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Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [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]

The University has quite a big campus located not far from the city center, in a quiet and convenient area with many facilities. The School was supported by Centre International de Mathématiques Pures et Appliquées (CIMPA), Agence Universitaire de la Francophonie (AUF), Institute for Research in Fundamental Sciences-Isfahan branch (IPM-Isfahan), Iran National Science Fundation (INSF) and University of Isfahan. Representation theory is a central branch of modern mathematics that studies realizations of abstract non-linear structures using classical linear and combinatorial concrete structures like matrices, linear operators and quivers. [Extracted from the article]

For a rooted cluster algebra (Q) over a valued quiver Q , a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver σ (Q) , for some permutation σ. The subalgebra of (Q) generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by ℬ (Q). In this paper, we identify the class of cluster algebras that satisfy ℬ (Q) = (Q) , which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given. [ABSTRACT FROM AUTHOR]

Deep clustering is an approach that uses deep learning to cluster data, since it involves training a neural network model to become familiar with a data representation that is suitable for clustering. Deep clustering has been applied to a wide range of data types, including images, texts, time series and has the advantage of being able to automatically learn features from the data, which can be more effective than using hand-crafted features. It is also able to handle high-dimensional data, such as time series with many variables, which can be challenging for traditional clustering techniques. In this paper, we introduce a novel deep neural network type to improve the performance of the auto-encoder part by ignoring the unnecessary extra-noises and labelling the input data. Our approach is helpful when just a limited amount of labelled data is available, but labelling a big amount of data would be costly or time-consuming. It also applies for the data in high-dimensional and difficult to define a good set of features for clustering. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of control laws for Discrete Event Dynamic Systems (DEDSs) represented by Timed Event Graphs (TEGs) under Generalized Marking Constraints (GMCs). The behaviour of TEGs is described using linear equations while GMCs are expressed by weighted inequalities in Min-Plus algebra. We formulate the problem in terms of control linear Min-Plus models. Hence, an algebraic method to calculate control laws ensuring the respect of GMCs is suggested. And once sufficient conditions are satisfied, we propose casual feedbacks to guarantee these marking specifications. Besides, the proposed control strategy is applied to a dual-armed cluster tool, a well-known industrial application. [ABSTRACT FROM AUTHOR]

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 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]

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]

Let W be the Weyl group corresponding to a finite dimensional simple Lie algebra g of rank ℓ and let m > 1 be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a W-action on Y m was constructed. Here Y m is the rational function field on c m ℓ commuting variables, where c ∈ { 1 , 2 , 3 } depends on g . This was motivated by the q-character map χ q of the category of finite dimensional representations of quantum affine algebra U q (g ^) . We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when q is a root of unity, Im χ q is a subring of the W-invariant subfield Y m W of Y m . In this paper, we give more detailed study on Y m W ; for each reflection r i ∈ W associated to the ith simple root, we describe the r i -invariant subfield Y m r i of Y m . [ABSTRACT FROM AUTHOR]

Ulam's spiral reveals patterns in the prime numbers by presenting positive integers in a right-angled whorl. The classic spatial prisoner's dilemma (PD) reveals pathways to cooperation by presenting a model of agents interacting on a grid. This paper brings these tools together via a deterministic spatial PD model that distributes cooperators at the prime-numbered locations of Ulam's spiral. The model focuses on a narrow boundary game variant of the PD for ease of comparison with early studies of the spatial PD. Despite constituting an initially small portion of the population, cooperators arranged in Ulam's spiral always grow to dominance when (i) the payoff to free-riding is less than or equal to 8/6 (≈1.33) times the payoff to mutual cooperation and (ii) grid size equals or exceeds 23 × 23. As in any spatial PD model, particular formations of cooperators spur this growth and here these formations draw attention to rare configurations in Ulam's spiral. [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]

