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Consider the square lattice Z 2 with vertices at points with integer-valued coordinates in R 2 = {(x 1 , x 2)| x k ∈ R, k = 1, 2} and complex (or real) scalar fields u on the lattice Z 2 , u : Z 2 → C, that are defined by their values u i 1 i 2 , u i 1 i 2 ∈ C, at each vertex of the lattice with the coordinates (i 1 , i 2), i k ∈ Z, k = 1, 2. Consider a… (More)

We consider a special class of two-dimensional discrete equations defined by relations on elementary N × N squares, N > 2, of the square lattice Z 2 , and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary N × N squares, N > 2, in the cubic lattice Z 3. For an arbitrary N we… (More)

The paper is devoted to complete proofs of theorems on consistency on cubic lattices for 3 × 3 determinants. The discrete nonlinear equations on Z 2 defined by the condition that the determinants of all 3×3 matrices of values of the scalar field at the points of the lattice Z 2 that form elementary 3 × 3 squares vanish are considered; some explicit concrete… (More)

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