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In this paper, we give an existence theorem about positive solutions for the Dirichlet boundary value problem of one dimensional Minkowski curvature equations. We apply the theorem to one parameter family of problems to investigate a constructive method for numerical range of parameters where positive solutions exist. Moreover, we establish a nonexistence theorem of positive solutions for the corresponding one parameter family of problems. The coefficient function may be singular at the boundary and nonlinear term satisfies a sublinear growth condition. Main argument for the proof of existence theorem is employed by Krasnoselskii's theorem of cone expansion and compression. We give a numerical algorithm and various examples to illustrate numerical information about ranges of the existence and nonexistence parameters which have been given only in a theoretical manner so far. [ABSTRACT FROM AUTHOR]

The Fibonacci cube of dimension n, denoted as Γ n , is the subgraph of the hypercube induced by vertices with no consecutive 1s. The Lucas cube Λ n is the cyclic version of Γ n . The irregularity of a graph G is the sum of | d (x) - d (y) | over all edges { x , y } of G. In two recent papers based on the recursive structure of Γ n , it is proved that the irregularity of Γ n and Λ n is two times the number of edges of Γ n - 1 and 2n times the number of vertices of Γ n - 4 , respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind, we give a bijective proof of both results. For these two graphs, we deduce also a constant time algorithm for computing the imbalance of an edge. In the last section using the same approach, we determine the number of edges and the sequence of degrees of the cube complement of Γ n . [ABSTRACT FROM AUTHOR]

Cardiorespiratory fitness (CRF) is not only an objective measure of physical activity, but also a useful diagnostic and prognostic health indicator for patients in clinical settings. There is a well-established inverse relationship between cardiorespiratory fitness (CRF) and mortality. However, the effect of CRF on mortality status might be different on subgroups of individuals and could be higher or lower than the estimated average effect of CRF. Thus, the objective of the study is to identify subgroups with higher or lower impact of CRF on mortality status. In addition, we evaluate and compare both tree-based and non-tree-based algorithms for identifying predictive features and subgroups. A penalized logistic regression with least absolute shrinkage and selection operator (LASSO) penalty is performed to identify the features that may be associated with low CRF and all-cause mortality. The algorithms considered are: virtual twins classification (VT(C)), generalized unbiased interaction detection and estimation (GUIDE) classification (Gc), GUIDE sum (Gs), GUIDE interaction (Gi) to find subgroups of participants where CRF exerts positive or negative association with all-cause mortality from the Ball State Adult Fitness Longitudinal Lifestyle Study (BALL ST) data. The overall result suggests that tree-based (VT and GUIDE) methods naturally define subgroups with fewer predictors and the non-tree-based method (logistic-LASSO) fails to find subgroups, only identify predictors that have impact on mortality status. In terms of predictive variable selection and subgroup identification, Gi is the best method compared to other tree-based and non-tree-based algorithms. Our study identifies subgroups that may be benefited from higher CRF. [ABSTRACT FROM AUTHOR]