Renormalized coupling constants for the three-dimensional scalar <Emphasis Type="Italic">λ</Emphasis><Emphasis Type="Italic">ϕ</Emphasis><Superscript>4</Superscript> field theory and pseudo-<Emphasis Type="Italic">ε</Emphasis>-expansion

The renormalized coupling constants g _2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g _2k ^* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R _2k = g _2k /g ₄ ^k-1 for the three-dimensional scalar λ ? ⁴ field theory. We derive pseudo-ε-expansions for g ₆ ^* , g ₈ ^* , R ₆ ^* , and R ₈ ^* in the five-loop approximation and present numerical estimates for R ₆ ^* and R ₈ ^* . The higher-order coefficients of the pseudo-ε-expansions for g ₆ ^* and R ₆ ^* are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives R ₆ ^* = 1.650, while the recent lattice calculation gave R ₆ ^* = 1.649(2). The pseudo-ε-expansions of g ₈ ^* and R ₈ ^* are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for R ₈ ^* gives the estimate R ₈ ^* = 0.890, differing only slightly from the values R ₈ ^* = 0.871 and R ₈ ^* = 0.857 extracted from the results of lattice and field theory calculations.     

Field theory and anisotropy of a cubic ferromagnet near the Curie point

It is known that critical fluctuations can change the effective anisotropy of a cubic ferromagnet near the Curie point. If the crystal undergoes a phase transition into the orthorhombic phase and the initial anisotropy is not too strong, then the effective anisotropy acquires the universal value A* = v*/u* at T _c, where u* and v* are the coordinates of the cubic fixed point of the renormalization group equations in the scaling equation of state and expressions for nonlinear susceptibilities. Using the pseudo-?-expansion method, we find the numerical value of the anisotropy parameter A at the critical point. Padé resummation of the six-loop pseudo-?-expansions for u*, v*, and A* leads to the estimate A* = 0.13 ± 0.01, giving evidence that observation of anisotropic critical behavior of cubic ferromagnets in physical and computer experiments is entirely possible.     

Critical exponents and the pseudo-<Emphasis Type="Italic">є</Emphasis>-expansion

We present the pseudo-?-expansions (τ-series) for the critical exponents of a λ?⁴-type three-dimensional O(n)-symmetric model obtained on the basis of six-loop renormalization-group expansions. We present numerical results in the physically interesting cases n = 1, n = 2, n = 3, and n = 0 and also for 4 ≤ n ≤ 32 to clarify the general properties of the obtained series. The pseudo-?-expansions or the exponents γ and α have coefficients that are small in absolute value and decrease rapidly, and direct summation of the τ -series therefore yields quite acceptable numerical estimates, while applying the Padé approximants allows obtaining high-precision results. In contrast, the coefficients of the pseudo-?-expansion of the scaling correction exponent ω do not exhibit any tendency to decrease at physical values of n. But the corresponding series are sign-alternating, and to obtain reliable numerical estimates, it also suffices to use simple Padé approximants in this case. The pseudo-?-expansion technique can therefore be regarded as a distinctive resummation method converting divergent renormalization-group series into expansions that are computationally convenient.     

N-point Padé approximants to real-valued Stieltjes series with nonzero radii of convergence

By employing special continued fractions to two Stieltjes series with nonzero radii of convergence we extend the inequalities for one-point Padé approximants reported by Baker (1975, Corollory 17.1) to the case of two-point Padé approximants. We prove that some convergents of the continued fractions form a monotone sequences of upper and lower bounds converging uniformly to Stieltjes function x₁(x) on compact subsets of (−R, ∞), where R is a radius of convergence of an expansion of x f₁(x) at x = 0. For an illustration of theoretical results we provide nontrivial numerical examples. As an application to real physical problems second order Padé approximants' bounds on the effective conductivity of a square array of cylinders are evaluated.     

Generalized Padé-type approximants to continuous functions

Summary We define generalized Padé-type approximants to continuous functions on a compact subset Eof Rⁿsatisfying the Markov's inequality and we show that the Fourier series expansion of a generalized Padé-type approximant to a u ∈ C ^∞(E ) matches the Fourier series expansion of uas far as possible. After studying the errors, we give integral representations and an answer to the convergence problem of a generalized Padé-type approximation sequence.     

Rational interpolation and best rational approximation

A relation between the degree of convergence (in capacity) of Padé approximants and the degree of best rational approximation is derived for functions in Gon?ar's class R₀. The results contain previous convergence results for Padé approximants proved by Nuttall, Pommerenke, and Gammel and Nuttall.     

