Separating convex sets in the plane

Given a setA inR ² and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane.We prove that, for any collectionF ofn disjoint disks inR ², there is a lineL that separates a disk inF from a subcollection ofF with at least (n–7)/4 disks. We produce configurationsH _n andG _n , withn and 2n disks, respectively, such that no pair of disks inH _n can be simultaneously separated from any set with more than one disk ofH _n , and no disk inG _n can be separated from any subset ofG _n with more thann disks.We also present a setJ _m with 3m line segments inR ², such that no segment inJ _m can be separated from a subset ofJ _m with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least n/3+1 elements ofF.     

Preparation of papers

LetF be a family of real-valued maps onR ⁿ, and letY be a subset ofR ⁿ. Denote byS(Y|F) the set of ally* Y such that, for somef F,f(y)f(y*) for ally inY. Let us say thatF is a scalarization family if, for any subsetY,S(Y|F) is equal to the set of properly efficient points inY. General conditions forF to be a scalarization family were given in Ref. 1. However, scalarization families must contain nondifferentiable functions. In this note, it is shown that, if the condition of Ref. 1 which forces nondifferentiability is dropped, thenS(Y|F) is dense in the set of properly efficient points.     

Bi-Lipschitz concordance implies bi-Lipschitz isotopy

We modify the proof of an earlier result of ours to deforming topological, bi-Lipschitz, and quasiconformal embeddings of an open subsetU ofR ⁿ which now are of small uniform distance from the inclusion map. As an application we show that two bi-Lipschitz homeomorphismsf ₀,f ₁:R ⁿRⁿ are bi-Lipschitz isotopic if and only ifd(f ₀,f ₁)<.Research supported in part by a grant from the Institut Mittag-Leffler.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

Study of linear information for classes of polynomial equations

Summary We study linear sequential (adaptive) information for approximating zeros of polynomials of unbounded degree and establish a theorem on constrained approximation of smooth functions by polynomials.For a positive we seek a pointx ^* such that|x ^* – _p | , where _p is a zero of a real polynomialp in the interval [a, b]. We assume thatp belongs to the classF ₁ of polynomials of bounded arbitrary seminorm and having a root in [a, b] or to the classF ₂ of polynomials which are nonpositive ata, nonnegative atb and have exactly one simple zero in [a, b]. The information onp consists ofn sequential (adaptive) evaluations of arbitrary linear functionals. The pointx ^* is constructed by means of an algorithm which is an arbitrary mapping depending on the information onp. We show that there exists no information and no algorithm for computingx ^* for everyp fromF ₁, no matter how large the value ofn is. This is a stronger result than that obtained by us for smooth functions.For the classF ₂ we can find a pointx ^* for arbitraryp and. Anoptimal algorithm, i.e., an algorithm with the smallest error, is thebisection of the smallest known interval containing the root ofp. We also exhibitoptimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is minimal. It turns out that in the class of nonsequential (parallel) information, i.e., when the functionals are given simultaneously, optimal information consists of the evaluations of a polynomial atn-equidistant points in [a, b]. In the class of sequential continuous information, optimal information consists of evaluations of a polynomial atn points generated by thebisection method. To prove this result we establish a theorem on constrained approximation of smooth functions by polynomials. More precisely, we prove that a smooth function can be arbitrarily well uniformly approximated by a polynomial which satisfies constrains given byn arbitrary continuous linear functionals.Our results indicate that the problem of finding an -approximation to a real zero of a real polynomial (of unknown degree) is essentially of the same difficulty as the problem of finding an -approximation to a zero of an infinitely differentiable function.     

Optimal control problems for distributed parameter systems in Banach spaces

We consider the infinite-dimensional nonlinear programming problem of minimizing a real-valued functionf ₀ (u) defined in a metric spaceV subject to the constraintf(u) Y, wheref(u) is defined inV and takes values in a Banach spaceE and Y is a subset ofE. We derive and use a theorem of Kuhn-Tucker type to obtain Pontryagin's maximum principle for certain semilinear parabolic distributed parameter systems. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations inL ¹ andL spaces.This work was supported in part by the National Science Foundation under Grant DMS-9001793.     

Calibrations and the size of Grassmann faces

Summary In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure ofm-dimensional area-minimizing surfaces inR ⁿ . Calibrations correspond to faces of the GrassmannianG(m,R ⁿ ) of orientedm-planes inR ⁿ , viewed as a compact submanifold of the exterior algebra ^m R ⁿ . Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands more general than area. One result implies that form-dimensional surfaces inR ⁿ , with 2 m n – 2, for any integrand , there are -minimizing surfaces with interior singularities.     

On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

A generalized conjugate gradient algorithm for minimization

Summary A generalized conjugate gradient algorithm which is invariant to a nonlinear scaling of a strictly convex quadratic function is described, which terminates after at mostn steps when applied to scaled quadratic functionsf: R _n R₁ of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andhC ¹ (R₁) an arbitrary strictly monotone functionh. The algorithm does not suppose the knowledge ofh orF but only off(x) and its gradientg(x).     

