The Invariance Concept

The Genomic Potential Hypothesis A Chemist s View of the Origins, Evolution and Unfolding of Life, by Christian Schwabe. 2001 Eurekah.com 

Presently the dart needle perspective makes sense. The basic material must have been the same to make a living creature. The early conditions that led to life are as restricting as the apertures the physicists use to analyze radiation except, as biology progresses, the holes get wider. The dart needle and its passenger, penetrating the skin, the cell and finally the nucleus, traveled backwards in evolutionary time through the apertures to the absolutely required initial structures, the nucleic acid core. 

Proteins are unstable yet their production is an inevitable step on the way from nucleic acid to life. While there is latitude in terms of the kind of proteins acquired, as concerns the functions of these proteins there are significant restrictions. Their catalytic activity must be supportive and not destructive for the parent organism. All conditions are de facto rectifiers that tend to spread uniformity. 

Whatever the outcome one can be sure that a physical contact type selection has led to the genomic code. Not only does the new hypothesis predict such contact, the status of evolution as a discipline of science depends upon it. Frozen accidents are not the stuff of hypotheses but more likely failures of insight. This means that the principle of code development is discoverable and with it, slowly to be sure, the whole mechanism of biochemical evolution. 

Clonal development of life means condensation of pieces of memory from a pool of limited volume until enough functions can be read from the nucleic acids core to support autonomy. A smooth transition of complex chemistry to complex biochemistry must proceed through stable (immortal) steps and products that last until the next 

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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. 

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. 

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. 

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. 

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... 

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

In Chap. 6 we learned that in the excluded volume limit ftc > 0,n —> oo, the cluster expansion breaks down, simply because it orders according to powers of z = j3enef2 —> oo. To proceed, we need a new idea, going beyond perturbation theory. The new concept is known as the Renormalization Group (RG), which postulates, proves, and exploits the fascinating scale invariance property of the theory. 

In this review we discuss the scaling properties of topological invariants and the relations between them. First, we recall the basic concepts of the chemical graph theory. Then, we introduce reduced topological invariants and discuss the problem of size extensivity. Following a brief overview of the known approximate relations between topological invariants, we move to considerations of their scaling properties. Finally, we discuss some practical aspects of the present formalism. 

The concept of a mass point remains valid, but a time interval dt can no longer be treated as a nondynamical parameter. Einstein s basic postulate [323, 393] is that the interval ds between two space-time events is characterized by the invariant expression... 

The molecular concept has become so central in chemistry that understanding of chemical events is commonly assumed to consist of relating experimental observations to micro events at the molecular level, which means changes in molecular structure. In this sense molecular structure is a fundamental theoretical concept in chemistry. As the micro changes are invariably triggered by electron transfer, the correct theory at the molecular level must be quantum mechanics. It is therefore surprising that a quantum theory of molecular structure has never developed. This failure stems from the fact that physics and chemistry operate at different levels and that grafting the models of physics onto chemistry produces an incomplete picture. 

Recently, Wiggins et al. [15] provided a firm mathematical foundation of the robust persistence of the invariant of motion associated with the phase-space reaction coordinate in a sea of chaos. The central component in RIT that is, unstable periodic orbits, are naturally generalized in many DOFs systems in terms of so-called normally hyperbolic invariant manifold (NHIM). The fundamental theorem on NHIMs, denoted here by M, ensures [21,53] that NHIMs, if they exist, survive under arbitrary perturbation with the property that the stretching and contraction rates under the linearized dynamics transverse to jM dominate those tangent to M. Note that NHIM only requires that instability in either a forward or backward direction in time transverse to M is much stronger than those tangential directions of M, and hence the concept of NHIM can be applied to any class of continuous dynamical systems. In the case of the vicinity of saddles for Hamiltonian problems with many DOFs, the NHIM is expressed by a set of all (p, q) satisfying both q = p = Q and o(Jb) + En=i (Jb, b) = E, that is. 

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