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This study introduces a novel cluster of hybrid polynomial kernel families, designed to achieve significantly lower asymptotic mean integrated squared error compared to traditional kernels. These hybrid kernels are developed by heuristically combining classical polynomial kernels using probability axioms. An in-depth analysis of error propagation within these kernels is conducted, utilizing both simulation experiments and real-life datasets, including the Life Span of Batteries and COVID-19 datasets. The findings consistently demonstrate that the proposed hybrid kernels outperform their classical counterparts in various density estimation tasks across different distribution types and sample sizes. This research highlights the potential of hybrid polynomial kernels to enhance accuracy in density estimation, advocating for their adoption in statistical modelling and analysis. [ABSTRACT FROM AUTHOR]

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

The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c -matrices for any quiver mutation-equivalent to a type A Dynkin quiver. [ABSTRACT FROM AUTHOR]

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

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

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, g-vectors, and g-vector fans. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a cluster algebra whose associated Cartan matrix is of affine type. [ABSTRACT FROM AUTHOR]

In this paper we study consequences of the results of Kang et al. [ Monoidal categorification of cluster algebras , J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$ , then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$ , then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [ Triangular bases in quantum cluster algebras and monoidal categorification conjectures , Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$. [ABSTRACT FROM AUTHOR]

Consider 2n points on the unit circle and a reference dissection D∘ of the convex hull of the odd points. The accordion complex of D∘ is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of D∘. In particular, this complex is an associahedron when D∘ is a triangulation and a Stokes complex when D∘ is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection D∘, generalizing known constructions arising from cluster algebras. [ABSTRACT FROM AUTHOR]

We introduce a generalization of the σ-SCF method to approximate noncollinear spin ground and excited single-reference electronic states by minimizing the Hamiltonian variance. The new method is based on the σ-SCF method, originally proposed by Ye et al. [J. Chem. Phys. 147, 214104 (2017)], and provides a prescription to determine ground and excited noncollinear spin states on an equal footing. Our implementation was carried out utilizing an initial simulated annealing stage followed by a mean-field iterative self-consistent approach to simplify the cumbersome search introduced by generalizing the spin degrees of freedom. The simulated annealing stage ensures a broad exploration of the Hilbert space spanned by the generalized spin single-reference states with random complex element-wise rotations of the generalized density matrix elements in the simulated annealing stage. The mean-field iterative self-consistent stage employs an effective Fockian derived from the variance, which is utilized to converge tightly to the solutions. This process helps us to easily find complex spin structures, avoiding manipulating the initial guess. As proof-of-concept tests, we present results for Hn (n = 3–7) planar rings and polyhedral clusters with geometrical spin frustration. We show that most of these systems have noncollinear spin excited states that can be interpreted in terms of geometric spin frustration. These states are not directly targeted by energy minimization methods, which are meant to converge to the ground state. This stresses the capability of the σ-SCF methodology to find approximate noncollinear spin structures as mean-field excited states. [ABSTRACT FROM AUTHOR]

Several papers also investigate the effectiveness of the TP model transformation in deriving alternative sets of parameters and components to describe the same polytopic model. The special issue focuses on advanced theories and design solutions that are based on the tensor product (TP) model transformation and enable improved control performance. This is of special interest to the field because - given the high degree of sensitivity that control design procedures for TS fuzzy and other polytopic models often display with respect to variations in the parameterizations of the underlying models - a given control design procedure can often be evaluated only by testing the effectiveness of the procedure for many different formulations of the same model. [Extracted from the article]

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]

In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the c-vectors associated with any seed defines a quasi-Cartan companion for the corresponding exchange matrix (i.e. they form a companion basis), and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers. [ABSTRACT FROM AUTHOR]

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

We first study a new family of graded quiver varieties together with a new t-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual Poincaré–Birkhoff–Witt basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras. [ABSTRACT FROM PUBLISHER]

It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc, and Schröer have recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper, we show that their parametrization coincides with Lusztig's parametrization of the canonical basis. [ABSTRACT FROM PUBLISHER]

