Schwartz Inequality

Proof.—If k-6 = 0, k°6° = k-. Applying Schwartz inequality to this last equation, we obtain... 

The a ssumption will provide that (l/Vz)X X converges to a finite matrix by virtue of the Cauchy-Schwartz inequality given above. If the assumptions made to ensure that plim (1/ )X e = 0 continue to hold, then consistency can be established by the Slutsky Theorem. 

In view of the experimental difficulties in determining p(y ) it may be helpful to note some rigorous inequalities resulting from the fact that p(y) > 0. by definition. Using the Schwartz inequality it is easily checked that... 

The quantity of excess entropy production is positive by the Cauchy-Schwartz inequality (similar to the inequality in Eq. (4.101)), indicating that P >... 

Proof TjCt us evaluate, first, the difi crcnce between the (7 .+ l)-th and -th iterations. According to formulae (4.74), (4.65), and the Schwartz, inequality... 

For p = q = 2. Holder s inequality is known as the Cauchy-Schwartz inequality. 

Cauchy-Schwartz inequality, it can be shown that this satisfies the criteria for a norm ([16], p. 93). 

This result may be viewed as an application of the Schwartz inequality. The problem with this formula is that it suffers from the same numerical troubles that plague the exact FEP perturbation formula the integrand might have significant contributions from configurations that are poorly sampled by the simulation. 

Schwartz Inequality For any two vectors belonging to the Euclidean space, the Schwartz... 

The Schwartz inequality agrees with what everyone recalls about the dot product of two vectors jc y) = jr II v COS0, where 6 is the angle between the two vectors. Taking the absolute value of both sides, we obtain... 

After the new inner product definition is introduced, the related quantities, the length of a vector and the distance between the vectors, ate defined in exactly the same way as in the Euclidean space. Also, the definitions of the orthogonality and the Schwartz inequality remain unchanged. 

Using here the Bunyakovsky-Schwartz inequality, we find... 

Applying the inequality (3.18) and the Schwartz inequality to the relation (3.42) taking into account the estimate (3.39), we find... 

In the field of quantum chemistry, many algorithms have been developed to improve the efficiency of the two-electron integral calculations over the years. One of the most frequently used algorithms is the screening based on the Schwartz (K.H.A. Schwartz) inequality (1888). That is, based on the inequality,... 

It can also be shown, using the Schwartz inequality, that... 

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