We call φ the root function of Φ. The continuity of Φ implies that (1.2) φ(γ) → |γ| as |γ| → ∞. Moreover the converse is true, if φ is continuous and satisfies this asymptotic formula, then the function Φ formed as in (1.1) is a multiplicative distance function. Theorem 1.1. A continuous function φ : C → [0,∞) is the root function of a multiplicative distance function iff it satisfies (1.2). Ar...

We study the r-th generalized minimum distance function (gmd function for short) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. If X is a set of projective points over a finite field and I(X) is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I(X) are equal to the r-th generalized Hamming weight of the...

In this paper, we obtain a generalized version of the wellknown distance function family Lp norm. We prove that the new functions satisfy distance function properties. By using these functions, convex symmetric shapes can be described as loci, the set of points which are in equal distance from a given point. We also show that these symmetric convex shapes can be easily parameterized. We also sh...

Instance-based learning techniques typically handle continuous and linear input values well, but often do not handle nominal input attributes appropriately. The Value Difference Metric (VDM) was designed to find reasonable distance values between nominal attribute values, but it largely ignores continuous attributes, requiring discretization to map continuous values into nominal values. This pa...

A standard way of presenting the semantics of default information is via preference relations on models. Under this view, the default information gives rise to a preference relation which orders models according to how well they satisfy it. Diierent ways of deening the relation in terms of the default information can be used to give diierent granularities to the default status the information h...

The objective of this paper is to present a technique for order preference by similarity to ideal solution (TOPSIS) algorithm to linear fractional bi-level multi-objective decision-making problem. TOPSIS is used to yield most appropriate alternative from a finite set of alternatives based upon simultaneous shortest distance from positive ideal solution (PIS) and furthest distance from negative ...

We characterize the denseness of singular set distance function from a $\mathcal{C}^1$-hypersurface in terms an inner ball condition and we address problem existence viscosity solutions Eikonal equation whose (i.e. non-differentiability points) is not no-where dense.