Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space  ^k ₊ = {x = (x ₁,…,x_k ): x_k > 0} is considered. It is assumed that the boundary values of a solution u = (u ₁,…,u _m) have the form ψ ₁ ξ ₁ + · · · + ψ _n ξ _n, 1 ≤ n ≤ m, where ξ ₁,· · ·,ξ _n is an orthogonal system of m-component normed vectors and ψ ₁,· · ·,ψ _n are continuous and bounded functions on ? ^k ₊. We study the mappings [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m and [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| _xk =₀∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.     

Description of polygonal regions by polynomials of bounded degree

We show that every (possibly unbounded) convex polygon P in \mathbbR²{\mathbb{R}^2} with m edges can be represented by inequalities p ₁ ≥ 0, . . ., p _n ≥ 0, where the p _i ’s are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m, k) ￡ n(m, k) ￡ (1+e_m) s(m, k){s(m, k) \leq n(m, k) \leq (1+\varepsilon_m) s(m, k)} with s(m,k) ≔ max {m/k, log₂ m} and e_m ? 0{\varepsilon_m \rightarrow 0} as m ? ￥{m \rightarrow \infty}. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p ₁ > 0, . . ., p _n >  0 is also given. For k ≤ m/log₂ m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k.     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

On a Class of Integer-Valued Functions

The paper deals with the class of entire functions that increase not faster than exp{γ∣z∣^6/5(ln∣z∣)^?1} and that, together with their first derivatives, take values from a fixed field of algebraic numbers at the points of a two-dimensional lattice of general form (in this case, the values increase not too fast). It is shown that any such functions is either a polynomial or can be represented in the form e^?mαzP(e^αz), where m is a nonnegative integer, P is a polynomial, and α is an algebraic number.

     

Estimates for Hyperbolic Equations of Space Dimension 3

We discuss the problem of boundedness fromL^p(Rⁿ) toL^p′(Rⁿ) (1/p+1/p′=1, 1?p?2) of operators of the typeM=F⁻¹e^i?(ξ)a(ξ) F, which is related to the study of hyperbolic equations with constant coefficients. The boundedness is dependent on a geometrical property ofΣ=?⁻¹(1), and its dependence has been exactly determined in the casesn=2, 1?p?2 andn?3,p=1, 2 (M. Sugimoto,Math. Z.215(1994), 519–531;222(1996), 521–531). This paper is devoted to the unsolved case 1<p<2, and a strange phenomenon is exhibited in the simplest casen=3.     

Effective Fast Algorithms for Polynomial Spectral Factorization

Let p(z) be a polynomial of degree n having zeros |ξ₁|≤???≤|ξ _m |<1<|ξ _m+1|≤???≤|ξ _n |. This paper is concerned with the problem of efficiently computing the coefficients of the factors u(z)=∏ _i=1 ^m (z?ξ _i ) and l(z)=∏ _i=m+1 ⁿ (z?ξ _i ) of p(z) such that a(z)=z ^?m p(z)=(z ^?m u(z))l(z) is the spectral factorization of a(z). To perform this task the following two-stage approach is considered: first we approximate the central coefficients x _?n+1,. . .x _n?1 of the Laurent series x(z)=∑ _i=?∞ ^+∞ x _i z ⁱ satisfying x(z)a(z)=1; then we determine the entries in the first column and in the first row of the inverse of the Toeplitz matrix T=(x _i?j ) _i,j=?n+1,n?1 which provide the sought coefficients of u(z) and l(z). Two different algorithms are analyzed for the reciprocation of Laurent polynomials. One algorithm makes use of Graeffe's iteration which is quadratically convergent. Differently, the second algorithm directly employs evaluation/interpolation techniques at the roots of 1 and it is linearly convergent only. Algorithmic issues and numerical experiments are discussed.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Asymptotic expansions in time for solutions of Schrödinger-type equations

Let ?iA = ?i(p(D) + V) be a dissipative operator in L₂(R_n), where p(D) is an elliptic differential operator of order m with real constant coefficients and V is a compact operator from the weighted Sobolev space H^m′,s (Rⁿ) to H^{0, p + s} (Rⁿ), s?R, for some m′ < m ? 1 and p > 1. Let R(z) be the resolvent of A. Then an asymptotic expansion of R(z) as z approaches a critical value of the polynomial p(ξ) is given; the coefficient operators in the expansion are computed explicitly. By using the resolvent expansion and the results of M. Murata [9], an asymptotic expansion of e^?itA as t → ∞ is given.     

Explicit Series Formulae for the Evaluation of Integrals by the Method of Steepest Descents

Explicit series formulae are given for use in the method of steepest descents, where it is required to find the contribution to an integral of the form ∫e^vw(z)φ(z)dz from a well-isolated saddle point z₀. Here v is a large positive parameter, and w^(N)(z0) ≠ 0 while lower-order derivatives at z₀ are equal to zero. Examples where the formulae are helpful are presented.     

О функциях, удовлетворяюших ослабленному условию асимптотической моногенности

We consider functionsf(z),z∈D, of one complex variable that satisfy the following weakened asymptotic monogeny condition: for some positiveσ<1/2,f(z) is monogenic at each pointξ∈D with respect to some setG(ξ) such that the lower density ofG(ξ) atξ is greater than 1/2+σ. We show that if for somep _σ ≥1 the function (log⁺|?(z)|) ^p _σ is locally integrable inD with respect to the plane Lebesgue measure, thenf(z) is holomorphic inD.     

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Oph(p)−z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p₀(x,ξ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp₀. First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C^∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

Reducibility of polynomials a0(x) + a1 (x)y + a2 (x) y2 modulo p

Let f=a₀(x)+a₁(x)y+a₂(x)y² ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 c_m·H(f)^e_mp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e₁ = 4,e₂ = 6, e₃ = 6 [2/3]{2}\over{3} and e_m = 2 m for m S 4. We also show that the exponents e_m are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

Anharmonic Oscillators in the Complex Plane, $\boldsymbol{\mathcal{PT}}$-symmetry,and Real Eigenvalues

For integers m ≥ 3 and 1 ≤ ℓ ≤ m − 1, we study the eigenvalue problems − u ^″(z) + [( − 1)^ℓ(iz) ^m  − P(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays argz=-\fracp2±\frac(l+1)pm+2\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2} in the complex plane, where P is a polynomial of degree at most m − 1. We provide asymptotic expansions of the eigenvalues λ _n . Then we show that if the eigenvalue problem is PT\mathcal{PT}-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when gcd(m,l)=1\gcd(m,\ell)=1, the eigenvalue problem has infinitely many real eigenvalues if and only if one of its translations or itself is PT\mathcal{PT}-symmetric. Also, we will prove some other interesting direct and inverse spectral results.     

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:

fⁿ(z)+P_n−3(f)=p₁e^α₁z+p₂e^α₂z