Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of O(m √ logm) vertices, and they conjectured that the bound of O( √ m) actually holds. We obtain separators with O( √ m logm) vertices. Let G = (V,E) be a graph with n vertices. A separator in G is a set S ⊆ V of vertices such that there is a parti...

We consider the Clay Institute Prize Problem asking for a mathematical analytical proof of existence, smoothness and uniqueness (or a converse) of solutions to the incompressible Navier-Stokes equations. We argue that the present formulation of the Prize Problem asking for a strong solution is not reasonable in the case of turbulent flow always occuring for higher Reynolds numbers, and we propo...

In his book on convex polytopes [2] A.D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in Rn+1, n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature [18]...

In this paper we focus on the stability of the Trojan asteroids for the planar Restricted Three-Body Problem (RTBP), by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on the numerical determination of the frequencies of the asteroid and the effective computation of the Kolmogorov normal form for the corresp...

In their paper [1], Tsoi and Tan present what they call a "canonical form", which they claim to be identical to that proposed in Nerrand et al [2]. They also claim that the algorithm which they present can be applied to any recurrent neural network. In the present comment, we disprove both claims. Back in 1993, Nerrand et al. [2] proposed a general approach to the training of recurrent networks...

Turing's BLOCKIN BLOCKIN present-­‐day BLOCKIN BLOCKIN and BLOCKIN BLOCKIN all-­‐time BLOCKIN BLOCKIN relevance BLOCKIN BLOCKIN arises BLOCKIN BLOCKIN from BLOCKIN BLOCKIN the BLOCKIN BLOCKIN timelessness BLOCKIN BLOCKIN of BLOCKIN BLOCKIN the issues he tackled, and the innovative light he shed upon them. Turing first defined the algorithmic limits of computability, when determined via effectiv...

In classical equilibrium analysis it is assumed that consumers may purchase a given amount of good in any fraction he (she) wishes from any of a xed number of rms. Sometimes, as with ski-lifts, at any one time it is only possible for consumers to purchase from one rm. An equilibrium price system is a complete price system. In this paper, a complete price system is a system of prices that varies...

an immediate corollary to the main result in 2] was the theorem of Furst, Saxe, and Sipser 6] and Ajtai 1] that there is a uniform sequence of polynomial size constant depth monotone Boolean circuits that is not equivalent to whatever non-uniform sequence of polynomial size constant depth positive Boolean circuits. The thing is, we can simply translate the point-sets deenable in nite models by ...

A new perceptron learning rule which works with multilayer neural networks made of multi-state units is obtained, and the corresponding convergence theorem is proved. The deenition of perceptron of maximal stability is enlarged in order to include these new multi-state perceptrons, and a proof of existence and uniqueness of such optimal solutions is outlined.