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March 2014. This deadline will allow the special issue to appear at the end of 2014. There is no strict regulation on article size, but as a guide the preferable size is 15–30 pages for contributed papers and 40–60 pages for reviews. Further advice on publishing your work in Journal of Physics A may be found at iopscience.iop.org/jphysa. Contributions to the special issue should be submitted by web upload via ScholarOne Manuscripts, quoting ‘JPhysA special issue on cluster algebras in mathematical physics’. Submissions should ideally be in standard LaTeX form. Please see the website for further information on electronic submissions. All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address. The special issue will be published in the print and online versions of the journal. [ABSTRACT FROM AUTHOR]

In this paper, we describe a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein-Fomin-Zelevinsky quivers for Schubert cells in a symmetric Kac-Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten, and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers. [ABSTRACT FROM AUTHOR]

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We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of "extended quivers," which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step towards understanding the notion of cluster superalgebra. [ABSTRACT FROM AUTHOR]

Following the proof of the purity conjecture for weakly separated collections, recent years have revealed a variety of wider examples of purity in different settings. In this paper we consider the collection AI,J of sets that are weakly separated from two fixed sets I and J. We show that all maximal by inclusion weakly separated collections W⊂AI,J are also maximal by size, provided that I and J are sufficiently “generic”. We also give a simple formula for the cardinality of W in terms of I and J. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables. [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]

A Delone (Delaunay) set is a uniformly discrete and relatively dense set of points located in space, and is a natural mathematical model of the set of atomic positions of any solid, whether it is crystalline, quasi‐crystalline or amorphous. A Delone set has two positive parameters: r is the packing radius and R is the covering radius. The value 2r can be interpreted as the minimum distance between points of the set. The covering radius R is the radius of the biggest 'empty' ball, i.e. the radius of the biggest ball containing no points from the set. The central concept of this article is the so‐called local group at a point of X which is defined as a group of a cluster (neighborhoods) around the point of radius 2R. This value 2R is notable because it is the minimum size of cluster that provides the finiteness of the cluster group at each point for any set X from the family of all Delone sets with the covering radius R. A few conjectures and theorems on the local groups for arbitrary Delone sets in the Euclidean plane and 3D space are discussed. Some of these statements significantly refine and generalize the famous Bravais theorem on the impossibility of fifth‐order axes in 2D and 3D lattices. A complete proof is given that, in a Delone set X in the 3D Euclidean space, the subset of all points at which the local groups contain rotations of order at most 6 is also a Delone set with a certain covering radius , where < 3R and R is the covering radius for X. [ABSTRACT FROM AUTHOR]

This paper concerns cluster algebras with principal coefficients associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [23]. Given any (generalized) arc or loop in the surface—with or without self-intersections—we associate an element of (the fraction field of) , using products of elements of PSL2. We prove an abstract combinatorial result which gives a formula for the number of matchings of a snake or band graph in terms of an appropriate product of 2×2 matrices. We then use this formula to prove that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [22, 23]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [13, 12, 14], who worked in the coefficient-free case. The results of this paper will be used in [23] in order to show that certain collections of arcs and loops comprise a vector-space basis for . [ABSTRACT FROM PUBLISHER]

Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi-Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface d P . In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables that are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the d P brane tiling for these formulas in most cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if is a clutter such that 〈〉 is a vertex decomposable simplicial complex or I() is squarefree stable, then is chordal. [ABSTRACT FROM AUTHOR]

In this paper we discuss, in terms of quiver with relations, sufficient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension at most two and we give a sufficient condition to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and combining both types of conditions, we are able to analyze if some given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin typeEp. [ABSTRACT FROM AUTHOR]

In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the N-clusters contained in an open bounded set $$\Omega $$ . Here with N-Cluster we mean a family of N sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any N-cluster attaining such a minimum a Cheeger N-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger N-clusters in a general ambient space dimension and we give a precise description of their structure in the planar case. The last part is devoted to the relation between the functional introduced here (namely the N-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin's conjecture. [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]

A call for papers on cluster algebras in mathematical physics along with editorial policy and guidelines for contributions' preparation for the journal is presented.

