Abscissa Of Point Research Articles

Generalization of Velu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, Velu’s formulae concern to a separable isogeny IG : E → E′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG(P) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstras coefficients of E′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.

Discussion of "An efficient search method for finding the critical circular slip surface using the Monte Carlo technique"

Greco 222 The authors have presented a method for locating the critical slip surface using the Monte Carlo technique. They must be congratulated for the simplicity of formulation that characterizes the work and the effective exposition of the paper, without obscure points and ambiguities. These features will certainly favor the circulation and use of the proposed method. In recent years many minimization procedures, of both deterministic and probabilistic type, have been proposed for locating critical slip surfaces in slope stability analysis. With respect to deterministic techniques, which are characterized by a higher degree of specialization and a greater complexity, the Monte Carlo methods are simpler in formulation and usage. This property is not of secondary importance for their performance.Geotechnical engineers, who are not specialists in optimization techniques, are naturally led to use methods that are more intuitive and more easily understood. The proposed optimization techniques were initially concerned with slip surfaces of general shape; however, circular slip surfaces were also examined. Circular slip surfaces are simpler to treat, because they are described by three variables only: the coordinates of the center and the radius (the abscissae of points A and B and the radius, in the method proposed by the authors). The critical circular slip surface is traditionally located using the grid method, where the centers of potential slip surfaces correspond to the nodes of a prefixed net and the radius r assumes many values, increasing from the minimum (rmin) to the maximum value (rmax) (Fig. D1). Using small increments of the radius, the grid method can also effectively analyze slopes with thin layers of weak material (such as those displayed in Figs. 11 and 12), although in this case the slip surface more suitable for the problem might be noncircular. Usually, a search with the grid method embraces a number of potential slip surfaces between 1000 and 10 000. For circular slip surfaces, slope stability analysis can be performed using Bishop’s (1955) method. This method, although simplified, gives results of a quality comparable with that of the equilibrated methods. Moreover, it is quick in the convergence of the iterative procedure for calculating the safety factor. In fact, a good code of automatic calculus is able to analyze even 1000 potential slip surfaces per second (with at least 50 slices) and of course this number will increase in the future with the improvement of personal computers and compilers. In this way, the search for a critical slip surface requires only a few seconds of computer time. The writer believes, therefore, that the search for a critical slip surface, circular in shape, does not require the use of a particular minimization procedure, because its use does not offer any significant advantage with respect to the computer time incurred using the grid method. The latter method has, moreover, another important advantage over the Monte Carlo method. There is a greater probability of finding the global minimum (or rather an acceptable approximation of it) in problems where more local minima are present. This criticism is not particularly addressed to the proposed method, which is one of the best for circular slip surfaces, but to an unjustified use of optimization methods. However, it would be interesting to know the number (on average) of potential slip surfaces examined with the proposed method in the illustrative examples that were presented. The problem of searching for critical slip surfaces that are general in shape is quite different. Here, an evaluation of potential slip surfaces by the grid method is very difficult. However, if a broken line with n vertices describes such a surface, then the problem is governed by 2n – 2 independent variables. If, still, in a simplified way, m is the number of positions that can be associated with each variable, the number of slip surfaces that should be considered is (2n – 2)m. It is evident that even for slip surfaces with 7–8 vertices and for m = 10, a search for a critical slip surface could require years of elaboration. It is thus undeniable that, for noncircular slip surfaces, the use of a minimization method is absolutely necessary. In this field, the same authors have recently presented another Monte Carlo method (Husein Malkawi et al. 2001) attractive for its effectiveness and ease

Number-theoretic properties of two-dimensional lattices

The notion of visible point in a lattice is extended to the notion of visible pair of points in a two-dimensional infinite lattice, normalized as a vertical strip of width 1, centered at the origin. The pair ( m, n) is shown to be visible iff | m( nd) − n( md)| = 1, where ( x) is the integer nearest to x, d (a value between 0 and 1 2 ) is the abscissa of point 1, while xd − ( xd) is that of point x. A general characterization of the visible and opposed pairs (with respect to the vertical axis of the lattice) will be put forward. Geometrical and computational algorithms delivering all the visible pairs for any value of d will be presented. The biological meanings of these and other results will be mentioned.