A survey on t-core partitions

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

Splitting the e-Abacus Diagram in the Partition Theory

In the partition theory, there is more then one form of representation of dedication, most notably the Abacus diagram, which gives an accurate and specific description. In the year 2019, Mahmood and Mahmood presented the idea of merging more than two plans, and then the following question was raised: Is the process of separating any somewhat large diagram into smaller schemes possible? The general formula to split e-abacus diagram into two or more equal or unequal parts was achieved in this study now.

Numerical Reconstruction Model and Simulation Study of Concrete Based on Damaged Partition Theory and CT Number

The applicability of mesoscopic models plays an important role in studying the mesoscopic mechanical properties of concrete. In this study, the computerized tomography (CT) test of concrete under uniaxial compression conditions is conducted using a portable dynamic loading equipment developed by Xi’an University of Technology and a medical Marconi M8000 spiral CT scanner. On the basis of damage partition theory, a probabilistic statistical method for determining threshold values is proposed, and a CT test images is obtained and divided into aggregate, hardened cement and hole-crack areas. A ‘structural random numerical concrete model’ is also established on the basis of the coordinates of each pixel unit in CT images. Uniaxial static compression and tensile numerical simulation tests are conducted. Results show that the structural random numerical concrete model can not only reflect the microscopic composition of concrete but also the interfacial transition zone (ITZ) between aggregate and mortar. The ITZ thickness is approximately 0.04 mm, which is close to the real concrete sample ITZ thickness (approximately 10–50 μm). In the two tests, the specimen damage starts from the initial defects, and the damage crack expands through the weaker ITZ around the aggregate. No matter under the action of static tension or compression load, the damage cracks of the sample almost never pass through the aggregate. Most of the many cracks in uniaxial compression are shear cracks. However, many cracks form at the beginning of uniaxial tension, and only one main crack, which is roughly perpendicular to the loading direction, exists in the end.