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Measurable Sets
Published in Hugo D. Junghenn, Principles of Analysis, 2018
An outer measure on a nonempty set X is a nonnegative, monotone, countably subadditive set function μ∗ $ \mu ^* $ on P(X) $ \boldsymbol{{ \fancyscript {P}}}(X) $ such that μ∗(∅)=0 $ \mu ^*(\emptyset ) = 0 $ . Clearly, every measure on P(X) $ \boldsymbol{{ \fancyscript {P}}}(X) $ is an outer measure. In particular, the set function that assigns 0 to the empty set and ∞ $ \infty $ to every nonempty set is an outer measure. By contrast, the set function that assigns 0 to the empty set and 1 to every nonempty set is an outer measure that is not a measure.
A performance-margin-based belief reliability model considering parameter uncertainty
Published in Stein Haugen, Anne Barros, Coen van Gulijk, Trond Kongsvik, Jan Erik Vinnem, Safety and Reliability – Safe Societies in a Changing World, 2018
Qingyuan Zhang, Meilin Wen, Rui Kang, Tianpei Zu
Definition 2.1. Uncertain measure, Liu (2007)). Let Γ be a nonempty set, and ℒ be a σ-algebra over Γ. A set function ℳ is called an uncertain measure if it satisfies the following axioms, Axiom 1. Normality Axiom ℳ{Γ}=1for the universal set Γ.Axiom 2.Duality Axiomℳ{Λ}+ℳ{Λx}=1for any eventΛ∈ℒ.Axiom 3. Subadditivity Axiom For every countable sequence of events Λ1,Λ2, we have ℳ{∪i=1∞Λi}≤∑i=1∞ℳ{Λi}.
Monotonic and fatigue behavior of cementitious composites modeled via a coupled sliding-decohesion-compression interface model
Published in Günther Meschke, Bernhard Pichler, Jan G. Rots, Computational Modelling of Concrete and Concrete Structures, 2022
A. Baktheer, M. Aguilar, R. Chudoba, M. Vořechovský
The shear limit, fT, follows the Mohr-Coulomb criterion, which represents the frictional, pressure-sensitive interface enhanced with kinematic and isotropic hardening as () fTσNeff,qTeff,Z;fcs,m=qTeff−fcs+Z+mσNeff where fcs denotes the shear stress limit and m is the pressure sensitivity factor. This form of threshold function has been used by the authors in (Baktheer & Chudoba 2018b) to simulate the fatigue behavior of the bond loaded in pull-out condition. Since the shear stress dominates in the pull-out problem, the threshold function given in Eq. (15) was sufficient to deliver realistic results. However, for structural configurations affected by an interaction of normal and tangential dissipative mechanisms, the threshold function must be extended to account for the tensile and compressive strength limits. The mathematical formulation of the multi-domain level set function satisfies the requirements of convexity, continuity and smoothness at an arbitrary level f(E,S)=ℓ. It consists of three parts corresponding to the tension-, compression-and shear-dominated subdomains within the effective stress half-space shown in Figure 1b. The subdomains associated with the tensile and compressive limits in normal direction are introduced using elliptical functions fNt and fNc, respectively, which are connected by the linear, pressure-sensitive shear threshold fT given in Eq. (15) as illustrated in Figure 1b.
A survey of fundamental operations on discrete convex functions of various kinds
Published in Optimization Methods and Software, 2021
A function that assigns a real number (or ) to each subset of is called a set function. A set function f is said to be submodular [9,11,23,48] if where it is understood that the inequality is satisfied if or is equal to .