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]

Density-based clustering is a widely explored domain in the field of machine learning, with numerous methods proposed to address challenges in clustering under varying density conditions. These methods primarily categorize samples based on their local density. However, challenges arise when dealing with boundary samples. The density of certain boundary samples is typically lower, which can lead to misclassification as noise. Moreover, in situations where two clusters have similar densities and are in close proximity, accurately classifying the boundary points between them becomes a challenging task. In this paper, we introduce an enhanced approach based on k -nearest neighbors to address this challenge within density-based clustering. In our method, upon the formation of a new cluster, we identify boundary samples by examining the spatial relationships between each sample and its k -nearest neighbors, as well as their connections to the newly established cluster. Once all clusters are formed, we refine the classification of these boundary samples by adjusting or retaining their assigned labels based on their k -nearest neighbors. Experimental evaluations on synthetic and real-world datasets demonstrate the effectiveness of our proposed method. • A new density clustering method based on boundary samples verification is proposed. • The method can handle datasets with different densities and shapes. • The method only requires one integer parameter, easy to use. • Outperforms existing methods on various benchmark datasets. [ABSTRACT FROM AUTHOR]

Prior clustering approaches of soccer players have employed a variety of methods based on various data categories, but none of them have focused on clustering by career paths characterized through a time series analysis of yearly performance quality. Therefore, this study aims to propose a methodology how a career path can be represented as a time series of a player's seasonal qualities and then be clustered with players that have a similar career path. The underlying data focuses on soccer players from the five largest European soccer nations (Big-5). This allows for the identification of different types of career paths of players and the investigation of significant disparities between career paths among the Big-5 nations. In line with our proposed methodological approach, we identified and interpreted 13 different clusters of player career paths. These range from the cluster with the highest player quality scores to the pattern comprising players with the weakest scores. Further, the detected clusters show significant differences regarding variables of soccer players' early career phase in adolescence (e.g., age of debut in professional soccer, years spent in a youth academy). The presented approach might represent a first step for stakeholders in soccer to get an objective insight in players' career by utilizing mainly freely available data sources. [ABSTRACT FROM AUTHOR]

We study the T -system of type A ∞ , also known as the octahedron recurrence/equation, viewed as a 2 + 1 -dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with 'flat' initial data, we consider initial data along parallel 'slanted' planes perpendicular to an arbitrary admissible direction (r , s , t) ∈ Z + 3 . The corresponding solutions of the T -system are interpreted as partition functions of dimer models on some suitable 'pinecone' graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The T -system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function. [ABSTRACT FROM AUTHOR]

We classify mutation-finite cluster algebras with arbitrary coefficients of geometric type. This completes the classification of all mutation-finite cluster algebras started in [Felikson, Shapiro, and Tumarkin, J. Eur. Math. Soc. 14 (2012)]. [ABSTRACT FROM AUTHOR]

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

There is a well-known analogy between cluster algebras and Kacmoody algebras: roughly speaking, Kac-Moody algebras are associated with symmetrizable generalized Cartan matrices while cluster algebras correspond to skew-symmetrizable matrices. In this paper, we study an interplay between these two classes of matrices. We obtain relations in the Weyl groups of Kac- Moody algebras that come from mutation classes of skew-symmetrizable matrices. More precisely, we establish a set of relations satisfied by the reflections of the so-called companion bases; these include c-vectors, which parametrize coefficients in a cluster algebra with principal coefficients. These relations generalize the relations obtained by Barot and Marsh for finite type. For affine type, we also show that the reflections of the companion bases satisfy the relations obtained by Felikson and Tumarkin. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices. [ABSTRACT FROM AUTHOR]

The local clustering coefficients of preferential attachment models are analyzed. Previously, a general approach to preferential attachment was proposed (the PA-class was introduced); it was shown that the degree distribution in all models of the PA-class obeys a power law. The global clustering coefficient was also analyzed, and a lower bound for the mean local clustering coefficient was found. In the paper, new results are obtained by analyzing the local clustering coefficients of models of the PA-class. Namely, the behavior of the mean value C ( n, d) of local clustering over vertices of degree d is studied. [ABSTRACT FROM AUTHOR]

