Reductio ad absurdum

At this stage, we are ready to prove that Kramers theorem holds also for the total angular momentum F. We will do it by reductio ad absurdum. Then, let Itte) be the eigenvector of H with eigenvalue E,... 

Joseph, Robert Joseph. "Ben Jonson a study of language and the reductio ad absurdum in three of his major comedies." M.A. thesis, Pennsylvania State University, 1971. 

Finally, we can demonstrate that the degeneracy is even when the total angular momentum quantum number F is half-integer, again via a reductio ad absurdum method. Suppose that the degeneracy of the eigenstates is k, then we... 

If there were consensus on all of these matters—and there is—you would have, at least, to concede that half of the objectivity battle—that of reliability and with if face validity—was won. Otherwise you would be forced to hypothesize that the consensus about formal properties was illusory and that the subjects (including me ) were just making up stories to satisfy expectations. This argument invites a reductio ad absurdum rejection. What motive could possibly be satisfied in this way And how could all of the subjects know what was expected of them, as the very notion of formal properties is recent and the specification of them an ongoing project Many people don t even understand what the term means when it is explained to them. 

This equation leads to a reductio ad absurdum that may provide a significant refinement of the helical conductor model. It will be noted that as the step helices (ya = 180°) approach the absolutely planar zigzag chain structure (ya = /b = 180°) they acquire very large cross sections. The index of helieity approaches a constant value of about 0.385 but the expected residue rotations approach infinity. (See Table 12). This result is, at least intuitively, absurd. 

We now prove statement (ii) of the lemma by reductio ad absurdum Assume that Cn,o 7 0- Then the asymptotic form of S iw a r) is given by (62) and we conclude that... 

In order to prove that the densities n(r, t) and n (r, t) will become different infinitesimally later than to, we have to demonstrate that the right-hand side of Eq. (25) cannot vanish identically. This is done by reductio ad absurdum Assume... 

We can now show that and t)2 yield dilferent densities reductio ad absurdum. We have... 

This equation, together with the variational principle, allowed Hohenberg and Kohn [1] to show by a trivial reductio ad absurdum that there is a one-to-one relationship between the density p and the external potential Oext- This implies that the energy E is a functional of the electronic density of the ground state i.e., for a given external potential u,... 

The form (128) suffers from the nonlinearity of H in the external vector potential The standard reductio ad absurdum of the HK-argument only works if H is linear in the external potentials. Thus no existence theorem can be proven with (128). On the other hand, such a proof is possible for the form (129), using the density n and the gauge dependent current jp - c/e)V x m as basic DFT... 

Carnot s theorem employs the method of reductio ad absurdum. It supposes that there... 

Salsburg D (1994) Intent to treat the reductio ad absurdum that became gospel. PharmacoepidemiolDrug Safety 3 329-35. 

Proof ([1], p. B865). The proof proceeds by reductio ad absurdum." We assume the existence of two external potentials Vi(r) and V2(r) such that... 

IV) A self-contradiction (ad absurdum) of Eq. (22) might also mean that the to-be-refuted assumptions (i) or/and (ii) of the Hohenberg-Kohn theorem are selfcontradictory with Eq. (19) and this is precisely the case of many-electron Coulomb systems with Coulomb-type class of external potentials. In other words, the original reductio ad absurdum proof of the Hohenberg-Kohn theorem based on the assumption (19) is incompatible with the ad absurdum assumption (ii) since the Kato theorem is valid for such systems [18]. 

Deb [31], Smith [32], and E. Bright Wilson (quoted by Lowdin [33] for the recent applications of the Kato theorem to the Hohenberg-Kohn theorem see also [34—36]). Therefore, if a given pair of iV-electron systems with the Hamiltonians Hi and H2 of the type (1) are characterized by the same groxmd-state one-electron densities (= to-be-refuted assumption (ii)), their external potentials Vi(r) and V2(r) of the form (24) are identical. The latter contradicts (19) and hence, the assumption (ii) cannot be used in the proof via reductio ad absurdum of the Hohenberg-Kohn theorem together with the assumption (19). In other words, they are Kato-type incompatible with each other. 

Consider a coset RiH with elements Rihx. For hx hy, Rihx must be different from Rihy, simply because two elements in the same row in the multiplication table can never be equal, as was proven in Eq. (3.9). Hence, the size of the coset will be equal to H. Then we consider an element Rj i RiH. This new element will in turn be the representative of a new coset, RjH, and we must prove that this new coset does not overlap with the previous one. This can easily be demonstrated by reductio ad absurdum. Suppose that there is an element Rjhx in the second coset that also belongs to the first coset, as Rihy. We then have ... 

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