Set Partitioning Research Articles

1.06 Tbit/s space division multiplexing self-homodyne coherent transmission with high spectral efficiency probabilistic shaping multidimensional modulation

We propose a probabilistic shaping multidimensional (MD) modulation format for self-homodyne coherent transmission systems. The redundancy of the MD signal is separated and transmitted together with the optical pilot tone (PT). The 4D/8D/12D modulation is constructed at the transmitter based on the amplitude translation (AT)-set partitioning (SP) principle and the probabilistic amplitude shaping (PAS) structure. Moreover, the hard-decision (HD) and soft-decision (SD) decoders for MD signals are introduced. We implement 20 GBaud 4D/8D/12D probabilistic shaping (PS) 16384-level quadrature amplitude modulation (QAM) transmission over 22.5 km uncoupled multicore fiber to analyze the performance. The experimental results show that it is possible for the MD signal to provide spectral efficiency (SE) gain up to 3.31 bit/s/Hz/4D-sym. The maximum SE transmitted is up to 57.29 bit/s/Hz, i.e., 17.5 bit/s/Hz/4D-sym, with a bit rate of 1.06 Tbit/s.

Modified Ascent Sequences and Bell Numbers

In 2011, Duncan and Steingrímsson conjectured that modified ascent sequences avoiding any of the patterns 212, 1212, 2132, 2213, 2231 and 2321 are counted by the Bell numbers. Furthermore, the distribution of the number of ascents is the reverse of the distribution of blocks on set partitions. We solve the conjecture for all the patterns except 2321. We describe the corresponding sets of Fishburn permutations by pattern avoidance, and leave some open questions for future work.

Repeated time-series cross-validation: A new method to improved COVID-19 forecast accuracy in Malaysia

Forecasting COVID-19 cases is challenging, and inaccurate forecast values will lead to poor decision-making by the authorities. Conversely, accurate forecasts aid Malaysian government authorities and agencies (National Security Council, Ministry of Health, Ministry of Finance, Ministry of Education, and Ministry of International Trade and Industry) and financial institutions in formulating action plans, regulations, and legal acts to control COVID-19 spread in the country. Therefore, this study proposes Repeated Time-Series Cross-Validation, a new data-splitting strategy to identify the best forecasting model that is capable of producing the lowest error measures value and a high percentage of forecast accuracy for COVID-19 prediction in Malaysia. Some of the highlights of the proposed method are:•A total of 21 models, five data partitioning sets, and four error measures to improve the forecast accuracy of daily COVID-19 cases in Malaysia.•The best model selected produces the lowest error measure value for the Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Mean Absolute Scaled Error (MASE).•The average 8-day forecast accuracy is 90.2 %. The lowest and highest forecast accuracy was 83.7 % and 98.7 %.

An Open-Source Implementation of the Scaffold Identification and Naming System (SCINS) and Example Applications.

Organizing and partitioning sets of chemical structures is of considerable practical significance, e.g., in compound library analysis and the postprocessing of screening hit lists. Approaches such as unsupervised clustering are computationally demanding and dataset-dependent; on the other hand, rule-based methods, such as those based on Murcko scaffolds, have linear time complexity but are often too fine-grained, leading to a large number of singletons or sparsely populated classes. An alternative rule-based method that seeks to achieve an optimal balance when grouping compounds into sets is the 'Scaffold Identification and Naming System' (SCINS). To facilitate public use of this previously published method, here, we provide an open-source Python implementation of SCINS, dependent only on RDKit. We show that SCINS can be useful in identifying sparsely and densely populated regions in chemical space in large databases, here exemplified with Enamine REAL Diverse and ChEMBL. We find that Enamine REAL Diverse covers a much smaller SCINS space relative to ChEMBL, whereas the opposite is true when Murcko and generic Murcko scaffolds are considered. Additionally, we show that SCINS can result in chemically intuitive grouping of medium-sized sets of bioactive compounds, which can be useful in compound selection from virtual screening campaigns as well as postprocessing of experimental hit lists. Hence, in this work, we provide both an open-source implementation of SCINS and its characterization with relevant use cases.

