Coding for locality in reconstructing permutations

The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with locality, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. Both approaches must present low query complexity to allow the user to find an element efficiently. We discuss both approaches, and give a particular focus to the combinatorial one. In the combinatorial approach, we provide upper and lower bounds for the maximal size of a set of permutations with locality, and provide several simple constructions which attain the upper bound. In cases where the upper bound is not attained, we provide alternative constructions using a variety of tools, such as Reed-Solomon codes, permutation polynomials, and multi-permutations. In addition, several low-rate constructions of particular interest are discussed. In the coding approach we discuss an alternative representation of permutations, present a paradigm for supporting arbitrary powers of the stored permutation, and conclude with a proof of concept that permutations may be stored more efficiently than ordinary strings over the same alphabet.     

On a Plus-Construction for Algebras over an Operad

We prove a analogous to Quillen's plus-construction in the category of algebras over an operad. For that purpose we prove that this category is a closed model category and prove the existence of an obstruction theory. We apply further this plus-construction for the specific cases of Lie algebras and Leibniz algebras which are a noncommutative version of Lie algebras: let sl(A) be the kernel of the trace map gl(A)A/[A,A], where A is an associative algebra with unit and gl(A) is the Lie algebra of matrices over A. Then the homotopy of slA)⁺ in the category of Lie algebras is the cyclic homology of A whereas it is the Hochschild homology of A in the category of Leibniz algebras.     

Real-time Fourier transformer based on fiber gratings

We use the well-known duality between paraxial diffraction in space and dispersion in time to propose a time-domain analog to spatial Fraunhofer diffraction. This analog permits the design of real-time optical Fourier-transformer systems. These systems are shown to be realizable by use of linearly chirped fiber gratings as dispersive media.     

Combined scanning optical coherence and two-photon-excited fluorescence microscopy

We demonstrate simultaneous imaging by optical coherence microscopy (OCM) and two-photon-excited (TPE) fluorescence microscopy. A mode-locked Ti:sapphire laser is focused and scanned in three dimensions through a fixed sample, generating both backscattered light and fluorescence light, which are independently detected. Both imaging modes provide rapid en-face imaging with submicrometer resolution. High-power delivery into the sample yields an OCM sensitivity in excess of 130 dB at 100-kHz pixel rates. Simultaneous imaging of cell nuclei with OCM and TPE is demonstrated in live drosophila embryos.     

Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings

We show that a temporal effect that is equivalent to the spatial self-imaging (Talbot) effect applies to the reflection of periodic signals from linearly chirped fiber gratings. The effect can be used for multiplying the repetition frequency of a given periodic pulse train without distorting the individual pulse characteristics. The practical limit on the frequency-multiplication factor depends only on the temporal width of the individual pulse. Thus we demonstrate that a suitable combination of well-known techniques for short-pulse generation, such as pulse mode locking, and the technique proposed here allows us to obtain short-pulse trains with ultrahigh repetition rates (in the terahertz regime). Results from simulations show good agreement with those predicted by theory.     

Non-defluorinative electrochemical silylation of ethyl trifluoroacetate: a practical synthesis of trifluoroacetyltrimethylsilane via its ethyltrimethylsilyl ketal

An efficient method for the preparation of original trifluoroacetyltrimethylsilane, CF₃COSiMe₃ (3), in two steps from readily available ethyl trifluoroacetate is described. Electrochemical reduction of this ester using a sacrificial anode and performed on a semimolar scale afforded the unprecedented corresponding ketal, CF₃C(SiMe₃)(OSiMe₃)OEt (2) in 30-56% isolated yield. Treated with concentrated sulphuric acid at room temperature, the latter directly led to pure acylsilane 3 in 86% yield.     

Iron-catalyzed electrochemical allylation of carbonyl compounds by allylic acetates

Homoallylic alcohols were synthesized from aldehydes or ketones and allylic acetates, using an electrochemical process catalyzed by iron complexes. We first studied the reactivity of allyl acetate, using N,N-dimethylformamide (DMF) or acetonitrile (AN) as solvent, FeBr(2) as catalyst, and Fe as the sacrificial anode. Then we tested the regioreactivity of crotyl acetate and other allylic derivatives.     

Spiroimidazolidinone library derivatives on SynPhase lanterns

Optimization and solid-phase synthesis of new spiroimidazolidinone derivatives as highly functionalized templates is reported. The synthesis of 1,4,8-triazaspiro[4.5]decan-2-one derivatives was performed on SynPhase lanterns from dipeptides anchored on the solid support and N-protected piperidone. A library of 180 discrete compounds was prepared.     

Connecting pi-chromophores by sigma-P-P bonds: new type of assemblies exhibiting sigma-pi-conjugation

To study the ability of sigma-P-P skeleton to mediate interaction between pi-chromophores, 1,1'-biphospholes bearing phenyl or thienyl substituents at the 2,2' and 5,5'-position have been prepared and studied. These air-stable derivatives are readily available via a "one-pot" synthesis starting from diynes. Theoretical studies and UV-vis data clearly establish that the two pi-systems interact via the sigma-P-P bridge. This through-bond interaction results in a lowering of the optical HOMO-LUMO gap of the assemblies. The nucleophilic sigma(3)-P centers of these 1,1'-biphospholes allow chemical modifications of the sigma-bridge. These modifications offer further tuning of the optical properties of the assembly. Electrooxidation of the thienyl-substituted 1,1'-biphosphole results in electroactive materials characterized by low optical band gap and reversible p-doping.