Piecewise Polynomial Approximation Methods in the Theory of Nikol’Skiĭ–Besov Spaces

Estimates for n-widths of two-weighted summation operators on trees

“…To this end, we argue similarly as in [4,Theorem 4.1], [5]. Denote by ϕ * : [0, µ n (X n )] → R + the non-increasing rearrangement of the function |ϕ|.…”

Equations of State That Describe the Phase Transition of Order–Disorder Type Substitution in Binary Alloys

“…In its disordered state, Cu x Au 1−x alloy has a facecentered A1 structure, the symmetry of which is with one atom in a primitive cell of the crystal. Ordering that is typical of (for example) stoichiometric composition CuAu, is described by the star of vector [5]. The star of this vector contains three rays, all of which are equivalent outside of the ordered phase.…”

“…where Γ i as γ i in (2) are associated by norming condition = 1. According to [1,4], the first variations of nonequilibrium potential (1) produce two independent equations that determine the symmetry and structure of ordered phases: (4) Equations ( 4) in [1,[5][6][7] were solved without considering the possible connection between γ i and η, and an entire class of solutions to equation of state corresponding to ordered phases was lost. This can be seen if we write (4) in and explicit form that assumes a low value of η.…”

“…This can be seen if we write (4) in and explicit form that assumes a low value of η. Restricting ourselves to terms proportional to η 2 , we use (4) to obtain (5) where…”

“…In addition to solutions = 1/3; = 0; = 1, which were considered in [1,[5][6][7], Eqs. (5) obviously have at least one more solution: ≠ 0;…”