SWMR Algorithm Approach for the Vendor Managed Inventory Problem when Static Demand Rates

In a two-phase supply chain with Vendor Managed Inventory (VMI) policies, the supplier takes on the duty of keeping its retailers' inventory up to date and making sure they never run out of stock. This is in addition to his inbound inventory. This study aims to combine the retailers in various set partitions or to maximize their distribution and inventory holding costs within the same set partition. This paper focuses on a two-stage supply chain made of a single warehouse and multiple retailers (SWMR) facing a static demand and implementing VMI. We proposed a model for coordinating the system with a two-phase solution method. The first phase is to minimize the overall inventory costs and, in the second phase, a constructive vehicle routing problem (VRP) heuristic is used to optimize the distribution costs by satisfying some additional restrictions. Results of this implementation of VMI in an illustrative supply chain case are shown and discussed in detail.

Partitioning the projective plane into two incidence‐rich parts

AbstractAn internal or friendly partition of a vertex set of a graph is a partition to two nonempty sets such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.

Solving the Problem of Fuzzy Partition-Distribution with Determination of the Location of Subset Centers

A large number of real-world problems from various fields of human activity can be reduced to optimal partitioning-allocation problems with the purpose of minimizing the partitioning quality criterion. A typical representative of such problem is an infinite-dimensional transportation problem and more generalized problems—infinite-dimensional problems of production centers placement along with the partitioning of the area of continuously distributed consumers with the purpose of minimizing transportation and production costs. The relevant problems are characterized by some kind of uncertainty level of a not-probabilistic nature. A method is proposed to solve an optimal fuzzy partitioning-allocation problem with the subsets centers placement for sets of n-dimensional Euclidean space. The method is based on the synthesis of the methods of fuzzy theory and optimal partitioning-allocation theory, which is a new science field in infinite-dimensional mathematical programming with Boolean variables. A theorem was proved that determines the form of the optimal solution of the corresponding optimal fuzzy partitioning-allocation problem with the subsets centers placement for sets of n-dimensional Euclidean space. An algorithm for solving fuzzy partitioning-allocation problems is proposed, which is based on the proved theorem and on a variant of Shor’s r-algorithm—a non-differential optimization method. The application of the proposed method is demonstrated on model tasks, where the coefficient of mistrust is integrated to interpret the obtained result—the minimum value of the membership function, which allows each point of the set partition to be assigned to a specific fuzzy subset.

Multi-phase weighted stator current tracking using a hyper-plane partition of the control set

This paper presents a new optimization technique for Finite Control Set Model Predictive Control (FCS-MPC) of multi-phase induction machines. The proposal allows for a fast computation of the control signal while simultaneously considering stator currents in the α−β (torque producing) plane and the x−y (harmonics) plane. The combinatorial search of the optimization phase is eliminated. Also, unlike previous approaches, the proposed region identification algorithm can be utilized in conjunction with weighting factors in the cost function. The optimization is done using a hyper-plane partition of the control set. The partition provides the weighted FCS-MPC solution with significantly less computation needs allowing for shorter sampling periods. The method is tested experimentally in a 5-phase induction motor showing enhanced results.

Decomposition of the λ -Fold Complete Equipartite Graph into Unicyclic Graphs of Order Five

For a graph G and a subgraph H of a graph G , an H -decomposition of the graph G is a partition of the edge set of G into subsets E i , 1 ≤ i ≤ k , such that each E i induces a graph isomorphic to H . In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the λ -fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

Tiling with monochromatic bipartite graphs of bounded maximum degree

AbstractWe prove that for any , there exists a constant such that the following is true. Let be an infinite sequence of bipartite graphs such that and hold for all . Then, in any ‐edge‐coloured complete graph , there is a collection of at most monochromatic subgraphs, each of which is isomorphic to an element of , whose vertex sets partition . This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi‐colour Ramsey numbers of bounded‐degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.

