By Abul Hasan Siddiqi
Consultant covers the most up-tp-date analytical and numerical equipment in infinite-dimensional areas, introducing fresh ends up in wavelet research as utilized in partial differential equations and sign and picture processing. For researchers and practitioners. comprises index and references.
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Extra resources for Applied functional analysis: numerical methods, wavelets, image processing
Let S be a monad in a category category [S] of S-algebras. Then the category [S][T] of T-algebras in [S] is isomorphic to the category [TS] of algebras over the composite monad TS in . Moreover, the unit of T defines a map S −→ TS of monads in . An analogous assertion holds for comonads. C C C C C Proof. Strictly speaking, in constructing TS, we are regarding S as the free Salgebra functor → [S], applying the functor T, and then applying the forgetful functor back to . We continue to neglect notation for forgetful functors and to write S and T ambiguously for both the given monads and the resulting free functors.
Ii) If R and R are A∞ or E∞ ring spectra, then so is R ∧L R ; if either R or R is an S-algebra, then so is R ∧L R . (iv) If R and R are A∞ ring spectra, M is an R-module and M is an R module, then M ∧L M is an R ∧L R -module. In particular, we have a functorial way to replace A∞ and E∞ ring spectra and their modules by S-algebras and commutative S-algebras and their modules. 6. For an A∞ ring spectrum R, S ∧L R is an S-algebra and λ : S ∧L R −→ R is a weak equivalence of A∞ ring spectra, and similarly in the E∞ case.
4, the argument iterates to prove the following statement. 6. For any j-tuple M1 , . . , Mj of L-spectra, there is a canonical isomorphism of L-spectra M1 ∧L · · · ∧L Mj ∼ = (j) L (1)j (M1 ∧ · · · ∧ Mj ), L where the iterated smash product on the left is associated in any fashion. I. PROLOGUE: THE CATEGORY OF L-SPECTRA 24 6. The equivalence of the old and new smash products We here show that the smash product ∧L does in fact realize the classical smash ¯ product of spectra up to homotopy, in the sense that the equivalence between h ¯ and h [L] preserves smash products.
Applied functional analysis: numerical methods, wavelets, image processing by Abul Hasan Siddiqi