One Day Workshop on Differential Geometry and its applications. On the memorial of late Professor Nabil Labib Youssef Feb., 7 2026 Mathematics Department Cairo University, Egypt.

Based on the generalized uncertainty principle (GUP) with the modification of the phase space, the partition function has recently been modified . In the present work, we analyze the self-consistency of the axiomatic thermodynamic derived from the modified partition function. This work studies the self-consistency for the thermodynamic quantities pressure, energy density, entropy, number density and some correlated quantities. The thermal parameters at the moment of freeze-out are extracted, as well as the occupation number of the fermionic states. We found that, the deformed phase space distribution does not obey the thermodynamic consistency conditions. On the other hand, the thermodynamic quantities and freeze-out parameters are well reproduced.

"GUP-corrected phase space and thermodynamics self consistency" Abstract Based on generalized uncertainty principle with the modication of the phase space, the partition function is modied recently. In the present work we analyze the self-consistency of the generic axiomatic thermodynamic derived from that modied partition function. This work studies the self consistency for all the thermodynamical quantities pressure, energy density, entropy and number density and some correlated quantities such as the speed of sound c^2s and the trace anomaly. First an deviation appeared between the hadron resonance gas model and the modied one due to GUP. But by studying the consistency, we found that the correlations are lack for axiomatic self-consistency.

"FREEZEOUT PARAMETERS AND DYNAMICAL NET-CHARGE FLUCTUATIONS AT NICA ENERGIES" The dependencies of different particle ratios on the nucleus-nucleus center-of-mass energy, which can be related to the chemical potential, are calculated by using hadron resonance gas (HRG) and Ultra-relativistic Quantum Molecular Dynamic (UrQMD) models. For UrQMD two different types of phase transitions are taken into consideration, namely crossover and first order, while the HRG model implements fully statistical aspects in describing the particle production and their correlations, especially in the hadron phase. The freezeout parameters, temperature (T) and baryon chemical potential (μ), are deduced by fitting various particle ratios estimated from UrQMD with HRG calculations at 7.7, 11.5, 19.6, 62.4 GeV. The results agree with the parameters which are independently determined from statistical fit of the measured particle ratios and the thermal-statistical approaches. Furthermore, the net-charge fluctuations for K/π, p/π and K/p determined UrQMD, and HRG are compared with the available STAR and NA49 measurements. The future NICA facility covers a wide range of low beam energies. The excellent agreement justifies the conclusion that both UrQMD and HRG are suitable approaches to explain both freezeout parameters and the dynamical net-charge fluctuations