Matrix-analytic solution of infinite,finite and level-dependent second-order fluid models

This paper presents a matrix-analytic solution for second-order Markov fluid models (also known as Markov-modulated Brownian motion) with level-dependent behavior. A set of thresholds is given that divide the fluid buffer into homogeneous regimes. The generator matrix of the background Markov chain, the fluid rates (drifts) and the variances can be regime dependent. The model allows the mixing of second-order states (with positive variance) and first-order states (with zero variance) and states with zero drift. The behavior at the upper and lower boundary can be reflecting, absorbing, or a combination of them. In every regime, the solution is expressed as a matrix-exponential combination, whose matrix parameters are given by the minimal nonnegative solution of matrix quadratic equations that can be obtained by any of the well-known solution methods available for quasi birth death processes. The probability masses and the initial vectors of the matrix-exponential terms are the solutions of a set of linear equations. However, to have the necessary number of equations, new relations are required for the level boundary behavior, relations that were not needed in first-order level dependent and in homogeneous (non-level-dependent) second-order fluid models. The method presented can solve systems with hundreds of states and hundreds of thresholds without numerical issues.     

Some asymptotically optimal one-sided embeddings of trees of similar formulas into rectangular lattices

We consider the problem of optimally placing trees of formulas in rectangular lattices. We construct and study two types of these trees and corresponding ways of placing (embedding) them into such lattices. The first is based on perfect binary trees, while the second is based on special binary trees. For the second type of tree embeddings, we prove asymptotic optimality among the trees of all formulas similar to the initial formula of no greater depth.     

Correction to “Queue-proportional rate allocation with per-link information in multihop wireless networks”

There is a gap in the proofs of both Propositions 3 and 4 in the correspondence “Queue-Proportional Rate Allocation with Per-Link Information in Multihop Wireless Networks,” previously published in the Queueing Systems (see Li and Srikant 2016). We filled this gap in the proofs for the multihop networks (see Li and Srikant in Queueing Syst 84:203–210, 2016), where each route at most has two links. In this note, we completely fill the gap in the proofs in our original paper (see Li and Srikant in Queueing Syst 83:329–359, 2016).     

Fast implementation of the Tukey depth

Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions \(p \ge 3\). In this paper, we address this computing issue by presenting two combinatorial algorithms. The first is naive and calculates the Tukey depth of a single point with complexity \(O\left( n^{p-1}\log (n)\right) \), while the second further utilizes the quasiconcave of the Tukey depth function and hence is more efficient than the first. Both require very minimal memory and run much faster than the existing ones. All experiments indicate that they compute the exact Tukey depth.     

A non-negative matrix factorization model based on the zero-inflated Tweedie distribution

Non-negative matrix factorization (NMF) is a technique of multivariate analysis used to approximate a given matrix containing non-negative data using two non-negative factor matrices that has been applied to a number of fields. However, when a matrix containing non-negative data has many zeroes, NMF encounters an approximation difficulty. This zero-inflated situation occurs often when a data matrix is given as count data, and becomes more challenging with matrices of increasing size. To solve this problem, we propose a new NMF model for zero-inflated non-negative matrices. Our model is based on the zero-inflated Tweedie distribution. The Tweedie distribution is a generalization of the normal, the Poisson, and the gamma distributions, and differs from each of the other distributions in the degree of robustness of its estimated parameters. In this paper, we show through numerical examples that the proposed model is superior to the basic NMF model in terms of approximation of zero-inflated data. Furthermore, we show the differences between the estimated basis vectors found using the basic and the proposed NMF models for \(\beta \) divergence by applying it to real purchasing data.     

Application of numerical schemes with singling out the boundary layer for the computation of turbulent flows using eddy-resolving approaches on unstructured grids

The use of eddy-resolving approaches to solving problems on arbitrary unstructured grids is investigated. The applications of such approaches requires the use of low dissipation numerical schemes, which can lead to numerical oscillations of the solution on unstructured grids. Numerical oscillations typically occur in domains with large gradients of velocities, in particular, in the near-wall layer. It is proposed to single out the boundary layer and use a numerical scheme with increased numerical dissipation in it. The algorithm for singling out the boundary layer uses a switching function to change the parameters of the numerical scheme. This algorithm is formulated based on the BCD scheme from the family NVD. Its validity and advantages are investigated using the zonal RANS–LES approach for solving some problems of turbulent flow of incompressible fluids.     

Approximation algorithm for the problem of partitioning a sequence into clusters

We consider the problem of partitioning a finite sequence of Euclidean points into a given number of clusters (subsequences) using the criterion of the minimal sum (over all clusters) of intercluster sums of squared distances from the elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is at the origin, while the center of each of the other clusters is unknown and determined as the mean value over all elements in this cluster. Additionally, the partition obeys two structural constraints on the indices of sequence elements contained in the clusters with unknown centers: (1) the concatenation of the indices of elements in these clusters is an increasing sequence, and (2) the difference between an index and the preceding one is bounded above and below by prescribed constants. It is shown that this problem is strongly NP-hard. A 2-approximation algorithm is constructed that is polynomial-time for a fixed number of clusters.     

A switching for all strongly regular collinearity graphs from polar spaces

We describe a general construction of strongly regular graphs from the collinearity graph of a finite classical polar spaces of rank at least 3 over a finite field of order q. We show that these graphs are non-isomorphic to the collinearity graphs and have the same parameters. For most of these parameters, the collinearity graphs were the only known examples, and so many of our examples are new.     

Competitive analysis of price online inventory problem with cost function

In this paper, a new price is given to the online decision maker at the beginning of each day. The trader must decide how many items to purchase according to the current price. We present three variants and an online algorithm based on cost function. The competitive ratio of the online algorithm is given for each variant, which is a performance measure of an online algorithm. More importantly, we show that the online algorithm is optimal.     

Applications of Hyphenated Liquid Chromatography Techniques for Polymer Analysis

We find polymers everywhere in our daily activities, for example, as a part of consumer electronics products, healthcare devices, vehicles, etc. Analytical characterization of such materials is an important step towards understanding their properties and behavior in various applications. The increase of material complexity driven by highly demanding requirements for many applications necessitates the use of sophisticated analytical techniques to obtain sufficient insight into the structure of these materials. Coupling of liquid chromatography with other information-rich instrumental techniques becomes more and more important in the field of polymer characterization. Such combination can enable simultaneous separation, identification, and quantification of polymer sample components. In addition, it can provide information on interdependence of two polymer properties, e.g., molecular weight and chemical composition. Different hyphenated systems may be applied to address different problems in polymer research and development and a selection of the right technique may not be an easy and straightforward task. In this paper, the applications of LC-NMR, LC-IR, LC-Raman, LC-MS, LC-MALDI, LC × LC, and LC × Py-GC for polymer analysis are reviewed, their advantages and limitations are discussed, and practical challenges for the implementation of these techniques in a lab are addressed. Different hyphenated options are compared to facilitate selection of a suitable instrument for the particular problem at hand.