Sentence examples for net of functions from inspiring English sources

Proposition 3.15 Under the same above notations and hypotheses, let f λ : X → R, λ ∈ Λ, be an ( F d L ) -Cauchy net of functions, globally continuous with respect to a single ( O ) -sequence.

If f λ : X → R, λ ∈ Λ, is a net of functions, globally continuous with respect to a single ( O ) -sequence independent of λ and ( U O F ) -convergent to f ∈ R X on X, then f is globally continuous and ( f λ ) λ is ℱ-exhaustive on X. Proof We begin with proving global continuity of f on X.

In this paper we investigate some properties of continuity of the limit of a net of functions, taking values in cone metric spaces, in terms of weak filter exhaustiveness, extending earlier results proved in [30 32], and relate filter exhaustiveness with filter uniform convergence (on compact subsets).

Meta-analysis performed on the complete set of differentially expressed genes using the GO database (Table 1), comprised a statistically strong cohesive net of functions as derived from the GO tree, thus further supporting the reliability of the adopted fold change expression thresholds.

Our approach is direct, simple and easy to handle in the context of our considered structures, that is, when it is dealt with nets of functions taking values in cone metric spaces, defined in general Hausdorff topological spaces and with filter exhaustiveness instead of metric spaces and equicontinuity, respectively, and it allows us to give direct necessary and sufficient conditions.

For a sequence or net of convex functions on a Banach space, we study pointwise convergence of their Lipschitz regularizations and convergence of their epigraphs.

As a by-product, slice convergence for an arbitrary net of convex functions can be deduced from the pointwise convergence of their regularizations precisely when the w* and the norm topologies agree on the dual sphere.

Point mutations falling within this zone, amounting to about 8% of the non-synonymous total, can accumulate indefinitely without net loss of function.

For example, the Hippo signaling pathway has an evolutionarily conserved role in controlling tissue growth rates and organ size during animal development and mutations that cause a net gain of function can lead to cancer.

Although, in general, OP exposure elicited a net AC gain-of-function, the effects differed among the various tissues, effectively ruling out a global effect on expression of receptors, G-proteins, or AC and instead pointing to selectively greater effects on specific aspects of hepatic function.

But in a general Dedekind complete lattice group it may happen the non-existence of Hausdorff topologies 'compatible' with the order (see also [40]), and we need an approach different from the classical one, by considering concepts like closedness and compactness by means of nets (or sequences) of functions rather than by topologies.