We prove that the localization Cw of the monoidal category Cw is rigid, and the category Cw,v admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R and an element w in the Weyl group, the subcategory Cw of the category R-gmod of finite-dimensional graded R-modules categorifies the quantum unipotent coordinate ring Aq(n(w)). In the previous paper, we constructed a monoidal category Cw such that it contains Cw and the objects {(M(wΛi,Λi) i ∈ 1} corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category Cw and (Cw-1)rev. Together with the already known left-rigidity of Cw, it follows that the monoidal category Cw is rigid. If v ≤ w in the Bruhat order, there is a subcategory Cw,v of Cw of that categorifies the doubly-invariant algebra N'(w)ℂ[N]N(v). We prove that the family [M(wΛi,vΛi))i∈I of simple R-module forms a real commuting family of graded central objects in the category Cw,v so that there is a localization Cw,v of Cw,v in which M(wΛi,vΛi) are invertible. Since the localization of the algebra N'(w)ℂ[N]N(v) by the family of the isomorphism classes of M(wΛi,vΛi) is isomorphic to the coordinate ring ℂ[Rw,v] of the open Richardson variety associated with w and v, the localization Cw,v categorifies the coordinate ring ℂ[Rw,v]. [ABSTRACT FROM AUTHOR]

This if the final paper in the series Continuous Quivers of Type A. In this part, we generalize existing geometric models of type A cluster structures for the new E-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use these geometric models and isomorphisms of cluster theories to begin classifying continuous type A cluster theories. We also introduce a continuous generalization of mutation. This encompasses mutation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure, and show that paths in this space are continuous mutations. [ABSTRACT FROM AUTHOR]

In this paper, by using a pinning impulse controller and a hybrid controller respectively, the research difficulties of asymptotic synchronization and finite time cluster synchronization of time-varying delayed neural networks are studied. On the ground of Lyapunov stability theorem and Lyapunov-Razumikhin method, a novel sufficient criterion on asymptotic cluster synchronization of time-varying delayed neural networks is obtained. Utilizing Finite time stability theorem and hybrid control technology, a sufficient criterion on finite-time cluster synchronization is also obtained. In order to deal with time-varying delay and save control cost, pinning pulse control is introduced to promote the realization of asymptotic cluster synchronization. Following the idea of pinning control scheme, we design a progressive hybrid control to promote the realization of finite time cluster synchronization. Finally, an example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We continue the work started in parts (I) and (II) of this series. In this paper, we classify which continuous quivers of type A are derived equivalent. Next, we define the new ${\mathcal {C}(A_{{\mathbb {R}},S})}$ , which we call weak continuous cluster category. It is a triangulated category, it does not have cluster structure but it has a new weaker notion of "cluster theory." We show that the original continuous cluster category of [15] is a localization of this new weak continuous cluster category. We define cluster theories to be appropriate groupoids and we show that cluster structures satisfy the conditions for cluster theories. We describe the relationship between different cluster theories: some new and some obtained from cluster structures. The notion of continuous mutation which appears in cluster theories (but not in cluster structures) appears in the next paper [20]. [ABSTRACT FROM AUTHOR]

Denote by N ⊂ SLk the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of ℂ[N] (and base affine spaces) can be parametrized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the dual canonical basis using the data of the corresponding semi-standard Young tableau. We apply our results to study cluster variables in ℂ[N]. [ABSTRACT FROM AUTHOR]

Rank 1 modules are the building blocks of the category CM(Bk,n) of Cohen-Macaulay modules over a quotient Bk,n of a preprojective algebra of affine type A. Jensen, King and Su showed in [8] that the category CM(Bk,n) provides an additive categorification of the cluster algebra structure on the coordinate ring C[Gr(k,n)] of the Grassmannian variety of k-dimensional subspaces in Cn. Rank 1 modules are indecomposable, they are known to be in bijection with k-subsets of {1,2, . . .,n}, and their explicit construction has been given in [8]. In this paper, we give necessary and sufficient conditions for indecomposability of an arbitrary rank 2 module in CM(Bk,n) whose filtration layers are tightly interlacing. We give an explicit construction of all rank 2 decomposable modules that appear as extensions between rank 1 modules corresponding to tightly interlacing k-subsets I and J. [ABSTRACT FROM AUTHOR]

