Computer-generated infrared depolarizer using space-variant subwavelength dielectric gratings

We present a novel method for the formation of a complete depolarizer that is based on a polarization-state scrambling procedure over the space domain. Such an element can be achieved by use of cascaded, computer-generated, space-variant subwavelength dielectric gratings. We introduce a theoretical analysis and experimentally demonstrate a depolarizer for infrared radiation at a wavelength of 10.6 microm.     

Towards generalization and optimization of implicit methods

A general implicit (GI) method for solving iteratively the algebraic system arising from a finite difference approximation of an elliptic partial differential equation is formulated. Under certain assumptions this method can be reduced to the already known implicit techniques. It is shown that the GI method has a very special physical meaning when solving fluid flow problems. It is shown also how this method can be optimized to achieve the maximum rate of convergence. Finally it is shown how this new strategy is applied by solving some classical numerical fluid dynamics problems.     

The first solid-phase synthesis of bis(oxazolinyl)pyridine ligands

Chiral Pybox (pyridine-2,6-bis(oxazoline)) ligands can be cleanly and efficiently prepared on polystyrene support via a five-step solid-phase synthetic sequence. Cu(I)-complexed polymer-bound Pybox was used as a catalyst in the first heterogeneously catalyzed asymmetric addition of alkynes to imines. Best enantioselectivity was observed with (t)Bu-substituted oxazolines.     

Rotating vectorial vortices produced by space-variant subwavelength gratings

A new class of vectorial vortex based on coherent addition of two orthogonal circularly polarized Bessel beams of identical order but with different propagation constants is presented. The transversely space-variant axially symmetric polarization distributions of these vectorial fields rotate as they propagate, while they maintain a propagation-invariant Bessel intensity distribution. These properties were demonstrated by use of discrete space-variant subwavelength gratings for 10.6 microm CO2 laser radiation. The polarization properties were verified by both full space-variant polarization analysis and measurements. Rotating intensity patterns are also demonstrated by transmitting the vectorial vortices through a linear polarizer.     

Thermal image encryption obtained with a SiO2 space-variant subwavelength grating supporting surface phonon-polaritons

Space-variant partially polarized thermal emission is investigated. We show that by coupling surface phonon-polaritons to a propagating field, large anisotropy of the emissivity is obtained within a narrow spectral range. We experimentally demonstrate this effect by fabricating a space-variant subwavelength grating on a SiO2 substrate to encrypt an image in the polarization state of a thermal radiation field.     

Additive representations of non-additive measures and the choquet integral

This paper studies some new properties of set functions (and, in particular, “non-additive probabilities” or “capacities”) and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results:

the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand;

the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures;

under fairly general conditions one may define a “Radon-Nikodym derivative” of one capacity with respect to another;

the “optimistic” pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space.

We also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.     

Propellanes—XXXIX : 13C NMR-spectra of configurationally different propellane isomers containing cyclopropane rings

The mono- and bis-adducts obtained in the reaction of carbethoxycarbene with 12-methyl-11,13-dioxo-12-aza[4.4.3]propella-3,8-diene have established configurations of cyclopropane rings and ester groups with respect to the hetero-ring. Thus ¹³C NMR measurements afford data which correlates the ¹³C absorptions of these isomers with their configurations.     

Symmetric LDPC codes and local testing

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides a new way of construction with better parameters and new lower bounds.Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates “look the same,” namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all “low-degree” codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a “con ict” between 1/rate and density (constraint length). In known constructions, if one is constant, then the other is almost the worst possible - n/poly(logn).Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log logn)², of constant rate and with constraints of length (logn)⁴. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes.Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka “good” codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by (log^(Ω(?)) n)(an Ω(?) iterated logarithm) if the group has a derived series of length ?. This negative result precludes natural local tests with constantly many queries for such solvable “good” codes.     

The standard proof,the elegant proof,and the proof without words of tasks in geometry,and their dynamic investigation

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.