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This paper proposes a completely non-centralized Volt/VAr control (VVC) algorithm for active distribution networks which are faced with voltage magnitude violations due to the high penetration of solar photovoltaics (PVs). The proposed VVC algorithm employs a two-stage architecture where the settings of the classical voltage control devices (VCDs) are decided in the first stage through a distributed optimization engine powered by the alternating direction method of multipliers (ADMM). In contrast, the PV smart inverters are instructed in the second stage through linear Q(P) decision rules - which are computed in a decentralized manner by leveraging robust optimization theory. The key to this non-centralized VVC routine is a proposed network partition methodology (NPM) which uses an electrical distance metric based on node Q􀀀jV j2 sensitivitiesfor computing an intermediate reduced graph of the network, which is subsequently divided into the final partitions using the spectral clustering technique. As a result, the final network partitions are connected, stable, close in cardinality, contain at least one PV inverter for zonal reactive power support, and are sufficiently decoupled from each other. Numerical results on the UKGDS-95 bus system show that the non-centralized solutions match closely with the centralized robust VVC schemes, thereby significantly reducing the voltage violations compared to the traditional deterministic VVC routines.

The scarcity of labels is common and great challenge in traditional supervised learning. Semi-supervised learning (SSL) leverages unlabeled samples to alleviate the absence of label information. Similar with annotation, label proportion is another type of prior information and plays a significant role in classification tasks. Compared with the acquisition of labels, label proportion can be obtained more easily. For example, only a small number of patients have been diagnosed with or not with cancers in hospital database, while the proportion with cancer can be generally estimated by historical records. How to incorporate such prior information of label proportion is crucial but rarely studied in literature. Traditional SSL methods often ignore this prior information and will lead to performance degradation inevitably. To solve this problem, we propose a novel SSL with Label Proportion (SSLLP). Our approach encourages to preserve label consistency and label proportion by imposing the cardinality bound constraints. Our formulated problem equals to a mixed-integer constrained submodular minimization and it is difficult to be solved directly. Therefore, we transformed the original problem into a convex one by Lov $\acute{\text{a}}$ sz extension and designed an efficient solving algorithm. Extensive experimental results present the improved performance of our method over several state-of-the-art methods.

Recently, stochastic hard thresholding (HT) optimization methods [e.g., stochastic variance reduced gradient hard thresholding (SVRGHT)] are becoming more attractive for solving large-scale sparsity/rank-constrained problems. However, they have much higher HT oracle complexities, especially for high-dimensional data or large-scale matrices. To address this issue and inspired by the well-known Gradient Support Pursuit (GraSP) method, this article proposes a new Relaxed Gradient Support Pursuit (RGraSP) framework. Unlike GraSP, RGraSP only requires to yield an approximation solution at each iteration. Based on the property of RGraSP, we also present an efficient stochastic variance reduction-gradient support pursuit algorithm and its fast version (called stochastic variance reduced gradient support pursuit (SVRGSP+). We prove that the gradient oracle complexity of both our algorithms is two times less than that of SVRGHT. In particular, their HT complexity is about Ks times less than that of SVRGHT, where Ks is the restricted condition number. Moreover, we prove that our algorithms enjoy fast linear convergence to an approximately global optimum, and also present an asynchronous parallel variant to deal with very high-dimensional and sparse data. Experimental results on both synthetic and real-world datasets show that our algorithms yield superior results than the state-of-the-art gradient HT methods.

Sketches have been widely applied to identify super hosts in an efficient and accurate way. However, most sketches cannot flexibly balance memory usage and accuracy in host cardinality estimation. In order to solve this issue, we propose a novel extensible and reversible sketch, named ExtendedSketch, to achieve accurate super host identification with high memory efficiency. The core idea of ExtendedSketch is to monitor low-cardinality hosts with small-sized counters while dynamically extend the size of counters when monitoring high-cardinality hosts by applying an adaptive extension strategy. Such the strategy can adaptively increase counter size according to network traffic status at runtime, which not only ensures the accuracy of high-cardinality host estimation but also avoids unnecessary memory consumption. We perform theoretical analysis and conduct a series of experimental evaluations on ExtendedSketch based on real world network traffic. Experimental results show that under same memory usage, compared to the state-of-the-art, ExtendedSketch achieves 1.47.5 times smaller error rate in measuring host cardinality with 1.926.7 times better accuracy on super host identification and 95215 times faster speed on abnormal address reconstruction. Its advance in accuracy and efficiency demonstrates the practical significance of ExtendedSketch for super host identification.

Many real-world datasets are given in the stream of entity-identifier pairs, and measuring data distribution on these datasets is fundamental for applications such as privacy protection. In this paper, we study the problem of computing the cumulative distribution for different cardinalities (i.e., the number of distinct entities owning the same identifier). However, previous sketch-based methods cost large memory space especially when there are a large number of identifiers, and sampling-based methods require much time for cardinality estimation. A recent work KHyperLogLog combines both sketch and sampling methods but it is wasteful to separately build a HyperLogLog sketch of large size for identifiers with small cardinalities. To address these challenges, we propose a memory-efficient method EV-HLL, which designs a shared structure to store all sampled identifiers and their entities and utilizes additional sketches to track value updates during the sampling procedure. Meanwhile, EV-HLL provides real-time unbiased estimations according to value changes whenever a new entity-identifier pair arrives. We evaluate the performance of EV-HLL and other state-of-the-arts on real-world available datasets. Experimental results demonstrate that comparing to other methods, EV-HLL effectively reduces their memory usage with the same estimation accuracy and has higher accuracy with the same memory usage.

