Disjoint Sets Research Articles

Multi-Input data ASsembly for joint Analysis (MIASA): A framework for the joint analysis of disjoint sets of variables.

The joint analysis of two datasets [Formula: see text] and [Formula: see text] that describe the same phenomena (e.g. the cellular state), but measure disjoint sets of variables (e.g. mRNA vs. protein levels) is currently challenging. Traditional methods typically analyze single interaction patterns such as variance or covariance. However, problem-tailored external knowledge may contain multiple different information about the interaction between the measured variables. We introduce MIASA, a holistic framework for the joint analysis of multiple different variables. It consists of assembling multiple different information such as similarity vs. association, expressed in terms of interaction-scores or distances, for subsequent clustering/classification. In addition, our framework includes a novel qualitative Euclidean embedding method (qEE-Transition) which enables using Euclidean-distance/vector-based clustering/classification methods on datasets that have a non-Euclidean-based interaction structure. As an alternative to conventional optimization-based multidimensional scaling methods which are prone to uncertainties, our qEE-Transition generates a new vector representation for each element of the dataset union [Formula: see text] in a common Euclidean space while strictly preserving the original ordering of the assembled interaction-distances. To demonstrate our work, we applied the framework to three types of simulated datasets: samples from families of distributions, samples from correlated random variables, and time-courses of statistical moments for three different types of stochastic two-gene interaction models. We then compared different clustering methods with vs. without the qEE-Transition. For all examples, we found that the qEE-Transition followed by Ward clustering had superior performance compared to non-agglomerative clustering methods but had a varied performance against ultrametric-based agglomerative methods. We also tested the qEE-Transition followed by supervised and unsupervised machine learning methods and found promising results, however, more work is needed for optimal parametrization of these methods. As a future perspective, our framework points to the importance of more developments and validation of distance-distribution models aiming to capture multiple-complex interactions between different variables.

Bipartite networks represent causality better than simple networks: evidence, algorithms, and applications.

A network, whose nodes are genes and whose directed edges represent positive or negative influences of a regulatory gene and its targets, is often used as a representation of causality. To infer a network, researchers often develop a machine learning model and then evaluate the model based on its match with experimentally verified "gold standard" edges. The desired result of such a model is a network that may extend the gold standard edges. Since networks are a form of visual representation, one can compare their utility with architectural or machine blueprints. Blueprints are clearly useful because they provide precise guidance to builders in construction. If the primary role of gene regulatory networks is to characterize causality, then such networks should be good tools of prediction because prediction is the actionable benefit of knowing causality. But are they? In this paper, we compare prediction quality based on "gold standard" regulatory edges from previous experimental work with non-linear models inferred from time series data across four different species. We show that the same non-linear machine learning models have better predictive performance, with improvements from 5.3% to 25.3% in terms of the reduction in the root mean square error (RMSE) compared with the same models based on the gold standard edges. Having established that networks fail to characterize causality properly, we suggest that causality research should focus on four goals: (i) predictive accuracy; (ii) a parsimonious enumeration of predictive regulatory genes for each target gene g; (iii) the identification of disjoint sets of predictive regulatory genes for each target g of roughly equal accuracy; and (iv) the construction of a bipartite network (whose node types are genes and models) representation of causality. We provide algorithms for all goals.

Tilings of [formula omitted] with multisets of distances

In this paper, we study tilings of Z, that is, coverings of Z by disjoint sets (tiles). Let T={d1,…,ds} be a given multiset of distances. Is it always possible to tile Z by tiles, for which the multiset of distances between consecutive points is equal to T? In this paper, we give a sufficient condition that such a tiling exists. Our result allows multisets of distances to have arbitrarily many distinct values. Our result generalizes most of the previously known results, all of which dealt with the cases of 2 or 3 distinct distances.

