Weak sense

Note that conditions (4.23), (4.24) hold in the weak sense. We see that boundary conditions considered at the crack faces have the equality type in this section. 

We want to prove that, if this is the case, then only solutions with Kg) < e will be produced by the minimization of the empirical risk, and convergence in this weak sense will be guaranteed. Let g be a function such that Kg ) > s. Then from Eq. (25)... 

In Section 2.2 we shall distinguish between sites that are identical in a strict or in a weak sense. Here, identical means that all Q k) are equal, independently of the specific set of k sites. 

We say that the sites are identical in a weak sense whenever the three PFs 0(1,0,0), 0(0,1,0), and 0(0,0, 1) have the same value. This is identical to the requirement that the single-site intrinsic constant is the same for any specific site. In this case we can replace these three PFs by three times one representative PF, as is done on the rhs of Eq. (2.2.22). We shall say that the sites are identical in a strict sense whenever the PF of any given occupation number is independent of the specific group of occupied sites. For instance, in an equilateral triangle all PFs with two sites occupied are equal. Hence we can replace the sum on the rhs of Eq. (2.2.23) by three times one representative PF. This cannot be done, in general, for a linear arrangement of the three sites, in which case 0(1,1,0) is different from 0(1,0,1), even when the sites are identical in the weak sense (see Chapter 5). Similarly, for... 

Since each site has a diiferent value of (a = a, b, c), the sign of the different correlations g(a, p) depends on the product (/ - 1) (Ji - 1). This is the same as in the two-site system, discussed at the end of Section 4.5. Perhaps the most important aspect of these correlations is their independence of the ligand-ligand distance. This is true for yia, b) and y(b, c), as well as for y(a, c). In fact, when the sites are identical in the weak sense, then h = hi, = h = h.ln this case all the indirect... 

If the system is known to have m identical sites, either in a strict or in a weak sense, then K.J. = mk, where k is the intrinsic binding constant. On the other hand, if we know that the system has m different binding sites, each having a different intrinsic binding constant k, then we must determine each of these from the limiting slope of the corresponding individual BI, i.e.. 

For all the above reasons we have defined g(C) without reference to any hypothetical, independent-site system. One simply extracts both 1(C) and all from the experimental data, and then constructs the quantity g(C). When the sites are identical in a weak sense, i.e., all k = k, some of the correlations for a given / might differ. For example, four identical subunits arranged in a square will have only one intrinsic binding constant k, but two different pair correlation functions. For this particular example we have four nearest-neighbor pair correlations g (2), and two second-nearest-neighbor pair correlations gJJ)- The average correlation for this case is... 

In the square model, the sites are identical only in a weak sense. This means that there is only one (first) intrinsic binding constant, but we have two different pair correlations, which are denoted by g and for the nearest and next-nearest... 

From now on we use only the GPF for the cyclic system and drop the subscript C. Since our system has m identical units, the sites will always be identical in the weak sense. There is always one intrinsic constant for the first site but, in general, we have more than one pair correlation, triplet correlation, etc. As in Section 7.1 we develop, for simplicity, the case of two states/= 2, but most of the results are quite general. 

We turn now to the finite open and closed chain and compare the pair correlations obtained in the different systems. First, we note that in the m —> °o limit all the sites become identical in the weak sense, i.e., there is only one intrinsic binding constant, but different pair (and higher-order) correlations as shown in Eq. (7.4.28). It should be noted, however, that owing to the translational invariance of the infinite system there is only one nn pair correlation, only one second nn pair correlation, etc. In other words, it does not matter where in the chain we choose the pair of nn neighbors, or the second nn neighbors, etc. This translational invariance is lost in the finite open system. 

Under quite general conditions on the geometry of the flow domain and the data we show that the model has a solution that satisfies the equations and boundary conditions in an integrated or weak sense. Clearly, the fluid velocity q, as well as the electrical charge c are solved independent of the chemistry. This part of the model ((81-3) and (9i 2)) is standard and its solution is straightforward. The challenging non-standard issue is the description of the chemistry ((84) and (93-5)), in particular the multi-valued dissolution rate in (95). Existence is demonstrated by regularization, where (94,5) are replaced by... 

Displays a weak sense of organization and/or focus, and may lack unity and/or flow of ideas Demonstrates an inadequate command of language, with limited or incorrect vocabulary, and incorrect or flawed sentence structure... 

Displays a weak sense of organization and/or focus, and may lack unity and/or flow of ideas... 

The interesting question concerns the justifications within the subset, the justifications of simple sentences which fix the reference of words and thereby constitute the structure of reality. Can sentences like This is a dog be justified and yet false Yes, in two different ways. The first way is when the justification procedure is not executed properly. I may fall victim to an illusion, or I may be too careless. I may then believe that the justification conditions are satisfied, but they are not. In a weak sense, I have justification, but my justified view is wrong. This does not present a difficulty for the view summarized in Figure 2, which maintains that the criteria of identity of entities are identical with justification conditions. One may misjudge whether a justification condition is satisfied. But this does not change the justification condition. Of course, a justification condition should be such that it can be easily and uncontroversially settled whether it is satisfied. (Otherwise it would not be a justification condition.) But it would be far too much to demand that no mistake should be possible about it. If the justification conditions are not affected by the occasional mistakes, they may fix the reference of words and provide criteria of identity. The occasional mistakes just do not matter. 

In a weak sense, yes, as long as we are careful to define "specificity." Imagine that A and B are two different materials. At each sampling frequency in the summation, compare the terms in the Hamaker coefficients for... 

Natural science proper is physics. It has a pure and applied part. The pure part is strictly apodeictic (proper in the strong sense). The applied part needs the assistance of principles of experience (4 469) and is proper in the weak sense. An example of strong/pure and weak/applied proper science would be pure and applied geometry. Physics is based on a priori principles, both from mathematics and philosophy (a metaphysics of nature) 3 natural science proper presupposes metaphysics of nature (4 469).4 The use of mathematics introduces the pme part of science, and at the highest level of abstraction there is the metaphysical a priori. 

Hence, it may be surmised that from 1795 onwards, under the influence of the work of Lavoisier, Kant considers chemistry to have achieved the status of proper science (in the weak sense). But the change was slow and Stahl was not simply dismissed he had prepared the way, because he tried to use a priori principles to order chemistry. Also Kant s transition from phlogiston to oxidation was slow for some time Kant seems to have used oxidation and dephlogistination as synonyms. 

An even stronger property than the ergodic property is the concept of a mixing system. For a mixing system, the finite time density p(z, t) converges, in the weak sense, to the invariant distribution Poo(z), as f 00. That is, we have, for all test functions (p in some chosen space... 

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