The spaces ((X,mathcal{P}_{C mathcal{A}})), in particular, generalize metric, ultrametric, quasi-metric, ultra-quasi-metric, b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra-quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra-quasi-gauge, and partial quasi-gauge spaces.
Fatigue tests of standardised uniform-gauge test specimens with duplex coating (plasma nitrided and hard PVD coating) were carried out in a low cycle regime at stress ratio R ≈ 0.05 at elevated temperatures of 150 °C and 500 °C.
Since the specimens are inevitably subject to structural plastic instabilities (necking in tension, barreling in compression) the strain and stress states are no longer uniform in the gauge section after the peak load.
The function (rho_{r}) is called the gauge of uniform convexity of g.
Dalla et al. [30] showed that the shapes of the tensile stress-strain curve and tensile elongation depend on specimen size, with uniform deformation along the gauge section observed in small specimens, but not in larger specimens.
Let s = sup { ∥ ∇ g ( x n ) ∥, ∥ ∇ g ( T j x n ) ∥ : j ∈ N ∪ { 0 }, n ∈ N ∪ { 0 } } and ρ s ∗ : E ∗ → R be the gauge of uniform convexity of the conjugate function g ∗.
Let s = sup { ∥ ∇ g ( T j n x n ) ∥ : j ∈ N ∪ { 0 }, n ∈ N ∪ { 0 } } and ρ s ∗ : E ∗ → R be the gauge of uniform convexity of the conjugate function g ∗.
Let (r_{3}=sup{Vert nabla g z Vert,Vert nabla g(S_{k}z Vert,Vert nabla g(S_{k}U_{n,k+1}z_{m} Vert :minBbb{N},k=1,2,ldots,n}) and (rho^_{r_{3}}:E^tomathbb{R}) be the gauge of uniform convexity of the conjugate function (g^).
Let (r_{1}=sup{Vert nabla g z Vert,Vert nabla g(S_{k}z Vert,Vert nabla g(S_{k}U_{n,k+1}z Vert :k=1,n}ldots,n}) and (rho^_{r_{1}}:E^to mathbb{R}) be the gauge of uniform convexity of the conjugate function (g^).
Let s = sup { ∥ ∇ g ( T j ( x n + e n j ) ) ∥ : j ∈ { 0, 1, 2, …, N }, n ∈ N ∪ { 0 } } and let ρ s ∗ : E ∗ → R be the gauge of uniform convexity of the conjugate function g ∗.
