EXPLICIT FACTORIZATION OF x^{2^{a}p^{b}r^{c}}-1 OVER A FINITE FIELD

Abstract

Let F q be a finite field of odd order q. In this paper, the irreducible factorization of x 2 a p b r c − 1 over F q is given in a very explicit form, where a, b, c are positive integers and p, r are odd prime divisors of q − 1. It is shown that all the irreducible factors of x 2 a p b r c − 1 over F q are either binomials or trinomials. In general, denote by v p (m) the degree of prime p in the standard decomposition of the positive integer m. Suppose that every prime factor of m divides q − 1, one has (1) if v p (m) ≤ v p (q − 1) holds true for every prime number p|q − 1, then every irreducible factor of x m − 1 in F q is a binomial; (2) if q ≡ 3(mod 4), then every irreducible factor of x m − 1 is either a binomial or a trinomial.

Synthesis, characterization and antioxidant evaluation of metal complexes derived from a dianil ligand with a flexible linkage: anomalous magnetic behavior of the nickel complex

“…The zinc complex exhibited the highest reducing activity (p \ 0.005), while cobalt metal showed the lowest reducing ability. The reducing powers of the present complexes are in good agreement with recent reports on related Schiff bases [22][23][24]. The reducing power of the ligand and its complexes both increase with concentration.…”

“…The reducing power of the ligand and its complexes both increase with concentration. The reducing ability of a compound generally depends on the development of reductones, which display antioxidative potential by terminating the free radical chain reaction [22][23][24]. These results are in agreement with the NO radical scavenging experiments, in revealing that the zinc complex exhibits the highest scavenging activity among the complexes.…”

“…The reducing power of the investigated compounds was measured using the K 4 Fe(CN) 6 /FeCl 3 method as described Transition Met Chem by Oyaizu [22][23][24]. Various concentrations, namely 0.05, 1.0, 0.25, 0.5 and 1.0 mg mL -1 of H 2 L and its metal(II) complexes were prepared.…”

“…An antioxidant may act as a common reductant but a reductant may not necessarily act as an antioxidant [22][23][24]. Thus, in the FRAP (Ferric reducing antioxidant power) method, the presence of reducing agent mediates the reduction in the Fe 3?…”

Biomarker detection for disease diagnosis using cost-effective microfluidic platforms

“…[16][17][18][19][20][21][22] An enormous number of microfluidic devices have been developed for biomedical applications. [23][24][25][26][27][28][29] These devices enable on-chip POC diagnosis and real-time monitoring of diseases from a small volume of body fluids. These microfluidic devices may act as a bridge to improve the global health care system with high efficiency and sensitivity, especially for remote areas with low-resource settings, such as the underdeveloped and developing countries, in home health care setting, and in emergency situations.…”