Given a graph G and a set S ⊆ V ( G ) , the Δ -interval of S , [ S ] Δ , is the set formed by the vertices of S and every w ∈ V ( G ) forming a triangle with two vertices of S . If [ S ] Δ = S , then S is Δ -convex of G ; if [ S ] Δ = V ( G ) , then S is a Δ -interval set of G . The Δ -interval number of G is the minimum cardinality of a Δ -interval set and the Δ -convexity number of G is the maximum cardinality of a proper Δ -convex subset of V ( G ) . In this work, we show that the problem of computing the Δ -convexity number is W[1]-hard and NP-hard to approximate within a factor O ( n 1 − ɛ ) for any constant ɛ > 0 even for graphs with diameter 2 and that the problem of computing the Δ -interval number is NP-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the Δ -convexity number of chordal graphs and for computing the Δ -interval number of block graphs. We also present results on the Δ -hull, Δ -interval and Δ -convexity numbers concerning the three standard graph products, namely, the Cartesian, the strong and the lexicographic products, in function of these and well-studied parameters of the operands.

A set of vertices S of a graph G is a (geodesically) convex set, if S contains all the vertices belonging to any shortest path connecting two vertices of S . The cardinality of a maximum proper convex set of G is called the convexity number, con ( G ) , of G . The complementary prism G G ¯ of G is obtained from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. In this work, we examine the convex sets of the complementary prism of a tree and derive formulas for the convexity numbers of the complementary prisms of all trees.

Hash bit selection determines an optimal subset of hash bits from a candidate bit pool. It is formulated as a zero-one quadratic programming problem subject to binary and cardinality constraints. In this article, the problem is equivalently reformulated as a global optimization problem. A collaborative neurodynamic optimization (CNO) approach is applied to solve the problem by using a group of neurodynamic models initialized with particle swarm optimization iteratively in the CNO. Lévy mutation is used in the CNO to avoid premature convergence by ensuring initial state diversity. A theoretical proof is given to show that the CNO with the Lévy mutation operator is almost surely convergent to global optima. Experimental results are discussed to substantiate the efficacy and superiority of the CNO-based hash bit selection method to the existing methods on three benchmarks.

A subset D ⊆ V of a graph G = ( V , E ) is called a global total k -dominating set of G if D is a total k -dominating set of both G and G ¯ , the complement of G . The Minimum Global Total k - Domination problem is to find a global total k -dominating set of minimum cardinality of the input graph G and the Decide Global Total k - Domination problem is the decision version of the Minimum Global Total k - Domination problem. The Decide Global Total k - Domination problem is known to be NP -complete for general graphs. In this paper, we study the complexity of the Minimum Global Total k - Domination problem. We show the Decide Global Total k - Domination problem remains NP -complete for bipartite graphs and chordal graphs. We also show that Minimum Global Total k - Domination cannot be approximated within a factor of ( 1 − ϵ ) ln | V | for any ϵ > 0 unless P = NP for bipartite graphs as well as for chordal graphs. Next, we show that Minimum Global Total k - Domination admits a constant approximation algorithm for bounded degree graphs. Finally, we show that Minimum Global Total k - Domination problem is APX -complete for bounded degree graphs. We also show that Decide Global Total k - Domination problem is W[2] -hard for bipartite graphs and for chordal graphs when we choose the size of the GTkD-set as the parameter.

The medical diagnostic decision support system uses machine learning and data mining algorithms to detect and diagnose diseases. Several deaths can be avoided if the diseases are detected and cured in the early stages of infection. Feature selection is a major pre-processing method used to obtain the most significant features, thereby enhancing the data mining model's classification accuracy. This work proposes a new feature selection algorithm to perform feature selection as a multi-objective optimization problem. The minimization of classification error rate and minimization of the feature subset's cardinality are the two contradictory objectives that need to be optimized simultaneously. The proposed work is applied for five clinical datasets such as lung cancer, breast cancer, diabetes, fertility, and immunotherapy and the results are compared with existing techniques based on 6 other datasets. This work converts the real-valued Jaya Optimization Algorithm into binary space. It also handles premature convergence and sensitivity–specificity trade-off. The proposed algorithm's efficiency is assessed and analyzed based on average classification accuracy, sensitivity, specificity, number of features selected, percentage feature selection, and CPU computation time. The proposed algorithm improves the effectiveness of data mining based medical diagnostic decision support system.

This paper presents and evaluates a unified cooperative perception framework that employs vehicle-to-vehicle (V2V) connectivity. At the core of the framework is a decentralized data association and fusion process that is scalable with respect to participation variances. The evaluation considers the effects of the communication losses in the ad-hoc V2V network and the random vehicle motions in traffic by adopting existing models along with a simplified algorithm for individual vehicle's on-board sensor field of view. Furthermore, a multi-target perception metric is adopted to evaluate both the errors in the estimation of the motion states of vehicles in the surrounding traffic and the cardinality of the fused estimates at each participating node/vehicle. The extensive analysis results demonstrate that the proposed approach minimizes the perception metric for a much larger percentage of the participating vehicles than a baseline approach, even at modest participation rates, and that there are diminishing returns in these benefits. The computational and data traffic trade-offs are also analyzed.

