Reconstructing Geometric Objects from the Measures of Their Intersections with Test Sets

Let us say that an element of a given family $\mathcal{A}$ of subsets of ? ^d can be reconstructed using n test sets if there exist T ₁,…,T _n ?? ^d such that whenever $A,B\in\mathcal{A}$ and the Lebesgue measures of A∩T _i and B∩T _i agree for each i=1,…,n then A=B. Our goal will be to find the least such n. We prove that if $\mathcal{A}$ consists of the translates of a fixed reasonably nice subset of ? ^d then this minimum is n=d. To obtain this we prove the following two results. (1) A translate of a fixed absolutely continuous function of one variable can be reconstructed using one test set. (2) Under rather mild conditions the Radon transform of the characteristic function of K (that is, the measure function of the sections of K), (R _θ χ _K )(r)=λ ^d?1(K∩{x∈? ^d :〈x,θ〉=r}) is absolutely continuous for almost every direction θ. These proofs are based on techniques of harmonic analysis. We also show that if $\mathcal{A}$ consists of the enlarged homothetic copies rE+t (r≥1,t∈? ^d ) of a fixed reasonably nice set E?? ^d , where d≥2, then d+1 test sets reconstruct an element of $\mathcal{A}$ , and this is optimal. This fails in ?: we prove that a closed interval, and even a closed interval of length at least 1 cannot be reconstructed using two test sets. Finally, using randomly constructed test sets, we prove that an element of a reasonably nice k-dimensional family of geometric objects can be reconstructed using 2k+1 test sets. An example from algebraic topology shows that 2k+1 is sharp in general.     

Algorithms for multiplayer multicommodity flow problems

We investigate the multiplayer multicommodity flow problem: several players have different networks and commodities over a common node set. Pairs of players have contracts where one of them agrees to route the flow of the other player (up to a given capacity) between two specified nodes. In return, the second player pays an amount proportional to the flow value. We show that the social optimum can be computed by linear programming, and we propose algorithms based on column generation and Lagrangian relaxation. In contrast, we prove that it is hard to decide if an equilibrium solution exists, although some natural conditions guarantee its existence.     

Lebesgue decomposition of contents via nonnegative forms

We present a new approach for the Lebesgue decomposition of finitely additive measures (or contents, for short). Using the recently proved results of Hassi, Sebestyén and de Snoo, we show that the Lebesgue decomposition of contents exists, and corresponds to the Lebesgue decomposition of their induced forms. Additionally, we also present some new results related to the almost dominated part of a form (resp., a content).     

Tail Asymptotics of Free Path Lengths for the Periodic Lorentz Process: On Dettmann’s Geometric Conjectures

In the simplest case, consider a \({\mathbb{Z}^d}\) -periodic (d ≥ 3) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann’s first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than t >  > 1 is \({\sim\frac{C}{t}}\) , where C is explicitly given by the geometry of the model. In its simplest form, Dettmann’s second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for \({\mathcal{L}}\) -periodic configuration of—possibly intersecting—convex bodies with \({\mathcal{L}}\) being a non-degenerate lattice. These questions are related to Pólya’s visibility problem (Arch Math Phys Ser 2:135–142, 1918), to theories of Bourgain et al. (Commun Math Phys 190:491–508,1998), and of Marklof–Strömbergsson (Ann Math 172:1949–2033,2010). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if d = 2 and the horizon is infinite.     

Revisiting a magneto-elastic strange attractor

We revisit an early example of a nonlinear oscillator that exhibits chaotic motions when subjected to periodic excitation: the magneto-elastically buckled beam. In the paper of Moons and Holmes (1980) [1] magnetic field calculations were outlined but not carried through; instead the nonlinear forces responsible for creation of a two-well potential and buckling were fitted to a polynomial function after reduction to a single mode model. In the present paper we compute the full magnetic field and use it to approximate the forces acting on the beam, also using a single mode reduction. This provides a complete model that accurately predicts equilibria, bifurcations, and free oscillation frequencies of an experimental device. We also compare some periodic, transient and chaotic motions with those obtained by numerical simulations of the single mode model, further illustrating the rich dynamical behavior of this simple electromechanical system.     

Design of Peptidic Foldamer Helices: A Stereochemical Patterning Approach

Assembly language : The programmed sequences of stereochemical building blocks lead to novel biomimetic helices. The rational design approach offers new possibilities for creating periodic secondary structures.

     

A relative approach for determining ring strain energies of heterobicyclic alkenes

Ring strain energies (RSEs) of heterocyclic compounds are predicted compared to their analogous homocyclic compounds using the G3 method. Suitable reference compounds are devised for row 2 and row 3 heterobicyclic alkenes. Any difference in energy between the ring and the reference is due to RSE. Row 3 heterobicyclic alkene RSEs are less than those of row 2. As the electronegativity of the heteroatom increases, the RSE increases. Substitutents at the bridgehead carbons cause a decrease in RSE. [3.2.1] Heterobicyclic alkenes are significantly less strained than their [2.2.1] counterparts. Relative and absolute RSEs are reported for heterobicyclic alkenes and their derivatives.     

Tempo-activated manganese-based catalase mimics

Catalase-like activity of the TEMPO-activated (TEMPO: 2,2,6,6-tetramethyl-1-piperidinyloxyl) manganese(II) complex Mn(IndH)Cl₂·CH₃OH (IndH: 1,3-bis[(2′-pyridylimino)isoindoline] is reported. The title compound was a suitable catalyst for the disproportionation of H₂O₂ into water and dioxygen under ambient conditions. Kinetic measurements resulted in the third-order rate equation d[O₂]/dt = k[H₂O₂][Mn(IndH)Cl₂·CH₃OH][TEMPO].     

Influence of Counteranion on the Chiral Supramolecular Assembly of Alkynylplatinum(II) Terpyridyl Metallogels That Are Stabilised by Pt⋅⋅⋅Pt and π–π Interactions

Helical superstructures : A series of luminescent chiral alkynylplatinum(II)–terpyridyl complexes has been successfully synthesised and shown to be capable of forming metallogels that show helical fibrous nanostructures. The chiral supramolecular structure and the spectroscopic properties are found to depend on the extent of aggregation through Pt???Pt and π–π interactions in various counteranions, as revealed by UV/Vis, circular dichroism (CD), and luminescence studies (see picture).