Approximation of bandlimited functions

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.     

3D discrete X-ray transform

The analysis of 3D discrete volumetric data becomes increasingly important as computation power increases. 3D analysis and visualization applications are expected to be especially relevant in areas like medical imaging and nondestructive testing, where elaborated continuous theory exists. However, this theory is not directly applicable to discrete datasets. Therefore, we have to establish theoretical foundations that will replace the existing inexact discretizations, which have been based on the continuous regime. We want to preserve the concepts, properties, and main results of the continuous theory in the discrete case. In this paper, we present a discretization of the continuous X-ray transform for discrete 3D images. Our definition of the discrete X-ray transform is shown to be exact and geometrically faithful as it uses summation along straight geometric lines without arbitrary interpolation schemes. We derive a discrete Fourier slice theorem, which relates our discrete X-ray transform with the Fourier transform of the underlying image, and then use this Fourier slice theorem to derive an algorithm that computes the discrete X-ray transform in O(n⁴logn) operations. Finally, we show that our discrete X-ray transform is invertible.     

Hydroxide ion initiated reactions under phase transfer catalysis conditions II. The roles of water and quat

The effects of water molecules and quat structure are shown to be significant in determining the behavior of alkylation reactions of weakly acidic carbon acids under PTC/OH^? conditions.     

Total synthesis of the marine alkaloid halitulin

The structure of the strongly cytotoxic marine alkaloid halitulin (1) has been confirmed by total synthesis and its absolute configuration determined as (15S). The synthesis follows a strategy previously reported by one of us and uses an efficient preparation of the quinoline-7,8-diol unit by modified Baeyer-Villiger and Skraup reactions. The O-benzyl protecting groups were removed in the last step of the synthesis by transfer hydrogenolysis without concomitant reduction of the quinoline ring. The method can be applied for the synthesis of halitulin analogues.     

A new diterpenoid related to eunicellin and cladiellin from a Muricellasp.

The isolation and structure elucidation of a new diterpenoid from a horny-coral (Gorgonaceae) are reported. The ¹H and ¹³C-NMR spectra are discussed.     

Effects of structure on properties of some new aromatic-aliphatic polybenzimidazoles

Polybenzimidazoles were prepared in poly(phosphoric acid) from isophthalic, m- and p-phenylene diacetic, succinic, adipic, suberic, and sebacic acids and 3,3′-diaminobenzidine, 3,3′,4,4′-tetraaminodiphenyl ether and 3,3′,4,4′-tetraaminodiphenylmethane. The thermal, mechanical, and bonding properties were studied. A 3:1 copolymer of isophthalic and m-phenylenediacetic acid with 3,3′-diaminobenzidine showed the best results as far as isothermal oxidation resistance and thermal and processing characteristics.     

Phosphorus heterocycle synthesis by RPX2,· AlX3 addition to [1,n]dienes VII.

The RPX₂·AlX₃ complex ($\text{1}$) reacts with unsaturated ketones and imines to give novel 7-oxa and 7-aza-2-phosphabicyclo[2.2.1]heptanes (compounds $\text{3}$ and $\text{6}$ respectively).A new 1,3-dipolar addition of (CH₃)₂ CCH₂P?XR to nArN=C=S was disclosed resulting in the formation of the 2-imino-1,3-thiaphospholanes ($\text{7}$).     

Complete modal decomposition for optical waveguides

Virtually all electromagnetic waveguiding structures support a multiplicity of modes. Nevertheless, to date, an experimental method for unique decomposition of the fields in terms of the component eigenmodes has not been realized. The fundamental problem is that all current attempts of modal decomposition do not yield phase information. Here we introduce a noninterferometric approach to achieve modal decomposition of the fields at the output of a general waveguiding structure. The technique utilizes a mapping of the two-dimensional field distribution onto the one-dimensional space of waveguide eigenmodes, together with a phase-retrieval algorithm to extract the amplitudes and phases of all the guided vectorial modes. Experimental validation is provided by using this approach to examine the interactions of 16 modes in a hollow-core photonic-band gap fiber.     

Optical bistability in axially modulated OmniGuide fibers

We demonstrate the feasibility of optical bistability in an axially modulated nonlinear OmniGuide fiber through analytical theory and detailed numerical experiments. At 1.55-microm carrier wavelength, the in-fiber devices that we propose can operate with only a few tens of milliwatts of power, can have a nearly instantaneous response and recovery time, and can be shorter than 100 microm.     

Performance of first-order methods for smooth convex minimization: a novel approach

We introduce a novel approach for analyzing the worst-case performance of first-order black-box optimization methods. We focus on smooth unconstrained convex minimization over the Euclidean space. Our approach relies on the observation that by definition, the worst-case behavior of a black-box optimization method is by itself an optimization problem, which we call the performance estimation problem (PEP). We formulate and analyze the PEP for two classes of first-order algorithms. We first apply this approach on the classical gradient method and derive a new and tight analytical bound on its performance. We then consider a broader class of first-order black-box methods, which among others, include the so-called heavy-ball method and the fast gradient schemes. We show that for this broader class, it is possible to derive new bounds on the performance of these methods by solving an adequately relaxed convex semidefinite PEP. Finally, we show an efficient procedure for finding optimal step sizes which results in a first-order black-box method that achieves best worst-case performance.