Geometry of mutation classes of rank 3 quivers

We present a geometric realization for all mutation classes of quivers of rank 3 with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by π-rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant p2 + q2 + r 2 − pqr, where p, q,r are the weights of arrows in a quiver. We also classify skew-symmetric mutation-finite real 3×3 matrices and explore the structure of acyclic representatives in finite and infinite mutation classes

Acyclic cluster algebras, reflection groups, and curves on a punctured disc

We establish a bijective correspondence between certain non-self-intersecting curves in an n-punctured disc and positive c-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, we obtain a proof of Lee–Lee conjecture [15] on the combinatorial description of real Schur roots for acyclic quivers with multiple arrows, and give a combinatorial characterization of seeds in terms of curves in an n-punctured disc

Coxeter groups, quiver mutations and geometric manifolds

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh, and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action of a finite (or affine) Coxeter group. These CW-complexes undergo mutations agreeing with mutations of quivers and diagrams. We also generalize the construction to quivers and diagrams originating from unpunctured surfaces and orbifolds

On hyperbolic Coxeter n-polytopes with n + 2 facets

A convex polytope admits a Coxeter decomposition if it is tiled by finitely many Coxeter polytopes such that any two tiles having a common facet are symmetric with respect to this facet. In this paper, we classify all Coxeter decompositions of compact hyperbolic Coxeter n-polytopes with n + 2 facets. Furthermore, going out from Schläfli‘s reduction formula for simplices we construct in a purely combinatorial way a volume formula for arbitrary polytopes and compute the volumes of all compact Coxeter polytopes in ℍ4 which are products of simplice

Cluster algebras from surfaces and extended affine Weyl groups

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types

Доменный эпитаксиальный рост сегнетоэлектрических пленок титаната бария-стронция на сапфире

The model of the crystal epitaxial growth of multicomponent films on single crystal substrates with domain corresponds are presented to an example of a solid solution of barium strontium titanate on sapphire substrates (r-cut). Domain matched epitaxial growth involves the matching of the film and substrate lattice planes having a similar structure, by matching domains. Varying the component composition of the solid solution allows to change the domain size in the range sufficient to reduce the mismatch of the lattice of barium strontium titanate and sapphire. Thus, it is possible to design an epitaxial film growth of various solid solutions on single-crystal substrates.Представлена модель эпитаксиального роста кристаллических многокомпонентных пленок на монокристаллических подложках с доменным соответствием на примере твердого раствора титаната бария-стронция на подложках сапфира (r-срез). Доменный эпитаксиальный рост предполагает согласование плоскостей решетки пленки и подложки, имеющих схожую структуру, путем сопоставления доменов, кратных целому числу межплоскостных расстояний. Варьирование компонентного состава твердого раствора позволяет изменять размер домена в диапазоне, достаточном для снижения рассогласования решеток титаната бария-стронция и сапфира до значения, достаточного для эпитаксиального роста. Таким образом, можно спроектировать эпитаксиальный рост пленок различных твердых растворов на монокристаллических подложках

Tri-critical behavior in rupture induced by disorder

We discover a qualitatively new behavior for systems where the load transfer has limiting stress amplification as in real fiber composites. We find that the disorder is a relevant field leading to tri--criticality, separating a first-order regime where rupture occurs without significant precursors from a second-order regime where the macroscopic elastic coefficient exhibit power law behavior. Our results are based on analytical analysis of fiber bundle models and numerical simulations of a two-dimensional tensorial spring-block system in which stick-slip motion and fracture compete.Comment: Revtex, 10 pages, 4 figures available upon reques

Electrocardiogram Monitoring Wearable Devices and Artificial-Intelligence-Enabled Diagnostic Capabilities: A Review

Worldwide, population aging and unhealthy lifestyles have increased the incidence of high-risk health conditions such as cardiovascular diseases, sleep apnea, and other conditions. Recently, to facilitate early identification and diagnosis, efforts have been made in the research and development of new wearable devices to make them smaller, more comfortable, more accurate, and increasingly compatible with artificial intelligence technologies. These efforts can pave the way to the longer and continuous health monitoring of different biosignals, including the real-time detection of diseases, thus providing more timely and accurate predictions of health events that can drastically improve the healthcare management of patients. Most recent reviews focus on a specific category of disease, the use of artificial intelligence in 12-lead electrocardiograms, or on wearable technology. However, we present recent advances in the use of electrocardiogram signals acquired with wearable devices or from publicly available databases and the analysis of such signals with artificial intelligence methods to detect and predict diseases. As expected, most of the available research focuses on heart diseases, sleep apnea, and other emerging areas, such as mental stress. From a methodological point of view, although traditional statistical methods and machine learning are still widely used, we observe an increasing use of more advanced deep learning methods, specifically architectures that can handle the complexity of biosignal data. These deep learning methods typically include convolutional and recurrent neural networks. Moreover, when proposing new artificial intelligence methods, we observe that the prevalent choice is to use publicly available databases rather than collecting new data