This article proposes a binary tree of classifiers for multi-label classification that preserves label dependencies and handles class imbalance. At each node, the input data is strategically split into two subsets for its subsequent child nodes, keeping the label correlations intact. A novel approach of partitioning the data based on label-set proximity has also been proposed. Various data appropriate classifiers are trained to learn the binary partition at every internal node. The tree of classifiers grows iteratively depending on two parameters – multi-label entropy and sample cardinality, computed on the data at the current node. During training, the decision at any node is based on these parameters and the branching out is restricted, if deemed unnecessary. Specific classifiers at the leaf nodes perform the final classification task and assign appropriate label-sets to the unlabelled data. The proposed system aims to appropriately split the data and build the hierarchical structure such that the training and classification tasks become simpler. Also, the problem of class imbalance leads to the irregular splitting of data and excessive branching out of the tree which is handled through the novel use of suitable classifiers and parameters at the intermediate and leaf nodes. The proposed method has shown significant performance improvement on fourteen datasets against fourteen existing multi-label classifiers. Two-tailed Wilcoxon signed rank test statistics show that for $T_{Wilcoxon}(14,0.2)=31$ the proposed method outperforms all the other comparison models.

The online social network has become an integral part of modern society and serves as an excellent platform for information diffusion, marketing, and so on. The minimum positive influence dominating set (MPIDS) problem aims to find a set of at least half neighbors of every individual with minimum cardinality and has lots of applications in the social network. Even though numerous algorithms have been proposed for this hard combinatorial problem, challenges such as poor quality and scalability have motivated the researchers for better solutions. In this article, we focus on designing a simple and effective method and propose an iterated carousel greedy (ICG) algorithm for the MPIDS problem. The destruction, carousel, and reconstruction (DCR) procedures are well-designed based on the problem-specific knowledge to improve the algorithm performance, while maintaining the simplicity of the ICG algorithm. The ICG algorithm uses a fast greedy algorithm to generate an initial solution with a certain quality. It then obtains a sequence of better solutions by iterating over DCR phases. A carefully designed experiment is carried out to determine the best parameter configuration. The experimental results on real-world networks indicate that the proposed ICG algorithm significantly outperforms the current state-of-the-art algorithms.

Problem definition: We study a joint product framing and order fulfillment problem with both inventory and cardinality constraints faced by an e-commerce retailer. There is a finite selling horizon and no replenishment opportunity. In each period, the retailer needs to decide how to “frame” (i.e., display, rank, price) each product on his or her website as well as how to fulfill a new demand. Academic/practical relevance: E-commerce retail is known to suffer from thin profit margins. Using the data from a major U.S. retailer, we show that jointly planning product framing and order fulfillment can have a significant impact on online retailers’ profitability. This is a technically challenging problem as it involves both inventory and cardinality constraints. In this paper, we make progress toward resolving this challenge. Methodology: We use techniques such as randomized algorithms and graph-based algorithms to provide a tractable solution heuristic that we analyze through asymptotic analysis. Results: Our proposed randomized heuristic policy is based on the solution of a deterministic approximation to the stochastic control problem. The key challenge is in constructing a randomization scheme that is easy to implement and that guarantees the resulting policy is asymptotically optimal. We propose a novel two-step randomization scheme based on the idea of matrix decomposition and a rescaling argument. Managerial implications: Our numerical tests show that the proposed policy is very close to optimal, can be applied to large-scale problems in practice, and highlights the value of jointly optimizing product framing and order fulfillment decisions. When inventory across the network is imbalanced, the widespread practice of planning product framing without considering its impact on fulfillment can result in high shipping costs, regardless of the fulfillment policy used. Our proposed policy significantly reduces shipping costs by using product framing to manage demand so that it occurs close to the location of the inventory.

New bounds are proposed for the subset selection problem which consists in minimizing the residual sum of squares subject to a cardinality constraint on the maximum number of non-zero variables. They rely on new convex relaxations providing both upper and lower bounds that are compared with others present in the literature. The performance of these methods is illustrated through computational experiments.

The scenario approach is a general data-driven algorithm to chance-constrained optimization. It seeks the optimal solution that is feasible to a carefully chosen number of scenarios. A crucial step in the scenario approach is to compute the cardinality of essential sets, which is the smallest subset of scenarios that determine the optimal solution. This paper addresses the challenge of efficiently identifying essential sets. For convex problems, we demonstrate that the sparsest dual solution of the scenario problem could pinpoint the essential set. For non-convex problems, we show that two simple algorithms return the essential set when the scenario problem is non-degenerate. Finally, we illustrate the theoretical results and computational algorithms on security-constrained unit commitment (SCUC) in power systems. In particular, case studies of chance-constrained SCUC are performed in the IEEE 118-bus system. Numerical results suggest that the scenario approach could be an attractive solution to practical power system applications.

A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources ("cakes") or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-floor(2n/3) maximin-share allocation of discrete bads among n agents., Comment: Appeared in Information Sciences, 587:164--187. But during the production, the main theorem text was deleted. The arXiv version is the correct one

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

Sparse optimization-fitting data with a low-cardinality linear model-is addressed through the minimization of a cardinality-penalized least-squares function, for which dedicated branch-and-bound algorithms clearly outperform generic mixed-integerprogramming solvers. Three acceleration techniques are proposed for such algorithms. Convex relaxation problems at each node are addressed with dual approaches, which can early prune suboptimal nodes. Screening methods are implemented, which fix variables to their optimal value during the node evaluation, reducing the subproblem size. Numerical experiments show that the efficiency of such techniques depends on the node cardinality and on the structure of the problem matrix. Last, different exploration strategies are proposed to schedule the nodes. Best-first search is shown to outperform the standard depth-first search used in the related literature. A new strategy is proposed which first explores the nodes with the lowest least-squares value, which is shown to be the best at finding the optimal solution-without proving its optimality. A C++ solver with compiling and usage instructions is made available.

