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In order to provide fault-tolerance and guarantee reliability, data redundancy should be introduced in distributed storage systems. The emerging coding techniques for storage such as fractional repetition codes, provide the required redundancy more efficiently than the conventional replication scheme. In this letter, we extend the construction of fractional repetition codes and present a new coding scheme, termed general fractional repetition codes. The proposed codes can be applied to storage systems in which the storage capacities of nodes may be different. Based on a combinatorial structure known as group divisible design, the new code construction is available for a large set of parameters. The performance of the proposed codes is evaluated by a metric called node repair alternativity, which measures the number of different subsets of nodes that enable the repair of a specific failed node.

We report here a polarization-independent drop filter (PIDF) based on a photonic crystal self-collimation ring resonator (SCRR). Despite of the large birefringence associated with the polarization-dependent dispersion properties, we demonstrate a PIDF based on multiple-beam interference theory and polarization peak matching (PPM) technique. The PIDF performance was also investigated based on finite-difference time-domain (FDTD) technique, with excellent agreement between the theory and the simulation. For the designed drop wavelength of 1550 nm, the polarization-independent free spectral range is about 36.1 nm, which covers the whole optical communication C-band window. The proposed PIDFs are highly desirable for applications in photonic integrated circuits (PICs).

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Let (G, Λ) be a self-similar k-graph with a possibly infinite vertex set Λ 0 . We associate a universal C*-algebra OG,Λ to (G, Λ). The main purpose of this paper is to investigate the ideal structures of OG,Λ. We prove that there exists a one-to-one correspondence between the set of all G-hereditary and G-saturated subsets of Λ 0 and the set of all gauge-invariant and diagonal-invariant ideals of OG,Λ. Under some conditions, we characterize all primitive ideas of OG,Λ. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar P -graph C*-algebras in depth.2010 Mathematics Subject Classification. 46L05.

The mining and application of data in sports training through intelligent computer-aided research, this paper proposes a system development and design method to improve the development and design efficiency of the data visualization system and reduce the development cost by using the Python language to handle the high efficiency of system programming and the convenience of charts. Thus, it further improves the efficiency and interactivity of the information management of training data and the convenience and stability of managing training data. This system is a preliminary exploration of Python language and Echarte technology in the management of training data of sports training programs. Because of the integration of training programs with computer application technology and data visualization technology, the system still needs to be strengthened in terms of data processing methods and the diversity of chart styles generated. In the future, the use of artificial intelligence, big data analysis, and other technologies in sports training data management and analysis has more room for development. The model only needs to measure 14 indicators of bipedal closed-eye standing and single-leg closed-eye standing movements to achieve the ability to predict the balance of motor training, and its accuracy meets the clinical requirements. This method has a shorter measurement time and fewer metrics than traditional methods.

Novel opal hydrogels with water-tunable photonic bandgap (PBG) exhibiting responses to external stimuli were self-assembled from polystyrene-co-poly(N,N-dimethylacrylamide) (PS-co-PDMAA) microspheres. The polymeric microspheres with narrow size distribution were successfully prepared in water, consisting of two regions. The inner region is rich in PS which is hard and hydrophobic; the outer region is rich in PDMAA which is soft and hydrophilic. The self-assembly of the PS-co-PDMAA hydrogel microspheres is ready induced by centrifugation and resulted in highly ordered three-dimensional (3D) photonic colloidal crystals (PCCs). With an increase of the amount of water, the PBG of the opal hydrogels shifted from the visible to near-infrared region of the electromagnetic spectrum. The maximum shift of diffraction peak positions could be larger than 500 nm with narrow full width at half maximum (FWHM) in the range of 20 to 40 nm only. The change in color was visible to the naked eye. The remarkable sensitivity to water of the lattice spacing of the opal hydrogels was repeatable after centrifugation. These observations are attributed to a reproducible degree of hydration of the hydrophilic outer region of the polymeric microspheres. Furthermore, the diffraction of the opal hydrogels was particularly sensitive to the presence of thiocyanate (SCN(-)) ions. The interaction between SCN(-) ions and DMAA repeat units is argued to block hydrogen bonds between DMAA and water molecules. Our PS-co-PDMAA opal hydrogels could be a practical system for diffraction-based detections.