A posteriori analysis of the finite element discretization of some parabolic equations

We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes.

RÉSUMÉ. Nous considérons la discrétisation d'équations paraboliques, soit linéaires soit semi-linéaires, par un schéma d'Euler implicite en temps et par éléments finis en espace. L'idée de cet article est de construire des indicateurs d'erreur liés à l'approximation en temps et en espace et de prouver leur équivalence avec l'erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d'éléments finis adaptés à la solution.

     

Noise sensitivity of Boolean functions and applications to percolation

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on ann+1 byn grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges ℓ with ω(ℓ)=1. By duality, the probability for having a crossing is 1/2. Fix an ɛ ∈ (0, 1). For each edge ℓ, let ω′(ℓ)=ω(ℓ) with probability 1 − ɛ, and ω′(ℓ)=1 − ω(ℓ) with probability ɛ, independently of the other edges. Letp(τ) be the probability for having a crossing in ω, conditioned on ω′ = τ. Then for alln sufficiently large,P{τ : |p(τ) − 1/2| > ɛ}<ɛ.     

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

An asymptotic model is found for the Neumann problem for the second-order differential equation with piecewise constant coefficients in a composite domain Ω∪ω, which are small, of order ε, in the subdomain ω. Namely, a domain Ω(ε) with a singular perturbed boundary is constructed, the solution for which provides a two-term asymptotic, that is, of increased accuracy O(ε²| log ε|^3/2), approximation to the restriction to Ω of the solution of the original problem. As opposed to other singularly perturbed problems, in the case of contrasting stiffness, the modeling requires the construction of a contour ∂Ω(ε) with ledges, i.e., with boundary fragments of curvature O(ε⁻¹). Bibliography: 33 titles.     

Algebraic polynomials least deviating from zero in measure on a segment

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment [–1, 1] with respect to a measure, or, more precisely, with respect to the functional μ(f) = mes{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1}. We also discuss an analogous problem with respect to the integral functionals ∫_–1¹ φ (∣f (x)∣) dx for functions φ that are defined, nonnegative, and nondecreasing on the semiaxis [0, +∞).     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

Self-injective rings and linear (weak) inverses of linear finite automata over rings

LetR be a finite commutative ring with identity and τ be a nonnegative integer. In studying linear finite automata, one of the basic problems is how to characterize the class of rings which have the property that every (weakly) invertible linear finite automaton ℳ with delay τ over R has a linear finite automaton ℳ′ over R which is a (weak) inverse with delay τ of ℳ. The rings and linear finite automata are studied by means of modules and it is proved that *-rings are equivalent to self-injective rings, and the unsolved problem (for τ=0) is solved. Moreover, a further problem of how to characterize the class of rings which have the property that every invertible with delay τ linear finite automaton ℳ overR has a linear finite automaton ℳ′ over R which is an inverse with delay τ′ for some τ′⩾τ is studied and solved. Project supported by the National Natural Science Foundation of China(Grant No. 69773015).     

Why vagueness is a mystery

This paper considers two “mysteries” having to do with vagueness. The first pertains to existence. An argument is presented for the following conclusion: there are possible cases in which ‘There exists something that is F’ is of indeterminate truth-value and with respect to which it is not assertable that there are borderline-cases of “being F.” It is contended that we have no conception of vagueness that makes this result intelligible. The second mystery has to do with “ordinary” vague predicates, such as ‘tall’. An argument is presented for the conclusion that although there are people who are “tall to degree 1”—definitely tall, tall without qualification—, no greatest lower bound can be assigned to the set of numbers n such that a man who is n centimeters tall is tall to degree 1. But, since this set is bounded from below, this result seems to contradict a well-known property of the real numbers.     

On the local convergence of a family of Euler-Halley type iterations with a parameter

This paper aims to study the local convergence of a family of Euler-Halley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r _α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ ( − ∞ , + ∞ ). For each α > 0, we get r _α  ≥ r _− α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r _α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α→0⁻. Results show that the method corresponding to α is better than the one corresponding to − α for each α > 0 and the Chebyshev-Euler method is the best among all methods in the family with α ∈ ( − ∞ , 0] from the view of both safe choice of the initial point and error estimation.     

Harnack Inequalities for some Lévy Processes

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ ^d . We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process in ℝ ^d with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr ^{− d − α} , ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent ϕ(λ) = λ ^α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).     