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) T 1 , T 2 , ... , T n . In this case, for 1 ≤ j ≤ n − 1 , two consecutive tiles T j and T j + 1 share exactly one edge, either the edge at the east (west) of T j ( T j + 1 ) or the edge at the north (south) of T j ( T j + 1 ). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras. [ABSTRACT FROM AUTHOR]

In a recent paper of Dokchitser-Dokchitser-Maistret-Morgan, the authors introduced the concept of a cluster picture associated to a hyperelliptic curve from which they are able to recover numerous invariants, including the inertia representation on the first étale cohomology group of the curve. The purpose of this paper is to explore the functionality of these cluster pictures and prove that the inertia representation of a hyperelliptic curve is a function of its cluster picture. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the universal PBW basis and HN system for the derived Hall algebra of a triangulated category satisfying some finiteness conditions. Furthermore, we study the related stability conditions on the module category A of the path algebra A of an acyclic quiver and on the bounded derived category of A. Moreover, we investigate the relation between the Ringel-Hall algebra and the HN system of A defined by Reineke. [ABSTRACT FROM AUTHOR]

In the present paper, we first give a characterization for Bongartz completion in |$\tau $| -tilting theory via |$c$| -vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using |$c$| -vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra |$\mathcal A$| whose vertices consist of the seeds of |$\mathcal A$| containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a |$Y$| -pattern over a universal semifield, each |$Y$| -seed (up to a |$Y$| -seed equivalence) is uniquely determined by the negative |$y$| -variables in this |$Y$| -seed. [ABSTRACT FROM AUTHOR]

This paper proposes a nonparametric approach with the purpose of estimating discrete wavelet transform (DWT) sub-band coefficients for high performance image interpolation. The number of clusters of defined statistical model that represents wavelet coefficients during the learning process is not fixed. The interpolating method is based on Hierarchical Dirichlet Process (HDP) where it uses the Blocked Gibbs Sampling method to obtain the optimum final values. The proposed HDP-HMM exploits statistical inter-scale, and intra-scale dependencies of image sub-bands of three-level decomposed 2D-DWT. It derives sub-bands of low resolution (LR) image, to obtain sub-bands of desired high resolution (HR) image. This research implements Hidden Markov model (HMM) to model the wavelet coefficients, and HDP to model the observations. It uses a very small size dataset that contains both LR and HR images of the dataset. The sophisticated statistical model introduced of the paper has excellent results in terms of Peak-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), Feature Similarity Index (FSIM), and Edge PSNR (EPSNR). It also has a great capability of repressing disturbing artifact, due to ability to model statistical dependencies of distant pixels. This method, and other compared state-of-the-art methods, have implemented on eighteen test-benches, with different statistical properties. [ABSTRACT FROM AUTHOR]

Here, for the first time, coauthor network and cluster analysis were utilized in the environmental engineering field to identify the driving force for scientific collaboration among individuals and the formation of clusters. Papers published in South Korean domestic environmental engineering journals from 2004 to 2018 were assessed, which enabled identification of unique network characteristics that represent not only the field of study, but also the regional boundaries of the data source. Despite being limited to a single country, the study identifies network characteristics, such as scale invariance, that are typically found in other coauthor networks. Nine clusters were identified, the identity of which could be defined by two variables: research interests and author affiliations. The clusters were divided by the sameness or geographical proximity of author affiliations and problem-oriented research topics. These also describe the inter-cluster relationships, validating the notion that the two variables are the major driving force for collaboration networks. This study substantially advances the understanding of scientific collaboration in the environmental engineering field and can guide future studies, such as the role of coauthor networks in environmental engineering within or outside of regional boundaries and the role of networks in domestic publications in other fields of study. [ABSTRACT FROM AUTHOR]

