The method of asymptotic stabilization to a given trajectory based on a correction of the initial data

Let S be an operator in a Banach space H and S ⁱ (u) (i = 0, 1, ..., u ∈ H) be the evolutionary process specified by S. The following problem is considered: for a given point z ₀ and a given initial condition a ₀, find a correction l such that the trajectory {S ⁱ (a ₀ + l)} approaches }S ⁱ (z ₀)} for 0 < i ≤ n. This problem is reduced to projecting a ₀ on the manifold ?^?(z ₀, f ⁽ⁿ⁾) defined in a neighborhood of z ₀ and specified by a certain function f ⁽ⁿ⁾. In this paper, an iterative method is proposed for the construction of the desired correction u = a ₀ + l. The convergence of the method is substantiated, and its efficiency for the blow-up Chafee-Infante equation is verified. A constructive proof of the existence of a locally stable manifold ?^?(z ₀, f) in a neighborhood of a trajectory of hyperbolic type is one of the possible applications of the proposed method. For the points in ?^?(z ₀, f), the value of n can be chosen arbitrarily large.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

An algebraic structure for Faber polynomials

Let Σ be the set of functions, convergent for all |z|>1, with a Laurent series of the form f(z)=z+∑_n?0a_nz^-n. In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell^(k) subgroup of the Riordan group, respectively. Further, a relationship between such Faber polynomial sequences and Lucas and Sheffer polynomial sequences is derived.     

Eigenvalue Intervals for 2mth Order Sturm—Liouville Boundary Value Problems

Values of λ are determined for which there exist positive solutions of the 2mth order differential equation on a measure chain, (-1)^m x ^?2m (t)=λa(t)f(u(σ(t))), y? [0,1], satisfying α _i+1 u ^?21(0)+0, γ _i+1 u ^?21(σ(1))=0, 0≤i≤m?1 with αi,βi,γi,δi≥0, where a and f are positive valued, and both lim _x-0+ (f(x)/x) and lim _x-0+ (f(x)/x) exist.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The inverse of a matrix polynomial

An explicit representation is obtained for P(z)^?1 when P(z) is a complex n×n matrix polynomial in z whose coefficient of the highest power of z is the identity matrix. The representation is a sum of terms involving negative powers of z?λ for each λ such that P(λ) is singular. The coefficients of these terms are generated by sequences u_k, v_k of 1×n and n×1 vectors, respectively, which satisfy u₁≠0, v₁≠0, ∑^k?1_h=0(1?h!)u_k?hP^(h)(λ)=0, ∑^k?1_h=0(1?h!)P^(h)(λ)v_k?h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at λ but not necessarily a polynomial, the terms in the representation involving negative powers of z?λ provide the principal part of the Laurent expansion for P(z)^?1 in a punctured neighborhood of z=λ.     

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.     

The algebraic structure of the set of solutions to the Thue equation

Let Fn be a binary form with integral coefficients of degree n?2, let d denote the greatest common divisor of all non-zero coefficients of Fn, and let h?2 be an integer. We prove that if d=1 then the Thue equation (T) Fn(x,y)=h has relatively few solutions: if A is a subset of the set T(Fn,h) of all solutions to (T), with r:=card(A)?n+1, then

(#)

h divides the numberΔ(A):=_∏1?k<l?rδ(ξk,ξl),

where ξk=〈xk,yk〉∈A, 1?k?r, and δ(ξk,ξl)=xkyl−xlyk. As a corollary we obtain that if h is a prime number then, under weak assumptions on Fn, there is a partition of T(Fn,h) into at most n subsets maximal with respect to condition (#).     

Some explicit formulae in simultaneous padé approximation

Using old results on the explicit calculation of determinants, formulae are given for the coefficients of P₀(z) and P₀(z)f_i(z) ? P_i(z), where P_i(z) are polynomials of degree σ ? ρ_i (i=0,1,…,n), P₀(z)f_i(z) ? P_i(z) are power series in which the terms with z^k, 0?k?σ, vanish (i=1,2,…,n), (ρ₀,ρ₁,…,ρ_n) is an (n+1)-tuple of nonnegative integers, σ=ρ₀+ρ₁+?+ρ_n, and {f_i}ⁿ_i=1 is the set of hypergeometric functions {₁F₁(1;c_i;z)}ⁿ_i=1$\text{(c}{}_{\mathrm{i}}\text{?}\text{Z}\text{z.drule;}\text{N}\text{, c}{}_{\mathrm{i}}\text{? c}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ or {₂F₀(a_i,1;z)}ⁿ_i=1$\text{(a}{}_{\mathrm{i}}\text{?}\text{Z}\text{?}\text{N}\text{, a}{}_{\mathrm{i}}\text{? a}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ under the condition ρ₀?ρ_i ? 1 (i=1,2,…,n).     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

Siegel’s Lemma and Sum-Distinct Sets

Let L(x)=a ₁ x ₁+a ₂ x ₂+???+a _n x _n , n≥2, be a linear form with integer coefficients a ₁,a ₂,…,a _n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a ₁,a ₂,…,a _n . The main result of this paper asserts that there exist linearly independent vectors x ₁,…,x _n?1∈? ⁿ such that L(x _i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a ₁,a ₂,…,a _n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.     

Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

Bilinear Relative Equilibria of Identical Point Vortices

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least n?m+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z ²+η ², where η is real, there is a unique p(z) of degree n, and a unique value of η ²=A _n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that $A_{n} \approx\frac{2}{3}n + \frac{1}{2}$ , as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.     

Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).