Unmixing problems in many areas such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of dictionary elements that match observed data. We show how aspects of these problems, such as misalignment of DOAS references and uncertainty in hyperspectral endmembers, can be modeled by expanding the dictionary wi...

This work describes a data structure, the Implicit Real-Vector Automaton (IRVA), suited for representing symbolically polyhedra, i.e., regions of n-dimensional space defined by finite Boolean combinations of linear inequalities. IRVA can represent exactly arbitrary convex and non-convex polyhedra, including features such as open and closed boundaries, unconnected parts, and non-manifold compone...

Some bounds on convex combinations of ω and χ for decompositions into many parts Abstract A k–decomposition of the complete graph K n is a decomposition of K n into k spanning subgraphs G 1 ,. .. , G k. For a graph parameter p, let p(k; K n) denote the maximum of k j=1 p(G j) over all k–decompositions of K n. It is known that χ(k; K n) = ω(k; K n) for k ≤ 3 and conjectured that this equality ho...

We study convex Hamilton-Jacobi equations H(x,Du) = a and ut+H(x,Du) = a in a bounded domain Ω of R with the Neumann type boundary condition Dγu = g in the viewpoint of weak KAM theory, where γ is a vector field on the boundary ∂Ω pointing a direction oblique to ∂Ω. We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex HJ equations, some...

The nearest point problem (NPP), i.e., finding the closest points between two disjoint convex hulls, has two classical solutions, the Gilbert–Schlesinger–Kozinec (GSK) and Mitchell–Dem’yanov–Malozemov (MDM) algorithms. When the convex hulls do intersect, NPP has to be stated in terms of reduced convex hulls (RCHs), made up of convex pattern combinations whose coefficients are bound by a mo1 val...

In this paper, we first characterize the convex $L$-subgroup of an $L$-ordered group by means of fourkinds of cut sets of an $L$-subset. Then we consider the homomorphic preimages and the product of convex $L$-subgroups.After that, we introduce an $L$-convex structure constructed by convex $L$-subgroups.Furthermore, the notion of the degree to which an $L$-subset of an $L$-ord...

‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

In this article, some new results on $M$-matrices, $H$-matrices and their inverse classes are proved. Specifically, we study when a singular $Z$-matrix is an $M$-matrix, convex combinations of $H$-matrices, almost monotone Cholesky factorizations $H$-matrices.