Properties of increasing positively homogeneous functions are studied; in particular, their representations by use of tropical inner products with coefficients chosen from tropical support sets are described. An application to a model of economic complementarityand weak links is developed. It is shown that weak links do not necessary bound total factor productivity from below but in some cases constraint it from above.
If g is a map from a space X into Rm and q is an integer, let B q,d,m(g) be the set of all planes Πd ⊂Rm such that |g−1(Πd)| ≥ q. Let also H(q,d,m,k) denote the set of all continuous maps g : X →Rm such that dim B q,d,m(g) ≤k. We prove that for any n-dimensional metric compactum X each of the sets H(3, 1,m, 3n + 1 −m) and H(2, 1,m, 2n) is dense and Gδ in the function space C(X, Rm) provided m ≥ 2n + 1 (in this case H(3, 1,m, 3n + 1 − m) and H(2, 1,m, 2n) can consist of embeddings). The same is true for the sets H(1,d,m,n+ d(m − d)) ⊂ C(X, Rm) if m ≥n + d, and H(4, 1, 3, 0) ⊂ C(X, R3) if dim X ≤ 1.
We prove that the bound from the theorem on ‘economic’ maps is best possible. Namely, for m > n + d we construct a map from an n-dimensional simplex to an m-dimensional Euclidean space for which (and for any close map) there exists a d-dimensional plane whose preimage has cardinality not less than the upper bound ⌈(dn + n + 1)/(m − n − d)⌉ + d from the theorem on ‘economic’ maps.
We develop a consensus clustering framework developed three decades ago in Russia and experimentally demonstrate that our least squares consensus clustering algorithm consistently outperforms several recent consensus clustering methods.
The new economic-mathematical model based on complex variables theory and the new approach to complex variables usage in economics are suggested in the article. The comparison of modeling results of actual production processes using Cobb-Douglass production function and complex variables production function is conducted. It is shown that the instrumental base of economicmathematical methods can be widen with usage of complex variables theory.