Zolotarev's first problem — the best approximation by polynomials of degree ≤n − 2 tox n − nσx n−1 in [−1, 1]

Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 2^1?n , which is attained when the polynomial is 2^1?n T _n(x), whereT _n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan²(π/2n) is quite simple, but that for |σ|> tan²(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem.     

Orthogonal polynomials for Jacobi-exponential weights (1−x 2) ρ e −Q(x) on (−1,1)

This paper deals with orthogonal polynomials for Jacobi-exponential weights (1?x ²) ^ρ e ^?Q(x) on (?1,1) and gives bounds on orthogonal polynomials, zeros, and Christofel functions. In addition, restricted range inequalities are also obtained.     

Regularity and explicit representation of (0,1,...,m−2,m) interpolation on the zeros of (1−x 2)P n −2/(α,β) (x)

Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x ²)P _n ^–2/(,) (x) (,>–1) in a manageable form are established, whereP _n ^–2/(,) (x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf ₀(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China.     

Some inequalities for polynomials satisfying p(z)≡znp(1/z)

Let be a polynomial degreen and let . Then according to Bernstein’s inequality ‖p’‖≤n‖p‖. It is a well known open problem to obtain inequality analogous to Bernstein’s inequality for the class II_n of polynomials satisfying p(z)≡zⁿp(1/z). Here we obtain an inequality analogous to Bernstein’s inequality for a subclass of II_n. Our results include several of the known results as special cases.     

A sharp error estimate of the best approximation by algebraic polynomials in the weighted space L 2(−1, 1)

We obtain an exact error estimate of the best approximation by algebraic polynomials in the Lebesgue space L ₂(?1, 1) with the weight 1 ? x ² of degree λ > ?1.     

Distribution of zeros of the Hermite-Padé polynomials for a system of three functions,and the Nuttall condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\Re _3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak{f}_1 ,\mathfrak{f}_2 ,\mathfrak{f}_3$ that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ?? \ B of the open (possibly, disconnected) set B ? ?? introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.     

A characterization of a class of hyperplanes of DW(2n−1,F)

A hyperplane of the symplectic dual polar space $DW(2n?1,\mathbb{F})$, $n\ge 2$, is said to be of subspace-type if it consists of all maximal singular subspaces of $W(2n?1,\mathbb{F})$ meeting a given $(n?1)$-dimensional subspace of $\text{PG}(2n?1,\mathbb{F})$. We show that a hyperplane of $DW(2n?1,\mathbb{F})$ is of subspace-type if and only if every hex $F$ of $DW(2n?1,\mathbb{F})$ intersects it in either $F$, a singular hyperplane of $F$ or the extension of a full subgrid of a quad. In the case $\mathbb{F}$ is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane $H$ of $DW(2n?1,\mathbb{F})$ is of subspace-type or arises from the spin-embedding of $DW(2n?1,\mathbb{F})?DQ(2n,\mathbb{F})$ if and only if every hex $F$ intersects it in either $F$, a singular hyperplane of $F$, a hexagonal hyperplane of $F$ or the extension of a full subgrid of a quad.     

A note on the diophantine equation (a n − 1)(b n − 1) = x 2

Let a, b be fixed positive integers such that a ≠ b, min(a, b) > 1, ν(a?1) and ν(b ? 1) have opposite parity, where ν(a ? 1) and ν(b ? 1) denote the highest powers of 2 dividing a ? 1 and b ? 1 respectively. In this paper, all positive integer solutions (x, n) of the equation (a ⁿ ? 1)(b ⁿ ? 1) = x ² are determined.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.     

Commutator identities of the homotopes of (−1, 1)-algebras

We study the commutator algebras of the homotopes of (?1, 1)-algebras and prove that they are Malcev algebras satisfying the Filippov identity h _a (x, y, z) = 0 in the case of strictly (?1, 1)-algebras. We also proved that every Malcev algebra with the identities xy ³ = 0, xy ² z ² = 0, and h _a (x, y, z) = 0 is nilpotent of index at most 6.