In this paper, we consider generating a representative subset of nondominated points at a prespecified precision in multi-objective mixed-integer programs (MOMIPs). The number of nondominated points grows exponentially with problem size and finding all nondominated points is typically hard in MOMIPs. Representing the nondominated set with a small subset of nondominated points is important for a decision maker to get an understanding of the layout of solutions. The shape and density of the nondominated points over the objective space may be critical in obtaining a set of solutions that represent the nondominated set well. We develop an exact algorithm that generates a representative set guaranteeing a prespecified precision. Our experiments on a variety of problems demonstrate that our algorithm outperforms existing approaches in terms of both the cardinality of the representative set and computation times.

We consider the maximum 2-club problem, which aims at finding an induced subgraph of maximum cardinality with the diameter at most two. Such subgraphs arise from a popular diameter-based clique relaxation concept, as a subgraph is a clique if and only if its diameter is one. In a 2-club every pair of non-adjacent vertices has a common neighbor; this “2-hop” property naturally arises in a variety of applications. In this paper, by exploiting a somewhat different interpretation of the problem, we provide two new mixed-integer programming (MIP) models for finding maximum 2-clubs. Our MIPs provide much tighter linear programming (LP) relaxations for sufficiently sparse graphs and have fewer constraints than the standard integer programming (IP) model at the expense of having slightly more continuous variables. We also consider feasibility versions of our MIPs that verify whether there exists a 2-club of some specified size. Then we incorporate them into a simple-to-implement “feasibility-check” algorithm that iteratively solves one of the feasibility MIPs for each possible 2-club size within some known lower and upper bounds. The upper bound is obtained from an LP relaxation of our new MIPs and is shown to be sharp. Furthermore, we show how to extend our approaches for solving some “robust” (attack- and failure-tolerant) generalizations of the maximum 2-club problem. Finally, we perform an extensive computational study with randomly generated and real-life graphs to support our theoretical results and to provide some empirical observations and insights.

Given a set X ⊆ V ( G ) , let [ X ] G denote the set of all vertices of G lying on some shortest path between two vertices of X . If [ X ] G = X , then X is a convex set of G . The convex hull of X , denoted by 〈 X 〉 G , is the smallest convex set containing X . We say that X is a hull set of G if 〈 X 〉 G = V ( G ) and that X is a geodetic set of G if [ X ] G = V ( G ) . The hull number h ( G ) of G is the minimum cardinality of a hull set of G ; and the geodetic number g ( G ) of G is the minimum cardinality of a geodetic set of G . The shadow graph of G , denoted by S ( G ) , is the graph obtained from G by adding a new vertex v ′ for each vertex v of G and joining v ′ to the neighbors of v in G . In this paper, we present sharp bounds for the geodetic and the hull numbers of shadow graphs. We characterize the classes of graphs for which some of these bounds are attained. We also prove for a fixed integer k , that the problem of deciding whether g ( S ( G ) ) ≤ n − k is NP -complete even if the diameter of G is 2; and that the problem of deciding whether h ( S ( G ) ) ≤ n − 1 belongs to P .

Constrained energy minimization (CEM) has shown great success in subpixel target detection. This paper develops a dummy variable trick (DVT) to extend CEM to CEM anomaly detector (CEM-AD) and shows that CEM-AD also enjoys the same success in animation detection (AD). Its idea converts a known specific target signature d imposed on CEM to an unknown specific target signature to develop a signal-to-background ratio (SBR)-constrained energy minimization (SBR-CEM) as an unknown specified-target constrained energy minimization (UST-CEM) which serves as a liaison to derive the desired CEM-AD without prior knowledge of d. Surprisingly, the derived CEM-AD turns out to be a sample correlation matrix R-based anomaly detector, R-AD in correspondence to the well-known sample covariance matrix K-based anomaly detector developed by Reed-Xiaoli, RX-AD. To further improve CEM-AD, a low rank and sparse matrix decomposition (LRaSMD) model introduced by go decomposition (GoDec) and its sparsity cardinality (SC) are further incorporated into CEM-AD where two new versions of SC, fixed sparsity cardinality (FSC) and variable sparsity cardinality (VSC), are particularly designed to enhance AD. Finally, to effectively evaluate AD performance, recently developed 3D receiver operating characteristic (ROC) curve-derived detection measures are used for comparative studies and analyses.

Online monitoring user cardinalities in graph streams is fundamental for many applications such as anomaly detection. These graph streams may contain edge duplicates and have a large number of user-item pairs, which makes it infeasible to exactly compute user cardinalities due to limited computational and memory resources. Existing methods are designed to approximately estimate user cardinalities, but their accuracy highly depends on complex parameters and they cannot provide anytime-available estimation. To address these problems, we develop novel bit/register sharing algorithms, which use a bit/register array to build a compact sketch of all users' connected items. Our algorithms exploit the dynamic properties of the bit/register arrays (e.g., the fraction of zero bits in the bit array) to significantly improve the estimation accuracy, and have low time complexity O(1) to update the estimations for a new user-item pair. In addition, our algorithms are simple and easy to use, without requirements to tune any parameter. Furthermore, we extend our methods to detect super spreaders with large cardinalities in real-time. We evaluate the performance of our methods on real-world datasets. The experimental results demonstrate that our methods are several times more accurate and faster than state-of-the-art methods using the same amount of memory.

We characterize coset spaces of topological groups which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups. Besides, it is shown that for a $G$-space $X$ with a $d$-open action there is a topological group $H$ of weight and cardinality less than or equal to the weight of $X$ such that $H$ admits a $d$-open action on $X$. This is further applied to show that if $X$ is a separable metrizable coset space then $X$ has a Polish extension which is a coset space of a Polish group.

