Narrow Results

This experiment has examined the corrosion and tribological properties of basalt fiber reinforced composite materials. There were slight changes of weight after the occurring of corrosion based on time and H₂SO₄ concentration, but in general, the weight increased. It is assumed that this happens due to the basalt fiber precipitate. Prior to the corrosion, friction-wear behavior showed irregular patterns compared to metallic materials, and when it was compared with the behavior after the corrosion, the coefficient of friction was 2 to 3 times greater. The coefficient of friction of all test specimen ranged from 0.1 to 0.2. Such a result has proven that the basalt fiber, similar to the resin rubber, shows regular patterns regardless of time and H₂SO₄ concentration because of the space made between resins and reinforced materials.

Using a special form of spanning surface for a knot, we give a formula for the coefficient of the $z^2$-term of the Alexander–Conway polynomial in terms of the sum of determinants of the blocks of $2\times 2$ submatrices of the Seifert matrix, from which the topological meaning of the coefficient is revealed.

Let l ≥ 1 be an arbitrary odd integer and p, q and r be primes. We show that there exist infinitely many ternary cyclotomic polynomials Φ_pqr(x) with l² + 3l + 5 ≤ p < q < r such that the set of coefficients of each of them consists of the p integers in the interval [-(p - l - 2)/2, (p + l + 2)/2]. It is known that no larger coefficient range is possible. The Beiter conjecture states that the cyclotomic coefficients a_pqr(k) of Φ_pqr satisfy |a_pqr(k)| ≤ (p + 1)/2 and thus the above family contradicts the Beiter conjecture. The two already known families of ternary cyclotomic polynomials with an optimally large set of coefficients (found by G. Bachman) satisfy the Beiter conjecture.

For a positive integer $n$, the $n$th cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ is called flat if its coefficients are $0$, $\pm 1$. Let $p<q<r$ be odd primes with $zr\equiv \pm 1(modpq)$ for some positive integer $z$. In this paper, we classify all the flat cyclotomic polynomials ${\mathrm{\Phi }}_{pqr}(x)$ when $z=3$, $4$ and $5$.