The notion of an extremal self-dual code has been introduced in 1 . As Gleason 2 remarks onemay use invariance properties of the weight enumerator of a self-dual code to deduce upper bounds on the minimum distance. Extremal codes are self-dual codes that achieve these bounds. The most wanted extremal code is a binary self-dual doubly even code of length 72 and minimum distance 16. One frequentl...

Even-cycle matroids are elementary lifts of graphic and even-cut cographic matroids. We present a polynomial algorithm to check if binary matroid is an even-cycle we matroid. These two algorithms rely on (to be described in pair follow-up papers) pinch-graphic.

The variational principle is used to build a model which describes open shell nuclei with ground state deformations. Hartree-Fock equations are solved by using single particle wave functions whose radial parts depend on the projection of angular momentum quantisation axis. Pairing effects taken into account solving Bardeen-Cooper-Schrieffer in each step minimization procedure. Gogny D1S finite-...

Fukuda and Handa [7] asked whether every even partial cube G is harmonic-even. It is shown that the answer is positive if the isometric dimension of G equals its diameter which is in turn true for partial cubes with isometric dimension at most 6. Under an additional technical condition it is proved that an even partial cube G is harmonic-even or has two adjacent vertices whose diametrical verti...

For a finite group $G$‎, ‎let $Cent(G)$ denote the set of centralizers of single elements of $G$‎. ‎In this note we prove that if $|G|leq frac{3}{2}|Cent(G)|$ and $G$ is 2-nilpotent‎, ‎then $Gcong S_3‎, ‎D_{10}$ or $S_3times S_3$‎. ‎This result gives a partial and positive answer to a conjecture raised by A‎. ‎R‎. ‎Ashrafi [On finite groups with a given number of centralizers‎, ‎Algebra‎ ‎Collo...