Search

The notion of generalized quantum cluster algebras was introduced as a natural generalization of Berenstein and Zelevinsky's quantum cluster algebras as well as Chekhov and Shapiro's generalized cluster algebras. In this paper, we focus on a generalized quantum cluster algebra of Kronecker type which possesses infinitely many cluster variables. We obtain the cluster multiplication formulas for this algebra. As an application of these formulas, a positive bar-invariant basis is explicitly constructed. Both results generalize those known for the Kronecker cluster algebra and quantum cluster algebra. [ABSTRACT FROM AUTHOR]

Since the late 20th century, Engineering Education Research (EER) has been expanding globally as a field, although its identity varies across institutions and countries around the world. This diversity in how EER is experienced across contexts impacts how capability is developed, including identities, knowledge, practices, agendas, funding, and pathways. The theme of 'Engineering Education Research Capability Development' was explored in papers, presentations, and workshops throughout the jointly held 2021 Research in Engineering Education Symposium and Australasian Association for Engineering Education conference. This special issue presents a continuation of that theme. The papers collectively contribute to expanding global understanding of the field, with implications for future capacity building efforts in engineering education practice and research. [ABSTRACT FROM AUTHOR]

The task of remote sensing image object detection in low latency scenes is of great research significance. To address the problem that the current high‐precision object detection algorithm based on a feature pyramid network is slow due to a large number of parameters and complicated computation, a fast remote sensing image object detection method based on a Single‐scale Feature Transformation Network (SFTN) is proposed. Firstly, based on the single‐scale remote sensing image features, the new channel features are quickly generated by a linear transformation of the original features and convolution kernel clustering optimization using cosine similarity; secondly, in order to obtain multi‐scale receptive fields, a parallel residual hole convolution module is designed to cover multi‐category remote sensing object scales on the feature map; finally, angle variables are introduced and optimized using angle similarity to effectively improve the object orientation accuracy. The experimental results on different datasets show that the method in this paper improves the detection speed rapidly while ensuring the accuracy of remote sensing image object detection, which is better than many remote sensing image object detection methods. The results demonstrate the reliability and robustness of the method. [ABSTRACT FROM AUTHOR]

In 1975, Symons [[5]] introduced a subsemigroup of HT ht , defined as HT ht , for a nonempty subset I Y i of I X i , determining all its automorphisms. The first paper [[21]] introduces a construction of a new graph associated with a semihypergroup, using the fundamental relation HT ht . In particular, it is shown that the Cayley graphs generated by transposition trees on the set HT ht are HT ht -extendable and their extendability number is HT ht for any integer HT ht . Connections between algebraic structure theory and graph theory have been established in order to solve open problems in one theory with the help of the tools existing in the other, emphasizing the remarkable properties of one theory with techniques involving the second. [Extracted from the article]

Few-shot Relation Classification identifies the relation between target entity pairs in unstructured natural language texts by training on a small number of labeled samples. Recent prototype network-based studies have focused on enhancing the prototype representation capability of models by incorporating external knowledge. However, the majority of these works constrain the representation of class prototypes implicitly through complex network structures, such as multi-attention mechanisms, graph neural networks, and contrastive learning, which constrict the model's ability to generalize. In addition, most models with triplet loss disregard intra-class compactness during model training, thereby limiting the model's ability to handle outlier samples with low semantic similarity. Therefore, this paper proposes a non-weighted prototype enhancement module that uses the feature-level similarity between prototypes and relation information as a gate to filter and complete features. Meanwhile, we design a class cluster loss that samples difficult positive and negative samples and explicitly constrains both intra-class compactness and inter-class separability to learn a metric space with high discriminability. Extensive experiments were done on the publicly available dataset FewRel 1.0 and 2.0, and the results show the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

