By Cannarsa P., Cardaliaguet P., Crasta G.

The method of partial differential equationsarises within the research of mathematical types for sandpile progress and within the context of the Monge-Kantorovich optimum mass shipping concept. A illustration formulation for the recommendations of a comparable boundary price challenge is the following received, extending the former two-dimensional results of the 1st authors to arbitrary area measurement. An software to the minimization of necessary functionals of the formwith f≥ zero, and h≥ zero most likely non-convex, is additionally incorporated.

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**Extra info for A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications**

**Example text**

Based on the above observations, the procedure to ﬁnd metapaths can be described as follows: The Algebraic Structure of Metagraphs 25 1. Select a candidate set of input rows I in A∗ such that xi ∈ B, ∀i ∈ I, B = i xi . Start with single rows, and repeat with larger sets progressively in successive iterations. 2. If ∃xj ∈ B such that aij∗ = φ, ∀i ∈ I , then there is no metapath from {xi | i ∈ I } to C. Return to step 1 and repeat with another set of rows. 3. Find a candidate set of triples in cells aij∗ such that i ∈ I, xj ∈ C that forms a cover for C (where a cover for C is a set of triples T such that C ⊆ t∈T output(t)).

Let i, j be such that aij = φ, but aij∗ = φ. Then there must be at least one simple path, p, from xi to xj in S consisting of a sequence of edges eij , . . , eik with each such eij ∈ E , k = 1, . . , K. The coinputs of xi in this path are all elements of X . Thus, there is a metapath from xi ∪ {x | x is a coinput of xi in p)} to xi in S (and also in S). 6), if there is a metapath in S from some V ⊆ X to some W ⊆ X , then amn = φ for Metagraph Transformations 37 all xm ∈ V and xn ∈ W . Since xi ∪ {x | x is a coinput of xi in p)} ⊆ X and xi ∈ X , it follows that aij = φ which contradicts the assumption.

If a proposition appears in the invertex of an edge, it must be true for the edge to be used in a metapath. Each edge may contain zero, one, or more assumptions, and each assumption may appear in one or more edges. Propositions appear in the generating set along with elements representing other types of variables. We distinguish metagraphs that contain propositions by the term conditional metagraph. We deﬁne a conditional metagraph as follows. 1. A conditional metagraph is a metagraph S = Xp ∪ Xv , E , in which Xp is a set of propositions and Xv is a set of variables, and: 1.