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 functionf(z,z ⁻¹) up to as high a degree inz, 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 off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z ⁻¹)B _n (z, z ⁻¹)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+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 degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+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.     

Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities

In this paper, the asymptotic behavior of solutions u _ε of the Poisson equation in the ε-periodically perforated domain Ω_ε ? $ {{\mathbb{R}}^n} $ , n ≥ 3, with the third nonlinear boundary condition of the form ? _ν u _ε + ε^?γσ(x, u _ε) = ε ^?γ g(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order ε^α with α > 1 and any γ. Here, all types of asymptotic behavior of solutions u _ε , corresponding to different relations between parameters α and γ, are studied.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

The existence and convergence of subsequences of Padé approximants

We prove the existence of an infinite number of Padé approximants, and thereby remedy a defect in Nuttall's theorem. It is proved that the sequences of Padé approximants shown by Perron, Gammel, and Wallin to be everywhere divergent contain subsequences which are everywhere convergent. It is further proved that there always exist, for entire functions, everywhere convergent [1, N] and [2, N] subsequences of Padé approximants. There must exist subsequences of [m, N] Padé approximants (N → ∞) which converge almost everywhere in $\text{¦z¦ ? ? < R}$ to functions f(z) which are regular except for a finite number (n ? m) of poles in $\text{¦z¦ < R}$. We prove convergence of the [N, N + j] Padé approximants in the mean on the Riemann sphere for meromorphic functions.     

Fast solution of toeplitz systems of equations and computation of Padé approximants

We present two new algorithms, ADT and MDT, for solving order-n Toeplitz systems of linear equations Tz = b in time O(n log²n) and space O(n). The fastest algorithms previously known, such as Trench's algorithm, require time $\text{Ω(n}{}^2\text{)}$ and require that all principal submatrices of T be nonsingular. Our algorithm ADT requires only that T be nonsingular. Both our algorithms for Toeplitz systems are derived from algorithms for computing entries in the Padé table for a given power series. We prove that entries in the Padé table can be computed by the Extended Euclidean Algorithm. We describe an algorithm EMGCD (Extended Middle Greatest Common Divisor) which is faster than the algorithm HGCD of Aho, Hopcroft and Ullman, although both require time O(n log²n), and we generalize EMGCD to produce PRSDC (Polynomial Remainder Sequence Divide and Conquer) which produces any iterate in the PRS, not just the middle term, in time O(n log²n). Applying PRSDC to the polynomials U₀(x) = x²ⁿ⁺¹ and U₁(x) = a₀ + a₁x + … + a_2nx²ⁿ gives algorithm AD (Anti-Diagonal), which computes any (m, p) entry along the antidiagonal m + p = 2n of the Padé table for U₁ in time O(n log²n). Our other algorithm, MD (Main-Diagonal), computes any diagonal entry (n, n) in the Padé table for a normal power series, also in time O(n log²n). MD is related to Schönhage's fast continued fraction algorithm. A Toeplitz matrix T is naturally associated with U₁, and the (n, n) Padé approximation to U₁ gives the first column of T^?1. We show how a formula due to Trench can be used to compute the solution z of Tz = b in time O(n log n) from the first row and column of T^?1. Thus, the Padé table algorithms AD and MD give O(n log²n) Toeplitz algorithms ADT and MDT. Trench's formula breaks down in certain degenerate cases, but in such cases a companion formula, the discrete analog of the Christoffel-Darboux formula, is valid and may be used to compute z in time O(n log²n) via the fast computation (by algorithm AD) of at most four Padé approximants. We also apply our results to obtain new complexity bounds for the solution of banded Toeplitz systems and for BCH decoding via Berlekamp's algorithm.     

A parabolic free boundary problem with double pinning

We study the Cauchy problem for the equation ∂_tu_ε−Δu_ε=−β_ε(u_ε) in (0,∞)×Rⁿ as $\text{ε→0}$, where the nonlinearity β_ε is assumed to converge to a measure concentrated at $\text{0}$. In this paper we allow for sign changes of β_ε and u_ε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of u_ε is a solution of the free boundary problem ∂_tu−Δu=0 in {u>0}∩(0,∞)×Rⁿ,|∇u⁺|²−|∇u⁻|²=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rⁿ) in the sense of domain variations. Depending on the structure of the nonlinearity $\text{β}{}_{\mathrm{\epsilon }}\text{,}$ the function g in the condition on the free boundary need not be a constant.     

