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

Stochastic differential equations have been intensively used to analyze data from physics, finance, engineering, medicine, biology, and forestry. This study proposes a general stochastic dynamical model of a forest stand development which includes random forces governing the dynamic of multivariate distribution of tree size variables. The dynamic of the multivariate probability density function of tree size components (diameter, height, crown base height, crown width and so on) in a stand is described by a mixed effect parameters Gompertz-type multivariate stochastic differential equation (SDE). The advantages of SDE method are that it do not need to choose many different equations to be tried, it relates the tree size components dynamic against the age dimension (time), and consider the underlying covariance structure driving changes in the tree (stand) size variables. SDE model allows us a better understanding of biological processes driving the dynamics of natural phenomena. The new derived multivariate probability density function and its marginal univariate, bivariate and trivariate distributions, and conditional univariate, bivariate and trivariate distributions can be applied for the modeling of stand attributes such as the mean diameter, height, crown base height, crown width, volume, basal area, slenderness ratio, their increments and much more. This study introduces general multivariate mutual information measures based on the differential entropy to capture multivariate interactions between state variables. The purpose of the present study is therefore to experimentally confirm the effectiveness of using multivariate mutual information measures to reconstruct multivariate interactions in state variables.

In general, a mathematical model that contains many linear/nonlinear differential equations, describing a phenomenon, does not have an explicit hierarchy of system variables. That is, the identification of the fast variables and the slow variables of the system is not explicitly clear. The decomposition of a system into fast and slow subsystems is usually based on intuitive ideas and knowledge of the mathematical model being investigated. In this study, we apply the singular perturbed vector field (SPVF) method to the COVID-19 mathematical model to expose the hierarchy of the model. This decomposition enables us to rewrite the model in new coordinates in the form of fast and slow subsystems and, hence, to investigate only the fast subsystem with different asymptotic methods. In addition, this decomposition enables us to investigate the stability analysis of the model, which is important in the case of COVID-19. We found the stable equilibrium points of the mathematical model and compared the results of the model with those reported by the Chinese authorities and found a fit of approximately 96%.

From the assumption that the slow roll parameter \epsilon has a Lorentzian form as a function of the
e-folds number N, a successful model of a quintessential inflation is obtained. The form corresponds
to the vacuum energy both in the inflationary and in the dark energy epochs. The form satisfies
the condition to climb from small values of \epsilon to 1 at the end of the inflationary epoch. At the
late universe \epsilon becomes small again and this leads to the Dark Energy epoch. The observables
that the models predicts fits with the latest Planck data: r ∼ 10−3 , ns ≈ 0.965. Naturally a large dimensionless factor that exponentially amplifies the inflationary scale and exponentially suppresses the dark energy scale appears, producing a sort of cosmological see saw mechanism. We find the
corresponding scalar Quintessential Inflationary potential with two flat regions - one inflationary
and one as a dark energy with slow roll behavior. Moreover, a reheating mechanism is suggested with numerical estimation for the homogeneous evolution of the universe. The suggested mechanism is consistent with the BBN bound.

Many molecules are naturally symmetric at their equilibrium state, but due to conformational flexibility, substitution, changing physical conditions, or during chemical processes, it is more likely to find them with different levels of distortion. Continuous symmetry and chirality measures quantify the level of distortion by estimating the distance between the input structure and the nearest symmetric (or achiral) structure with the same atoms and connectivity. Recently we developed algorithms for calculating these measures that utilize the connectivity map of the input molecule to reduce the number of possible permutations by scanning only structure-preserving permutations. This strategy resulted with significantly increased accuracy and dramatic reduction of the running time, by up to tens orders of magnitude. Consequently, a wide variety of molecules can now be analyzed, turning the measures into powerful and robust molecular descriptors. The methodology will be discussed along with various applications of measuring symmetry and chirality to explore structural effects of organic and inorganic systems as well as protein structure.

The cosmological constant is now a fundamental ingredient of the standard ΛCDM model and its value is constrained by concordance with empirical data. Despite its importance in modern cosmology, we still do not understand its origin. A naive calculation of the contribution of the fluctuations of the quantum vacuum to vacuum energy (considering it to be the source of the cosmological constant) yields predictions 120 orders of magnitude larger than observations [1]. This poses one of the most celebrated unsolved problems in physics and cosmology, in particular. In this work, we discuss a model of quantum thermal fluctuations of the cosmic microwave background with a Boltzmann factor. Fluctuations of a bosonic field are studied and we show that they could match the vacuum energy density if they correspond to an axionic field with a particle rest mass in the range of a few meV. This mass range is in agreement with present bounds on the mass of the Peccei-Quinn’s axions arising from the spontaneous symmetry breaking that explainsCP conservation in weak interactions [2]. The relevance of this model to the Hubble tension debate [3], i. e., the statistically significant discrepancy among measurements of the Hubble parameter based upon the cosmic microwave background and those using the cosmic distance ladder, is also discussed. Our model also predicts a decreasing influence of the cosmological parameter or quintessence in the future of the Universe allowing for the possibility of recollapse. These scenarios are also studied and the options to distinguish them from ΛCDM are discussed.

(1) Weinberg, S.Reviews of Modern Physics1989,61, 1–23.

(2) Berenji, B.; Gaskins, J.; Meyer, M.Phys. Rev. D2016,93045019, 045019.

(3) Efstathiou, G.arXiv e-prints2020, arXiv:2007.10716.