Graphs and Ideals generated by some 2-minors

Let G be a finite graph on [n] = {1,2,3,...,n}, X a 2 times n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this paper, we study about ideals I_G of S generated by 2-minors [i,j] of X which correspond to edges {i,j} of G. In particular, we construct a Groebner basis of I_G as a set of paths of G and compute a primary decomposition.Comment: 14 pages, to appear in Communications in Algebr

Equivariant Matlis and the local duality

Generalizing the known results on graded rings and modules, we formulate and prove the equivariant version of the local duality on schemes with a group action. We also prove an equivariant analogue of Matlis duality.Comment: 32pages, corrected a typ

Turbulent jet interaction with a long rise-time pressure signature

A sonic boom signature with a long rise time has the ability to reduce the sonic boom, but it does not necessarily minimize the sonic boom at the ground level because of the real atmospheric turbulence. In this study, an effect of the turbulence on a long rise-time pressure signature was experimentally investigated in a ballistic range facility. To compare the effects of the turbulence on the long and short rise-time pressure signatures, a cone-cylinder projectile that simultaneously produces these pressure signatures was designed. The pressure waves interacted with a turbulent field generated by a circular nozzle. The turbulence effects were evaluated using flow diagnostic techniques: high-speed schlieren photography, a point-diffraction interferometer, and a pressure measurement. In spite of the fact that the long and short rise-time pressure signatures simultaneously travel through the turbulent field, the turbulence effects do not give the same contribution to these overpressures. Regarding the long rise-time pressure signature, the overpressure fluctuation due to the turbulence interaction is almost uniform, and a standard deviation 1.5 times greater than that of the no-turbulence case is observed. By contrast, a short rise-time pressure signature which passed through the same turbulent field is strongly affected by the turbulence. A standard deviation increases by a factor of 14 because of the turbulence interaction. Additionally, there is a non-correlation between the overpressure fluctuations of the long and short rise-time pressure signatures. These results deduce that the length of the rise time is important to the turbulence effects such as the shock focusing/diffracting