We consider whether the “inequality-splitting” property established in the Brøndsted– Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions...

With this norm, K(W) is a Banach space and even a Banach lattice, i.e., a vector lattice with the property that \g\ ^/, fEk(W) for a measurable g implies gGA(JF) and ||g|| á||/||. Condition (3) ensures [l, Theorem 5] that the norm (4) is continuous with respect to monotone limits. One obtains examples of such functions W as follows. Let (S, B, p.) be a measure space with measure p, which we sha...

A simple connected graph G is said to be interval distance monotone if the interval I (u, v) between any pair of vertices u and v in G induces a distance monotone graph. Aïder and Aouchiche [Distance monotonicity and a new characterization of hypercubes, Discrete Math. 245 (2002) 55–62] proposed the following conjecture: a graph G is interval distance monotone if and only if each of its interva...

BASIC DEFINITIONS: crossing pair of curves and common interior point between two arcs. crossing number cr(G) minimum number of crossings in a drawing of G l-grid is a pair two disjoint edge subset E1, E2 ⊂ E of a drawing of a graph G = (V, E), such that |E1| = |E2| = l and every edge in E1 crosses every edge in E2 bi-clique complete bipartite graph where vertex classes differ in size by at most...

A networked dynamical system is composed of subsystems interconnected via prescribed interactions. In many engineering applications, however, one subsystem can also affect others “unintended” interactions that significantly change the intended network's behavior. Although unintended be modeled as disturbance inputs to subsystems, these disturbances depend on states. Thus, a attenuation property...

In a drawing of a graph, two edges form an odd pair if they cross each other an odd number of times. A pair of edges is independent if they share no endpoint. For a graph G, let ocr(G) be the smallest number of odd pairs in a drawing of G and let iocr(G) be the smallest number of independent odd pairs in a drawing of G. We construct a graph G with iocr(G) < ocr(G), answering a question by Széke...

Let $mathcal{X}$ be a partially ordered set and $d$ be a generalized metric on $mathcal{X}$. We obtain some results in coupled and coupled coincidence of $g$-monotone functions on $mathcal{X}$, where $g$ is a function from $mathcal{X}$ into itself. Moreover, we show that a nonexpansive mapping on a partially ordered Hilbert space has a fixed point lying in  the unit ball of  the Hilbert space. ...

1. Overview This paper talks about that monotone circuit is P-Complete. Decision problem that include P-Complete is mapping that classify input with a similar property. Therefore equivalence relation of input value is important for computation. But monotone circuit cannot compute the equivalence relation of the value because monotone circuit can compute only monotone function. Therefore, I make...

A metric graph is a pair (G, d), where G and $$d:E(G) \rightarrow \mathbb {R}_{\ge 0}$$ distance function. Let $$p \in [1,\infty ]$$ be fixed. An isometric embedding of the d) in $$\ell _p^k = (\mathbb {R}^k, d_p)$$ map $$\phi :V(G) {R}^k$$ such that $$d_p(\phi (v), \phi (w)) d(vw)$$ for all edges $$vw\in E(G)$$ . The _p$$ -dimension least integer k there exists an _p^k$$ functions d has _p^K$$...