Survey on A Smart Medical System on Drug Analysis using ML Algorithms

Online health communities continue to offer huge variety of medical information useful for medical practitioners, system administrators and patients alike. In this system we collect real time health posts from reputed websites, where patients express their views, including their experiences and side-effects on drugs used by them. We propose to perform Summarization of user posts per drug, and come out with useful conclusions for medical fraternity as well as patient community at a glance. Further, we propose to classify the users based on their ‘emotional state of mind’. Also, we shall perform knowledge discovery from user posts, whereby useful ‘patterns’ about the triad ‘drugs-symptoms-medicine’ is done by Association Rule Mining. Association rule mining is a popular and widely-known machine learning task. It is used to find out interesting relations between variables in large database. Rules generated by association have two disjoint set of items having form LHS (Left Hand Side) => RHS (Right Hand Side). The rule says that RHS is likely to occur whenever the LHS set occurs.

A generalization of Bondy’s pancyclicity theorem

AbstractThe bipartite independence number of a graph $G$ , denoted as $\tilde \alpha (G)$ , is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$ , there is an edge between $A$ and $B$ . McDiarmid and Yolov showed that if $\delta (G)\geq \tilde \alpha (G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $\delta (G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$ . In this paper, we show that $\delta (G)\geq \tilde \alpha (G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$ , thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.

Random Sampling Method of Large-Scale Graph Data Classification

Graph data appears in broad real-world applications in modelling complex objects in big data. Effective analysis of graph data provides a deeper understanding of the data in data mining tasks, including classification, clustering, prediction, and recommendation systems. Mining a large number of graphs becomes a challenging task because state-of-the-art methods are not scalable due to the memory limit. To address this issue, we propose a novel approximate random sampling method for large-scale graph data classification. In this approach, we applied a representation method to encode each graph as a record of a vector string and a set of graphs as a set of N records in a file. Then, we partition the set of records into disjoint subsets of data blocks, making each data block a random sample of the data file. After that, we randomly select a subset of data blocks, each being a random sample of the graph dataset, and compute the different graph property distributions. Since the data blocks in this model are much smaller than the entire data set, it is more efficient to analyze them on a standalone small machine, and multiple data blocks can be analyzed on multiple nodes of the cluster in parallel. Finally, we classified the graphs of data blocks using the SVM algorithm. In experimental evaluation, our proposed method outperformed state-of-the-art graph kernels on graph classification datasets in terms of accuracy.

Testability of positive integers

Given a set [Formula: see text] of some positive integers, a positive integer [Formula: see text] is [Formula: see text]-testable if there are two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]. The testability of the set [Formula: see text] of positive integers not exceeding [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there is a set [Formula: see text] with [Formula: see text] positive integers satisfying that each [Formula: see text] is [Formula: see text]-testable. In this short paper, we prove that [Formula: see text] and propose an algorithm to output the representation of an arbitrary positive integer by the form [Formula: see text], where [Formula: see text] and [Formula: see text] are two disjoint sets of some non-negative integers not exceeding [Formula: see text].

E3WD: A three-way decision model based on ensemble learning

Three-way decision model is an effective way to deal with complex decision problems. However, since the three-way decision models now proposed are all based on a single decision criterion, the decision results typically reflect only one preference of decision-makers. Thus, these models may also not effectively deal with complex decision-making problems. To solve the above problems, this paper proposes a new three-way decision model based on ensemble learning. Specifically, we first obtain different three-way decision results by employing different decision criteria. Then, we can acquire the core and candidate sets of the positive and negative regions through set operations. Next, we use the K-means algorithm to divide the candidate sets into three disjoint subsets based on similarities. After that, we adopt a hierarchical filtering method to select suitable objects from the candidate sets and add them to the core sets. Finally, we employ four three-way decision models with different decision criteria as examples to conduct experiments on eight datasets. Experimental results show that our proposed model can obtain higher classification accuracy and lower deferment rate than other traditional three-way decision models under most experimental conditions.

