Sentence examples for random derivations from inspiring English sources

In Section 3, we prove the Hyers-Ulam stability of random derivations in complete random normed algebras associated with the Cauchy-Jensen additive functional inequality (1.3).

In this paper, we prove the generalized Hyers-Ulam stability of random homomorphisms and random derivations associated with the generalized additive functional equation (1.3) in random Banach algebras.

In this section, using the fixed point method, we prove the Hyers-Ulam stability of random derivations on complete random normed algebras associated with the Cauchy-Jensen additive functional inequality (1.3).

Random derivation box (RDB).

An additive mapping D : X → Y is called a random derivation if D ( x y ) = D ( x ) y − x D ( y ) for all x, y ∈ X.

An R -linear mapping f : X → X is called a random derivation if f ( x y ) = f ( x ) y + x f ( y ) for all x, y ∈ X. Definition 1.6 Let ( X, μ, T ) be an RN-space.

An R -linear mapping f : X → Y is called a random homomorphism if f ( x y ) = f ( x ) f ( y ) for all x, y ∈ X.   (2) An R -linear mapping f : X → X is called a random derivation if f ( x y ) = f ( x ) y + x f ( y ) for all x, y ∈ X.  .

for all x, y ∈ Y and all t > 0. Thus the mapping D : Y → Y satisfies D ( x y ) = D ( x ) y + x D ( y ) for all x, y ∈ Y. Therefore, there exists a unique random derivation D : Y → Y satisfying (3.3).

Multivariable logistic regression models based on FA profiles (FA) and standard risk factors (SRF) were developed on a random 2/3rds derivation set and validated on the remaining 1/3rd.

Remark 1: The CFR of each subcarrier of hop-1 and hop-2 are considered as i.i.d random variables for derivation of close-form expressions.

The original random pore model derivation given by Bhatia and Perlmutter is extended to account also for these peculiarities and the resulting kinetic relation described our reaction rate data well over the entire conversion range.