How number is presented? Is it represented difference for different species? We first review behavioural and neuropsychological studies for the representation of different level of mathematical ability for animals, infants, lesion cases and images studies to demonstrate the relationship of mathematics and language. It can be included that 1) both humans and animals share an elemental number quantification system, which is without supporting by language ability; 2) the language ability is crucial to get higher mathematical ability. We then summarize the main research status for each line of studies. Finally, we outline recommendations for future research directions.
Based on the thermodynamic potential function of Fermi gas in a strong magnetic field, using the thermodynamics method, the integrated analytical expressions of thermodynamic quantities of the system at low temperatures are derived, and the effects of the magnetic field on the statistic properties of the system are analysed. It is shown that, as long as the temperature is not zero, the effects of the magnetic field on the thermodynamic quantities of the system contain both oscillatory and non-oscillatory parts. For the non-oscillatory part, compared with the situation of Fermi gas in a weak magnetic field, the influence of the magnetic field on the thermodynamic quantities is not exactly the same. For the oscillatory part, the period and amplitude of the oscillation are all related to the magnetic field. Due to the oscillation, the chemical potential may be greater than Ferim energy of the system, but the oscillation does not affect the thermodynamic stability of the system.
Based on the analytical expression of relativistic free energy for a weakly interacting Fermi gas in a weak magnetic field, by using the method of quantum statistics, the stability conditions of the system at both high and low temperatures are given, and the effects of magnetic field and interparticle interactions on the stability of the system are analysed. It is shown that at high temperatures, the stability conditions of the system are completely the same, no matter whether it is the ultrarelativistic case or nonrelativistic case. At extremely low temperatures, the mechanical stability conditions of the system show a similar rule through a comparison between the ultrarelativistic case and nonrelativistic case. At the same time, thermal stability of a relativistic Bose gas in a weak magnetic field is discussed, and the influence of the effect of relativity on the thermal stability of the system is investigated.
Using the methods of numerical simulation and quantum statistics, based on the thermodynamic quantities of Fermi gas trapped in a general external potential, we studied the thermodynamic properties of a relativistic Fermi gas trapped in hard-sphere potential, gave the analytical expressions of the energy and heat capacity of the relativistic Fermi system, and analyzed the influence of the relativistic effect on the energy and heat capacity. The results showed the energy and heat capacity of the relativistic Fermi system is larger than those of the nonrelativistic case. The larger the relativistic eigenvalue, the lower the change temperature of heat capacity. As the temperature rises, the larger the relativistic eigenvalue, the greater the energy.
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