On an approximation property of Pisot numbers

Let 1<q<2 be a real number, m≥1 be a rational integer and l_m(q)={|P(q)|,P∈Z[X],P(q)≠0,H(P)≤m}, where Z[X] denotes the set of polynomials P with rational integer coefficients and H(P) is the height of P. The value of l_m(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers l_m(q) for any fixed m. We also determine the greatest limit point for the case m=1. This revised version was published online in June 2006 with corrections to the Cover Date.     

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

Comments on the height reducing property

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ? of the rational integers such that ?[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

Approximation by polynomials with bounded coefficients

Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.     

Bannai–Ito algebras and the universal R-matrix of $$\pmb {\mathfrak {osp}}(1|2)$$osp(1|2)

Letters in Mathematical Physics - The Bannai–Ito algebra BI(n) is viewed as the centralizer of the action of $$\mathfrak {osp}(1|2)$$ in the n-fold tensor product of the universal algebra of...