Sentence examples for Continuous valuation from inspiring English sources

It is proved that the classical Laplace transform is a continuous valuation which is positively GL(n) covariant and logarithmic translation covariant.

They proposed an AL algorithm to learn a continuous valuation model from discrete preferences.

The Steiner point map s : K n → R n is the unique vector valued rigid motion equivariant and continuous valuation.

307] The Steiner point map s : K n → R n is the unique vector valued rigid motion equivariant and continuous valuation.

As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations.

In particular, we recall an important embedding of Klain  [31] of even translation invariant continuous valuations in the space of continuous functions on the Grassmannian.

However, the starting point for a systematic investigation of general valuations was Hadwiger's  [26] fundamental characterization of the linear combinations of intrinsic volumes as the continuous valuations that are rigid motion invariant (see  [1,2,6,38] for recent important variants).

Obviously, a Blaschke-Minkowski homomorphism is a continuous Minkowski valuation which is SO ( n ) equivariant and ( n − 1 ) -homogeneous.

We also note that since translation invariant continuous Minkowski valuations of degree one are linear with respect to Minkowski addition (see e.g.  [26]), inequality (7.1) also holds in the case j = 1 (this follows from (5.5)).

To this end, we use Theorem 2 to define the derivation operator Λ for translation invariant continuous Minkowski valuations: Corollary 5.4 Suppose that Φ ∈ MV al. Then there exists a Λ Φ ∈ MV al such that for every K ∈ K n and u ∈ S n − 1, h (K ), u ) = d d t | t = 0 h (Φ (K + t B n ), u ).

Moreover, the coefficient of λ 1 i 1 ⋯ λ m i m, where i 1 + ⋯ + i m = i, is a continuous translation invariant valuation of degree i j in K j, called a mixed valuation derived from φ. Clearly, we have φ (K, …, K ) = φ (K ).