Invariance concept

Similar invariance concepts for anisotropic materials were also developed by Tsai and Pagano [2-7]. For anisotropy, the following definitions... 

Discussion of invariance concepts for laminates will be deferred until Chapter 7 after the development of lamination concepts in Chapter 4. 

Can one be so sure about this invariance concept only because we can never test this prediction Not really. The Jovian moons are within our reach and Mars is still in contention as a life-supporting rock, so one must consider the idea in principle testable. Life will be the same and everybody who is searching for life elsewhere knows it intuitively, because they are looking with the methods it takes to detect our kind of life. They are all unwitting supporters of the invariance concept of the Genomic Potential Hypothesis. 

Furthermore, (Filippini Zio 2013, Angelo Filippini 2013) represent the infrastructure systems in a directed graph and a state dynamics is associated to each subsystem for describing the failure and recovery processes and their interdependent effects. A first attempt of resilience analysis is proposed based on invariance concepts and asymptotic stability. However, a thorough characterization of the resilience and a metric of the system resilience are not provided. 

This section presents the necessary conditions on the design parameters for the existence of a resilience region. The region is identified using the invariance concepts introduced in Section 2.2 above. 

The Morgan Algorithm classifies all the congeneric atoms of a compound and selects invariant-labeled atoms (see Section 2.5.3.1). The classification uses the concept of considering the number of neighbors of an atom (connectivity), and does so in an iterative manner (extended connectivity, EC). On the basis of certain rules. 

The advantage of writing the tran forriTation eqi tions in the form of Equation (2.98) is that parts of Q.,., Q.,2, Q22. and Qgg are then obviously invariant under rotations about the z-axis (perpendicular to the lamina). This concept of invariance is useful when examining the prospect of orienting a lamina at various angles to achieve a certain stiffness profile. For example,... 

Invariant stiffness concepts as developed by Tsai and Pagano [7-16 and 7-17] can be used as an aid to understanding the stiffnesses of laminates of arbitrary orientation and how those stiffnesses can be varied. The concepts and their use are discussed in Sections 7.7.3.1 through 7.7.3.3. 

The invariant stiffness concepts for a iamina will now be extended to a laminate. All results in this and succeeding subsections on invariant laminate stiffnesses were obtained by Tsai and Pagano [7-16 and 7-17]. The laminate is composed of orthotropic laminae with arbitrary orientations and thicknesses. The stiffnesses of the laminate in the x-y plane can be written in the usual manner as... 

The analytical tools to accomplish laminate design are at least twofold. First, the invariant laminate stiffness concepts developed by Tsai and Pagano [7-16 and 7-17] used to vary laminate stiffnesses. Second, structural optimization techniques as described by Schmit [7-12] can be used to provide a decision-making process for variation of iami-nate design parameters. This duo of techniques is particularly well suited to composite structures design because the simultaneous possibility and necessity to tailor the material to meet structural requirements exists to a degree not seen in isotropic materials. 

This novel manifestation of the gas chromatographic separation demands that Our fundamental understanding of the GC method-invariably of single-dimensional scope-is challenged as follows concepts of column efficiency and separation are now Supplanted by a need to compare the performances of two columns operating... 

In the case of non-metallic materials, the term corrosion invariably refers to their-deterioration from chemical causes, but a similar concept is not necessarily applicable to metals. Many authorities consider that the term metallic corrosion embraces all interactions of a metal or alloy (solid or liquid) with its environment, irrespective of whether this is deliberate and beneficial or adventitious and deleterious. Thus this definition of corrosion, which for convenience will be referred to as the transformation definition. 

The mechanism of Self-organized criticality, a concept first introduced by Bak, Tang and Wiesenfeld [bak87a], may possibly provide a fundamental link between such temporal scale invariant phenomena and phenomena exhibiting a spatial scale invariance - familiar examples of which are given by fractal coastlines, mountain landscapes and cloud formations [mandel82],... 

To obtain Polya s Theorem from this we take the "weight" of a figure to be x where k is its content. (We should note here that the term "weight" is regarded by some authors as synonymous with "content" however, it is preferable to use them for the two different concepts just given.) Suppose that a configuration is invariant under... 

Invariance Properties.—Before delving into the mathematical formulation of the invariance properties of quantum electrodynamics, let us briefly state what is meant by an invariance principle in general. As we shall be primarily concerned with the formulation of invariance principles in the Heisenberg picture, it is useful to introduce the concept of the complete description of a physical system. By this is meant at the classical level a specification of the trajectories of all particles together with a full description of all fields at all points of space for all time. The equations of motion then allow one to determine whether the system could, in fact, have evolved in the way... 

In a quantum mechanical framework, Postulate 1 remains as stated. It implies that there exists a well-defined connection and correspondence between the labels attributed to the space-time points by each observer, between the state vectors each observer attributes to a given physical system, and between observables of the system. Postulate 2 is usually formulated in terms of transition probabilities, and requires that the transition probability be independent of the frame of reference. It should be stated explicitly at this point that we shall formulate the notion of invariance in terms of the concept of bodily identity, wherein a single physical system is viewed by two observers who, in general, will have different relations to the system. 

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