The batch clustering algorithm for classification application requires the initial parameters and also has a drifting phenomenon for the stochastic process. The initial parameters are critical for the clustering to converge to the partial optimum. The drifting phenomenon in original batch clustering still has space to be improved thus to speed up the convergence based on the initial parameters. This paper proposes an unsupervised clustering method by addressing these two issues. Firstly, the estimation method for the initial parameters has been given in preliminary with a hierarchical manner of principal component analysis (PCA). The nonlinear parameters have been estimated based on a mathematical connection between PCA and clusters membership. With initial parameters, the drifting issue is addressed by combing the gradient descent and the batch clustering on an auxiliary objective to refine the initial parameters. The efficiency of the clustering process is proved based on the relationship between two quadratic functions followed by a justification. In addition, the effectiveness of the proposed method has been validated with the statistical F measure in classification application. The validation results show that the efficiency of the proposed gradient descent batch clustering has been improved significantly with trade-off to the accuracy in comparison of the original algorithms under the mean squared error (MSE) criterion. [ABSTRACT FROM AUTHOR]

• Cluster energy-saving optimization is described as two MILP problems by two different variable definitions. • The switching cost of servers is considered in the two MILP problems to prevent server switching jitter. • A variable division method is proposed to transform the MINLP problem into an MILP problem. • We present the detailed experimental results from several perspectives. The issue of how to dynamically optimize the deployment of an application server cluster according to the changing load to reduce energy consumption is an important problem that must be urgently solved. In this paper, we propose an energy-saving optimization strategy for application server clusters, whose optimization content includes the on/off state, CPU frequency, and load size of each server. Compared with existing research, our strategy is not only more accurate in power and load models but also considers the switching cost of servers to avoid server switching jitter. The strategy includes two schemes, which both formulate the cluster energy-saving optimization as a mixed integer linear programming (MILP) problem and then adopt a toolkit to solve the problem. One scheme defines variables for each server, and the resulting programming problem is called the MILP4PH problem. The other scheme defines variables for each server type, resulting in a programming problem called the MILP4GH problem. The experimental results reveal that for clusters with poor homogeneity, the MILP4PH problem has fewer variables and can be solved in real time, while for clusters with good homogeneity, the MILP4GH problem has fewer variables and can be solved in real time. [ABSTRACT FROM AUTHOR]

The terahertz (THz) waves with enormous bandwidth can be used along with the existing sub-6GHz and millimeter wave (mmWave) bands to achieve the ever evolving ecosystem of applications that need to be supported by the modern wireless networks. This paper investigates a user-centric dynamic base station (BS) clustering design for a hybrid network where THz, mmWave, and sub-6GHz BSs coexist. Invoking the proposed clustering model, the BS cooperation within each band is made possible by considering long term channel variations and all the surrounding BSs within a cluster with tunable size. A typical user is associated with the best BS cluster, from either a sub-6GHz, mmWave or THz tier based on the maximum signal-to-interference-plus-noise-ratio (SINR) criterion or the maximum rate criterion. Using tools from stochastic geometry, we assess the performance of the proposed user-centric hybrid system in terms of SINR and rate coverage performances, while accounting for: band specific propagation models, directional beamfroming, and BSs random locations. The accuracy of the analytical results is validated with Monte-Carlo simulations. The obtained results recognize a clear coverage/rate trade-off where a high fraction of THz BSs improves the rate significantly but may degrade the coverage performance. Thus, with carefully planned networks, enabling user-centric BS cooperation for hybrid wireless systems can achieve ultra-high rates while maintaining sufficient coverage in sixth-generation (6G) networks. [ABSTRACT FROM AUTHOR]