Analyzing Russia–Ukraine War Patterns Based on Lanchester Model Using SINDy Algorithm

In this paper, we present an effective method for analyzing patterns in the Russia–Ukraine war based on the Lanchester model. Due to the limited availability of information on combat powers of engaging forces, we utilize the loss of armored equipment as the primary data source. To capture the intricate dynamics of modern warfare, we partition the combat loss data into disjoint subsets by examining their geometric properties. Separate systems of ordinary differential equations for these subsets are then identified using the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm under a generalized formulation of the historical Lanchester model. We provide simulations of our method to demonstrate its effectiveness and performance in analyzing contemporary warfare dynamics.

K-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities

Abstract In this article, our main aim is to investigate the existence of radial k k -convex solutions for the following Dirichlet system with k k -Hessian operators: S k ( D 2 u ) = λ 1 ν 1 ( ∣ x ∣ ) ( − u ) p 1 ( − v ) q 1 in ℬ ( R ) , S k ( D 2 v ) = λ 2 ν 2 ( ∣ x ∣ ) ( − u ) p 2 ( − v ) q 2 in ℬ ( R ) , u = v = 0 on ∂ ℬ ( R ) . \left\{\begin{array}{ll}{S}_{k}\left({D}^{2}u)={\lambda }_{1}{\nu }_{1}\left(| x| ){\left(-u)}^{{p}_{1}}{\left(-v)}^{{q}_{1}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ {S}_{k}\left({D}^{2}v)={\lambda }_{2}{\nu }_{2}\left(| x| ){\left(-u)}^{{p}_{2}}{\left(-v)}^{{q}_{2}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ u=v=0& {\rm{on}}\hspace{1em}\partial {\mathcal{ {\mathcal B} }}\left(R).\end{array}\right. Here, u p 1 v q 1 {u}^{{p}_{1}}{v}^{{q}_{1}} is called a Lane-Emden type nonlinearity. The weight functions ν 1 , ν 2 ∈ C ( [ 0 , R ] , [ 0 , ∞ ) ) {\nu }_{1},{\nu }_{2}\in C\left(\left[0,R],\left[0,\infty )) with ν 1 ( r ) > 0 < ν 2 ( r ) {\nu }_{1}\left(r)\gt 0\lt {\nu }_{2}\left(r) for all r ∈ ( 0 , R ] r\in \left(0,R] , p 1 , q 2 {p}_{1},{q}_{2} are nonnegative and q 1 , p 2 {q}_{1},{p}_{2} are positive exponents, ℬ ( R ) = { x ∈ R N : ∣ x ∣ < R } {\mathcal{ {\mathcal B} }}\left(R)=\left\{x\in {{\mathbb{R}}}^{N}:| x| \lt R\right\} , N ≥ 2 N\ge 2 is an integer, N 2 ≤ k ≤ N \frac{N}{2}\le k\le N . In order to achieve our main goal, we first study the existence of radial k k -convex solutions of the above-mentioned systems with general nonlinear terms by using the upper and lower solution method and Leray-Schauder degree. Based on this, by constructing a continuous curve, which divides the first quadrant into two disjoint sets, we obtain the existence and multiplicity of radial k k -convex solutions for the system depending on the parameters λ 1 {\lambda }_{1} , λ 2 {\lambda }_{2} and the continuous curve.

Progressive Multiple Alignment of Graphs

The comparison of multiple (labeled) graphs with unrelated vertex sets is an important task in diverse areas of applications. Conceptually, it is often closely related to multiple sequence alignments since one aims to determine a correspondence, or more precisely, a multipartite matching between the vertex sets. There, the goal is to match vertices that are similar in terms of labels and local neighborhoods. Alignments of sequences and ordered forests, however, have a second aspect that does not seem to be considered for graph comparison, namely the idea that an alignment is a superobject from which the constituent input objects can be recovered faithfully as well-defined projections. Progressive alignment algorithms are based on the idea of computing multiple alignments as a pairwise alignment of the alignments of two disjoint subsets of the input objects. Our formal framework guarantees that alignments have compositional properties that make alignments of alignments well-defined. The various similarity-based graph matching constructions do not share this property and solve substantially different optimization problems. We demonstrate that optimal multiple graph alignments can be approximated well by means of progressive alignment schemes. The solution of the pairwise alignment problem is reduced formally to computing maximal common induced subgraphs. Similar to the ambiguities arising from consecutive indels, pairwise alignments of graph alignments require the consideration of ambiguous edges that may appear between alignment columns with complementary gap patterns. We report a simple reference implementation in Python/NetworkX intended to serve as starting point for further developments. The computational feasibility of our approach is demonstrated on test sets of small graphs that mimimc in particular applications to molecular graphs.

