Independence axiom

The nature of these conditions is still under dispute. The fact that too small particles cannot be ordered derives from quantum mechanics [21]. But the question whether the Pauli principle is an independent axiom of quantum mechanics or not is still unclear [34]. 

Two axioms underlie axiomatic design—the independence axiom and the information axiom. 

The independence axiom asserts that all FRs and their associated DPs remain independently attached therefore, if you adjust a DP to satisfy an FR, you do this without affecting other FRs. Designs that don t satisfy the independence axiom are called coupled. Designs that do satisfy the independence axiom are called uncoupled or decoupled. 

The information axiom is based on information theory, which essentially says that the best design is the one with the least information content—while also satisfying the independence axiom. Information content is defined in terms of probability the more likely the design is to reduce the influence of variation from process parameter changes, different customer-usage conditions and repeated use, the better it meets the information axiom. 

We can illustrate the independence axiom by showing the relationship of two DPs and two FRs— knowing that for such a simple 2x2 design, one would not need axiomatic design. But since axiomatic design is so complex in practice, this is the best way to illustrate how it works at a high level. The examples that follow are highly stylized for illustration purposes in reality, they would be more extensive and detailed. 

The first axiom, the Independence Axiom, requires an uncoupled or at least a decoupled design, which guarantees independent control of the functionality of the product. The FRs must be translated into DPs without affecting other FRs to satisfy the Independence Axiom. That means the set of DPs has to be chosen so that they satisfy the FRs as well as maintain their independence (Suh 1990). 

Time-independent real complexity, when a design is coupled (Independence Axiom violation)... 

Functionally coupled design to make a physical integration Many designers often misunderstand the Independence Axiom by confusing functional independence with physical independence. The physical integration is desirable as long as their functional requirements are independent and uncoupled. 

Kim (2004) illustrated the four causalities of complexity with respect to the design axioms (Fig. 2). Type 1 complexity is a result of heavy coupling of the functional requirements, which is a violation of the independence axiom. Time-independent complexity is a type II complexity and is a result of the information axiom violation. 

Axiomatic design (AD) is a systems design methodology using matrix methods to systematically analyze the transformation of customer needs into functional requirements, design parameters, and process variables [25]. Hereby, the attempt is made to build on the development of new products based on a system of axioms, which are based on mathematics or physics sciences. The formalization is supposed to lead to the design of technical systems. Starting from two axioms (the independence axiom and the information axiom) a system of theorems is set up. 

The so-called laws of Nature are scientific generalizations of regularities observed in the behaviour of a system under specified conditions. Behaviour in this sense implies, almost invariably, the way in which a system of interest develops as a function of time. More basic still, more than law, call it axiom, is the all but universally accepted premise that the outcome of any scientific experiment is independent of its location and orientation in three-dimensional space, provided the experimental conditions can be replicated. A moment s reflection shows that this stipulation defines a symmetry which is equivalent to the conviction that space is both homogeneous and isotropic. The surprising conclusion is that this reproducibility, which must be assumed to enable meaningful experimentation, dictates the nature of possible observations and hence the laws that can be inferred from these observations. The conclusion is father to the thought that each law of Nature is based on an underlying symmetry. 

Your first objective with axiomatic design is to make your design as independent as possible with reference to functionai requirements. After this you wouid make it as robust as possible, as per the information axiom, using such techniques as Robust Design (Technique 38), Design FMEA (Technique 40), and Mistake Proofing (Technique 49). 

While zigging and zagging, the designer ensures that each set of lower-order FRs, DPs, and PVs retains as much independence as possible and also fulfills the information axiom. 

Despite the general acceptance of Euclidean geometry, there appeared to be a problem with the parallel postulate as to whether or not it really was a postulate or that it could be deduced from other definitions, propositions, or axioms. The history of these attempts to prove the parallel postulate lasted for nearly 20 centuries, and after numerous failures, gave rise to the establishment of Non-Euclidean geometry and the independence of the parallel postulate. 

In our case the quotation marks framing the word fundamental imply that the hypotheses considered in the initial stage of development of the theory as postulates can later themselves become the subject of independent investigations and appear as consequences of a more fundamental system of axioms. 

The second law of thermodynamics is believed to be a fact that cannot boil down to more basic statements. It is somewhat like a postulate, or in mathematics, an axiom. However, from the formal view, the enttopy can reach only a maximum, if a maximum in fact exists. Otherwise, entropy cannot reach a maximum. Here we confine ourselves to nonpathological functions, as encountered in physics. From elementary mathematics, there are two conditions necessary for a maximum, namely d5 = 0 and d 5 < 0. If we are somewhat away from the maximum, we have the condition d5 > 0. In this case, the sign of the independent variable tells the direction where to find the maximum. 

Projective geometry has developed independent of affine and absolute geometries and in a sense is a combination of the two by avoiding all but the hrst two of Euclid s axioms. The relation of intermediacy therefore also falls away and segments are not dehned. 

Sometimes there are some situations that the knowledge of an event B does not give us any more information about A than what we already had. Like this situation the event A is called independent of the event B if P(A B) = P A). In this case, we have the following axiom ... 

The advantage of the above axiom is that it treats the events symmetrically and will be easier to generalize to more than two events. Many gambling games provide models of independent events. The spins of a roulette wheel and the tosses of a pair of dice are both series of independent events. 

The derivation of eq. (44) is to a large extent independent of the choice of v(p) and 0/(p) this derivation would hold true even for different isotherms. It is however noted that the validity of the above derivation requires that if another vertical isotherm is intended to replace the BET equation, it must satisfy condition (40). If axioms (C M7) and (C M8) are accepted, the general C M equation is specialized substituting eqs. (34) and (35) for v(p) and (p), respectively, in eq. (44). The resulting equation, whose exphcit expression will be given later (eq. (51)), is referred to as special C M isotherm . 

One checks readily that the axioms I-III hold tme as well if the operations are restricted to V . Hence, with these operations, the set V is also a vector space. In addition, being generated by linear independent vectors (polynomial functions) x(x+l) and x(x-l), the space P is of dimension 2. Generally... 

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