It is shown that the two component Fermi or Bose many-body Hamiltonian, such as the two-orbit fermion pairing and two-site Bose-Hubbard model with arbitrary finite higher order terms can always be solved exactly by using Bethe ansatz vector construction based on the permutation of two components of bosons or fermions involved. As examples of the solution, the extended one-dimensional dimer Bose -Hubbard model with multi-body interactions and the mean-fifield plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits are demonstrated with the eigenstates and the eigen-energy and the related Bethe ansatz equations. It is shown that the main feature of the solutions lies in the fact that the Bethe ansatz vectors can be expressed in terms of binomials of the boson or fermion operators times the related symmetric functions. As the consequence, two-component quantum many-body systems , such as the extended Lipkin-Meshkov-Glick model with higher-order interactions, can be solved in a similar way.

We propose to deploy limits that arise from different tests of the Pauli Exclusion Principle in order: i) to provide theories of quantum gravity with an experimental guidance; ii) to distinguish among the plethora of possible models the ones that are already ruled out by current data; iii) to direct future attempts to be in accordance with experimental constraints. We firstly review experimental bounds on nuclear processes forbidden by the Pauli Exclusion Principle, which have been derived by several experimental collaborations making use of different detector materials. Distinct features of the experimental devices entail sensitivities on the constraints hitherto achieved that may differ one another by several orders of magnitude. We show that with choices of these limits, renown examples of flat noncommutative space-time instantiations of quantum gravity can be heavily constrained, and eventually ruled out. We devote particular attention to the analysis of theκ-Minkowski and θ-Minkowski noncommutative spacetimes. These are deeply connected to some scenarios in string theory, loop quantum gravity and noncommutative geometry. We emphasize that the severe constraints on these quantum spacetimes, although cannot rule out theories of top-down quantum gravity to whom are connected in various way, provide a powerful limitations of those models that it will make sense to focus on in the future.

Noether's theorem connects symmetries to conservation laws in various physical systems. Among the unique symmetries of continuous matter are labelling symmetries which are manifested by Arnold's diffeomorphism group. A special symmetry subgroup of the diffeomorphism is the translation of labelling. This subgroup is connected to conservation laws which suffer a topological interpretation. For example in ideal barotropic fluids the metage translation symmetry subgroup is connected through Noether's theorem to the conservation of helicity. Helicity is a measure of the knottiness of vortex lines and thus a topological constant of motion. The same is true for barotropic or incompressible magnetohydrodynamics (MHD) in which the same subgroup leads to the conservation of cross helicity. Although standard cross helicity is not conserved in non-barotropic MHD it was shown that a new kind of cross helicity which is conserved in the non barotropic case can be introduced. This conservation law was deduced from the variational principle using the Noether’s theorem. The symmetry subgroup associated with the new cross helicity was magnetic metage translations. Also we show that additional labelling translations symmetries exist which are connected to new and different topological conservation laws.

Symmetry contributes to processes of perceptual organization in biological vision and influences the quality and time of goal directed decision making in animals and humans, as discussed in recent work on the examples of 'symmetry of things in a thing' and bilateral shape symmetry (Dresp-Langley, Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing. Symmetry 2016, 8, 127; Dresp-Langley, Bilateral Symmetry Strengthens the Perceptual Salience of Figure against Ground. Symmetry 2019, 11, 225). The present study was designed to show that selective chromatic variations in geometric shape configurations with mirror symmetry can be exploited to highlight functional properties of 'symmetry of things in a thing' in human vision. The experimental procedure uses a psychophysical two-alternative forced choice technique, where human observers have to decide as swiftly as possible whether two shapes presented simultaneously on a computer screen are symmetrical or not. The stimuli are computer generated 2D shape configurations consisting of multiple elements, with and without systematic variations in local color, color saturation, or contrast to manipulate 'symmetry of things in a thing'. All stimulus pairs presented had perfect geometric mirror symmetry. The results show that altering the color saturation of local shape elements selectively in multi-chromatic and mono-chromatic shapes significantly slows down perceptual response times, which are a direct measure of uncertainty. It is concluded that local chromatic variations may produce functionally important variations in 'symmetry of things in thing', increase stimulus uncertainty, and affect the perceptual salience of mirror symmetry and the time course of goal-relevant human decisions.

The asymmetric phenotypic graphs of codons and anticodons of the Standard Genetic code determine the mode of evolution of proteins

Marco V. José^* and Gabriel S. Zamudio

Theoretical Biology Group, Instituto de Investigaciones Biomédicas, Universidad Nacional Autónoma de México, CDMX CP 04510, Mexico City

The Standard Genetic Code (SGC) is written in an alphabet of four letters (C, A, U, G), grouped into words three letters long, called triplets or codons. Each of the 64 codons specifies one of the 20 amino acids or else serves as a punctuation mark signaling the end of a message. The SGC is implemented via the transfer RNAs that bind each codon with its anticodon. These molecules define the genetic code, by linking the specific amino acids and tRNAs with the corresponding anticodons. To understand the meaning of symmetrical/asymmetrical properties of the Standard Genetic Code (SGC), we designed synthetic genetic codes with known symmetries and with the same degeneracy of the SGC. We determined their impact on the substitution rates for each amino acid under a neutral model of evolution. We prove that the phenotypic graphs of the SGC for codons and anticodons for all the possible arrangements of nucleotides are asymmetric. Both the SGC and symmetrical synthetic codes exhibit a proportional probability of occurrence of the amino acids according to their degeneracy. Unlike the SGC, the synthetic codes display a constant probability of occurrence of the amino acid according to their codonicity. The asymmetry of the phenotypic graphs of codons and anticodons of the SGC, has important implications on the evolutionary processes of proteins by preferring specific amino acids irrespective of their codonicity.

Keywords: Standard Genetic code; Anticodon code; Phenotypic graphs; Protein evolution

* Presenting author: marcojose@biomedicas.unam.mx