Application of the integral operator method for multispin correlation function calculations in the one-dimensional Ising model.

The one-dimensional (1D) Ising model is revisited. The integral operator method is used for the model to derive the general statistical identity for multispin correlation functions. As an application of this identity to generate various correlations, recursive formulas are derived for the correlations of pairs, threes, fours, and fives of spins. Both compact clusters and disjoint sets of spins are considered. It is shown that multispin correlations for the 1D Ising model can be exactly expressed in terms of pairwise correlations using recursive procedures. For clusters with different structures, numerical calculations of multispin correlations as a function of temperature and external magnetic field are performed. The results of these calculations are presented in the figuresand discussed.

Robust H2 filtering of phase‐type semi‐Markov jump linear systems with cluster observations

AbstractIn this work we study the design of filters for phase‐type (PH) semi‐Markov jump linear systems, considering partial information on the system's operating mode and possible parameter uncertainties. With respect to the mode of operation, it is assumed that the state space of the semi‐Markov chain can be written as the union of disjoint sets, called clusters, and the only information available to the filter is which cluster the state of the semi‐Markov chain belongs to. A new linear matrix inequality (LMI) parameterization employing the slack variable technique is introduced for the design of switching full‐order filters according to the cluster observations so that suitable bounds on the norm of the estimation error are guaranteed for the uncertain system. If the system is restricted to be a Markov jump linear system and the Markov chain is assumed to be perfectly measured, the design conditions are shown to be also necessary leading to the optimal full‐order filter. Furthermore, the general filter can be particularized into an observer form for the case in which the system matrices are the same within each cluster and there are no parameter uncertainties except for possible uncertainties affecting the transition rates of the jumping process. The paper concludes with an illustrative example in the context of networked control systems.

A multi-neighborhood tabu search for solving multi-budget maximum coverage problem

Many practical decision-making problems involve selecting k distinct subsets of objects from a set of candidate objects such that the chosen objects optimize a given goal while satisfying some constraints. The multi-budget maximum coverage problem (MMCP) is a general model that facilitates formulating such decision-making problems. Given a budget of k, a set of elements with profits, and a set of items with costs where each item is composed of a subset of the elements, the MMCP aims to select k disjoint subsets of items that maximize the total profits of the elements covered by these subsets, while their costs do not exceed their respective budgets. In this work, we present a multi-neighborhood tabu search (MTS) for this NP-hard problem. MTS comprises three essential components that employ four efficient neighborhoods to collaboratively perform the neighborhood exploration to obtain high-quality local optima, select neighborhood solutions with added perturbations, and generate offspring solutions based on population information exchange. To evaluate the effectiveness of MTS, we performed computations on 30 benchmark instances of the extended BMCP. We compared the results with the approximation algorithm, genetic algorithm, particle swarm optimization algorithm, and CPLEX solver. We also provide experiments to highlight the beneficial effect of the essential components in the MTS algorithm. The experimental results demonstrate that MTS not only performs efficiently but also yields solutions of superior quality.

Three remarks on W2 graphs

Let k≥1. A graph G is Wk if for any k pairwise disjoint independent vertex subsets A1,…,Ak in G, there exist k pairwise disjoint maximum independent sets S1,…,Sk in G such that Ai⊆Si for i∈[k]. Recognizing W1 graphs is coNP-hard, as shown by Chvátal and Slater (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing Wk graphs is coNP-hard for k≥2. On the positive side, we show that recognizing Wk graphs is, for each k≥2, FPT parameterized by clique-width and by tree-width. Finally, we construct graphs G that are not W2 such that, for every vertex v in G and every maximal independent set S in G−N[v], the largest independent set in N(v)∖S consists of a single vertex, thereby refuting a conjecture of Levit and Tankus.

