Internal Path Length Research Articles

Single-mode waveguides for GRAVITY

The second generation Very Large Telescope Interferometer (VLTI) instrument GRAVITY is a two-field infrared interferometer operating in the K band between 1.97 and 2.43 µm with either the four 8 m or the four 1.8 m telescopes of the Very Large Telescope (VLT). Beams collected by the telescopes are corrected with adaptive optics systems and the fringes are stabilized with a fringe-tracking system. A metrology system allows the measurement of internal path lengths in order to achieve high-accuracy astrometry. High sensitivity and high interferometric accuracy are achieved thanks to (i) correction of the turbulent phase, (ii) the use of low-noise detectors, and (iii) the optimization of photometric and coherence throughput. Beam combination and most of the beam transport are performed with single-mode waveguides in vacuum and at low temperature. In this paper, we present the functions and performance achieved with weakly birefringent standard single-mode fiber systems in GRAVITY. Fibered differential delay lines (FDDLs) are used to dynamically compensate for up to 6 mm of delay between the science and reference targets. Fibered polarization rotators allow us to align polarizations in the instrument and make the single-mode beam combiner close to polarization neutral. The single-mode fiber system exhibits very low birefringence (less than 23°), very low attenuation (3.6–7 dB km−1 across the K band), and optimized differential dispersion (less than 2.04 µrad cm2 at zero extension of the FDDLs). As a consequence, the typical fringe contrast losses due to the single-mode fibers are 6% to 10% in the lowest-resolution mode and 5% in the medium- and high-resolution modes of the instrument for a photometric throughput of the fiber chain of the order of 90%. There is no equivalent of this fiber system to route and modally filter beams with delay and polarization control in any other K-band beamcombiner.

Design parameter study to extend the capacity range of dry twin screw compressors

The possibility for extending the capacity range for oil-free dry twin screw compressors has been studied. Basic geometry parameters influence on compressor performance has been derived. The two parameters shaft speed and rotor diameter are critical. The lower capacity limit for dry screw compressors depends on maximum acceptable shaft speed, considering bearings, gearbox or power electronics. The higher capacity limit depends on rotor size, considering machining costs and tolerances for large rotor diameters. Thermodynamic efficiency and optimal rotor tip speed are dependent on internal throttling and leakage losses. Simplified analytical modelling has been applied, neglecting detailed geometry with minor influence on compressor performance. Basic kinematic relations give shaft speed as dependent on screw geometry non-dimensional specific displacement, on rotor tip speed (moderately dependent of machine size), and on capacity. Basic kinematic relations also define the rotor diameter dependence on these three parameters, specific displacement; tip speed, and capacity. The rotor profile geometry, i.e. number of lobes, rotor lobe depth/diameter, male tip width/diameter; as well as rotor length/diameter ratio and wrap angle determine the specific displacement. The specific displacement is non-dimensional and hence independent of the machine size or capacity. Also the specific internal leakage path lengths in the rotor mesh and between rotors and casing, as well as the specific outlet port size will be determined by the rotor profile major geometry parameters. Here specific means in relation to the male rotor diameter. A fixed internal pressure ratio (volume ratio) has been assumed.A simplified computerized model was developed to calculate the necessary geometry data and in particular the specific displacement and the optimal tip speed, from basic geometry parameters. For the case of constant specific clearances (non-dimensional), the optimal tip speed is constant, independent of compressor size and capacity. For the case of constant absolute clearances, the optimal tip speed is slightly dependent on the compressor size and capacity. A simplified, non-female addendum, point generated, asymmetric profile was assumed. Rotor profile details with minor influence on performance have then been omitted. A cylindrical part of the male tip would allow a bigger outlet port while maintaining the built-in volume ratio.The theoretical results from this study indicate that optimal compressor data should primarily be dependent on the number of male lobes and the lobe depth, i.e. the specific displacement. A screw profile with a lower number of lobes will result in a lower optimal shaft speed and should hence enable the design of dry screw compressors for lower capacity than the commonly used 4:6 lobe configuration, without exceeding the drive train high speed constraint. A screw profile with a higher number of lobes would result in a higher optimal shaft speed and in particular reduced rotor size (rotor diameter). It should hence enable the design of dry screw compressors for higher capacity than for the commonly used 4:6 lobe configuration, without having the machining problems of too large a rotor diameter.Considering both the potential capacity range extension and the efficiency, low capacity dry screw compressors with fewer rotor lobes, e.g. 2:4, might be the most promising option.

