any pair of vertices u and v , d S ( u , v ) ≤ d G ( u , v ) + c where d G and d S are the usual distance functions in G and S , respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs in multi-dimensional tori. We show that each two-dimensional torus has a set of two edge-disjoint spanners of delay approximately the size of the smaller dimension. Moreover, we show that this delay is close to the best possible. In three-dimensional tori, we find a set of three edge-disjoint spanners with delay approximately the sum of the sizes of the two smaller dimensions when all dimensions are of even size. Surprisingly, we also find a set of two edge-disjoint spanners in three-dimensional tori of constant delay. In d -dimensional tori, we show that for any k ≤ d / 5 , there is a set of k edge-disjoint spanners with delay depending only on k and the size of the smaller k dimensions.

Scaling and multifractal fields in the solid earth and topography

Abstract. Starting about thirty years ago, new ideas in nonlinear dynamics, particularly fractals and scaling, provoked an explosive growth of research both in modeling and in experimentally characterizing geosystems over wide ranges of scale. In this review we focus on scaling advances in solid earth geophysics including the topography. To reduce the review to manageable proportions, we restrict our attention to scaling fields, i.e. to the discussion of intensive quantities such as ore concentrations, rock densities, susceptibilities, and magnetic and gravitational fields. We discuss the growing body of evidence showing that geofields are scaling (have power law dependencies on spatial scale, resolution), over wide ranges of both horizontal and vertical scale. Focusing on the cases where both horizontal and vertical statistics have both been estimated from proximate data, we argue that the exponents are systematically different, reflecting lithospheric stratification which – while very strong at small scales – becomes less and less pronounced at larger and larger scales, but in a scaling manner. We then discuss the necessity for treating the fields as multifractals rather than monofractals, the latter being too restrictive a framework. We discuss the consequences of multifractality for geostatistics, we then discuss cascade processes in which the same dynamical mechanism repeats scale after scale over a range. Using the binomial model first proposed by de Wijs (1951) as an example, we discuss the issues of microcanonical versus canonical conservation, algebraic ("Pareto") versus long tailed (e.g. lognormal) distributions, multifractal universality, conservative and nonconservative multifractal processes, codimension versus dimension formalisms. We compare and contrast different scaling models (fractional Brownian motion, fractional Levy motion, continuous (in scale) cascades), showing that they are all based on fractional integrations of noises built up from singularity basis functions. We show how anisotropic (including stratified) models can be produced simply by replacing the usual distance function by an anisotropic scale function, hence by replacing isotropic singularities by anisotropic ones.

Edge-disjoint spanners in Cartesian products of graphs

A spanning subgraph S = ( V , E ′ ) of a connected graph G = ( V , E ) is an ( x + c ) -spanner if for any pair of vertices u and v, d S ( u , v ) ⩽ d G ( u , v ) + c where d G and d S are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs, focusing on graphs formed as Cartesian products. Our approach is to construct sets of edge-disjoint spanners in a product based on sets of edge-disjoint spanners and colorings of the component graphs. We present several results on general products and then narrow our focus to hypercubes.

Minimally separating sets, mediatrices, and Brillouin spaces

Brillouin zones and their boundaries were studied in [J.J.P. Veerman et al., Comm. Math. Phys. 212 (3) (2000) 725] because they play an important role in focal decomposition as first defined by Peixoto in [J. Differential Equations 44 (1982) 271] and in physics [N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rhinehart, and Winston, 1976; L. Brillouin, Wave Propagation in Periodic Structures, Dover, 1953]. In so-called Brillouin spaces, the boundaries of the Brillouin zones have certain regularity properties which imply that they consist of pieces of mediatrices (or equidistant sets). The purpose of this note is two-fold. First, we give some simple conditions on a metric space which are sufficient for it to be a Brillouin space. These conditions show, for example, that all compact, connected Riemannian manifolds with their usual distance functions are Brillouin spaces. Second, we exhibit a restriction on the Z 2 -homology of mediatrices in such manifolds in terms of the Z 2 -homology of the manifolds themselves, based on the fact that they are Brillouin spaces. (This will used to obtain a classification up to homeomorphism of surface mediatrices in forthcoming paper [J. Bernhard, J.J.P. Veerman, The topology of surface mediatrices, Portland State University].)

CLUSTERING PANEL DATA VIA PERTURBED ADAPTIVE SIMULATED ANNEALING AND GENETIC ALGORITHMS

Non-hierarchical cluster analysis for panel data is known to be hampered by structural preservation, computational complexity and efficiency, and dependency problems. Resolving these issues becomes increasingly important as efficient collection and maintenance of panel data make application more conducive. To address some computational issues and structural preservation, Bonzo [3] presented a stochastic version of Kosmelj and Batagelj's approach [16] to clustering panel data. The method used a probability link function (instead of the usual distance functions) in defining cluster inertias with the aim of preserving the clusters' probabilistic structure. Formulating clustering as an optimization problem, the objective function allows the application of heuristic and stochastic optimization techniques. In this paper, we present a modified heuristic for adaptive simulated annealing (ASA) by perturbing the state vector's sampling distribution, specifically, by perturbing the drift of a diffusion process. Such an approach has been used to hasten convergence towards global optimum at equilibrium for diversely complex, combinatorial, and large-scale systems. The perturbed ASA (PASA) heuristic is then embedded in a genetic algorithm (GA) procedure to hasten and improve the stochastic local search process. The PASA-GA hybrid can be further modified and improved such as by explicit parallel implementation.

