Equivalence theorems

We have already seen (in section B of Chapter III) that (7) is partially decidable and that whenever (7) holds for P we can convert P into an always halting tree scheme and decide strong equivalence and total equivalence (Theorems 3.6 and 3.8 and Corollaries 3.7 and 3.9). Hence (7) - (11) are partially decidable. 

D. R. Alcoba, Equivalence theorems between the solutions of the fourth-order modified contracted Schrodinger equation and those of the Schrodinger equation. Phys. Rev. A 65, 032519 (2002). 

Therefore we get our functor Ms C(l)s0 — M(l)s0 a pair ( >/s) as above exists. Suppose that So is the spectrum of a Noetherian complete local ring whose residue field has a finite p-basis. In this case the functor Ms induces an equivalence of the full subcategory of C(l)s0 consisting of group schemes which have multiplicative part equal to zero and a corresponding subcategory of At (1) s0 (Theorem 9.3). If the residue field is perfect then we can show under some additional hypotheses on (5,/s) that Ms C(l)s0 At(l)s0 is an equivalence (Theorem 10.2). 

Probably the most important theoretical result from optimal design theory is the general-equivalence theorem [14, 15], which states the following three equivalent assertions ... 

The Mueller-Jones matrix provides a complete description of the anisotropy properties of an object [9,10]. However, the information in the matrix is in implicit form. The history of the problem of analysis of the Jones and Mueller-Jones matrix goes back to the derivation of three equivalence theorems by Hurwitz and Jones [17]. According to the first theorem, an optical system (object) composed of any number of retardation plates (that is an object with linear phase anisotropy) and rotators (circular phase anisotropy) is optically equivalent to a system containing only two elements a retardation plate, and a rotator. The second theorem is analogous to the first and but is concerned with partial polarizers (linear amplitude anisotropy) and rotators. The third theorem claims that an optical system composed of any number of partial polarizers, retardation plates, and rotators is optically equivalent to a system containing only four elements two retardation plates, a partial polarizer, and rotator. 

In the context of the numerical solution of the Boltzmann equation and the numerical solution of partial differential equations, stability ensures that roundoff and other errors in the calculation are not amplified to cause the calculation to blow up. An algorithm for solving an evolutionary (e. g., a time-dependent) partial differential equation is stable if the numerical solution at a fixed time remains bounded as the time stepsize goes to zero. An important theorem in mathematics, the so-called Lax equivalence theorem, states that an algorithm converges if it is consistent and stable. 

As a special case, we get the celebrated revenue-equivalence theorem [100, 65], which states that the most popular auction formats, i.e. English, Dutch, first-price sealed-bid and second-price sealed-bid, all yield the same price on average in a single item allocation problem with symmetric agents. This is an immediate consequence because these auctions are all efficient in the simple private values model. 

Tjoy of free space that is, Hya = Exa/ o- Applying the equivalence theorem, it can be shown that the far field from this elementary aperture is... 

---