The dichromatic number of a graph G is the maximum integer k such that there exists an orientation of the edges of G such that for every partition of the vertices into fewer than k parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and NeumannLara conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number. We m...

and Applied Analysis 3 3. Numerical Solution Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (6). Solving diffi...

A new fractional order seminorm, ICTV r , r ∈ R , r ≥ 1 , is proposed in the one-dimensional setting, as a generalization of the standard ICTV k -seminorms, k ∈ N . The fractional ICTV r -seminorms are shown to be intermediate between the standard ICTV k -seminorms of integer order. A bilevel learning scheme is proposed, where under a box constraint a simultaneous optimization with respect to t...

We derive the hypoelliptic estimates for a kinetic equation of the form ∂tf + k · ∇xf = (−∆d)h, for (t, x, k) ∈ R× R × S, where d ≥ 1, β > 0, Sd is the unit sphere in Rd+1 and ∆d is the Laplace-Beltrami operator on Sd. Such equations arise in the modeling of high frequency waves in random media with long-range correlations. Assuming some (fractional) Sobolev regularity in the momentum variable ...

New fractional r -order seminorms, TGV r , r ∈ R , r ≥ 1 , are proposed in the one-dimensional (1D) setting, as a generalization of the integer order TGV k seminorms, k ∈ N . The fractional r -order TGV r -seminorms are shown to be intermediate between the integer order TGV k -seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect...

We state and prove various new identities involving the ZK parafermion characters (or level-K string functions) cn for the cases K = 4, K = 8, and K = 16. These identities fall into three classes: identities in the first class are generalizations of the famous Jacobi θ-function identity (which is the K = 2 special case), identities in another class relate the level K > 2 characters to the Dedek...