In large-scale multiprocessor systems, it is significant to estimate the reliability of the interconnection networks. The κ k and λ k are two generalized measurements of reliability for networks. The κ k ( λ k ) of G is the minimum cardinality of a set of vertices (edges) of G , if any, whose deletion disconnects G , and every remaining component has minimum degree at least k . For the modified bubble-sort networks M B n , this paper proves κ k ( M B n ) = λ k ( M B n ) = 2 k ( n − k ) for k ≤ n / 2 . The conclusions extend some known results in the literature, imply that to make a modified bubble-sort network M B n disconnected with all remaining vertices have at least k neighbors, at least 2 k ( n − k ) vertices or edges have to be faulty.

This paper concentrates on tracking multiple targets using an asynchronous network of sensors with different sampling rates. First, a timely fusion approach is proposed for handling measurements from asynchronous sensors. In the proposed approach, the arithmetic average fusion of the estimates provided by local cardinalized probability hypothesis density filters is recursively carried out according to the network-wide sampling time sequence. The corresponding intersensor communication is conducted by a partial flooding protocol, in which either cardinality distributions or intensity functions pertinent to local posteriors are disseminated among sensors. Moreover, both feedback and non-feedback fusion-filtering modes are provided to meet the performance and real-time requirements, respectively. Second, an extension of the timely fusion approach referred to as robust bootstrap approach is presented, which can deal with unknown clutter and detection parameters by exploiting a local bootstrap filtering scheme. Finally, numerical simulations are performed to test the proposed approaches.

We consider here a scenario where a team of agents want to switch from a fully connected communication network to a network with limited bandwidth because of a cyber attack. The goal is to identify which communication link should still be active in the new network for near-optimal overall performance. This is formulated as a cardinality constrained quadratic minimization problem, which is NP-hard in general. To obtain an approximately optimal solution efficiently, we used random projection (RP) for dimensionality reduction. We show that the value of the sketched team is bounded above by a constant factor times the optimal value of the team with high probability.

We study a geometric property related to spherical hyperplane tessellations in R d . We first consider a fixed x on the Euclidean sphere and tessellations with M ≫ d hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of d log ⁡ ( d ) log ⁡ ( M ) such that the radius of the cell containing x induced by these hyperplanes is bounded above by, up to constants, d log ⁡ ( d ) log ⁡ ( M ) / M . We extend this result to hold for all cells in the tessellation with high probability. Up to logarithmic terms, this upper bound matches the previously established lower bound of Goyal et al. (1998) [6] .

Low-rank and sparse representation (LRaSR)-based approaches have been widely used for anomaly detection (AD). Their central ideas are to minimize the rank of the low-rank space constrained to predetermined values, while using various regularization parameters to control the sparse representation. Three key issues arise from LRaSR. The first is how to determine the constrained rank. The second is an appropriate selection of regularization parameters. The third one is the detector used for AD. This article presents a new but rather simple competing model, called component decomposition analysis (CDA) which represents a data space X as a linear orthogonal decomposition of three components, X = PC $^{{m}}$ + IC $^{{j}}$ + N with m principal components, PC $^{{m}}$ , generated by principal component analysis (PCA) and j independent components, IC $^{{j}}$ , generated by independent component analysis (ICA) plus a noise component N. CDA offers several advantages over LRaSR. First, CDA uses well-known component analysis techniques to decompose the dataset without solving constrained optimization problems. Second, the values of m and j can be automatically determined by virtual dimensionality (VD) and a minimax-singular value decomposition (MX-SVD). To better extract anomalies from the IC $^{{j}}$ component space, the concept of sparsity cardinality (SC) is further incorporated into CDA to derive a CDASC anomaly detector (CDASC-AD). The experimental results demonstrate that CDASC-AD is very competitive against the LRaSR-based models and performs well in hyperspectral AD.

Fil: Carpio, María Belén. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Investigaciones Geohistóricas; Argentina Fil: Carpio, María Belén. Universidad Nacional del Nordeste; Argentina Fil: Gómez, Cecilia Paula. Pontificia Universidad Católica Argentina. Instituto de Investigaciones; Argentina Fil: Gómez, Cecilia Paula. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Resumen: Este artículo analiza las distintas maneras en que los tobas del oeste de Formosa, qomléʔk (Argentina), también conocidos como tobas de Sombrero Negro (Métraux 1937a y b), se refieren a las zonas cardinales y a sus contextos de uso. Postulamos que las definiciones que describen y nombran dichos espacios son vivenciadas corporalmente y están vinculadas a los vientos, al curso del sol y del río Pilcomayo, con diferencias entre generaciones. Describimos, además, cómo algunos de los términos que designan zonas cardinales se utilizan para expresar la posición/ orientación de una entidad respecto de otra en un espacio no solo geográfico (a escala amplia) sino también manipulable (por ejemplo, cómo se ubican los objetos sobre una mesa). Planteamos, a modo de hipótesis, que el lexema cháʔhema “allá hacia arriba (Sur)” podría hacer referencia al “polo sur celeste”, identificable por los tobas mediante determinados asterismos. Résumé: Zones cardinales et orientation chez les Qomléʔk (Tobas de l’ouest de Formosa, Argentine). Cet article traite des différentes façons dont les Tobas de l’ouest de Formosa, Qomléʔk (Argentine), aussi connus comme Tobas de Sombrero Negro (Métraux 1937a et b), font référence aux zones cardinales et à leurs contextes d’utilisation. Nous montrons que les définitions qui décrivent et nomment ces espaces sont vécues corporellement, qu’elles sont liées aux vents, à la course du soleil et au cours de la rivière Pilcomayo, et qu’elles présentent des différences entre les générations. Nous décrivons en outre comment certains de ces termes qui désignent les zones cardinales sont utilisés pour désigner la position/orientation d’une entité par rapport à une autre, dans un espace non seulement géographique (à grande échelle), mais aussi manipulable (par exemple, des objets disposés sur une table). Nous suggérons enfin, à titre d’hypothèse, que le lexème cháʔhema « làhaut (Sud) » pourrait se référer au « pôle sud céleste », identifiable chez les Tobas par certains astérismes. Abstract: Cardinal zones and orientation among the Qomléʔk (Tobas of Western Formosa, Argentina). This paper analyzes the different ways in which the Tobas of Western Formosa, Qomléʔk (Argentina), also known as Tobas de Sombrero Negro (Métraux 1937a and b), refer to cardinal zones and their contexts of use. We suggest that definitions describing and naming these areas are bodily experienced and are linked to the winds, the course of the sun, and the Pilcomayo River, with differences between people of different generations. We also describe how some of the lexemes that encode cardinal zones are used to encode position/orientation between entities not only in a geographic (large-scale) space, but also in a manipulable one (e.g. objects located on a table). We propose, as a hypothesis, that the lexeme cháʔhema “up there (South)” may refer to the “south celestial pole,” identifiable by the Tobas’ use of certain asterisms.