An algorithm widely used in hotel companies for demand analysis is the so-called K-means. The aforementioned algorithm is based on the use of the Euclidean distance as a dissimilarity measure and this fact can cause a main handicap. Concretely, the Euclidean distance provides a global difference measure between the values of the descriptive variables that can blur the relative differences in each component separately and, hence, the cluster algorithm can assign a custom to an incorrect cluster. In order to avoid this drawback, this paper proposes an application of the use of Ordered Weighted Averaging (OWA) operators and an OWA-based K-means for clustering customers staying at a real five-star hotel, located in a mature sun-and-beach area, according to their propensity to spend. It must be pointed out that OWA-based distance calculates relative distances and it is sensitive to the differences in each component separately. All experiments show that the use of the OWA operator improves the performance of the classical K-means up to 21.6 % and reduces the number of convergence iterations up to 48.46 % . Such an improvement has been tested through a ground truth, designed by the marketing department of the firm, which states the cluster to which each tourist belongs. Moreover, the customer classification is achieved regardless of the season in which the customer stays at the hotel. All these facts confirm that the OWA-based K-means could be used as an appropriate tool for classifying tourists in purely exploratory and predictive stages. Furthermore, the new methodology can be implemented without requiring radical changes in the implementation of the classical methodology and in data processing which is crucial so that it can be incorporated into the control panel of a real hotel without additional implementation costs. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of erasure coding in distributed storage. We consider a model of storage that assumes that nodes are organized into equally sized groups, called racks, that within each group the nodes can communicate freely without taxing the system bandwidth, and that the only information transmission that counts is the one between the racks. This assumption implies that the nodes within each of the racks can collaborate before providing information to the failed node. The main emphasis of the paper is on code construction for this storage model. We present an explicit family of maximum distance separable (MDS) array codes that support recovery of a single failed node from any number of helper racks using the minimum possible amount of inter-rack communication(such codes are said to provide optimal repair). The codes are constructed over finite fields of size comparable to the code length. We also derive a bound on the number of symbols accessed at helper nodes for the purposes of repair, and construct a code family that approaches this bound, while still maintaining the optimal repair property. Finally, we present a construction of scalar Reed-Solomon codes that support optimal repair for the rack-oriented storage model. [ABSTRACT FROM AUTHOR]

A topological index of graph G is a numerical quantity which describes its topology. If it is applied to the molecular structure of chemical compounds, it reflects the theoretical properties of the chemical compounds. A number of topological indices have been introduced so far by different researchers. The Wiener index is one of the oldest molecular topological indices defined by Wiener. The Wiener index number reflects the index boiling points of alkane molecules. Quantitative structure activity relationships (QSAR) showed that they also describe other quantities including the parameters of its critical point, density, surface tension, viscosity of its liquid phase, and the van der Waals surface area of the molecule. The Wiener polarity index has been introduced by Wiener and known to be related to the cluster coefficient of networks. In this paper, the Wiener polarity index W p G and Wiener index W G of certain triangular networks are computed by using graph-theoretic analysis, combinatorial computing, and vertex-dividing technology. [ABSTRACT FROM AUTHOR]

Let g be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D on the coordinate ring C[N] related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g is of type An or D4, the map D takes similar distinguished values on the set of all flag minors of C[N], raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D on flag minors belonging to the same standard seed, and we show that in any ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of C[N] via representations of quiver Hecke algebras. [ABSTRACT FROM AUTHOR]

Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of G(4, n) as well as certain algebraic functions of cluster variables. In this paper we solve the C Z = 0 matrix equations associated to several cells of the totally non-negative Grassmannian, combining methods of arXiv:2012.15812 for rational letters and arXiv:2007.00646 for algebraic letters. We identify sets of parameterizations of the top cell of G+(5, 9) for which the solutions produce all of (and only) the cluster variable letters of the 2-loop nine-particle NMHV amplitude, and identify plabic graphs from which all of its algebraic letters originate. [ABSTRACT FROM AUTHOR]

Generalised bi-adjoint scalar amplitudes, obtained from integrations over moduli space of punctured ℂℙk − 1, are novel extensions of the CHY formalism. These amplitudes have realisations in terms of Grassmannian cluster algebras. Recently connections between one-loop integrands for bi-adjoint cubic scalar theory and D n cluster polytope have been established. In this paper using the Gr (3, 6) cluster algebra, we relate the singularities of (3, 6) amplitude to four-point one-loop integrand in the bi-adjoint cubic scalar theory through the D 4 cluster polytope. We also study factorisation properties of the (3, 6) amplitude at various boundaries in the worldsheet. [ABSTRACT FROM AUTHOR]

