Shape Optimization for Stokes Flows using Conformal Metrics

We derive a shape optimization approach for a two-dimensional Stokes flow problem. Our goal is to compute a geometry whose wall shear stress is L₂-close to a desired target stress. Computations are carried out on a fixed domain, while shape variations are described through conformal metric changes. Tools from differential geometry are used to handle metric dependent operators. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous dependence on the time and spatial geometry for the equations of thermoelasticity

In this paper we establish the stability of the solution of the standard initial-boundary value problem of linear anisotropic thermoelasticity under perturbations of the initial time geometry and of the spatial geometry. This is done by deriving appropriate explicit a priori inequalities which permit us to bound in particular the L₂ integral of the perturbation in terms of some well defined measure of the perturbation in the geometry.     

Hilbert metrics and Minkowski norms

It is shown that the Hilbert geometry (D, h_D) associated to a bounded convex domain is isometric to a normed vector space if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert’s metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.     

Faber-hypercyclic operators

Let Ω be a bounded domain of the complex plane whose boundary is a closed Jordan curve and (F _n ) _n≥0 the sequence of Faber polynomials of Ω. We say that a bounded linear operator T on a separable Banach space X is Ω-hypercyclic if there exists a vector x of X such that {F _n (T)x: n ≥ 0} is dense in X. We show that many of the results in the spectral theory of hypercyclic operators involving the unit disk or its boundary have Ω-hypercyclic counterparts which involve the domain Ω or its boundary. The influence of the geometry of Ω or the smoothness of its boundary on Faber-hypercyclicity is also discussed.     

On the Volume of a Domain Obtained by a Holomorphic Motion Along a Complex Curve

Let D be a domain obtained by a holomorphic motion of a domain D _p M _p ^n–1 along a complex curve P in a complex space form M ⁿ . We prove that, if n= 2, the volume of D depends only on the geometry of D _p and the intrinsic geometry of P, but not on the extrinsic geometry of P. When M is closed (compact without boundary), then the dependence on P is only through its topology. When n > 2, and for arbitrary domains D _p, a similar result holds only for Frenet motions, but when D _p has certain integral symmetries (and only in this case) this result is still true for any motion .     

Asymptotic analysis and scaling of friction parameters

We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale collinear faults periodically disposed. If β₀^* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic problem behaves as β₀^*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result is that the macroscopic critical slip D_c scales with D_c^∈/∈ (here D_c^∈ is the small scale critical slip).     

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ ₁^Ω(α) corresponding to Δu = λ (α) u in W, \frac?u?v = au \Omega, \frac{\partial u}{\partial v} = \alpha u on ∂Ω, with α a fixed real, and W ì Rⁿ\Omega \subset {\mathcal{R}}^n a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ₁^Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.     

Special domains and nonmanipulability

We investigate domains on which a nonmanipulable, nondictatorial social choice function exists, having at least three distinct values. We do not make the assumptions of Kalai and Muller (1977). We classify all such 2-person functions on the domain which is the cyclic group Z_m. We show that for any domain containing Z_m, existence for 2 voters and existence for some n > 2 voters are equivalent. We show that for an n-person, onto, nonmanipulable social choice function F on Z_m, F(P₁, P₂,…, P_n) {x₁, x₂,…, x_n} always, x_i being the most preferred alternative under preference P_i. We show that no domain containing the dihedral group admits such a social choice function. We show that there exists a domain on which all k-tuples are free for arbitrarily large k, for which such a social choice function does exist.     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-estimation for the location parameters in stochastic volatility models

L ₁-estimation of a location parameter is studied for the “product type” stochastic volatility models. The asymptotic distribution of the L ₁-estimator is established under general conditions on the behavior of the distribution function of the errors near zero.     

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ ₁^Ω(α) corresponding to Δu = λ (α) u in on ∂Ω, with α a fixed real, and a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ₁^Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable. Robert Smits: This author was partially supported by a grant of the National Security Agency, grant #H98230-05-1-0060.     

Totally ordered sets and the prime spectra of rings

Let T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain O_v such that T is order isomorphic to two special subsets of Spec(O_v). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U?Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.     

Geometry of Mean Value Sets for General Divergence Form Uniformly Elliptic Operators

In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point x₀ in the domain, there exists a nested family of sets {D_r(x₀)} where the average over any of those sets is related to the value of the function at x₀. Although it is known that the {D_r(x₀)} are nested and are comparable to balls in the sense that there exists c,C depending only on L such that B_cr(x₀) ? D_r(x₀) ? B_Cr(x₀) for all r >?0 and x₀ in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.

     

Valuation Extensions of Filtered and Graded Algebras

In this note we relate the valuations of the algebras appearing in the noncommutative geometry of quantized algebras to properties of sublattices in some vector spaces. We consider the case of algebras with PBW-bases and prove that under some mild assumptions, the valuations of the ground field extend to a noncommutative valuation. Later we introduce the notion of F-reductor and graded reductor and reduce the problem of finding an extending noncommutative valuation to finding a reductor in an associated graded ring having a domain for its reduction.     

Straight Rings

A (commutative integral) domain is called a straight domain if A ? B is a prime morphism for each overring B of A; a (commutative unital) ring A is called a straight ring if A/P is a straight domain for all P ∈ Spec(A). A domain is a straight ring if and only if it is a straight domain. The class of straight rings sits properly between the class of locally divided rings and the class of going-down rings. An example is given of a two-dimensional going-down domain that is not a straight domain. The classes of straight rings, of locally divided rings, and of going-down rings coincide within the universe of seminormal weak Baer rings (for instance, seminormal domains). The class of straight rings is stable under formation of homomorphic images, rings of fractions, and direct limits. The “straight domain" property passes between domains having the same prime spectrum. Straight domains are characterized within the universe of conducive domains. If A is a domain with a nonzero ideal I and quotient field K, characterizations are given for A ? (I: _K I) to be a prime morphism. If A is a domain and P ∈ Spec(A) such that A _P is a valuation domain, then the CPI-extension C(P) := A + PA _P is a straight domain if and only if A/P is a straight domain. If A is a going-down domain and P ∈ Spec(A), characterizations are given for A ? C(P) to be a prime morphism. Consequences include divided domain-like behavior of arbitrary straight domains.     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of H?lder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane f(z) = ?_n=0^￥ a_n X_n zⁿf(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}, where the X _n ’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a _n are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and Ef(z)[`(f(w))]\mathbf{E}f(z)\overline{f(w)} is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.     

p-harmonic functions on graphs and manifolds

We show that the LiouvilleD _p -property is invariant under rough isometries between a Riemannian manifold of bounded geometry and a graph of bounded degree. The first author was supported partly by the EU HCM contract No. CHRX-CT92-0071. This article was processed by the author using the Springer-Verlag TEX P Jourlg macro package 1991.     

Aubry-mather theory and the inverse spectral problem for planar convex domains

The inverse spectral problem is concerned with the question to what extent the spectrum of a domain determines its geometry. We find that, associated to a convex domain Ω in ℝ², there is a convexfunction which is a length spectrum invariant under continuous deformations. It includes several geometric quantities, such as the lengths and Lazutkin parameters of caustics, as well as the asymptotic invariants discovered by Marvizi and Melrose. Via a Poisson relation, we also find invariants determined by the Laplace spectrum of Ω.