Maximizing a congruence with respect to its partition of idempotents

For a congruence σ on a semigroupS a congruence μ(σ) onS, containing σ, is defined such that the semigroupS/σ is fundamental if and only if σ=μ(σ). The congruence μ(σ) is shown to possess maximality properties and for idempotent-surjective semigroups, μ(σ) is the maximum congruence with respect to the partition of the idempotents determined by σ. Thus μ is the maximum idempotent-separating congruence on any idempotent-surjective semigroup. It is shown that μ(μ(σ))=μ(σ). If ρ is another congruence onS, possibly with the same partition of the idempotents as σ, then it is of interest to know when ρ⊆σ (or ρ⊆μ(σ)) implies μ(ρ)⊆μ(σ) or even μ(ρ)=μ(σ). These implications are not true in general but if σ⊆ρ⊆μ(σ) then μ(ρ)⊆μ(σ). IfS is an idempotent-surjective semigroup and ρ and σ have the same partition of the idempotents then μ(ρ)=μ(σ).     

On the embedding problem with non-Abelian kernel of orderp 4. IV

In the present paper, the embedding problem is considered for number fields with p-groups whose kernel is either of two groups with two generators α and β and with the following relations: (1) α^ρ=1, α^ρ=1, [α,β,β]=1, [α,β,α,α]=1, or (2) α^ρ=[α, β α], β^ρ=1, [α,β,β]=1. It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 120–126. Translated by V. V. Ishkhanov.     

Constructing greedy linear extensions by interchanging chains

A natural way to prove that a particular linear extension of an ordered set is ‘optimal’ with respect to the ‘jump number’ is to transform this linear extension ‘canonically’ into one that is ‘optimal’. We treat a ‘greedy chain interchange’ transformation which has applications to ordered sets for which each ‘greedy’ linear extension is ‘optimal’.     

W 2,p -a priori estimates for the emergent Poincaré Problem

We derive W ^2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.      

Frequency-Domain Bounds for Nonnegative,Unsharply Band-Limited Functions

In this paper we present a technique for proving bounds of the Boas-Kac-Lukosz type for unsharply restricted functions with nonnegative Fourier transforms. Hence we consider functions F(x) ≥ 0, the Fourier transform f(u) of which satisfies |f(u)| ≤ ε for all u in a subset of (-∞,-1] ⋃ [1,∞), and are interested in bounds on |f(u)| for |u| ≤ 1. This technique gives rise to several "epsilonized" versions of the Boas-Kac-Lukosz bound (which deals with the case f(u) = 0, |u| ≥ 1). For instance, we find that |f(u)| ≤ L(u) + O(ε^2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. The technique also turns out to be sufficiently powerful to yield the best bound as ε ↓ 0 in various other cases with less severe restrictions on f.     

Selfadjoint extensions of the operator of the Dirichlet problem in a 3-dimensional region with an edge

Taking various viewpoints, we study the selfadjoint extensions $ \mathcal{A} $ \mathcal{A} of the operator A of the Dirichlet problem in a 3-dimensional region Ω with an edge Γ. We identify the infinite dimensional nullspace def A with the Sobolev space H ^−ϰ(Γ) on Γ with variable smoothness exponent −ϰ ∈ (−1, 0); while the selfadjoint extensions, with selfadjoint operators $ \mathcal{T} $ \mathcal{T} on the subspaces of H ^−ϰ(Γ). To the boundary value problem in the region with a “smoothed” edge we associate a concrete extension, which yields a more precise approximate solution to the singularly perturbed problem.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

The duality of quantum Liouville field theory

It has been found empirically that the Virasoro center and three-point functions of quantum Liouville field theory with the potential exp(2bϕ(x)) and the external primary fields exp(αϕ(x)) are invariant with respect to the duality transformations ℏα→q−α, where q=b⁻¹+b. The steps leading to this result (via the Virasoro algebra and three-point functions) are reviewed in the path-integral formalism. The duality occurs because the quantum relationship between the α and the conformal weights Δ_α is two-to-one. As a result, the quantum Liouville potential can actually contain two exponentials (with related parameters). In the two-exponential theory, the duality appears naturally, and an important previously conjectured extrapolation can be proved. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 299–307, May, 2000.     

A generalization of Steiner symmetrization for immersed surfaces and its applications

We generalize the classical Steiner symmetrization to surfaces with self-intersections. Then we apply the generalized Steiner symmetrization to several isoperimetric problems. For example, let Г⊂ℝ³ be an analytic plane Jordan curve which is symmetric with respect to a plane ϖ (ϖ⊅Г). LetS be a compact immersed surface bounded by Λ which has the smallest area among all compact surfaces bounded by Λ with a fixed volume. In this situation, under some additional assumptions, the wholeS is proved to be symmetric with respect to ϖ. When Λ is a round circle,S is proved to be a spherical cap or the flat disk bounded by Λ without any additional assumptions. Dedicated to Professor Hideki Ozeki on his 60th birthday This article was processed by the author using the Springer-Verlag T_EX P Jourlg macro package 1991.     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.