A construction is presented which gives a possibility of describing ternary relations. Graphical representation of ternary relations is noted.

On representation of

We generalize a well-known separation condition of Everitt and Giertz to a class of weighted symmetric partial differential operators defined on domains in $\Bbb R^n$ Also, for symmetric second-order ordinary differential operators we show that $\limsup_{t\to c} (pq')'/q^2=\theta<2$ where $c$ is a singular point guarantees separation of $-(py')'+qy$ on its minimal domain and extend this criterion to the partial differential setting As a particular example it is shown that $-\Delta y+qy$ is separated on its minimal domain if $q$ is superharmonic For $n=1$ the criterion is used to give examples of a separation inequality holding on the domain of the minimal operator in the limit-circle case

/pdf/two-separation-criteria-for-second-order-ordinary-or-partial-1ym462z9oj.pdf

Two separation criteria for second order ordinary or partial differential operators

/pdf/on-representation-of-cyclically-ordered-sets-2zcuxwfm1s.pdf

On representation of cyclically ordered sets

The orientation problem for ternary cyclic order relations has been attacked in the literature from combinatorial perspectives, through rotations , and by connection with Petri nets. We propose here a twofold characterization of orientable cyclic orders in terms of symmetries of partial orders as well as in terms of separating sets (cuts). The results are inspired by properties of non-sequential discrete processeses, but also apply to dense structures of any cardinality.

https://hal.inria.fr/hal-01360144/document

Cyclic Ordering through Partial Orders

It is shown that every 3-connected planar graph with a large number of vertices has a long induced path.

Long induced paths in 3-connected planar graphs

Several classes of cyclic orders arising from geometrical, algebraical, and combinatorial structures are introduced, and their extendability to total cyclic orders is studied. By analogy to Dushnik–Miller dimension for partial orders we define, for circular orders, intersection and product dimension that may differ up to a factor of two. A class of cyclic orders that allow a graphic representation similar to Hasse diagrams is also studied.

Extendability, dimensions, and diagrams of cyclic orders

We present some lower and upper bounds on the length of the maximum induced paths and cycles in Kneser graphs

Long induced paths and cycles in Kneser graphs

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Extendability, dimensions, and diagrams of cyclic orders

Long induced paths and cycles in Kneser graphs