Motivation Tumours are composed of distinct cancer cell populations (clones), which continuously adapt to their local micro-environment. Standard methods for clonal deconvolution seek to identify groups of mutations and estimate the prevalence of each group in the tumour, while considering its purity and copy number profile. These methods have been applied on cross-sectional data and on longitudinal data after discarding information on the timing of sample collection. Two key questions are how can we incorporate such information in our analyses and is there any benefit in doing so? Results We developed a clonal deconvolution method, which incorporates explicitly the temporal spacing of longitudinally sampled tumours. By merging a Dirichlet Process Mixture Model with Gaussian Process priors and using as input a sequence of several sparsely collected samples, our method can reconstruct the temporal profile of the abundance of any mutation cluster supported by the data as a continuous function of time. We benchmarked our method on whole genome, whole exome and targeted sequencing data from patients with chronic lymphocytic leukaemia, on liquid biopsy data from a patient with melanoma and on synthetic data and we found that incorporating information on the timing of tissue collection improves model performance, as long as data of sufficient volume and complexity are available for estimating free model parameters. Thus, our approach is particularly useful when collecting a relatively long sequence of tumour samples is feasible, as in liquid cancers (e.g. leukaemia) and liquid biopsies. Availability and implementation The statistical methodology presented in this paper is freely available at github.com/dvav/clonosGP. Supplementary information Supplementary data are available at Bioinformatics online. [ABSTRACT FROM AUTHOR]

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height $1$ to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes and defining ideals of graded lower bound cluster algebras. [ABSTRACT FROM AUTHOR]

Ensemble clustering has been a popular research topic in data mining and machine learning. Despite its significant progress in recent years, there are still two challenging issues in the current ensemble clustering research. First, most of the existing algorithms tend to investigate the ensemble information at the object-level, yet often lack the ability to explore the rich information at higher levels of granularity. Second, they mostly focus on the direct connections (e.g., direct intersection or pair-wise co-occurrence) in the multiple base clusterings, but generally neglect the multiscale indirect relationship hidden in them. To address these two issues, this paper presents a novel ensemble clustering approach based on fast propagation of cluster-wise similarities via random walks. We first construct a cluster similarity graph with the base clusters treated as graph nodes and the cluster-wise Jaccard coefficient exploited to compute the initial edge weights. Upon the constructed graph, a transition probability matrix is defined, based on which the random walk process is conducted to propagate the graph structural information. Specifically, by investigating the propagating trajectories starting from different nodes, a new cluster-wise similarity matrix can be derived by considering the trajectory relationship. Then, the newly obtained cluster-wise similarity matrix is mapped from the cluster-level to the object-level to achieve an enhanced co-association matrix, which is able to simultaneously capture the object-wise co-occurrence relationship as well as the multiscale cluster-wise relationship in ensembles. Finally, two novel consensus functions are proposed to obtain the consensus clustering result. Extensive experiments on a variety of real-world datasets have demonstrated the effectiveness and efficiency of our approach. [ABSTRACT FROM AUTHOR]

We study the distribution of line-of-sight velocities of galaxies in the vicinity of Sloan Digital Sky Survey (SDSS) red-sequence Matched-filter Probabilistic Percolation (redMaPPer) galaxy clusters. Based on their velocities, galaxies can be split into two categories: galaxies that are dynamically associated with the cluster, and random line-of-sight projections. Both the fraction of galaxies associated with the galaxy clusters, and the velocity dispersion of the same, exhibit a sharp feature as a function of radius. The feature occurs at a radial scale R edge ≈ 2.2 R λ, where R λ is the cluster radius assigned by redMaPPer. We refer to R edge as the 'edge radius'. These results are naturally explained by a model that further splits the galaxies dynamically associated with a galaxy cluster into a component of galaxies orbiting the halo and an infalling galaxy component. The edge radius R edge constitutes a true 'cluster edge', in the sense that no orbiting structures exist past this radius. A companion paper tests whether the 'halo edge' hypothesis holds when investigating the full three-dimensional phase-space distribution of dark matter substructures in numerical simulations, and demonstrates that this radius coincides with a suitably defined splashback radius. [ABSTRACT FROM AUTHOR]

In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respectively. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups and Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. Besides, we show that semigroup algebras of Boolean reflection monoids are cellular algebras. [ABSTRACT FROM AUTHOR]

Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point. [ABSTRACT FROM AUTHOR]

Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces. [ABSTRACT FROM AUTHOR]

The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type A 2 (2). As corollaries, we obtain bar-invariant Z [ q ± 1 2 ] -bases established in [6] , and naturally deduce the positivity of the elements in these bases. One bar-invariant basis as the triangular basis of this quantum cluster algebra is also explicitly described. [ABSTRACT FROM AUTHOR]

This document is a preface to a special issue of the Journal of Computational Biology, which features a selection of articles from the 27th Annual International Conference on Research in Computational Molecular Biology (RECOMB). The conference aims to connect the fields of computational, mathematical, statistical, and biological sciences. The conference took place in Istanbul, Turkey, from April 16 to April 19, 2023, and the chair of the Program Committee oversaw the review process that resulted in the selection of 44 papers out of 186 submissions. Accepted articles were also invited to be submitted to the Journal of Computational Biology, which has a longstanding partnership with RECOMB. [Extracted from the article]

This research introduces a new strategy in cluster ensemble selection by using Independency and Diversity metrics. In recent years, Diversity and Quality, which are two metrics in evaluation procedure, have been used for selecting basic clustering results in the cluster ensemble selection. Although quality can improve the final results in cluster ensemble, it cannot control the procedures of generating basic results, which causes a gap in prediction of the generated basic results’ accuracy. Instead of quality, this paper introduces Independency as a supplementary method to be used in conjunction with Diversity. Therefore, this paper uses a heuristic metric, which is based on the procedure of converting code to graph in Software Testing, in order to calculate the Independency of two basic clustering algorithms. Moreover, a new modeling language, which we called as “Clustering Algorithms Independency Language” (CAIL), is introduced in order to generate graphs which depict Independency of algorithms. Also, Uniformity, which is a new similarity metric, has been introduced for evaluating the diversity of basic results. As a credential, our experimental results on varied different standard data sets show that the proposed framework improves the accuracy of final results dramatically in comparison with other cluster ensemble methods. [ABSTRACT FROM AUTHOR]

Highlights • There is a need for methods for dealing with missing data in proximity matrices. • New method called Proximity Matrix Completion algorithm addresses this need. • The PMC algorithm is shown to be effective compared to benchmarks. • A case study from plant breeding established practical relevance. Abstract In most applications of data clustering the input data includes vectors describing the location of each data point, from which distances between data points can be calculated and a proximity matrix constructed. In some applications, however, the only available input is the proximity matrix, that is, the distances between each pair of data point. Several clustering algorithms can still be applied, but if the proximity matrix has missing values no standard method is directly applicable. Imputation can be done to replace missing values, but most imputation methods do not apply when only the proximity matrix is available. As a partial solution to fill this gap, we propose the Proximity Matrix Completion (PMC) algorithm. This algorithm assumes that data is missing due to one of two reasons: complete dissimilarity or incomplete observations; and imputes values accordingly. To determine which case applies the data is modeled as a graph and a set of maximum cliques in the graph is found. Overlap between cliques then determines the case and hence the method of imputation for each missing data point. This approach is motivated by an application in plant breeding, where what is needed is to cluster new experimental seed varieties into sets of varieties that interact similarly to the environment, and this application is presented as a case study in the paper. The applicability, limitations and performance of the new algorithm versus other methods of imputation are further studied by applying it to datasets derived from three well-known test datasets. [ABSTRACT FROM AUTHOR]

We introduce and study the algebras of generalized quaternion type, being natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these algebras, with 2-regular Gabriel quivers, are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles. The main result of the paper forms an important step towards the Morita equivalence classification of all periodic symmetric tame algebras of non-polynomial growth. Applying the main result, we establish existence of wild periodic algebras of period 4, with arbitrary large number (at least 4) of pairwise non-isomorphic simple modules. These wild periodic algebras arise as stable endomorphism rings of cluster tilting Cohen-Macaulay modules over one-dimensional hypersurface singularities. [ABSTRACT FROM AUTHOR]

