The aim of this paper is to present a new class of learning models for linear as well as non-linear neural layers called Orthonormal StronglyConstrained (SOC or Stiefel). They allow to solve orthonormal problems where orthonormal matrices are involved. After general properties of the learning rules belonging to this new class are shown, examples derived independently or by reviewing learning th...

A new nonparametric test for two sample problem based on order statistics is suggested. This test can be successful in the case of small samples. The probabilities of the statistic are given. Comparisons with KolmogorovSmirnov, Mann Whitney-Wilcoxon, and Wilks’ empty block test for small sample sizes are made by the simulation.

We calculate from first principles the motivic cohomology of Gl(n) and Stiefel manifoldsW(n,m). We also demonstrate a comparison map ΣtP → Gl(n). CONTENTS

Using a combinatorial approach, we introduce the translated Whitney numbers. This seems to be more natural than to write a product of anarithmetical progression in terms of a power variable and conversely. We also extend our ideas to translated r-Whitney numbers of both kinds and to translated Whitney-Lah numbers.

For sums of squares formulas of the form (x1 + · · · + xr)(y 1 + · · · + y s) = z 1 + · · · + z t where the zi are bilinear functions of the xi and yi, we have two wellknown lower bounds on the size of t given r and s. One was obtained independently by Hopf and Stiefel, and another by Atyiah. These bounds are given by requiring certain binomial coefficients be divisible by certain powers of 2. ...

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce a single eigenenergy. For spinless particles, the stability triple point requires two ingredients: rotational symmetry order three, four, or six; combined with mirror space-time-inversion symmetry. However, despite ample studies their classification, robust boundary signatures ...