If the rural economy wants to be in line with the development of the times, it must improve the production-marketing model of agricultural products, grasp the market dynamics, and use the advantages of the Internet to conduct marketing and promotion. Moreover, the networked social background also provides a better platform for the expansion and development of agricultural production-marketing channels, which also creates better opportunities. This paper proposes an evaluation method of vertical cooperative algebra connectivity in agricultural production-marketing chain based on control parameters, making full use of the advantages of both community discovery algorithm and Ward hierarchical clustering algorithm, which is applied to solve numerical nonlinear equations. It can be seen from the simulation example that the algorithm not only has good stability, but also has high computational speed and precision. It is a feasible and effective optimization algorithm. The modification method can be applied not only in the agricultural production-marketing network, but also it provides a new way for the algorithm to be applied in other aspects. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional algebra, and M be a d -cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d -cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as an essentially small d -abelian category, known to be equivalent to a d -cluster tilting subcategory of an abelian category A. The completion of M , denoted by Ind (M) , is defined as the universal completion of M with respect to filtered colimits. We explore Ind (M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M , comprising left d -exact functors. Notably, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. For a d -cluster tilting subcategory M of mod A , M → consists of all filtered colimits of objects from M , is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M → is a d -rigid subcategory remains unanswered. However, if it is indeed d -rigid, it qualifies as a d -cluster tilting subcategory. In the case d = 2 , employing cotorsion theory, we establish that M → is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether M → is a d -cluster tilting subcategory of Mod A appears to be equivalent to Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]

In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster variable in an acyclic cluster algebra coincides with the dimension vector of the corresponding module. In this paper, we give analogues of their results for cluster algebras from triangulated surfaces by comparing two kinds of intersection numbers of tagged arcs. [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]

We advance the exploration of cluster-algebraic patterns in the building blocks of scattering amplitudes in N = 4 super Yang-Mills theory. In particular, we conjecture that, given a maximal cut of a loop amplitude, Landau singularities and poles of each Yangian invariant appearing in any representation of the corresponding leading singularities can be found together in a cluster. We check these adjacencies for all one-loop amplitudes up to 9 points. Along the way, we also prove that all (rational) N²MHV Yangian invariants are cluster adjacent, confirming original conjectures. [ABSTRACT FROM AUTHOR]

An accurate troubled-cell indicator (TCI) is always desired by the discontinuous Galerkin methods for solving nonlinear hyperbolic conservation laws. In this paper, we will design a new TCI using K-means clustering. The values of a troubled-cell indication variable are in general of different magnitudes between smooth and discontinuous regions, so the values from a local region are grouped into clusters by a K-means clustering algorithm so that each cluster consists of either the troubled cells or the untroubled cells. A classification criterion is also proposed to discriminate what kind of cells that each cluster contains. The main advantage of the new TCI is its great potential of extensibility. It works for structured meshes and is expected to work for unstructured meshes and irregular meshes, using various indication variables. Also, it is free of problem-dependent parameters. Extensive numerical tests in both one and two dimensions show that the new TCI can detect the troubled cells accurately using the KXRCF indication variable, and it is in general better than the original KXRCF TCI. [ABSTRACT FROM AUTHOR]

Hernandez and Leclerc proved that the Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional modules of any untwisted quantum affine algebra U q (g ̂) has a cluster algebra structure. This makes it possible to compute q -characters of real prime simple modules associated to cluster variables as a result of a sequence of cluster mutations. In this paper, for minimal affinizations of type G2, we describe a cluster algebra algorithm to compute their q -characters and prove the geometric q -character formula conjectured by Hernandez and Leclerc. [ABSTRACT FROM AUTHOR]

