Inversions, cuts, and orientations

8.997 Topics in Combinatorial Optimization 18 Orientations, Directed Cuts and Submodular Flows 18.1 Graph Orientations

Proof: ⇐: Fix a strongly-connected orientation D. For any non-empty U ⊂ V , we may choose u ∈ U and v ∈ V \U . Since D is strongly connected, there is a directed u-v path and a directed v-u δ in path. Thus |δ out (U)| ≥ 1 and | D (U)| ≥ 1, implying |δG(U)| ≥ 2. D ⇒: Since G is 2-edge-connected, it has an ear decomposition. We proceed by induction on the number of ears. If G is a cycle then we m...

On the Theory of Pfaffian Orientations. II. T-joins, k-cuts, and Duality of Enumeration

This is a continuation of our paper \A Theory of Pfaaan Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of T-joins and k-cuts of graphs. As a consequence, we show that the computational problem to nd the maximum weight of an edge-cut is polynomially solvable for the instances (G; w) where G is a graph embedded on an arbit...

Type Two Cuts, Bad Cuts and Very Bad Cuts

Type two cuts, bad cuts and very bad cuts are introduced in [KL] for studying the relationship between Loeb measure and U-topology of a hyper nite time line in an !1-saturated nonstandard universe. The questions concerning the existence of those cuts are asked there. In this paper we answer, fully or partially, some of those questions by showing that: (1) type-two cuts exist, (2) the @1-isomorp...

About Folding-Unfolding Cuts and Cuts Modulo

Study of Temperature and Salinity Inversions in the Oman Sea

In this research, observational data are analyzed to explain temperature and salinity inversions and double diffusion in the Oman Sea. The results of this paper show that intrusive structures extend over a depth of about 100-450 meters in the Oman Sea. Temperature and salinity inversions observed depth of 250 m in winter and summer in the Oman Sea. Steps thickness is about 10 to 20 meters. A wa...