Embedding and unsolvability theorems for modular lattices

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ? be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ?(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ? ?, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ^*) having five generators, where δ^*=δ for infinite δ and δ^*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ?(R)-lattice on ω generators is embeddable in the free ?(R)-lattice on five generators. THEOREM. IfL has an (m, n), ?(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ?(R)-presentation.     

Symmetry and-commutativity of topological*-algebras

An advertibly complete locallym-convex (lmc)^*-algebraE is symmetric if and only if each normed (inverse limit) factorE/N , A, ofE is symmetric in the respective Banach factorE , A, ofE. Every locally C^*-algebra is symmetric. If denotes the continuous positive functionals on an lmc^*-algebraE and withL _f ={x E: f(x ^* x) =0}, thenE is, by definition,-commutative if for anyx, y E.-commutativity and commutativity coincide in lmcC ^*-algebras, so that an lmc^*-algebra with a bounded approximate identity is-commutative if and only if its enveloping algebra is commutative. Several standard results for commutative lmc^*-algebras are also obtained in the-commutative case, as for instance, the nonemptiness of the Gel'fand space of a suitable-commutative lmc^*-algebra, the automatic continuity of positive functionals when the algebras involved factor, as well as that the spectral radius is a continuous submultiplicative semi-norm, when the algebras considered are moreover symmetric. An application of the latter result yields a spectral characterization of-commutativity.     

On the limit behaviour of the joint distribution function of order statistics

Fork ₀ fixed we consider the joint distribution functionF _n ^k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF _n ^k (x ₁,,x _n-k) in terms of the limit behaviour ofn(1-F(x _n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution functionF.     

An implicit function theorem: Comment

In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:R ⁿ ×R ^m R ⁿ, withF(x ⁰,y ⁰)=0, that requires neither differentiability ofF nor nonsingularity of _x F(x ⁰,y ⁰). In the proof, the local one-to-one condition forF(·,y):A R ⁿ R ⁿ for ally B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that the condition is truly if and only if.     

Univalent mappings associated with the Roper-Suffridge extension operator

The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfεH(U) to a (locally) biholomorphic mappingF∈H(B_n). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iff∈S ^*, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB _n. Finally, we investigate some open problems. Partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A9221.     

Gaussian kernels have only Gaussian maximizers

A Gaussian integral kernelG(x, y) onR ⁿ ×R ⁿ is the exponential of a quadratic form inx andy; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound ofG as an operator fromL ^p (R ⁿ ) toL ^p (R ⁿ ) and to prove that theL ^p (R ⁿ ) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1<pq< and also forp>q in some special cases. Besides greatly extending previous results in this area, the proof technique is also essentially different from earlier ones. A corollary of these results is a fully multidimensional, multilinear generalization of Young's inequality.Oblatum 19-XII-1989Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A03     

Near-rings with no non-zero nilpotent two-sidedR-subsets

It has been proved that, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and if the annihilator of any non-zero ideal is contained in some maximal annihilator, thenR is a subdirect sum of strictly prime near-rings. Moreover, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and satisfying a.c.c. or d.c.c. on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite subdirect sum of strictly prime near-rings. It is also proved that, ifR is a regular and right duo near-ring that satisfies a.c.c. (or d.c.c.) on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite direct sum of near-ringsR _i (1 i n) where eachR _i is simple and strictly prime.     

The projection theorem for spectral sets

LetG be a locally compact group with polynomial growth and symmetricL ¹-algebra andN a closed normal subgroup ofG. LetF be a closedG-invariant subset of Prim_* L ¹(N) andE={ker; with |N(k(F))=0}. We prove thatE is a spectral subset of Prim_* L ¹(G) ifF is spectral. Moreover we give the following application to the ideal theory ofL ¹(G). Suppose that, in addition,N is CCR andG/N is compact. Then all primary ideals inL ¹(G) are maximal, provided allG-orbits in Prim_* L ¹(N) are spectral.Dedicated to Professor Elmar Thoma on the occasion of his 60th birthday     

A note on reflexivity in Banach spaces

For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX ^**. This says that for any ε>0, for anyF in the unit ball ofX ^**, and for anyf ₁,…,f _n inX ^*, the system of inequalities |f _i(x)?F(f _i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.     

On bending of a convex surface to a convex surface with prescribed spherical image

We prove the following theorem. LetF be a regular convex surface homeomorphic to the disk. Suppose the Gaussian curvature ofF is positive and the geodesic curvature of its boundary is positive as well. LetG be a convex domain on the unit sphere bounded by a smooth curve and strictly contained in a hemisphere. LetP be an arbitrary point on the boundary ofF andP ^* be an arbitrary point on the boundary ofG. If the area ofG is equal to the integral curvature of the surfaceF, then there exists a continuous bending of the surfaceF to a convex surfaceF such that the spherical image ofF coincides withG andP ^* is the image of the point inF corresponding to the pointP F under the isometry.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 295–300, August, 1995.