An exchange matrix B dominates an exchange matrix B' if the signs of corresponding entries weakly agree, with the entry of B always having weakly greater absolute value. When B dominates B', interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for B often refines the mutation fan for B'; the scattering (diagram) fan for B often refines the scattering fan for B'; and there is often an injective homomorphism from the principal-coefficients cluster algebra for B' to the principal-coefficients cluster algebra for B, preserving \mathbf {g}-vectors and sending the set of cluster variables for B' (or an analogous larger set) into the set of cluster variables for B (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena. [ABSTRACT FROM AUTHOR]

For any simply-laced type simple Lie algebra g and any height function ξ adapted to an orientation Q of the Dynkin diagram of g , Hernandez–Leclerc introduced a certain category C ≤ ξ of representations of the quantum affine algebra U q (g ^) , as well as a subcategory C Q of C ≤ ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring C [ N ] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism D ~ ξ on a torus Y ≤ ξ containing the image of K 0 (C ≤ ξ) under the truncated q-character morphism. We prove that the restriction of D ~ ξ to K 0 (C Q) coincides with the morphism D ¯ recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in C Q , as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of D ¯ allows us to prove a conjecture by the first author on the distinguished values of D ¯ on the flag minors of C [ N ] . We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh's generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park's recent constructions to define a cluster algebra A ¯ Q as a subquotient of K 0 (C ≤ ξ) naturally containing C [ N ] , and suggest the existence of an analogue of the Mirković–Vilonen basis in A ¯ Q on which the values of D ~ ξ may be interpreted as certain equivariant multiplicities. [ABSTRACT FROM AUTHOR]

The fusion of two Mirković--Vilonen cycles is a degeneration of their product, defined using the Beilinson--Drinfeld Grassmannian. In this paper, we put in place a conceptually elementary approach to computing this product in type A.We do so by transferring the problem to a fusion of generalized orbital varieties using the Mirković--Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of GL4, confirming that all the cluster variables are contained in the Mirković--Vilonen basis. [ABSTRACT FROM AUTHOR]

Metaheuristic algorithms have been hybridized with the standard K-means to address the latter's challenges in finding a solution to automatic clustering problems. However, the distance calculations required in the standard K-means phase of the hybrid clustering algorithms increase as the number of clusters increases, and the associated computational cost rises in proportion to the dataset dimensionality. The use of the standard K-means algorithm in the metaheuristic-based K-means hybrid algorithm for the automatic clustering of high-dimensional real-world datasets poses a great challenge to the clustering performance of the resultant hybrid algorithms in terms of computational cost. Reducing the computation time required in the K-means phase of the hybrid algorithm for the automatic clustering of high-dimensional datasets will inevitably reduce the algorithm's complexity. In this paper, a preprocessing phase is introduced into the K-means phase of an improved firefly-based K-means hybrid algorithm using the concept of the central limit theorem to partition the high-dimensional dataset into subgroups of randomly formed subsets on which the K-means algorithm is applied to obtain representative cluster centers for the final clustering procedure. The enhanced firefly algorithm (FA) is hybridized with the CLT-based K-means algorithm to automatically determine the optimum number of cluster centroids and generate corresponding optimum initial cluster centroids for the K-means algorithm to achieve optimal global convergence. Twenty high-dimensional datasets from the UCI machine learning repository are used to investigate the performance of the proposed algorithm. The empirical results indicate that the hybrid FA-K-means clustering method demonstrates statistically significant superiority in the employed performance measures and reducing computation time cost for clustering high-dimensional dataset problems, compared to other advanced hybrid search variants. [ABSTRACT FROM AUTHOR]

