Linear hierarchy

FIG. 2. The formation of the apical protein complex involves two distinct steps. Bazooka is localized apically in the epithelium from which neuroblasts are derived. In the interphase (G2), delaminating neuroblast formation of the apical complex is initiated. It is thought that Baz acts to allow neuroblasts to retain the apical/basal polarity inherent in the epithelium. Baz recruits Insc to the neuroblast apical stalk during delamination before Pins becomes part of the complex. During this initiation step Baz, Insc and Pins are part of a linear hierarchy. However following delamination and during mitosis, the maintenance of the apical localization of each of these proteins requires all three proteins. 

Rossler, O. E. Hudson, D. L. (1985). A piecewise linear hierarchy. Symp. on Math. Biol., Kyoto (Abstract) pp. 18-20. 

Decision/action charts are linear descriptions of the task and provide no information on the hierarchy of goals and objectives that the worker is trying to achieve. 

By insisting on isolating systems from an outside, and by treating fundamental processes as though they are merely linearly additive strings of simple two-point interactions, conventional physics thus strips away much of the universe s nonlinear richness. Certainly, such a program fails to capture subtle nuances of behavior, that can be revealed only when the full integrity of system complexity - with all of its interconnected hierarchy of interactions - is retained and respected. 

Where Western science has heretofore been predicated on (1) static partitions of S2 (modulo our co-evolved senses and language), and (2) simple, linear chains of cause o effect, the generalized CA-based physics represents a paradigm shift to (1) causal webs, and (2) fully coevolving object interaction hierarchies and dynamic partitions. 

Abstract This chapter gives an overview of the research on the self-assembly of amorphous block copolymers at different levels of hierarchy. Besides the influence of composition and topology on the morphologies of block copolymers with linear, cyclic and branched topologies blends of block copolymers with low molecular weight components, other polymers or block copolymers and nanoparticles will also be presented. 

Fig. 1. The four types of linear protein domain relationships. They have been ordered by degree of complexity in a manner analogous to the Chomsky finite automaton language hierarchy. Only the first three have been identified in nature thus far. This is no doubt that, in the fourth case, multiple simultaneous mutations would be required. If the linear order of the amino acid sequence is viewed as generated by a finite automaton, there is an equivalence with the Chomsky hierarchy. 

An important characteristic of ab initio computational methodology is the ability to approach the exact description - that is, the focal point [11] - of the molecular electronic structure in a systematic manner. In the standard approach, approximate wavefunctions are constructed as linear combinations of antisymmetrized products (determinants) of one-electron functions, the molecular orbitals (MOs). The quality of the description then depends on the basis of atomic orbitals (AOs) in terms of which the MOs are expanded (the one-electron space), and on how linear combinations of determinants of these MOs are formed (the n-electron space). Within the one- and n-electron spaces, hierarchies exist of increasing flexibility and accuracy. To understand the requirements for accurate calculations of thermochemical data, we shall in this section consider the one- and n-electron hierarchies in some detail [12]. 

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

The recognition of the two fundamental mechanisms of reptation and arm fluctuation for linear and branched entangled polymers respectively allows theoretical treatment of the hnear rheology and dynamics of more complex polymers. The essential tool is the renormahsation of the dynamics on a hierarchy of timescales, as for the case of star polymers. It is important to stress that experimental checks on well-controlled architectures of higher complexity are still very few due to the difficulty of synthesis, but the case of comb-polymers is an example where good data exists [7]. 

It is straightforward to solve the hierarchy of equations iteratively. However, the simplest iteration schemes converge only linearly (i.e., poorly). One should better consider quadratic iteration schemes. 

In the usual coupled-cluster hierarchy, MP2 energy functional, while linearized coupled-cluster single-doubles (L-... 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

While in normal combinatorial peptide libraries (either chemical or phage display) each component has a unique sequence that is different from all others, in the cycloscan libraries all components have the same sequence, but differ in their conformation. This conformational diversity is generated in a dendrimeric hierarchy as shown exemplarily in Scheme 27 for the parent linear heptapeptide A-B-C-D-E-F-G. The diversity of the 1st order sublibrary (this nomenclature was adopted from Furka[468l) is based on the mode of cyclization. Excluding the head-to-tail cyclization there are seven different modes of cyclization that can be used for cycloscan three natural modes of cyclization and four modes of N-backbone cyclization. In addition there are five theoretical modes of C-backbone cyclization (see Scheme 1) which are not included in Scheme 27. 

Separability between electronic and nuclear states is fundamental to get a description in terms of a hierarchy of electronic and subsidiary nuclear quantum numbers. Physical quantum states, i.e. wavefiinctions 0(q,Q), are non-separable. On the contrary, there is a special base set of functions Pjt(q,Q) that can be separable in a well defined mode, and used to represent quantum states as linear superpositions over the base of separable molecular states. For the electronic part, the symmetric group offers a way to assign quantum numbers in terms of irreducible representations [17]. Space base functions can hence be either symmetric or anti-symmetric to odd label permutations. The spin part can be treated in a similar fashion [17]. The concept of molecular species can be introduced this is done at a later stage [10]. Molecular states and molecular species are not the same things. The latter belong to classical chemistry, the former are base function in molecular Hilbert space. 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

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