Comparison of Birch with electrochemical reduction of 2,3-diphenyl-1,4-diazaspiro[4.5]deca-1,3-diene

Reduction of 2,3-diphenyl-1,4-diazaspiro[4.5]deca-1,3-diene is investigated both voltammetrically (glassy-carbon electrode, 20 °C, non-aqueous-solvents) and using dissolving-metals (sodium, ?78 °C, tetrahydrofuran (THF)/NH₃(l)). Remarkably, electro-reduction furnishes two two-electron processes, whilst only a four-electron product results from Birch synthesis.     

Equivariant geometry in EuclideanG-spheres of cohomogeneity two,I

The techniques of equivariant geometry are applied to a specific problem in Riemannian geometry, namely the study of constant mean curvature (and minimal) hypersurfaces of the Euclidean sphereS ⁿ , subject to the constraint of being invariant with respect to a suitable connected symmetry groupG Iso(S ⁿ ), dimS ⁿ /G=2.The surfaces are studied by deriving and solving the reduced differential equation, which is an ODE with singularities on the boundary of the orbit space. ForG U (1) the orbit space is (topologically) a 2-disk with 0, 1, 2 or 3 vertices. The case of no vertex was settled around 1970 (Hsiang-Lawson) and the case of two fixed points as vertices was analyzed in the 1980s by Hsiang and his collaborators in their study of the spherical Bernstein problem.The case of one vertex is settled in this paper. The main problem is the determination of volume function and the orbital distance metric, leading to the associated ODE. For convenience, the authors also employ numerical simulation to describe the qualitative behavior of solution curves. However, this part of the proof can be avoided by a more careful but lengthy analysis which is roughly similar to previous works in this direction.     

Space-time domain decomposition for parabolic problems

We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.     

Antioxidant capacity of novel amine derivatives of buckminsterfullerene: Determination of inhibition rate constants in a model oxidation system

The radical scavenging efficiency of fullerenes can be significantly activated by means of a connection with hydrogen donating groups of antioxidants such as phenolic, amine and sulfhydryl. The developed system of conjugated σ–π bonds arranged in the fullerene molecule in a closed shape can promote a strong resonance effect on the grafted units and increase the hydrogen atom abstraction efficiency. In this case the known ability of fullerene to trap alkyl radicals might be combined with the strong chain breaking functionality to afford a new class of antioxidants with bimodal action. Explorations in the field of fullerene derivatives on their antioxidant performance provide novel information on the potential stabilization properties of this type of molecular structure. A series of amine derivatives of buckminsterfullerene (C₆₀) with tethered aliphatic chain, cycloaliphatic and aromatic fragments were synthesized and their antioxidant activity was determined. The antioxidant activity of the investigated derivatives was studied by measuring the inhibition rate constants for their reaction with alkyl and peroxy radicals in a model cumene initiated (2,2′-azobisisobutyronitrile, AIBN) oxidation experiments and compared to that recorded under identical experiments for buckminsterfullerene itself and commercial primary aromatic amine stabilizers. The results indicate that linking the amine moieties groups directly to the fullerene core gives rise to a new chain breaking antioxidant mode for the buckminsterfullerene while cyclic fragments containing the same but distant amine group do not reveal this ability. The inhibition rate constants for trapping of peroxy radicals by the amine derivatives were found to be higher than that of known aromatic amine antioxidants Neozone-D and Naugard 445. In addition the C₆₀ part of these molecules acts synergistically by trapping alkyl radicals with inhibition rate constants which exceed that of underivatised fullerene. The amine derivative containing sterically hindered piperidine and pyrrolidine fragments also heighten the inherent rate constant of buckminsterfullerene for scavenging alkyl radicals due to the additional antioxidant contribution promoted by the radical-quenching ability of the formed nitroxyl intermediates. These novel C₆₀–amine conjugates may be considered as promising molecules for broad-spectrum radical scavenging antioxidants.     

Global Geometry of Planary 3-Body Motions

The planary 3-body problem is investigated in the framework of equivariant Riemannian geometry, where the global geometry of the trajectories of the 3-body motion are reduced to that of their moduli curves. These curves record the change of size and shape, in the 3-dimensional moduli space of oriented triangles with a given mass distribution. However, it is shown that the moduli curve, with some obvious exceptions, is already determined by the associated shape curve on the shape space M ^* ≃S ², which only records the change of the similarity class of the triangle. In this way the 3-body motion is encoded into the relative geometry between the shape curve γ ^* and the gradient field ∇ U ^* of the induced Newtonian potential function U ^* on the 2-sphere M ^* . In particular, a separation of size and shape is achieved, the size function can be reconstructed from γ ^* and the latter is a solution of a 3rd order ODE on the 2-sphere.      

Efficient Testing of Large Graphs

Let P be a property of graphs. An e\epsilon -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than en²\epsilon n^2 edges to make it satisfy P. The property P is called testable, if for every e\epsilon there exists an e\epsilon -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability, admit an e\epsilon -test. In this paper we make a first step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type ``"$\forall \exists ' are always testable, while we show that some properties containing this alternation are not.     

A Constructive Inversion Framework for Twisted Convolution

In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences supported on corresponding cosets. The invertibility condition then follows from Cramer’s rule and Wiener’s lemma for this special class of matrices. The problem results from a well known approach of studying the invertibility properties of the Gabor frame operator in the rational case. The presented approach gives further insights into Gabor frames. In particular, it can be applied for both the continuous (on ) and the finite discrete setting. In the latter case, we obtain algorithmic schemes for directly computing the inverse of Gabor frame-type matrices equivalent to those known in the literature.     

Testing of matrix-poset properties

Combinatorial property testing, initiated by Rubinfeld and Sudan [23] and formally defined by Goldreich, Goldwasser and Ron in [18], deals with the following relaxation of decision problems: Given a fixed property P and an input f, distinguish between the case that f satisfies P, and the case that no input that differs from f in less than some fixed fraction of the places satisfies P. An (ε, q)-test for P is a randomized algorithm that queries at most q places of an input f and distinguishes with probability 2/3 between the case that f has the property and the case that at least an ε-fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e. as a product order of total orders). The main result here presents an (ε, poly(1/ε))-test for every property of 0/1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the “monotonicity” property studied in [9,8,13], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less efficient) test for such properties of labeled grids with larger fixed size alphabets. All the above tests have in addition a 1-sided error probability. This class of properties is related to properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations. We also show that with one quantifier alternation, a certain property can be defined, for which no test with query complexity of O(n ^1/4) (for a small enough fixed ε) exists. The above results identify new classes of properties that are defined by means of restricted logics, and that are efficiently testable. They also lay out a platform that bridges some previous results. A preliminary version of these results formed part of [14]. Research supported in part by grant 55/03 from the Israel Science Foundation.