One of most central issues in various interconnection networks of parallel and distributed systems is to find edge-disjoint paths concerned with an information transmission through links. It is universally acknowledged that an interconnection network can be modeled as an undirected simple graph G = ( V , E ) with processors and physical links between the processors represented as vertices and edges of the G, respectively. The studies on the edge-disjoint paths are closely related to the edge-connectivity. By the well-known Menger theorem, the maximum number of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g-extra edge-connectivity of G λ g ( G ) . It is the cardinality of the minimum set of edges in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The k-ary n-cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. This article intends to show that for 3 ⌈ n 2 ⌉ + r − ⌊ 3 2 r + e + 1 2 ⌋ ≤ g ≤ 3 ⌈ n 2 ⌉ + r , the g-extra edge-connectivity of ternary n-cube (also called 3-ary n-cube) ( n ≥ 3 ) exists a concentration behavior, and prove that these corresponding g-extra edge-connectivities of ternary n-cubes are the constants 2 ( ⌊ n 2 ⌋ − r ) 3 ⌈ n 2 ⌉ + r , where r = 0 , 1 , 2 , ⋯ , ⌊ n 2 ⌋ − 1 , and e = 1 if n is odd and e = 0 if n is even. The above upper and lower bounds of g are sharp. As the integer g varies within the range between 3 ⌈ n 2 ⌉ + r − ⌊ 3 2 r + e + 1 2 ⌋ and 3 ⌈ n 2 ⌉ + r , a necessary and sufficient condition of λ g -optimality is found. Our results improve the several previous results.

A subset S ⊆ V ( G ) is called an F -isolating set of a graph G if G − N [ S ] contains no subgraph isomorphic to any F ∈ F , where F is a family of connected graphs. The F -isolation number of G , denoted by ι ( G , F ) , is the minimum cardinality of an F -isolating set in G . Denote ι ( G , { K 1 , k + 1 } ) simply by ι k ( G ) . Thus, ι 1 ( G ) is the cardinality of a smallest set S such that G − N [ S ] consists of isolated vertices and isolated edges only. In this paper, we prove that if G is a connected graph of order n and different from P 3 , C 3 or C 6 , then ι 1 ( G ) ≤ 2 7 n . Further, if g i r t h ( G ) ≥ 7 , then this bound can be improved to n 4 unless G is P 3 , C 7 or C 11 . Both two bounds above are sharp. Two related problems are proposed.

Aggregation functions, which are at the heart of a number of information fusion processes, allow summarization of multiple inputs into a single representative value. Extended aggregation functions are defined such that the input data can be of varying cardinality, with the implication that there is some consistency across the methods of calculation. This article formalizes an approach to extended aggregation such that contributions of repeated inputs or regions of high density are limited in their ability to influence the final value. We establish important definitions and properties, in particular around whether such functions will be monotone or directionally monotone. We then propose a powerful construction method for extended pre-aggregation functions. Illustrative examples are provided throughout.

In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$ . In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$ . In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node.

Restricted connectivity is a measure of fault-tolerance in multiprocessing systems. The h -restricted connectivity of a graph G , denoted by κ h ( G ) , is the minimum cardinality of vertex set F such that G − F is disconnected and the degree of each component is at least h . In this paper, we study the h -restricted connectivity of n -dimensional balanced hypercube B H n and prove that κ 2 h ( B H n ) = κ 2 h − 1 ( B H n ) = 4 h ( n − h ) , where 1 ≤ h ≤ ⌊ n 2 ⌋ and n ≥ 2 .

