scholarly journals $(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables

10.37236/7188 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Bolor Turmunkh
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Cluster Algebra ◽  
Quiver Varieties ◽  
Quantum Cluster ◽  
T System

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.

Identification of gene interactions in fungal–plant symbiosis through discrete dynamical system modelling

2009 ◽  
Vol 3 (5) ◽  
pp. 414-428 ◽  
Author(s):  
J.G.C. Angeles ◽  
Z. Ouyang ◽  
A.M. Aguirre ◽  
P.J. Lammers ◽  
M. Song
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Gene Interactions ◽  
System Modelling

scholarly journals Positivity of the T-System Cluster Algebra

10.37236/229 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Philippe Di Francesco ◽  
Rinat Kedem
Keyword(s):  
Continued Fraction ◽  
Cluster Algebra ◽  
Path Model ◽  
Partition Functions ◽  
Weighted Graphs ◽  
Generic Boundary ◽  
Seed Data ◽  
Q System ◽  
Extra Parameter ◽  
T System

We give the path model solution for the cluster algebra variables of the $T$-system of type $A_r$ with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the $Q$-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weights are "time-dependent" where "time" is the extra parameter which distinguishes the $T$-system from the $Q$-system, usually identified as the spectral parameter in the context of representation theory. The path model is alternatively described on a graph with non-commutative weights, and cluster mutations are interpreted as non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.

Geometry and Kinematics of Cylindrical Waterbomb Tessellation

2021 ◽  
Author(s):  
Rinki Imada ◽  
Tomohiro Tachi
Keyword(s):  
Dynamical System ◽  
Recurrence Relation ◽  
Discrete Dynamical System ◽  
Kinematic Model ◽  
Theoretical Reason ◽  
Finite Solution ◽  
Geometry And Kinematics ◽  
Folded Surfaces

Abstract Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, the theoretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a kinematic model of waterbomb tube by parameterizing the geometry of a module of waterbomb tessellation and derive a recurrence relation between the modules. Through the visualization of the configurations of waterbomb tubes under the proposed kinematic model, we classify solutions into three classes: cylinder solution, wave-like solution, and finite solution. Furthermore, we give proof of the existence of a wave-like solution around one of the cylinder solutions by applying the knowledge of the discrete dynamical system to the recurrence relation.

scholarly journals The Limit Point of the Pentagram Map

2018 ◽  
Vol 2020 (9) ◽  
pp. 2818-2831 ◽  
Author(s):  
Max Glick
Keyword(s):  
Dynamical System ◽  
Limit Point ◽  
Convex Polygon ◽  
Discrete Dynamical System ◽  
Explicit Description ◽  
Cartesian Coordinates ◽  
The Subject ◽  
Pentagram Map

Abstract The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the 1st paper on the subject, Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.

EFFECT OF STEP SIZE ON BIFURCATIONS AND CHAOS OF A MAP-BASED BVP OSCILLATOR

2010 ◽  
Vol 20 (06) ◽  
pp. 1789-1795 ◽  
Author(s):  
HONGJUN CAO ◽  
CAIXIA WANG ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Dynamical System ◽  
Neuron Model ◽  
Discrete Dynamical System ◽  
Period Doubling ◽  
Vital Role ◽  
Bifurcation Parameter ◽  
Decomposition Technique ◽  
Step Size ◽  
Discrete Step ◽  
Slow Decomposition

The continuous Bonhoeffer–van der Pol (BVP for short) oscillator is transformed into a map-based BVP model by using the forward Euler scheme. At first, the bifurcations and chaos of the map-based BVP model are investigated when the step size varies as a bifurcation parameter. By using the fast-slow decomposition technique, a two-parameter bifurcation diagram is obtained to give insight into the effect of the step size on bifurcations and chaos of the map-based BVP model. The investigation shows that the period-doubling bifurcation is dependent on the step size, while the saddle-node bifurcation is independent of the step size. Second, when the fast–slow decomposition technique cannot be used, we rigorously prove that in the map-based BVP model there exists chaos in the sense of Marotto when the discrete step size varies as a bifurcation parameter. These results show that the discrete step sizes play a vital role between the continuous-time dynamical system and the corresponding discrete dynamical system. Much attention should be paid on the step size when a map-based neuron model is used as an alternative to a continuous neuron model.

scholarly journals Bar-invariant bases of the quantum cluster algebra of type A 2 (2)

2011 ◽  
Vol 61 (4) ◽  
pp. 1077-1090 ◽  
Author(s):  
Xueqing Chen ◽  
Ming Ding ◽  
Jie Sheng
Keyword(s):  
Cluster Algebra ◽  
Type A ◽  
Quantum Cluster Algebra ◽  
Quantum Cluster
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