On convergence of wavelet packet expansions

It is well known that the-Walsh-Fourier expansion of a function from the block space ([0, 1 ) ), 1 ＜q≤∞, converges pointwise a.e. We prove that the same result is true for the expansion of a function from in certain periodixed smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of Lp-functions, 1＜p＜∞, converges in norm and pointwise almost everywhere.     

Imaging and quantifying transverse flow velocity with the Doppler bandwidth in a phase-resolved functional optical coherence tomography

The Doppler bandwidth extracted from the standard deviation of the frequency shift in phase-resolved functional optical coherence tomography (F-OCT) was used to image the velocity component that is transverse to the optical probing beam. It was found that above a certain threshold level the Doppler bandwidth is a linear function of flow velocity and that the effective numerical aperture of the optical objective in the sample arm determines the slope of this dependence. The Doppler bandwidth permits accurate measurement of flow velocity without the need for precise determination of flow direction when the Doppler flow angle is within +/-15 degrees perpendicular to the probing beam. Such an approach extends the dynamic range of flow velocity measurements obtained with the phase-resolved F-OCT.     

Investigation of Brillouin effects in small-core holey optical fiber: lasing and scattering

We demonstrate for what is believed to be the first time a Brillouin laser based on a holey fiber (HF). Using a simple Fabry-Perot resonator scheme containing a 73.5-m-long highly nonlinear HF with an effective area of 2.85 microm(2) , we obtain a threshold of 125 mW and a slope efficiency of ~70% . Stimulated and spontaneous Brillouin scattering effects are investigated in the HF, and we show that the high lasing threshold is due mainly to reduction of the effective gain coefficient caused by structural nonuniformity along the fiber length.     

Synthesis of linear and tripoidal oligo(phenylene ethynylene)-based building blocks for application in modular DNA-programmed assembly

Rigid linear and tripoidal organic modules based on the oligo(phenylene ethynylene) backbone having salicylaldehyde-derived termini are synthesized. A highly functionalized 5-iodosalicyl aldehyde was prepared and coupled to each ethynyl group of 1,4-diethynylbenzene or 1,3,5-triethynylbenzene in Sonogashira couplings. The two or three termini of the compounds are functionalized for incorporation in linear and branched oligonucleotide strands. For the linear module (LM), the two termini are equipped with amide spacers, and one of these was functionalized with a DMTr (dimethoxytrityl)-protected hydroxy group and the other with a phosphoramidite. One of the tripoidal modules is prepared with DMTr groups in two of its three termini. A tripoidal module is also synthesized with three different groups on its hydroxy termini: a phosphoramidite, a DMTr group, and an Fmoc group. Extended studies have shown that these rigid linear and tripoidal organic modules can be incorporated into short oligonucleotides. Several of these modules can be applied for DNA-directed assembly and covalent coupling into structures of predetermined connectivity. Such structures have potential application for molecular electronics and nanotechnology.     

Nonlinear Approximation with Dictionaries I. Direct Estimates

We study various approximation classes associated with m-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in L^p spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss some new estimates they provide.     

Fiber Bragg gratings for pure dispersion-slope compensation

Single-fiber Bragg gratings designed to compensate for pure third-order dispersion are demonstrated. The devices are designed by use of a layer-peeling inverse-scattering technique and exhibit dispersion profiles across their reflection bandwidths of -20, +/- 40, and -225 ps/nm2 over bandwidths of 4.5, 6, and 3 nm, respectively. Furthermore, they have almost constant peak reflectivities of approximately 75% and are, respectively, only 8, 23, and 40 mm long.     

Internet protocol network design with uncertain demand

We present a case study concerning the design and dimensioning of the internet protocol network of TDC, the largest Danish network operator. Due to historical reasons the number of points of presence (POPs) in the network has reached a level, believed to be too high. To point out potential POPs for dismantling, we consider a network planning problem concerning dimensioning of the POPs and capacity expansion of the transmission links of the network. This problem is formulated as a two-stage stochastic program using a finite number of scenarios to describe the uncertain outcome of future demand. The problem is then solved by an L-shaped algorithm, and we report results of our computational experiments.     

On a cycle partition problem

Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c₁,c₂ of positive integers satisfying c₁+c₂=c(G), the vertex set of G admits a partition into two sets V₁ and V₂, such that V_i induces a graph of circumference at most c_i, i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture.This work is inspired by a well-known, long-standing, analogous conjecture involving paths.