Supervised learning of classifiers often resorts to the minimization of a quadratic error, even if this criterion is more especially matched to nonlinear regression problems. It is shown that the mapping built by a quadratic error minimization (QEM) tends to output the Bayesian discriminating rules even with nonuniform losses, provided the desired responses are chosen accordingly. This property...

The transformation of a data set using a second-order polynomial mapping to find statistically independent components is considered (quadratic independent component analysis or ICA). Based on overdetermined linear ICA, an algorithm together with separability conditions are given via linearization reduction. The linearization is achieved using a higher dimensional embedding defined by the linear...

KP = { z ∈ C | the sequence P ◦k(z) is bounded }. We will write Pc(z) = z2 + c, and Kc, etc., when discussing quadratic polynomials speciˇcally. If P is monic of degree d with KP connected, there is then a unique conformal mapping φP : C − KP → C − $ D which satisˇes φP (P (z)) = (φP (z))d and tangent to the identity at ∞. We call RP (θ) = φ−1 P ({re2π iθ , r > 1}) the external ray of KP at ang...

In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x+ 1), f(x+ 2), ..., f(x+ κ)) are quadratic non-residue is ≈ 1 2 . In particular for f(x) = x + ax+ b corresponding to the elliptic curve y = x + ax+ b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly dis...

The heights of simple closed loops with respect to a holomorphic quadratic diierential play an important role on compact Riemann surfaces. Here, the analogue is developed for quadratic diierentials of nite norm in the disk. The height of a loop is replaced by the height of a cross cut, which is the same as the vertical distance, with respect to the q.d., of its end points.

A vehicle hitting a traffic barrier creates a major threat to traffic safety as a result of displacement of components of the barrier by the forces transferred from the vehicle. We built a theoretical model of the dynamics of such processes and behaviour of concrete barriers set on various types of surfaces. We calculated the displacement as a function of vehicle impact energy and barrier base....

An approximate error function for the discretization error on a given mesh is obtained by projecting (via the energy inner product) the functional residual onto the space of continuous, piecewise quadratic functions which vanish on the vertices of the mesh. Conditions are given under which one can expect this hierarchical basis error estimator to give efficient and reliable function recovery, a...

Teichmüller was the first person to point out explicitly the connection between quadratic differentials and the solutions of certain extremal problems in Function Theory. He enunciated the principle that if a point is required to be fixed the quadratic differential will have a simple pole there, if in addition fixed values are required for the first n derivatives of competing functions the quad...

The paper deals with solving of contact problems with Coulomb friction for a system of 3D elastic bodies. The iterative method of successive approximations is used in order to find a fixed point of certain mapping that defines the solution. In each iterative step, an auxiliary problem with given friction is solved that is discretized by the FETI method. Then the duality theory of convex optimiz...

The Hyers-Ulam stability, the Hyers-Ulam-Rassias stability, and also the stability in the spirit of Gǎvru̧ta for each of the following quadratic functional equations f(x+y)+ f(x−y) = 2f(x)+ 2f(y), f(x+y + z)+ f(x−y)+ f(y − z)+ f(z−x) = 3f(x)+3f(y)+3f(z), f (x+y+z)+f(x)+f(y)+f(z)= f(x+y)+f(y+z)+f(z+x) are investigated. 2000 Mathematics Subject Classification. Primary 39B52, 39B72, 39B82.