For interconnection network losing processors, usually, when the surviving network has a large connected component, it can be used as a functional subsystem without leading to severe performance degradation. Consequently, it is crucial to characterize the interprocessor communication ability and efficiency of the surviving structure. In this article, we prove that when a subset $D$ of at most $6n-17$ processors is deleted from an $n$ -dimensional alternating group graph $\text{AG}_n$ , there exists a largest component with cardinality greater or equal to $|V(\text{AG}_n)|-|D|-3$ for $n\geq 6$ in the remaining network, and the union of small components is, first, an empty graph; or, second, a 3-cycle, or an edge, or a 2-path, or a singleton; or, third, an edge and a singleton, or two singletons. Then, we prove that when a subset $D$ of at most $8n-25$ processors is deleted from $\text{AG}_n$ , there exists a largest component with cardinality greater or equal to $|V(\text{AG}_n)|-|D|-5$ for $n\geq 6$ in the remaining network, and the union of small components is, first, an empty graph; or, second, a 5-cycle, or a 4-path, or a 4-claw, or a 4-cycle, or a 3-path, or a 3-claw, or a 3-cycle, or a 2-path, or an edge, or a singleton; or, third, a 4-cycle and a singleton, or a 3-path and a singleton, or a 3-claw and a singleton, or a 2-path and a singleton, two edges, an edge and a singleton, or two singletons; or, fourth, two edges and a singleton, or a 2-path and two singletons, or an edge and two singletons, or three singletons.

Let F q denote the finite field with q = p λ elements. Maximum Rank metric codes (MRD for short) are subsets of M m × n ( F q ) whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2015) presented a new class of MRD codes over F q called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes H k , s ( L 1 , L 2 ) . The equivalence and duality of twisted Gabidulin codes were discussed by Lunardon, Trombetti, and Zhou (2016). A new class of MRD codes in M 2 n × 2 n ( F q ) was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case L 1 ( x ) = x , where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of H k , s ( x , L ( x ) ) and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of some twisted Gabidulin codes.

Distributed Arithmetic Coding (DAC) is a practical realization of Slepian-Wolf coding that partitions source space into cosets. Coset Cardinality Spectrum (CCS) is an important property of DAC that was defined in our previous work. In this paper, we give two applications of CCS. First, we find that DAC bitstream is not compact. The rate loss of DAC bitstream is caused by two factors: unequal coset partitioning and bit indivisibility . It is proved that as code length goes to infinity, the expected value of bit-indivisibility rate loss will tend to 0.5 for any irrational Rate Change Step (RCS), where the RCS refers to the rate change when one source bit is flipped. With the help of CCS, the bit-indivisibility rate loss can be compensated to some extend. Especially, for any irrational RCS, as code length goes to infinity, the expected value of the remaining bit-indivisibility rate loss after compensation will tend to about 0.47. The second application of CCS is DAC decoder design. We derive the formula of path metric and find that in the original paper on DAC, the intuitive formula of path metric is not correct. The backward-replacing algorithm is proposed to make full use of memory. Experimental results confirm the correctness of theoretical analyses.

Herein, I describe several novel branches of mathematics that can be synthesized such that the analogies between the varying forms of the differentiated branches yield something called, "logic vectors." These novel vector spaces can be used to generate new kinds of meta-mathematical spaces as well as exotic materials.

Understanding the way in which counting represent numerosities was shown to be a long-lasting process. As shown in the Give-a-number task, acquiring the meanings of verbal number words goes through successive developmental stages in which children first learn the cardinal meanings of small number words one at a time before generalizing the cardinal principle they have induced from the first three number words to all number words within their counting range. This acquisition would take about a year, and would be completed by the age of 3 ½ years. The aim of the present study was to provide a conceptual replication of the developmental sequence described in Wynn’s study nearly 30 years ago using the Give-a-number task. A first cross-sectional study was conducted on 213 Belgian children aged between 39 and 74 months using the Give-a-number task to examine the developmental pattern and the influence of age on this acquisition. The time span of acquisition was examined in a second study in which 34 children were tested five times every months between the age of 36 to 52 months. Results showed that acquiring the cardinal meanings of number words spread out over a protracted period, far more extended than assumed by Wynn. Furthermore, children do not generalize all-at-once to large number words, the cardinal knowledge they learned on small number words. Rather, number words were found to be learned one at a time in a really progressive manner. Results were discussed with regard to their implications for the existing theories and in relation with other tasks assessing the acquisition of verbal number symbols.

The approximate Caratheodory theorem states that given a compact convex set $${\mathcal {C}}\subset {\mathbb {R}}^n$$ and $$p\in [2,+\infty [$$ , each point $$x^*\in {\mathcal {C}}$$ can be approximated to $$\epsilon $$ -accuracy in the $$\ell _p$$ -norm as the convex combination of $${\mathcal {O}}(pD_p^2/\epsilon ^2)$$ vertices of $${\mathcal {C}}$$ , where $$D_p$$ is the diameter of $${\mathcal {C}}$$ in the $$\ell _p$$ -norm. A solution satisfying these properties can be built using probabilistic arguments or by applying mirror descent to the dual problem. We revisit the approximate Caratheodory problem by solving the primal problem via the Frank-Wolfe algorithm, providing a simplified analysis and leading to an efficient practical method. Furthermore, improved cardinality bounds are derived naturally using existing convergence rates of the Frank-Wolfe algorithm in different scenarios, when $$x^*$$ is in the interior of $${\mathcal {C}}$$ , when $$x^*$$ is the convex combination of a subset of vertices with small diameter, or when $${\mathcal {C}}$$ is uniformly convex. We also propose cardinality bounds when $$p\in [1,2[\cup \{+\infty \}$$ via a nonsmooth variant of the algorithm. Lastly, we address the problem of finding sparse approximate projections onto $${\mathcal {C}}$$ in the $$\ell _p$$ -norm, $$p\in [1,+\infty ]$$ .

