Orders and Their Representations

2020 ◽  
pp. 65-86
Author(s):  
Robert G. Chambers
Keyword(s):  
Structural Properties ◽  
Binary Relation ◽  
Distance Function ◽  
Distance Functions

An order concept, ≽(y), is introduced and interpreted as a correspondence. Some common structural properties imposed on ≽(y) are discussed. A distance function, d(x,y;g), is derived from ≽(y) and interpreted as a cardinal representation of the underlying binary relation expressed in the units of the numeraire g∈ℝ^{N}. Properties of distance functions and their superdifferential and subdifferential correspondences are treated. The chapter closes by studying the structural consequences for d(x,y;g) of different convexity axioms imposed on ≽(y).

scholarly journals KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES

2006 ◽  
Vol 02 (03) ◽  
pp. 431-453
Author(s):  
M. M. DODSON ◽  
S. KRISTENSEN
Keyword(s):  
Distance Function ◽  
Diophantine Approximation ◽  
Direct Proof ◽  
Distance Functions ◽  
Transference Principle ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

scholarly journals On convergence of closed sets in a metric space and distance functions

1985 ◽  
Vol 31 (3) ◽  
pp. 421-432 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Distance Functions ◽  
Convergence Theorems ◽  
Closed Set ◽  
Closed Sets ◽  
The Family ◽  
Kuratowski Convergence ◽  
General Convergence

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

scholarly journals Metric spaces with nice closed balls and distance functions for closed sets

1987 ◽  
Vol 35 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Metric Spaces ◽  
Closed Ball ◽  
Distance Functions ◽  
Entire Space ◽  
Closed Sets ◽  
Bounded Sets

A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.

On various semiconvex relaxations of the squared-distance function

1999 ◽  
Vol 129 (6) ◽  
pp. 1309-1323 ◽  
Author(s):  
K. Zhang
Keyword(s):  
Distance Function ◽  
Distance Functions ◽  
Dimensional Plane ◽  
Rank One ◽  
The Difference

For the Euclidean squared-distance functionf(·) = dist2(·, K), withK ⊂ MN×n, we show thatKis convex if and only iff(·)equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i)Kis compact and contractible or (ii) dimC(K) = k < Nn, Kis convex if and only iffequals one of the semiconvex relaxations when dist2(P, K)is sufficiently large, and for case (i),P ∈MNxn; for case (ii),P ∈ Ek—a k-dimensional plane containingC(K). We also give some estimates of the difference between dist2(P, K)and its semiconvex relaxations. Some possible extensions to more generalp-distance functions are also considered.

scholarly journals A Theoretical Framework for Modeling Asymmetric, Nonpositive Definite and Nonuniform Distance Functions on R exp. n

2010 ◽  
Vol 8 (03) ◽  
Author(s):  
H. Sánchez-Larios ◽  
S. Guillén-Burguete
Keyword(s):  
Euclidean Space ◽  
Distance Function ◽  
Transportation Cost ◽  
Theoretical Framework ◽  
Positive Definite ◽  
Distance Functions ◽  
Fundamental Function ◽  
Uniform Distance ◽  
Lp Norms ◽  
Theoretical Foundations

In this paper, we give theoretical foundations for modeling distance functions on the usual Euclidean space R exp. n, where distance may refer to physical kilometers, liters of fuel consumed, time spent in traveling, or transportation cost. In our approach, a distance function d is derived from a function F0 called the fundamental function of d. Our distance functions, unlike metrics, can be asymmetric and non-positive definite, and unlike the Lp norms, they can be nonuniform. We illustrate our theoretical framework by modeling an asymmetric and non-uniform distance function on R2 which can take negative values.

scholarly journals Selective Image Segmentation Models Using Three Distance Functions

