Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N.G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^card(Ω) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality card(Ω), the group Sym(Ω) contains a coproduct of 2^card(Ω) copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set V. From this one easily obtains analogs of the results stated above with “group” and Sym(Ω) replaced by “monoid” and the monoid Self(Ω) of endomaps of Ω, by “associative K-algebra” and the K-algebra End_K (V) of endomorphisms of a K-vector-space V with basis Ω, and by “lattice” and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K(V) contains a coproduct of 2^card(Ω) copies of itself.That paper also gave an example of a group of cardinality 2^card(Ω) that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_K(V), and Equiv(Ω), and also for the monoid Rel(Ω) of binary relations on Ω.Many open questions and areas for further investigation are noted.     

Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments

We study complex damped and undamped dynamics and targeted energy transfers (TETs) in systems of coupled oscillators, consisting of single-degree-of-freedom primary linear oscillators (LOs) with vibro-impact attachments, acting, in essence, as vibro-impact nonlinear energy sinks (VI NESs). First, the complicated dynamics of such VI systems is demonstrated by computing the VI periodic orbits of underlying Hamiltonian systems and depicting them in appropriate frequency–energy plots (FEPs). Then, VI damped transitions and distinct ways of passive TETs from the linear oscillators to the VI attachments for various parameter ranges and initial conditions are investigated. As in the case of smooth stiffness nonlinearity [Y. Lee, G. Kerschen, A. Vakakis, P. Panagopoulos, L. Bergman, D.M. McFarland, Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment, Physica D 204 (1–2) (2005) 41–69], both fundamental and subharmonic TET can be realized in the VI systems under consideration. It is found that the most efficient mechanism for VI TET is through the excitation of highly energetic VI impulsive orbits (IOs), i.e., of periodic or quasiperiodic orbits corresponding to zero initial conditions except for the initial velocities of the linear oscillators. In contrast to NESs with smooth essential nonlinearities considered in previous works, VI NESs are capable of passively absorbing and locally dissipating significant portions of the energies of the primary systems to which they are attached, at fast time scale. This renders such devices suitable for applications, like seismic mitigation, where dissipation of vibration energy in the early, highly energetic regime of the motion is a critical requirement.     

BDD-based heuristics for binary optimization

In this paper we introduce a new method for generating heuristic solutions to binary optimization problems. We develop a technique based on binary decision diagrams. We use these structures to provide an under-approximation to the set of feasible solutions. We show that the proposed algorithm delivers comparable solutions to a state-of-the-art general-purpose optimization solver on randomly generated set covering and set packing problems.     

Ozonolysis-based route to the in situ formation of aldehyde-bearing self-assembled monolayer surfaces

While ozonolysis of a terminal carbon-carbon double bond to produce aldehydes is a well-established synthetic strategy for conventional solution chemistry, exposure of vinyl-terminated self-assembled monolayers to ozone has been reported to yield carboxylic acids. By using a cold solution of ozone in methanol and then adding a reducing agent to this solution, acid formation is minimized and near-quantitative aldehyde formation is achieved. The aldehyde-bearing surface is characterized by its physical and chemical properties and by ATR-FTIR spectroscopy showing a characteristic aldehyde C-H peak at 2715 cm(-1) and carbonyl peak at 1729 cm(-1). The reactivity of the aldehyde-bearing surface is shown by its reaction with amines and amine derivatives to give surface-bound imines and by the reversible cycling between aldehyde and acetal. The acetal also provides a useful way to mask the aldehyde and store readily released aldehyde surface functionality for subsequent surface elaboration.     

cis-3,5-Dibenzylidene-1,2,4-tritellurole,a novel organotellurium heterocycle

The reaction of sodium phenylethynyltellurolate with ethereal hydrogen chloride afforded, in addition to $\text{trans}$-2,4-dibenzylidene-1,3-di telluretane, a product shown to be the title compound by an X-ray crystallographic analysis.     

Pi-stacking induced NMR spectrum splitting in enantiomerically enriched Ru(II) complexes: evaluation of enantiomeric excess

Several chiral octahedral complexes of the general formula [Ru(bpy)2 (Lig)][PF6]2 (Lig = a ligand that can participate in pi-stacking interactions such as eilatin, isoeilatin, and tpphz) were synthesized in both the racemic and enantiomerically pure/enriched forms. Nonracemic mixtures of enantiomers of all these complexes exhibit splitting of the 1H NMR spectra (NMR nonequivalence); i.e., each spectrum contains a major and a minor set of peaks. The origin of this phenomenon is attributed to a fast equilibrium between monomers and discrete dimers held together by pi-stacking interactions, and it is observed for a wide range of pi-stacking interaction strengths. The NMR spectrum splitting exhibited by these complexes can be exploited for the evaluation of their enantiomeric excess simply from the integral ratio, without addition of chiral shift reagents.     

Gln-Gly cleavage: a dominant dissociation site in the fragmentation of protonated peptides

An understanding of the gas-phase dissociation of protonated peptides within the mass spectrometer is essential for automated high-throughput protein identification. In this communication we describe a facile cleavage of the Gln-Gly peptide bond under low-collisional energy conditions. A variety of synthetic peptides have been analysed where key amino acids have been substituted within the sequence PQGPPQQGGR, which is a consensus repeat present in the tryptic peptides of acidic proline-rich protein 1 (PRP-1). The collision-induced dissociation spectra obtained from the PRP-1 tryptic peptides and the synthetic peptides indicate that facile Gln-Gly cleavage occurs when an X-Gln-Gly-Y sequence is present in a peptide, where X is any amino acid and Y any amino acid other than Gly.