Copulas in Classical Probability Sense

Copulas are simply equivalent structures to joint distribution functions. Then, we propose modified structures that depend on classical probability space and concepts with respect to copulas. Copulas have been presented in equivalent probability measure forms to the classical forms in order to examine any possible modern probabilistic relations. A probability of events was demonstrated as elements of copulas instead of random variables with a knowledge that each probability of an event belongs to [0,1]. Also, some probabilistic constructions have been shown within independent, and conditional probability concepts. A Bay's probability relation and its properties were discussed with respect to copulas. Moreover, an extension of multivariate constructions of each probabilistic copula has been presented. Finally, we have shown some examples that explain each relation of copula in terms of probability space instead of distribution functions.

FORMAL QUALITATIVE PROBABILITY

Choices rarely deal with certainties; and, where assertoric logic and modal logic are insufficient, those seeking to be reasonable turn to one or more things called “probability.” These things typically have a shared mathematical form, which is an arithmetic construct. The construct is often felt to be unsatisfactory for various reasons. A more general construct is that of a preordering, which may even be incomplete, allowing for cases in which there is no known probability relation between two propositions or between two events. Previous discussion of incomplete preorderings has been as if incidental, with researchers focusing upon preorderings for which quantifications are possible. This article presents formal axioms for the more general case. Challenges peculiar to some specific interpretations of the nature of probability are brought to light in the context of these propositions. A qualitative interpretation is offered for probability differences that are often taken to be quantified. A generalization of Bayesian updating is defended without dependence upon coherence. Qualitative hypothesis testing is offered as a possible alternative in cases for which quantitative hypothesis testing is shown to be unsuitable.

Static Fatigue of SiC Multifilament Tows at Temperatures up to 1200 °C in Air

SiC-based fibers are sensitive to delayed failure under constant load at high temperatures in air. Static fatigue at intermediate temperatures < 800 °C was attributed to slow crack growth from flaws located at the surface of fibers, driven by the oxidation of free carbon at grain boundaries. The present paper examines the static fatigue behavior of SiC-based Hi-Nicalon fibers at high temperatures up to 1200 °C and Hi Nicalon S fibers at intermediate temperatures (500–800 °C). The degradation of stress- rupture time relation of multifilament tows with increasing temperature was investigated. Predictions of tow lifetime based on critical filament-based model of tow failure were compared to experimental stress-rupture time diagrams. Critical filaments are characterized by strength–probability relation. The critical filament-based model was found to describe satisfactorily the static fatigue behavior of fiber tows at these temperatures. The influence of various factors on lifetime as well as the origins of variability is analyzed.

The Reinforced Concrete Structural Vulnerability Analysis under Earthquake Loads

The structural vulnerability analysis decides the probability of the structural failure under the earthquake loads. According to the seismic failure and the failure mechanism of the structure, this paper defines the structural resistance with the limit states of the structural function and builds the probability relation between the structural failure and the structural resistance with which the failure probability of a structure under the load of some intensity earthquake can be decided and the curves of structure vulnerability can be drawn. With this method the structural tolerance of earthquake loads can be evaluated and the safe reliability of the structure can be predicted.