Abstract Snake modules introduced by Mukhin and Young form a family of modules of quantum affine algebras. The aim of this paper is to prove that the Hernandez–Leclerc conjecture about monoidal categorifications of cluster algebras is true for prime snake modules of types A n and B n. We prove that prime snake modules are real. We introduce S -systems consisting of equations satisfied by the q -characters of prime snake modules of types A n and B n. Moreover, we show that every equation in the S -system of type A n (respectively, B n) corresponds to a mutation in the cluster algebra A (respectively, A ′) constructed by Hernandez and Leclerc and every prime snake module of type A n (respectively, B n) corresponds to some cluster variable in A (respectively, A ′). In particular, this proves that the Hernandez–Leclerc conjecture is true for all prime snake modules of types A n and B n. [ABSTRACT FROM AUTHOR]

In this paper we give an overview of a generalization, introduced by R. Bautista and the author, of the theory of mutation of quivers with potential developed in 2007 by Derksen-Weyman-Zelevinsky. This new construction allows us to consider finite dimensional semisimple F-algebras, where F is any field. We give a brief account of the results concerning this generalization and its main consequences. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove that positivity of denominator vectors holds for any skew-symmetric cluster algebra. [ABSTRACT FROM AUTHOR]

For cluster algebras from surfaces, there is a known formula for cluster variables and F -polynomials in terms of the perfect matchings of snake graphs. If the cluster algebra has trivial coefficients, there is also a known formula for cluster variables in terms of continued fractions. In this paper, we extend this result to cluster algebras with principal coefficients by producing a formula for the F -polynomials in terms of continued fractions. [ABSTRACT FROM AUTHOR]

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan FB called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of FB for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces. [ABSTRACT FROM AUTHOR]

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization. [ABSTRACT FROM AUTHOR]

This paper shows the polytopality of any finite type g -vector fan, acyclic or not. In fact, for any finite Dynkin type Γ, we construct a universal associahedron Asso un ( Γ ) with the property that any g -vector fan of type Γ is the normal fan of a suitable projection of Asso un ( Γ ) . [ABSTRACT FROM AUTHOR]

The double Pieri algebra, constructed by Howe, Lee, and the author in [10,12] , encodes information on the decomposition of Pieri type tensor products of irreducible representations for complex classical groups. In this paper we study its finite presentation in terms of generators and relations, and then prove that it admits the structure of a cluster algebra. We also study its sl 2 -module structure. [ABSTRACT FROM AUTHOR]

In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers of finite mutation type. In particular, we show that a mutation finite cluster quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of X 7 . We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with boundary has a maximal green sequence. We also compute explicit maximal green sequences for exceptional quivers of finite mutation type. [ABSTRACT FROM AUTHOR]

Parabolic-trough solar collector fields are large-scale systems, so the application of centralized optimization-based control methods to these systems is often not suitable for real-time control. As such, this paper formulates a novel coalitional control approach as an appropriate alternative to the centralized scheme. The key idea is to split the overall solar collector field into smaller subsystems, each of them governed by a local controller. Then, controllers are clustered into coalitions to solve a local optimization-based problem related to the corresponding subset of subsystems, so that an approximate solution of the original centralized problem can be obtained in a decentralized fashion. However, the operational constraints of the solar collector field couple the optimization problems of the multiple coalitions, thus limiting the ability to solve them in a fully decentralized manner. To overcome this issue, a novel population-dynamics-assisted resource allocation strategy is proposed as a mechanism to decouple the local optimization problems of the multiple coalitions. The proposed coalitional methodology allows to solve the multiple local subproblems in parallel, hence reducing the overall computational burden, while guaranteeing the satisfaction of the operational constraints and without significantly compromising the overall performance. The effectiveness of proposed approach is shown through numerical simulations of a 10- and 100-loop version of the ACUREX solar collector field of Plataforma Solar de Almería , Spain. • Decentralized control of large-scale parabolic-trough solar collector fields. • Reducing the computational burden of model predictive controllers for solar plants. • Splitting large-scale solar plants by clustering model predictive controllers. • Improving the scalability of model predictive control for large-scale solar plants. • Evolutionary game theoretical assistance for model predictive control in solar plants. [ABSTRACT FROM AUTHOR]

