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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]

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]

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]

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]

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]

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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]

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]

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]

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]

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]

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]

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]

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 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]

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.

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]

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]

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]

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]

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]

The SL (2 , Z) -symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the t → ∞ q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the "Fourier transformed" picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further. [ABSTRACT FROM AUTHOR]

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type. [ABSTRACT FROM AUTHOR]

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds. [ABSTRACT FROM AUTHOR]

The mixed spin Ising nanowire with random and dilute crystal fields has been explored based on the finite cluster approximation. Much concern has been addressed in treating the effects of the randomness with two distribution types of the crystal field on the phase diagrams. It has been demonstrated that for a low random crystal field D marked by the fluctuation d, all transition lines are of second order and are not particularly sensitive to the fluctuation; whereas, for high values of D, the transition temperature depends on the fluctuation and may display tricritical points. Under the effect of the dilute crystal field, according to the surface shell coupling strength, the tricritical behavior occurs exclusively for higher values of the concentration p where a large majority of spins are influenced by the crystal field. All these behaviors have been confirmed by examination of magnetizations. [ABSTRACT FROM AUTHOR]

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y -variables in terms q -Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity. [ABSTRACT FROM AUTHOR]

We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. The evolution for different choices of the graph coincides with many notable dynamical systems including the pentagram map, Q-nets, and discrete Darboux maps. On the other hand, for plabic graphs we prove unique extendability of a configuration from the boundary to the interior, an elegant illustration of the fact that Postnikov's boundary measurement map is invertible. In all cases there is a cluster algebra operating in the background, resolving the open question for Q-nets of whether such a structure exists. [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 prove that the quivers with potentials associated with triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly five punctures. This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related to a flip, their associated QPs are related to the corresponding QP-mutation. As a by-product, for (arbitrarily punctured) surfaces with non-empty boundary, we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series. The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called Popping Theorem, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence. [ABSTRACT FROM AUTHOR]

Multiple graph clustering is an important tool in data integration and data mining for graph-based data. The prediction and classification accuracy can be significantly improved by integrating information from multiple sources and data sets. The aim of multiple graph clustering is to partition objects into several clusters such that clusters in each graph are well-separated and clusters across different graphs are consistent. Existing methods assume that the degrees of association among different graphs are the same. However, some graphs may be strongly or weakly associated with the other graphs due to high or low correlation of their associated cluster structures. When their cluster structures are (or are not) similar, their degree of association should be high (or low). Accurate clustering results can be obtained by integrating such association information in multiple graphs. The main aim of this paper is to study multiple graph semi-supervised clustering by considering a large amount of multiple graph data and a small amount of labeled data. We propose a constrained optimization problem that can determine cluster structures and degrees of association simultaneously in multiple graph clustering. In our formulation, we make use of orthogonality for cluster structure indicator and cosine correlation for degree of association. With orthogonality constraint in the clustering process, we develop a gradient flow method to solve the resulting optimization problem. And the convergence of the proposed iterative method is also shown. Numerical examples including synthetic data and real data sets with few known labels are tested, and are presented to show the efficiency and effectiveness of our proposed method compared with the testing methods in the literature. [ABSTRACT FROM AUTHOR]

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

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

This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated. We include several examples of presentations produced by these methods. [ABSTRACT FROM AUTHOR]

Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassman- nians. In recent years plabic graphs have found numerous applications in mathematics and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this paper we undertake a comprehensive study of a certain semilocal transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing recent work of Inoue-Lam-Pylyavskyy. Special cases of transformations we consider include ge- ometric R-matrices appearing in Berenstein-Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov-Shen. [ABSTRACT FROM AUTHOR]

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