Uncertainty measures can help people to effectively analyze data and to reveal the essential characteristics of data sets. The roughness, accuracy and approximation accuracy are effective evaluations of uncertainty measure based on the lower and upper approximation sets in the classical rough set. However, due to the lack of consideration of the size of the granularity and the granulation distribution of approximation sets, the classical uncertainty measures and its improved methods still have suffered from some shortcomings, such as inaccurate measurement in some special cases, inconsistent with people's cognition and so on, which motivate the study in this paper. In view of the fact that the upper and lower approximation sets are the basic concepts which are used to express the imprecision of knowledge. The paper introduces a more appropriate measure for evaluating the uncertainty, which combines the granularity of knowledge, the cardinality of the lower and upper approximation sets and the granularity distribution of the lower and upper approximation sets. Firstly, the roughness and accuracy based on the granularity distribution of the approximation sets are proposed, and the corresponding properties of the new roughness and accuracy are discussed. Two types of definitions of the approximation accuracy are investigated in the decision systems and their some properties are induced. Theoretical analyses indicate that the proposed measurements can be used to reasonably evaluate the uncertainty of information systems and decision systems. Finally, some experiments on nine UCI data sets are conducted and experimental results demonstrate that the proposed methods of uncertainty measurements are effective for evaluating the uncertainty of rough sets.

Summary We consider the problem of testing for the presence of linear relationships between large sets of random variables based on a postselection inference approach to canonical correlation analysis. The challenge is to adjust for the selection of subsets of variables having linear combinations with maximal sample correlation. To this end, we construct a stabilized one-step estimator of the Euclidean norm of the canonical correlations maximized over subsets of variables of prespecified cardinality. This estimator is shown to be consistent for its target parameter and asymptotically normal, provided the dimensions of the variables do not grow too quickly with sample size. We also develop a greedy search algorithm to accurately compute the estimator, leading to a computationally tractable omnibus test for the global null hypothesis that there are no linear relationships between any subsets of variables having the prespecified cardinality. We further develop a confidence interval that takes the variable selection into account.

This paper investigates shortcomings that limit the performance of optical code division multiple access (OCDMA) systems including the low cardinality and data rate as well as the high power at reception. The main drawback for such systems known as multiple access interference accompanying by phase induced intensity noise is also investigated to effeciencly propose a novel two dimensional cyclic shift (2D-CS) code to be implemented in non-coherent OCDMA systems. The developed code is based on a one dimensional cyclic shift (1D-CS) code previously provided by research works processing spectral amplitude coding for optical code division multiple access (SAC-OCDMA) systems. Numerical results obtained by this study are therefore compared to previous studies employing different codes like two dimensional extended double weight (2D-EDW), two dimensional flexible cross correlation/modified double weight (2D-FCC/MDW), two dimensional perfect difference (2D-PD), two dimensional diluted perfect difference (2D-DPD), two dimensional multi service (2D-MS) and two dimensional zero cross correlation/multi diagonal (2D-ZCC/MD) codes. Accordingly, it is demonstrated that the proposed 2D-CS code outperforms all codes given previously in terms of system capacity where the small increasing percentage is about 40% compared to 2D-ZCC/MD and 2D-MS. Systems using 2D-CS code can support until 203 simultaneous users with a total code length equal to 171. System performance investigation leads to a BER and Q-Factor closely to1.0E−12 and 1.0E−27, and 6.6 dB and 10.6 dB at 20 km of single mode fiber length using white light source and Laser, respectively. Furthermore, such a code can be easily adopted by OCDMA systems for a long distance up to approximately 55 and 100 km.

This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α =1/2. Several realization problems are also generated in this paper.

Let G be a connected graph. A subset S of V (G) is a connected co-independent hop dominating set in G if the subgraph induced by S is connected and V (G)\S is an independent set where for each v ∈ V (G)\S, there exists a vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of such an S is called the connected co-independent hop domination number of G. This paper presents the characterizations of the connected co-independent hop dominating sets in the join, corona and lexicographic product of two graphs. It also discusses the corresponding connected co-independent hop domination numbers of the aforementioned graphs.

A set S ⊆ V (G) is a restrained strong resolving dominating set in G if S is a strongresolving dominating set in G and S = V (G) or ⟨V (G) \ S⟩ has no isolated vertex. The restrained strong resolving domination number of G, denoted by γrsR(G), is the smallest cardinality of a restrained strong resolving dominating set in G. In this paper, we present characterizations of the restrained strong resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the exact value of the restrained strong resolving domination number of each of these graphs.

Formal reasoning about finite sets and cardinality is an important tool for many applications, including software verification, where very often one needs to reason about the size of a given data structure and not only about what its elements are. The Constraint Logic Programming tool {log} provides a decision procedure for deciding the satisfiability of formulas involving very general forms of finite sets, without cardinality. In this paper we adapt and integrate a decision procedure for a theory of finite sets with cardinality into {log}. The proposed solver is proved to be a decision procedure for its formulas. Besides, the new CLP instance is implemented as part of the {log} tool. In turn, the implementation uses Howe and King's Prolog SAT solver and Prolog's CLP(Q) library, as an integer linear programming solver. The empirical evaluation of this implementation based on +250 real verification conditions shows that it can be useful in practice., Comment: Under consideration in Theory and Practice of Logic Programming (TPLP)

Let (Formula presented.) be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras (Formula presented.) such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.). For example, we shall prove that, in the case where (Formula presented.) is an infinite-dimensional, unital Banach function algebra, (Formula presented.) is a maximal abelian subalgebra in infinitely-many closed subalgebras of (Formula presented.) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples (Formula presented.) that are local algebras. We shall also give examples of uniform algebras of the form (Formula presented.), where (Formula presented.) is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras (Formula presented.) such that each (Formula presented.) contains (Formula presented.) as a closed subalgebra and is such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.).