The Parity of the Number of Irreducible Factors for Some Pentanomials

It is well known that Stickelberger-Swan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on. We discuss this problem for type II pentanomials namely x^m +x^{n+2} +x^{n+1} +x^n +1 \in\ IF_2 [x]. Such pentanomials can be used for efficient implementing multiplication in finite fields of characteristic two. Based on the computation of discriminant of these pentanomials with integer coefficients, it will be characterized the parity of the number of irreducible factors over IF_2 and be established the necessary conditions for the existence of this kind of irreducible pentanomials.

Parity of the Number of Irreducible Factors for Composite Polynomials

Various results on parity of the number of irreducible factors of given polynomials over finite fields have been obtained in the recent literature. Those are mainly based on Swan’s theorem in which discriminants of polynomials over a finite field or the integral ring Z play an important role. In this paper we consider discriminants of the composition of some polynomials over finite fields. The relation between the discriminants of composed polynomial and the original ones will be established. We apply this to obtain some results concerning the parity of the number of irreducible factors for several special polynomials over finite fields.

Influence of occupation number of single particle levels on K - K charge transfer in collisions of 90 keV - Ne{^9+} on Ne

The influence of the occupation of the single particle levels on the impact parameter dependent K - K charge transfer occuring in collisions of 90 keV Ne{^9+} on Ne was studied using coupled channel calculations. The energy eigenvalues and matrixelements for the single particle levels were taken from ab initio self consistent MO-LCAO-DIRAC-FOCK-SLATER calculations with occupation numbers corresponding to the single particle amplitudes given by the coupled channel calculations.

Effect of rosemary (Rosmarinus officinalis) extract on weight, hematology and cell-mediated immune response of newborn goat kids

This study aimed at evaluating the effects of different levels of rosemary (Rosmarinus officinalis) extract on growth rate, hematology and cell-mediated immune response in Markhoz newborn goat kids. Twenty four goat kids (aged 7 +/- 3 days) were randomly allotted to four groups with six replicates. The groups included: control, T1, T2 and T3 groups which received supplemented-milk with 0, 100, 200 and 400mg aqueous rosemary extract per kg of live body weight per day for 42 days. Body weights of kids were measured weekly until the end of the experiment. On day 42, 10 ml blood samples were collected from each kid through the jugular vein. Cell-mediated immune response was assessed through the double skin thickness after intradermal injection of phyto-hematoglutinin (PHA) at day 21 and 42. No significant differences were seen in initial body weight, average daily gain (ADG) and total gain. However, significant differences in globulin (P <0.05), and white blood cells (WBC) (P <0.001) were observed. There were no significant differences in haemoglobin (Hb), packed cell volume (PCV), red blood cells (RBC), lymphocytes and neutrophils between the treatments. Skin thickness in response to intra dermal injection of PHA significantly increased in the treated groups as compared to the control group at day 42 (P< 0.01) with the T3 group showing the highest response to PHA injection. In conclusion, the results indicated that aqueous rosemary extract supplemented-milk had a positive effect on immunity and skin thickness of newborn goat kids.