It is conjectured by Assem, Schiffler and Shramchenko in [3] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of Dynkin type or rank 2, see [3] . The aim of this paper is to prove it for algebras of type A ˜ . We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type A ˜ . We also prove the automorphism conjecture from [3] for algebras of type A ˜ as a direct consequence. [ABSTRACT FROM AUTHOR]

The sine-Gordon Y-systems and the reduced sine-Gordon Y- systems were introduced by Tateo in the 1990's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and only a part of them have been proved so far. In this paper we formulate these Y-systems by the polygon realization of cluster algebras of types A and D and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems. [ABSTRACT FROM AUTHOR]

In this paper we propose a frameworks for identifying patterns and regularities in the pseudo-anonymized Call Data Records (CDR) pertaining a generic subscriber of a mobile operator. We face the challenging task of automatically deriving meaningful information from the available data, by using an unsupervised procedure of cluster analysis and without including in the model any a priori knowledge on the applicative context. Clusters mining results are employed for understanding users' habits and to draw their characterizing profiles. We propose two implementations of the data mining procedure; the first is based on a novel system for clusters and knowledge discovery called LD-ABCD, capable of retrieving clusters and, at the same time, to automatically discover for each returned cluster the most appropriate dissimilarity measure (local metric). The second approach instead is based on PROCLUS, the well-know subclustering algorithm. The dataset under analysis contains records characterized only by few features and, consequently, we show how to generate additional fields which describe implicit information hidden in data. Finally, we propose an effective graphical representation of the results of the data-mining procedure, which can be easily understood and employed by analysts for practical applications. [ABSTRACT FROM AUTHOR]

Video stabilization technique is often used in handheld multimedia devices, whereas the difficulties in the accurate extraction aspect of global motion vectors restrict its development. This paper proposes a novel video stabilization approach that is based on the shortest spanning path clustering algorithm for effective and reliable estimation of the global motion vectors. As demonstrated in our experimental results, the proposed approach achieves superior stabilized effectiveness compared with the other state-of-the-art approaches based on both qualitative and quantitative measurements. [ABSTRACT FROM AUTHOR]

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group $ W$ $ \pi_\downarrow^c$ to the unique maximal $ c$ $ \pi_\downarrow^c$-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of $ \pi_\downarrow^c$ [ABSTRACT FROM AUTHOR]

This paper introduces post-MORDM, a decision-support framework that augments Many Objective Robust Decision Making (MORDM). MORDM often creates an intractable number of environmental management policies, characterized by decision variable, objective, and robustness values. This large number of policies inhibits decision support, causing disagreements among decision-makers. Post-MORDM addresses these challenges via the Self-Organizing Map (SOM), synthesizing MORDM data as layers organized in a map-like coordinate system. It uses the SOM to cluster policies, discover salient characteristics, and assess cause-effect relationships between decision-maker choices (i.e. decision variable values) and performance (objective and robustness values). Overall, the goal of post-MORDM is to create a structured platform that encourages negotiation and compromise. We demonstrate post-MORDM with a case study of two illustrative decision-makers for reservoir operation policy in the Colorado River Basin, USA. Post-MORDM helps communicate tradeoffs between storage and delivery objectives, relate tradeoffs to shortage policies, and identify mutually feasible policies. [Display omitted] • Paradigm shift: MORDM decision variables, objectives, robustness as map layers. • Self-Organizing Map for policy clustering, dimension reduction, and visualization. • Topology maps for decision-maker tradeoff analysis, negotiation, and compromise. • Case study: multi-objective reservoir operation policy in the Colorado River Basin. • Lake Mead: high magnitude, low frequency vs low magnitude, high frequency shortage. [ABSTRACT FROM AUTHOR]

