By Suzanne Foley
Catalog of an exhibition of a similar identify held on the Whitney Museum of yankee artwork, Dec. nine, 1981-Feb. 7, 1982, and consequently on the San Francisco Museum of recent paintings. Works through Peter Voulkos, John Mason, Kenneth expense, Robert Arneson, David Gilhooly, and Richard Shaw
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N). Hence, dϕν |M ≡ dxk ∂ϕν ∂ϕν ∂gij + ∂xk ∂gij ∂xk = 0 (k = 1, . . , n; ν = 1, . . , µ). M Multiplying by ξ k and summing over k from 1 to n we obtain: ξk ν ∂ϕν k ∂ϕ ∂gij + ξ ∂xk ∂gij ∂xk The invariance test Xϕν ν k ∂ϕ ξ ∂xk M = 0 (ν = 1, . . , µ). 1) is written now: ∂ξ k ∂ξ k − gik j + gjk i ∂x ∂x ∂ϕν ∂gij = 0 (ν = 1, . . , µ). H. IBRAGIMOV SELECTED WORKS, VOL. 3). Conversely, Eqs. 5). The lemma is proved. 3. 1). 3. Let us introduce the notation: hij = ξ k ∂gij ∂ξ k ∂ξ k + g + g ik jk ∂xk ∂xj ∂xi (i, j = 1, .
1. For brevity, we term as the determining manifold of the space Vn the determining manifold of the manifold given by Eqs. e. 1). 3 is based on the following lemma. 1. 1) is invariant under the group H if and only if ∂gij ξk k ∂x ∂ξ k ∂ξ k + gik j + gjk i ∂x ∂x ∂ϕν ∂gij = 0 (ν = 1, . . , µ). 3) M Proof. Since µ ≤ 12 n(n+1), all quantities xk (k = 1, . . 1) can be considered as independent parameters on which depend the quantities gij (i, j = 1, . . , n). Hence, dϕν |M ≡ dxk ∂ϕν ∂ϕν ∂gij + ∂xk ∂gij ∂xk = 0 (k = 1, .
This section describes some invariant solutions of the rank two obtained from one-parameter subgroups. Let us introduce the notation H1 for one-parameter subgroups and H2 for two-parameter subgroups. The variables U, V, P, R are functions of two arguments λ, µ with values being defined for different subgroups in every separate case. 1. Let us consider the system (10). The invariant H1 -solution in the form u = U, v = V, p = P, ρ = R; λ = x, µ = y, corresponds to the operator X1 . It is a stationary well investigated case, therefore we shall not take it into consideration.
Ceramic Sculpture Six Artists by Suzanne Foley