Two parameterized deformed Poisson type operator and the combinatorial moment formula

In this paper, we shall introduce two parameterized deformation of the classical Poisson random variable from the viewpoint of noncommutative probability, namely (q,s)-Poisson type operator (random variable) on the two parameterized deformed Fock space, namely, the (q,s)-Fock space constructed by the weighted q-deformation approach in [11,4] (see also [6]). The recurrence formula for the orthogonal polynomials of the (q,s)-deformed Poisson distribution is determined. Moreover we shall also give the combinatorial moment formula of the (q,s)-Poisson type operator by using the set partitions and the card arrangement technique with their statistics. Our method presented in this paper provides nice combinatorial interpretations to parameters, q and s. The deformation presented in this paper can be regarded as a generalization of the Al-Salam-Carlitz type, because the restricted case s=q recovers the q-Charlier polynomials of Al-Salam-Carlitz type appeared in combinatorics [17].

Machine Learning for K-Adaptability in Two-Stage Robust Optimization

Two-stage robust optimization problems constitute one of the hardest optimization problem classes. One of the solution approaches to this class of problems is K-adaptability. This approach simultaneously seeks the best partitioning of the uncertainty set of scenarios into K subsets and optimizes decisions corresponding to each of these subsets. In a general case, it is solved using the K-adaptability branch-and-bound algorithm, which requires exploration of exponentially growing solution trees. To accelerate finding high-quality solutions in such trees, we propose a machine learning-based node selection strategy. In particular, we construct a feature engineering scheme based on general two-stage robust optimization insights, which allows us to train our machine learning tool on a database of resolved branch-and-bound trees and to apply it as is to problems of different sizes and/or types. We experimentally show that using our learned node selection strategy outperforms a vanilla, random node selection strategy when tested on problems of the same type as the training problems as well as in cases when the K-value or the problem size differs from the training ones. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grants OCENW.GROOT.2019.015 and VI.Veni.191E.035]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0314 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0314 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Markov chains generating random permutations and set partitions

The Chinese Restaurant Process may be considered to be a Markov chain which generates permutations on n elements proportionally to absorption probabilities θ|π|, θ>0, where |π| is the number of cycles of permutation π. We prove a theorem which provides a way of finding Markov chains, restricted to directed graphs called arborescences, and with given absorption probabilities. We find transition probabilities for the Chinese Restaurant Process arborescence with variable absorption probabilities. The method is applied to an arborescence constructing set partitions, resulting in an analogue of the Chinese Restaurant Process for set partitions. We also apply our method to an arborescence for the Feller Coupling Process. We show how to modify the Chinese Restaurant Process, its set partition analogue, and the Feller Coupling Process to generate derangements and set partitions having no blocks of size one.

Image Compression Based on DWT Using EZW/SPIHT Encoders

this paper reviews recent research on image compression techniques utilizing the Embedded Zero-Tree Wavelet (EZW) and Set Partitioning in Hierarchical Trees (SPIHT) algorithms. The study addresses the question of which algorithm, EZW or SPIHT, provides superior compression performance for various image types. The research method involves a comprehensive analysis of published research papers that employed these algorithms for image compression. The findings indicate that SPIHT generally outperforms EZW in terms of compression ratio, computational efficiency, and image quality, making it a more versatile and effective choice for various image compression applications.

Stable recovery of piecewise constant conductance on spider networks

ABSTRACT We address the discrete inverse conductance problem for well-connected spider networks, that is, to recover the conductance function on a well-connected spider network from the Dirichlet-to-Neumann map. It is well-known that this inverse problem is exponentially ill-posed, requiring the implementation of a regularization strategy for numerical solutions. Our focus lies in exploring whether prior knowledge of the conductance being piecewise constant within a partition of the edge set comprising a few subsets enables stable conductance recovery. To achieve this, we propose formulating the problem as a polynomial optimization problem, incorporating a regularization term that accounts for the piecewise constant hypothesis. We show several experimental examples in which the stable conductance recovery under the aforementioned hypothesis is feasible.

