The 1:1 resonance in Hamiltonian systems

Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances $k:l$, but not the semisimple cases $1:1$ and $1:?1$. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to $S^1$-symmetric systems. The question we wish to address is about the co-dimension of such a system in $1:1$ resonance with respect to left-right-equivalence, where the right action is $S^1$-equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.     

Synthesis and structure of new composite hydrogels based on poly(N-vinyl caprolactam) with nanosized anatase

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Trimetallic NiCoM catalysts (M = Mn,Fe, Cu) for methane conversion into synthesis gas

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Efficient and stereoselective synthesis of (S)-α-propargylglycine derivatives from allenylboronic acid

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Structure and quantum chemical study of crystalline platinum(II) acetate

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On percolation and ‐hardness

The edge‐percolation and vertex‐percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational complexity of problems whose inputs are obtained by applying percolation to worst‐case instances. Specifically, we show that a number of classical ‐hard problems on graphs remain essentially as hard on percolated instances as they are in the worst‐case (assuming ). We also prove hardness results for other ‐hard problems such as Constraint Satisfaction Problems and Subset‐Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number and the chromatic number are robust to percolation in the following sense. Given a graph G, let be the graph obtained by randomly deleting edges of G with some probability . We show that if is small, then remains small with probability at least 0.99. Similarly, we show that if is large, then remains large with probability at least 0.99. We believe these results are of independent interest.     

Synthesis and Spatial Structure of the Derivatives of 2,5-Dioxo-3,3-Bis(Trifluoromethyl)-6,7-Benzo-1,4,2-and 2,5-Dioxo-4,4-Bis(Trifluoromethyl)-6,7-Benzo-1,3,2-Oxazaphosphepines

Abstract

Hexafluoroacetone imine easily interacts with compounds (I, R = OMe, OCH₂CF₂CHF₂, NEt₂, Ph) in two directions unlike hexafluoroacttone and gives 1,4,2-oxazaphosphepines (II) (pathway I) or 1,3,2-oxszaphosphepines (III) (pathway 2). The compound (II) (R = NEt₂) lightly hydrolyzes to yield the salt (IV). The structure of heterocycles II-IV) has been confirmed by X-ray analysis (see fig. I, II, R = OMe; fig. 2, IV). The detail structural peculiarities of the compounds am discussed.     

Enumeration of non-labile oxygen atoms in dissolved organic matter by use of 16O/18O exchange and Fourier transform ion-cyclotron resonance mass spectrometry

We report a simple approach for enumeration of non-labile oxygen atoms in individual molecules of dissolved organic matter (DOM), using acid-catalyzed ¹⁶O/¹⁸O exchange and ultrahigh-resolution Fourier-transform ion-cyclotron-resonance mass spectrometry (FTICR-MS). We found that by dissolving DOM in H₂ ¹⁸O at 95 °C for 20 days it is possible to replace all oxygen atoms of DOM molecules (excluding oxygen from ether groups) with ¹⁸O. The number of exchanges in each molecule can be determined using high-resolution FTICR. Using the proposed method we identified the number of non-labile oxygen atoms in 231 molecules composing DOM. Also, using a previously developed hydrogen–deuterium (H/D)-exchange approach we identified the number of labile hydrogen atoms in 450 individual molecular formulas. In addition, we observed that several backbone hydrogen atoms can be exchanged for deuterium under acidic conditions. The method can be used for structural and chemical characterization of individual DOM molecules, comparing different DOM samples, and investigation of biological pathways of DOM in the environment.