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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Berenstein and Zelevinsky introduced quantum cluster algebras [3] and the triangular bases [4]. The support conjecture in [12] asserts that the support of a triangular basis element for a rank-2 cluster algebra is bounded by an explicitly described region that is possibly concave. In this paper, we prove the support conjecture for all skew-symmetric rank-2 cluster algebras. [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 this paper, we consider two applications of the pentagon equation. The first deals with actions of flips on edges of triangulations labeled by rational functions in some variables. The second can be formulated as a system of linear equations with variables corresponding to triangles of a triangulation. The general method says that if there is some general data (say, edge lengths or areas) associated with states (say, triangulations) and a general data transformation rule (say, how lengths or areas are changed under flips) then after returning to the initial state we recover the initial data. [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]

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]

The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C [ G / P K − ] contains a cluster algebra if G is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure A of the coordinate ring C [ N K ] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure A ˆ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra A ˆ is equal to C [ G / P K − ] after localizing some special minors. [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 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]

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]

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]

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]

In this paper, we consider Poisson thinning Integer-valued time series models, namely integer-valued moving average model (INMA) and Integer-valued Autoregressive Moving Average model (INARMA), and their relationship with cluster point processes, the Cox point process and the dynamic contagion process. We derive the probability generating functionals of INARMA models and compare to that of cluster point processes. The main aim of this paper is to prove that, under a specific parametric setting, INMA and INARMA models are just discrete versions of continuous cluster point processes and hence converge weakly when the length of subintervals goes to zero. [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]

The finite-dimensional symmetric algebras over algebraically closed fields, based on surface triangulations, motivated by the theory of cluster algebras, have been extensively investigated and applied. In particular, the weighted surface algebras and their deformations were introduced and studied in [14–18] , and it was shown that all these algebras, except a few singular cases, are symmetric tame periodic algebras of period 4. In this article, using the general form of a weighted surface algebra from [17] , we introduce and study so called virtual mutations of weighted surface algebras, which constitute a new large class of symmetric tame periodic algebras of period 4. We prove that all these algebras are derived equivalent but not isomorphic to weighted surface algebras. The results of this paper form an essential step towards a classification of all tame symmetric periodic algebras. [ABSTRACT FROM AUTHOR]

Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We also show that upper bounds coincide with the corresponding generalized quantum upper cluster algebras under the "coprimality" condition. [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]

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]

• 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]

This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots of non-self-intersecting curves in an n -punctured disc and the set of real Schur roots of acyclic (valued) quivers with n vertices. [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra and U q (g ˆ) be the corresponding untwisted quantum affine algebra. In this paper, we give a path description of the q -characters of Hernandez-Leclerc modules, show that up to spectral parameter shift, the equivalent classes of Hernandez-Leclerc modules in the Grothendieck ring of the category of finite-dimensional U q ( sl n + 1 ˆ) -modules are cluster variables in the cluster algebra introduced by Hernandez and Leclerc, and finally prove that the geometric q -character formula conjecture is true for Hernandez-Leclerc modules. [ABSTRACT FROM AUTHOR]

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of 2-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the D ˜ 4 -type GDAHA to representations of the E ˜ 6 -type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type D ˜ 4 and E ˜ 6 into matrix algebras over quantum cluster X -varieties, thus linking to the theory of higher Teichmüller spaces. For E ˜ 6 , the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations. [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]

Let C be a 2-Calabi-Yau triangulated category, and let T be a cluster tilting subcategory of C. An important result from Dehy and Keller tells us that a rigid object c ∈ C is uniquely defined by its index with respect to T. The notion of triangulated categories extends to the notion of (d + 2)-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a (d + 2)-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate that this result applies to higher dimensional cluster categories of Dynkin-type A. [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]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra Λ such that Λ has non-trivial open Frobenius structures. [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]

In 2011, Musiker-Schiffler-Williams proved the positivity for cluster algebras from surfaces by giving the expansion formulas for the cluster variables, in terms of perfect matching, γ -symmetric perfect matching and γ -compatible perfect matching. In this paper, we introduce two lattices L (T o , γ (q)) and L (T o , γ (p , q)) to give a unified expansion formula for cluster algebras from surfaces. [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]

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group G ∨ , and its Poisson–Lie dual group G ∗ , respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell G ∨ ; w 0 , e ⊂ G ∨ is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of K ∗ ⊂ G ∗ (the Poisson–Lie dual of the compact form K ⊂ G ). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of K ∗ are equal to symplectic volumes of the corresponding coadjoint orbits in Lie (K) ∗ . To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells G w 0 , e ⊂ G and G ∨ ; w 0 , e ⊂ G ∨ . [ABSTRACT FROM AUTHOR]

Let Λ be an Artin algebra. In this paper, the notion of n Z -Gorenstein cluster tilting subcategories will be introduced. It is shown that every n Z -cluster tilting subcategory of mod-Λ is n Z -Gorenstein if and only if Λ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an n Z -Gorenstein cluster tilting subcategory of mod-Λ is an n Z -cluster tilting subcategory of the exact category Gprj - Λ , the subcategory of all Gorenstein projective objects of mod-Λ. Some basic properties of n Z -Gorenstein cluster tilting subcategories will be studied. In particular, we show that they are n -resolving, a higher version of resolving subcategories. [ABSTRACT FROM AUTHOR]

We show that associated with any |$n$| -dimensional Bott–Samelson variety of a complex semi-simple Lie group |$G$|⁠ , one has |$2^n$| Poisson brackets on the polynomial algebra |$A={\mathbb{C}}[z_1, \ldots , z_n]$|⁠ , each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of |$G$|⁠. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction. [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]

Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points. Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers. This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra. We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes. [ABSTRACT FROM AUTHOR]