Edge-disjoint spanners of complete bipartite graphs

A spanning subgraph S=(V,E′) of a connected simple graph G=(V,E) is an (x+c)- spanner if for any pair of vertices u and v, d S(u,v)⩽d G(u,v)+c where d G and d S are the usual distance functions in graphs G and S, respectively. The parameter c is called the delay of the spanner. We investigate the number of edge-disjoint spanners of a given delay that can exist in complete bipartite graphs. We determine the exact number of such edge-disjoint spanners of delay 4 or larger. For delay 2, we obtain many exact values of and some general bounds on this number.

Edge-disjoint spanners of complete graphs and complete digraphs

A spanning subgraph S=( V, E′) of a connected simple graph (or digraph) G=( V, E) is an f( x)- spanner if for any pair of vertices u and v, d S ( u, v)⩽ f( d G ( u, v)) where d G and d S are the usual distance functions in graphs (digraphs) G and S, respectively. The delay of the f( x)-spanner is f( x)− x. We investigate the number of edge-disjoint spanners of a given delay that can exist in complete graphs and in complete digraphs.

Degree-constrained spanners for multidimensional grids

A spanning subgraph S = ( V, E′) of a connected simple graph G = ( V, E) is a f (x) -spanner if for any pair of nodes u and v, d s ( u, v) ⩽ f ( d G ( u, v)) where d G and d s are the usual distance functions in graphs G and S, respectively. The delay of the f (x) -spanner is f(x) − x. We construct four spanners with maximum degree 4 for infinite d-dimensional grids with delays 2 d − 4, 2⌈ d 2 ⌉ + 2[(d − 2)/4] + 2, 2⌈(d − 6)/8⌉ + 4⌈d + 1)/4⌉+ 6, and ⌈(⌈d/2⌉ + 1)/ (1 + 1)rl + 2⌈ d 2 ⌉ + 21 + 2. All of these constructions can be modified to produce spanners of finite ( d-dimensional grids with essentially the same delay. We also construct a (5 d + 4 + x) -spanner with maximum degree 3 for infinite d-dimensional grids. This construction can be used to produce spanners of finite d-dimensional grids where all dimensions are even with the same delay. We prove an Ω(d) lower bound for the delay of maximum degree 3 or 4 spanners of finite or infinite d-dimensional grids. For the particular cases of infinite 3- and 4-dimensional grids, we construct (6 + x) -spanners and (14 + x) -spanners, respectively. The former can be modified to construct a (6 + x) -spanner of a finite 3-dimensional grid where all dimensions are even or where all dimensions are odd and a (8 + x) -spanner of a finite 3-dimensional grid otherwise. The latter yields (14 + x) -spanners of finite 4-dimensional grids where all dimensions are even.

New derivation of the geometric phase

Using the usual distance function (the Fubini-Study metric) and a new distance function on the projective Hilbert space, the geometric phase is defined for non-cyclic, non-unitary and non-Schrödinger evolutions of quantum systems.

Degree-Constrained Network Spanners with Nonconstant Delay

A spanning subgraph $S = ( V,E^\prime )$ of a connected simple graph $G = ( V,E )$ is a $f( x )$-spanner if for any pair of nodes u and $v $, $d_S ( u,v ) \leq f ( d_G ( u,v ) )$ where $d_G $ and $d_S $ are the usual distance functions in graphs G and S, respectively. The delay of the $f( x )$-spanner is $f( x ) - x$. In this paper $( 2.5\sqrt {( 3x + 6 )/4} + 6 + x )$-spanners for two-dimensional grids with maximum degree 3 are found and it is proven that the delay of these spanners is within a constant factor of optimal. A $( \frac{1}{k}x + k + 8 - \frac{7}{k} + x )$-spanner of the X-tree with maximum degree 3 is described, and it is proven that the delay of this spanner is within a constant factor of optimal. In addition, a $( 2 + x )$-spanner of the pyramid with maximum degree 6 and a $( \frac{1}{k}x + k + 8 - \frac{7}{k} + x )$-spanner of the pyramid with maximum degree 5 are described, and it is proven that the delay of the latter spanner is within a constant factor of optimal.

Additive graph spanners

AbstractA spanning subgraph S = (V, E′) of a connected simple graph G = (V, E) is a f(x)‐spanner if for any pair of nodes u and v, dS(u, v) ≦ f(dG(u, v)), where dG and dS are the usual distance functions in graphs G and S, respectively. We are primarily interested in (t + x)‐spanners, which we refer to as additive spanners. We construct low‐degree additive spanners for X‐trees, pyramids, and multidimensional grids. We prove, for arbitrary t > 0, that to determine whether a given graph G has an additive spanner with no more than m edges is NP‐complete. © 1993 by John Wiley & Sons, Inc.

