Abstract. Let φm be a polynomial satisfying some mild conditions. Given a set R ⊂ C, a continuous function f on R and its best approximation p n−1 from Πn−1 with respect to the maximum norm, we show that p ∗ n−1 ◦φm is a best approximation to f ◦ φm on the inverse polynomial image S of R, i.e. φm(S) = R, where the extremal signature is given explicitly. A similar result is presented for constra...

Suppose that f satisfies the following: (1) polyharmonic equation Δmf=Δ(Δm−1f) =φm (φm∈C(Bn¯,Rn)), (2) boundary conditions Δ0f=φ0,Δ1f=φ1,…,Δm−1f=φm−1 on Sn−1 (φj∈C(Sn−1,Rn) for j∈{0,1,…,m−1} and denotes of unit ball Bn), (3) f(0)=0, where n≥3 m≥1 are integers. Initially, we prove a Schwarz type lemma use it to obtain Heinz inequality mappings satisfying with above Dirichlet value conditions. Fu...

Given two cyclotomic polynomials Φn(x) and Φm(x), n 6= m, we determine the minimal natural number k such that we can write k = a(x)Φn(x) + b(x)Φm(x), with a(x) and b(x) integer polynomials. In June 2000 the author of this note gave a talk at the conference ”Combinatorics 2000” in Gaeta (Italy), titled Dividing Cyclotomic Polynomials. The content of that talk was proposed to the American Mathema...

      i∂tΦ1 +∆Φ1 + g1 (|x|, |Φ1| , . . . , |Φm| ) Φ1 = 0, .. i∂tΦm +∆Φm + gm (|x|, |Φ1| , . . . , |Φm| )Φm = 0, Φi(0, x) = Φ 0 i (x) for 1 ≤ i ≤ m. For 1 ≤ i ≤ m : Φi : R N → C and gi : R ∗ + × R m + → R, Φi : R+ × R N → C, has numerous applications in physical problems. It appears in the study of spatial solitons in nonlinear waveguides [30], the theory of Bose-Einstein condensates [12],...

Given a digraph G, a lot of attention has been deserved on the maximum number φ(G) of fixed points in a Boolean network f : {0, 1} → {0, 1} with G as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the classical upper bound φ(G) ≤ 2 , where τ is the minimum size of a feedback vertex set of G. In this paper, we study the maximum number...

Under various assumptions on A and αj s, Pisot and Schoenberg [8] solved for monotone solutions of (1.1). In a subsequent paper [9], they treated the case where the domain of solutions is a subset of Rn with the following main result. Let α1,α2, . . . ,αA be elements of Rn (n < A) satisfying the following conditions: (1) every set of n elements among the αi is linearly independent over R, (2) t...

Let p and r be two primes and n, m be two distinct divisors of pr. Consider Φn and Φm, the n-th and m-th cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of Φn modulo Φm.

We answer a question of T. Shioda and show that, for any positive integer m prime to 6, the Picard group of the Fermat surface Φm is generated by the classes of lines contained in Φm. A few other classes of surfaces are also considered.

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by (φm)m∈N0 . The system (φm)m∈N0 is defined in the space C(Zp,Cp) of Cp-valued continuous functions on Zp. J. Rutkowski has also considered some questions concerning expansions of functions from C(Zp,Cp) with respect to (φm)m∈N0 . This paper is a remark to Rutkowski’s paper....

Data from SABLES98 experimental campaign have been used in order to study the influence of stability (from weak to strong stratification) on the flux-profile relationships for momentum and heat: φm and φh. Standard instrumentation, 14 thermocouples and 3 sonic anemometers at three levels (5.8, 13.5 and 32 m) were available from 10 to 28 September 1998 and calculations are done in order to extra...