2021 ◽  
Vol 21 (No.1) ◽  
pp. 95-116
Author(s):  
Abdul Kadir Jumaat ◽  
Siti Aminah Abdullah
Keyword(s):  
Image Segmentation ◽  
Distance Function ◽  
Execution Time ◽  
Euclidean Distance ◽  
Operator Splitting ◽  
Splitting Method ◽  
Distance Functions ◽  
Similarity Coefficients ◽  
City Block ◽  
Euclidean Distance Function

Image segmentation can be defined as partitioning an image that contains multiple segments of meaningful parts for further processing. Global segmentation is concerned with segmenting the whole object of an observed image. Meanwhile, the selective segmentation model is focused on segmenting a specific object required to be extracted. The Convex Distance Selective Segmentation (CDSS) model, which uses the Euclidean distance function as the fitting term, was proposed in 2015. However, the Euclidean distance function takes time to compute. This paper proposed the reformulation of the CDSS minimization problem by changing the fitting term with three popular distance functions, namely Chessboard, City Block, and Quasi-Euclidean. The proposed models were CDSSNEW1, CDSSNEW2, and CDSSNEW3, which applied the Chessboard, City Block, and Quasi-Euclidean distance functions, respectively. In this study, the Euler-Lagrange (EL) equations of the proposed models were derived and solved using the Additive Operator Splitting method. Then, MATLAB coding was developed to implement the proposed models. The accuracy of the segmented image was evaluated using the Jaccard and Dice Similarity Coefficients. The execution time was recorded to measure the efficiency of the models. Numerical results showed that the proposed CDSSNEW1 model based on the Chessboard distance function could segment specific objects successfully for all grayscale images with the fastest execution time as compared to other models.

scholarly journals Fixed and common fixed point theorems in frame of quasi metric spaces under contraction condition based on ultra distance functions

2018 ◽  
Vol 23 (5) ◽  
pp. 724-748 ◽  
Author(s):  
Wasfi Shatanawi
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Distance Function ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Distance Functions ◽  
Type I ◽  
Type Ii ◽  
Contraction Condition ◽  
Common Fixed Point Theorems

In this paper, we introduce the notion of ultra distance function. Based on the notion of ultra distance function, we introduce the definitions of (k, ψ, L)-quasi contractions of type (I) and type (II) in the frame of quasi metric spaces. We employ our new definitions to construct and prove many fixed and common fixed point results in the frame of quasi metric spaces. Our results extend and improve many exciting results in the literatures. Also, we introduce some examples and some applications in order to support the usability of our work.

Positive points in polar lattices II

1980 ◽  
Vol 29 (4) ◽  
pp. 504-510
Author(s):  
R. T. Worley
Keyword(s):  
Distance Function ◽  
Distance Functions

AbstractThe problem of positive points in polar lattices, discussed by Hossain and Worley for the distance functions Ft(x1, x2) = │x1│+│tx2 │ and , is considered for a general distance function F. Best possible results are obtained.

Constructive Heterogeneous Object Modeling Using Signed Approximate Real Distance Functions

2005 ◽  
Vol 6 (3) ◽  
pp. 221-229 ◽  
Author(s):  
Pierre-Alain Fayolle ◽  
Alexander Pasko ◽  
Benjamin Schmitt ◽  
Nikolay Mirenkov
Keyword(s):  
Distance Function ◽  
Model Material ◽  
Shape Modeling ◽  
Distance Functions ◽  
Material Distribution ◽  
Object Modeling ◽  
Smooth Approximation ◽  
Model Framework ◽  
Heterogeneous Object ◽  
Heterogeneous Object Modeling

We introduce a smooth approximation of the min∕max operations, called signed approximate real distance function (SARDF), for maintaining an approximate signed distance function in constructive shape modeling. We apply constructive distance-based shape modeling to design objects with heterogeneous material distribution in the constructive hypervolume model framework. The introduced distance approximation helps intuitively model material distributions parametrized by distances to so-called material features. The smoothness of the material functions, provided here by the smoothness of the defining function for the shape, helps to avoid undesirable singularities in the material distribution, like stress or concentrations. We illustrate application of the SARDF operations by two- and three-dimensional heterogeneous object modeling case studies.

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