Vine copulas are a flexible way for modeling dependences using only pair-copulas as building blocks. However if the number of variables grows the problem gets fastly intractable. For dealing with this problem Brechmann at al. proposed the truncated R-vine copulas. The truncated R-vine copula has the very useful property that it can be constructed by using only pair-copulas and a lower number of conditional pair-copulas. In our earlier papers we introduced the concept of cherry-tree copulas. In this paper we characterize the relation between cherry-tree copulas and truncated R-vine copulas. It turns out that the concept of cherry-tree copula is more general than the concept of truncated R-vine copula. Although both contain in their expressions conditional independences between the variables, the truncated R-vines constructed in greedy way do not exploit the existing conditional independences in the data. We give a necessary and sufficient condition for a cherry-tree copula to be a truncated R-vine copula. We introduce a new method for truncated R-vine modeling. The new idea is that in the first step we construct the top tree by exploiting conditional independences for finding a good-fitting cherry-tree of order k. If this top tree is a tree in an R-vine structure then this will define a truncated R-vine at level k and in the second step we construct a sequence of trees which leads to it. If this top tree is not a tree in an R-vine structure then we can transform it into such a tree at level k+1 and then we can again apply the second step. The second step is performed by a backward construction named Backward Algorithm. This way the cherry-tree copulas always can be expressed by pair-copulas and conditional pair-copulas.
In this note we show that 1-skeletons and 2-skeletons of n-pseudomanifolds with full boundary are (n+ 1)-connected graphs and n-connected 2-complexes, respectively. This generalizes previous results due to Barnette and Woon.
Recently, the topic of construction methods for triangular norms (triangular conorms), uninorms, nullnorms, etc. has been studied widely. In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods given by Ertuğrul, Karaçal, Mesiar [15] and Çaylı [8] as results. Also, we give some illustrative examples. Finally, we conclude that the introduced construction methods can not be generalized by induction to a modified ordinal sum for t-norms and t-conorms on bounded lattices.
Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on an appropriate bounded lattice, respectively. And we provide some illustrative examples.
In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by $\log \log (1/|x|)^{-1}$. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116.
For general interacting particle systems in the sense of Liggett, it is proven that the class of cylinder functions forms a core for the associated Markov generator. It is argued that this result cannot be concluded by straightforwardly generalizing the standard proof technique that is applied when constructing interacting particle systems from their Markov pregenerators.
Let $T$ be an operator acting on a Banach space $X$. We show that between extensions of $T$ to some Banach space $Y\supset X$ which do not increase the defect spectrum (or the spectrum) it is possible to find an extension with the minimal possible defect spectrum.