Derivatives of Computable Functions

As is well known the derivative of a computable and C¹ function may not be computable. For a computable and C∞ function f, the sequence {f⁽ⁿ⁾} of its derivatives may fail to be computable as a sequence, even though its derivative of any order is computable. In this paper we present a necessary and sufficient condition for the sequence {f⁽ⁿ⁾} of derivatives of a computable and C^∞ function f to be computable. We also give a sharp regularity condition on an initial computable function f which insures the computability of its derivative f′.     

On Computability Theoretic Properties of Structures and Their Cartesian Products

In this paper we show that for any set X ⊆ ω there exists a structure 𝒜 that has no presentation computable in X such that 𝒜² has a computable presentation. We also show that there exists a structure 𝒜 with infinitely many computable isomorphism types such that 𝒜² has exactly one computable isomorphism type.     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

Computable choice functions for computable linear orderings

A choice set for a computable linear ordering is a set which contains one element from each maximal block of the ordering. We obtain a partial characterization of the computable linear order‐types for which each computable model has a computable choice set, and a full characterization in the relativized case; Every model of the linear order‐type α of degree ≤ d has a choice set of degree ≤ d iff α can written as a finite sum of order‐types, each of which either has finitely many blocks, or has order‐type n · η for some integer n.     

Controllability of the wave equation with moving point control

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given timeinterval [0,T] the existence of the curves providing approximate controllability inH _D ^–[n/2]–1 ()×H _D ^–[n/2]–1 () (wheren stands for the space dimension) is established with controls fromL ²(0,T; R ⁿ +1). The same curves ensure exact controllability inL ²() × H^–1() if controls are allowed to be selected in [L (0,T; R ⁿ+1)]. Required curves can be constructed to be continuous on [0,T).This work was supported in part by NSF Grant ECS 89-13773 and NASA Grant NAG-1-1081.     

A generalization of a conservativity theorem for classical versus intuitionistic arithmetic

A basic result in intuitionism is Π⁰₂‐conservativity. Take any proof p in classical arithmetic of some Π⁰₂‐statement (some arithmetical statement ?x.?y.P(x, y), with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ?x.?y.P(x, y), with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, we get the original conservativity result as particular case. This result was a by‐product of a semantical construction. J. Avigad of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman's A‐translation. His proof is included here with his permission. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strichartz Estimates for Wave Equations with Coefficients of Sobolev Regularity

Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. We use such methods to show that the existing results for C ^{1, 1} and C ^{1, α} coefficients can be improved when the coefficients of the wave operator lie in a Sobolev space of sufficiently high order.     

Upper Bounds for Neumann–Schatten Norms

Let H and H _aux be Hilbert spaces, H a nonnegative self-adjoint operator in H,,s>0 and J a bounded linear transformation from the Hilbert space D(H ^s/2) (equipped with the graph scalar product of H ^s/2) to H _aux. It is shown that the operator J(H+)^–t belongs to the Neumann–Schatten class of order p=2+2(u–t)/(t–s/2) provided s/2<t<u,t–s/2<u–t and J(H+)^–u is Hilbert–Schmidt operator. An upper bound for the pth order Neumann–Schatten norm of J(H+)^–t is derived. If J is a closed operator from D(H ^1/2) to H _aux and D(J)D(H) then there exists a unique self-adjoint operator H ^J in H such that D(H ^J )D(J) and ( . Conditions which are sufficient in order that the operator (H ^J +)^–1–(H+)^–1 is compact and conditions which are sufficient in order that the wave operators W ^±(H ^J ,H) exist and are complete are derived. Instead of (Jf,Jg)_aux also certain other perturbation terms, not by necessity nonnegative, are considered. The special case when H equals the operator (–) ^r in L ²(R ^d ) for any strictly positive real number and H ^J equals (–) ^r + for some suitably chosen measure is discussed in detail. In particular, new results on existence and completeness of the wave operators W ^±(–+,–) are obtained.     

Inverse Obstacle Problem for the Non-Stationary Wave Equation with an Unknown Background

We consider boundary measurements for the wave equation on a bounded domain M ? ?² or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in M. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For τ ∈C(?M) the domain of influence M(τ) is the set of those points on the manifold from which the distance to some boundary point x is less than τ(x).     

Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

Let A = k^[3] be the polynomial ring in three variables over a field k, and let D be a nontrivial locally finite iterative higher derivation on A. Let A^D denote the kernel of D. In this note, we prove that, if chark > 0 and ML(A^D) ≠ A^D, then A^D ? k^[2]. As a consequence of this result, we give another proof of the cancellation theorem for k^[2] over any field k of positive characteristic.     

Complexity of Categorical Theories with Computable Models

M. Lerman and J. Scmerl specified some sufficient conditions for computable models of countably categorical arithmetical theories to exist. More precisely, it was shown that if T is a countably categorical arithmetical theory, and the set of its sentences beginning with an existential quantifier and having at most n+1 alternations of quantifiers is _n+1 ⁰ for any n, then T has a computable model. J. Night improved this result by allowing certain uniformity and omitting the requirement that T is arithmetical. However, all of the known examples of theories of₀-categorical computable models had low level of algorithmic complexity, and whether there are theories that would satisfy the above conditions for sufficiently large n was unknown. This paper will include such examples.     

On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces

Let D??³ be a bounded domain with connected boundary δD of class C². It is shown that Herglotz wave functions are dense in the space of solutions to the Helmholtz equation with respect to the norm in H¹(D) and that the electric fields of electromagnetic Herglotz pairs are dense in the space of solutions to curl curl E=k²E with respect to the norm in H_curl(D). Two proofs are given in each case, one based on the denseness of the traces of Herglotz wave functions on δD and the other on variational methods. Copyright © 2001 John Wiley & Sons, Ltd.     

The isotropy groups of bounded homogeneous domains are nontrivial

It will be shown in this paper that the automorphism group of a bounded homogeneous domain D in ℂ ⁿ can never act freely on $D$. An equivalent statement is that the isotropy groups of bounded homogeneous domains always contain at least two elements. Received: 7 May 1998     

Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials

In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L² norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L^∞ superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011     

Moment Propagation for Weak Solutions to the Vlasov–Poisson System

We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ?³. In the periodic case, we show that velocity moments of order > 14/3 are propagated.     

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.     

Complete surfaces in the hyperbolic space with a constant principal curvature

In this paper we study complete orientable surfaces with a constant principal curvature R in the 3‐dimensional hyperbolic space H ³. We prove that if R² > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H ³. When R² ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Projective Valuations,D-Relations,and Ends of Λ-Trees

Abstract

A projective valuation on a set Eis a mapping w : E ⁴ → Λ ∪ {±∞}, where Λ is an ordered abelian group, satisfying certain axioms. A D-relation on Eis a four-place relation on E, again with certain properties. There is a projective valuation on the set of ends of a Λ-tree (and on any subset, by restriction) and we show, using a construction suggested by Tits in the case Λ = ?, that every projective valuation arises in this way. Every projective valuation wdefines a D-relation, and there is a simple geometric interpretation of the D-relation, given a Λ-tree defining w. Our main result is a converse, that any D-relation can be defined by a projective valuation, hence arises from an embedding into the set of ends of a Λ-tree.     

Generalized convex kernels of simply connected <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets in the plane

Let D be a compact, simply connected subset of ℜ². Assume that every two points of D can be connected by a polygonal line with at most n edges within D. Then there is a point q ∈ D that can be connected to any other point in D by a polygonal line with at most n edges. This is best possible for all n.