Über die Zetafunktionen gewisser algebraischer Zahlkörper

Ohne Zusammenfassung     

A transformation based method to design ultrasound array

In medical ultrasound imaging, the desired lateral field distribution at each focal distance can be obtained by optimal apodization. On the other hand, the lateral field is a function of focal distance. Hence, finding the optimal apodization is a very arduous process. To overcome this, we have introduced a suboptimal method by which optimal apodization can be calculated in any distance through a nonlinear transformation by the knowledge of the optimal one at a distance. This transformation is established on a fact that the lateral field distribution at focal distance can be expressed as the Fourier transform of a nonlinear function of the aperture weighting, instead of direct expression as the Fourier transform of the above. We have applied this method to map the apodization which obtains the desired beam pattern into the apodization which maintains the same properties on the lateral field distribution. For example, applying this method on a 50-elements λ/2 spaced linear array with length D has resulted in apodization that is optimal at distances D or D/2 by precision better than 9%. This method is useful especially in optimization problems, having no explicit constraint on the main lobe width, such as minimizing the sidelobe levels or minimizing main lobe width constrained to a predetermined value of sidelobe level. However, as the results show, this technique provides acceptable results in other cases.     

On the Jacobians of plane cubics

Let S be a scheme and f a ternary cubic form whose ten coefficients are sections of O_S without common zero. The equation f=0 defines a family of plane cubic curves parametrized by S. We prove that the family of generalized Jacobians of those cubic curves is a group scheme J/S which is the locus of smoothness of a scheme f*=0, where f* is a Weierstrass cubic formf*=f*(x,y,z)=y²z+a₁xyz+a₂yz²-x³-a₂x²z-a₄xz²-a₆z³, in which the coefficient a_i is a homogeneous polynomial with integral coefficients, of degree i in the ten coefficients of f, which we give explicitly. A key ingredint of the proof is a characterization, over sufficiently nice bases, of group algebraic spaces which can be described by such a Weierstrass equation.     

Quadratische Körper im Gebiete der höheren Kongruenzen. II.

Mathematische Zeitschrift -