For a suitable triangulated category with a Serre functor S and a full precovering subcategory closed under summands and extensions, an indecomposable object C in is called Ext-projective if Ext 1 (C ,) = 0. Then there is no Auslander–Reiten triangle in with end term C. In this paper, we show that if, for such an object C , there is a minimal right almost split morphism β : B → C in , then C appears in something very similar to an Auslander–Reiten triangle in : an essentially unique triangle in of the form Δ = X → ξ B → β C → Σ X , where X is an indecomposable not in and ξ is a -envelope of X. Moreover, under some extra assumptions, we show that removing C from and replacing it with X produces a new subcategory of closed under extensions. We prove that this process coincides with the classic mutation of with respect to the rigid subcategory of generated by all the indecomposable Ext-projectives in apart from C. When is the cluster category of Dynkin type A n and has the above properties, we give a full description of the triangles in of the form Δ and show under which circumstances replacing C by X gives a new extension closed subcategory. [ABSTRACT FROM AUTHOR]

In this paper, we mainly answer a Lampe's question [3] about the solutions of a Diophantine equation, that is, we give a criterion to determine which solutions of the Diophantine equation are in the orbit of the initial solution (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of the group G which is defined by mutations of a cluster algebra. In order to do this, using a rational map φ , we transform the Diophantine equation to a related equation whose all positive integral solutions form the orbit of an initial solution φ (ϵ , ϵ , ϵ , ϵ , ϵ) = (3 , 4 , 4) under the actions of the group G ˜ , and the set S (3 , 4 , 4) is shown to be the orbit of (ϵ , ϵ , ϵ , ϵ , ϵ) under the actions of a subgroup of G. Then the criterion is proved as the main conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. For a quantum Grassmannian cluster algebra C q [ Gr (k , n) ] , we show that there is an associated braid group and each generator σ i of the braid group preserves the quasi-commutative relations of quantum Plücker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that σ i also preserves r -term (r ≥ 4) quantum Plücker relations of C q [ Gr (k , n) ] and other relations which cannot be derived from quantum Plücker relations (if any). Up to this conjecture, we show that σ i is a quasi-automorphism of C q [ Gr (k , n) ] and the braid group acts on C q [ Gr (k , n) ]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce |$n\mathbb{Z}$| -abelian and |$n\mathbb{Z}$| -exact categories by axiomatizing properties of |$n\mathbb{Z}$| -cluster tilting subcategories. We study these categories and show that every |$n\mathbb{Z}$| -cluster tilting subcategory of an abelian (resp. exact) category has a natural structure of an |$n\mathbb{Z}$| -abelian (resp. |$n\mathbb{Z}$| -exact) category. Also, we show that every small |$n\mathbb{Z}$| -abelian category arises in this way, and discuss the problem for |$n\mathbb{Z}$| -exact categories. [ABSTRACT FROM AUTHOR]

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh's result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$ -Calabi–Yau. [ABSTRACT FROM AUTHOR]

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ -lengths in a marked disk, generalizing Schiffler's T -path formula. In the present paper, we give an alternate combinatorial expression for these super λ -lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams. [ABSTRACT FROM AUTHOR]

Fomin and Zelevinsky's definition of cluster algebras laid the foundation for cluster theory. The various categorifications and generalizations of the original definition led to Iyama and Yoshino's generalized cluster categories T / T f d coming from positive-Calabi-Yau triples (T , T f d , M). Jin later defined simple minded collection quadruples (T , T p , S , S) , where the special case S = Σ − d is the analogue of Iyama and Yang's triples: negative-Calabi-Yau triples. In this paper, we further study the quotient categories T / T p coming from simple minded collection quadruples. Our main result uses limits and colimits to describe Hom-spaces over T / T p in relation to the easier to understand Hom-spaces over T. Moreover, we apply our theorem to give a different proof of a result by Jin: if we have a negative-Calabi-Yau triple, then T / T p is a negative cluster category. [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]

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]

A new approach has been recently developed to study the arithmetic of hyperelliptic curves y2=f(x)$y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of f$f$. Since its introduction, numerous papers have used this machinery of 'cluster pictures' to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self‐contained fashion, complemented by an abundance of examples. [ABSTRACT FROM AUTHOR]