Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Semidiscrete and asymptotic approximations for the nonstationary radiative–conductive heat transfer problem in a periodic system of grey heat shields

We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε ^3/4) for asymptotic approximations. Bibliography: 9 titles.     

On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

THE ASYMPTOTIC THEORY OF INITIAL VALUEPROBLEMS FOR SEMILINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

This paper deals with the asymptotic theory of initial value problems for semilinear waveequations in three space dimensions. The well-posedness and validity of formal approximations ona long time scale of order |ε|^-1 are discussed in the classical sense of C^2 This result describes aceu-ratively the approximations of solutions. At the end of this paper an application of the asymptotictheory is given to analyze a special model for a perturbed wave equation,     

Random evolutions with locally independent increments on increasing time intervals

Three main schemes of limit theorems for random evolutions are discussed: averaging, diffusion approximation, and the asymptotics of large deviations. Markov stochastic evolutions with locally independent increments on increasing time intervals T _ε  = t/ε → ∞, ε → 0, are considered. The asymptotic behavior of random evolutions is investigated with the use of solutions of the singular perturbation problems for reducibly invertible operators.     

Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order e = O(N^-1).\varepsilon =\mathcal{O}(N^{-1}). The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are ε-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as ε→0.     

The central connection problem at turning points of linear differential equations

A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x ₀, 0 < ε ⩽ ε₀ , with an asymptotic expansionsA (x, ε) ∼ ∑ _r=0 ^∞ A _r (x) ε ^r , as ε→0. The eigenvalues ofA ₀(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.     

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σ_ε(u _ε) + εκ _m (u _ε) = εg ^(m) _ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.     

Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

On homogenization of the mixed boundary-value problem for the heat equation in a domain whose part contains channels arranged in a periodic way

In this paper, we study the asymptotic behavior of the solutionsu _ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ω_ε=Ω⁻∪Ω _ε ⁺ ∪γ one part of which (Ω _ε ⁺ ) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω⁺) is a homogeneous medium; γ=∂Ω _ε ⁺ ∩∂Ω⁺. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ω_ε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission condition on γ. The estimates for the difference betweenu _ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997.     

Estimation and detection of a function from tensor product spaces

We observe an unknown function of d variables ƒ(t), t ∈ [0, 1]^d, in the white Gaussian noise of level ε > 0. We assume that {ie4526-01}, where {ie4526-02} is a ball in the Hilbert space {ie4526-03} of tensor product structure. Under minimax setup, we consider two problems: estimate ƒ (for quadratic losses) and detect ƒ, i.e., test the null hypothesis H₀: ƒ = 0 against the alternatives {ie4526-04}. We are interested in the case {ie4526-05}. We study sharp, rate, and log-asymptotics (as ε → 0 and d → ∞) in the problems. In particular, we show that log-asymptotics are essentially different for d ≪ log ε⁻¹ and d ≫ log ε⁻¹. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 180–218.     

The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type

Convergence theorems and asymptotic estimates (as ε → 0) are proved for the eigenvalues and the eigenfunctions of the Neumann problem in a dense singular junction Ω _ɛ of a domain Ω₀ and a large number N of thin cylinders with thickness of order ε=lN⁻¹, where l is the total length of common boundaries for Ω₀ and the cylinders in question. Bibliography: 27 titles. We dedicate the present paper to Olga Arsenievna Oleinik, as a symbol of our deep respect and gratitude Translated from Trudy Seminara imeni I G. Petrovskogo, No. 19. pp. 000-000. 0000.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

Su una questione di statica proposta dal Malfatti

Riassunto Si considera un corpo pesante appoggiato inm punti ad un piano orizzontale; è noto che le equazioni cardinali della statica, salvo nel casom≤3, non sono sufficienti, in generale, per determinare le reazioni Φ_i (i=1, 2, …,m) degli appoggi. Valendosi di analogia col casom=3Malfatti (1731–1807) ha proposto formule [4] per Φ_i valide per ognim. Queste formule soddisfano l'equazione per il risultante, ma non sembra che Malfatti verifichi quella per il momento, verifica che non risulta immediata. In questa nota, scritte le formule di Malfatti in forma moderna, si osserva che esse si possono suddividere in due parti ε₁ ed ε₂ (ε₂ fino a 4 lati), e si dimostra che i termini di ε₁ soddisfano per ognim anche l'equazione dei momenti mentre, per i termini di ε₂ (meno intuitivi dal punto di vista fisico) si è limitata la verifica al caso del pentagono e dell'esagono. In sostanza, almeno considerando i corpi rigidi, si conferma la intuizione di Malfatti.

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Summary We consider a heavy body leaned against a horizontal plane onm pointsA _i; it is known that the statics equations, except the casem≤3, are not sufficient in general to determine the reaction forces reazioni Φ_i(i=1,…,m) at pointsA _i. Generalizing the casem=3,Malfatti (1731–1807) have the formulas [4] to obtain ε_i for everym. He proves that these formulas verify the resultant equation, but he does not show clearly that the moment equation is fulfilled too. In this paper we write the Malfatti formulas in a modern form, and we observe that it is possible to decompose them in two parts ε₁ and ε₂ (ε₂ form→-4). Moreover we demonstrate that ε₁ satisfies also the moment equation for everym, while we show that ε_i is solution also of this equation when one consider the pentagon and hexagon cases.

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Ricerca svolta con i fondi del M.P.I.—quota 40%. Gruppo di ricerca: Problemi di evoluzione nei solidi e nei fluidi.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

Moduli of non-dentability and the radon-nikodým property in banach spaces

We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given.     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.