Sentence examples for gauge function and from inspiring English sources

We define the new measure arising from the gauge function and denote it by G α ( F, a, b ).

where W = ( W 1, …, W m ), B ( x, y ) is the gauge function, and W α ∈ A, then X is called a partial Noether operator corresponding to L, and X is the α th prolongation of the generalized operator (2.7).

Corollary 3 Let ( X, p ) be a partial metric space such that the induced weightable T 0 -qpm d p is complete, let φ : [ 0, ∞ ) → [ 0, ∞ ) be a Bianchini-Grandolfi gauge function and let T : X → Cl ( d p ) s ( X ) be a multivalued map. If one of the following two conditions is satisfied, then T has a fixed point.

Corollary 1 Let ( X, d ) be a complete T 0 -qpm space, q a Q-function on ( X, d ), φ : [ 0, ∞ ) → [ 0, ∞ ) a Bianchini-Grandolfi gauge function and T X → ClCl d s ( X ) a multivalued map such that for each x, y ∈ X and u ∈ T x, there is v ∈ T y satisfying q ( u, v ) ≤ φ ( q ( x, y ) ). Then T has a fixed point. If we take φ ( t ) = r t where r ∈ [ 0, 1 ) we get one of the main results in [2].

In [1] Proinov proved his main results by assuming Bianchini-Grandolfi gauge functions and the mapping f satisfying the contractive condition (3.5) when the underlying space is endowed with a metric (see Corollary 3.9).

In fact, since, and is a gauge function, then for, and (318).

It is easy to check (see [30, Page 8]) that if is a Bianchini-Grandolfi gauge function, then, for all, and hence.

In Section 'Metric and field equations', with kink metric the field equations and its solutions are obtained, when the gauge function β = β(r) and β = β(t).

In the formulation of scale invariant theory of gravitation, Wesson assumed (1) the metric is diagonal and spherically symmetric; (2) the gauge function β depends only one coordinate; and (3) the energy momentum tensor is that of a perfect fluid.

Such functions are called gauge functions or Minkowski functionals, and are well studied in convex analysis and functional analysis.

In Theorem 3.1, we use (hinmathcal{C}), and (mathcal{C}) is a class of more general functions than the gauge function used in Theorems 2.1 and 2.2 of [24].