Given a nite set of non-collinear points in the plane there exists a line that passes through exactly two points. Such a line is called an ordinary line. An e cient algorithm for computing such a line was proposed by Mukhopadhyay et al [MAH97]. In this note we extend this result in two directions. We rst show how to use this algorithm to compute an ordinary conic, that is, a conic passing throu...

The uniqueness has been proved by Piatetskiî-Sapiro and Safarevic. The proof of existence is outlined below. Let o be the closed point of A = Spec C[[t] ]. Let An be the finite covering of A obtained by extracting an nth. root of t; let on be the closed point of An. A family of surfaces over An is a flat, projective map ƒ: Xn —> An such that the generic fiber is smooth, connected, and two dimen...

This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point source. In one space dimension, solutions of these problems with zero initial data are known to approach the stationary solution in an asymptotically self-similar...

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in d ≥ 1 dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, [19], and used in our earlier work, [6].

where T1, . . . , Tn, S1, . . . , Sn : [a, b] → R. (2) Suppose K 6= 0. Then we may consider each of the systems {T1, . . . , Tn}, {S1, . . . , Sn} linearly independent. Indeed, starting from an expression of kind (1), we consequently reduce the number of items while it is needed. Assume that we need to express the functions (2) in terms of K. In order to do it, we find such points (a proof of e...

In this survey, we review fundamental properties of coalgebras and their representation theory. Following J.A. Green we present the block theory of coalgebras using indecomposable injectives comodules. Using the cohom and cotensor functors we state Takeuchi-Morita equivalence and use it to sketch the proof of existence of “basic” coalgebras, due to the author and S. Montgomery. This leads to a ...

Abstract. The approach, referred to as ”monodromy transform”, provides some general base for solution of all known integrable space time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of gravitationally interacting massless fields, as well as for some string theory induced gravity models. In this communication we present the key point...

the existence is a categorical concept according to kant’s philosophy. kant, going to be a subjectivist philosopher, negates the existence as a perfection and concludes the separation between the mind and extra-mental world. but the externality is not an origin which we deny it easily, even if it is faced with multiple problems such as the issue of simplicity and multiplicity and also being his...