By Kerimov N. B.
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Additional info for A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions
Am ]iι (x1 , x2 ) (56) = (λµ)ιk+κ K. where, of course, [a0 , . . , am ]iι (x1 , x2 ) considered as a function, is different from (a0 , . . , am )iι (x1 , x2 ) . But (56) is a covariant relation for a covariant of f . This proves the theorem. 44 CHAPTER 2. PROPERTIES OF INVARIANTS The proof holds true mutatis mutandis for concomitants of an n-ary form and for simultaneous concomitants. The index of K is 1 1 ρ = ι · (im − ω) + (ιω − ) 2 2 1 = (iιm − ), 2 and its weight, 1 (iιm + ). 2 Illustration.
SYMBOLICAL INVARIANT PROCESSES55 f = a0 x31 + 3a1 x21 x2 + 3a2 x1 x22 + a3 x32 is a particular case, bears a close formal resemblance to a power of linear form, here the third power. This resemblance becomes the more noteworthy when ∂f bears the same formal resemblance to the we observe that the derivative ∂x 1 derivative of the third power of a linear form: ∂f = 3(a0 x21 + 2a1 x1 x2 + a2 x22 ). ∂x1 That is, it resembles three times the square of the linear form. When we study the question of how far this formal resemblance may be extended we are led to a completely new and strikingly concise formulation of the fundamental processes of binary invariant theory.
Since φ is the same function of ai , x1 x2 , that φ is of ai , x1 x2 we have, by dropping primes, the result: Theorem. A set of necessary and sufficient conditions that a homogeneous function, φ, of the coefficients and variables of a binary form f should be a concomitant is O − x1 ∂ ∂x2 φ = 0, Ω − x2 ∂ ∂x1 φ = 0. 3. SPECIAL INVARIANT FORMATIONS 31 In the case of invariants these conditions reduce to Oφ = 0, Ωφ = 0. These operators are here written again, for reference, and in the un-primed variables: ∂ ∂ ∂ + (m − 1)a2 + · · · + am , ∂a0 ∂a1 ∂am−1 ∂ ∂ ∂ Ω = a0 + 2a1 + · · · + mam−1 .
A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions by Kerimov N. B.