A new formula of the determinant tensor with symmetries

In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base field is not of characteristic $2$. Considering some symmetries in that formula, we found a new formula so that \begin{equation*} \operatorname{Crank}(det_n) \leq \operatorname{rank}(det_n) \leq \frac{n!}{2^{\lfloor(n-2)/2 \rfloor}} \end{equation*} when the base field is not of characteristic $2$.Comment: 13 page

A new formula for the determinant of 4×44 \times 4 matrices

In this paper, we present a new formula for the determinant of a $4 \times 4$ matrix. We approach via the sparse optimization problem and derive the formula through the Least Absolute Shrinkage and Selection Operator (LASSO). Our formula has the potential to advance understanding of the algebraic structure of determinants, such as an upper bound of the tensor rank, various notions to measure complexity, and effective computational tools in exterior algebras. We also address several numerical experiments which compare our formula with built-in functions in a computer-algebra system.Comment: 11 pages, 1 figur

Flexible/functional porous ceramic membranes for high-performance battery separators

Department of Energy Engineering(Battery Science and Technology)There is no doubt that rechargeable lithium-ion batteries occupy predominant position as a competent power source in a wide variety of industry fields including portable electronics, electric vehicles (EVs) and grid-scale energy storage systems (ESSs). For example, spinel lithium manganese oxide (LiMn2O4, LMO) materials, which are widely used in large-scale batteries for applications in EVs and ESSs, are struggling with dissolution of Mn2+ ions at elevated temperatures. The Mn2+ dissolutiontriggered disruption/contamination of electrodes are known to provoke serious capacity fading during charge/discharge cycling. As a ceramic-driven material/architecture strategy to develop an ultimate battery separator far beyond traditional polymeric separators, we have demonstrated flexible/functional porous ceramic membranes (referred to as “2F ceramic separators”) with chemical traps for chelating heavy metal ions (here, Mn2+). The 2F ceramic separator, which was comprised of the densely-packed F-silica particles spatially besieged by the PVP/PAN nanofiber skeleton, was fabricated using the simultaneous electrospraying/electrospinning process. The superlattice crystals-mimic structural uniqueness of the close-packed ceramic particles, in combination with the well-designed electrospun nanofiber skeleton, provided remarkable advances in the thermal/dimensional tolerance, mechanical flexibility and other separator properties. Furthermore, both the F-silica particles and PVP/PAN nanofibers possessed the Mn2+-chelating ability, which served as chemical traps for Mn2+ ions during their passage through the liquid electrolyte-filled interstitial voids of the 2F ceramic separator. As a consequence, the 2F ceramic separator enabled unprecedented improvements in the high-performance lithium-ion batteries. The 2F ceramic separator will hold a great deal of promise as a chemically-active separator for high-performance batteries that are eager to adopt high-energy (but, struggling with metal dissolution) electrode materials and also open a new ceramic opportunity for next-generation multifunctional membranes that are in strong pursuit of selectivity removing heavy metal ions.ope

Characteristics and treatments of large cystic brain metastasis: radiosurgery and stereotactic aspiration.

Brain metastasis represents one of the most common causes of intracranial tumors in adults, and the incidence of brain metastasis continues to rise due to the increasing survival of cancer patients. Yet, the development of cystic brain metastasis remains a relatively rare occurrence. In this review, we describe the characteristics of cystic brain metastasis and evaluate the combined use of stereotactic aspiration and radiosurgery in treating large cystic brain metastasis. The results of several studies show that stereotactic radiosurgery produces comparable local tumor control and survival rates as other surgery protocols. When the size of the tumor interferes with radiosurgery, stereotactic aspiration of the metastasis should be considered to reduce the target volume as well as decreasing the chance of radiation induced necrosis and providing symptomatic relief from mass effect. The combined use of stereotactic aspiration and radiosurgery has strong implications in improving patient outcomes