Introduction to the improved functional epsilon algorithm

This paper introduces the improved functional epsilon algorithm. We have defined this new method in principle of the modified Aitken Δ² algorithm. Moreover, we have found that the improved functional epsilon algorithm has remarkable precision of the approximation of the exact solution and there exists a relationship with the integral Padé approximant. The use of the improved functional epsilon algorithm for accelerating the convergence of sequence of functions is demonstrated. The relationship of the improved functional epsilon algorithm with the integral Padé approximant is also demonstrated. Moreover, we illustrate the similarity between the integral Padé approximant and the modified Aitken Δ² algorithm; thus we have shown that the integral Padé approximant is a natural generalisation of modified Aitken Δ² algorithm.     

Chebyshev expansions and rational approximations

Let A(z) = A_m(z) + a_mz^mB(z,m) where A_m(z) is a polynomial in z of degree m-1. Suppose A(z) and B(z,m) are approximated by main diagonal Padé approximations of order n and r respectively. Suppose that the number of operations needed to evaluate both sides of the above equations by means of the Padé approximations and polynomial noted are the same. Thus 4n = 3m + 4r. We address ourselves to the question of which procedure is more efficient? That is, which procedure produces the smallest error? A variant of this problem is the situation where A(z) and B(z,m) are approximated by their representations in infinite series of Chebyshev polynomials of the first kind truncated after n and r terms respectively. Here n = m + r.Let F(z) have two different series type representations in overlapping or completely disjoint regions of the complex z-plane. Suppose that for each representation there is a sequence of rational approximations of the same type, say of the Padé class, which converge for |arg z| < π except possibly for some finite set of points. Assume that the number of machine operations required to make evaluations using the noted approximations are the same. Again, we ask which procedure is best? Other variants are studied.General answers to the above questions are not known. Instead, we illustrate the ideas for a number of the rather common special functions.     

Homogenization of Schrödinger-type equations

We consider a self-adjoint elliptic operator A_ε, ε> 0, on L₂(R^d; Cⁿ) given by the differential expression b(D)*g(x/ε)b(D). Here \(b(D) = \sum\nolimits_{j = 1}^d {b_j D_j }\) is a first-order matrix differential operator such that the symbol b(ξ) has maximal rank. The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice. We study the operator exponential \({e^{ - i\tau {A_\varepsilon }}}\), where τ ∈ R. It is shown that, as ε → 0, the operator \({e^{ - i\tau {A_\varepsilon }}}\) converges to \({e^{ - i\tau {A^0}}}\) in the norm of operators acting from the Sobolev space H^s(R^d;Cⁿ) (with suitable s) to L₂(R^d;Cⁿ). Here A⁰ is the effective operator with constant coefficients. Order-sharp error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation i?_τu_ε(x, τ) = A_εu_ε(x, τ).     

Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition

In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation ut=ε²Δu+f(u) in a smooth bounded domain Ω⊂R³ under the boundary condition εν_∂u=δεg(u). It is assumed that Ω has a cross-section which locally minimizes area and limε→0εlnδε=κ, with 0?κ<∞ and δε>1 when κ=0. The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior.     

The epsilon algorithm an a non-commutative algebra

In the case of a non-commutative algebra, the epsilon algorithm is deduced from the Padé approximants at t = 1, and from the use of the cross rule; their algebraic properties are a consequence of those verified by the Padé approximants. The computation of the coefficients is particularly studied. It is shown, that it does not exist any non-invertible needed elements if and only if the Hankel matrices M_k (Δ′S_n) = (Δ′S_n+i+j)_i=j=0^k−1 for l =1, 2 and 3, have an inverse. Some results of convergence and convergence acceleration are also given.     

Two-Phase Stefan Problem as the Limit Case of Two-Phase Stefan Problem with Kinetic Condition

Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule u_ε(R_ε(t),t)=εR_ε′(t) at the moving boundary are considered. We prove, when ε approaches zero, R_ε(t) converges to R(t) in C^1+δ/2[0,T] for any finite T>0, 0<δ<1.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

On non-autonomous strongly damped wave equations with a uniform attractor and some averaging

In this paper we study the existence of a uniform attractor for strongly damped wave equations with a time-dependent driving force. If the time-dependent function is translation compact, then in a certain parameter region, the uniform attractor of the system has a simple structure: it is the closure of all the values of the unique, bounded complete trajectory of the wave equation. And it attracts any bounded set exponentially. At the same time, we consider the strongly damped wave equations with rapidly oscillating external force gε(x,t)=g(x,t,t/ε) having the average g⁰(x,t) as ε→⁰⁺. We prove that the Hausdorff distance between the uniform attractor Aε of the original equation and the uniform attractor A₀ of the averaged equation is less than O(ε1/2). We mention, in particular, that the obtained results can be used to study the usual damped wave equations.