Proof by contradiction

The derivations presented below illustrate the logical technique of proof by contradiction. In this method of proof, we begin by assuming that Carnot s principle is untrue, then demonstrate that we could easily produce crazy consequences that contradict experience if this assumption were valid. That is, we conclude that Carnot s principle must be true, because the contrary assumption leads to inconsistencies with inductive experience. 

Show that there are no quasiperiodic solutions of the Lorenz equations. Solution We give a proof by contradiction. If there were a quasiperiodic solu-... 

Proof Consider a compact, unordered subset L of K". Let t be a unit vector satisfying 0 orthogonal projection onto that is, Qx — x-(x-v)v. Since L is unordered, Q is one-to-one on L (this could fail only if L contains two ordered points). Therefore, Qi, the restriction of Q to L, is a Lipschitz homeomorphism of L onto a compact subset of //y. A straightforward argument by contradiction establishes the existence of wt > 0 such that Qz.Xi —Q X2 > wt X —X2I... 

Solution The proof is by contradiction. If X were countable, we could list all the real numbers between 0 and 1 as a set XpXj.x,.... Rewrite these numbers in decimal form ... 

In their article from 1964, Hohenberg and Kohn proved that the density uniquely determines the potential up to a constant, which does not matter since the potential is always determined up to a constant in any way. This means that the density can be used as the basic variable of the problem, since the potential determines all ground state properties of the system, as can be seen from the Schrodinger equation. In the article, the proof is very simple, and done by contradiction ... 

Proof of sufficiency is obvious and necessity follows by contradiction if > 0 at some real a then such real X may be found at which inequality is invalid. Analogously for b <0. The remaining follows. Q.E.D. 

Proof of sufficiency is obvious and necessity follows by contradiction If it would be that trBXi 0 for some fixed matrix X = Xi then (A.93) must be valid for X = / Xi with any real / . But this is impossible because a + pfr Xi < 0 cannot be valid for some real / . 

Proof of Sufficiency. Suppose that the conditions (1) — (3) are fulfilled for a normal single coronoid G. Then we shall prove by contradiction that G cannot be regular. Assume that the set K of G can be divided into K and K2 such that K (z = 1, 2) contains some fixed single bonds iti an edge cut of type 2, and R, R2 is a standard combination. Suppose... 

Proof We will prove by contradiction. Assume there exists an allocation oiiower t) < oic/it) such that no resource conflicts will arise if aiower(t) resources of type t are allocated. Since aj<, /er(<) < cfactor G,0 t)), there exists at least one resource binding that is derived from the allocation aiower(t) where two operations bound to the same hardware resource may execute in parallel. This results in a resource conflict and hence contradicts the previous assertion that Qiower t) is a conflict-free allocation of t. Therefore, acj t) = cfactor(G,0 t)) is the conflict-free allocation of t. ... 

Proof We wUl prove by contradiction. Assume G is well-posed but there exists a cycle with data-dependent length. Let the cycle be denoted by C. Since C has data-dependent length, this implies that there exists an anchor a on the cycle such that the length of the cycle is greater than or equal to the execution delay 6(a). Consider now the next vertex v that follows a on the cycle C. By definition of anchor sets, a is in the anchor set of v, i.e. a A(v). From Lemma 6.2.2, the anchor sets of all vertices on the cycle must be identical, implying that a is also in the anchor set of a itself. This results in a contradiction. Therefore, we conclude that no cycle of data-dependent length exists in G. ... 

Belkner et al. [32] demonstrated that 15-LOX oxidized preferably LDL cholesterol esters. Even in the presence of free linoleic acid, cholesteryl linoleate continued to be a major LOX substrate. It was also found that the depletion of LDL from a-tocopherol has not prevented the LDL oxidation. This is of a special interest in connection with the role of a-tocopherol in LDL oxidation. As the majority of cholesteryl esters is normally buried in the core of a lipoprotein particle and cannot be directly oxidized by LOX, it has been suggested that LDL oxidation might be initiated by a-tocopheryl radical formed during the oxidation of a-tocopherol [33,34]. Correspondingly, it was concluded that the oxidation of LDL by soybean and recombinant human 15-LOXs may occur by two pathways (a) LDL-free fatty acids are oxidized enzymatically with the formation of a-tocopheryl radical, and (b) the a-tocopheryl-mediated oxidation of cholesteryl esters occurs via a nonenzymatic way. Pro and con proofs related to the prooxidant role of a-tocopherol were considered in Chapter 25 in connection with the study of nonenzymatic lipid oxidation and in Chapter 29 dedicated to antioxidants. It should be stressed that comparison of the possible effects of a-tocopherol and nitric oxide on LDL oxidation does not support importance of a-tocopherol prooxidant activity. It should be mentioned that the above data describing the activity of cholesteryl esters in LDL oxidation are in contradiction with some earlier results. Thus in 1988, Sparrow et al. [35] suggested that the 15-LOX-catalyzed oxidation of LDL is accelerated in the presence of phospholipase A2, i.e., the hydrolysis of cholesterol esters is an important step in LDL oxidation. 

The first viewpoint contradicts the autocatalytic character of the reaction, conductometric measurements in the polymerization system and some other facts (see below). Scheme (33) can be considered as completely experimentally substantiated. The following important proofs were obtained A direct experimental discovery of a quaternary ammonium alcoholate in the reaction system, 42) a full agreement of the nature of the active propagating site with all the existing kinetic and structural data l4,149 153 157 I58) establishment of the ionic behaviour of the propagating sites by comparison of the kinetic curves of the process with the character of the electric... 

Suppose that Conn(G) contains a vertex of degree 3 (either a component C , or a face Fj), then there is a vertex of degree 1, which should be a component CV and so we reach a contradiction by the same method as in the proof for spheres. So, vertices of Conn(G) are of degree 2. This means that Conn(G) is of the form ... 

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