Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets Λ⊂ℝ ^d with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. For our main result, we introduce the notion of algebraic Delone sets Λ⊂ℝ² and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.     

Platonic polyhedra,topological constraints and periodic solutions of the classical <Emphasis Type="Italic">N</Emphasis>-body problem

We prove the existence of a number of smooth periodic motions u _∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group R\mathcal{R} of one of the five Platonic polyhedra. The number N coincides with the order |R||\mathcal{R}| of R\mathcal{R} and the particles have all the same mass. Our approach is variational and u _∗ is a minimizer of the Lagrangian action A\mathcal{A} on a suitable subset K\mathcal{K} of the H ¹ T-periodic maps u:ℝ→ℝ^3N . The set K{\mathcal {K}} is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group R\mathcal{R}. There exist infinitely many such cones K{\mathcal {K}}, all with the property that A|_K{\mathcal {A}}|_{{\mathcal {K}}} is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure.     

Polygons inscribed in a closed curve and a three-dimensional convex body

Here are samples of results obtained in the paper. Let γ be a centrally symmetric closed curve in ℝ ⁿ that does not contain its center of symmetry, O. Then γ is circumscribed about a square (with center O), as well as about a rhombus (also with center O) whose vertices split γ into parts of equal length. If n is odd, then there is a centrally symmetric equilateral 2n-link polyline inscribed in γ and lying in a hyperplane. Let K ⊂ ℝ³ be a convex body, and let x ∈ (0; 1). Then K is circumscribed about an affine-regular pentagonal prism P such that the ratio of the lateral edge l of P to the longest chord of K parallel to l is equal to x. Bibliography: 7 titles.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Partial convexity

We consider a generalization of the classical notion of convexity, which is calledpartial convexity. LetV ∋ ℝ ⁿ be some set of directions. A setX ∋ ℝ ⁿ is calledV- convex if the intersection of any line parallel to a vector inV withX is connected. Semispaces and the problem of the least intersection base for partial convexity is investigated. The cone of convexity directions is described for a closed set in ℝ ⁿ . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 406–413, September, 1996. The authors wish to express their gratitude to V. V. Gorokhovik and E. A. Barabanov for useful remarks and discussions. This research was supported in part by the Belorussian Foundation for Basic Research under grant No. F95-016.     

Restricted Isometry Property of Matrices with?Independent Columns and Neighborly Polytopes by?Random Sampling

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X ₁,…,±X _N ∈ℝ ⁿ , (N≥n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a subexponential tail inequality possess the restricted isometry property with overwhelming probability. We show that such “sensing” matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ ₁ minimization with m≤Cn/log ²(cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K⊂ℝ ⁿ is a convex body and X ₁,…,X _N ∈K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m∼n/log ²(cN/n).     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

The present paper studies the following constrained vector optimization problem: min  _C f(x), g(x)∈−K, h(x)=0, where f:ℝ ⁿ →ℝ ^m , g:ℝ ⁿ →ℝ ^p and h:ℝ ⁿ →ℝ ^q are locally Lipschitz functions and C⊂ℝ ^m , K⊂ℝ ^p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x ⁰ to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.     

Convex integration with constraints and applications to phase transitions and partial differential equations

We study solutions of first order partial differential relations Du∈K, where u:Ω⊂ℝ ⁿ →ℝ ^m is a Lipschitz map and K is a bounded set in m×n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du and second we replace Gromov’s P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite. Received April 23, 1999 / final version received September 11, 1999