Layered double hydroxide uniformly coated on mesoporous silica with tunable morphorlogies for catalytic transesterification of glycerol with dimethyl carbonate

Layered double hydroxides (LDH) is an attractive solid base catalyst for transesterification of glycerol with dimethyl carbonate. A major challenge in enhancing the catalytic performance of LDH for this process is to overcome the internal diffusion limitations and increase the availability of basic sites. Herein we present ultrasonic and hydrothermal post-synthesis strategies in uniformly coating of LDH on mesochannels of mesoporous silica. The effects of precursors, preparation methods and morphologies on preparing LDH/SBA-15 nano-composites were investigated. Rice-shaped LDH/SBA-15 nano-composite catalysts exhibited better performance with a 78% glycerol conversion and 90% selectivity of glycerol carbonate than the catalysts with morphologies of short-rods, and long-rods. The results revealed that hierarchical nanostairs at the edge and well-ordered mesochannels stacked along the long axis direction reduced internal diffusion path lengths and increased availability of basic sites. This approach in morphology controlling synthesis of LDH based nano-composite catalysts shed light on the uniformly coating of other diatomic and triatomic metal LDH within mesochannels of mesoporous silica to fabricate novel nano-composite catalysts.

Relative time multiplexing of heralded telecom-band single-photon sources using switchable optical fiber delays.

We experimentally demonstrate a time-multiplexing method for pulse-pumped heralded single-photon sources using switchable optical fiber delay lines. The optical fibers constitute a 32-channel binary-division photon buffer that can multiplex time slots into single temporal mode to enhance single-photon generation rate without increasing multi-photon noise. Relative time multiplexing using the same actively controlled variable delay realizes Hong-Ou-Mandel interferometry by two photons produced in arbitrary two time-bins. The interference dip depth in good agreement with theory verifies path length matching within the photonic coherence length and dispersion compensation of the optical fiber delay sections. We independently confirm by low-coherence interferometry that the internal optical path lengths of the optical switches are constant during the switching operations.

Dependence between path-length and size in random digital trees

AbstractWe study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

Modeling the impact of paste additives and pellet geometry on paste utilization within lead acid batteries during low rate discharges

When designing a lead acid battery, there are many factors to consider in order to obtain the best compromise of cost, performance, and ease of manufacturability. We use a modeling approach to study some of the key factors which affect the amount of active material which can be utilized during a low rate discharge. We investigate the effects of pellet size, pellet geometry, disconnected grid mesh borders, and inert paste additives. Furthermore, we look at how the internal path length resistance within a pellet is dependent on those features. Our findings correlate well with earlier works, and help to explain some of the previously observed phenomenon. It is observed that utilization is indeed affected by pellet size, but small grid mesh sizes on the order of ∼4 mm edge lengths are necessary in order to realize a significant benefit. Utilization is presented as a function of pellet size, aspect ratio of the pellets, and the loading level of the inert additives in the pellets up to ten percent by volume.

A General Central Limit Theorem for Shape Parameters of $m$-ary Tries and PATRICIA Tries

Tries and PATRICIA tries are fundamental data structures in computer science with numerous applications. In a recent paper, a general framework for obtaining the mean and variance of additive shape parameters of tries and PATRICIA tries under the Bernoulli model was proposed. In this note, we show that a slight modification of this framework yields a central limit theorem for shape parameters, too. This central limit theorem contains many of the previous central limit theorems from the literature and it can be used to prove recent conjectures and derive new results. As an example, we will consider a refinement of the size of tries and PATRICIA tries, namely, the number of nodes of fixed outdegree and obtain (univariate and bivariate) central limit theorems. Moreover, trivariate central limit theorems for size, internal path length and internal Wiener index of tries and PATRICIA tries are derived as well.

The preparation of MnFe2O4 decorated flexible graphene wrapped with PANI and its electrochemical performances for hybrid supercapacitors

A modest, cost effective and eco-friendly method is employed for the preparation of MnFe2O4 nanocube decorated, flexible graphene sheets, followed by polyaniline (PANI) which is wrapped by an in situ chemical oxidative polymerization method. The formation of the hybrid composites and their individual constituents are realized through X-ray diffraction (XRD), Fourier transforms infrared spectroscopy (FT-IR) and Raman spectroscopy. The MnFe2O4 nanocube decorated flexible graphene and PANI wrapped composite is visualised through transmission electron microscopic (TEM) images. The decorated MnFe2O4 particles have sizes in the range of 75–100 nm. The capacitive properties of the ternary composite is investigated and it has the obtained specific capacitance of 338 F g−1 at 0.5 mA cm−2 in 1 M NaCl aqueous electrolyte, which is 10 times greater than the pristine MnFe2O4 (32 F g−1). Similarly, the fabricated hybrid supercapacitor (MGP‖AC) provides the specific capacitance of about 51.87 F g−1 at 5 mV s−1 and its energy density is 10.25 W h kg−1 at 10 mA cm−2. Thus the obtained significant properties of the ternary composite are due to the distinctive characteristics of the individual constituents which leads to enhanced electrochemical properties like reducing the internal resistance and diffusion path length via the synergistic effect.

