We find the polynomials of the best one-sided approximation to the Heaviside and sign functions. The polynomials are obtained by Hermite interpolation at the zeros of some Jacobi polynomials. Also we give an estimate of the error of approximation and characterize the extremal points of the convex set of the best approximants. c ⃝ 2012 Elsevier Inc. All rights reserved.

In this paper the superheating of electron plasma by femtosecond laser pulses is investigated. With Heaviside thermal equation (Lasers in Engineering, 12, (2002), p. 17) the generation of superhot electrons is described. It is shown that in hot electron plasma (i.e. with electron energies > 5 MeV) the thermal shock waves can be generated.

<b>Background:</b> Pulsed field ablation (PFA) is a new type of physical energy source in the fields tumor and atrial fibrillation ablation, based on irreversible electroporation with non-thermal, clear boundaries, selective killing, rapid advantages. PFA induces changes electrical conductivity zone, which can be described by step function used to predict zone. However, current rese...

The ring of locally-constant integer-valued functions on the dominant chamber of the Shi arrangement is endowed with a filtration and a new basis, compatible with this filtration, is found. This basis is compared to the trivial basis. The ring is given a presentation by generators and relations.

Included among these is the sum of the squares of the characteristic numbers of P1, i.e., the sum of ihe characteristic numbers of N1 = AA*; this is the well-known unitary invariant Eapdpq of Frobenius. p,q When A is normal AA* = A*A or PU.UI*P1 = U*"P.P,U so that I= U*PU = (U*PU)2. Hence P1 = U*P,U or UP1 = POU. Conversely if UP1 = P1U we have A*A = A*A so that a matrix A is normal. when and o...