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The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions W_Q on X₀(N) for N ≤ 50 are Weierstrass points or not.

The design is by using Lawittes's and Schoeneberg's theorems.

Finding all Weierstrass points on X₀(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.

The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X₀(N).

Magnetic fluid has excellent function used as lubricants in bearings and mechanical seals, and the purpose of this study is to investigate the sealing performance in a spiral groove mechanical seal lubricated by magnetic fluid.

The sealing characteristic parameters of the lubricating film between the end faces of two sealing rings were calculated based on the Muijderman narrow groove theory for a spiral groove mechanical seal lubricated by magnetic fluid. The film thickness was determined according to the balanced forces on the rotating ring, and the effects of operating conditions, intensity of the magnetic field and diameter of nanoparticles on the sealing characteristics were investigated.

It has been found that the intensity of magnetic field has a great effect on the viscosity of magnetic fluid, film thickness and friction torque while has a little effect on the mass flux of magnetic fluid. The film thickness, mass flux of magnetic fluid and friction torque increase with the increasing volume fraction, rotating speed and diameter of magnetic nanoparticles in magnetic fluid. The mass flux of magnetic fluid decrease with the increasing closing force, and the friction torque decreases with the increase of media pressure.

The change of intensity of magnetic field can affect the viscosity of magnetic fluid and then changes the sealing performance in a mechanical seal lubricated by magnetic fluid. To reduce the mass flux of magnetic fluid and friction torque, the volume fraction, diameter of solid magnetic particles and film thickness should be 5%–7%, 8–10 nm and 2–9.3 µm, respectively.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-02-2023-0032/

This study aims to acquire the influence mechanism of gas film adaptive adjustment (GFAA) acted on the dynamic characteristics of spiral groove dry gas seal (S-DGS) and then propose a sealing stability enhancement measure.

The gas film dynamic stiffness and damping of S-DGS are obtained by numerically solving the transient Reynolds equation based on perturbation method and finite difference method. The dynamic coefficients in GFAA model and constant gas film thickness (CGFT) model are compared and analyzed.

There is the risk to misestimate the instability of DGS with rotational speed or medium pressure grows under the condition of CGFT assumption. Based on GFAA model, increasing balance ratio B properly is an effective measure to improve the stability of DGS. The balance ratio can stimulate the sensitivity of gas film dynamic coefficients to the variation of rotational speed. Increasing medium pressure in small balance ratio range will be conducive to reducing the risk of angular instability.

The influence mechanism of GFAA on S-DGS dynamic characteristics is analyzed. The interactions between rotational speed and balance ratio, medium pressure and balance ratio acted on gas film dynamic characteristics are explored based on the GFAA model.

The purpose of this paper is to investigate the hydrodynamic performance of liquid film seals with oblique grooves (OGs) and spiral grooves (SGs), considering cavitation, compare and analyze the differences between them.

Considering cavitation effect, the incompressible steady-state Reynolds equation was solved to obtain the sealing performance parameters of the liquid film seal with oblique groove and spiral groove.

The hydrodynamic performance of oblique groove seal (OGS) and spiral groove seal (SGS) shows a similar trend with the change of operating parameters. When the groove angle is less than 20°, the load-carrying capacity of SGS is better than that of OGS, while when the groove angle continues to increase, the hydrodynamic performance of OGS is slightly better than that of SGS, and more suitable for use under small differential pressure and high speed.

The hydrodynamic characteristics of liquid film seals with oblique grooves and spiral grooves considering cavitation effect were studied, which provides a theoretical reference for the application of oblique groove seal.

The purpose of this study is to investigate the heat transfer characteristics in a spiral groove mechanical seal lubricated by magnetic fluid.

The viscosity relationship of magnetic fluid in external electromagnetic field was deduced. The temperature distribution of sealing ring was calculated by the method of separation variables.

It has been found that the rotating ring absorbs most friction heat. The temperatures on the end faces of rotating ring and stationary ring decrease from inner radius to outer radius, the temperature of magnetic fluid film decreases from rotating ring to stationary ring and the highest temperature of the sealing system is at the junction of the inner radius and the end face of rotating ring.

Selecting the sealing rings with higher thermal conductivity and reducing the volume fraction of solid particles in magnetic fluid can reduce the temperature of sealing system effectively.

With the rapid development in wired and wireless networks, the demand for network security system is rising rapidly due to more and more new applications introduced. The main factors that rate the encryption algorithms are its ability to secure and protect data against attacks, its speed and efficiency. In this paper, a reconfigurable network security design using multi-mode data encryption standard (DES) algorithm has been implemented with low complexity and low cost, which will also reduce the speed. The paper aims to discuss these issues.

The design can be easily reconfigured to 3DES (triple DES) which is more secure and more powerful in encryption and decryption, as one of the trick in designing 3DES is to reuse three instances of DES. The design can be used for wired and wireless network applications, and it has been described using VHDL and implemented in a reconfigurable Programmable System-on-Chip (PSoC). The hardware implementation has targeted Xilinx Spartan XC3S700-AN FPGA device.

The main idea of reducing the complexity for the hardware implementation is by optimizing the number of logic gates and LUTs of the design. The number of logic gates can be decreased by changing the way of writing the VHDL code and by optimizing the size of the chip.

The design has been tested in simulation and hardware levels, and the simulation results and performance are discussed.

The purpose of this paper is to study the dynamic characteristics of mechanical face seals with liquid-lubricated inclined elliptical grooves.

The steady-state and perturbation Reynolds control equations of liquid films were established. The film pressure and the liquid film dynamic coefficients were obtained, impacts of groove structures on the liquid film dynamic characteristic coefficients were analyzed.

The analysis results indicate that the axial dynamic stiffness and damping coefficients of the liquid film seal with inclined elliptical grooves are far greater than those of the angular directions. Furthermore, the dynamic stiffness coefficient of the liquid film with the nonclosed inclined elliptical grooves is higher than those with the closed grooves, whereas the dynamic damping coefficient of the liquid film is lower.

The effects of inclined elliptical groove structures on the dynamic characteristics of the liquid film seal are investigated. The results presented are expected to enrich the theoretical basis of optimizing the dynamic performance of liquid film seals with textures.