Asymptotic Analysis of Hoppe Trees

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n→∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

Asymptotic Analysis of Hoppe Trees

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n→∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

The -Version of Binary Search Trees: An Average Case Analysis

Following a suggestion of Cichoń and Macyna, binary search trees are generalized by keeping (classical) binary search trees and distributing incoming data at random to the individual trees. Costs for unsuccessful and successful search are analyzed, as well as the internal path length.

Graphs with diameter [formula omitted] minimizing the spectral radius

The spectral radius ρ(G) of a graph G is the largest eigenvalue of its adjacency matrix A(G). For a fixed integer e⩾1, let Gn,n-emin be a graph with minimal spectral radius among all connected graphs on n vertices with diameter n-e. Let Pn1,n2,…,nt,pm1,m2,…,mt be a tree obtained from a path of p vertices (0∼1∼2∼⋯∼(p-1)) by linking one pendant path Pni at mi for each i∈{1,2,…,t}. For e=1,2,3,4,5, Gn,n-emin were determined in the literature. Cioabaˇ et al. [2] conjectured for fixed e⩾6, Gn,n-emin is in the family Pn,e={P2,1,…,1,2,n-e+12,m2,…,me-4,n-e-2∣2<m2<⋯<me-4<n-e-2}. For e=6,7, they conjectured Gn,n-6min=P2,1,2,n-52,⌈D-12⌉,D-2 and Gn,n-7min=P2,1,1,2,n-62,⌊D+23⌋,D-⌊D+23⌋,D-2. In this paper, we settle their conjectures positively. Note that any tree in Pn,e is uniquely determined by its internal path lengths. For any e-4 non-negative integers k1,k2,…,ke-4, let T(k1,k2,…,ke-4)=P2,1,…,1,2,n-e+12,m2,…,me-4,n-e-2 with ki=mi+1-mi-1, for 1⩽i⩽e-4. (Here we assume m1=2 and me-3=n-e-2.)Let s=∑i=1e-4ki+2e-4. For any integer e⩾6 and sufficiently large n, we proved that Gn,n-emin must be one of the trees T(k1,k2,…,ke-4) with the parameters satisfying ⌊s⌋-1⩽kj⩽⌊s⌋⩽ki⩽⌈S⌉+1 for j=1,e-4 and i=2,…,e-5. Moreover, 0⩽ki-kj⩽2 for 2⩽i⩽e-5,j=1,e-4; and |ki-kj|⩽1 for 2⩽i,j⩽e-5. These results are best possible as shown by cases e=6,7,8, where Gn,n-emin are completely determined here. Moreover, if n-6 is divisible by e-4 and n is sufficiently large, then Gn,emin=T(k-1,k,k,…,k,k,k-1) where k=n-6e-4-2.

On Tail Bounds for Random Recursive Trees

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

On Tail Bounds for Random Recursive Trees

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of randomb-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Limit theorems are established for some functionals of the distances between two nodes in weighted randomb-ary recursive trees. We consider the depth of thenth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Limit theorems are established for some functionals of the distances between two nodes in weighted random b-ary recursive trees. We consider the depth of the nth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

On the Asymptotic Internal Path Length and the Asymptotic Wiener Index of Random Split Trees

The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.

Formula omitted]: The internal extent formula for compacted tries

It is well known (Knuth, 1997 [5, pp. 399–400]) that in a binary tree the external path length minus the internal path length is exactly 2 n − 2 , where n is the number of external nodes. We show that a generalization of the formula holds for compacted tries, replacing the role of paths with the notion of extent, and the value 2 n − 2 with the trie measure, an estimation of the number of bits that are necessary to describe the trie.

On the parity of the Wiener index

It is a known fact that the Wiener index (i.e. the sum of all distances between pairs of vertices in a graph) of a tree with an odd number of vertices is always even. In this paper, we consider the distribution of the Wiener index and the related tree parameter “internal path length” modulo 2 by means of a generating functions approach as well as by constructing bijections for plane trees.

On [formula omitted]-functional equations and excursion moments

We analyse q -functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive integer k . We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of k th powers of the standard Brownian excursion. Our approach yields a recursion for the moments of the limit distribution. It can be used to analyse asymptotic expansions of the moments, and it admits an extension to other types of singularity.