Robust image tamper detection and recovery with self-embedding watermarking using SPIHT and LDPC

In today’s digital landscape, the pervasive use of digital images across diverse domains has led to growing concerns regarding their authenticity and reliability. The potential for malicious manipulation of these images underscores the critical need to develop robust methods for detecting tampering and ensuring their integrity. Fragile watermarking has been found to have extensive applications for tamper detection and recovery. An image watermarking technique for tamper detection, correction, and recovery is presented in this study. The proposed method employs a self-embedding method to generate the reference watermark from the original image, which has the advantage of superior tamper detection, localization, recovery capabilities, and robustness against attacks. Set Partitioning in the Hierarchical Tree (SPIHT) algorithm is applied to generate a reference watermark of the original image. Low-Density Parity Check (LDPC) is employed for error correction, providing higher-quality reconstruction to recover the original image. Schur decomposition processed the watermarked image blocks to generate authentication bits for each block to enhance tampering detection. The proposed watermarking method was evaluated using PSNR, SSIM, BER, and NC metrics for grayscale and colored images. The technique demonstrated high robustness in tamper detection and recovery against various malicious attacks. Comparative analysis with existing methods shows the efficacy of the proposed method.

A Class of Polynomial Recurrences Resulting in (n/logn,n/log2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {n}}/{\ ext {log}}\\,\\,{\ ext {n}}, {\ ext {n}}/{\ ext {log}}^2{\ ext {n}})$$\\end{document}–Asymptotic Normality

We consider sequences of polynomials that satisfy differential–difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. It is, therefore, of interest to understand the properties of such polynomials and their probabilistic consequences. We identify a class of polynomial recurrences that lead to a normal law with the expected value and the variance proportional to n/logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/\\log ~n$$\\end{document} and n/log2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/\\log ^2n$$\\end{document}, respectively. Examples include Stirling numbers of the second kind and other polynomials concerning set partitions as well as polynomials related to Whitney numbers of Dowling lattices.

Quantitative determination of oxides in cement raw meal based on near-infrared spectroscopy and hybrid feature selection strategy

Determining the content of components in cement raw meal is vital in Portland cement manufacturing. Near-infrared spectroscopy (NIRS)enables rapid, safe, and accurate quantitative analysis of the primary constituents of cement raw meal (CaO, SiO2, Al2O3, and Fe2O3), thereby assisting the cement industry in reducing the time delay in raw meal proportioning control and enhancing the quality of the raw meal. This study employed a hybrid feature selection strategy to identify effective feature points in the NIRS of raw meal samples. Effective feature points from the raw meal were integrated with partial least squares (PLS) to develop a regression model. Initially, NIRS data from raw meal samples were obtained using a Fourier NIR spectrometer. Subsequently, sample set partitioning based on joint X-Y distance (SPXY) was utilized to divide the samples into a calibration set (70 samples) and a validation set (22 samples). Next, A backward interval partial least squares (BiPLS) approach, along with various preprocessing algorithms, was then employed to initially screen the optimal waveband combinations of the NIRS. Within these combinations, effective feature points were further identified using competitive adaptive reweighted sampling (CARS), variable combined population analysis (VCPA) and weighted fusion variable space shrinkage approach (WFVSSA). Finally, we developed and compared PLS models based on the full spectrum and three hybrid feature selection methods. The results demonstrated that.WFVSSA offers a comprehensive assessment of wavelength importance, effectively preserving feature points while eliminating redundant information. BiPLS-WFVSSA-PLS outperformed BiPLS-CARS-PLS and BiPLS-VCPAPLS on the validation set, with R2, RMSEP, and RPD values of 0.8676, 0.1431, and 2.8129 for CaO; 0.9212, 0.1216, and 3.6461 for SiO2; 0.9198, 0.0607, and 3.5618 for Al2O3; and 0.8717, 0.0166, and 2.8614 for Fe2O3, respectively. These findings underscore the superior accuracy and stability of BiPLS-WFVSSA-PLS in determining cement raw meal component contents.

Set partitions with colored singleton blocks

Set partitions with colored singleton blocks

Set partitions avoid a four-letter pattern

We present an explicit formula for the generating function for the number of set partitions of {1,2,…,n} that avoid a four-letter pattern 1abc.