The present study involves a novel computational technique, regarding simultaneous use of the pseudo particle method, Poisson integral method and a special-purpose computer originally designed for molecular dynamics simulations (MDGRAPE-3). In the present calculations, the dynamics of two colliding Vortex rings have been studied using the Vortex method. The present acceleration technique allows the calculation of 107 Vortex elements. The reconnection of the Vortex rings was clearly observed, and the discretization error was nearly negligible.

Let R be a ring, and let n, d be non-negative integers. A right R-module M is called (n, d)-projective if Extd+1R (M, A)=0 for every n-copresented right R-module A. R is called right n-cocoherent if every n-copresented right R-module is (n+1)-copresented, it is called a right co-(n, d)-ring if every right R-module is (n, d)-projective. R is called right n-cosemihereditary if every submodule of a projective right R-module is (n, 0)-projective, it is called a right n-V-ring if it is a right co-(n, 0)-ring. Some properties of (n, d)-projective modules and (n, d)-projective dimensions of modules over n-cocoherent rings are studied. Certain characterizations of n-copresented modules, (n, 0)-projective modules, right n-cocoherent rings, right n-cosemihereditary rings, as well as right n-V-rings are given respectively.

We observe some new characterizations of n-presented modules. Using the concepts of (n, 0)-injectivity and (n, 0)-flatness of modules, we also present some characterizations of right n-coherent rings, right n-hereditary rings, and right n-regular rings.

Let R, S be rings with identity and M be a unitary (R, S)-bimodule. We characterize homomorphisms and derivations of the generalized matrix ring T =(or sm), and provide a triangular representation of the differential polynomial ring T[b; d].

One of the first constructions of algebra is the quotient field of a commutative integral domain, constructed as a set of fractions, which can lead to a very useful technique in commutative ring theory. In this article the researchers considered rings of fractions for gamma rings and some new characterizations were developed in gamma rings of fractions.

Vortex rings can maintain their structure during motion and achieve long-distance transport with low energy consumption, which is a fluid transport method with great energy-saving potential. In this paper, a reciprocating Vortex ring generator structure is designed, which can generate two Vortex rings during the reciprocating motion of one piston, making full use of the thrust in the reciprocating motion period of the piston and improving the Vortex ring generation frequency compared with traditional Vortex ring generators. For the characteristics of long-distance transport of Vortex rings, an experimental platform is designed and built, and 277 sets of experiments are carried out with different geometric parameters. The results show that the effect of generating two Vortex rings could be achieved under other parameter conditions, except for some parameter conditions where the diameter ratio D1/D2 = 4. By analyzing the influence of baffle width ratio, length ratio, and diameter ratio on the moving distance of Vortex rings, the performance of the Vortex ring generator is preliminarily studied. In 277 sets of experiments, the maximum moving distance ratio x1 of Vortex ring 1 is 13. 7 when L1/L2 = 2. 4, D1/D2 = 2, and w1 = 0. 2. And the maximum moving distance ratio x2 of Vortex ring 2 is 20 when L1/L2 = 2, D1/D2 = 2. 5, and w2 = 0. 2.

To discover the nonlinear characteristics of pipe flow, we simulated the flow as a sum of many Vortex rings. As a first step, we investigated the nonlinear interaction among a maximum of three Vortex rings. The pipe wall was replaced by many bound vortices. A free Vortex ring moves right or left according to the radius, and that of a particular radius keeps the initial position. The energy of a free Vortex ring, except when it is close to a wall, coincides with that without boundaries. Two Vortex rings of equal radii always show a repeated overtaking process. In the case of three Vortex rings, they show a wide variety of behavior. For certain combinations of radii and the axial spaces among them, the motion, which seems to be very complex, is limited on a curved surface in three-dimensional space whose axes correspond to the three radii. It was found that this simplicity comes from the momentum conservation law, and also that its direction depends on the energy contribution calculated from free Vortex rings. Our results elucidate the nonlinear behavior among many vortices.