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

In this paper, pedestrian evacuation considering different human behaviors is studied by using a cellular automaton (CA) model combined with the snowdrift game theory. The evacuees are divided into two types, i.e. cooperators and defectors, and two different human behaviors, herding behavior and independent behavior, are investigated. It is found from a large amount of numerical simulations that the ratios of the corresponding evacuee clusters are evolved to consistent states despite 11 typically different initial conditions, which may largely owe to self-organization effect. Moreover, an appropriate proportion of initial defectors who are of herding behavior, coupled with an appropriate proportion of initial defectors who are of rationally independent thinking, are two necessary factors for short evacuation time. [ABSTRACT FROM AUTHOR]

In this paper, the authors are devoted to design a new segmentation approach based on I-Ching operators in the framework of shadowed fuzzy C-means clustering. The I-Ching operators are innovative operators, which are evolved from ancient Chinese I-Ching philosophy. I-Ching operators include three kinds of operators, intrication operator, turnover operator, and mutual operator. These new operators are very flexible and efficient in evolution procedure. In this paper, the new operators are specifically designed to search for the optimal cluster centers of shadowed fuzzy C-means. Considering the local spatial information in image segmentation procedure, a new segmentation algorithm called I-Ching spatial shadowed fuzzy C-means (ICSSFCM) is proposed. Traditional segmentation approaches based on fuzzy C-means, shadowed fuzzy C-means, and spatial shadowed fuzzy C-means are compared with the proposed method. The experimental results show that the proposed ICSSFCM is very efficient approach not only in tackling the overlapping segments but also in suppressing the noise in images. [ABSTRACT FROM AUTHOR]

We study f-vectors, which are the maximal degree vectors of F-polynomials in cluster algebra theory. For a cluster algebra of finite type, we find that positive f-vectors correspond with d-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of d-vectors, we prove that cluster variables in a cluster are uniquely determined by their f-vectors when the cluster algebra is of finite type or rank 2. [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]

Edge-bridged halide tantalum clusters based on the {Ta6Br12}4+ core have been the topic of many physicostructural investigations both in solution and in the solid-state. Despite a large number of studies, the fundamental correlations between compositions, local symmetry, electronic structures of [{Ta6Bri12}La6]m+/n− cluster units (L = Br or H2O, in solution and in the solid-state), redox states, and vibrational and absorption properties are still not well established. Using K4[{Ta6Bri12}Bra6] as a starting precursor (i: inner and a: apical), we have investigated the behavior of the [{Ta6Bri12}Bra6]4− cluster unit in terms of oxidation properties and chemical modifications both in solution (water and organic solvent) and after recrystallization. A wide range of experimental techniques in combination with quantum chemical simulations afford new data that allow the puzzling behavior of the cluster units in response to changes in their environment to be revealed. Apical ligands undergo changes like modifications of interatomic distances to complete substitutions in solution that modify noticeably the cluster physical properties. Changes in the oxidation state of the cluster units also occur, which modify significantly their physical properties, including optical properties, which can thus be used as fingerprints. A subtle balance exists between the number of substituted apical ligands and the cluster oxidation state. This study provides new information about the exact nature of the species formed during the transition from the solid-state to solutions and vice versa. This shows new perspectives on optimization protocols for the design of Ta6 cluster-based materials. [ABSTRACT FROM AUTHOR]

This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras E over a field F , such that E ⊗ F E o p is semisimple. We assume that E contains a certain type of F -basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the tensor algebra over E , of any finite-dimensional E - E -bimodule on which F acts centrally. In this case, we introduce a cyclic derivative and to each potential we associate a Jacobian ideal. Finally, we develop a mutation theory of potentials, which in the case that the bimodule is Z -free, it behaves as the quiver case; but allows us to obtain realizations of a certain class of skew-symmetrizable integer matrices. [ABSTRACT FROM AUTHOR]