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 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.

On Cognitive Computing

2012 ◽  
pp. 83-97
Author(s):  
Yingxu Wang
Keyword(s):  
Computational Intelligence ◽  
Autonomic Computing ◽  
Theoretical Framework ◽  
Autonomous Agent ◽  
Cognitive Computing ◽  
Cognitive Informatics ◽  
Intelligent Computing ◽  
Agent Systems ◽  
The Brain ◽  
Theoretical Foundations

Inspired by the latest development in cognitive informatics and contemporary denotational mathematics, cognitive computing is an emerging paradigm of intelligent computing methodologies and systems, which implements computational intelligence by autonomous inferences and perceptions mimicking the mechanisms of the brain. This article presents a survey on the theoretical framework and architectural techniques of cognitive computing beyond conventional imperative and autonomic computing technologies. Theoretical foundations of cognitive computing are elaborated from the aspects of cognitive informatics, neural informatics, and denotational mathematics. Conceptual models of cognitive computing are explored on the basis of the latest advances in abstract intelligence and computational intelligence. Applications of cognitive computing are described from the aspects of autonomous agent systems and cognitive search engines, which demonstrate how machine and computational intelligence may be generated and implemented by cognitive computing theories and technologies toward autonomous knowledge processing.

Conceptual and Theoretical Foundations of Social Capital

2009 ◽  
pp. 18-42
Author(s):  
Ben Kei Daniel
Keyword(s):  
Social Sciences ◽  
Social Capital ◽  
Theoretical Framework ◽  
Analytical Tool ◽  
The Social ◽  
Scientific Legitimacy ◽  
Basic Understanding ◽  
Work Done ◽  
Conceptual Foundations ◽  
Theoretical Foundations

Social capital is a complex multifaceted and litigious theory, discussed in the Social Sciences and the Humanities. It is a theory increasingly researchers questioned its scientific legitimacy and yet paradoxically many other researchers continuously use it as a conceptual and theoretical framework to explain the structural and functional operations of communities. This Chapter discusses work done on the theory. It covers some of the theoretical controversy with a goal of aligning its conceptualization and distinguishing it from other types of capitals. The Chapter is organized first the basic theoretical and conceptual foundations of social capital are described. The aim is to present the reader with a basic understanding of what constitutes social capital, by opening discussion about various forms of capital(s)—as discussed in the disciplines of Economics and Sociology. Second, the Chapter discusses the origin of the theory as well as the work of key scholars who have contributed to the development of the theory. Furthermore, in order to identify the strengths and the weaknesses of the theory, the Chapter provides the reader with analysis of benefits and shortcomings of social capital both as a theoretical and analytical tool for studying communities.

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 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.

Tibetan Diaspora, Mobility and Place: ‘Exiles in Their Own Homeland’

2016 ◽  
Vol 34 (1) ◽  
pp. 115-136
Author(s):  
Chris Vasantkumar
Keyword(s):  
Euclidean Space ◽  
Recent Work ◽  
Theoretical Framework ◽  
Theoretical Orientations ◽  
Tibetan Diaspora ◽  
Making Sense ◽  
Critical Scrutiny ◽  
Social Topology

This article elaborates a theoretical framework for making sense of Tibetans in Tibet who live as ‘exiles in their own homeland’. Placing questions of mobility at the centre of anthropological approaches to diaspora, it subjects ‘the fact of movement’ to critical scrutiny. In so doing it calls into question three fundamental assumptions of recent work in both ‘new mobilities’ and the study of diaspora more broadly: first, that people move and territory does not; second, that ‘place(s)’ and ‘movement(s)’ are different sorts of things, and clearly distinguishable; and, third, that movement takes places only in Euclidean space. Beginning by placing recent Tibetan experiences of exile and diaspora in comparative context, it then works through recent deconstructions of the boundary between movement and place, a critique of Western ethno-epistemologies of movement, and Law and Mol’s work on social topology as theoretical orientations that might allow us to make sense of mobile homelands and diasporas in situ.

scholarly journals A characterisation of the ellipsoid

1970 ◽  
Vol 11 (4) ◽  
pp. 385-394 ◽  
Author(s):  
P. W. Aitchison
Keyword(s):  
Euclidean Space ◽  
Quadratic Forms ◽  
Convex Bodies ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Major Result

The ellipsoid is characterised among all convex bodies in n-dimensional Euclidean space, Rn, by many different properties. In this paper we give a characterisation which generalizes a number of previous results mentioned in [2], p. 142. The major result will be used, in a paper yet to be published, to prove some results concerning generalizations of the Minkowski theory of reduction of positive definite quadratic forms.

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.

Sign in / Sign up

Export Citation Format

Share Document