Hidden symmetry

The Jacobian is not symmetric. However, we can rewrite this equation in a way that takes advantage of a hidden symmetry. This allows faster computation as only the upper triangular part of the symmetric matrix J needs to be computed ... 

The energy spectrum of the nucleus according to the semi-empirical shell model [23] appears not at zero ratio, but in two disjoint parts at ratios 0.22 and 0.18. This shift relates to the appearance of the symmetric arrangement at ratio 1.04 rather than 1. An Aufbau procedure based on this result fits the 8-period table derived from the number spiral, but like the observed periodic table, at ratio r, the shell-model result also has hidden symmetry. At ratio zero, the inferred energy spectrum not only fits the 8-period table but also... 

Figure 2.16 Atomic positions superimposed on the sine wave that describes the hidden symmetry of periodic classification. Short vertical lines define allowed positions for closed electronic sub-shells the horizontal axis represents symmetry numbers. Atomic numbers are shown in parentheses.

The universally observed flow of time is another example of a broken symmetry. A theoretical formulation of this proposition is not known, but in principle it should parallel the theory of superconductivity. A high-symmetry state could be associated with Euclidean Minkowski space that spontaneously transforms into a curved manifold of lower symmetry. In this case the hidden symmetry emerges from a Lagrangian which is invariant under the temporal evolution group... 

This equation is identical to the classical Kelvin equation (5.13) and the main conclusions are consistent with the nucleation model of section 5.10.1. As pointed out there, nanosized clusters, with the hidden symmetry of bulk solid, exist in the critical fluid, available to be captured as quantum dots in a glassy matrix. At this stage the crystalline structure of the bulk material has already been developed and the effects of quantum confinement can be observed. 

According to Noether s theorem (Arnold (1989)) symmetries of a mechanical system are always accompanied by constants of the motion. According to Section 3.1, system symmetries can be obvious (e.g. geometric) or hidden . Examples for obvious symmetries that lead to constants of the motion are invariance with respect to time translations, spatial translations and rotations. Invariance with respect to time leads to the conservation of energy, spatial and rotational symmetries lead to the conservation of linear and angular momentum, respectively (see, e.g., Landau and Lifschitz (1970)). Hidden symmetries cannot be associated with... 

A famous example for the presence of a hidden symmetry is the Toda Hamiltonian (see, e.g., Lichtenberg and Lieberman (1983)). In its simplest form it describes the dynamics of three particles moving on a ring subjected to repulsive two-body forces according to... 

SMR revision and practice at judging the states of compounds at different pHs. Hidden symmetry nom conjugation. 

More difficult is the differentiation of tip convolution effects if there is no hidden symmetry. In the images shown in Fig. 2.46, the tip was altered in the course of an experiment. In fact, the images show the same area of a vertically stretched... 

V.I. Pupyshev, A.V. Scherbinin, Hidden symmetry in the confined hydrogen atom problem, Phys. Lett. A 299 (4) (2002) 371-376. 

The synthetic plan followed by the authors is presented in Scheme 24 which shows strategic bond formations in the retrosynthetic sense. For the construction of 181, a ketophosphonate aldehyde condensation was used as the macrocyclization step which led to compound 182, an ester derived from hydroxyaldehyde 183 and ketophosphonate acid 184. The remaining C-C bond formations are indicated on 183 and 184, which then conducted to the key building blocks 127,185-188. The authors took advantage of hidden symmetry elements which led them to select both enantiomers of xylose and tartaric acid as chiral starting materials and/or chiral auxiliaries. 

Scheme 3.26 Target molecules incorporating hidden symmetry...

G.P. Alexander, B.G.G. Chen, E.A. Matsumoto, R.D. Kamien, Power of the poincare group elucidating the hidden symmetries in focal conic domains. Phys. Rev. Lett. 104, 257802 (2010)... 

A novel and ingenious approach to the stereochemically frightening macrolide erythronolide A (M, Scheme 15) is being pursued by Hannes-siAN, Rancourt and Guindon. The approach is based on the recognition of hidden symmetry in the molecule (33) a conception which enables... 

These equations can be shown to have a hidden symmetry about the plane z = 0 similar to that of the equations for uniform states, (5.78) and (5.80). Define a function f(z) by... 

This is reason enough to look very closely for every type of hidden symmetry in synthetic targets. 

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