Constructing projective resolution and taking cohomology for gentle algebras in the geometric model

The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present paper firstly unifies these two realizations of objects in module category and derived category via same surface for any gentle algebra, by the rotation of permissible curves corresponding to the objects in the module category. Secondly, the geometric characterization of the cohomology of complexes over gentle algebras is established by the truncation of projective permissible curves. It is worth mentioning that the rotation of permissible curves and the truncation of projective permissible curves are mutually inverse processes to some extent. As applications, an alternative proof of ``no gaps" theorem as to cohomological length for the bounded derived categories of gentle algebras is provided in terms of the geometric characterization of the cohomology of complexes. Moreover, we obtain a geometric proof for the strong Nakayama conjecture for gentle algebras. Finally, we contain two examples to illustrate our results.Comment: 29 pages, comments are welcom

Transformer Volume Reduction: A New Analysis and Design of an SSSA Control Based 20kW High Power Density Wide Range Resonant Converter

Isolated DC/DC converters play a pivotal rolein the realm of power electronics, particularly in the contextof electric vehicle (EV) fast charging. These converters areresponsible for delivering high-voltage direct current toEVs, sourced from a 3-phase power factor correction (PFC)converter, and exhibit compatibility with both low-voltageand high-voltage vehicle batteries. However, in manyinstances, the demand for constant power charging invarious applications results in a significant portion of thetransformer volume, thereby leading to a decrease inconverter power density. This paper presents a newanalysis and design for a converter based on secondaryside semi-active (SSSA) control. This analysis providestheoretical support for transformer volume reduction andpower density increase. It employs SSSA control totransfer stored energy from the transformer to the resonantnetwork during boost operation, even when fs > fr, with theexcitation inductance participating in resonance. Based onthis analysis, the design of a 20kW 650-850V input to300-900V with 66.7A max output prototype is discussed.The objective is to achieve the highest feasible converterpower density. The designed results confirm that the2*PQ6535 (or 1*PQ6549) core can effectively serve the20kW transformer, resulting in an ultra-high power densityof 14.36 kW/L (235 W/in3)

Accelerated computation of the genetic algorithm for energy-efficient virtual machine placement in data centers

Energy efficiency is a critical issue in the management and operation of cloud data centers, which form the backbone of cloud computing. Virtual machine (VM) placement has a significant impact on energy-efficiency improvement for virtualized data centers. Among various methods to solve the VM-placement problem, the genetic algorithm (GA) has been well accepted for the quality of its solution. However, GA is also computationally demanding, particularly in the computation of its fitness function. This limits its application in large-scale systems or specific scenarios where a fast VM-placement solution of good quality is required. Our analysis in this paper reveals that the execution time of the standard GA is mostly consumed in the computation of its fitness function. Therefore, this paper designs a data structure extended from a previous study to reduce the complexity of the fitness computation from quadratic to linear one with respect to the input size of the VM-placement problem. Incorporating with this data structure, an alternative fitness function is proposed to reduce the number of instructions significantly, further improving the execution-time performance of GA. Experimental studies show that our approach achieves 11 times acceleration of GA computation for energy-efficient VM placement in large-scale data centers with about 1500 physical machines in size

Sub-GeV millicharge dark matter from the U(1)XU(1)_X hidden sector

We conduct a comprehensive study on the sub-GeV millicharge dark matter produced through the freeze-in mechanism. We discuss in general the mixing mechanism, encompassing both kinetic mixing and mass mixing, between the $U(1)_X$ hidden sector and the standard model, which can generate millicharge carried by the dark fermions from the hidden sector. We discuss in depth how such millicharge is generated, and clarify several misunderstandings regarding this subject in the literature. Without employing an effective field theory approach, where the photon field directly mixed with the additional $U(1)$, we analyze a general renormalizable model and investigate the complete evolution of the hidden sector particles. Due to the substantial self-interactions among hidden sector particles, the evolution of the hidden sector temperature plays a crucial role, which is addressed concurrently with the number densities of hidden sector particles by solving a set of coupled Boltzmann equations. We thoroughly examine eight benchmark models from six distinct cases. Some of our key findings from the analysis of these benchmark models may be generalizable and applicable to broader freeze-in scenarios. We also explore the possibility that the $\mathcal{O}$(keV) $U(1)_X$ dark photon is a viable dark matter candidate, even though it can contribute at most $\sim 5\%$ to the total observed dark matter relic density.Comment: 55 pages, 13 figure

(E)-3,5-Dimeth­oxy­benzaldehyde oxime

In the title compound, C9H11NO3, the oxime grouping is twisted by 12.68 (6)° with respect to the dimethoxyl­benzene ring. In the crystal, mol­ecules are linked into an infinite [100] chain via O—H⋯N hydrogen bonds, instead of the more common oxime packing motif of dimers with an R 2 2(6) graph-set motif