Structure preserving transformations in hyperkähler Euclidean spaces

Linear Transformations of Euclidean Topological Spaces

For simplicity, we adopt the following rules: X, Y denote sets, n, m, k, i denote natural numbers, r denotes a real number, R denotes an element of RF, K denotes a field, f , f1, f2, g1, g2 denote finite sequences, r1, r2, r3 denote real-valued finite sequences, c1, c2 denote complex-valued finite sequences, and F denotes a function. Let us consider X, Y and let F be a positive yielding partial...

Determinant preserving transformations on symmetric matrix spaces

Let Sn(F) be the vector space of n × n symmetric matrices over a field F (with certain restrictions on cardinality and characteristic). The transformations φ on the space which satisfy one of the following conditions: 1. det(A+ λB) = det(φ(A) + λφ(B)) for all A,B ∈ Sn(F) and λ ∈ F; 2. φ is surjective and det(A+ λB) = det(φ(A) + λφ(B)) for all A,B and two specific λ; 3. φ is additive and preserv...

Euclidean Structure in Finite Dimensional Normed Spaces

Linear Transformations of Euclidean Topological Spaces. Part II

For simplicity, we follow the rules: X denotes a set, n, m, k denote natural numbers, K denotes a field, f denotes an n-element real-valued finite sequence, and M denotes a matrix over RF of dimension n × m. One can prove the following propositions: (1) X is a linear combination of the n-dimension vector space over RF if and only if X is a linear combination of En T. (2) Let L2 be a linear comb...

On Dummy Variables of Structure-Preserving Transformations

A method is given for obtaining equivalence subgroups of a family of differential equations from the equivalence group of simpler equations of a similar form, but in which the arbitrary functions specifying the family element depend on fewer variables. Examples of applications to classical equations are presented, some of which show how the method can actually be used for a much easier determin...