It is an important aspect of cluster theory that cluster categories are “categorifications” of cluster algebras. This is expressed formally by the (original) Caldero–Chapoton map X which sends certain objects of cluster categories to elements of cluster algebras. Let τ c → b → c be an Auslander–Reiten triangle. The map X has the salient property that X ( τ c ) X ( c ) − X ( b ) = 1 . This is part of the definition of a so-called frieze, see [1] . The construction of X depends on a cluster tilting object. In a previous paper [14] , we introduced a modified Caldero–Chapoton map ρ depending on a rigid object; these are more general than cluster tilting objects. The map ρ sends objects of sufficiently nice triangulated categories to integers and has the key property that ρ ( τ c ) ρ ( c ) − ρ ( b ) is 0 or 1. This is part of the definition of what we call a generalised frieze. Here we develop the theory further by constructing a modified Caldero–Chapoton map, still depending on a rigid object, which sends objects of sufficiently nice triangulated categories to elements of a commutative ring A . We derive conditions under which the map is a generalised frieze, and show how the conditions can be satisfied if A is a Laurent polynomial ring over the integers. The new map is a proper generalisation of the maps X and ρ . [ABSTRACT FROM AUTHOR]

Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in connection to finite type cluster algebras. Following recent work of L. Pournin in types A and B , this paper focuses on geodesic properties of generalized associahedra. We prove that the graph diameter of the n -dimensional associahedron of type D is precisely 2 n − 2 for all n greater than 1. Furthermore, we show that all type B C D associahedra have the non-leaving-face property, that is, any geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. This property was already proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type A . In contrast, we present relevant examples related to the associahedron that do not always satisfy this property. [ABSTRACT FROM AUTHOR]

This paper proposes a method for designing base station (BS) clusters and cluster patterns for pair-wise BS coordination. The key idea is that each BS cluster is formed by using the second-order Voronoi region, and the BS clusters are assigned to a specific cluster pattern by using edge-coloring for a graph drawn by Delaunay triangulation. The main advantage of the proposed method is that the BS selection conflict problem is prevented, while users are guaranteed to communicate with their two closest BSs in any irregular BS topology. With the proposed coordination method, analytical expressions for the rate distribution and the ergodic spectral efficiency are derived as a function of relevant system parameters in a fixed irregular network model. In a random network model with a homogeneous Poisson point process, a lower bound on the ergodic spectral efficiency is characterized. Through system level simulations, the performance of the proposed method is compared with that of conventional coordination methods: dynamic clustering and static clustering. Our major finding is that, when users are dense enough in a network, the proposed method provides the same level of coordination benefit with dynamic clustering to edge users. [ABSTRACT FROM AUTHOR]

Abstract: In Long Term Evolution-Advanced (LTE-A) networks, Device-to-device (D2D) communications can be utilized to enhance the performance of multicast services by leveraging D2D relays to serve nodes with worse channel conditions within a cluster. For traditional D2D relay schemes, D2D links with poor channel condition may be the bottleneck of system sum data rate. In this paper, to optimize the throughput of D2D communications, we introduce an iterative combinatorial auction algorithm for efficient D2D relay selection. In combinatorial auctions, the User Equipments (UEs) that fails to correctly receive multicast data from eNodeB (eNB) are viewed as bidders that compete for D2D relays, while the eNB is treated as the auctioneer. We also give properties of convergency and low-complexity and present numerical simulations to verify the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Conway and Coxeter introduced frieze patterns in 1973 and classified them via triangulated polygons. The determinant of the matrix associated to a frieze table was computed explicitly by Broline, Crowe and Isaacs in 1974, a result generalized 2012 by Baur and Marsh in the context of cluster algebras of type A. Higher angulations of polygons and associated generalized frieze patterns were studied in a joint paper with Holm and Jørgensen. Here we take these results further; we allow arbitrary dissections and introduce polynomially weighted walks around such dissected polygons. The corresponding generalized frieze table satisfies a complementary symmetry condition; its determinant is a multisymmetric multivariate polynomial that is given explicitly. But even more, the frieze matrix may be transformed over a ring of Laurent polynomials to a nice diagonal form generalizing the Smith normal form result given in [3] . Considering the generalized polynomial frieze in this context it is also shown that the non-zero local determinants are monomials that are given explicitly, depending on the geometry of the dissected polygon. [ABSTRACT FROM AUTHOR]

A correction to this paper has been published: https://doi.org/10.1007/JHEP12(2021)110 [ABSTRACT FROM AUTHOR]