How Analytic Functions Preserve Closeness

Topological dynamics is an area of topology in which one studies the geometric properties of certain groups of transformations which had their origins in classical dynamics and systems of differential equations. Henri Poincare is generally credited with originating the subject. He was probably the first to solve dynamics problems as problems in topology. Poincare's work was later abstracted, systematized, and carried on by G. D. Birkhoff. An area of current research in Topological Dynamics is the study of the somewhat elusive expanisve property. Roughly speaking, the property deals with the ability of functions to spread out points by means of repeated application of the functions, i.e., iterations. This concept will be precisely defined later. In this paper, various types of expansiveness will be defined, and it will then be shown that in most cases, analytic functions are not The results are both new and easy to prove, and the paper gives the reader with a limited background in topology an introduction to an area of current research. In addition the proofs are good exercises in complex analysis. If f is a homeomorphism of a metric space (X, d) onto itself, then f is said to be if there exists 8 >0 such that x,y E (X, d), x t=y implies d(fn(x),fJ(y)) > 5 for some integer n. Iff is only assumed to map (X, d) continuously into itself, the property can be adjusted by calling f if the integer n is non-negative. The concept of expansiveness can also be weakened by requiring that for each x E (X, d), there is a S(x) >0 such that if y E (X, d), y =x, then d(fn(x), ff(y)) >5(x) for some integer n. In this case, f is called A similar modification allows one to weaken to Clearly, and positively are stronger than pointwise and pointwise respectively, and for homeomorphisms onto, i.e. bijections, positively is stronger than expansive. We now illustrate these concepts with several examples. Let us take (X, d) to be the real line with the usual distance function in the following two examples. First, if f(x) = 2x, f is a homeomorphism of (X, d) onto itself, and since.fn(x) = 2nx, and d(f(x),fn(y)) = 2 I x -ye, we see that f is both and (Choose 8 to be any positive real number.) If f(x) = 2x for x> 0 and x/2 for x 0 such that x,y E (X, d), x t=y implies d(f(x),f(y)) > d for some f E F. A similar generalization allows one to define families. Clearly expansive family implies pointwise family, and these two concepts are implied by their respective predecessors, since if f is either or expansive, we can take F= { fn: n = 0, ? 1,...), and if f is either or expansive, we can take F= { fn: n = 0, 1,.. . ). These definitions are easily understood by anyone who knows what a metric space is, but some very simple questions about these properties remain unanswered. For instance, it has been known for over 18 years (see [7] and [8]) that there does not exist an homeomorphism on the closed unit disk, but the existence or non-existence of an homeomorphism on the closed unit ball in three dimensions is still an open question.

Logical processing of set inclusion relations in meaningful text

The present experiments attempted to resolve some recent conflicting findings in cognitive structure research between processing linear ordering and set inclusion relations described in meaningful paragraphs of text. In a self-paced study-test paradigm, college students studied such paragraphs. Definite processing differences were found for the two set-theoretic relations. Set inclusion test results were found to be due to faulty logical processing and not to memory retrieval problems. Subjects made invalid conversions of universally quantified statements and failed to make valid transitive inferences between such statements. This failure was found to be an increasing function of the distance separating the two terms in the set inclusion. No such problems were observed for linear orderings. The usual distance function was found: Accuracy on test questions about the ordering was an increasing function of the distance between the terms in the question. Results similar to those for linear orderings were observed for set inclusions when subjects were given special instructions about the validity and invalidity of symmetric and transitive inferences.

On a theorem of Klee

In 1953 and 1956, Klee [3], [4] proved that for E any infinitedimensional normed linear space and K any compact subset of E, E\K is homeomorphic to E. Klee's argument used sequences of bounded convex sets. In [5], Lin has some extensions of Klee's results using modifications of his methods. In this paper we give a short and elementary proof of a somewhat more general result' using only simple set-theoretic properties. A space S is said to be an a-space provided (1) S is an infinite-dimensional topological linear space, i.e., an infinite dimensional real vector space with a Hausdorff topology in which vector addition and scalar multiplication are jointly continuous, (2) S has a Schauder basis, i.e., a sequence {xi i>o of elements of S such that for each sES there is a unique sequence of scalars ai with s= aixi (convergence being in the topology of S) such that the function fi defined by fi(s) =ai is continuous for each i, and (3) there is a neighborhood U of the origin such that the elements xi I of the Schauder basis above are not in U. Henceforth, all spaces under discussion are to be a-spaces. For each i, let Mi denote the product of i copies of the reals with usual distance function di referring to distance between points, between a point and a set or between two sets. Let fi be as defined in condition (2) of the definition of an a-space and let gi be the map of S onto M, defined by gi(s) = (fi(s), f2(s), . . . , fi(s)). Since, by hypothesis, fi is continuous (for each i), then so is gj for each j. A set K CS is said to be projectible provided (1) K is closed, (2) for any pES\K, there is a j such that gj(p) is not an element of the closure of gj(K), and (3) there exist infnitely many i such that fi(K) is bounded above or below. The proof of the following lemma is trivial and is therefore omitted.