N-bipolar hypersoft sets: Enhancing decision-making algorithms.

This paper introduces N-bipolar hypersoft (N-BHS) sets, a versatile extension of bipolar hypersoft (BHS) sets designed to effectively manage evaluations encompassing both binary and non-binary data, thereby exhibiting heightened versatility. The major contributions of this framework are twofold: Firstly, the N-BHS set introduces a parameterized representation of the universe, providing a nuanced and finite granularity in perceiving attributes, thereby distinguishing itself from conventional binary BHS sets and continuous fuzzy BHS sets. Secondly, this model signifies a new area of research aimed at overcoming limitations inherent in the N-bipolar soft set when handling multi-argument approximate functions. Through the strategic partitioning of attributes into distinct subattribute values using disjoint sets, the N-BHS set emerges as a powerful tool for effectively addressing uncertainty-related problems. In pursuit of these objectives, the paper outlines various algebraic definitions, including incomplete N-BHS sets, efficient N-BHS sets, normalized N-BHS sets, equivalence under normalization, N-BHS complements, and BHS sets derived from a threshold, exemplified through illustrative examples. Additionally, the article explores set-theoretic operations within the N-BHS sets framework, such as relative null/whole N-BHS sets, N-BHS subsets, and two distinct approaches to N-BHS extended/restricted union and intersection. Finally, it proposes and compares decision-making methodologies regarding N-BHS sets, including a comprehensive comparison with relevant existing models.

Approximating subset sum ratio via partition computations

We present a new FPTAS for the Subset Sum Ratio problem, which, given a set of integers, asks for two disjoint subsets such that the ratio of their sums is as close to 1 as possible. Our scheme makes use of exact and approximate algorithms for Partition, and clearly showcases the close relationship between the two algorithmic problems. Depending on the relationship between the size of the input set n and the error margin ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, we improve upon the best currently known algorithm of Melissinos and Pagourtzis [COCOON 2018] of complexity O(n4/ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O} (n^4 / \\varepsilon )$$\\end{document}. In particular, the exponent of n in our proposed scheme may decrease down to 2, depending on the Partition algorithm used.

A critical probability for biclique partition of Gn,p

The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G) with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take G=Gn,p, where p is constant and less than a certain threshold value p0≈0.312. This verifies a conjecture of Chung and Peng for these values of p. We also show that if p0<p<1/2 then bp(Gn,p)=n−(1+Θ(1))α(Gn,p) with high probability.

Morph-KGC: Scalable knowledge graph materialization with mapping partitions

Knowledge graphs are often constructed from heterogeneous data sources, using declarative rules that map them to a target ontology and materializing them into RDF. When these data sources are large, the materialization of the entire knowledge graph may be computationally expensive and not suitable for those cases where a rapid materialization is required. In this work, we propose an approach to overcome this limitation, based on the novel concept of mapping partitions. Mapping partitions are defined as groups of mapping rules that generate disjoint subsets of the knowledge graph. Each of these groups can be processed separately, reducing the total amount of memory and execution time required by the materialization process. We have included this optimization in our materialization engine Morph-KGC, and we have evaluated it over three different benchmarks. Our experimental results show that, compared with state-of-the-art techniques, the use of mapping partitions in Morph-KGC presents the following advantages: (i) it decreases significantly the time required for materialization, (ii) it reduces the maximum peak of memory used, and (iii) it scales to data sizes that other engines are not capable of processing currently.

Magnetic fractional Poincaré inequality in punctured domains

We study Poincaré-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb et al. (2003) [19] showed that, if a bounded domain Ω is the union of two disjoint sets Γ and Λ, then the Lp-norm of a function calculated on Ω is dominated by the sum of magnetic seminorms of the function, calculated on Γ and Λ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete.