Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix $A$, for any index $k$, one can define an automorphism associated with $A,$ of the field $\mathbf{Q}(u_1, >..., u_n)$ of rational functions of $n$ independent indeterminates $u_1,..., u_n.$ It is an isomorphism between two cluster algebras associated to the matrix $A$ (see section 4 for precise meaning). When $A$ is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.Comment: revised versio

Radiation of a relativistic electron with non-equilibrium own Coulomb field

The condition and specific features of the non-dipole regime of radiation is discussed in the context of the results of the recent CERN experiment NA63 on measurement of the radiation power spectrum of 149 GeV electrons in thin tantalum targets. The first observation of a logarithmic dependence of radiation yield on the target thickness that was done there is the conclusive evidence of the effect of radiation suppression in a thin layer of matter, which was predicted many years ago, and which is the direct manifestation of the radiation of a relativistic electron with non-equilibrium own Coulomb field. The special features of the angular distribution of the radiation and its polarization in a thin target at non-dipole regime are proposed for a new experimental study

Discrete integrable systems, positivity, and continued fraction rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.Comment: 24 pages, 2 figure

Q-systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure

Hospital mortality in diffuse liver diseases, complicated by portal hypertension

Departamentul de chirurgie N3, Universitatea Naţională de Medicină “A.Bogomoletz”, Kiev, Ucraina, Al XII-lea Congres al Asociației Chirurgilor „Nicolae Anestiadi” din Republica Moldova cu participare internațională 23-25 septembrie 2015Introducere: Recurența hemoragiei variceale (RHV) în maladiile hepatice difuze (MHD), ca complicație a hipertensiunii portale (HTP) agravează prognosticul și este considerată un factor de risc independent de deces. Scopul lucrării: De a studia cauzele mortalității intra-spitalicești în cazul MHD cu HTP și episoade de RHV. Materiale și metode: Am analizat datele clinice și cele ale autopsiei a 525 pacienți cu MHD și HTP cu hemoragie variceală în Centrul de Hemoragii Gastrointestinale (HGI) din Kiev pe perioada anilor 2007-2014. Toate cazurile au fost divizate în 2 grupe: A). 388 pacienți (73,9%) care au decedat în timpul hemoragiei continue sau recurente; B). 137 pacienți (26,1%) care au decedat după stoparea hemoragiei. Rezultate: La toți pacienții din momentul internării au fost efectuate măsuri adecvate de diagnosticare, terapie intensivă cu hemostază endoscopică și resuscitare. În grupul A, după internare au decedat: în primele 24h - 186 (35,4%) pacienți, în 24-48h - 113 (21,5%) pacienți și în 48-72h - 89 (17%) pacienți, în timpul hemoragiei variceale continue (61,1%) sau recurente (38,9%). Rezultatele autopsiei au relevat varice esofagiene și gastrice de un grad sever cu mai multe leziuni (41,0%), combinații de eroziuni esofagiene și gastrice (31,2%), ulcer acut esofagian și gastric (15,2%), ulcere peptice (7,2%), sindrom Mallory-Weiss (5,4%). Principalele cauze ale deceselor ne-asociate cu continuarea sau recurența hemoragiei (grupul B) au fost: insuficiența renală progresivă și poliorganică (70,9%), edemul cerebral (12,4%), insuficiența cardiacă acută cu edem pulmonar (10,9%), sepsisul (5,8%). Concluzii: Pacienții cu MHD și HTP necesită o abordare multidisciplinară, folosind metode endovasculare de reducere a fluxului de sînge portal cu scop de prevenire a RHV secundare.Introduction: Recurrence of varicose bleeding (RVB) in diffuse liver diseases (DLD), complicated portal hypertension (PH) worsen prognosis and is considered as an independent risk factor for death. The aim: To study the causes of hospital mortality in DLD with PH and episodes of RVB. Material and methods: We analyzed clinical data with the data of the autopsies of deceased 525 patients (pts) with DLD and PH with varicose bleeding in Kiev center of GIB from 2007 to 2014 yrs. All cases were divided in 2 groups: A) 388 (73.9%) pts died amid continued or recurrent bleeding B) 137 (26.1%) pts died after stopping bleeding later. Results: All the patients from the time of hospitalization were conducted adequate diagnostic complex, intensive therapy with endoscopic hemostasis, resuscitation. In group A after admission 186 (35.4%) died during first 24 hours, next 24-48 hours – 113 (21.5%) and 48-72 hours – 89 (17%) pts from continued varix bleeding (61.1%) or recurrent bleeding (38.9%). Autopsy showed sever grade esophageal and upper part of stomach varix with multiple veins ruptures (41.0%), combinations with multiple esophageal and stomach erosions (31.2%), acute esophageal and stomach ulcer (15.2%), peptic ulcers (7.2%), MVS (5.4%). The main causes of deaths not associated with continued or recurrent bleeding (group B) were progressive renal and multiple organ failure (70.9%), edema and swelling of the brain (12.4%), acute heart failure with pulmonary edema (10.9%), sepsis (5.8%). Conclusions: Patients with DLD and PH need multidisciplinary approach using endovascular portal blood flow reduction methods for secondary prevention RVB

The solution of the quantum A1A_1 T-system for arbitrary boundary

We solve the quantum version of the $A_1$ $T$-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum $A_1$ $Q$-system and generalize it to the fully non-commutative case. We give the relation between the quantum $T$-system and the quantum lattice Liouville equation, which is the quantized $Y$-system.Comment: 24 pages, 18 figure

Cluster algebras of type A2(1)A_2^{(1)}

In this paper we study cluster algebras \myAA of type $A_2^{(1)}$. We solve the recurrence relations among the cluster variables (which form a T--system of type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of \myAA (which form a Y--system of type $A_2^{(1)}$). In \myAA there is a natural notion of positivity. We find linear bases \BB of \myAA such that positive linear combinations of elements of \BB coincide with the cone of positive elements. We call these bases \emph{atomic bases} of \myAA. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of \BB are parameterized by \ZZ^3 via their $\mathbf{g}$--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of \myAA restricts to a bijection between \BB and \ZZ^3. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations to express every element of \myAA as linear combinations of elements of \BB.Comment: Latex, 40 pages; Published online in Algebras and Representation Theory, springer, 201