On the Structure of the Classes of Copulas and Quasi-Copulas with a Given Diagonal Section

In this paper we provide an alternative proof to that given in Ref. 1 to the fact that the class of multivariate copulas with a given diagonal section δ — denoted by 𝒞δ — is a singleton if, and only if, the unique element of the class is the copula of comonotonicity among random variables. This will be used to prove that, for any diagonal δ different from the diagonal of the comotonic copula, 𝒞δ is strictly included in the class of all multivariate quasi-copulas whose diagonal section is δ.

On the lower bound of Spearman’s footrule

AbstractÚbeda-Flores showed that the range of multivariate Spearman’s footrule for copulas of dimension d ≥ 2 is contained in the interval [−1/d, 1], that the upper bound is attained exclusively by the upper Fréchet-Hoeffding bound, and that the lower bound is sharp in the case where d = 2. The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure but also via the copula’s diagonal section.

Analysis of the Arrangement of Prestressed Steel in Web of Continuous Concrete Box-Girder Bridges

Concrete continuous box- girder bridges have a large proportion in small span and long span bridges, and it has very broad prospects for development. Along with the large-scale construction of this kind of bridge, various problems have also emerged, especially the damage of RC beam’s diagonal section in bridge girder deflection. It always appears the inclined cracks in webs which cause by principal tensile stress, so it will be the potential damage to the whole bridge. However, the existence of the vertical prestress, will make the principal tensile stress of box girder in greatly reduced. So as to make cracking resistance performance of diagonal section is better than the ordinary reinforced concrete bridge. For an engineering example, this paper puts forward several different vertical prestressed steel arrangements. Based on different decorate a form of vertical pretressd bridge girder under stress numerical simulation, it is concluded that bridge’s web principal tensile stress distribution and its variation law, and then optimize the vertical reinforcing steel arrangement, improve the overall safety of bridge structure. It can provide the reference in the process of construction when we face the similar problems in the future.

ON THE CLASSES OF COPULAS AND QUASI-COPULAS WITH A GIVEN DIAGONAL SECTION

Let [Formula: see text] and [Formula: see text] be, respectively, the classes of all copulas and quasi-copulas whose diagonal section is δ. We determine under which conditions on δ we have: (a) both [Formula: see text] and [Formula: see text] consist of a singleton; (b) [Formula: see text]. Moreover, a simple construction of copulas with a given convex diagonal section is introduced.